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Doctoral Thesis

Optical investigations in quantum spin systems

Author(s): Caimi, Giulio

Publication Date: 2005

Permanent Link: https://doi.org/10.3929/ethz-a-005084655

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Diss. ETH No. 16248

Optical investigations in quantum spin systems

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGY

ZURICH

for the degree of

Doctor of Natural Sciences

presented by

GIULIO CAIMI

Dipl. Phys. ETH Zürich

born on the 14 th of April, 1977

citizen of Ligornetto, Ti

accepted on the recommendation of

Prof. Dr. Leonardo Degiorgi, examiner

Prof. Dr. Hans-Rudolf Ott, co-examiner

Prof. Dr. Thierry Giamarchi, co-examiner

2005

A Emanuela,

il tocco di colore nella mia vita.

Contents

Glossary v

Abstract vii

Riassunto ix

1 Introduction 1

2 The low dimensional quantum spin systems 3

2.1 The spin-Peierls transition . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Phenomenology of the spin-Peierls transition . . . . . . . . . . . . 5

2.1.2 Theory of spin-Peierls transition . . . . . . . . . . . . . . . . . . . 9

2.2 Spin density waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Mean-field treatment of 1D SDW transition . . . . . . . . . . . . . . 13

2.2.2 The electrodynamic response of a 1D SDW . . . . . . . . . . . . . 16

2.2.3 The Fermi surface nesting . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Frustrated systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1 Lattice properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.2 Frustration parameter f . . . . . . . . . . . . . . . . . . . . . . . . 22

Bibliography 25

3 Experimental technique 29

3.1 The optical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Reflectivity measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 The Lorentz-Drude model . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3.1 The generalized Drude analysis . . . . . . . . . . . . . . . . . . . 37

i

ii Contents

3.4 Fano’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Bibliography 43

4 TiOX (X= Cl and Br) 45

4.1 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.1.1 Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.1.2 Band structure calculations . . . . . . . . . . . . . . . . . . . . . . 48

4.1.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Optical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3.1 Theoretical treatment of the phonon modes in TiOX compounds . . 66

4.3.2 Temperature evolution of the fit parameters . . . . . . . . . . . . . 73

Bibliography 85

5 LiCu 2O2 89


5.2 Optical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Bibliography 103

6 Na0.7CoO2 105


6.1.1 Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.1.2 The superconductor NaxCoO2•yH2O . . . . . . . . . . . . . . . . 108

6.1.3 The non-hydrated NaxCoO2 compounds . . . . . . . . . . . . . . 111

6.1.4 Band structure calculations . . . . . . . . . . . . . . . . . . . . . . 115

6.2 Optical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.2.1 Optical investigation in Na0.7CoO2 . . . . . . . . . . . . . . . . . . 122

6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.3.1 The Ruvalds and Virosztek approach for a nested Fermi liquid . . . 133

Bibliography 139

7 Conclusion and outlook 145

Contents iii

Acknowledgments 149

Curriculum vitae 151

Publications and presentations 153

Glossary

nD n dimensional, n∈ NBCS Bardeen Cooper Schrieffer

CDW charge density wave

SDW spin density wave

SP spin-Peierls

FL Fermi liquid

NFL nested Fermi liquid

T temperature

Tc critical temperature

AFM anti-ferromagnetic

FM ferromagnetic

LD Lorentz-Drude

IR infrared

FIR far infrared

MIR mid infrared

UV ultraviolet

nn nearest neighbor

nnn next nearest neighbor

LDA local density approximation

DOS density of states

KK Kramers Kroning

v

Abstract

In the last decade, magnetic properties of low dimensional spin 1/2 systems attracted

great interest. From the investigation of their magnetic properties, one hopes to get some

understanding on the possible mechanism responsible for high Tc superconductivity. Fur-

thermore, in low dimensional spin 1/2 systems, numerous different ground states have

been predicted theoretically. In this thesis, we report on optical properties of three spin

S= 1/2 systems.

The first TiOX (X=Cl, Br) is a possible inorganic spin-Peierls system. Each compound

of the TiOX series shows two magnetic transitions accompanied by a lattice distortion

along the chain direction. Our optical measurements detected strong phonon anomalies

and our results reflect the strong role played by quantum fluctuations in these compounds.

Additionally, combining results from Raman spectroscopy and infrared reflectivity, we esti-

mated the magnetic energy gap of 2∆/kB ≈ 430K in TiOCl.

Second, we studied LiCu2O2, which is characterized by three different magnetic

phases. At 9 K, LiCu2O2 undergoes an anti-ferromagnetic transition upon cooling, while

between 9 and 23 K the adopted phase is characterized by a spin helix arrangement.

Above 23 K, the lowest-energy magnetic configuration is separated from the first excited

one by an energy gap of ∆/kB≈ 72K. A puzzling temperature dependence of the phonon

mode at 30 meV was established. We also found that the spectral weight distribution in

the absorption spectrum of LiCu2O2 maps the three magnetic phases. Moreover, we de-

tected a strong interband transition from the O2− 2p into the Cu2+ upper Hubbard 3d

band at 3.1 eV. This transition is compatible with the electronic band-structure established

by photoemission spectroscopy.

vii

viii Abstract

Finally, we investigated the cobaltate Na0.7CoO2. We found an electrodynamic re-

sponse that bears several analogies with that of the high Tc superconductors. The scatter-

ing rate of the electronic response is found to vary linearly as a function of both tempera-

ture and photon energy. This result is interpreted on the basis of a nested Fermi-surface

approach and suggests that Na0.7CoO2 is close to a spin density wave metallic phase.

Riassunto

Le proprietà magnetiche dei sistemi a bassa dimensionalità con spin 1/2 hanno attratto

negli ultimi anni l’interesse di molti ricercatori che sperano dal loro studio di ricavare im-

portanti informazioni, utili per rivelare ad esempio i meccanismi responsabili della super-

conduttività ad alta temperatura. In aggiunta, un notevole numero di differenti stati fonda-

mentali sono stati predetti teoricamente in questi sistemi. Motivati da queste peculiarità, in

questa tesi studiamo le proprietà ottiche di tre materiali con spin S= 1/2.

Anzitutto TiOX (X=Cl, Br), materiali inorganici che hanno parecchie analogie con sis-

temi spin-Peierls. I cristalli TiOX soggiaciono a due transizioni magnetiche accompagnate

da una distorsione del reticolo cristallino in direzione delle catene. Le nostre misure ottiche

hanno ravvisato in questi materiali forti anomalie nello spettro fononico, che ne sottolinea

il forte regime di fluttuazioni. Combinando i risultati di Raman e riflettività infrarossa, siamo

stati in grado di stimare il gap dell’eccitazione magnetica in 2∆/kB ≈ 430K per TiOCl.

LiCu2O2 è caratterizzato da tre differenti stati fondamentali. Al di sotto di 9 K,

LiCu2O2 transisce in uno stato anti-ferromagnetico, mentre tra 9 e 23 K, lo stato fonda-

mentale magnetico è caratterizzato da un’elica di spin. Al di sopra di 23 K, un’energia

∆/kB ≈ 72 K separa lo stato magnetico con minore energia dal primo stato eccitato.

In LiCu2O2, il fonone piccato a 30 meV ha un’intrigante dipendenza dalla temperatura.

Dall’analisi del peso spettrale nello spettro di assorbimento, abbiamo evidenziato come

esso riflette le tre fasi presenti in LiCu2O2. Ad alte frequenze, abbiamo rivelato un forte

assorbimento a 3.1 eV, attribuito ad una transizione tra le bande O2− 2p e la banda di

Hubbard 3d di Cu2+. Questi risultati sono compatibili con la struttura di banda misurata in

fotoemissione.

Da ultimo, abbiamo indagato le prorietà ottiche di Na0.7CoO2. Abbiamo trovato che

ix

x Riassunto

la risposta elettrodinamica presenta diverse analogie con quelle dei superconduttori ad

alta temperatura. In più la frequenza di diffusione della risposta elettronica risulta essere

lineare in funzione sia della temperatura che dell’energia dei fotoni. Questi risultati sono

interpretati con una teroria basata sulle proprietà di annidamento alla superficie di Fermi,

e suggeriscono che Na0.7CoO2 è prossimo ad uno stato metallico con onde di densità di

spin.

1 Introduction

The study of strongly correlated electron systems is one of the most stimulating branches in

solid state physics. These systems are often described on the basis of the Hubbard model,

which is characterized by two energy terms in the Hamiltonian. The first one considers the

probability t for an electron to hop from its site onto a neighboring one, while the second

one describes the repulsion U between electrons on the same site. This last term favors

the formation of local magnetic moments, since it suppresses the possibility of a second

electron at the same site. Thus in the case where U is dominant (U t), the electrons

are strongly correlated, and are localized at a specific lattice site. As a consequence of

this localization, the material turns out to be an insulator (Mott insulator). However these

electrons still exhibit some degrees of freedom, such as orbital and spin.

We are here interested in those systems where the low temperature physics is dom-

inated by the spin degrees of freedom. By considering the hopping term t as a small per-

turbation (i.e., t U ), the Hubbard Hamiltonian transforms to a spin Heisenberg Hamilto-

nian H = J∑i j SiSj , with the magnetic exchange constant given by J = 4t2/U . In the last

decades, spin-systems of this type attracted researchers’ interest due to the numerous

different magnetic ground states, which may be realized. Of course, the main motivation to

study the magnetic properties of materials with spin S= 1/2 is also intimately connected

to the discovery of high Tc superconductivity. In fact, it has been suggested that the driving

force for the Cooper pair formation in high Tc materials could be based on magnetic inter-

actions in the CuO2 planes and not on the electron-phonon coupling, as in conventional

superconductors. At present, the detailed mechanism for high Tc superconductivity is not

fully understood.

The work presented in this thesis is based on optical investigations of three prototype

1

2 1. Introduction

materials with spin S= 1/2, where the concomitance of the low spin value, strong quan-

tum fluctuation and frustration effects leads to particular magnetic ground states. These

systems have indeed attracted in the last couple of years the attention of the scientific

community and are at the research’s forefront in solids state physics. Optical methods are

a powerful spectroscopic technique in order to shed light on relevant mechanisms, govern-

ing magnetic transitions or broken symmetry ground states. Indeed, beside informations

about the dynamics of the transport properties, optical techniques are an ideal experi-

mental tool in order to investigate the electronic excitations and the lattice dynamics (e.g.,

possible phonon’s anomalies) connected to the magnetic transitions.

We study the titanium oxyhalides systems (TiOX, X=Cl, Br), where the single d elec-

tron occupies the dxy orbital. The dxy orbitals are split from the others t2g orbitals and order

forming a chain along the b axis. At low temperatures, TiOX crystals show a magnetic tran-

sition associated to a lattice dimerization along the b axis. These latter properties have a

lot of analogies with a spin-Peierls transition. Furthermore, of interest is the theoretical pre-

diction that TiOX might exhibit unconventional superconductivity upon metallization if suit-

ably intercalated (electron doping). We also present our optical investigation on LiCu2O2,

a cuprate characterized by chains of Cu2+ atoms and CuO4 plaquettes. These plaque-

ttes are the building units of the superconducting plane of high Tc superconductors. The

arrangement of these CuO4 plaquettes in LiCu2O2 leads to bond frustration with strong

competing magnetic interactions. LiCu2O2, in the high temperature regime, presents a

gapped magnetic phase while by lowering the temperature it undergoes a phase transition

into a long range ordered state. Finally, we investigate the Na0.7CoO2 compound, which

is characterized by a triangular arrangement of the Co ions. Na0.7CoO2 is a metal and

appears in the phase diagram of NaxCoO2 at the border of a metallic spin density waves

region.

The thesis is organized as follow: first we start with a short theoretical introduction

about quantum spin systems characterized by a gap in the magnetic excitation spectrum.

Thereafter, we briefly introduce the optical experimental techniques and the analysis used

to account for the experimental data. We will then concentrate on the experimental results

collected for TiOX, LiCu2O2 and Na0.7CoO2. In the final chapter, we summarize the main

results and we will give an outlook on future (optical) investigations in the field of quantum

spin systems.

2 The low dimensional quantum spinsystems

In the last decade, the low dimensional quantum spin systems have attracted a lot of in-

terest both from a theoretical as well as from an experimental point of view. In this context,

several new materials exhibiting a variety of novel phenomena have been discovered. The

field of low-dimensional quantum magnetism provides a fertile ground for rigorous the-

ory. The powerful available theoretical techniques, like the Bethe Ansatz and bosonization,

allow to study the ground and excited state properties. Some models developed for inter-

acting spin systems lead indeed to an exact solution of the ground state and in some cases

of the low energy excitation spectrum [1].

A quantum spin system is obtained in those materials which show a spin degree of

freedom at low temperature. Therewith a new physics at low energies scale appears which

is described by a spin-Hamilton operator, like the Heisenberg Hamiltonian [2]:

H = ∑<i j>

Ji j (Si ·Sj). (2.1)

Si is the spin operator acting on the lattice site i and Ji j denotes the strength of the ex-

change interaction. Real materials are three dimensional (3D), but if the exchange inter-

action is restricted to lower dimensions, spin-chains and ladders or systems with a more

complex exchange geometry are realized. These systems exhibit a number of unusual

properties and are strongly influenced by quantum fluctuations [3].

As an example, we first take the 1D S= 1/2 spin chain with uniform nearest neigh-

bor exchange coupling. This simple system shows, according to the Lieb-Schultz-Mattis

theorem, a degeneracy of the singlet ground state with the triplet state [4]. Assuming

3

4 2. Theory

Figure 2.1: Spin excitations on a homogeneous chain. a) Homogeneous chain. b) Gener-

ation of two spinons (vertical bars) by a spin flip. c) Propagation of the spinon excitation

due to consecutive exchange processes [3].

negligible spin anisotropies even for T = 0, the ground state is gapless and not mag-

netically ordered [5]. Triplet excitations in such a system are described as domain wall like

S= 1/2 spinons (fermions). As illustrated in Fig. 2.1, these mass-less spinons are created

as pairs, e.g., by an exchange process, and can propagate due to consecutive exchange

processes [3].

A quantum phase transition from a gapless critical state into a gapped state can be

introduced for example by dimerization, i.e. an alternation δ of the coupling constants J±nn =

(1± δ)Jnn between nearest neighbors (nn) along the chain, or by a sufficient frustration

α = Jnnn/Jnn due to the next nearest neighbor exchange Jnnn [6]. With dimerization the

spinons are confined into massive triplet excitations. We note in Fig. 2.2 that the singlet

ground state is composed of spin dimers. The resulting quantum disordered ground state

is characterized by short-ranged exponentially decaying spin-spin correlations, where the

resulting lifted degeneracy of triplet and singlet excitations leads to an energy gain of the

system. An excitation in this situation of strong dimerization corresponds to the breaking

of one dimer. The energy related to this process is the singlet-triplet energy gap ∆.

In the following sections we will review in more details some cases of anti-

ferromagnetic arrangement, characterized by a spin gap, which are of interest in the devel-

opment of this thesis, namely the spin-Peierls, spin density waves (SDW) and the frustrated

systems.

2.1 The spin-Peierls transition 5

Figure 2.2: Sketch of spin excitations on a dimerized chain. a) Starting point is the dimer-

ized chain. b) Breaking a dimer corresponds to supply to the system the energy of the

singlet-triplet gap ∆ [3].

2.1 The spin-Peierls transition

2.1.1 Phenomenology of the spin-Peierls transition

The spin-Peierls (SP) transition is an unusual kind of magneto-elastic transition occurring

in a limited number of quasi-1D insulating systems. This transition was predicted almost 40

years ago [7,8] but the experimental evidence came solely with the discovery of the quasi-

1D organic conductors, among which (TTF)(TCNQ) is by far the best known example [9].

In 1993, the first SP transition in inorganic materials was discovered in CuGeO3 [10],

while three years later α′-NaV2O5 has been reported to show a transition affine to a SP

transition [11]. Nowadays, the α′-NaV2O5 transition is understood as a charge ordering

accompanied by a spin-gap opening [12]. Other compounds with possible SP transition

are the titanium-oxyhalide, discussed later in this thesis (chapter 4) [13,14].

A SP transition is characterized by the elastic distortion of the lattice accompanied by

a 1D magnetic ordering. This ordering is called spin-Peierls transition in anti-ferromagnetic

materials since it has strong similarities to the Peierls transition in a quasi-1D metal (Fig.

2.3). In those materials, Peierls demonstrated that, as a consequence of the electron-

phonon interaction, it is energetically favorable to introduce a periodic lattice distortion

of period λ = π/kF , kF being the Fermi wave vector. This distortion introduces a gap at

the Fermi surface, as illustrated for an half-filled band in Fig. 2.3a. The result is a filled

valence band, an empty conduction band and an overall lowering of the energy of the

system. Furthermore, the distorted lattice introduces a periodic potential which is screened

6 2. Theory

Figure 2.3: Sketch of the Peierls transition, comparing the energy dispersion E(k) for a

half-filled band of an uniform chain with that of a dimerized chain. The electronic energy

is lowered as the system undergoes a metal-insulator transition. As a consequence of the

lattice distortion, a periodic potential develops which modulates the electronic charge [15].

by the free electrons, resulting in a charge modulation dubbed as charge density wave

(CDW) (Fig. 2.3b). The spin-Peierls scenario is similar to that of a CDW, where in a chain

of quantum spins, the lattice distortion leads to the formation of spin dimers (Fig. 2.4).

Spin-Peierls systems are insulating at all temperatures contrary to CDW, which might be

characterized by a metal-insulator transition [17].

In a qualitative description, the SP materials can be considered to consist of an


a)

b)

J J JJ . . .

Uniform

Dimerized

J1 J2 J1J2 . . .

Figure 2.4: Sketch of the spin-Peierls transition. a) Uniform Heisenberg AF chain. b) After

the SP transition one obtains an alternating chain (J2/J1 < 1), which is related to a dimer-

ization of the atom’s position (black dots). In this simple model, the dimerization doubles

the unit cell [17].

assembly of quantum spin chains described by the spin Hamiltonian given in eq. (2.1),

with just nearest neighbor (nn) exchange coupling of AFM type. The chains are stacked

parallel to one another and interchain magnetic coupling will be neglected. However, since

the exchange energy of the spin chains is a function of the separation between adjacent

lattice sites, an elastic distortion of the lattice will influence the spin Hamiltonian of the

chains. This effect can be represented by adding to the 1D Heisenberg Hamiltonian (eq.

(2.1)) a rigid lattice term [17]:

H = ∑<i>

Ji,i+1(Si ·Si+1)+ ∑<q,α>

ω0(q,α)b†qαbqα. (2.2)

The sum over the lattice sites j includes nn interaction only, and b†qα(bqα) are the creation

(annihilation) operators for the 3D phonon with wave vector q on branch α. ω0 is the

phonon energy. Since phonons have a 3D character and the exchange energy Ji,i+1 is a

function of 3D spatial separation of the sites j and j +1, the 1D spin interaction depends

on the 3D motion of the lattice sites.

In the simplified picture of Fig. 2.4, as the temperature is lowered, the uniform spin

chain transforms into a dimerized state at a critical temperature TSP. As illustrated in Fig.

8 2. Theory

2.4b, the dimerized state can be visualized as a state where neighboring pair of ions are

displaced by a small amount from their uniform distribution (Fig. 2.4a). Experiments show

that in SP systems, the dimerization proceeds continuously and progressively below TSP

to a maximum value at T = 0 K. Hence, the SP transition is of second order in zero mag-

netic field. The onset of a SP distortion precludes additional magnetic ordering at lower

temperatures. The order parameter is therefore given by the degree of lattice distortion

or equivalent by the magnitude of the magnetic gap. Another important experimental fea-

ture in the organic SP systems is the observed precursive 3D softening of the phonon.

This softening reduces the phonon frequency to about an order of magnitude below its

usual value, and appears to be associated to a structural transition at temperatures much

higher than TSP. For example in TTF(CuBDT) the structural transition occurs at about 225

K, whereas TSP is only 11 K [18]. The preexisting mode softening greatly facilitates the

magnetic ordering of the spins with the consequent dimer formation. As a result, the SP

transition sets-in at relatively high temperatures, due to its advantages with respect to other

magnetic ordering. This is of course true in zero magnetic fields, since high fields might

hamper the dimer formation and lead to a different magnetic state.

Infrared (IR) reflectivity is sensitive to new phonon modes, which might be activated

by the SP transition. Moreover with IR reflectivity, one can probe an eventual softening of

the phonon modes at the Γ point, even though a well defined softening of the modes is

expected just at the boundary of the Brillouin zone. Magnetic excitations are in principle

not detectable by R(ω) since in a typical two magnon excitation the presence of a centre of

inversion inhibits any asymmetric displacement of charge and thus the associated dipole

moment is zero. However, the situation is different if a static or dynamic breaking of the

symmetry is present. For example in 2D cuprates, phonon assisted two magnon excitations

were found [19], while in α′-NaV2O5 a charged magnon was detected [20]. It should be

noted that due to the spin conservation in these latter two cases only those excitations with

∆S= 0 can be probed in R(ω) experiments. It follows that singlet-triplet excitations cannot

in principle be detected by optical experiments. Just in one case, a singlet-triplet transition

was observed in the SP system CuGeO3 at 44.3 cm−1. The nature of this transition was

revealed by a Zeeman splitting in a magnetic field [21].


E (k)

kπ/a− π/a

UniformE (k)

kπ/2a− π/2a

Dimerized

Figure 2.5: Schematic representation of the elementary excitation in a uniform Heisen-

berg AFM chain with lattice period a (left) and an alternating chain (right) (periodicity now

is 2a (see Fig. 2.4)). The heavy dot at k = 0 in the right panel indicates the singlet ground

state [15].

2.1.2 Theory of spin-Peierls transition

In this subsection, we treat the SP transition in a more mathematical way by illustrating

the solution of the Hamiltonian given in eq. (2.2) as proposed by E. Pytte [22]. This simple

approach can be refined by considering the four fermion interactions as done by Cross and

Fisher [23]. First, we ask why it is energetically favorable for an AFM chain to dimerize. The

answer lies in the nature of the excitation spectrum and in the concept of quantum fluctu-

ations. As depicted in Fig. 2.5, the excitation spectrum of an infinite 1D AFM Heisenberg

chain is degenerate with the ground state at q = 0,±π/a. This degeneracy brings some

excited states infinitely close to the ground state. Therefore, quantum zero-point fluctua-

tions of the chain will populate the low-lying excited states. The real state at T = 0 K is

thus a composite of the singlet ground state and excited triplet states. The consequence

is that the Néel state is not a true eigenstate of the Hamiltonian and there is no long range

order. If the chain is dimerized, a gap develops in the excitation spectrum, which lifts the

aforementioned degeneracy (see Fig. 2.5). Now the zero-point fluctuations can no longer

populate the excited states and the net magnetic energy is lowered [17]. In this context

we note that quantum fluctuations decrease very rapidly as |S| increases to values greater

than 1/2. So, one would not expect high spin materials to show a SP transition [17].

For solving the Hamiltonian given in eq. (2.2), one has to describe the spin system

10 2. Theory

with the fermion operators, using the Jordan-Wigner transformation [24,25]:

Ψl = (−2)l−1 Sz1Sz

2 · · ·Szl−1S−l , (2.3)

where S± = Sx± iSy. The fermion operators are defined such that the exchange relation

holds,

Ψl ,Ψ†m= δlm. (2.4)

In terms of these operators, one has:

S+l S−l+1 = Ψ†

l Ψl (2.5)

Szl =

12−Ψ†

l Ψl . (2.6)

The spin exchange can be assumed to depend on the instantaneous position of the mag-

netic ions, so that we can expand it in terms of the lattice displacement of the magnetic

ions u j [22]:

Jj, j+1 = J+∑j[u j −u j+1] ·∇ jJj, j+1 + · · · . (2.7)

It is important that the lattice displacement u j is 3D, so that it couples the 1D chain to the

higher dimension. We remember that the 1D spin systems themselves cannot undergo a

phase transition at non zero temperature [26]. Proceeding with the linear J approximation

(eq. (2.7)), one can Fourier transform u j in terms of the phonon normal modes Q(α,q)

u j = (mN)−1/2∑α,q

e(α,q)eiq·aQ(α,q) (2.8)

where

Q(α,q) = ((2ω0(α,q))−1/2(bα,q +b†−α,q). (2.9)

a is the lattice constant along the chain, e(α,q) is the phonon polarization vector, m is the

mass of the magnetic lattice site, and N is the number of chain sites.

Inserting equations (2.3)-(2.9) in the Hamiltonian (eq. (2.2)), and using the Hartree-

Fock approximation for the fermion-fermion interaction, the Hamiltonian for the uniform

chain is then approximated by [17]:

H = ∑k

EkΨ†kΨk + ∑

kα,q

g(α,q,k)Ψ†kΨk−q(bα,q +b†

−α,q)+∑α,q

ω(q,α)b†q,αbq,α, (2.10)


where

Ek = pJcos(ka)

g(α,q,k) =ip

((2mω0(α,q))−1/2·e(α,q) · (∇ j ·Jj, j+1) · sin(ka)−sin((k−q)a)

p = 1−2/N∑k

nk cos(ka)

nk =⟨

Ψ†kΨk

⟩=

1

(eEk/kT +1).

Equation (2.10) is identical to the Fröhlich Hamiltonian, commonly used to model the con-

ventional Peierls transition in a gas of electrons, with the difference that here the spin is

also considered. This Hamiltonian is usually solved by applying the random phase ap-

proximation (RPA) to the fermion-phonon coupling term. This gives an expression for the

renormalization of the phonon frequency ω [22]:

ω2 = ω20(α,q)+

1N ∑g(α,q,k)

g(α,q,k−q)nk−g∗(α,q,k)nk−q

ω−Ek−q +Ek. (2.11)

As in the case of a conventional Peierls transition, the phonon frequency renormalization is

most significant for q = 2kF [15]. With decreasing temperature, the renormalized phonon

frequency goes to zero and this defines the transition temperature at which the frozen-in

distortion occurs:

ω20(q,q = 2kF) =

g2(α,q,q = 2kF)J2

2π

Z pJ

−pJdE

((pJ)2−E2)1/2

Etanh

E2kBTSP

. (2.12)

In the weak coupling limit kBTSP pJ the expression for the transition temperature (eq.

(2.12)) correspond to the form obtained in the Bardeen Cooper Schrieffer (BCS) theory [22]

kBTSP= 1.14· p·J ·e1/λ, (2.13)

where λ is given by

λ =4g2(α,q,q = 2kF)p2N0

ω20

. (2.14)

N0 = 1/pJπ is the density of states at kF for the fermionic band. Note that pJ here plays

the role of the Fermi energy εF in the BCS formula for the conventional Peierls or CDW

transition (see below eq. (2.26)) and that the effect of the fermion-fermion interaction is

contained in the parameter p. When the weak coupling is not valid, then eq. (2.12) must

be solved numerically in order to obtain TSP. Of course if the coupling is so strong that the

12 2. Theory

resulting lattice distortion is very large, then the assumptions of linearized J dependence

and harmonic phonons have to be abandoned.

The magnetic gap is also given in the weak coupling regime (i.e., kBTSP pJ) by the

BCS formula:

∆(T = 0) = 1.756·kBTSP. (2.15)

∆(T) follows the BCS temperature dependence for the energy gap [17].

An approximative description of the system for temperatures below TSP may be ob-

tained by using an alternating Heisenberg chain [27]:

J1,2 = J(1±δ). (2.16)

In equation (2.9) the normal phonon mode coordinate is replaced by its thermal average

Q(α,q) = 〈Q〉δq,2kF (2.17)

and then one obtains [22]

δ =2g(α,q,q = 2kF)

J〈Q〉 . (2.18)

In the limit of small distortions, the gain in magnetic energy E is proportional to −δ2 ln2δ

[27]. Because the cost in lattice energy is proportional to ∝ δ2, the dimerized state will

have a lower energy.

2.2 Spin density waves

Spin Density Waves (SDW) are broken-symmetry ground states of metals that arise as

a consequence of electron-electron interactions. In contrast to the SP transition, a SDW

does not involve any lattice distortion and the ground state is characterized by a periodic

modulation of the spin density with the period related to the Fermi wave vector kF . For

simplicity, we concentrate in this short theoretical introduction to the one-dimensional metal

case of SDW, where the spin modulation along a chain direction x may be written as [31]

∆S(x) = ∆S0cos(2kFx+φ). (2.19)

2.2 Spin density waves 13

The ground state can be described in the framework of a mean-field approach, where the

SDW is treated as a second order phase transition with thermodynamics similar to that of

a BCS superconducting ground state [32]. The ground state of a SDW is a coherent su-

perposition of hole-electron pairs with opposite spins. A gap develops in the single particle

excitation spectrum. The amplitude of the gap at zero T is related to the SDW transition

temperature TSDW through the BCS relation of eq. (2.15), already observed in SP systems.

The opening of the gap may lead to a metal-insulator transition in the case of a complete

removal of the Fermi surface.

2.2.1 Mean-field treatment of 1D SDW transition

The SDW ground state is thought to arise as a consequence of the electron-electron inter-

action. The simplest possible description of this interaction together with the kinetic energy

term results in the so called Hubbard Hamiltonian which is given in 1D by [31]:

H = ∑k,σ

εk a†k,σak,σ +

UN ∑

k,k′,q

a†k,σak+q,σa†

k′,−σak′−q,−σ. (2.20)

a†k,σ(ak,σ) are the creation (annihilation) operators for an electron state with momentum

k, spin σ and energy εk. U is the on-site Coulomb interaction and N is the number of

electrons per unit length.

We now want to consider the electrons’ response to an external magnetic field H,

in the framework of a mean-field approximation. Considering an external magnetic field,

which varies along the chain direction as H(x) = ∑qHqeiqx, we have to add an extra term

to the Hamiltonian (2.20):

H ′ =−∑q

MqH−q. (2.21)

The additional term (2.21) to eq. (2.20) describes the coupling of the electrons to H(x).

Here, Mq is the qth component of the magnetization. Assuming that H is applied along an

arbitrary direction (z), we denote with ↑ (↓) the spin direction parallel (opposite) to the H

direction. The expectation value for the magnetization is then [15]

⟨Mq⟩

= µB(⟨nq,↑⟩−⟨nq,↓⟩) = Nχ0(q)

(Hq +

U(⟨nq,↑⟩−⟨nq,↓⟩)

2µB

)= Nχ0(q)He f f

q ,

(2.22)

14 2. Theory

where nq,↑ is the electron density and χ0(q) is the susceptibility in the absence of Coulomb

interactions. Equation (2.22) has to be solved self-consistently for the difference ∆nq =⟨nq,↑⟩−⟨nq,↓⟩. The magnetization reduces to:

⟨Mq⟩

= Nχ0(q)

1−Uχ0(q)/2µ2B

Hq = Nχ(q)Hq. (2.23)

For a uniform magnetization (i.e., q = 0) and with χ0(0) = 2µ2Bn(εF), the static suscepti-

bility

χ(0) =2µ2

Bn(εF)1−Un(εF)

(2.24)

turns out to be enhanced by the Stoner factor (1−Un(εF))−1.

For a 1D electron gas, χ(q) is strongly peaked at q = 2kF and the enhancement is

more important for perturbations with this wave vector. The temperature dependence of

χ0(q = 2kF ,T) is given by χ0(q = 2kF ,T) = n(εF) ln(ε0/kBT) [15] with a cut off energy

ε0 of the order of the Fermi energy. Neglecting the fluctuations, one can define the critical

temperature (TSDW) for a SDW as the temperature where the denominator in eq. (2.23)

diverges, i.e.

Uχ0(2kF ,T)2µ2

B

= Un(εF) ln(ε0/kBTSDW) = 1. (2.25)

This gives

kBTSDW = 1.14εFe−1/λ. (2.26)

The dimensionless electron-electron coupling in eq. (2.26) is given by λ = Un(εF). Equa-

tion (2.26) has the same form as the BCS relation for Tc of superconductors [32].

Below the transition temperature TSDW a spatially varying magnetization develops

which may be described by introducing the spatially dependent operators [15,31]

ψσ(x) =1√V

∑k

eikxak,σ, (2.27)

so that the spin density turns out to be:

S(x) =12

(ψ†↑(x)ψ↑(x)−ψ†

↓(x)ψ↓(x))

=1

2V ∑k,k′

(a†

k,↑ak′,↑−a†k,↓ak′,↓

)e−i(k−k′)x. (2.28)


Figure 2.6: a) The energy dispersion relation for a one dimensional SDW material below

the phase transition. The opening of the gap at ±kF is clearly visible. b) A SDW viewed

as two CDWs, one for the spin-up and another for the spin-down sub-band, which are

spatially out of phase by π [15,31].

Since in 1D the response at q = 2kF is divergent, we assume that only terms with k′ =

k±2kF are important. Thus, one obtains for the spin expectation value [15,31]

〈S(x)〉 =1

2V ∑k

(⟨a†

k,↑ak+2kF ,↑

⟩−⟨

a†k,↓ak+2kF ,↓

⟩)e+i2kFx +c.c

= 2|S|cos(2kFx+φ), (2.29)

where φ and |S| are defined by:

S≡ |S|eiφ =1V ∑

k

⟨a†

k,↑(↓)ak+2kF ,↑(↓)

⟩. (2.30)

We see from eq. (2.29) that the spin density wave is periodically modulated with a period

λ0 = π/kF .

In a more phenomenological manner, we can represent the SDW ground state as

two charge density waves, one for the "spin-up" and one for the "spin-down", as shown in

16 2. Theory

Figure 2.7: Temperature dependence of the dc conductivity as a function of the inverse

temperature for three organic compounds showing a SDW transition at low temperature

[31].

Fig. 2.6b. Each charge density wave has a modulation given by [15,31]:

ρ↑(x) = ρ0

(12

+∆

VFkFλcos(2kFx+φ)

)(2.31)

ρ↓(x) = ρ0

(12

+∆

VFkFλcos(2kFx+φ+π)

). (2.32)

The resulting spin density variation ρ↑(x)−ρ↓(x) is given by eq. (2.29), with |S|= N∆/U ,

while the charge variation ρ↑(x)+ρ↓(x) = ρ0 is constant as shown in Fig. 2.6. Both mod-

ulations (ρ↑(x) and ρ↓(x)) are tied to the Fermi surface and this will have important conse-

quences on the excitations and on the dynamics of a SDW ground state.

2.2.2 The electrodynamic response of a 1D SDW

SDW condensates couple to electromagnetic fields and the fluctuation of the phase φ (eqs.

(2.31)-(2.32)) leads to an electric current. Since the development of a SDW leads to the


Figure 2.8: The frequency dependent σ1(ω) in (TMTSF)2PF6. Above TSDW = 11.5 K, the

low frequency σ1(ω) corresponds to a Drude term, while below TSDW the low frequency

peak is due to the pinned mode. The inset shows the frequency dependence of σ1(ω) and

dielectric constant ε1 for a SDW. The inset is appropriate for ω0 < 2∆ and for the clean

limit 1/τ < 2∆ [31].

opening of a gap at the Fermi surface and in the case of a total removal of the Fermi

surface to a metal-insulator transition, the SDW transition can be detected by measuring

the dc conductivity (σdc(T)). Figure 2.7 shows ρ(T) = 1/σdc(T) of three organic com-

pounds which undergo a SDW transition. Below TSDW, the conductivity is well described

by σdc(T) = σ0exp(−∆/kBT).

Carrier excitations across the gap lead to an electromagnetic absorption with the on-

set frequency ω = 2∆/~, similarly to what is observed in a superconductor, where the BCS

theory predicts zero absorption for photon energies smaller than the energy gap value.

Contrary to an s-wave superconductor, where Anderson demonstrated that superconduc-

tivity is not affected by non-magnetic impurities [33], in a SDW state, impurities interact

with the SDW collective mode. Such an interaction leads to a collective mode pinned to

the impurity with a resonance frequency ω0 and 1/τ damping. The interaction between the

18 2. Theory

collective mode and the impurities is described in an oversimplified manner by a restoring

force k = ω20 ·m∗. The equation of motion for the related φ phase is:

d2φd2t

+1τ

dφdt

+ω20φ =

n·em∗ E(t). (2.33)

In the presence of an acelectric field E(t) = E0eiωt , one obtains for the contribution of the

pinned mode to the optical conductivity:

Re σ(ω) =n·em∗

ω2/τ(ω2

0−ω2)2 +(ω/τ)2(2.34)

Im σ(ω) =n·em∗

ω(ω20−ω2)

(ω20−ω2)2 +(ω/τ)2

. (2.35)

These formulae correspond to the Lorentz model (see below eq. (3.14)) and are illustrated

in the inset of Fig. 2.8. In the clean limit 1/τ < 2∆ (where 2∆ is the gap of the system),

which is appropriate in the quasi-1D organic conductors like (TMTSF)2PF6, the optical

conductivity is zero for ω < 2∆. Thus for ω < 2∆, one expects in the absorption spectrum

σ1(ω) only contributions from the pinned collective mode at ω = ω0. Contrary to a CDW

where the many body interactions normalize the electron mass, in a SDW system, the

effective mass should be equal to the band mass. Therefore, all the spectral weight should

totally transfer into the collective mode [31]. However, optical data for (TMTSF)2PF6 are

suggestive of a moderate enhancement of the electron mass [34].

Experimental results in (TMTSF)2PF6 (Fig. 2.8) are in good agreement with the phe-

nomenological description of the electrodynamic response of a SDW . Above the transition

temperature TSDW= 11.5 K [35], the system behaves like a Drude metal, while in the SDW

state, the spectral weight is suppressed below the energy gap of 21 cm−1, as determined

by the analysis of σdc(T). Simultaneously in the microwave region, one recovers a reso-

nance which is associated to a pinned SDW mode, with a pinning frequency ω0 smaller

than the single particle gap.

2.2.3 The Fermi surface nesting

The description of the SDW transition presented until now is based on a strict 1D scenario.

However, materials that develop a SDW have higher dimensions. Of basic importance in

the SDW systems is the topology of the Fermi surface (FS) and its "nesting" capability

(see Fig. 2.9 and below). The "nesting" vector Q connects a part of the Fermi surface

2.3 Frustrated systems 19

BZ

Q

Quasi-1D

BZ

Q

1D

BZ

Q

Chromium (Cr)

Hole FS Electron FS

H

Γ.

.

Figure 2.9: Possible configurations for the nesting process in a perfect 1D material, in a

quasi-1D system and in the case of chromium, a 3D material [2].

with another one and is defined by the nesting condition −εk = εk+Q. In Fig. 2.9, some

examples of nesting are presented: in the left panel one sees the case of a perfect nesting

in an ideal 1D metal, where nesting is always present with Q = 2kF . The center of Fig.

2.9 illustrates the more realistic case of the quasi-1D metal, where the small interchain

coupling t⊥ (perpendicular to the chain) causes a warping of the Fermi surface, reducing

the fraction of the nested Fermi surface. On the right of Fig. 2.9, the Fermi surface nesting

of Chromium (Cr) is displayed. Cr is a 3D metal where the nesting occurs between a hole

type FS in the corner of the Brillouin-zone (BZ) and an electron like FS around the Γ-point.

At nesting, one assumes a linear energy dispersion around the Fermi energy. With

this assumption, the spin susceptibility χ0(q) is characterized by a logarithmic divergence

[2], which manifests for q→ Q. Like the 1D case [2], the divergence of χ0(q) at q = Q

leads via eq. (2.23) to a magnetic instability, which results in a magnetic ordering with

wave vector Q. This magnetic ordering is a SDW of the form S(r) = |S|cos(Q· r) [2].

2.3 Frustrated systems

Initially, condensed matter physics was about systems with a small number of accessible

ground states, because this feature made the calculations easier. However, in his work on

ice, Pauling presaged that some systems could have a thermodynamically-large number

of accessible ground states [39]. "Thermodynamically-large" means that the number of the

20 2. Theory

available states grow exponentially with N, the number of atoms in the system. The discov-

ery of spin glasses and the realization that its slow dynamics is due to relaxation among

a large number of nearly degenerated ground states changed the concept of the ground

state. From that time on, the competition among different ground states was dubbed as

"frustration" [40]. Also the so called bond frustration in structurally-periodic systems can

lead to a thermodynamically-large number of ground states. Bond frustration means that

a finite fraction of two body interaction cannot simultaneously exist in their lowest energy

configuration.

An ideal context to test ideas of frustration is provided by magnetism in solid, due to

the unique attributes of magnetic materials, namely:

– the variety in the size and dimension of the effective atomic moment. Low spin values

lead to enhanced quantum effects.

– The variety in spin-spin interactions. These include dipole-dipole coupling, direct ex-

change, indirect exchange, super-exchange, itinerant exchange and anisotropic ex-

change. The range of those interactions extends beyond nearest neighbor distances

and can be anisotropic, leading to low effective spatial dimensionality.

– The ability to couple directly to individual moments with magnetic field.

This large variety of model parameters allows systematic studies of several interesting

problems, which also have analogues outside the domain of magnetism.

The frustrated materials form a class on their own, in the sense that they share com-

mon macroscopic and microscopic properties. Macroscopically, they display properties

characteristic of a thermodynamically-large number of ground states. Microscopically, they

are characterized by particular magnetic lattices and isotropic spins. Despite the advances

in the understanding of the insulating magnets, and contrary to the non frustrated magnetic

insulators, cooperative effects in frustrated systems are still less well understood [41].

2.3.1 Lattice properties

In the following section we analyze in more details some lattice properties and geometries,

which are responsible for geometrically frustrated systems. The geometry of the lattice


a) c)b)

AFM

AFM

AFM AFM

AFM AFMFM

AFM

AFM AFM

Figure 2.10: a) Frustration due to a defect. The ferromagnetic bond is representative of

a ligand defect in an otherwise periodic anti-ferromagnetic (AFM) lattice. It follows that

the upper right spin has two equal-energy directions. b) Geometrical frustration on a

triangular lattice. There are six equivalent energy configurations per triangle. c) A 120

state on a triangular lattice.

is an essential component in realizing frustration, since frustration often arises from the

incompatibility of the local anti-ferromagnetic (AFM) interaction with the global symmetry

imposed by the crystal structure.

Figure 2.10a illustrates the role of disorder by producing geometrical frustration on

a square lattice of spins. If all spins are coupled with an AFM interaction, each bond’s

energy can be individually minimized. However, if one of the bonds is made ferromagnetic

(FM), then one spin is frustrated as it is unable to satisfy the constrains imposed by the

neighbors. When FM bonds are introduced randomly, one obtains a prototype of frustration

from quenched disorder. Such a system has been realized, for example, in the spin-glass

La1−xSrxCuO4 [42].

In the triangular geometry, illustrated in Fig. 2.10b, each interaction is AFM, giving

rise to a frustration of the third spin. Once two spins are accommodated on a triangular

lattice, the third cannot realize a perfect AFM arrangement, as required by the magnetic

interactions. This kind of frustration occurs due to the incompatibility of the two body inter-

action with the lattice’s geometry. The basic difference with respect to the squared lattice

is that in the triangular geometry it is not necessary to invoke impurities to produce frus-

tration. It is entirely given by the special geometry of the lattice.

If the minimum energy for the two spin interaction occurs at angles of 0, 120 and

240, then an ordered state can form. This state is characterized by a difference in the

22 2. Theory

orientation of neighbors spins of 120, as illustrated in Fig. 2.10c. Often the two spin inter-

action is not Ising-like as assumed in Figs. 2.10a and 2.10b but has rather a continuous

symmetry (Heisenberg or XY), which enables a discretionary orientation of the spin in

the plane. The 120 state, depicted in Fig. 2.10c, is important from a theoretical point of

view, since it is used to test spin-wave calculations and furthermore has been observed

experimentally in some triangular systems, like CsMnBr3 [43].

The triangular lattice is the basic situation for illustrating geometrical frustration. How-

ever, frustration can be modeled on many different lattices, simply by adjusting the range

and sign of the magnetic interactions. An example is the axial next nearest neighbor Ising

chain in one dimension (1D), where J1 and J2 are the nearest neighbor (nn) and next

nearest neighbor (nnn) couplings in a Hamiltonian expressed as:

H = J1∑i

SiSi+1 +J2∑i

SiSi+2. (2.36)

Although a 1D chain cannot exhibit long-range order at finite temperature, such a model

is frustrated for J1/J2 = −1/2 [44], where the lowest energy ground states do not si-

multaneously satisfy all individual magnetic constraints. Other examples, with the same

Hamiltonian (eq. (2.36)), are the isotropic Heisenberg- or XY chain for AFM interac-

tions (i.e., J1,J2 > 0). The XY model is obtained from the Heisenberg Hamiltonian by

turning off the coupling between the z component of the spins [45]. Depending on the

ratio between J1 and J2, this class of materials may exhibit a gapless collinear phase

(J2/J1 < αC1), a gapped phase ( αC1 < J2/J1 < αC2), or a quasi-long-range order spiral

phase (αC2 < J2/J1). For the XY chain, the critical values are found to be αC1 ' 0.33and

αC2 ' 1.26 [46–48], while for the isotropic Heisenberg chain one obtains αC1 ' 0.24 [49].

Moreover, the gapped state persists up to high αC2 [50]. On the contrary, the model of

eq. (2.36) with the FM nn and the AFM nnn interactions (i.e., J1 < 0, J2 > 0) is less well

understood [51]. It is established that the ground state is ferromagnetic for J2/ |J1|< 1/4,

while it is an incommensurate singlet state for J2/ |J1|> 1/4 [51].

2.3.2 Frustration parameter f

The results on Ising systems suggest that the suppression of the ordering temperature

may be used as an empirical measurement of frustration. In theoretical models for the

triangular AFM, no order is found at non zero temperatures. However in real systems,


AFM

FM

PM

1/χ(T )

TTC

Figure 2.11: Characteristic behavior of the inverse susceptibility versus temperature for

non frustrated spin systems with anti-ferromagnetic (AFM), ferromagnetic (FM) and zero

nearest neighbor coupling (paramagnetic) [40].

anisotropy and long-range interactions can overcome frustration and produce long range

order. We can thus view the occurrence of a long range order as a failure of the systems

to support the frustrated state. A useful parameter for judging the degree of frustration

is the quality factor, defined as the ratio between the expected ordering temperature and

the observed ordering temperature. For the expected ordering temperature, one takes the

Weiss constant θW, obtained from the fit of the experimental data with the Curie-Weiss

expression for the susceptibility [52]:

χ(T) =CCurie

T−θW. (2.37)

CCurie = (µBg(S(S+ 1)))2/3kB, µB is the Bohr magneton and g is the Landé factor. Ex-

amples of 1/χ(T) for conventional magnets are shown in Fig. 2.11 for different magnetic

interactions (AFM or FM). The type of the interactions determines the sign of θW (i.e., -

for AFM and + for FM interactions). In the FM case the expected critical temperature is

TC = θW, since χ(T) diverges at TC. For AFM usually TN, the Néel temperature, is lower

than |θW|. We can thus define the frustration parameter f as:

f =θW

TN. (2.38)

24 2. Theory

TN represents the transition temperature of either an AFM or a spin glass transition. Typical

values for the f parameter is of a few units for common AFM compounds, like transition

metals oxides (e.g. fFeO∼ 2.9 or fMnO ∼ 5) [53]. Some 2D magnets with a triangular

magnetic lattice, like VCl2, NaTiO2 or the 3D magnets with a pyrochlore lattice, show a

frustration parameter of the order of tens. The 2D magnet Na0.75CoO2 (see chapter 6)

shows a frustration factor of f ≈ 7.5−10 [54, 55], while f ≈ 4 in LiCu2O2 (see chapter

5) [56]. The mean-field solution for anti-ferromagnets with two sublattices yields f = 1 [57].

Taking into account a larger number of sublattices and adding higher order of nearest

neighbor interactions, the f parameter increases. However, Haar and Lines suggest that

mean field theory breaks down at f ∼ 10, where the number of sublattices and the order

of nearest neighbor exchange is no longer realistic [58].

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3 Experimental technique

A powerful experimental tool for studing the elementary excitations of an N-particle sys-

tem is optical spectroscopy including ellipsometry, Kerr rotation, Raman scattering, optical

transmission and reflectivity. This last technique consists in shining electromagnetic ra-

diation of different frequencies onto the sample, and by observing at which frequencies

there is absorption or reflection by the material. The obtained reflectivity contains a lot of

useful informations on the elementary excitations of the system under investigation. At low

photon energies, the optical spectroscopy technique allows to investigate low energy col-

lective modes, the lattice dynamics, as well as the behavior of the electronic states close to

the Fermi energy. At high enough energies (of the order of eV), reflectivity gives valuable

informations on the band structure, since the incoming light may excite an electron from

an occupied state below the Fermi energy to a band above it. The multiplicity of effects

observable and the extremely broad investigated spectral range connected with the fact

that one gets different informations simultaneously, make reflectivity technique more and

more attractive for investigations in solid state physics.

In the context of analyzing reflectivity data, a brief sketch of the theoretical back-

ground is in order. First we concentrate on theoretical aspects, giving some definitions

about optical functions followed by an overview of the experimental set-up and facilities.

Thereafter, we discuss briefly the Lorentz-Drude model and its extension for the Drude

term (the so called "generalized Drude model"). At the end of the chapter we concen-

trate on a phenomenological model based on Fano’s calculation, which takes into account

anharmonicities of the spectra [1].

29

30 3. Experimental technique

3.1 The optical functions

In order to interpret and analyze the reflectivity data we need to calculate dielectric

functions which are mostly related to the electronic structure of the solid [2]. The focus

here is primarily on definitions, as detailed derivations may be found elsewhere [2,3]. The

dielectric response of the investigated materials under the action of an electromagnetic

wave is described by the Maxwell equations [2,4]:

∇ ·D = 0

∇ ·B = 0

∇×E = −1c

B

∇×H =1c

(D+4π j

). (3.1)

In eqs. (3.1) the standard definitions D = E + 4πP, P = αE, B = H + 4πM, M = χH

and j = σE are used. α, χ and σ describe the polarizability, susceptibility and electrical

conductivity, respectively. These equations (eqs. (3.1)) can be combined to give a wave

equation for the electric field:

∇2E =εµc2

∂2

∂2tE +

4πσµc2

∂∂t

E. (3.2)

Analogously, an equivalent differential equation can be obtained for the magnetic field B.

The differential equation (3.2) can be solved with the plane wave assumption

E(t, r) = E0ei(ωt−q·r), (3.3)

whereby q is the complex wave vector. Inserting eq. (3.3) in eq. (3.2) leads to

q2 = µω2

c2

(ε+ i

4πσω

). (3.4)

We introduce the complex refractive index n via

q =ωc· n =

ωc· (n+ ik). (3.5)

Using n, we can define the complex dielectric function as ε = ε1− iε2 = n2 and the complex

optical conductivity σ = σ1 + iσ2, which is related to ε by ε(ω) = 1− i4πω σ(ω). Separating

3.1 The optical functions 31

the real and the imaginary part for ε and σ, one obtains:

ε1(ω) = 1+4πω

σ2(ω)

ε2(ω) =4πω

σ1(ω). (3.6)

The complex electric field Er reflected on the sample surface is given by Er = rE i

where Ei is the incoming field and r is complex reflectance:

r = ρ ·ei∆ =n−1n+1

. (3.7)

In equation (3.7), ∆ describes the phase and ρ the reflectance amplitude. Of interest is a

measurable quantity, namely the optical reflectivity, defined as the ratio of the intensities of

the light beam reflected by the samples IR and by a reference mirror I0:

R=IRI0

= |r|2 = ρ2 =(n−1)2 +k2

(n+1)2 +k2. (3.8)

Evaluating the optical functions in eq. (3.6) via eq. (3.7) requires the knowledge of

both the phase ∆ and the reflectance amplitude ρ. The phase ∆ can be calculated starting

from the measured reflectivity R. To this end, one makes use of the Kramers-Kronig (KK)

relations, which are integral relations connecting the real and the imaginary part of com-

plex functions, such as ε, n and r . The physical basis of the KK relations is the causality

principle. Causality means that a response cannot occur before an external stimulus. In

our experiment the stimulus is the E field of the incident light and the response is the elec-

trons’ and lattice motions. Following Ref. [2], the transformation of the reflectivity R(ω),

determining the phase ∆, is:

∆(ω) =ωπ

Z ∞

0

ln [R(ω′)/R(ω)]ω2−ω′2 dω′. (3.9)

Once the phase has been determined, it is straightforward to calculate the components of

the dielectric function [2]:

ε1(ω) =(1−R(ω))2−4R(ω)sin2∆(ω)(1+R(ω)−2

√R(ω)cos∆(ω)

)2

ε2(ω) =4(1−R(ω))

√R(ω)sin∆(ω)(

1+R(ω)−2√

R(ω)cos∆(ω))2 . (3.10)


Figure 3.1: Electromagnetic spectrum: our experiments use infrared, visible and ultravio-

let radiation.

The limits of the integration in (3.9) are obviously well beyond experimental reach, so

it is necessary to extrapolate measured data to (very) low and high frequencies. Clearly

the broader the spectral range of data is, the more accurate the transformation will be.

Towards the dc limit (ω = 0) either a constant value for an insulator or the Hagen-Rubens

extrapolation for a metal

R(ω) = 1−2

√268.67·ω

σDC, (3.11)

with the units [ω] = eV and [σDC] = (Ω cm)−1, is assumed. At large energies, the reflec-

tivity is assumed to decay as ω−s, s= 2 up to twice the frequency of the last data point.

This part should account for further interband contributions. At higher frequency, s= 4 is

intended to simulate excitations into the continuum.

3.2 Reflectivity measurements

This thesis contains reflectivity data collected over a broad spectral range varying from

the far infrared (FIR) (4 meV ≡ λ ∼ 4 ·10−2 cm (Fig. 3.1)) up to the ultraviolet (UV) (12.4

eV ≡ λ ∼ 10−5 cm (Fig. 3.1)). Additional informations are obtained performing optical

experiments with linearly polarized light, in order to probe the possible crystallographic

anisotropies of the samples. The spectral range is covered using different spectrometers

and measurement techniques, with overlapping frequency ranges. In this section we review

in more details each technique.

3.2 Reflectivity measurements 33

DETECTOR

SOURCE

Sample

Figure 3.2: Sketch of a Michelson interferometer.

In the infrared (IR) region, R(ω) was measured with a Bruker IFS 113v Fourier spec-

trometer. The Fourier transform spectroscopy is based on the Michelson interferometer

principle. Light is split at the beam splitter in two different paths. A movable mirror changes

the path’s length for one way (see Fig. 3.2). The two different paths merge then together at

the beamsplitter. The resulting modulated intensity as a function of the optical path differ-

ence δ is called interferogram (I(δ)). δ is measured with a laser. The interferogram I(δ) is

thus sent onto the sample or onto the reference mirror (see Fig. 3.2) and the reflected in-

terferogram IR(δ) is detected. The recorded IR(δ) is Fourier-transformed in order to obtain

the frequency dependence of the reflected intensity IR(ω) [2].

Our Fourier spectrometer is equipped with an Oxford cryostat filled with liquid He

and equipped with a magnet, which permits to apply a magnetic field up to 8 T perpen-

dicular to the optical surface.The temperature can be varied from 250 down to 1.5 K and

is maintained constant ±0.1 K during the measurement with an electronic control system.

The samples are mounted on a sample holder together with a piece of tungsten as the

reference mirror. The optical surface to be irradiated might be varied by choosing different

aperture’s of the sample holder (diameter varies from 1 to 2 mm).

The cryostat is equipped with two sets of windows in order to cover the whole far and


mid infrared spectral range:

– In the far infrared (FIR), the electromagnetic radiation is provided by a Hg lamp.

The beamsplitter in the Michelson interferometer is made of Mylar and the cryostat

optical windows of Polytene, which ensures high transmission in FIR. The reflected

light is detected with a He cooled bolometer.

– The mid infrared (MIR) radiation is generated by a glow-bar light source and the

optical windows are made of KBr (potassium bromide) which is the most commonly

used material for IR spectrometers. The MIR light is detected with a MCT (HgCdTe)

detector cooled by liquid nitrogen.

Furthermore as a check, the measurements are repeated at room temperature with a

Bruker IFS 48 Fourier spectrometer in the IR range (40-600 meV), in order to verify the

consistency of the overlaps among different spectral regions.

Reflectivity in the higher energy range (0.5-5 eV) is measured with a home made

spectrometer based on a Zeiss monochromator, where a turning light guide allows to

measure the intensity of the reflected beam and the intensity of the incident light (see

Fig. 3.3). This mechanism allows a direct measurement of the reflectivity without the use

of a reference mirror. The high energies reflectivity spectra (4.13-12.4 eV) are measured

with a commercial McPherson 225 spectrometer, where a double beam technique avoids

the measurement of the reference mirror, as well. The last two spectrometers provide room

temperature data only. However, by experience, the magnetic field and temperature depen-

dence of R(ω) is negligible at such high photon energies. In fact, the reflectivity spectra at

different temperatures and fields already merge in the MIR spectral range for all materials

treated in this thesis.

3.3 The Lorentz-Drude model

The Lorentz-Drude (LD) model is a simple phenomenological approach based on the clas-

sical dispersion theory. Even though it predates quantum mechanics, it remains useful due

to direct parallels in quantum mechanical theory. In this model, electrons are bound to the

lattice with a restoring force mω 2j r j at position r j and are characterized by a damping

3.3 The Lorentz-Drude model 35

DetectorTurning

light guide

Sampleholder

light-

source

a)

Detector

Sampleholder

light-

source

b)

Figure 3.3: Sketch of the measurement’s principle of the home built spectrometer for the

IR and visible spectral range. a) Measurement of the reference light intensity: the sample

holder is in a low position so that the light goes directly into the light guide and detector. b)

The sample holder is in the up position and the light guide is turned to collect the reflected

intensity.

terms Γ, which accounts for energy loss due to various scattering mechanisms. Further-

more, the electrons are driven by the local electric field E, so that the resulting equation of

motion is:

m∗j

d2r j

dt2+m∗

j Γ jdr j

dt+m∗

j ω2j r j = eE. (3.12)

m∗j is the effective mass of the jth electron and e is the electron charge. The electric field

can be taken to vary in time as E ∼ eiωt , thus the solution of eq. (3.12) is:

r(ω) =−eE

m1

ω 2j −ω2 + iΓ jω

(3.13)

The electron’s displacement generates a dipole moment which multiplied by Nj , the density

of electrons bound with resonance frequency ω j , generates a macroscopic polarization


σ1 (ω) σ2 (ω)

ωωj

Figure 3.4: Frequency dependence of σ1(ω) and σ2(ω) for one harmonic oscillator at

resonance frequency ω j .

P =−Njer = 14π(1− ε)E. Using eqs. (3.6) one obtains for the optical conductivity:

σ j(ω) =ω 2

p j

4πiω


. (3.14)

The plasma frequency, ωp j =√

4πNje2/m ∗j is defined as a mode strength of the jth

harmonic oscillator with resonance frequency ω j and is bound by the condition ∑ j Nj = N,

where N is the total number of electrons per unit volume. Figure 3.4 shows the frequency

dependence of the real and imaginary part of the optical conductivity obtained from eq.

(3.14).

Since free electrons have no restoring force, we can easily derive the Drude model

for free electrons by placing ω j = 0 in equation (3.14) [2]. Summing over all contributions

j , one obtains the LD formula for fitting the optical conductivity:

σ(ω) =14π

(ω2

pD

ΓD− iω+∑

jω 2

p jiω


), (3.15)

where the first term on the right hand side is the Drude contribution and the second one

accounts for all finite energy excitations.

3.3 The Lorentz-Drude model 37

3.3.1 The generalized Drude analysis

Although the classical Drude model turns out to be appropriate for the electrodynamic

response of the vast majority of simple metals, the Drude formula is applicable only at

low frequencies and low temperatures, where elastic scattering from impurities and weak

quasielastic scattering from thermally activated excitations, such as phonons, dominates

[5, 6]. In order to extend the Drude formula and make it applicable also for other more

complex systems, the optical conductivity for free charge carriers can be generalized by

making the scattering rate complex and frequency dependent. Therefore, one introduces

the memory function M(ω,T) = Γ(ω,T)− iωλ(ω,T) [8] which takes the place of Γ in

the Drude term of eq. (3.15). The Γ(ω,T) is the new frequency dependent scattering

rate while λ(ω,T) describes the mass enhancement of the quasiparticles due to many

body interactions. Dropping the temperature dependence and defining the charge carriers

effective mass m∗(ω)/mb = 1+λ(ω) one obtains for the Drude optical conductivity:

σD(ω) = σ1D(ω)+ iσ2D(ω) =ω2

p

4π1

Γ(ω)− iωm∗(ω)mb

. (3.16)

As a result the optical conductivity is now composed by an infinite set of Drude peaks, each

describing σD(ω) in the vicinity of a particular frequency ω with a set of parameters Γ(ω)

and m∗(ω)/mb. By inverting eq. (3.16), one recovers the scattering rate and the effective

mass ratio:

Γ(ω) =ω2

p

4πσ1D

|σD|2(3.17)

m∗(ω)mb

=ω2

p

4πσ2D

ω |σD|2. (3.18)

We see from eqs. (3.17) and (3.18) that an exact knowledge of the Drude component

is necessary in order to calculate Γ(ω) and m∗(ω)/mb. σD(ω) can be isolated from the

optical conductivity σ(ω) in eq. (3.15) by writing [7]:

σ(ω) = σD(ω)+iω4π

(ε∞−1). (3.19)

The term iω4π(ε∞−1) accounts for all the contributions at higher energies (interband transi-

tions), and the constant ε∞ is the value of the real part (ε1(ω)) of the dielectric function at

ω ' ωc. The cut-off frequency ωc coincides with the onset of electronic interband transi-

tions. The generalized Drude optical conductivity is obtained by subtracting this term from


σ(ω), obtained from the Kramers-Kronig transformations. As a final remark, we mention

that the correction of the Drude part (eq. (3.19)) was neglected in the early nineties, when

this model was widely employed in the analysis of the optical response of the high Tc su-

perconductors. Only recently it was realized that this small correction has to be considered

as well.

3.4 Fano’s problem

A basic assumption in the derivation of the Lorentz-Drude model is that the dielectric func-

tion is the sum of contributions of independent oscillators, where no interaction between

them is considered. We will now try to extend the model within Fano’s treatment [1] by

taking into account the interaction between a localized state and a continuum of states,

and derive a modified function to fit the optical conductivity. This treatment can describe

the asymmetric shape of some lines in the absorption spectrum of several material.

The starting point is the unperturbed system characterized by a localized state φ and

a continuum of states ψE′ at energies E′. φ and ψE′ are eigenstates of the unperturbed

Hamiltonian H0 = Hcontinuum+ Hlocalized. If we switch on the interaction between the lo-

calized state and the continuum (H = H0 +Hinteraction), the eigenstates of the resonance

takes the form [1]

Φ = φ+PZ

dE′VE′ψE′

E−E′ , (3.20)

where VE defines the interaction strength between the localized state and the continuum

(〈ψE|H|φ〉=VE). P denotes the Cauchy principal value of the integral. The form of the new

resonance Φ indicates that the state φ is modified by an admixture of continuum states.

Note that this mixing shifts the resonance position and adds an imaginary part to Φ.

Analogously the continuum states, which diagonalize the Hamiltonian H, are also modified

upon interaction as follows:

ΨE = a φ+Z

dE′bE′ψE′ =sin∆πVE

Φ+cos∆ ψE (3.21)

3.4 Fano’s problem 39

with

a =sin∆πVE

(3.22)

bE′ =VE′

πV∗E

sin∆E−E′ −cos∆ δ(E−E′) (3.23)

∆ = −arctanπ |VE|2

E− (Eφ +F(E)). (3.24)

Eφ is the energy of the unperturbed localized state φ and F(E) describes the shift in the

energy of the resonance Φ with respect to φ. ∆ is the phase shift due to interaction and de-

scribes the mixing of the ψE and Φ states to generate ΨE. Indeed, the new continuum ΨE

in eq. (3.21) is given by a projection of the resonance Φ and the unperturbed continuum

ψE.

We can analyze two special cases:

– VE → 0 i.e, no interaction. Following eq. (3.24) one approximates ∆ ∝ |VE|2, sin∆πVE

∝

VE → 0 and cos∆ → 1, so that in eq. (3.21) the new continuum reduces to the

unperturbed continuum ΨE = ψE. Also the resonance Φ (eq. (3.20)) reduces to the

localized state φ.

– At E ∼ Eφ + F(E) one recovers ∆ ∼ π/2 and the continuum eigenstate is ΨE =

Φ/πVE, i.e., the eigenstate of the new continuum is entirely given by the resonance

Φ.

We can now consider the variation of the probability of excitation from an initial state i

in the stationary state ΨE. Whatever the excitation mechanism is, this probability may be

represented as the squared matrix element of a suitable transition operator T between the

states i and ΨE. The ratio of the transition probability into the new perturbed state with

respect to the old unperturbed one is given by [1]:

|〈ΨE|T|i〉|2

|〈ψE|T|i〉|2=

(q+ ε)2

1+ ε2 . (3.25)

This formula is obtained with some mathematics and using the following definitions:

ε = −cot∆ =E−Eφ−F(E)

π |VE|2=

E−Eφ−F(E)Γ/2

(3.26)

q =(Φ|T|i)

πVE (ψE|T|i), (3.27)


where ε defines a reduced energy and q is the power of absorption of the resonance Φ.

This parameter depends only on the characteristic resonance Φ, on the transition operator

T and on the interaction VE.

Fano’s formula (eq. (3.25)) was used by Davis and Feldkamp to describe the changes

in the dielectric tensor ε2 due to the interaction between a localized state and the conti-

nuum [9]. After the subtraction of a background, they obtained for the optical conductivity

of a single oscillator [9,10]:

σ1 j(ω) = ωRj

((q j + ε j)2

1+ ε2j

−1

)(3.28)

where analogously to eq. (3.26), ε j = (ω−ω0 j )Γ j/2 is defined as the difference between the

photon- and the resonance frequency. Γ j is the scattering rate and Rj is a constant pro-

portional to the squared plasma frequency. The imaginary part of the conductivity tensor

is obtained through KK transformation of eq. (3.28) [9,10]:

σ2(ω) = ωRj

((q2

j −1)(−ε j)+2q j

1+ ε2j

)(3.29)

Fano’s formula proposed by Davis and Feldkamp has unfortunately a different power

law decay at higher energies (i.e., σ1(ω) ∝ 1/ω for ω→∞) than the Lorentz-Drude formal-

ism. The consequence is that the spectral weight integral (SW=R

σ1(ω)dω = ∑ j ω2p j/8)

does not converge. To overcome this inconvenience, Damascelli et al. proposed to re-

place ε j in eqs. (3.28) and (3.29) with x j = (ω2−ω20 j)/Γω [11]. We see that in the region

around the resonance (i.e., ω ≈ ω0), x j reduces to ε j . Furthermore the complex optical

conductivity of eqs. (3.28) and (3.29) may be written in a more compact way [12]:

σ(ω) = σ1(ω)+ iσ2(ω) = ∑j

iσ0 j(q j + i)2

i +x(ω), (3.30)

where σ0 is a constant of the form σ0 = ω2p/(Γ ·q2). The factor 1/q2 in σ0 is introduced

with the aim to norm the maximum of σ1(ω) and in order to recover the Lorentz-Drude

formalism of eq. (3.15) in the limit q→ ∞. Indeed, after eq. (3.27) one expects q to tends

to infinity in an unperturbed system (i.e, VE → 0).

Figure 3.5 displays the lineshapes obtained from eq. (3.30) after separating the real

and imaginary part of σ(ω). The strong asymmetry of the lineshape as a function of q is

3.4 Fano’s problem 41

σ1 (ω

) σ

2 (ω

)

ω

ω

q = -2 q = +2 q = -4 q = +4 q = -8 q = +8 q = -15 q = +15

LD model

Figure 3.5: Real and imaginary part of the optical conductivity of eq. (3.30) for different

values of q.

evident in σ1(ω) (upper panel). For negative value of q, the asymmetry develops at low

energies, indicating that the continuum lies at energies lower than the resonance itself,

while a positive q would indicate a continuum at energies predominantly higher than the

resonance [12]. The asymmetry is more pronounced at low values of |q| which accordingly

to eq. (3.27) indicates a strong interaction between continuum and localized state (i.e., VE

is large). On the other hand at higher |q| values, the lineshapes for σ(ω) converge into


the Lorentz-Drude behavior of eq. (3.15). Indeed one sees in Fig. 3.5 that already for

q = 15 the lineshape is almost symmetric. This is of great practical interest because Fig.

3.5 demonstrates that the obtained Fano’s formula (3.30) is suitable for the fitting of any

symmetric and asymmetric shape. In this thesis, all the measured curves are fitted with

eq. (3.30) and in the case of a symmetric shape we set q = 107.

Bibliography

[1] U. Fano, Phys. Rev. 124, 1886 (1961).

[2] F. Wooten, Optical properties of solids (Academic Press, New York, 1972).

[3] M. Dressel and G. Grüner, Electrodynamics of Solids (Cambridge University Press,

Cambridge, 2002).

[4] J. D. Jackson, Classical Electrodynamics,, 2nd ed. (Wiley, New York, 1975).

[5] T. Timusk and D. B. Tanner, in Physical Properties of High Temperature

Superconductors, edited by D. M. Ginsberg (World Scientific, Singapore, 1989).

[6] S. V. Shulga, O. V. Dolgov, and M. E. G., Physica C 178, 266 (1991).

[7] J. Hwang, T. Timusk, and G. D. Gu, Nature 427, 714 (2004).

[8] A. V. Puchkov, D. N. Basov, and T. Timusk, J. Phys.: Condens. Matter 8, 10049 (1996).

[9] L. C. Davis and L. A. Feldkamp, Phys. Rev. B 15, 2961 (1977).

[10] S. Lupi, M. Capizzi, P. Calvani, B. Ruzicka, P. Maselli, P. Dore, and A. Paolone, Phys.

Rev. B 57, 1248 (1998).

[11] A. Damascelli, K. Schulte, D. van der Marel, and A. A. Menovsky, Phys. Rev. B 55,

4863 (1997).

[12] A. Damascelli, Ph.D. thesis, University of Groningen, 1999.

43

4 TiOX (X= Cl and Br)

The research on layered high Tc superconductors raised considerable interest in related

low dimensional transition metal oxides. The aim is to understand the interplay of topologi-

cal aspects, strong electronic correlation and magnetism in low dimension. Since quantum

magnetism with spin-12 is characterized by strong fluctuations and suppression of long

range magnetic order [1, 2], a variety of ground states with exceptional properties may be

realized. Apart from systems with Cu2+ in the 3d9 configuration with a hole in the eg or-

bitals, S= 1/2 quantum magnets are also achieved with Ti3+ and V4+, namely in the 3d1

configuration and one single d electron occupying a t2g orbital.

Initially, the TiOX compounds, with X=Cl and Br, were considered as 2D anti-

ferromagnets and candidates for resonance valence bond ground states [3]. Recently, new

experimental findings in TiOCl jeopardized, however, this picture and point towards a one

dimensional character of the electronic system. The magnetic properties of TiOX also at-

tracted great interest due to two magnetic transitions. The one at lower temperature is

characterized by a spin-gap [4] accompanied by a crystallographic distortion along the b

axis [5]. This has analogies with a spin-Peierls (SP) transition. The second one is mainly

related to an energy gain in the spin system. The title compounds thus seem to be ideal

materials to investigate a broken symmetry ground state with orbital degrees of freedom

but without charge ordering. The relevant role played by large electronic energy scales,

associated to the orbital degrees of freedom, differentiates TiOX from CuGeO3, an inten-

sively studied spin-Peierls systems characterized by a state without orbital and charge

degrees of freedom [6,7]. In this context, it is worth mentioning that the coupling of orbital

to spin degrees of freedom in a chain system may establish a novel route to the formation

of spin gap states and spin-orbital excitations [8–10].

45

46 4. TiOX (X= Cl and Br)

O

Ti

X

c

a

b

Figure 4.1: Schematic representation of the crystal structure of the TiOX compounds. The

octahedron surrounding each Ti ions is traced out.

In the following, the recent experimental and theoretical findings (obtained mainly

through experiments on TiOCl) are reviewed and summarized, and the current under-

standing of the titanium oxyhalide is presented. We will focus our attention on those find-

ings which are significant in analyzing and interpreting our optical data. In section 4.1,

after a short description of the structural properties, we concentrate on the band structure

calculations using the LDA+U methods and on the correlated orbital ordering of the Ti d

electrons. Thereafter, we describe the magnetic properties of this spin 1/2 quantum mag-

net. The section 4.2 is dedicated to the presentation of Raman spectroscopy and infrared

(IR) optical data, followed (section 4.3) by an analysis based on the Lorentz-Drude model

and on the Fano’s approach, as described in section 3.4. A thorough discussion about the

nature of the magnetic transitions and about the role of the phonons in TiOX will conclude

this chapter.

4.1 Material properties

4.1.1 Crystal structure

The TiOX single crystals were synthesized by F. C. Chou at the Center for Material Science

and Engineering MIT, using vapor-transport techniques from TiO2 and TiX3 (X= Cl or Br),

4.1 Material properties 47

Figure 4.2: Orbital ordering and chain formation along the b axis (left panel) and zigzag

chain formation along the a axis (right panel) [4]. Note that only the Ti atoms are displayed

on this projection of the ab plane. The fat circles refer to the Ti atoms on the upper layer,

while the thin ones refer to the underlying layer.

as reported in Ref. [12]. The structure of the oxyhalogenide TiOX (Fig. 4.1) is the same as

that of FeOCl and belongs to the space group Pmmn(59) adopting the D132h crystal sym-

metry. The unit cell contains two groups of TiOX atoms (Z=2). TiOX is formed by a double

layer of Ti3+O2− intercalated by a X− bilayer. The layers extend in the ab plane and are

stacked along the c direction with c = 8.03Å, in the case of X=Cl [4]. The X− bilayers me-

diate only weak van der Waals interactions between successive Ti3+O2− bilayers, inside

which Ti and O form a buckled double plane [3]. Each Ti ion is surrounded by a distorted

octahedron of O and X ions. The TiO4X2 octahedron has the apexes along the a axis oc-

cupied by two O ions. The sides parallel to the b axis are formed either by two O or by two

X ions (Fig. 4.1).

Figure 4.2 shows a top view of the lattice in the ab plane, where for clarity only the

Ti positions are indicated. The fat circles refer to Ti atoms in the top layer, while the thin

ones to the underlying layer. The Ti sublattice consists of two rectangular layers with lattice

parameters a= 3.79Å and b= 3.38Å in the case of TiOCl. The top layer is shifted laterally

and displaced vertically from the bottom layer by 1.96 Å. Thus, the shortest Ti-Ti bond

length turns out to be the distance of 3.21 Å between Ti in different layers. The important

exchange path in the electronic structure is given by the direct t2g orbital overlap. The t2g

dxy orbitals form a linear chain running along the b axis (left panel of Fig. 4.2), linking Ti

ions in the same plane [4]. The dxz orbitals are rotated by 45 so that two of the lobes point


Figure 4.3: Band structure calculations using the LDA+U method with the split of t2g bands

(see text). The oxygen and chlorine p levels are drawn in blue, the eg bands are colored

in red, the t2g associated with the orbital dxy in black and the remaining t2g in green [4].

towards the Ti atoms of the neighboring layer, forming a zigzag chain along the a axis, as

it is shown in the right panel of Fig. 4.2. Note that such a state is degenerate with a similar

state derived from the dyz orbitals, where the latter are connecting a different set of pairs

of neighboring chains.

4.1.2 Band structure calculations

The band structure of the TiOCl compound [4] was calculated using the full potential lin-

ear muffin-tin orbital method [14] within the Local Density Approximation (LDA) [13] and

taking into account a strong on-site interaction U (Hubbard term), which may induce an

orbital ordering. The calculations yield a magnetic moment of 1 µB per formula unit. An

overview of the calculated bands is given in Fig. 4.3. The oxygen and chlorine p levels

(blue bands) form well-separated bands from the Ti d levels (black, green and red bands)

with only small hybridizations between the two. The octahedral crystal field has clearly

split the d states into t2g (black and green) and eg (red) contributions. Seidel et al. [4] cal-

culate that the repulsive Hubbard term of U = 3.3 eV causes the splitting of two nearly

degenerate one dimensional bands (derived from dxy orbitals and drawn in black in Fig.

4.3), from the rest of the t2g bands. This splitting, just at the Fermi energy, accounts for the


Figure 4.4: LDA + U calculations results for AFM Ti spin arrangement along b. The O-2p

and Cl-3p states are mainly occupied but show sizable hybridization with the Ti 3d states.

The narrow peak close to the Fermi level is predominantly Ti-3dxy [15].

insulating behavior of TiOX, which is characterized, in this calculation, by a band gap of

about 1 eV. The dxy bands have a width of about 0.9 eV. If one identifies this width with 4t,

where t is the nearest-neighbor hopping in a one-dimensional tight-binding model, the ex-

change constant in an effective Heisenberg model may be estimated via J = 4t2/U ≈ 714

K. Furthermore, refined calculations using LDA+U as well as GGA (generalized gradient

approximation) of Saha-Dasgupta et al. [15] hint to an AFM Ti-Ti interaction with sizable

super-exchange with neighboring O and Cl p orbitals. The small differences between the

two calculations [4, 15] arise from the fact that Saha-Dasgupta et al. considered a spin-

polarized state of the Ti atoms, whilst Seidel et al. considered only a ferromagnetic ar-

rangement of the Ti spins. The density of states (DOS) calculated by Saha-Dasgupta et

al. is plotted in Fig. 4.4. Besides the confirmation of the band structure calculations for the

d electrons, the DOS indicates that below the Fermi energy, the Cl-3p and O-2p bands

are mainly occupied, with a predominance of Cl-3p at the higher energies (i.e., from −3

to ∼ −5 eV) and of the O-2p at lower energies. An analysis of the TiOBr band struc-


−4.0 −3.0 −2.0 −1.0 0.0 1.0 2.0 3.0

ω (eV)

0.0

0.2

0.4

0.6

0.8

1.0

nt2g=1.0

nt2g=1.8

nt2g=1.9

−3.0 −1.5 0.0 1.5 3.0

ω (eV)

0.0

0.2

0.4

Figure 4.5: t2g partial DOS for the 3dxz,yz and 3dxy (inset) orbital for different values of the

total electron number [17].

ture shows that there is basically no difference between the LDA+U bands of TiOCl and

TiOBr at the Fermi level [16].

Recently, the LDA+U calculations in TiOCl have been refined using dynamical mean

field theory (DMFT) in order to account for the dynamical effects of strong multi-orbital

electronic correlations in the 3d1 case [17, 18]. Usually, LDA+U gives the correct ground

state for insulating systems, but generically overestimates localization, leading to a charge

gap too large if compared with experiments [17]. Craco et al. found that the t2g complex

is split with a gap of 0.57 eV as demonstrated in Fig. 4.5. Craco et al. investigated also

the possible DOS upon hole- and electron-doping. With hole-doping (x) no instability to-

wards a metallic phase is found up to x = 0.1. Simulation of small pressure found no

insulating-metallic (I-M) transition in the case of a hole-doping [17]. On the other hand,

electron-doping affects dramatically the conduction properties. An I-M transition is pre-

dicted with an electron concentration in the t2g band of nt2g = 1+x≈ 1.9. Looking at Fig.

4.5, one sees that only the 3dxz,yz bands show a metallic behavior, while the 3dxy DOS


Figure 4.6: Magnetic susceptibility χ(T) of TiOCl over a broad temperature range. The

line refers to the fit with the Bonner-Fisher formula (see text). Inset: χ(T) of TiOCl and

TiOBr in the low temperature region where the magnetic phase transitions appear [4,11].

The data are taken from Refs. [4,11].

displays at all doping levels an insulating behavior. One concludes that electron-doping

mainly affects the 3dxz,yz bands, while hole-doping influences mainly the 3dxy band [17].

The DOS at the Fermi energy as a function of the 3dxz,yz orbital occupation (nxz,yz) shows a

jump by increasing nxz,yz. This indicates an electron-doping transition of first order. Based

on these calculations, Craco et al. suggest that suitably doped TiOCl may also exhibit su-

perconductivity, for example with intercalation similarly to LixZrNCl or MxHfNCl (M=Li, Na)

systems [17].

4.1.3 Experimental results

The magnetic susceptibility has been measured by Chou [11] on single-crystals of both

TiOCl and TiOBr over a broad temperature range extending from a few Kelvin up to 800

K. The results are plotted in Fig. 4.6 after subtraction of a small Curie tail and correc-


tion for trace amounts of ferromagnetism. The high temperature (above 130 K) part of

the data measured on TiOCl can be fitted [4] to the Bonner-Fisher-curve [19] using the

nearest-neighbor exchange J as the only free parameter in the Heisenberg Hamiltonian

(eq. (2.1)). The experimental data are fitted with an exchange constant J = 660 K [4], in

good agreement with the crude estimate of J made above. In TiOBr, χ(T) shows a max-

imum at T ∼ 210 K but, contrary to TiOCl, is not well described by eq. (2.1) [16]. At low

temperatures (inset Fig. 4.6), one observes a kink and a sharp drop in the χ(T), denot-

ing two phase transitions: the 1st at Tc1 (first order transition) and the 2nd at Tc2 (second

order transition). It is worth to note that both compounds share the same features, even

though in TiOCl the critical temperatures (Tc1 ∼ 67 K and Tc2 ∼ 92 K, respectively) are

higher than in TiOBr (Tc1∼ 28K and Tc2∼ 47K). The reduced Tc in TiOBr with respect to

TiOCl may be phenomenologically explained as a consequence of the increased distance

between the Br and TiO bilayers, which reduces the interplane coupling and enhances the

low dimensionality character of TiOBr. At lower dimensions, the more pronounced fluctu-

ations hamper the formation of long-range order. At T < Tc1, the TiOX compounds have

a non magnetic state, which is associated to the opening of a spin-gap and to a related

dimerization of the chain with dxy orbitals. This scenario is reminiscent of a spin-Peierls

transition [4]. The two phase transitions in both TiOX are also observed in the tempera-

ture dependence of the specific heat (Cp) [20, 21] and in the thermal expansion along the

c axis [21]. Nevertheless, the small anomalies of Cp show that only a small amount of

entropy is released at both phase transitions [20,21].

The presence of lattice distortion below Tc1 was demonstrated for both TiOX com-

pounds [5,22,23]. In TiOCl, one detects below Tc2≈ 92K a lowering of the symmetry while

below Tc1 ≈ 67 K a two fold superstructure develops, which unambiguously evidences the

lattice distortion below Tc1 [5]. The superlattice reflections in TiOCl are detected at the

(h, k+1/2, l) positions, indicating a doubling of the unit cell along b, and forming a mon-

oclinic supercell of point group P21/m. TiOBr has displacements along the b axis similar to

TiOCl, but with amplitudes only half of that in TiOCl [23]. These results support the picture

of TiOX as spin–Peierls systems.

Nuclear magnetic resonance (NMR) experiments on TiOCl provides microscopic in-

formations on the spin degree of freedom and lattice dynamics. The spin-lattice relaxation

time T1, measured by Imai et al. and plotted in Fig. 4.7, turns out to be identical for both

measurements on 47Ti and 49Ti. Analyzing the nuclear magnetization M(t) spectra of both


Figure 4.7: 1/T1 and 1/T1T at 47,49Ti sites in TiOCl. Solid and dashed curves are guides to

the eyes. On the left panel, the same 1/T1T data are plotted in a semi-logarithmic scale.

Solid line is the best exponential fit [24]. The data are taken from Ref. [24].

47,49Ti sites, Imai et al. conclude that 1/T1 is dominated entirely by magnetic fluctuations

and this is true at all temperatures [24]. We can now follow the temperature evolution of

the low frequency spin fluctuations, described by 1/T1T. 1/T1T starts to increase by low-

ering T and reaches a maximum around T∗ ' 135± 10 K. By further lowering T, the

spin fluctuations are suppressed by almost two orders of magnitude between T∗ and Tc1.

The observed behavior of 1/T1T around T∗ reminds that of the pseudo-gap phase seen

in underdoped high Tc cuprates. Below Tc1, 1/T1T can be fitted with an exponential form

1/T1T ∼ e−Eg/kBT which suggests the existence of an energy gap Eg/kB = 430±60 K

(see right panel of Fig. 4.7). The lattice involvement in the phase transitions is testified

by the 1/T1 of the 35Cl (Fig. 4.8), indicating that the dynamic lattice distortion gradually

develops below ∼ 200K (Fig. 4.8a). The peak’s frequency of the 35Cl NMR lineshape is

presented as a function of the temperature in Fig 4.8b. At Tc2 the 35Cl (Fig. 4.8b) and

47,49Ti (not shown) lineshapes broaden into a continuum (denoted by the dashed lines

in Fig. 4.8b). This implies the presence of numerous inequivalent Ti and Cl sites in the

TiOCl lattice. Thus, the intermediate phase between Tc2 and Tc1 is not a simply dimerized

state in 1D. At Tc1 the continuum suddenly collapses into a doublet, suggesting two in-


Figure 4.8: a) 35Cl 1/T1. b) 35Cl NMR frequencies observed at 9 Tesla. Dashed lines

represent the continuum. Solid error bars represent the frequency range corresponding

to the half-intensity of the resonance mode [24].

equivalent Cl and Ti positions. These results are consistent with the formation of a singlet

ground state by lattice dimerization at Tc1. On the other hand, some experimental details,

such as the intermediate phase between Tc1 and Tc2 and the discontinuous change of 1/T1

for 35Cl at Tc1 (Fig. 4.8a), are at odds with a second order spin-Peierls transition. Also the

reduced gap ratios 2∆/kBTSP' 10−15are unusually large compared with the mean field

SP gap ratio (2∆/kBTSP' 3.5) and are not consistent with a SP mechanism in the weak

coupling limit [25]. Such a large magnitude of the energy gap ∆ strongly suggests that the

spin excitations from the singlet ground state are dressed by other electronic degrees of

freedom, most likely of orbital origin.

4.2 Optical results 55

4.2 Optical results

In this section we present the optical investigations performed with Raman spectroscopy

and reflectivity measurements. Raman investigations were performed by P. Lemmens et

al. at the Max Planck Institute for Solid State Research in Stuttgart. Since in TiOX the point

group symmetry is D2h and due to the presence of an inversion center, optical reflectivity

and Raman spectroscopy complement each other. In the scattering geometry presented

here, IR reflectivity and Raman spectroscopy exclusively probe in-plane and out-of-plane

(c axis) displacements, respectively. The data were collected in TiOCl and TiOBr. Sc doped

Ti1−xScxOCl with x≈ 0.025−0.04, where the role of Sc is to introduce holes into the t2g

complex similarly to the high Tc superconductors, were also investigated with measure-

ments of the IR reflectivity. However, the results are not reported here, since they are

equivalent to the undoped TiOCl compound. This is consistent with the LDA+DMFT band

calculation of Craco et al. (see subsection 4.1.2), which predicts no insulating-metallic

transition when hole doping these systems [17].

Raman spectroscopy is based on inelastic scattering of the photons with exitations

in the material. In this scattering process, a phonon or other kinds of excitations may be

created or annihilated modifying the energy of the scattered photons with respect to the

energy of the incoming ones. The difference in energy between the incoming and the scat-

tered photons (Raman shift) corresponds to the excitation energy of the scattering partner.

Therefore, Raman spectroscopy is a suitable technique for the investigation of all sorts

of excitations in bulk materials. In the experiment presented below, the sample is illumi-

nated with light with wavelengths of 514.5 nm as well as 488 nm, and the Raman shift of

the scattered light is recorded in the Stokes configuration. No resonance effects were de-

tected comparing the 514.5 and 488 nm excitation wavelength, so that the presented data

refer to the 514.5 nm excitation wavelength. The platelet-like single crystals of TiOCl are

investigated at different temperatures varying from 200 down to 5 K and with either bb po-

larization (incoming light polarized along b and scattered also along b) or aa polarization.

The main results of the Raman investigations of TiOCl are summarized in Figs. 4.9

and 4.10. The Raman spectra with (aa) and (bb) polarization, shown in the lower inset of

Fig. 4.9, display three modes at 203, 365 and 430 cm−1, denoted by β, γ and δ, respec-

tively. Since the major changes in temperature are observed for a light polarization in the

bb direction, all the spectra at all temperatures, presented in the main panel of Fig 4.9,


0 200 400 600

0

500

1000

Inte

nsity (

arb

. units)

Raman shift (cm )-1

300K *1/3a g

d

b

5K

50K

100K

200K

(bb)(aa)

(bb)

0

a g

Phonon e

nerg

y (

cm

)-1

0 100

100

120

140

160

T (K)0 100

T*360

370

380

390

Figure 4.9: Raman scattering intensity of TiOCl as a function of temperature in (bb) polar-

ization and with an offset for clarity. The lower inset compares (bb) and (aa) polarizations

at T = 300K with the intensity reduced by a factor 1/3. The four most important modes

are denoted by α to δ. The upper inset shows mode energies on heating the sample. The

arrows mark Tc1, Tc2, and T∗.

refer to this polarization. The response in (bb) polarization, parallel to the chain direction of

the t2g orbitals, is dominated by quasi elastic scattering (E≈ 0) and a very broad scattering

continuum with a maximum at about 160 cm−1, denoted by α. With decreasing tempera-

ture the linewidth of the α mode strongly decreases and the maximum of the absorption

softens down to 130 cm−1, i.e. by 20%. For T > Tc2, this softening is accompanied by a

reduction of the quasielastic background of the α mode. This very large softening occurs

in the fluctuation regime between 200 K and Tc1 and the energy of the α mode is compa-

rable to the spin fluctuation temperature T∗. Finally, for T < Tc1 only well defined modes


with sharp sidebands exist. All modes change appreciably by splitting into several sharp

components. In the inset of Fig. 4.9, the frequency dependence of the α and γ modes

is shown as a function of temperature together with arrows that denote the characteristic

temperatures. The splitting of the γ mode is definitely observed for temperatures T > Tc2.

Below Tc1 all transition induced modes have a sharp and comparable linewidth and any

additional anomaly is observed.

The large nonlinear softening and the strong fluctuations of the α mode suggest to a

phonon as its origin. As the three symmetry-allowed Ag modes (see section 4.3.1) have all

been observed (β, γ, δ) and since the large intensity of the α mode is not consistent with

a small local symmetry breaking, we attribute the α mode to the Brillouin zone boundary.

The β mode is its related zone center Raman-allowed phonon due to its close proximity in

energy. Lattice shell model calculations show that the respective displacement is a pure c

axis in-phase Ti-Cl mode [26]. A projection of the corresponding zone-boundary displace-

ments onto two adjacent Ti chains leads to an alternating deflection of the Ti sites out of

the b axis chain (see inset of Fig. 4.10). A band structure calculation [15] using frozen

phonons has shown that exactly these displacements strongly couple to the Ti and hy-

bridized oxygen and chlorine states and locally modify the scheme presented in section

4.1.1 where the Ti dxy orbital ordering forms the linear chain. Now the dxz and dyz orbitals,

that mediate the exchange perpendicular to the chains, are admixed to the ground state

and the respective hopping matrix elements are enhanced [15]. Such a coupling of low

and high energy scales via the quasi-degenerate orbitals is a general property of these t2g

systems [4, 27–29]. This result supports therefore the concept of possible orbital fluctua-

tions in this system, induced by strong electron-phonon interactions. Even though in the

calculations with frozen phonons [15], only a few chosen cases of distorted structures were

considered, the new state suggests that lattice, charge and orbital degrees of freedom are

intimately related in this system.

Figure 4.10 shows the high energy part of the Raman scattering spectra, where the

energy range is comparable to the exchange coupling constant J. Two maxima are ob-

served, a symmetric one at 2J and another one, asymmetric, at frequencies corresponding

to 3J. The magnetic exchange constant is J = 660K, as determined from magnetic sus-

ceptibility measurement [4]. This scattering has also a quite remarkable low-energy onset

at 1.3J and a cutoff at 3.4J. Interestingly, the onset energy of 1.3J corresponds to twice the

energy gap revealed by the NMR 1/T1 response for 47,49Ti (i.e., 2∆ = 860K). The shape


Figure 4.10: High energy Raman scattering intensity of TiOCl in the (bb) polarization.

Data have an offset for clarity. The inset shows a Ti3+ displacement corresponding to

the α phonon mode. The dxy orbitals with direct overlap form a chain, while dashed lines

represent dxz and dyz orbitals.

of the first maximum 2J resembles the two-magnon continuum of the spin tetrahedra sys-

tem Cu2Te2O5Br2, which is in the proximity of a quantum critical point [30,31]. As a result

strong magnon-magnon interactions lead to a renormalization of the spectral weight and a

line shape not compatible with classical long-range magnetic order. The second maximum

3J is not expected within a simple spin Hamiltonian and might be related to the competi-

tion of direct Ti-Ti and more 2D super-exchange paths via oxygen and chlorine. Its higher

energy 3J would be consistent with a larger coordination number of the involved magnetic

sites expected in 2D. In the stripe ordered phase of (La,Sr)2NiO4 a similar two-peak struc-

ture has been observed and attributed to exchange processes along and across stripe

domains [32, 33]. The onset of magnetic Raman scattering at 600 cm−1≈ 860 K ≈ 1.3J

is identified as the lowest energy of local spin-pair excitations, similar to observations in

other quantum magnets.

The optical reflectivity R(ω) (section 3.2) was measured in a broad spectral range

(30-105 cm−1) as a function of temperature ranging from 10 to 300 K and at selected

magnetic fields 0-7 T. No magnetic field dependence is observed, neither in TiOCl nor in


TiOBr at any temperature. This is not surprising since, within the experimental uncertainty,

the specific heat between 300 and 1.8 K is not affected by magnetic field up to 5 T [20]. We

will therefore focus our attention on the temperature dependence only. Light was linearly

polarized along the chain b axis and the transverse a axis. In both TiOX compounds,

the samples were too thin to allow an investigation of the electrodynamic response along

the c axis. In order to avoid leakage effects of the polarizer, the polarization of light in

our experiment always coincides with the vertical axis of the sample mounting, so that

the investigated crystallographic direction was perfectly parallel to the polarization of the

light beam. Therefore, the polarization dependence was obtained by rotating the sample

(instead of the polarizer) by 90 degrees inside the cryostat. This assures that no undesired

projections of the light polarization along any transverse crystallographic direction occur in

our experiment.

Figure 4.11 summarizes our results on TiOCl, by focusing the attention on the tem-

perature dependence of R(ω) in the infrared spectral range and for both polarization di-

rections [34]. Experimental R(ω) curves for TiOBr are plotted in Fig. 4.12 at selected tem-

peratures. In both figures, the upper panel shows the R(ω) spectra measured with light

polarized along the transverse a axis, while the lower one shows the spectra taken with

light polarized along the chain b axis. The first obvious observation, by comparing the elec-

trodynamic response along different axes, is the strong anisotropy of the optical response

within the ab plane and for photon energies below ∼ 104 cm−1. The insets of Figs. 4.11

and 4.12 display the entire R(ω) spectra at 300 K on a logarithmic energy scale.

Figure 4.13 shows σ1(ω) of TiOCl in the far infrared spectral range. Note the log-

arithmic scale on the σ1(ω) axis, which emphasizes the temperature dependence. The

features in σ1(ω) below 50 cm−1 for both polarization directions are most likely spurious

effects induced by the KK transformations. These features turn out to be temperature inde-

pendent and are not considered further in our analysis. The FIR frequency part of σ1(ω) in

TiOBr shows only a weak temperature dependence, however with similar trends as those

of TiOCl (Fig. 4.13). It is of interest to compare the optical conductivities of the two com-

pounds measured in the same configuration. Such comparisons are displayed in Figs. 4.14

and 4.15. Several absorptions characterize σ1(ω) in the FIR spectral region and they are

listed in Tables 4.1 and 4.2 for the a and b axis, respectively.

Along the a axis in TiOBr (Fig. 4.14), a strong absorption is seen at 417 cm−1, with

a broad high frequency tail defining a shoulder at about 481 cm−1. At low frequencies, we


100

80

60

40

20

0

Ref

lect

ivity

(%

)

800600400200

Energy (cm-1)

80

60

40

20

0

Ref

lect

ivity

(%

)

80

60

40

20

0102 103 104 105

80

60

40

20

0102 103 104 105

T=300K T=220K T=150K T=100K T=10K

T=300K T=200K T=150K T=100K T=10K

a-axis

b-axis

TiOCl

Fit

Fit

Figure 4.11: Optical reflectivity R(ω) in the infrared spectral range of TiOCl along the a

axis (upper panel) and b axis (lower panel). The insets show the entire spectra at 300 K

up to the ultraviolet spectral range and the Lorentz-Drude fit as described in the text.

detect a sharp absorption at 77 cm−1 and a small one at 65 cm−1. Similarly in TiOCl, there

is a strong peak at 438 cm−1 and less intensive absorptions at 68, 104, 294, 347 and 387

cm−1, as well as a very broad feature around 200 cm−1.


100

80

60

40

20

0

Ref

lect

ivity

(%

)

800600400200

Energy (cm-1)

80

60

40

20

0

Ref

lect

ivity

(%

)

80

60

40

20

0102 103 104 105

100

80

60

40

20

0102 103 104 105

T=300 K T=100 K T=40 K T=10 K

a-axis

b-axis

Fit

Fit

TiOBr

Figure 4.12: R(ω) of TiOBr at selected temperatures for light polarized along the a axis

(upper panel) and the b axis (lower panel). The insets show the spectra in the entire

measured energy interval and their Lorentz-Drude fit (see text).

Along the chain b axis in TiOBr (Fig. 4.15) a strong absorption is observed at 275

cm−1, with a shoulder at 303 cm−1 defining its high frequency tail. On the other hand,

in TiOCl there is a strong peak at 294 cm−1 with additional absorptions, overlapped to


101

102

103

σ 1(ω

) (Ω

•cm

)-1

5004003002001000

Energy (cm-1)

101

102

σ 1(ω

) (Ω

•cm

)-1

T=300K T=220K T=150K T=100K T=10K

T=300K T=200K T=150K T=100K T=10K

TiOCl

a-axis

b-axis

Figure 4.13: Real part σ1(ω) of the optical conductivity of TiOCl as a function of temper-

ature along the b and the a axis.

its low frequency tail, at 251 and 231 cm−1. Moreover, we recognize broad absorptions

at 177 cm−1 and around 90 cm−1. Few absorptions are furthermore characterized by an

asymmetric shape (see below). The strongest modes at 275 and 294 cm−1 along the b

as well as at 417 and 438 cm−1 along the a axis, respectively, display rather broad high

frequency tails, which might be indicative of some anharmonicity.

One notes the overall similarity between the spectra of both compounds. Nonethe-


1000

800

600

400

200

0

σ1(ω

) (Ω

⋅cm

)-1

6005004003002001000

Energy (cm-1)

500

400

300

200

100

0

σ 1

(ω)

(Ω⋅c

m)-1

TiOBr 300 K Fit

65 cm-1

77 cm-1

153 cm-1

227 cm-1

274 cm-1

417 cm-1

481 cm-1

a-axis

TiOCl 300 K Fit

68 cm-1

104 cm-1

200 cm-1

294 cm-1

347 cm-1

387 cm-1

438 cm-1

497 cm-1

Figure 4.14: The optical conductivity of TiOBr and TiOCl in FIR along the a axis. The total

fit and its components, identified in the legend by their respective resonance frequency,

are also shown.

less, in TiOBr a generalized red-shift of the phonon spectrum with respect to TiOCl is

observed as may be expected when replacing Cl by a heavier element such as Br. There

are, however, some differences, like the number and shape of the absorption modes be-

tween the two TiOX compounds. For example, the strong absorption peak at 417 cm−1 in

TiOBr along the a axis shows a quite pronounced asymmetry which is not observed in

the 438 cm−1 absorption of TiOCl. Similarly, the mode at 387 cm−1 is well resolved in

TiOCl but not in TiOBr. Along the b axis, the strong absorption at 275 cm−1 and the


500

400

300

200

100

0

σ1(ω

) (Ω

⋅cm

)-1

4003002001000

Energy (cm-1)

400

300

200

100

0

σ 1

(ω) (

Ω⋅c

m)-1

b-axis

TiOCl 300 K Fit

90 cm-1

177 cm-1

231 cm-1

251 cm-1

294 cm-1

320 cm-1

TiOBr 300 K Fit

88 cm-1

131 cm-1

170 cm-1

196 cm-1

233 cm-1

275 cm-1

303 cm-1

Figure 4.15: The optical conductivity of TiOBr and TiOCl in FIR along the b axis. The total

fit and its components, identified in the legend by their respective resonance frequency,

are also shown.

weaker mode at 233 cm−1 are well distinguished in TiOBr, while in TiOCl one finds a

single asymmetric mode at 294 cm−1.

Since the high frequency part of the σ1(ω) in TiOCl is similar to that of TiOBr (see

inset of Fig. 1 in Ref. [36]), we present here the high frequency part of σ1(ω) in TiOBr only

(Fig. 4.16). It highlights the absorption spectrum associated with the electronic interband

transitions. The high frequency part of σ1(ω) is characterized by five main absorptions

(evidenced by the arrows in Fig. 4.16). The first two are at ∼ 0.8−1 eV and ∼ 1.5 eV, the


a axis

TiOBr 65 77 153 227 274 417 481

TiOCl 68 104 200 294 347 387 438 497

ω0Br/ω0Cl 0.96 0.74 0.77 0.77 0.79 0.95 0.97

Table 4.1: Resonance frequencies of the FIR absorptions in TiOBr [35] and TiOCl [34]

(i.e., peaks in σ1(ω)) along the a axis, determined by the Fano approach (eq. (3.30)).

The table also reports the ratio of the resonance frequencies between the Cl and Br

compound, to be compared with eq. (4.2) and (4.5). The bold frequencies refer to the B3u

modes.

b axis

TiOBr 88 131 170 196 233 275 303

TiOCl 90 177 231 251 294 320

ω0Br/ω0Cl 0.97 0.74 0.74 0.78 0.94 0.95

Table 4.2: Resonance frequencies of the FIR absorptions in TiOBr [35] and TiOCl [34]

(i.e., peaks in σ1(ω)) along the b axis, determined by the Fano approach (eq. (3.30)).

The table also reports the ratio of the resonance frequencies between the Cl and Br

compound, to be compared with eq. (4.2) and (4.5). The bold frequencies refer to the B2u

modes.

third at ∼ 3.8 eV and the fourth at ∼ 9.3 eV. For the sake of completeness we mention

that recent transmission measurements found an absorption at ∼ 0.7 eV along the a axis

and ∼ 1.5 eV along b [21]. The LDA+U calculations on TiOCl [4], presented in section

4.1.2, predict a charge gap of about ∼ 1 eV due to the splitting of the one dimensional t2g

orbitals, while this splitting is about ∼ 0.6 eV in the LDA+DMFT calculation (Figs. 4.3 and

4.5). This agrees with the optical gap measured in TiOBr as well as in TiOCl (i.e. the onset

of absorption at about 1 eV). Furthermore, band structure as well as DOS calculations

indicate an energy difference between the t2g and eg of about 2 eV (Figs. 4.3 and 4.4),

consistent with the absorption feature at about 1.5 eV. The band structure calculations

also hint to the scenario where the high energies absorptions seen in our spectra (i.e.,

3.8 and 9.3 eV) may be associated to the interband transitions between the Cl-3p/O-2p

and Ti-d bands. It is of no surprise that both titanium oxyhalides display similar absorption

features [34,35], since the atomic distances within the ab plane vary only slightly between


1400

1200

1000

800

600

400

200

0

σ1(

ω)

(Ω⋅c

m)-1

0.1 1 10

Energy (eV)

103 104 105

Energy (cm-1)

300 K a-axis b-axis

TiOBr

Figure 4.16: High frequency part of σ1(ω) in TiOBr at 300 K for light polarized along the

a and b axis. The arrows indicate the characteristic absorption features (see text).

the two compounds [3] and the band structure is similar in both TiOX compounds, as

demonstrated by the LDA+U calculation [16].

4.3 Discussion

4.3.1 Theoretical treatment of the phonon modes in TiOX com-pounds

4.3.1.1 Analysis of the lattice vibrations: the correlation method

As stated in Ref. [3] the TiOX crystals have Pmmnas space group and thus the irreducible

representation of the lattice vibrations corresponds to the point group D132h. The character

table of this group is reproduced in Table 4.3 together with the translational and rotational

modes. The last column of Table 4.3 describes the Raman activity.

In the Bravais cell there are 2 units of TiOX, so that 2Ti, 2O and 2Cl atoms occupy

an equivalent position in the Bravais cell. The possible site symmetries allowed for the

symmetry group D132h are: 2C2v(2), 2Ci(4), 2Cs(4) and C1(8). Only the site symmetry

4.3 Discussion 67

D2h ≡ Vh E C2(z) C2(y) C2(x) i σ(xy) σ(zx) σ(yz)

Ag 1 1 1 1 1 1 1 1 αxx, αyy, αzz

B1g 1 1 -1 - 1 1 1 -1 -1 Rz αxy

B2g 1 -1 1 -1 1 -1 1 -1 Ry αxz

B3g 1 -1 -1 1 1 -1 -1 1 Rx αyz

Au 1 1 1 1 -1 -1 -1 -1

B1u 1 1 -1 -1 -1 -1 1 1 Tz

B2u 1 -1 1 -1 -1 1 -1 1 Ty

B3u 1 -1 -1 1 -1 1 1 -1 Tx

Table 4.3: Character table of the D2h representation. The second last column gives the

species of translation (T) and rotation (R). The last column describes the species of the

polarization tensor (Raman activity).

D2h Ag B1g B2g B3g Au B1u B2u B3u

C2v (C2(z)) A1 A2 B1 B2 A2 A1 B2 B1

Table 4.4: Correlation between the group D2h and the site C2v (C2(z)) symmetries.

C2v(2) can accommodate two atoms and thus this is the correct site symmetry for the

Ti, O and X atoms. The correlation between the site symmetry of each single atom to

the group symmetry is presented in Table 4.4 (see page 203 in Ref. [37]). The correct

correlation symmetry is chosen looking at the crystal structure (Fig. 4.1), where one sees

that the crystal is invariant under a rotation of π along the z axis (C2(z)) [37].

The translational active mode in the site symmetry C2v are the A1 (Tz), B1 (Tx) and

B2 (Ty) and each of these has the number of translation equal to 1 (i.e., tγ = 1). It follows

that the degree of vibrational freedom (fγ) is 2. This knowledge allows to complete the

correlation Table 4.5 and to recover the correct modes for atoms belonging to the C2v

group representation. With Γeq. set = ∑ζ aζ ·ζ as the modes of one set of atoms, one has:

ΓC2v = Ag +B2g +B3g +B1u +B2u +B3u.


fγ tγ C2v site species → D2h factor group species ζ Cζ aζ

2 1 A1 → Ag 1 1

B1u 1 1

2 1 B1 → B2g 1 1

B3u 1 1

2 1 B2 → B3g 1 1

B2u 1 1

Table 4.5: Correlation for the lattice vibrations of the atoms between the site group C2v

and the factor group D2h.

Summing over the modes for each atoms, one gets:

Γcryst = 3ΓC2v = 3Ag +3B2g +3B3g +3B1u +3B2u +3B3u.

In TiOX there are 3 acoustic modes (see Table 4.3) namely Γacoustic= B1u+B2u+B3u, so

that one obtains for the vibrational modes:

Γcrystvib = Γcryst−Γacoustic= 3Ag +3B2g +3B3g +2B1u +2B2u +2B3u.

Only the modes B1u, B2u, B3u are infrared active (IR) and they show a well defined polar-

ization (Table 4.3):

ΓIR = 2B1u (E//c)+2B2u (E//b)+2B3u (E//a).

The Raman active modes are:

ΓRaman= 3Ag (aa,bb,cc,)+3B2g (ac)+3B3g (bc).

4.3.1.2 Shell-model

The above result may be extended performing classical shell-model calculations in order

to extract the eigenfrequencies and eigenvectors for both Raman- and IR-active phonons

in TiOCl [26]. As far as the IR phonons are concerned, the calculations predict the B3u

4.3 Discussion 69

a)

d)c)

b)

Figure 4.17: Schematic representation of the eigenvectors for the B3u and B2u normal

modes in TiOCl. The atom displacements for the IR active phonons occur within the

ab plane. The calculated normal frequencies in cm−1 are compared with the observed

values for TiOCl (in brackets).

(a axis) phonons at 91 and 431 cm−1 and the B2u (b axis) at 198 and 333 cm−1. Figure

4.17 shows the normal mode eigenvector patters for the IR active phonons. The Raman

as well as the IR eigenfrequencies calculated by the shell-model are listed in Table 4.6

together with the so far experimentally determined phonon frequencies [34, 36]. Unfortu-

nately, for light polarized within the abplane, Raman spectroscopy [36] can detect only the

3Ag modes, which include displacements along the c axis [26]. The B2g and B3g modes


Infrared Raman

B1u B2u B3u Ag B2g B3g

308 198 91 248 84 126

(177) (104) (203)

433 333 431 333 219 237

(294) (438) (365)

431 491 390

(430)

Table 4.6: The infrared and Raman active phonons for TiOCl, calculated with shell model

[26], are compared to the experimentally obtained values (in brackets). Ag modes are

observable in (aa) or (bb) polarization. B2g and B3g modes are only accessible in (ac)

and (bc) polarization, respectively, with one polarization vector parallel to the c axis.

are only accessible in (ac) and (bc) polarizations, which however cannot be measured in

our crystals due to the small extension of the samples along the c axis. The agreement

between the calculated phonon frequencies and the experimental observations is good,

particularly, for the two high frequencies of the IR active phonons with B2u and B3u sym-

metry, which can be identified as the most pronounced features in the spectra along the a

and b axis (Figs. 4.14 and 4.15).

4.3.1.3 Frequency shift of the phonon modes

We can describe the phonon modes of the TiOX compound [34, 35] using the known

phonon displacements from the shell model calculation and taking into account the renor-

malization of the phonon frequencies due to the corresponding ion mass and to the change

in the lattice coupling. This latter parameter directly correlates with the relative volume

variation of the unit cell when substituting Br with Cl. Within the linear harmonic approxi-

mation, the shift of the eigenfrequencies of the phonons is obtained by a renormalization

of the oscillator strength constant f , when Br and Cl ions are not involved in the oscillatory

displacements:

ω0Br =

√fBr

m=

√fCl · fBr

fCl

m= ω0Cl ·

√fBr

fCl. (4.1)

4.3 Discussion 71

We may estimate√

fBrfCl

for both directions (a and b axis) from the IR optical active phonons,

whose frequencies are predicted to be independent from mBr/Cl (Fig. 4.17b and 4.17d):

√fBr

fCl=

ω0Brω0Cl

= 417438 = 0.9521 for a axis

ω0Brω0Cl

= 275294 = 0.9354 for b axis.

(4.2)

On the other hand, for the IR phonons, where the mass of the Cl/Br ions is involved (Fig.

4.17a and 4.17c), one should consider also the renormalization due to the reduced mass

ω0Br =

√fBr

µBr=

√√√√ fCl · fBrfCl

µCl · µBrµCl

= ω0Cl ·√

µCl

µBr·

√fBr

fCl. (4.3)

According to the eigenvector illustrated in Fig. 4.17 one has one atom of Ti and O in

each unit cell, moving together against the Br atom. Therefore, the Ti+O ensemble has a

resulting atomic mass m= mTi +mO = 63.87 au. The reduced mass µ of the eigenmode

is then:

µCl =m

1+ mmCl

= 22.80 au

µBr =m

1+ mmBr

= 35.50 au (4.4)

for the Cl and Br compound, respectively. Inserting these reduced masses in eq. (4.3) and

making use of the values in eq. (4.2), the expected red-shift of the IR active phonons in

TiOBr with respect to those of TiOCl can be estimated as:

ω0Br = ω0Cl ·√

µCl

µBr·

√fBr

fCl= ω0Cl ·

0.7630 for a axis

0.7496 for b axis.(4.5)

This simple approach accounts very well for the generalized red-shift of the phonon

spectrum in TiOBr with respect to TiOCl. There is an excellent agreement between the

scaling following eq. (4.2) and (4.5) and the measured red-shift of the B2u and B3u modes,

as may be seen by the ratios ω0Br/ω0Cl in Tables 4.1 and 4.2. Applying this argument to

the eigenfrequencies, calculated for the B2u and B3u modes of TiOCl, we can anticipate

the expected (theoretical) eigenfrequencies for TiOBr. These results are listed in Table 4.7.

The agreement with the experimentally determined values (Tables 4.1 and 4.2) is rather

compelling.


B2u ( b axis) B3u (a axis)

TiOBr Theory 148 311 69 410

TiOCl Theory 198 333 91 431

Table 4.7: IR phonon mode frequencies for TiOCl after the shell model calculations and

the estimated ones for TiOBr using the renormalization factors calculated by eq. (4.2) and

(4.5).

In both TiOX compounds, more modes are detected than expected by symmetry.

Nonetheless, we can extend the above analysis to these additional modes, as well. In

Tables 4.1 and 4.2, we have reported all mode frequencies measured along the a and b

axis. It is worth noting that the modes ranging from 104 up to 347 cm−1 in TiOCl along the

a axis shift upon Br substitution of∼ 77−79%, indicating that for these modes the halogen

mass plays an active role. The high frequency tail of the strong phonon at 438 cm−1 as

well as the lowest phonon at 68 cm−1, scale by ∼ 96−97% in TiOBr, suggesting that only

the renormalization of the lattice strength plays here a relevant role. The mode scaling

along the b axis is similar to that of the a axis. Those modes between the two predicted

IR phonons (177-294 cm−1 in TiOCl) show a renormalization of ∼ 74− 78% in TiOBr,

face a prediction of ∼ 75%. The shoulder at high frequency of the B2u mode, generated

by the displacement of the Ti and O ions, and the lowest phonon at 90 cm−1 in TiOCl,

display a weak softening of ∼ 95−97% in TiOBr, confirming that in these two modes the

halogen displacement is generally not relevant. Even though interference effects cannot

be excluded a priori, the fact that all FIR absorptions for both polarization directions scale

following eq. (4.2) or eq. (4.5) supports the lattice dynamics origin for these excitations. The

larger number of phonon mode might indicate either inhomogeneities of the sample or a

lower crystallographic symmetry. A lowering of the symmetry enlarges indeed the number

of infrared modes by activating e.g. purely Raman ones, which otherwise would be silent

or even forbidden. We remark that more phonon modes than expected from the nominal

space group have been also detected in other related TiX2 (X=I, Br, and Cl) compounds

[38].

4.3 Discussion 73

4.3.2 Temperature evolution of the fit parameters

Looking at the temperature evolution of the phonon spectrum, we observe first of all that

the number of phonon modes does not change going through both phase transitions. A

change of the modes’ number would be an optical evidence of the lattice distortion de-

tected by x-ray spectroscopy in both TiOX compounds [5,22,23]. Indeed, lattice distortion

might increase the number of IR active modes, for example by backfolding modes at the

Brillouin zone center. As complement, we mention that an additional phonon mode along

the a axis at 21.8 meV (∼ 176cm−1) is found in recent transmission data of TiOBr. This

additional mode progressively disappears above 28.8 K [21]. This result is in our opin-

ion somehow unreliable. First, this additional mode is only detected for TiOBr but not for

TiOCl which, according to x-ray data, shows an even bigger crystal distortion along the b

axis [23]. Second, the additional phonon mode is polarized along the a and not along the

b axis, direction where the lattice distortion is found.

In order to highlight the temperature dependence of the phonon spectrum, we apply

the phenomenological Fano approach, described in section 3.4, to fit the optical conductiv-

ity σ(ω). For the TiOCl compound the total fit of σ1(ω), covering the entire spectral range

from FIR up to UV, is obtained by summing over eleven and ten contributions in eq. (3.30)

for the a and b axis, respectively (the oscillators pertinent for the FIR spectral range are

shown in Figs. 4.14 and 4.15) [39]. In TiOBr the fit of the complete σ1(ω) is obtained by

summing over twelve contributions (the seven oscillators shown in Figs. 4.14 and 4.15 and

five more for the high frequency spectral range) for both the a and b axis [39]. For both

TiOX compounds the fit of the low frequency part of σ1(ω) at 300 K as well as its single

components are shown in Figs. 4.14 and 4.15. The reproduction of the experimental data

is surprisingly good and the same fit quality is obtained at all temperatures. The same

set of fit parameters also allows us to reproduce the measured R(ω) spectra with good

fit quality as demonstrated by the insets in Figs. 4.11 and 4.12. Only the oscillators with

the lowest frequencies (Figs. 4.14 and 4.15) display a temperature dependence and are

discussed further. The oscillators exhibiting a temperature dependence are characterized

by the resonance frequencies listed in Tables 4.1 and 4.2.

It turns out that most of the absorptions, seen in our spectra (Fig. 4.14 and 4.15) and

described by the j-components in eq. (3.30), adopt a Lorentzian (i.e., q j → ∞) shape. In

TiOCl, only the peaks at 104 cm−1 along the a axis and at 294 cm−1 along the b axis


0 100 200 3000

100

200

300

400

∆q

j(T

)/q

j(3

00

K)

(%)

(b)

0

20

40

60

T ( K )

(a)

|qj|

294 cm-1

104 cm-1

104 cm-1294 cm-1

eq. 3.30

eq. 3.28

Figure 4.18: Temperature dependence of the asymmetry factor∣∣q j∣∣ in TiOCl for the 104

cm−1 mode along the a axis and the 294 cm−1mode along the b axis, calculated after

eq. (3.30). Note that q < 0 for both polarization directions. b) Temperature dependence

of the corresponding percentage changes with respect to 300 K (i.e., ∆q j(T)/q j(300

K), with ∆q j(T) = q j(T)−q j(300K)) for the asymmetry factor q j of the a and b axes. In

addition, we also display the q j factor and its percentage change for both modes obtained

with the Fano formalism, based on the approach of Davis and Feldkamp (eq. (3.28)). The

equivalence of the two approaches (eq. (3.30) and eq. (3.28)) is obvious.

display a Fano-like asymmetry. The asymmetry (Fig. 4.18) of the mode at 104 cm−1 along

the a axis in TiOCl gradually decreases (i.e.,∣∣q j∣∣ gets larger) with decreasing tempera-

ture, although the mode remains considerably asymmetric at all temperatures [39]. This

indicates that there is a predominant interaction with the continuum both above and below

4.3 Discussion 75

Tc1. Spectacular is the temperature dependence of the asymmetry for the 294 cm−1 mode

along the b axis in TiOCl, which displays a clear crossover between 200 K and Tc2 from

an asymmetric Fano-type shape (i.e.,∣∣q j∣∣ small) to a Lorentzian oscillator (i.e.,

∣∣q j∣∣ very

large) [39]. As far as TiOBr is concerned, the q j factor, describing the asymmetry of the

phonon at 77 cm−1 along the a axis, increases from ∼ −8 at T > 150 K to values of

about −30 at low temperatures [39]. On the other hand, the mode at 417 cm−1 along the

a axis and at 275 cm−1 along the b axis in TiOBr display an asymmetric line shape which

however does not change in temperature.

The distinct behavior in the temperature dependence of the q j -factors within the ab

plane is an additional indication for the anisotropy of the lattice dynamics as well as of the

coupling between phonon and a continuum. The clear Fano-Lorentz crossover along the

chain b axis in TiOCl suggests the suppression of the interaction with the continuum with

decreasing temperature [40, 41]. Since q j < 0 for both asymmetric modes, the relevant

continuum of excitations covers an energy interval below the phonon frequencies [41]. For

the chain b axis, this identifies a characteristic energy scale of the order of∼ 430K. In Fig.

4.18, we also report, for comparison, the q j -factor for both modes as calculated using the

Fano model based on the approach of Davis and Feldkamp (eqs. (3.28) and (3.29)) [42].

Even though the two Fano approaches eq. (3.28) and (3.30) are characterized by different

energy power-law decays of the absorption coefficient (see section 3.4), the corresponding

q j -factors are identical both in absolute value [39] and in the relative percentage change

(Fig. 4.18). This stresses the equivalence of the Fano asymmetry concept (parameterized

by the q j -factors) for both fits.

The temperature dependence of the fit parameters ω0 j ,Γ j ,ωp j is shown either in

Figs. 4.19 and 4.20 for TiOCl or in Figs. 4.21 and 4.22 for TiOBr [39] . The tempera-

ture dependence is plotted as percentage variation with respect to the 300 K data (e.g.,

∆ω0 j(T)/ω0 j(300K), with ∆ω0 j(T) = ω0 j(T)−ω0 j(300K)). The overall temperature de-

pendence of the fit parameters develops in a broad temperature interval extending from

200 K down to 10 K in TiOCl, while in TiOBr develops below 150 K and tends to saturate

below 30 K. The fact, that this temperature interval extends, in both compounds, well above

Tc1, underlines the presence of an extended fluctuation regime, already pointed out in the

description of the NMR data of TiOCl (see section 4.1.3). Also electron-spin resonance

(ESR) parameters in TiOCl highlight the strong coupling between spin and lattice degree

of freedom, and show a progressive evolution in a temperature interval ranging from 200


2.0

1.5

1.0

0.5

0.0

-0.5

∆ω0j(T

)/ω

0j(3

00 K

) (

%)

-60

-40

-20

0

20

∆Γj(T

)/Γ j(3

00 K

) (

%)

-50

-40

-30

-20

-10

0

10

20

∆ωpj(T

)/ω

pj(3

00 K

) (

%)

300250200150100500

T (K)

68 cm-1

104 cm-1

200 cm-1

294 cm-1

347 cm-1

387 cm-1

438 cm-1

TiOCl

a-axis

Figure 4.19: Temperature dependence along the a axis in TiOCl of the percentage vari-

ation with respect to 300K (see text) for the resonance frequencies (ω0 j ), the dampings

(Γ j ) and the oscillator strengths (ωp j) of the phonon modes (identified in the legend by

their respective resonance frequency in cm−1). The dotted vertical lines mark the transi-

tion temperatures Tc1 and Tc2.

K to Tc1 [43].

Looking at the fit parameters for TiOCl (Figs. 4.19 and 4.20) one sees that the res-

onance frequencies (ω0 j ) of almost all phonons tend to increase (Fig. 4.19 and 4.20),

4.3 Discussion 77

4

3

2

1

0

-1

-2

∆ω0j(T

)/ω

0j(3

00 K

) (

%)

-40

-30

-20

-10

0

10

∆Γj(T

)/Γ j(3

00 K

) (

%)

-30

-20

-10

0

10

∆ωpj(T

)/ω

pj(3

00 K

) (

%)

300250200150100500

T (K)

90 cm-1

177 cm-1

231 cm-1

251 cm-1

294 cm-1

320 cm-1

b-axis

TiOCl

Figure 4.20: Temperature dependence along the b axis in TiOCl of the percentage vari-





though moderately (i.e., the change does not exceed 4 %), with decreasing temperature.

This indicates a progressive hardening of the modes. The resonance at 320 cm−1 along

the b axis, accounting for the broad high frequency tail of the mode at 294 cm−1, dis-


-20

-10

0

10

20

∆ωpj(T

)/ω

pj(3

00 K

) (

%)

300250200150100500

T (K)

20

15

10

5

0

∆Γj(T

)/Γ j(3

00 K

) (

%)

-2

-1

0

1

2

∆ω0j(T

)/ω

0j(3

00 K

) (

%)

65 cm-1

77 cm-1

153 cm-1

227 cm-1

274 cm-1

417 cm-1

481 cm-1

TiOBr

a-axis

Figure 4.21: Temperature dependence along the a axis in TiOBr of the percentage vari-





plays on the contrary a weak softening. Looking at TiOBr (Figs. 4.21 and 4.22), ω0 j of

almost all phonons along the a axis (upper panel in Fig. 4.21) tends to increase with de-

creasing temperature with changes not exceeding 2 %. Only the highest mode at 481

4.3 Discussion 79

-40

-30

-20

-10

0

10

∆Γj(T

)/Γ j(3

00 K

) (

%)

-2

-1

0

1

2

∆ω0j(T

)/ω

0j(3

00 K

) (

%)

-20

-10

0

10

∆ωpj(T

)/ω

pj(3

00 K

) (

%)

300250200150100500

T (K)

88 cm-1

131 cm-1

170 cm-1

196 cm-1

233 cm-1

275 cm-1

303 cm-1

b-axis

TiOBr

Figure 4.22: Temperature dependence along the b axis in TiOBr of the percentage vari-





cm−1 shows a weak softening. Along the b axis, the low energy phonons display a weak

hardening, while the three modes around the strong absorption feature peaked at 275

cm−1 show a weak softening (upper panel in Fig. 4.22). A softening of the phonon modes


is in principal expected in models for a conventional spin-Peierls transition [44], where the

structural deformation is driven by a linear coupling between the lattice and the magnetic

degrees of freedom (see section 2.1). However, since the dimerization must be related

to normal modes away from the zone center, the softening of one or more modes across

a spin-Peierls transition should be expected at the boundary of the Brillouin zone. Opti-

cal techniques can only probe the phonon branch at the Γ point (q = 0). Evidence for a

phonon softening at finite wave vector can therefore only be obtained by neutron scattering.

Nonetheless, one can hope to gain interesting insights on the temperature dependence of

the phonon spectrum, if the dispersion and the mixing of the branches are not too strong.

In that case, the presence of a soft mode at the boundaries of the Brillouin zone would

result in an overall softening of the branch it belongs to. Thus, the weak softening of the

modes at about 275 cm−1 below 100 K might be ascribed to a general red-shift of the

B2u branch. We remember at this point, that in Raman scattering of TiOCl (Fig. 4.9) the α

mode, identified as a Ag phonon at the Brillouin zone boundary, shows a 20 % softening.

As far as the temperature dependence of the scattering rate (Γ j ) in TiOCl is con-

cerned (Figs. 4.19 and 4.20), most of the phonon modes get narrow with decreasing

temperature. Only the mode at 294 cm−1 along the a axis displays a broadening with

decreasing temperature. On the contrary, the modes at 417 and 481 cm−1 along the a

axis in TiOBr show a broadening with decreasing temperature, while the remaining modes

do not change the width with temperature (Fig. 4.21). Along the b axis (Fig. 4.22) almost all

phonons but the one at 303 cm−1 in TiOBr narrow with decreasing temperature. The pro-

nounced narrowing of the modes in TiOCl occurs in the temperature interval between 150

and 100K. It seems, therefore, natural to relate this phonon narrowing to the suppression

of low-frequency spin fluctuations.

The mode strength ωp j for the great majority of phonons decreases at low temper-

ature for both polarizations. Only ωp j of the phonons at 438 in TiOCl and 481 cm−1 in

TiOBr for the a axis, and at 320 in TiOCl and 303 cm−1 in TiOBr for the b axis increases.

The mode at 196 cm−1 along the b axis in TiOBr also increases its strength, though mod-

erately. The spectral weight (SW), defined by

SW=∑ j ω2

p j

8=

Zσ1(ω)dω, (4.6)

displays an overall depletion with decreasing temperature at frequencies below the strong

modes at 417 and 387 cm−1 (for TiOBr and TiOCl, respectively) along the a axis, and

4.3 Discussion 81

30

20

10

0

-10

-20

-30

Spe

ctra

l wei

ght v

aria

tion

(%)

300250200150100500

T (K)

-40

-30

-20

-10

0

10

20

30

Spe

ctra

l wei

ght v

aria

tion

(%)

TiOBrSW(T) = (ωp7 (T))2 SW(T) = Σi=1

6 (ωpi (T))2

b-axis

TiOClSW(T) = (ωp6 (T))2

SW(T) = Σi=15 (ωpi (T))2

a-axis

TiOClSW(T) = (ωp7 (T))2

SW(T) = Σi=16 (ωpi (T))2

TiOBrSW(T) = (ωp7 (T))2

SW(T) = Σi=16 (ωpi (T))2

Figure 4.23: Temperature dependence of the spectral weight variation(

SW(T)−SW(300K)SW(300K)

)for both polarizations and both samples. The figure highlights the redistribution of spectral

weight between low (decreasing SW with decreasing temperature) and high (increasing

SWwith decreasing temperature) frequency (see text).

at 275 and 294 cm−1 (TiOBr and TiOCl) along b. In both compounds, the suppressed

spectral weight is redistributed to higher frequencies.

This is shown in Fig. 4.23, which visualizes the temperature dependence of the spec-

tral weight redistribution. The decreasing SW is obtained by summing the squared oscil-

lator strength of each mode at energies smaller than either 417 (TiOBr) and 387 (TiOCl)

cm−1 along the a axis or 275 (TiOBr) and 294 (TiOCl) cm−1 along b. The increasing SW


is encountered in the high frequency tail of the strong IR phonons. An equivalent analysis

(eq. (4.6)) may be performed by integrating σ1(ω) either from 0 to an appropriate cut-off

energy or from such a cut-off energy up to energies where the σ1(ω) spectra at different

temperatures are no more distinguishable. The cut-off energy can be chosen in such a way

to differentiate between energy intervals where a depletion, respectively a gain in spectral

weight has been established (i.e., either 417 (a axis) and 275 cm−1 (b axis) for TiOBr,

or 387 (a axis) and 294 cm−1 (b axis) in TiOCl). Both analyses show that the total spec-

tral weight is fully conserved at all temperatures from ∼ 1000cm−1 on. The variation of

SW in both TiOX happens at temperatures extending well above Tc1 and Tc2, pointing out

again the importance of fluctuation effects. Comparing the temperature evolution of SW in

TiOBr with that of TiOCl, one notes that the redistribution of SW in TiOBr develops at lower

temperature than in TiOCl. This goes hand in hand with the temperature dependence of

χ(T) (Fig. 4.6), signaling lower critical temperature for the spin-gap phase.

We remark at this point that in TiOCl the spectral weight is progressively removed

below 300 cm−1 along the b axis, as may also be inferred intuitively from σ1(ω) (Fig.

4.13). This suggest an energy scale of ∼ 430K, which is of the same magnitude than the

energy scale recovered by the crossover from the Fano lineshape to the symmetric-Lorentz

one of the 294 cm−1 phonon in TiOCl along the chain direction (Fig. 4.18). Furthermore,

in Raman spectroscopy a depletion of a weak continuum of scattering is visible, in the

low energy range and for T < Tc1. This effect is better noticeable when extrapolating the

scattering intensity from higher energies, as shown by the dashed line for the T = 5 K

data (Fig. 4.9). It has an approximate onset at 300 cm−1≈ 430K, in agreement with the

energy scale obtained from spectral changes in the IR absorption spectra. Since similar

effects have been observed in α′-NaV2O5 and Sr14Cu24O41 at the double energy of the

system’s spin-gap [1], we also attribute this onset in TiOCl to 2∆spin = 300 cm−1. The

spin-gap 2∆spin is about a factor of two smaller than the pseudogap 2∆ f luct determined

by NMR [24] and would lead to gap ratios of 2∆spin/kBTC = 4.6 and 6.7, for Tc2 and Tc1,

respectively. With respect to the mean-field results, our larger gap ratios reveal competing

exchange paths or electronic degrees of freedom, but they are more reasonable than those

obtained from NMR study [24].

Finally, combining our results with those in the literatures, we can discuss the phase

diagram of TiOCl, with respect to the sequence of the characteristic temperatures Tc1, Tc2

and T∗. At high temperature T T∗ the layered quantum spin system TiOCl behaves

4.3 Discussion 83

as a spin 1/2 chain running along the b axis and the spin is localized in the Ti dxy or-

bitals. The LDA+U calculations of Saha-Dasgupta et al. [15] attest that the ground state

is completely determined by these dxy orbitals. This theoretical result is supported by the

high temperature data of the spin susceptibility, which can be fitted with a spin 1/2 1D

Heisenberg model. By decreasing T below T∗ the coherence length of the structural dis-

tortion slowly increases while the admixture of the t2g orbitals changes as hinted by the

Raman spectroscopy results [36] and supported by frozen-in phonon calculations [15].

For T∗ > T > Tc2, a more 2D character of the magnetic correlation with an enlarged role

of the dyz and dxz orbitals appears. This phase is characterized by possible orbital fluc-

tuations induced by strong electron-phonon interactions. The energy gain for T < Tc2 is

mainly related to the spin system. The anomalies in the specific heat are small [20] and in

conventional x-ray scattering no sign of a coherent structural distortion can be found [5].

By further lowering the temperature the spin and orbital fluctuations are suppressed and

below Tc1 there is a long range structural distortion. This is supposed to be the conse-

quence of an order-disorder transition based on spin-lattice coupling, which enlarges the

splitting between dyz and dxz, while the ground state remains described by the dxy or-

bital [15]. TiOCl below Tc1 can be regarded as a 1D dimerized spin chain system with a

global spin-gap 2∆spin. In the short-range-ordered phase (T < Tc1), there exists a larger

spin-gap 2∆ f luct, detected by NMR [24], as a smallest energy for a local double-spin-flip.

Therefore, 2∆ f luct, extracted from the NMR data, is not related to the transitions at Tc1 and

Tc2. TiOBr, even though far less investigated both theoretically and experimentally, reflects

the above proposed scenario for TiOCl, with however lower critical temperatures.

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5 LiCu 2O2

As pointed out in the introduction and in the theory chapter, low dimensional frustrated

S= 1/2 spin systems are of major relevance in solid state physics, since in their phase

diagram a manifold of fascinating phenomena like spin and charge ordering, dimerization,

or superconductivity have been predicted. In this context, LiCu2O2 is of interest. Indeed,

a great experimental attention was recently devoted to several cuprates characterized by

CuO4 plaquettes, which are the structural basic unit of high Tc superconductors. The differ-

ent way these plaquettes are arranged in the 2D plane (i.e. either corner- or edge-sharing)

influences strongly the magnetic configuration of the investigated systems [1–3]. The edge-

sharing CuO4 plaquettes in LiCu2O2 leads to a reduced nearest neighbor (nn) and a next

nearest neighbor (nnn) exchange of similar size, which allows for frustration effects [4].

LiCu2O2 crystals can thus be regarded as a realization of a S= 1/2 spin chain with

competing nn and nnn interactions (see section 5.1). Above 23 K, high-field electron spin

resonance (ESR) gives evidence for a spin singlet state with a spin gap ∆/kB ∼ 72 K.

Interestingly, upon cooling the spin gapped state evolves at TC ' 22.5 K into a long-range

ordered state with a helimagnetic structure [5]. Furthermore, some bulk measurements

point to the presence of a second low temperature transition to a collinear AFM structure

at Tc2 ' 9 K [6]. In all these phase transitions, an important role is played by the chem-

ical disorder which influences the exchange interactions responsible for the long range

magnetic order [5]. These magneto-structural peculiarities of LiCu2O2 provide a good op-

portunity to investigate the influence of chemical disorder on the magnetic properties of a

frustrated spin chain system [7].

89

90 5. LiCu2O2

c

ab

a) b) J4

J2J

1

Figure 5.1: a) Crystal structure of the LiCu2O2 compound [8]. b) Magnetic exchanges

between the different Cu2+ atoms in the zigzag ladder-like structure.


LiCu2O2 is a compound with a copper mixed valence (i.e., Cu+ and Cu2+) and has an

orthorhombic crystal structure of a space group Pnma(62). The unit-cell parameters are

a = 5.72 Å, b = 2.86 Å and c = 12.42 Å [8, 9], and since the a/b ratio is close to 2,

LiCu2O2 crystals are predestinated to show micro-twinning, i.e. they have domains with

different growth orientation. Furthermore, x-ray diffraction studies reveal local lattice distor-

tions with the consequent deviation from the Pnmasymmetry [8,10]. The crystal structure

is illustrated in Fig. 5.1a and is formed by a double layer of magnetic Cu2+ (d9 configura-

tion), O and Li. The double layers are separated by a layer of nonmagnetic Cu+ (d10). In

LiCu2O2, there is an equal amount of Cu+ and Cu2+ atoms.

Cu2+ ions form two linear chains along the b axis which are arranged in such a way

to form a zigzag ladder-like structure (see Fig. 5.1a). The ladders are isolated from each

other by both Li ions and layers of nonmagnetic Cu+. The distance between the magnetic

nearest-neighbor Cu2+ ions along the chain is of about 2.86 Å, while the distance between

Cu2+ ions on different chains is 3.08 Å [5]. The shortest distance between the Cu2+ on

neighboring ladders is 4.79 Å along the c axis, while along the a direction is 5.72 Å. These


distances are almost twice as large as the distance between Cu2+ ions along the ladders,

so that magnetic exchange between different zigzag ladders is small, even though a super-

exchange Cu2+-Cu2+ coupling is possible via oxygen bridges along the a axis.

Looking at the crystal structure of LiCu2O2 (Fig. 5.1), one can remark that the Cu+

atoms are coordinated by two oxygen atoms along c axis, whereas in the ab plane, each

Cu2+ atom has four O disposed in a square, with the Cu2+ atoms lying in the middle of this

square. These CuO4 units are connected by the square’s edge along the chain b direction,

so that the resulting Cu-O-Cu bond angle is nearly 90. The arrangement of the CuO4

units is of great importance, since the magnetic properties of cuprates are known to be

crucially dependent upon the Cu-O-Cu bond angle, which links nearest-neighboring Cu

atoms. When this angle approaches 90, the strength of the super-exchange interaction

diminishes and changes from anti-ferromagnetic (AFM) to ferromagnetic (FM) [1–3]. The

peculiarity of LiCu2O2 crystals arises from the fact that the Cu-O-Cu angle in LiCu2O2 is

close to the crossover region between AFM and FM ordering. Consequently, frustration

effects are anticipated as a result of competing interactions. Carrying on with structural

considerations, we note that the two Cu2+ chains are staked in such a manner that a

fifth oxygen atom is shared with the adjacent chain. This O is the apex of a O5 pyramid

around the Cu2+ atom. Furthermore, the Li+ ions are embedded in similar O5 pyramid

(Fig. 5.1a) [9,11].

The resulting space group Pnma(62) corresponds to a D162h point group. X-ray inves-

tigations reveal two inequivalent positions for the O atoms [8, 11]. Each atom has a site

symmetry Cs so that the total number of the expected modes after the correlation method

turns out to be [12]

Γcryst = 5ΓCs = 10Ag +5B1g +10B2g +5B3g +5Au +10B1u +5B2u +10B3u.

After subtracting the 3 acoustic modes, namely Γacustic= B1u+B2u+B3u, one obtains for

the vibrational modes:

Γcrystvib = Γcryst−Γacoustic= 10Ag +5B1g +10B2g +5B3g +5Au +9B1u +4B2u +9B3u.

Among them some modes are Raman active while others are infrared (IR) active:

ΓRaman = 10Ag (aa,bb,cc)+5B1g (ab)+10B2g (ac)+5B3g (bc)

ΓIR = 9B1u (E‖c)+4B2u (E‖b)+9B3u (E‖a). (5.1)

92 5. LiCu2O2

(10

-3 e

mu

/mo

l)χ

Figure 5.2: Magnetic susceptibility χ(T) of LiCu2O2 measured in a magnetic field H =

100 Oe parallel or perpendicular to the c axis. The numerical derivative of the χ(T) is

plotted in the inset and reveals a phase transition at Tc ≈ 22.5 K [5].

At high temperatures, Raman experiments detected 12 of the 15 predicted phonon in the

ab configuration, while due to lattice rearrangement, two additional phonons are detected

below 55 K [7]. In this context, it is worth mentioning that early x-ray investigation reported

the tetragonal crystal structure P42/nmc (137) with point group D154h [13]. Applying the

correlation methods [12] to this point group, one obtains the same number of phonons

along the c axis (9A2u (E‖c)) while in the abplane one has 13Eu (E‖ab) independently of

light polarization. Furthermore, high-resolution x-ray spectroscopy shows a substantially

increase (by a factor of 2) of the orthorhombic strain defined as (a−2b)/(a+ 2b) while

the temperature is increased from 10 K up to 300 K [10]. This increase might influence

the lattice dynamics. An increase of the lattice strain is unusual and one would expect a

decrease, since materials tend to approach higher symmetries by increasing temperatures.

Indeed, the increase of the lattice vibrations leads to lattice’s relaxation and to strain’s

reduction.


Figure 5.3: Specific heat of LiCu2O2. The dashed line indicates the phonon contribution

for ΘD = 400K, while the solid one indicates the magnetic entropy. The inset shows the

9 K peak in more details for an applied magnetic field of H = 0 and 12 T. [6].

The magnetic susceptibility χ(T) of LiCu2O2, presented in Fig. 5.2, is measured in

the temperature range 5−350 K with a magnetic field H = 100 Oe applied parallel (χ‖)

and perpendicular (χ⊥) to the ab plane [5]. χ(T) is characterized by a broad maximum

at T ∼ 36 K typical of low dimensional anti-ferromagnets [6]. The temperature derivative

of the magnetic susceptibility (inset of Fig. 5.2) reveals a sharp anomaly at Tc ∼ 22.5 K,

which is attributed to the onset of long-range magnetic order. The high-temperature part

of the χ(T) curve is fitted above T ∼ 100 K with the Curie-Weiss formula for an AFM

(eq. (2.37)) with fitting constants CCurie' 0.373emu·K and ΘW ' 81 K for magnetic field

along c, and CCurie ' 0.334 emu·K and ΘW ' 93 K in the ab plane [6]. For the sake of

completeness, we mention that a different χ(T) measurement reports a second magnetic

transition at T ≈ 9 K [6], not detected in the data of Fig. 5.2. It is argued that this feature

might arise due to an impurity phase, most likely Li2CuO2, which is known to undergo an

AFM transition at 9 K [14].

The specific heat (Cp(T)) measurements of three independent groups [5,6,10] show

two small maxima at Tc ∼ 22.5 K and T1 ∼ 24 K, as it is illustrated in Fig. 5.3 [6]. The

transition at T1 ∼ 24 K is interpreted as a precursor effect of the magnetic transition at

Tc ∼ 22.5 K, which contrary to the transition at T1 is also detected in χ(T). One sees in

94 5. LiCu2O2

Figure 5.4: Crystallographic unit cell of LiCu2O2 showing the magnetic Cu2+ sites and

the planar helimagnetic spin structure determined by neutron scattering [5].

Fig. 5.3 that at 9 K there is another feature which is connected to the phase transition, found

in the susceptibility data of Ref. [6]. This peaks shifts to 7.3 K by applying a magnetic field

of 12 T, indicating that high magnetic fields disturb the AFM ordering at low temperatures.

The dashed line in Fig. 5.3 represents a crude estimate for the phonon contribution with a

Debye temperature of ΘD = 400K. This estimation of the phonon contribution is removed

from the Cp(T) data and the obtained curve is integrated in order to achieve the magnetic

entropy removed at each transition. The resulting magnetic entropy is depicted by the solid

line in Fig. 5.3.

Neutron scattering is an appropriate experiment to shed light on the nature of the

magnetic ordered state. It is found that below Tc ∼ 22.3 K, LiCu2O2 acquires incommen-

surate magnetic long-range order [5], which has been also detected in NMR (nuclear

magnetic resonance) and Raman spectroscopy studies [4, 7]. These results point to the

presence of frustrated magnetic interactions, on the nature of which there is a lot of con-

troversy [15,16]. Masuda et al. assume in the analysis of the magnetic Bragg peaks [5] that


the exchange interactions J1, J2 are of AFM nature while J4 is negligible (see Fig. 5.1b).

In this way, under the assumption that all the magnetic Cu2+ sites carry the same moment

(i.e., no spin-density wave), the recovered magnetic structures corresponds to an uniform

planar spin helixes propagating along the double zigzag chains (see Fig.5.4) [5,15].

On the other hand, Ref. [4] and [16] propose an alternative scenario where J1 ≈ 0,

so that the zigzag ladder reduces to a couple of parallel chains. The chain is characterized

by a FM nearest neighbor exchange J2 ≡ Jnn, as it would be expected from the Kanamori-

Goodenough rule1, and by an AFM next nearest neighbor exchange J4 ≡ Jnnn [4, 16].

Besides a better account of the experimental data, this FM-AFM picture is furthermore

supported by the LDA calculation, which estimates exchange values of the order Jnn ∼−7.6 meV and Jnnn∼ 7.3 meV [4]. This estimation of the exchange parameters leads to a

frustration parameter α = |Jnnn||Jnn| ≈ 1, well above the critical value of ∼ 1/4 for the isotropic

Heisenberg chain (see section 2.3.1). Gippius et al. emphasize furthermore that for a 3D

arrangement of the chains such as in LiCu2O2 and for the FM J2 ≡ Jnn exchange, the in-

chain frustration is the only source which is strong enough to drive the system into a spiral

spin arrangement [4].

Following the proposal of Ref. [4] and [16] with a frustration parameter α ≈ 1, one

expects a gapped disordered phase (see section 2.3.1). Indeed, electron spin resonance

(ESR) experiment shows that a magnetic energy gap opens above Tc ≈ 23 K, separating

the low energy spin singlet from the first excited triplet (Fig. 5.5) [6]. In Fig. 5.5, the in-

tensity of the ESR resonance is plotted as a function of the temperature together with the

high temperature fit based on a Bolzmann distribution [6]. The best fit of the experimental

data gives an energy gap ∆/kB ≈ 72 K [6]. The drop below Tc in the ESR spectra might

be associated to a collapse of the magnetic phase characterized by spin-singlet states,

with a transition into the long-range-ordered magnetic state discussed above [6]. In the

same temperature range (i.e., above 23 K), Raman spectroscopy observes a two magnon

continuum from 102 to 120 cm−1 [7], which is assigned to a double spin-flip process of two

singlets into a higher singlet state [17]. Since the onset of the continuum corresponds to

twice the gap, one obtains an energy gap ∆/kB≈ 73K, in agreement with ESR results [7].

1 FM exchange interaction is predicted for Cu-O-Cu bond angle close to 90 [1,2].

96 5. LiCu2O2

Figure 5.5: Integrated resonance intensity of the ESR signals versus temperature. The

solid line is a fit with the energy gap ∆/kB ≈ 72 K [6]. The inset shows the singlet-triplet

schema.

5.2 Optical results

We measured the optical reflectivity R(ω) from the far-infrared (FIR) up to the ultra-

violet (UV), i.e., 5 meV-12 eV, as a function of temperature (T) and magnetic field. We

did not find, however, any magnetic field dependence in our spectra. Light was polarized

along the a and b axis. Figure 5.6 presents the R(ω) data for both polarizations in the low

frequency spectral range, where a temperature dependence is detected. For the sake of

completeness, the inset of Fig. 5.6 displays the whole R(ω) spectra at 200 K. The real

part of the optical conductivity (σ1(ω)) is plotted in Fig. 5.7. The inset of Fig. 5.7 is a

blow-up of σ1(ω) spectra at 200 K in the far infrared (FIR) spectral range for both polariza-

tion directions, emphasizing the phonon modes of LiCu2O2. The R(ω) and σ1(ω) optical

spectra clearly establish the insulating nature of LiCu2O2 at each temperature and for both

polarizations, in agreement with the resistivity data [19,20].

5.3 Discussion 97

100

80

60

40

20

0

Ref

lect

ivity

(%

)

70x10-36050403020100Energy (eV)

80

60

40

20

0

Ref

lect

ivity

(%

)

200 K 150 K 100 K 50 K 10 K

LiCu2O2 a axis

b axis

100

80

60

40

20

00.01 0.1 1 10

200 K a axis b axis

Figure 5.6: Temperature dependence of R(ω) at ω < 70 meV, for light polarized along

the a (upper panel) and b axis (lower panel). Inset: R(ω) spectrum at 200 K up to the UV

spectral range for both polarization directions. [18].

5.3 Discussion

At high energies the interband transitions set in smoothly with a spectacular and unusual

huge absorption at 3.1 eV (for both polarizations), which is followed by other weaker inter-

band absorptions at 3.8 and 7.8 eV. The huge resonance at 3.1 eV is related to an optical

98 5. LiCu2O2

6000

4000

2000

0

σ1(

ω) (

1/Ω

cm)

0.1 1 10

Energy (eV)

200 K a axis fit a axis b axis

300

200

100

0604020

x10-3

a axis fit a axis b axis fit b axis

Figure 5.7: High frequency part of σ1(ω) at 200 K for both polarizations. The Fano-

Lorentz fit is also shown for the a polarization. The inset is a blow up of the energy

region pertinent for the phonon modes. The outstanding fit quality of the σ1(ω) spectra is

appreciable for both polarizations (see text). [18].

excitation from the O 2p nonbonding bands to the upper 3d Hubbard band. We may iden-

tify this prominent absorption peak with the transition from the narrow band observed in

the ARPES spectra at 2.1 eV [18]. A similar feature has been identified at the same energy

in the absorption spectrum of Li2CuO2 [21]. Furthermore by performing a Hubbard model

calculation in LiCu2O2, Mizuno et al. found a very small spectral weight associated to the

optical transition from the Zhang-Rice local singlet [22], which is formed by an antibonding

combination of Cu 3d and O 2p hole states on the same plaquette, into the upper 3d Hub-

bard band. The small spectral weight is the result of the reduced hopping probability of a

hole along edge-sharing chains. The absorption feature associated to this transition cannot

be identified in the optical conductivity of LiCu2O2 below 3.1 eV (Fig. 5.7). Nevertheless

5.3 Discussion 99

with the help of ARPES data [18], which locate the top of the Zhang-Rice local singlet band

at 0.95 eV, we can determine the charge transfer energy as ∆E = 1.95 eV. Interestingly,

this value is substantially larger than the LDA+U band structure calculations, which have

predicted an insulating energy gap of 0.66 eV [9]. In this calculation, the insulating gap is

opened mainly between the oxygen 2p state and a hybrid 3d−2p state. We further assign

the interband transitions observed in our spectra at 3.8 and 7.8 eV to transitions from the

remaining nonbonding and bonding O 2p and Cu 3d states to the upper Hubbard band.

We now move to the discussion of the phonon modes. The electrodynamic response

along the b axis, is characterized by two sharp phonons at 0.03 and 0.052 eV, and two

weak modes at 0.036 and 0.039 eV. Along the a axis, four major phonons are detected

at 0.03, 0.04, 0.049 and 0.055 eV with an additional weak mode at 0.036 eV (Fig. 5.6

and 5.7). The four phonons observed along the b axis agree with the number of the B2u

modes predicted by group theory (eq. (5.1)). On the other hand along the a axis, our

measurement only detects five modes, less than the predicted nine B3u. The energies of

the modes along the a axis are similar to those observed along b axis and such a similarity

in the mode energies as well as in their number might indicate that twinning in our sample

cannot be neglected. We remember that twinning is common in LiCu2O2 single crystals

and is favored by the crystal structure geometry. Furthermore, other techniques reported

similarities between the a and b axes. For example, the first x-ray experiment established

for LiCu2O2 a D4h point group, where the rotation by π/2 along the c axis is a symmetry of

the system [13]. This implies the uniformness of the electrodynamic response, particularly

as far as the lattice dynamics is concerned.

In order to quantitatively analyze the electrodynamic response at different temper-

atures and polarizations, each σ1(ω) curve has been fitted with the Fano’s formula (eq.

(3.30)). The asymmetric line shape for the (sharp) phonon modes derives from an interac-

tion between lattice vibrations and a continuum, usually given by an electronic background.

The interaction with a magnetic continuum may also lead to a Fano lineshape. The fit’s

quality at any temperature is astonishingly good, as demonstrated by the fit of σ1(ω) at

200 K (Fig. 5.7). Furthermore, the same set of fit parameters used to account for the real

part of the optical conductivity fits equally well the reflectivity curves.

The phonons of both polarizations have a weak temperature dependence, with fit

parameters changing smoothly. Of particular interest is the temperature evolution of the

phonon mode peaked at 0.03 eV (242 cm−1) along the b axis. This phonon gradually

100 5. LiCu2O2

8x10-3

6

4

2

0

1/q2

200150100500T (K)

1.4x10-3

1.2

1.0

0.8

0.6

0.4

Γ (eV)

1/q2 (left axis) Γ (right axis)

3.0x10-3

2.9

2.8

2.7

2.6

2.5

SW

(eV2)

1 10 100T (K)

3.6x10-3

3.5

3.4

3.3

3.2

3.1

SW

(eV

2 )

a axis b axis

Figure 5.8: Scattering rate Γ and interaction strength 1/q2 of the phonon mode at 30 meV

along the b axis. The inset shows the temperature dependence of the spectral weight

(SW) for both polarizations. Note the different y-scales for SW along a and b axis. The

two y-scales have been shifted in such a way that the maximum of both SWaround 10 K

coincides.

hardens by 1.2 % as the temperature decreases from 200 K down to 2 K. This harden-

ing might be due to a freezing of the lattice motion. Furthermore, this phonon shows an

asymmetric line shape fitted with a negative q value in the Fano’s approach. The negative

value of the asymmetry factor q indicates an interaction between the phonon mode and an

electronic or magnetic continuum, extending at lower energies. In the case of LiCu2O2, we

propose a continuum, obtained by populating states with excitation energy ∆/kB≈ 72−73

K. This would be consistent with findings from the electron spin resonance and Raman

spectroscopy [6,7].

The temperature dependence of the interaction strength 1/q2 is depicted in Fig. 5.8.

The asymmetry is strongly reduced with decreasing temperature, until it reaches a satura-

5.3 Discussion 101

tion around 30 K, where an almost symmetric line shape of the mode at 30 meV develops.

Simultaneously, the phonon mode interacting with the continuum gets narrow by lowering

the temperature, as demonstrated by the decrease of its damping Γ at low T (Fig. 5.8). It

is quite of interest that these two parameters show an analogous temperature dependence

(Fig. 5.8): the phonon’s life time seems to be directly correlated with the increase of the

phonon-continuum interaction. The increase of 1/q2 indicates that the interaction with the

continuum is more important at high than at low temperatures. The continuum is thermally

populated at high T and therefore can interact with the phonon mode. On the other hand, at

low T, the continuum is not populated and the interaction between continuum and phonon

disappears (Fig. 5.8). The reduction of the scattering rate Γ at low T obviously supports

this picture, since the phonon’s lifetime increases upon the suppression of the continuum.

Finally, we briefly address the temperature dependence of the spectral weight (SW),

obtained by integrating σ1(ω) from 0 up to 33 meV (i.e., a frequency just above the phonon

mode), which is plotted in the inset of Fig. 5.8 for both polarizations. One notes that SW

has a broad peak between 5 and 20 K, while below 5 K SW has a sudden drop. Above

20 K, there is a reduction of the SWby increasing the temperature along the a axis, while

along b the SWreduction is softer. Such a temperature dependence of SWgives evidence

of the three different phases in the LiCu2O2 samples. The SW reduction below 5 K might

be connected with the antiferromagnetic phase transition detected at 9 K with other ex-

perimental methods in different LiCu2O2 samples. Since this transition might be due to

Li2CuO2 impurity phases, our data would then signal the presence of some Li2CuO2 im-

purity domains. The broad peak in the SWextending from 5 to 20 K reflects the long range

ordered magnetic phase between 9 and 23 K, while once the temperature increases above

23 K and the system enters in the gapped spin singlet state phase, LiCu2O2 reacts with a

decrease of SW.

Bibliography

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Bartkowski, M. Neumann, and R. Berger, Phys. Rev. B 57, 4377 (1998).

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[13] S. J. Hibble, J. Köhler, and A. Simon, J. Solid State Chem. 88, 534 (1990).

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6 Na0.7CoO2

Since the discovery of high Tc superconductivity in layered copper oxides [1], many re-

searchers tried to find similar behaviors in other layered metal oxides involving 3d transi-

tion metals, such as cobaltates and nickelates. Before the discovery of superconductivity in

NaxCoO2•yH2O such attempts had failed, with the result that the copper oxide layer was

thought to be essential for superconductivity. This is one of the reasons why the reported

superconductivity in NaxCoO2 intercalated with water induced such a great excitement in

the scientific community [2]. Another reason for the great interest in NaxCoO2 is the pe-

culiar geometry of the Co layers, which are characterized by a triangular lattice (see Fig.

6.1 below). This gives a unique possibility to study the interplay between spin frustrated

systems and superconductivity. Moreover, NaxCoO2 (with no H2O intercalation) allows to

address interesting problems related to the frustration of the magnetic order, which is one

of the major issue in the research on strongly correlated materials [3].

In the following chapter, the major findings on the NaxCoO2 compounds will be sum-

marized and reviewed. First of all, we will present the crystal structure and the most rel-

evant physical properties, unfolding the NaxCoO2 phase diagram. Emphasis will be dedi-

cated to those aspects which are important in the discussion of our data collected on the

x= 0.7 sample. Thereafter, we will review some band structure calculations, and compare

them with the experimental data. In the second part of the chapter we will present our op-

tical results on the x = 0.7 sample, complemented with results on samples with other Na

content. This will give an overview of the electrodynamic response across the entire phase

diagram. The discussion will mainly focus on the interpretation of our data, analyzed in

the framework of the generalized Drude term. Our data reveal a non Fermi-liquid behavior

over a broad interval in both temperature T and frequency ω. The implications of our find-

ings with respect to the formation of a Spin Density Wave (SDW) state will conclude the

105

106 6. Na0.7CoO2

Figure 6.1: Schematic drawing of the layered structures for a) non-hydrated Na0.61CoO2

and b) fully hydrated Na0.35CoO2•1.3H2O [8].

chapter.


6.1.1 Crystal structure

NaxCoO2 is a highly hygroscopic material, which makes it very unstable under ambient

conditions. This and the unavoidable Na evaporation during the high-temperature synthe-

sis reduces the control over the final stoichiometry and prevents a good characterization

of the intrinsic and structural properties for a wide range of x. The most stable phase,

x ≈ 0.67, is quoted to be hexagonal with space group P63/mmc (194) [8, 9]. However


recent x-ray data in the compound with a Na content of x = 0.67 reveal two new minor-

ity phases of the hexagonal space groups, P63/m (176) and P6/m (175), which must be

considered in addition to the P63/mmcmajority phase [10]. In this context it is worth men-

tioning that first experiments found a different symmetry for the NaxCoO2 (e.g., P63/22 in

x≈ 0.5 in Ref. [11]). In the case of the superconducting samples (x≈ 0.35), the complexity

and controversy is even bigger, because a mixture of fully and partially hydrated phases

normally coexist in the same specimen [12].

The crystal structure of NaxCoO2 consists of two dimensional Co sheet octahedrally

coordinated with O above and below the Co planes (Fig. 6.1). The Co atoms on the layer

are arranged on a triangular geometry so that the resulting crystal structure turns out to

be hexagonal [13]. As seen in Fig. 6.1a, also the O above and below the Co layer form

a triangular lattice which however is slightly shifted with respect to that of Co. It results

that each Co is surrounded by six O that form a distorted octahedra [14]. The CoO2 layers

are stacked along the hexagonal c axis with layer of Na intercalated between them. The

resulting structure is then characterized by the D46h point group, considering the space

group P63/mmc(194). The reported crystal structure has two inequivalent positions for

the Na ion, the first with Wickof index 2d and the second 2b [14,15]. The positions (Wickof

index) for the Co and O atoms are 2a and 4 f , respectively. Each unit cell contains two

groups of NaCoO2 and using the correlation method [16], the whole phonon spectrum for

the NaCoO2 is:

ΓD6hcryst = ΓD3d +ΓC3v +2ΓD3h = A1g +3B1g +E1g +3E2g +4A2u +2B2u +4E1u +2E2u.

Among them, some phonon modes are Raman active:

ΓRaman= A1g(aa,bb,cc)+E1g(aa,bb,ab)+3E2g(ac,bc)

in agreement with previous published calculations [14,15]. We remark that in all the Raman

active modes, the Co atoms is not involved. Indeed the A1g and the E1g modes arise

entirely from Oxygen vibrations, while both O and Na atoms are involved in the E2g modes

[17]. The IR active modes are obtained after subtraction of the acoustic modes (Γacustic=

A2u +E1u) and turn out to be:

ΓIR = 3A2u(E ‖ c)+3E1u(E ‖ ab plane).

108 6. Na0.7CoO2

Figure 6.2: Variation of Tc as a function of Na content in NaxCoO2•yH2O. The water

content is constantly y≈ 1.3 [19].

As a final remark we note that this phonon calculation has been performed on the assump-

tion that all Na sites are occupied. However, the different stoichiometry in the Na content

introduces Na vacancies, which might reduce the intensity of the detected phonon modes.

6.1.2 The superconductor Na xCoO2•yH2O

The report by Takada et al. [2] that sodium-cobaltate intercalated with water (see Fig. 6.1b)

develops superconductivity below 4.5 K increased the interest on this material. The final

composition of the first reported superconductor NaxCoO2•yH2O was a x,y content of

x≈ 0.35 and y≈ 1.3. Initially, Schaak et al. reported that the critical temperature Tc ver-

sus Na content (x) displays the same kind of dome shape behavior that is observed in

the high Tc copper oxides, with the highest superconducting Tc in a narrow range around

x≈ 0.35 [18]. That raised speculations on a possible non conventional superconductivity

behavior in NaxCoO2•yH2O. However, further investigations by Milne et al. [19] and Chen

et al. [12] indicate instead that superconductivity manifests over a broad range of x with

almost constant Tc, as illustrated in Fig. 6.2 [19]. We remark at this point that although

the water content is difficult to be controlled, it appears to be always constant at y≈ 1.3.

The Tc in Fig. 6.2 varies from 4.3 to 4.8 K in the region 0.28< x < 0.37, close to the op-

timum doping (x≈ 0.35) reported for this material. Similarly, single crystal measurements


Figure 6.3: The variation of Tc as a function of the c/a ratio. Closed circles represent

measurements from Milne et al. (Ref. [19]), the open ones refer to the data of Schaak et

al. (Ref. [18]) and open diamonds to the those of Jin et al. (Ref. [20]).

display an even broad superconducting regime with x extending from 0.28 to 0.42 [12].

The weak dependence of Tc upon doping x suggests a more conventional mechanism for

superconductivity.

The reason why water inclusion triggers superconductivity is not fully understood

yet. However the intercalation of H2O between the CoO2 layers increases dramatically

the lattice spacing and thus reduces the electronic dimensionality of the structure (the c

axis increases more than 50 % over its original value). In this respect, it is interesting to

compare the critical temperatures with the ratio of the c and a axis as shown in Fig. 6.3,

where the Tc of three different works [18–20] are plotted as a function of the c/a ratio. One

sees that the highest Tc is reached in those samples with a c/a ratio of ∼ 6.96− 7.02

irrespective of x content [19]. Samples with a c/a ratio below ∼ 6.96 shows a lower Tc,

which may be indicative of under hydrated samples [19].

Among various scenarios advanced for the role of water, the most plausible one is

that H2O has only a structural role and acts as a passive lattice spacer. Hydrated samples

with lower Na content (i.e., x∼ 0.35) accept more H2O and become more 2D than crystals

with higher Na content. Thus, superconductivity is favored in a sample with x∼ 0.35. The

role of H2O as passive spacer is also supported by band structure calculations of Johannes

110 6. Na0.7CoO2

and Singh [21], who found that from an electronic point of view the hydrated and the non-

hydrated compounds are identical, indicating that the effect of water is overwhelmingly

structural and not electronic. Another possibility is that H2O posses a chemical role and

modify the doping of the CoO2 planes via unusual chemistry. For example thermoelectric

data of Banobre-Lopez et al. point to an active role of H2O in the determination of the

number of charge carriers in the CoO2 layer [10]. So far an accepted consensus on this

issue has not been reached.

The hint, that superconductivity in NaxCoO2•yH2O may be of unconventional type,

fueled speculations about the possible kind of symmetry pairing in the superconduct-

ing channel. Several theoretical works were dedicated to this issue with a multiplicity

of ideas. The density-functional calculations of Singh predict a weak itinerant ferromag-

netic (FM) state with consequent speculation about a triplet superconductivity, similarly

to Sr2RuO4 [22]. In fact, Nuclear Quadrupole Resonance (NQR) experiments of Ihara et

al. found a weak ferromagnetic order in a non superconducting NaxCoO2•yH2O, with a

magnetic transition temperature TM ∼ 5.5 K. Ihara et al. suggest that superconductivity

is realized near the magnetic phase, and thus that magnetic correlations are essential for

superconductivity [23]. Similarly, Michizuki et al. propose that triplet pairing is favored by

the ferromagnetic fluctuations on the hole-pocket band, caused by the Hund’s-rule cou-

pling between the Co t2g orbitals [24]. An early theoretical proposal of Tanaka and Hu

also predicts a spin triplet superconductivity triggered however by AFM fluctuations [25].

Johannes et al. assert that the nesting structures at the Fermi surface are most compat-

ible with an odd-gap triplet state [26]. Tanaka et al. propose a new mechanism based on

charge fluctuations, which in the vicinity of a charge density wave instability in a triangular

lattice may induce f -wave triplet superconductivity [27]. However, for the time being, no

evidence of charge fluctuations has been found. A similar idea is used also by Foussats et

al. to show that the interplay between electronic correlations and electron-phonon interac-

tions (which may be enhanced by charge fluctuations) leads to unconventional pairing [28].

Kuroki et al. suggest, instead, that possible spin-triplet f -wave pairing may be realized due

to disconnected Fermi surfaces [29]. From the experimental point of view, the majority of

results hint towards an unconventional spin triplet superconductivity [23,30–33]. The NQR

measurements of Fujimoto et al. are most consistent with non s-wave superconductivity

and furthermore suggest the existence of a nodal line in the gap function [31]. The same

conclusions are inferred from the specific heat study, which evidenced a Cv ∝ T2 depen-

dence at low T [30]. The 59Co NMR study of Kato et al. gives evidence for a p-wave paring


Figure 6.4: The phase diagram of non-hydrated NaxCoO2. The charge-ordered insulating

state at x = 1/2 is sandwiched between the paramagnetic metal at x ' 0.3 and the

Curie-Weiss metallic state at x ' 0.65/0.75. Above x ≈ 0.75, there is a spin density

wave metallic phase. The superconducting state is obtained on intercalation with H2O at

low x content (see section 6.1.2).

symmetry [32]. The measurement of the upper critical field HC2 performed by Maska et

al. showed an unusual temperature dependence with an abrupt change of the slope of

HC2 versus T. These findings are consistent with a crossover from a singlet supercon-

ducting state at low magnetic field (H < 0.9 T) into a triplet superconductivity at higher

fields [33].

6.1.3 The non-hydrated Na xCoO2 compounds

In order to better understand the superconducting properties of NaxCoO2•yH2O, a care-

fully study of the non hydrated system NaxCoO2 is of basic importance. This leads to

extensive studies of the physical properties in the NaxCoO2 systems (0.2 6 x 6 1), which

change abruptly as a function of Na content. This multiplicity of properties is mainly due to

the changes in the electronic configuration. Indeed, the metallic NaxCoO2 can be viewed

as a doped band insulator with a hole concentration of (1−x). In principle, with maximum

112 6. Na0.7CoO2

Na content (x= 1), there are no holes in the lattice. As the Na content is reduced, the holes

increase in proportion until every lattice site is unoccupied at x = 0. The phase diagram

for NaxCoO2 at different Na content is illustrated in Fig. 6.4, as proposed by Foo et al. [34]

based on spin susceptibility χ(T) and resistivity ρ(T) measurements (Fig. 6.5). In these

experiments the Na content is verified by the inductive coupled plasma-atomic emission

spectroscopy, while the unit cell parameters are determined by powder x-ray diffraction.

The recovered variation of the c axis versus the x content is plotted in the inset of Fig.

6.5c [34].

The region of the phase diagram with Na content x< 0.5 is characterized by a param-

agnetic metal response, where the profile of χ(T) is relatively featureless with a magnitude

which is large compared with the Pauli susceptibility of conventional metals [34]. This sus-

ceptibility indicates that all the Co ions are identical with an average valence Co(4−x)+.

The resistivity Na0.31CoO2 reflects a high-conductivity metallic state and at T below 30 K,

ρ(T) shows a temperature dependence distinctive of a Fermi liquid (i.e., ρ ∝ T2).

The case with x = 0.5 is of particular interest, since one has an equal amount of

Co3+ and Co4+ ions, and this may lead to exotic physical properties of this system. Indeed,

χ(T) reveals the existence of two cusps at Tc1 = 88K and Tc2 = 53K (arrows in Fig. 6.5a).

The specific heat measurement indicates that the transition at Tc1 = 88 K has a structural

component since a distinct peak is observed, whereas at Tc2 = 53 K the anomaly is more

subtle [35]. This is confirmed also by nuclear magnetic resonance (NMR) measurements,

where just at 88 K a magnetic rearrangement is detected, while at 53 K any anomaly is

observed [36]. The second transition at Tc2 signals the onset of an insulating state, which is

confirmed by the behavior of the resistivity ρ(T). In Na0.5CoO2, ρ(T) increases gradually

as T falls towards Tc1 and Tc2 and it exhibits only weak anomalies (note the change in

scale in Fig. 6.5). Below Tc2, however, ρ(T) rises rapidly. The nature of this insulating

state, which is confined into a narrow interval of x [34], is elucidated by electron diffraction

studies which reveal that Na ions order in a superstructure with lattice vectors a√

3x and

2ay in the basal plane, with a being the hexagonal lattice parameter and x, y the cartesian

unit vectors. By contrast at other doping levels, the superstructure Bragg spots are either

much weaker or even absent [34]. Additional experiments confirm that Na ions order in

a zigzag chain [35, 37]. Also thermal conductivity measured parallel to the layers rises

steeply below Tc2 only in Na0.5CoO2, indicating that with this Na content the scattering of

phonons by the Na disorder in the sublattice is strongly suppressed. This implies that the


Figure 6.5: The susceptibility χ(T) a) and in-plane resistivity ρ(T) b) of NaxCoO2 single

crystals. In panel a), χ(T) is measured with an in-plane magnetic field H ≈ 5 T. Panel

b) shows the T dependence of ρ(T) at selected x. The variation of c axis at different Na

content, obtained through x-ray measurements, is illustrated in panel c).

Na ions at x= 1/2 order with very long-range correlation [34]. The Na ordering, although it

does not affect directly the electrons on the Fermi surface, might be essential to explain the

insulating behavior of Na0.5CoO2. Since the Na vacancies couple more strongly to Co4+

than to Co3+, the Na ordering might be thus responsible for a Co4+/Co3+ charge ordering

at low temperatures [37].

At just higher Na content (x = 0.57), the resistivity is again metallic and follows the

unusual power law of 3/2 (i.e., ρ ∝ T3/2) below 175 K [38]. For Na content x' 2/3, the

susceptibility χ(T) follows the Curie-Weiss law χ(T) = C/(T + θ) with θ ≈ 70 K. This

114 6. Na0.7CoO2

Figure 6.6: Phase diagram of NaxCoO2 determined by the muon spin spectroscopy [39].

Solid and open circles represent data on NaxCoO2 single crystals and polycrystalline

samples, respectively. The point at x = 1 is extrapolated from the data on the related

compound LiCoO2 [40]

susceptibility resembles that of an insulator that is frustrated from attaining the ordered

Néel state. In this region, the observed Curie-Weiss susceptibility might be explained with

(1−x) Co4+ ions with spin S= 1/2 and x Co3+ with S= 0. The resistivity is metallic with

a characteristic T linear profile below 100 K [7, 34]. This ambivalence, metallic in charge

conduction but insulator-like in spin alignment, has been dubbed a “Curie-Weiss metal” [3].

For the crystal with x≈ 0,75, χ(T) is slightly rounded below 20 K, consistent with an

anti-ferromagnetic ordering [34]. The ρ(T) displayed in Fig 6.5 shows a distinct change in

slope near 20 K. This is in agreement with the measurements of Motohashi et al. which

first reported a kink in the resistivity at 22 K and a large positive magneto-resistance [41].

The same authors measured a small jump in the specific-heat at the same temperature.

Refined thermal as well as transport experiments by Sales et al. [42] and Wooldridge et

al. [43] indicate that this magnetic transition is consistent with the formation of a spin den-

sity wave (SDW) metallic state. Furthermore, muon spin spectroscopy on both polycrystral

and single crystals with 0.6 6 x 6 0.9 detected a transition from a high T paramagnetic to

a low T commensurate or incommensurate SDW state [39,44]. The recovered phase dia-

gram, describing the transition temperatures of the SDW versus the x content, is illustrated

in Fig. 6.6 [39].


Figure 6.7: LDA band structure calculation of Na0.5CoO2 [13].

6.1.4 Band structure calculations

The first published band structure calculation was performed in Na0.5CoO2 using den-

sity functional calculations within the local density approximation (LDA). The employed

method was the general potential linearized augmented plane wave (LAPW) with local or-

bital extensions, which is well suited to materials with low site symmetries and open crystal

structures, like Na0.5CoO2 [13]. The space group of the crystal used in that calculation is

P63/22which is only slightly different from P63/mmc, the today’s accepted NaxCoO2 crys-

tal space group. The Na ions were treated using the virtual crystal approximation on the

trigonal prismatic site 2b (0.5 occupation), neglecting the 2d symmetry. This approximation

is reasonable to the extent that the only role of the Na ions is to donate charge to the CoO2

framework [13].

The paramagnetic band structure is shown in Fig. 6.7, while the corresponding elec-

tronic density of states (DOS) is shown in Fig. 6.8. The O 2p bands extend from approx-

imately -7 eV to -2 eV, relative to the Fermi energy (EF ), and are clearly separated from

the transition metal d bands, which lie above. The Co d–O p hybridization is weak. The

Co d bands are crystal field split as usual in the octahedral O environment into a lower

lying t2g and an upper lying eg bands, separated by approximately 2.5 eV. The bands have

narrow bandwidths, of ∼ 1.6 eV in t2g complex and of ∼ 1.2 eV in eg. EF is in the top of

116 6. Na0.7CoO2

Figure 6.8: LDA DOS (N(E)) of Na0.5CoO2. The solid curve is the total DOS while the

dotted curve is the DOS of the bands with d-character [13].

the t2g complex, 0.22 eV below the band edge, corresponding to 0.5 holes per Co ion. This

agrees with the metallic behavior of Na0.5CoO2 (at high T).

The six bands composing the t2g complex in Na0.5CoO2 are further split in the rhom-

bohedral crystal field into two a1g and four e′g bands. The amount of splitting is sensitive to

the rhombohedral distortion of the O octahedra. The t2g band complex, calculated by Lee

et al. [45] for the case x = 1/3, is shown in detail in Fig. 6.9, where the band with primarily

a1g character is drawn in the “fatbands” representation with the corresponding DOS. This

band dispersion agrees well with that calculated by Rosner et al. [46]. The a1g character

is strong at the bottom of the t2g complex along the Brillouin zone boundary as well as

around the Γ point at the Fermi energy. Doping holes enter only a1g states until x≈ 0.6,

whereupon an e′g Fermi surface begins to form. The a1g DOS lies higher than that of e′g

due to the particular dispersion and to a substantially larger effective bandwidth. Judging

from the dispersion curves themselves, the a1g and e′g bands differ little in width. However,

nearly all of the e′g states lie within a 1 eV region, whereas the a1g DOS extends over 1.5

eV [45].

The a1g band crosses the Fermi level along both the Γ−K and Γ−M directions,

while one of the two e′g bands crosses it only along the Γ−K direction close to the K point

(Fig 6.9). The resulting Fermi surface is plotted in Fig. 6.10 for the case x= 1/3 [45], which

however is similar to the result obtained for x = 0.5 [13]. The Fermi surface show a large


Figure 6.9: Band structure (in the virtual crystal approximation) along high symmetry lines

(left panel) and the density of states (right panel) for the x = 1/3 cobaltate in the local

density approximation. The a1g symmetry band is emphasized with circles proportional to

the amount of a1g character. The a1g density of states is indicated by the darker line [45].

Γ-centered hole cylinder with some flattening perpendicular to the Γ−K direction. This

cylinder contains 0.43 a1g holes/Co. In addition, there are six additional hole cylinders,

primarily e′g in character, lying along the Γ−K direction, containing 0.04 holes in each

cylinders. The total is 0.67 holes necessary to account for x = 0.33 electrons, which was

the starting point for this calculation [45].

This Fermi surface geometry has important consequences for the different nesting

vectors. A nesting vector translates one portion of the Fermi surface into another one (see

section 2.2). There are three distinct inter-cylinder nesting vectors (Q1, Q2 and Q3) as well

as a main nesting vector on the central cylinder (Fig. 6.10). If the small six cylinder were

exactly circular, all three nesting vectors Q1, Q2 and Q3, shown in Fig. 6.10, would nest

perfectly. Neither symmetry requirement forces these pockets to be circular nor there is a

restriction imposed on their distance from the Γ point. However, they are nearly perfectly

elliptical so that only opposite pockets (i.e., nesting vector Q1 in Fig. 6.10) nest almost ex-

actly. Accidentally, the distance between opposite pockets is just slightly less than half of

the reciprocal lattice vector G [26]. Also the flattening of the Fermi surface along the Γ−K

may enhance the nesting properties in NaxCoO2. This Fermi surface structure can lead

118 6. Na0.7CoO2

Figure 6.10: Fermi surface for NaxCoO2, x = 0.30, in the two-dimensional Brillouin zone.

The large cylinder contains ag holes, whereas the six small cylinders contain holes that

are primarily e′g-like [45].

in the charge channel to a Peierls-type charge density wave instability, i.e., a structural

transition. Indeed, various superstructures have been reported, especially at x = 1/2, that

are ascribed to charge ordering of Na ions [35,37,47] (see section 6.1.3). The calculations

of Johannes et al. suggest that also the CoO2 planes have a tendency to form superstruc-

tures, independently from Na contributions. The observed superstructures, presumably,

are generated by both Co and Na ordering. Finally, we remarks that an anti-ferromagnetic

ordering with the presented nesting properties would result in the formation of a spin den-

sity wave [26].

For the sake of completeness, we mention that the local spin density approximation

calculations with an electron-electron interaction U = 5.5 eV (LSDA +U) found a slightly

different Fermi surface topology [48]. Zhang et al. calculated the Fermi surface at 06 x6 1

and found a strong doping dependence of the Fermi-surface and of the DOS at the Fermi

energy, as shown in Fig. 6.11. The DOS at the EF increases sharply with increasing doping

for x 6 0.15, reaches a maximum at about x≈ 0.2, and then decreases with increasing x.

The narrow doping range 0.1 6 x 6 0.3 beyond which DOS decreases rapidly is closely


Figure 6.11: Density of states of CoO2 layer at the Fermi energy as a function of doping

x, obtained by LSDA+U band structure calculations [48].

related to the Fermi surface structure of the system. The Fermi surface quickly extends and

then shrinks with increasing x. However, compared with LDA [13] or LDA+U [45] results

(Fig. 6.10), Zhang et al. found only a large Fermi surface centered at the Γ point, without

the six small pockets [48].

Experimentally, the Fermi surface has been measured with angle-resolved photo-

electron spectroscopy (ARPES) [49–51]. In Fig. 6.12, the intensity of the second deriva-

tives of the measured energy distribution curves is plotted in order to display the band

dispersion in Na0.6CoO2 [49]. From Fig. 6.12 it is evident that the extracted band dis-

persion has some similarities with the band structure calculated by Singh with the LDA

method [13]. This band calculation, superimposed in white on the ARPES data, shows

that the O 2p bands (-2 to -7 eV) and the Co 3d bands are well separated due to a weak

Co d-O p hybridization. This separation is clearly detected in the ARPES measurements

and is depicted by the black spots in Fig. 6.12, which represent the lack of bands. Integrat-

ing the spectra over a large k space one can mimic the DOS. Four peaks can be identified

at binding energies of 5.9, 4.6, 2.8, and 0.7 eV, respectively, matching well with the DOS

calculation shown in Fig. 6.8. Also the full valence band spectrum measured by Hasan et

al. in Na0.7CoO2 shows five prominent features at 0.7, 3, 4.1, 6 and 11 eV [50].

The experimentally measured spectral distribution (n(k)), which indicates the shape

120 6. Na0.7CoO2

Figure 6.12: Intensity plots of second derivatives of spectra along Γ-M and Γ-K directions

in Na0.6CoO2 [49] compared with the LDA calculated bands [13].

of the Fermi surface of Na0.7CoO2 over the complete Brillouin zone, is illustrated in Fig.

6.13a. The Fermi surface has been measured experimentally also for Na contents x =

0.3, 0.48, 0.6 and 0.72. In all these compounds, a single hexagonal hole like Fermi surface

centered at the Γ point is found and the Fermi surface size shrinks as the Na content x

increases. The parallel Fermi surface edges can be connected by a nesting vector QΓ

(see Fig. 6.10) of magnitude ∼ 1.41, 1.40, 1.20, 1.18 Å−1 (±0.1 Å−1) in the different

Na contents x = 0.3, 0.48, 0.6 and 0.72 [51]. The measured Fermi surface shape is

similar to the one calculated for Na0.5CoO2 using LDA [13]. A comparison is presented

in Fig. 6.13b. However, any of the small satellite pockets predicted by LDA calculations

is observed around the large hexagonal Fermi surface (Fig. 6.10) for a wide range of x

(0.3 6 x 6 0.72) [51]. In this respect the LSDA+U calculation of Zhang et al. seems to be

more consistent with the experimental data [48].

6.2 Optical results

A review of the reflectivity R(ω) in NaxCoO2 with x = 0.25, 0.5, 0.75at room temperature

as measured by Hwang et al. is offered in Fig. 6.14a [52]. The measured R(ω) is com-

pleted with the data of Wang et al. (full circles) at nominally x = 0.7 (Fig. 6.17 [53]) and


Figure 6.13: a) Spectral distribution n(k) in Na0.7CoO2 reflecting the Fermi surface. A hole

pocket is centered around the Γ point. The Fermi surface exhibits a hexagonal anisotropy.

b) Comparison with LDA calculation on Na0.5CoO2 [13] and measured data in Na0.7CoO2.

Red dots indicate the locations of measured Fermi crossings [50].

those of Lupi et al. (open circles) at x= 0.57 (Fig. 6.18 [54]). One notes that the reflectivity

increases by increasing the hole doping in the CoO2 layers (i.e., lower x) and the data mea-

sured by three different groups confirms the same trend. The R(ω) reported in Fig. 6.14a

was measured between 50 and 40 000 cm−1 [52]. In order to perform reliable KK transfor-

mation, the data were extended in the low frequency range with the Hagen-Rubens (HR)

extrapolation and at high frequencies (40 000 to 100 000 cm−1) with our experimental data

on Na0.7CoO2 (see below) [55]. The recovered optical conductivity for the three different

Na contents is plotted in Fig. 6.14b [52]. The dc resistivity used for the HR extrapolations

is marked on the ordinate axis.

The inset in Fig. 6.15 illustrates the number of effective carriers (Ne f f(ω)) as a func-

tion of the Na content (1− x). Ne f f(ω) is obtained by integrating the optical conductivity

and it grows monotonically with hole doping, indicating an increase of the number of charge

carriers. This is the optical evidence that the Na ions introduce holes at the Fermi surface

(see section 6.1.4). Viewed in this high frequency and high T region, the system becomes

more metallic with doping, even though, at low frequency and low temperature the x= 0.50

and 0.25 samples are actually insulators [47,52].

122 6. Na0.7CoO2

Figure 6.14: a) The ab plane reflectivity of NaxCoO2 at three doping levels, at room tem-

perature. The symbols at 1000 and 2000 cm−1 are data points from Lupi et al. [54] (open

circles) and Wang et al. [53] (filled circles). b) The corresponding optical conductivity in

the ab plane [52]. The data are taken from Ref. [52].

6.2.1 Optical investigation in Na 0.7CoO2

We have investigated a NaxCoO2 single crystals with nominally Na content x = 0.7 and

a size 2 x 2 mm. The crystals were grown by H. Berger et al. at the “Institut de physique

de la matière complexe” EPF Lausanne, using the flux methods as described in detail in

Ref. [56]. X-ray scattering measurements on these crystals confirm the Na content to be

0.700 ± 0.016. Samples from the same batch were furthermore characterized by the dc

transport measurements1. The temperature dependence of the resistivity ρ(T) (inset of

Fig. 6.16b) within the ab plane displays a linear behavior in temperature from 300 down to

1 The dc resistivity ρ(T) was measured within the abplane and along the c axis at EPF-Lausanne, with theconventional four points contact method.


Figure 6.15: Effective number of the charge carriers (Ne f f), obtained by integration of

σ1(ω) for the three different Na contents. The legend is the same as in Fig. 6.14. The

inset shows Ne f f at 7500 cm−1. The error on the x content is ±0.08 [52]. The data are

taken from Ref. [52].

100 K, as well as below 100 K, however with a smaller slope, in agreement with data of

Refs. [7,34]. We performed optical reflectivity measurements as a function of temperature

between 10 and 300 K and in a magnetic field up to 7 T. No changes in the spectra were

however found as a function of the magnetic field. Worth noting is that our investigations

covers the largest spectral range addressed so far on NaxCoO2.

Figure 6.16a displays the optical reflectivity R(ω) of Na0.7CoO2. Beside some ab-

sorptions at about and above 1 eV, one can recognize the quite sharp plasma edge fea-

ture with an onset at ' 0.7 eV. R(ω) increases with decreasing temperature below 0.2 eV,

indicative for the metallic character of Na0.7CoO2. At 0.07 eV, we clearly see the infrared-

active phonon mode, as in other compounds with different Na content. In the inset of Fig.

6.16a the reflectivity is plotted on linear scales at low energy, in order to highlight the linear

frequency dependence of R(ω) (see below).

The real part (σ1(ω)) of the optical conductivity, shown in Fig. 6.16b, is obtained

through Kramers-Kronig (KK) transformations of R(ω). The good agreement between

σ1(ω→ 0) and σdc = 1ρdc

is emphasized by the inset in Fig. 6.16b. It is easily verified that

the main conclusions of our work are fully independent from the employed extrapolations

due to the extremely broad measured spectral range. As expected from the R(ω) spec-

tra, the effective intraband metallic component in σ1(ω) is enhanced below 0.03 eV and

the σ1(ω → 0) limit increases with decreasing temperature, typical for a metallic system.

124 6. Na0.7CoO2

100

80

60

40

20

0

R (%

)

0.001 0.01 0.1 1 10

Energy (eV)

4000

3000

2000

1000

0

σ 1 (Ω·c

m)-1

A

B

C D

E

Na0.7CoO2

T=200 K T=150 K T=100 K T=60 K T=10 K

1.0

0.8

0.6

0.4

0.2

0.0

ρ (Τ

) /ρ

(300

Κ)

3002001000

T (K)

Optical Transport

100

75

500.20.10.0

T = 10 K Linear fit

a)

b)

Figure 6.16: a) Reflectivity and b) real part σ1(ω) of the optical conductivity of

Na0.7CoO2 at selected temperatures. The four high frequency absorptions in σ1(ω) are

labeled (see text). Inset a): R(ω) at 10 K between 0 and 0.2 eV, emphasizing the linear

behavior of R(ω) at low energies. Inset b): Comparison between ρdc(T) and the estima-

tion of the dc resistivity from the optical experiment (i.e., ρopticaldc (T) = 1/σ1(ω→ 0,T)).

At higher frequencies we recognize a weak feature at 0.4 eV (A) followed by more pro-

nounced and well defined features at 1.4 eV (B), at 2.8 eV (C), at 4.8 eV (D) and at 10 eV

(E).

An optical investigation in Na0.7CoO2 was also performed by Wang et al. [53] and

their results are plotted in Fig. 6.17. However, our optical data bear only a rough similarity


Figure 6.17: Frequency dependence of the in-plane R(ω) spectra at different tempera-

tures in Na0.7CoO2. The inset shows the σ1(ω) obtained trough KK transformations [53].

with the data of Fig. 6.17, even though the reflectivity was measured at the same nominally

Na content. A first astonishing difference is the crossing of the reflectivity spectra around

0.1 eV (Fig. 6.17), which is totally absent in our data (Fig. 6.16). Furthermore the spectra

of Wang et al. differ from ours by the bump at ∼200 cm−1 (0.024 eV), which is reflected

in a localized absorption in σ1(ω). This bump shifts to lower energy as the temperature

decreases and might be interpreted as a signature of charge localization such as the

formation of a polaronic band [59] or of a pinned collective phase mode [60]. The optical

measurement of Wang et al. shows many analogies with the optical data of Bernhard et

al. in Na0.82CoO2 [17]. We thus argue that the real sodium content in the sample of Wang

et al. is possibly higher than what reported by the authors.

On the other hand, our optical data and their trends in temperature are confirmed by

the data of Hwang et al. in NaxCoO2 with x = 0.75±0.08. Furthermore, our optical data

are also in good agreement with the optical data of Lupi et al., which measured the optical

reflectivity on a mosaic of Na0.57CoO2 crystals (Fig. 6.18) [54]. Indeed, the reflectivity of

126 6. Na0.7CoO2

Figure 6.18: Reflectivity of a mosaic of coplanar Na0.57CoO2 crystals at different temper-

atures, measured in the ab plane. In the inset, the R(ω) of the mosaic at 295 K (dotted

line) is compared with R(ω) measured on a single crystal through infrared microscopy

(solid line) [54].

Fig. 6.18 is similar to our results by not showing either the crossing at 1300 cm−1 nor

the bump at 200 cm−1. One notes that Lupi’s reflectivity is metallic at each temperature

in agreement with the ρdc measurement [38]. The measured metallic regimes with no

hint of charge localization in NaxCoO2 (x = 0.57,0.7) give an optical confirmation of the

phase diagram proposed in Fig. 6.4, where for those concentrations a metallic behavior is

predicted.

The same phase diagram also predicts that Na0.5CoO2 is an insulator for a narrow

regime around x = 0.5. This has been confirmed optically by Wang et al. [47] (Fig. 6.19)

and Hwang et al. [52]. The temperature dependence of R(ω) after Ref. [47] is reported

in Fig. 6.19. The inset shows the dc resistivity in the ab plane [52]. The optical reflectivity

displays clearly the phase transition from the metallic state at high temperature to the insu-

lating one below ∼ 53 K. In fact, the reflectivity in the mid-infrared region increases by de-

creasing temperature, but at low frequencies (∼ 600cm−1) crosses the high temperatures

reflectivity and saturates into a constant value at zero frequency [47]. The insulating state


Figure 6.19: Frequency dependence of the in-plane reflectivity spectra for Na0.5CoO2 at

different temperatures. The inset shows the dc resistivity ρ(T) for the same Na0.5CoO2

crystal [47].

is even more evident looking at the optical conductivity of Fig. 6.20, where σ1(ω) clearly

shows the opening of a charge gap of 2∆≈ 125cm−1below ∼ 47 K.

The origin of this metal-insulator phase transition at TMI ≈ 53 K in Na0.5CoO2 is still

very much debated, however many groups believe that the ordering of the Na ions plays a

central role [34,35,37] by influencing the distribution of the Co3+/Co4+ ions. In this respect

the optical data of Wang et al. and Hwang et al. might give indications that the origin of

the insulating state in Na0.5CoO2 is due to a Charge Density Wave (CDW) state [47]. The

magnitude of the optical gap 2∆ gives a reduced gap value of 2∆/kBTMI = 3.5, which is

in good agreement with the BCS gap ratio [61]. Furthermore for T < 53 K, a small peak

is perceptible at 230 cm−1(see Fig. 6.20), while if we look at the low temperature data of

Hwang in Na0.5CoO2 (Fig. 3a Ref. [52]) this peak is even more pronounced. Theoretically,

a peak has been predicted by Lee et al. for CDW systems in 1D [62] at energies just above

the gap. Such peak has been observed in many other CDW systems, as K0.3MoO3 [63] or

the organic conductor TEA(TCNQ)2 [64]. These observations provide a rather convincing

evidence for the formation of an insulating CDW ground state in Na0.5CoO2 below 50

K [47].

128 6. Na0.7CoO2

Figure 6.20: The σ1(ω) of Na0.5CoO2 at different temperatures as obtained by KK trans-

formations of the R(ω) of Fig. 6.19. The arrow at 125 cm−1 indicates the gap position;

the arrow at the higher frequency side indicates the hump, where the spectral weight is

recovered [47].

Finally, a few remarks about the feature at 800 cm−1 at low temperatures (Fig. 6.20),

which Wang et al. relate to an energy scale where charge carriers become frozen or

bounded. Although at present Wang et al. cannot fully explain the origin of this bump,

it is speculated that it must be associated with the formation of the charge ordered state. In

fact, this feature recovers the spectral weight suppressed by the opening of the gap. In this

respect the appearance of the aforementioned bump at temperature (∼ 100K) higher than

Tc may be explained as due to fluctuation effect. Another possibility is that the bump under-

line the polaronic nature of the charge carriers, because of the enhanced electron-phonon

interactions at low temperatures. This is supported by the temperature dependence of the

hump, which is in good agreement with the expected behavior of polarons [59,65].

6.3 Discussion

We now focus the attention on our data collected for x = 0.7 Na content, the absorption

spectrum of which was previously illustrated in Fig. 6.16. As already mentioned, at high

6.3 Discussion 129

frequencies several absorptions are detected in our spectra and are labeled from A to

E. These absorptions (i.e., 0.4 eV (A), 1.4 eV (B), 2.8 eV (C), 4.8 eV (D), and 10 eV

(E)) are in good agreement with the observations in other compounds with different Na

content [17,47,52,53]. Similar absorption energies were also detected in ARPES [49,50].

The A hump at 0.4 eV, observed at almost the same energy by Wang et al. [53] and Hwang

et al. [52], might be associated to an interband transition between different t2g bands (see

Fig. 6.9).

The B absorption at 1.4 eV is tentatively ascribed to the charge transfer between

the lower lying O 2p states and the Co 3d levels, in fair agreement with band structure

calculations (Fig. 6.8). After the calculated band structure (Figs. 6.7 and 6.8), the O p

bands develops from −7 to −2 eV with a predominance of states around −4 and −6 eV.

We therefore assign the remaining absorptions, particularly the two at 2.8 eV (C) and at

4.8 eV (D) to interband transitions involving O 2p states. Furthermore, the C absorption

could also be assigned to a transition from the t2g bands at the Fermi surface to the eg

bands, which lies 2.5 eV above it.

The metallic component of σ1(ω) cannot be fully reproduced by a simple Drude term

(see section 3.3), the most common description for simple metals and also successfully

applied in several oxides. The effective metallic component of the optical conductivity can

be alternatively described in terms of an "anomalous or generalized Drude" model (see

section 3.3.1), where both the effective mass m∗(ω)/mb and the scattering rate Γ(ω) of

the itinerant charge carriers are allowed to depend on frequency. Therefore, we analyze

the complex optical conductivity using equations (3.17) and (3.18). In those equations, ω2p

is the spectral weight associated with the itinerant charge carriers, and it can be estimated

by integrating σ1(ω) from zero frequency up to a cut-off frequency ωc coinciding with the

onset of the electronic interband transitions. We choose ωc ≈ 0.63 eV, giving a value of

ωp ' 1.17 eV, while for the constant ε∞ in eq. (3.19), we get ε∞ = 3.75. Figure 6.21a

displays the frequency dependence of the scattering rate Γ below 0.15 eV. In order to cal-

culate Γ(ω) we have first interpolated the R(ω) between 0.055 and 0.09 eV (i.e., around

the phonon at 0.07 eV) with a straight line, with the aim to eliminate the phonon contribu-

tion. The resulting σ1(ω) spectrum was then transformed after eq. (3.19) in order to extract

Γ(ω) and m∗(ω)/mb. However this interpolation does not affected our discussion.

m∗(ω)/mb (Fig. 6.21b) weakly increases with decreasing frequency at all tempera-

tures, reaching a value of about 5 in the limit ω → 0 [55]. An analogous treatment of the

130 6. Na0.7CoO2

0.3

0.2

0.1

0.00.150.100.050.00

10K (full curve) Fit. eq. (5)

0.4

0.3

0.2

0.1

0.0

Γ(ω

) (eV

)

10 K 60K 100K 150K 200K Fit eq. (5)

4

3

2

1

0

m(ω

)/mb

0.150.100.050.00

Energy (eV)

10K 60K 100K 150K 200K

a)

b)

Figure 6.21: a) Frequency dependence of the scattering rate (eq. (3.17)) and its fit ac-

cording to eq. (6.1) at selected temperatures. Note that the IR active phonon has been

subtracted in order to better highlight the linear or sub-linear fit. Inset: the original curve of

Γ(ω) (i.e., comprehensive of the IR phonon at 0.07 eV) at 10 K is shown with the fit after

eq. (6.1) with α = 1. The phonon subtraction does not affect the fit of Γ(ω). This is true at

all temperatures. b) Frequency dependence of m∗(ω)/mb at selected temperatures (eq.

(3.18)).

metallic component in σ1(ω) is proposed by Lupi et al. in Na0.57CoO2, and the recovered

effective mass at zero frequency coincides with our result [54].

As expected for a conducting system, Γ(ω) over a broad spectral range decreases

6.3 Discussion 131

0.14

0.12

0.10

0.08

0.06

0.04

Γ(ω

→ 0

) (e

V)

200150100500

Temperature (K)

Data

1.05

1.00

0.95

0.90

0.85

Exp

onen

t α

a)

b)

Figure 6.22: a) Temperature dependence of the exponent α in eq. (6.1). b) Temperature

dependence of Γ(ω) in the static limit ω → 0. Γ(ω → 0) can be well approximated by a

linear T fit.

with decreasing temperature. Over a very broad spectral range, extending up to about 0.12

eV, Γ(ω) can be however fitted with the power law expression:

Γ(ω)∼ ωα. (6.1)

We establish that α' 1 for temperatures below about 50 K. Γ(ω)∼ ω might be indicative

of a non-Fermi liquid behavior in Na0.7CoO2. The exponent α (Fig. 6.22a) tends neverthe-

less to decrease at higher temperatures (e.g., α = 0.88 at 200 K). We emphasize once

again at this point that our fit after eq. (6.1) is not affected by the ad-hoc phonon sub-

traction. The inset of Fig. 6.21 shows indeed that even the original curve of Γ(ω) at 10 K

with the phonon contribution included can be well fitted by eq. (6.1) with α = 1. The same

132 6. Na0.7CoO2

applies at all temperatures, making our analysis of Γ(ω) robust. The linear frequency de-

pendence of Γ(ω) at ω > T pairs with the linear temperature dependence of ρdc(T) for

T < 100K (inset of Fig. 6.16).

The linear behavior of Γ(ω) has a lot of analogies with the electrodynamic re-

sponse of the high Tc superconductors such as, for example, Bi2Sr2CaCu2O8 [66] and

YBa2Cu3O7 [67, 68]. The analogy of NaxCoO2 systems with the high Tc superconduc-

tors has been also highlighted by subsequent data on samples with different Na con-

tent [52, 69]. In Na0.75CoO2, a linear dependence of the scattering rate was detected at

high frequencies and T > 200K, while at low frequencies and temperatures Γ(ω) drops to

zero. At ∼ 600cm−1, Γ(ω) of Na0.75CoO2 is furthermore dominated by a bosonic mode,

whose contribution weakens as the temperature increases. These latter features in Γ(ω)

are moreover observed in compound with low Na content (0.186 x 6 0.36) [69]. It is also

worth noting that the recent report of Wang et al. shows a suppression of the optical con-

ductivity below 2000 cm−1 at low temperatures for 0.186 x 6 0.36 [69]. This suppression

in σ1(ω) is automatically reflected in a suppression of Γ(ω) at low frequencies and temper-

atures. Similar findings have been observed in many underdoped high Tc superconductors

(see for example Ref. [66]), and were interpreted as a signature for the opening of the

pseudogap [68].

For the sake of completeness we mention that the analysis in terms of the gener-

alized Drude in Na0.57CoO2 leads to Γ(ω) following the unusual frequency dependence

Γ(ω) ∝ ω3/2, consistent nevertheless with an equivalent temperature dependence of the

resistivity [38]. In this case and like in our data (Fig. 6.21), Γ(ω) converges to a constant

value at low frequencies and does not show the drop at low frequency and temperature

observed in NaxCoO2 for x = 0.75 and 0.186 x 6 0.36. The difference in the power law

of Γ(ω) could be explained by the different stoichiometry of the samples. Na0.57CoO2 is

quite close to the charge-ordered insulating phase (at x = 0.5) [34]. On the contrary, our

sample Na0.7CoO2 is located in the phase diagram [34] well within the Curie-Weiss metal

sector and is close to the boundary (at low temperatures) of the SDW metallic phase. It

remain to be seen how one can (theoretically) reconcile these variegated power laws of

Γ(ω) as a function of the Na content x with the phase diagram.

6.3 Discussion 133

Figure 6.23: Reflectivity data of YBa2Cu3O7 [70] and Bi2Sr2CaCu2O8 [71] compared with

the NFL theory results. By contrast, the dashed line gives the conventional Drude be-

havior for a constant damping Γ [72].

6.3.1 The Ruvalds and Virosztek approach for a nested Fermi liquid

The generalized Drude term eq. (3.16), illustrated in section 3.3.1, is an empirical formula

which works for many compounds, such as the high Tc superconductors and the heavy

fermion materials [68]. Of interest is the frequency dependence of the scattering rate and

of the effective mass. Usually the responses are classified as a Fermi liquid (FL) (i.e.,

Γ(ω) ∼ ω2) or non-FL (like, e.g., Γ(ω) ∼ ω). The case for non-FL is quite extended and

accounts for different responses. Consequently, manifold conjectures and theories have

been developed in order to the explain the different properties observed in the non-FL

systems.

Ruvalds and Virosztek explain the uncommon linear frequency dependence of the

reflectivity, observed in the high Tc superconductors (Fig. 6.23), with the assumption that

a peculiar Fermi surface facilitates the nesting properties. A nested Fermi liquid (NFL)

134 6. Na0.7CoO2

Figure 6.24: Scattering rate for Drude functions in both NFL and for ordinary FL. The

curves are calculated using g' 1.04, W = 4 eV. At low frequencies ω < T, eq. (6.4)

reduces to 1/τNFL = β ·α ·T ' 3.52 T [72]. This calculation simulates the real behavior

found in YBa2Cu3O7.

is obtained when the Fermi surface satisfies the nesting condition ε(k) + ε(k+ Q) ' 0

for the quasiparticle energy ε(k). Ruvalds and Virosztek demonstrate that in the pres-

ence of NFL one can obtain an anomalous frequency variation of the dielectric functions,

showing considerable deviation from the standard Drude behavior [72, 73]. Ruvalds and

Virosztek start from the Hamiltonian H = ∑k,sε(k)c†k,sck,s+U ∑i ni↑ni↓, where U denotes

the on-site Coulomb repulsion and c†k,s the creation operator for an itinerant electron or

hole. They find that the Fermi surface nesting modifies the region of momentum space for

electron scattering. This may be valid down to very low frequencies in the case of nearly

perfect nesting [72,73].

Starting from the above Hamiltonian and calculating the susceptibility χ(q,ω) in the

case of a NFL, Ruvalds and Virosztek [72] demonstrate that the generalized Drude form

(eq. (3.16)) is justified for ω > T. Thereafter they concentrate on the frequency and tem-

perature dependence of the scattering rate ΓNFL(ω,T). Assuming that the susceptibil-

6.3 Discussion 135

ity χ(q,ω) is enhanced at the nesting vector Q and thus that the contribution χ(Q,ω)

dominates, Ruvalds and Virosztek find that the quasiparticle damping is of the form

ΓNFL(ω) ∼ ω ln(ω), which is approximately linear in ω except the negligible correction

ln(ω). Furthermore, Ruvalds and Virosztek calculate the asymptotic behavior for T → 0

ΓNFL(ω) = 1/τNFL(ω) = α ω, (6.2)

and the static ω→ 0 counterpart:

ΓNFL(T) = 1/τNFL(T) =4π2α

3γT = β ·α ·T. (6.3)

α is an amplitude factor determined from the electron-electron coupling g=U/W (W being

the bandwidth), and γ is a constant which ranges from 4 in the weak-coupling regime

(α 1) to π in the strong-coupling regime (α 1) [72, 73]. These equations might be

merged together in a reasonable representation:

ΓNFL(ω,T) = 1/τNFL(ω,T)' αmax(βT,ω). (6.4)

By comparison a conventional Fermi-liquid (FL) in 3D will exhibit [74]:

ΓFL(ω,T) = 1/τFL(ω,T) =πg2

2W[(2πT)2 +ω2]. (6.5)

For electrons in 2D with an effective-mass dispersion, one has to add a weak logarithmic

correction to the conventional FL behavior of eq. (6.5). Figure 6.24 represents the unusual

frequency variation of the scattering rate in a NFL, which is compared with a weak damp-

ing in standard FL (eq. (6.5)) [74]. The linear increase at higher frequencies in NFL must

terminate with a cut-off frequency which is at most [72]:

ωc = W/(1+α). (6.6)

By performing Kramers-Kronig transformation on the frequency dependent scattering

rate one obtains the renormalization of the charge carriers mass:

m∗

mb= 1+

2απ

lnωc

max(βT,ω). (6.7)

Figure 6.25 illustrated the obtained frequency variation of the effective mass that enters

in the optical properties at different temperatures. The low temperature logarithmic di-

vergence of the quasiparticle mass m∗/mb might rise some doubts on the validity of the

136 6. Na0.7CoO2

Figure 6.25: Plot of the effective mass m∗/mb from the NFL analysis as a function of

frequency at two different temperatures. The curves are calculated using eq. (6.7) with

g' 1.04 and W = 4 eV [72].

proposed treatment. However below a critical temperature, the occurrence of a phase tran-

sition may include attractive components which are not considered in this NFL model. The

phase transition into a spin density wave or charge density wave or even superconducting

state may be dictated by the specific nature of the nesting and quasiparticle interactions.

In the case of an imperfect nesting, the NFL behavior might revert at low temperatures to

a conventional Fermi liquid one.

In order to approximate the reflectivity R(ω), one inserts the obtained expression for

the scattering rate (eq. (6.4)) and the renormalized mass (eq. (6.7)) in the generalized

Drude optical conductivity (eq. (3.16)). Unphysical structures at the cut-off frequency ωc

(eq. (6.6)) can be avoided by using a quadratic smoothing interpolation formula, so that

the expected R(ω) in the wide frequency range T < ωωp (ωp being plasma frequency)

and ω < ωc reduces to the approximate function:

RNFL(ω)' 1− 2√

2ωωp

[(

m∗

mb

)2

+α2

]1/2

− m∗

mb

1/2

. (6.8)

6.3 Discussion 137

Due to the weak logarithmic frequency variation of m∗ in the considered frequency range,

one recovers a quasilinear drop in the reflectivity for a nested Fermi surface. This formula

accounts well for the experimental behavior observed in the high Tc superconductors as

demonstrated in Fig. 6.23.

The effective Drude component in Na0.7CoO2 is characterized by a relaxation rate

that is linear in frequency for ω > T (Fig. 6.21a), as in the Ruvalds and Virosztek’s cal-

culation (eq. (6.2)). Performing the ω → 0 limit on our data for the scattering rate (Fig.

6.21a), we see that Γ(ω → 0,T) in Na0.7CoO2 displays an approximate linear behavior

in T, as shown in Fig. 6.22b. This is in perfect agreement with the NFL-predictions of

Ruvalds and Virosztek (eq. (6.3)). As a last point, the linear dependence of the reflectivity

calculated by Ruvalds and Virosztek (eq. (6.8)) is also verified in the experimental data

of Na0.7CoO2 (inset of Fig. 6.16a) over a broad spectral range. The good agreement be-

tween the Ruvalds and Virosztek’s calculations and the experimental results supports the

Fermi-surface nesting scenario. Nesting of the Fermi surface has been also verified exper-

imentally by ARPES measurements [51]. The unusual geometry of the Fermi surface and

the consequent nesting channel, illustrated in Fig. 6.10, may be an important ingredient

for the theoretical explanation of our optical results. The NFL scenario proposed by Ru-

valds and Virosztek is applicable for the charge and spin density wave state where nesting

is an essential ingredient [72]. From an optical point of view, Na0.7CoO2 seems therefore

to be at the verge of a SDW metallic phase. This is consistent with the phase diagram

reported in Figs. 6.4 and 6.6. We remark finaly that the electronic structure with perfect

nesting exhibits several analogies with that of the 1D systems, where, depending on the

type of interaction and on the order of the expansion, a linear ω and T dependence of the

scattering rate might also be found [75,76].

In conclusion, we shall note that after Ruvalds and Virosztek a non Fermi liquid-Fermi

liquid crossover is possible at very low temperatures in the case of an imperfect Fermi-

surface nesting. This seems to be the case in Na0.7CoO2 as well, since the dc-transport

data of Li et al. displays a typical Fermi liquid T2 behavior for T < 1 K [77]. Fascinating is

that the FL region is enhanced up to 4 K by applying a magnetic field of 16 T [77]. This

might suggest that the Fermi surface is modified by the strong magnetic field. As a result,

the Fermi surface nesting in strong magnetic field is reduced and the system shows a

conventional Fermi-liquid behavior up to temperatures higher than in zero magnetic field.

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7 Conclusion and outlook

Our optical investigations of selected quantum spin systems lead to a manifold of inter-

esting results. In TiOX, we have thoroughly investigated the phonon spectra with optical

methods (Raman and IR reflectivity), and our results were supported by correlation method

and shell model calculations. The agreement between measurements and calculations is

astonishingly good. We account for the red shift of the IR phonon modes upon substituting

Cl with the heavier Br atom by considering the mass renormalization in the phonon eigen-

vectors. Furthermore, we have emphasized the temperature dependence of the phonon

spectrum. The temperature dependence of all relevant parameters, including the phonon

linewidth and the spectral weight, develops over a broad temperature interval extending

far above Tc1. This is a fingerprint for the important role played by fluctuation effects. The

pronounced narrowing of the IR modes with decreasing temperature coincides with the

suppression of low-frequency spin fluctuations, recognized in the spin-gap phase of the

NMR spectra. The behavior of the IR spectral weight with temperature establishes the

presence of a characteristic energy scale associated with the opening of a spin gap. Al-

though similar findings are frequently observed in 1D systems, the case of TiOX is different

due to the unusually high energy scale involved. The temperature dependence of the SW

redistribution mimics the behavior of the spin susceptibility for both compounds, implying

a spin-gap phase at lower temperatures in TiOBr than in TiOCl. Additionally, the phonon

analysis of our spectra does not give any evidence of a structural transitions, with the

number of phonon modes remaining constant down to the lowest measured temperature.

This might be due to the fact that the measured lattice distortion along the b axis does

not activate additional modes, for example through a back-folding of the Brillouin zone.

Therefore, our optical data indicates that in the TiOX systems the transition at Tc1 is not a

conventional spin-Peierls one. The large phonon anomalies as well as the presence of an

145

146 7. Conclusion and outlook

intermediate phase between Tc2 and Tc1 hint to competing lattice, spin and orbital degrees

of freedom as the driving force for the transition. As a future outlook, it would be of inter-

est to investigate intercalated TiOX systems in order to verify the theoretical prediction of

superconductivity upon metallization, like in the LixZrNCl or MxHfNCl (M=Li, Na) systems.

For LiCu2O2, we illustrated the temperature dependence of the phonon at 30 meV,

whose asymmetry indicates the presence of a high temperature magnetic continuum. Our

results suggest that this magnetic continuum is suppressed by lowering the temperature in

agreement with Raman results. Analyzing the temperature dependence of SW, we identify

three distinct behaviors, which are correlated to the three different magnetic phases of

LiCu2O2. Moreover, in the high frequency spectral range, an unusual strong absorption

peak is measured at 3.1 eV, which is assigned to an interband transition from the O 2p to

the Cu2+ 3d band. For the future, it would be important to investigate LiCu2O2 samples

without twinning, or the isostructural and untwined NaCu2O2, in order to unambiguously

distinguish between the electrodynamic response along different axes. Also of interest

would be to dope the samples either with holes or magnetic impurity and study their effect

on the magnetic as well as on the conduction properties.

The optical properties of Na0.7CoO2 have highlighted the numerous analogies be-

tween this frustrated system and the high Tc superconductors. The frequency dependence

of the scattering rate of the itinerant charge carriers is extracted from the complex opti-

cal conductivity, and the recovered scattering rate behaves linearly as a function of both

temperature and photon energy. Na0.7CoO2 seems to be in the proximity of a spin-density-

wave metallic state. Such a marginal or non Fermi liquid behavior was found to be consis-

tent with a so called nested Fermi liquid approach. It turns out that the exact stoichiometry

plays an essential role in defining the intrinsic physical properties of NaxCoO2. As a future

outlook, it would be of great interest to investigate with optical methods crystals where

Cobalt is substituted with Gallium, Iridium, Manganese and Titanium. These substitutions

allow to introduce holes and magnetic impurities in the system and are found to suppress

the superconducting region and influence the charge ordering in Na0.5Co1−xXxO2.

In conclusion the investigations of spin 1/2 systems have been very profitable and

IR reflectivity technique lead to valuable results. This motivates us to carry on with inves-

tigation of comparable compounds, as for example Cu3TeO6. Cu3TeO6 has an hexagonal

arrangement of the Cu2+ atoms with a novel type of magnetic lattice formed by a 3 dimen-

sional arrangement of the hexagons, dubbed "spin web". Cu3TeO6 shows a magnetic tran-

7. Conclusion and outlook 147

sition at TN = 61K with the formation of an anti-ferromagnetic collinear spin arrangement.

Neutron scattering seems to indicate an alignment of the spins along the [111] direction

which would introduce a magneto elastic strain. This strain may lead to a structural phase

transition, which affect the optical response. While the preliminary study of Cu3TeO6 ap-

pear to be very promising, a complete analysis of the optical spectra is however left for the

future.

Acknowledgments

Foremost I would like to thank Prof. Dr. L. Degiorgi for giving me the opportunity to perform

my PhD. in his laboratory. He gave me the impetus to study the quantum spin systems

and his constant presence and dedication were the prerequisite in order to reach the inter-

esting results presented in this dissertation. My gratitude goes also to my first co-referee,

Prof. Dr. H. R. Ott, for providing the financial support and for supplying the hexaborides

samples. Europium hexaborides have been an important research topic during my PhD.,

even though they are not reported here. I would also like to thank Prof. Dr. T. Giamarchi for

accepting to be my second co-referee.

With P. Lemmens, we had a profitable collaboration which leaded to several joint

publications. I want to thanks him for motivating us to study the TiOX systems as well

as other systems. I’m also grateful to M. Grioni for providing us photoemission data on

LiCu2O2 prior publication. My deep thanks go also to the people who supplied us the

crystals for our investigations: F. Chou, H. Berger, L. Forró, A. D. Bianchi and Z. Fisk.

In the laboratory, I benefited of the friendship and technical support of Jürg Müller.

I’ll miss our discussions during the measurement-breaks. Andrea Perucchi has been a

colleague and a friend, with whom I spent the PhD time chatting on physics and italian

politics. In the first year, I have shared the lab with Sam Broderick who introduced me to

the world of the Kerr rotation. I would like to thank him for his numerous tips. To Andrea

Sacchetti, the new researcher of our group, I would like to wish good luck and to achieve

meaningful results. As last but not least, I am indebted with Ingrid Heer and Gaby Strahm

for helping me with administrative stuff.

During this three and an half years I have enjoyed the fellowship of many researchers

at the ETH Zurich. Although I’ll not mention them personally, I want to thank them for their

149

150 Acknowledgments

humor, presence and friendship. In particular I would like to thank Prof. Dr. B. Batlogg and

his group for the nice moments spend together, like the hiking-days, the skiing-week-ends

and the numerous suppers. My last thanks goes to the whole Ott’s group with which I

shared pleasant moments at the MaNEP’s meetings.

Curriculum vitae

Name Giulio Caimi

Born April 14, 1977 in Zurich

Citizen of Ligornetto TI, Switzerland

Education and Employment

1983 - 88 Scuola elementare di Ligornetto, Ti (primary school).

1988 - 92 Scuola media di Stabio, Ti (secondary school).

1992 - 96 Liceo Cantonale di Mendrisio, type A, liceo classico, (high scool),

graduated with “Attestato di Maturità Cantonale" (June, 1996).

1996 - 01 Physics study at the Swiss Federal Institute of Technology, Zurich,

diploma with distinction in experimental physics (October, 2001).

2002 - 05 Teaching assistant and Ph.D. student under Prof. Dr. L. Degiorgi at the

Laboratory for Solid State Physics, ETH Zurich.

151

Publications and presentations

Publications

– D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Gunter, A. A. Grabar, I. M.

Stoika, and Y. M. Vysochanskii,

“Electro-optical properties of Sn2P2S6”,

Optics Communications 215, 333 (2003).

– G. Caimi, S. Broderick, H. R. Ott, L. Degiorgi, A. D. Bianchi, and Z. Fisk,

“Magneto-optical Kerr effect in Eu1−xCaxB6”,

Phys. Rev. B 69, 012406 (2004).

– A. Perucchi, G. Caimi, H. R. Ott, L. Degiorgi, A. D. Bianchi, and Z. Fisk,

“Optical evidence for a spin-filter effect in the charge transport of Eu1−xCaxB6”,

Phys. Rev. Lett. 92, 067401 (2004).

– G. Caimi, L. Degiorgi, N. N. Kovaleva, P. Lemmens, and F. C. Chou,

“Infrared optical properties of the spin-(1/2) quantum magnet TiOCl”,

Phys. Rev. B 69, 125108 (2004).

– G. Caimi, L. Degiorgi, H. Berger, N. Barisic, L. Forró, and F. Bussy,

“Optical evidence for the proximity to a spin-density-wave metallic state in

Na0.7CoO2”,

Eur. Phys. J. B 40, 231 (2004).

– G. Caimi, L. Degiorgi, P. Lemmens, and F. C. Chou,

“Analysis of the phonon spectrum in the titanium oxyhalide TiOBr ”,

J. Phys.: Condens. Matter 16, 5583 (2004).

153

154 Publications and presentations

– P. Lemmens, K. Y. Choi, G. Caimi, L. Degiorgi, N. N. Kovaleva, A. Seidel, and F. C.

Chou,

“Giant phonon anomalies in the pseudo-gap phase of TiOCl”,

Phys. Rev. B 70, 134429 (2004).

– G. Caimi, A. Perucchi, H. R. Ott, L. Degiorgi, V. M. Pereira, A. H. Castro Neto, A. D.

Bianchi, and Z. Fisk,

“Magneto-optical evidence of double exchange in a percolating lattice”,

cond-mat/0510155,

to be published in Phys. Rev. Lett. (2005).

– G. Caimi, L. Degiorgi, H. Berger, and L. Forró

“Optical evidence for a magnetically driven structural transition in the spin web

Cu3TeO6”,

cond-mat/0510186 (2005).

– M. Papagno, D. Pacile, G. Caimi, H. Berger, L. Degiorgi, and M. Grioni,

“Experimental study of the electronic structure of copper-oxide 1D chains in

LiCu2O2”,

in preparation (2005).

– G. Caimi, L. Degiorgi, H. Berger, and L. Forró

“Phonon analysis in the S=1 quantum spin system Ni5Te4O12Cl2”,

in preparation (2005).

Conferences

– MaNEP Topical Meeting and Review Panel

Neuchatel (CH) - June 25-26, 2003

Ferromagnetism and Superconductivity in Boride Systems.

– 2003 Swiss Workshop on Materials with Novel Electronic Properties (MaNEP)

Les Diablerets (CH) - September 29-October 1, 2003

Magneto-optical response of Eu1−xCaxB6.

Publications and presentations 155

– 2004 Swiss Physical Society - (MaNEP Meeting)

Neuchatel (CH) - March 3-4, 2004

Magneto-optical response of Eu1−xCaxB6.

– 2005 Swiss Workshop on Materials with Novel Electronic Properties (MaNEP)

Les Diablerets (CH) - September 26-28, 2005

Magnetically driven structural transition in Cu3TeO6.

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In Copyright - Non-Commercial Use Permitted Rights ... · GIULIO CAIMI Dipl. Phys. ETH Zürich born on the 14th of April, 1977 citizen of Ligornetto, Ti accepted on the recommendation