Nikolay Tsyrulin- Neutron Scattering Studies of Low-Dimensional Quantum Spin Systems

diss. eth no. 18845

Neutron Scattering Studies of Low-DimensionalQuantum Spin Systems

A dissertation submitted to

ETH ZURICH

for the degree of

Doctor of Science

presented by

NIKOLAY TSYRULIN

Dipl. Phys. MSU

born 19.11.1983

Russian citizen

accepted on the recommendation of

Prof. Dr. M. Troyer, examinerProf. Dr. J. Mesot, co-examinerDr. M. Kenzelmann, co-examiner

2010

ii

Contents

Abstract vii

Kurzfassung ix

1 Theoretical introduction and motivation 1

1.1 Quantum magnetism in two dimensions . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Ground state properties of S = 1/2 Heisenberg antiferromagnets . . . 2

1.1.2 Effects of applied field on S = 1/2 Heisenberg antiferromagnets . . . . 6

1.2 Quantum phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.1 Bose-Einstein condensation of magnons in quasi-1D systems . . . . . 10

1.2.2 Quantum phase transitions in 2D square-lattice antiferromagnets . . 13

1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Neutron scattering 19

2.1 Main properties of the neutron . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 Neutron sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.2 Energy and time scales . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Elastic neutron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.1 Fermi’s Golden rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.2 Nuclear scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.3 Magnetic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Neutron spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.1 Inelastic nuclear scattering . . . . . . . . . . . . . . . . . . . . . . . . 27

iv Contents

2.3.2 Inelastic magnetic scattering . . . . . . . . . . . . . . . . . . . . . . . 28

2.4 Experimental technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.1 Three-axis spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.2 The experimental resolution function . . . . . . . . . . . . . . . . . . 31

2.5 Sample environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.5.1 Split coil magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.5.2 Dilution refrigerator . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 2D S = 1/2 antiferromagnet on a square lattice Cu(pz)2(ClO4)2 37

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Bulk properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1 Crystal structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.2 Magnetic susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.3 Specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3 Neutron diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.1 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.2 Magnetic order parameter . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.3 Ordered magnetic structure . . . . . . . . . . . . . . . . . . . . . . . 48

3.4 Neutron spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4.1 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4.2 Spin dynamics in zero field . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4.3 Spin dynamics in applied magnetic fields . . . . . . . . . . . . . . . . 58

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4 Magnetism in S = 1 quasi-1D antiferromagnet NiCl2 · 4SC(NH2)2 67

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3.1 Ordered magnetic structure . . . . . . . . . . . . . . . . . . . . . . . 72

4.3.2 Magnetic phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . 76

Contents v

4.3.3 Bose-Einstein condensation of magnons . . . . . . . . . . . . . . . . . 79

4.3.4 Spin dynamics in the fully magnetized phase . . . . . . . . . . . . . . 83

4.3.5 Spin dynamics at the first critical field . . . . . . . . . . . . . . . . . 87

4.3.6 Spin dynamics deep in the antiferromagnetic phase . . . . . . . . . . 88

4.3.7 Spin dynamics in the antiferromagnetic phase with H 6‖ c . . . . . . . 95

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Outlook 99

Appendix A 101

Appendix B 105

Appendix C 109

Acknowledgments 125

Publications and conferences 127

Curriculum Vitae 129

vi Contents

Abstract

This thesis is devoted to experimental studies of magnetism in low dimensions. Being of great

fundamental interest, low dimensional magnets support strong quantum fluctuations which

can result in novel quantum excitations and novel ground states. We used neutron scat-

tering technique to study two macroscopic quantum systems: (A) two-dimensional weakly-

frustrated S = 1/2 antiferromagnet on a square lattice, Cu(pz)2(ClO4)2, and (B) strongly

anisotropic quasi-one-dimensional S = 1 chain, NiCl2 · 4SC(NH2)2.

(A) A comprehensive experimental study of the two-dimensional S = 1/2 square-lattice

antiferromagnet, Cu(pz)2(ClO4)2, was performed up to one third of the magnetic saturation

field. The spin Hamiltonian of the system is determined precisely. Our experiments provide

evidence for the presence of a small antiferromagnetic next-nearest neighbor interactions

which enhances quantum fluctuations associated with resonating valence bonds in the square-

lattice S = 1/2 antiferromagnet. We show that magnetic fields of the order of one third of

the saturation value lead to a qualitative change of the quantum fluctuations that inverts the

zone-boundary dispersion.

(B) NiCl2 · 4SC(NH2)2 is a quasi-one-dimensional anisotropic S = 1 quantum magnet

which undergoes a field-induced quantum phase transition from a quantum paramagnetic to

a three-dimensional antiferromagnetic state. This process can be treated theoretically as a

Bose-Einstein condensation of magnons. We directly observe the field-induced closing of the

energy gap using inelastic neutron scattering. The phase diagram and the symmetry of the

antiferromagnetic order are determined, showing the order of the transverse spin components.

The study of the spin dynamics in the fully saturated ferromagnetic phases revealed nearest

neighbor exchange interactions. The study in the antiferromagnetic phase, on the other

hand, revealed additional spin interactions that are not present outside this phase, providing

viii Abstract

evidence of a coupling between magnetism and the crystal structure and the presence of

additional spin anisotropies in the material.

Kurzfassung

In dieser Arbeit wird eine experimentelle Studie uber Magnetismus in niedrigen Dimensio-

nen vorgestellt. Es besteht ein fundamentales Interesse an niedrig-dimensionierten Mag-

neten, da diese enorme Quantenfluktuationen begunstigen, was wiederum in neuartigen

Quantenanregungs- und Grundzustunden resultieren kann. Es wurden verschiedene Tech-

niken der Neutronenstreuung eingesetzt, um zwei makroskopische Quantensysteme zu studie-

ren: (A) ein zweidimensional, schwach-frustrierter S = 1/2 Antiferromagnet in einem Quadrat-

gitter, Cu(pz)2(ClO4)2, und (B) eine stark anisotrope quasi-eindimensionale S = 1 Kette,

NiCl2 · 4SC(NH2)2.

(A) Eine umfassende experimentelle Studie wurde an dem System des zweidimensionalen

S = 1/2 Quadratgitter Antiferromagneten, Cu(pz)2(ClO4)2, durchgefuhrt, in magnetischen

Feldern von einer Starke bis zu einem Drittel des Sattigungswertes. Der Spin Hamiltonian

des Systems wurde prazise bestimmt. Unsere Experimente erbringen den Beweis der Existenz

einer kleinen antiferromagnetischen Wechselbeziehung zwischen nachsten Nachbarn, welche

Quantenfluktuationen begunstigt, die mit Resonanz-Valenz-Bindungen im Quadratgitter des

S = 1/2 Antiferromagneten in Verbindung stehen. Wir zeigen, dass magnetische Felder in der

Grossenordnung von einem Drittel des Sattigungswertes zu einer qualitativen Veranderung

der Quantenfluktuationen fuhren, die die Zonengrenzen Dispersion umkehren.

(B) Das quasi-eindimensionale S = 1 Kettensystem ist in NiCl2 ·4SC(NH2)2 realisiert und

untergeht einer Feld-induzierten Quantenphasenumwandlung von einem quanten-paramagne-

tischen zu einem dreidimensionalen antiferromagnetischen Zustand. Dieser Prozess kann the-

oretisch als Bose-Einstein Verdichtung der Magnonen gesehen werden. Wir beobachten direkt

das Feld-induzierte Schliessen der Energielucke bei Verwendung von inelastischer Neutronen-

streuung. Das Phasendiagramm und die Symmetrie der antiferromagnetischen Ordnung wur-

x Kurzfassung

den untersucht und zeigen die Anordnung der querverlaufenden Spin Komponenten. Studien

der Spin-Dynamik in der vollig gesattigten ferromagnetischen Phase zeigen die Prasenz von

nachste-Nachbarn Wechselwirkungen. Messungen in der Feld-induzierten antiferromagnetis-

chen Phase, auf der anderen Seite, zeigen die Prasenz von Wechselwirkungen, die sonst nicht

da sind. Diese Messungen sind daher Indikationen fur eine Kopplung von Magnetismus und

der Kristallstruktur sowie die Prasenz von zusatzlichen Spin-Anisotropien in dieser Phase.

Chapter 1

Theoretical introduction and

motivation

1.1 Quantum magnetism in two dimensions

Low-dimensional magnets can have unusual ground states and spin correlations, especially

for small spin values such as S = 1/2. A prime historical example is the class of the

high-temperature superconducting cuprates whose undoped insulating phases consist of two-

dimensional (2D) antiferromagnetic planes of spins S = 1/2 [1]. Superconductivity emerges

upon doping of the 2D lattices. While anisotropic 2D antiferromagnets order at finite temper-

atures, isotropic 2D systems can adopt long range magnetic order only at zero temperature

[2]. But even then a ground state is not completely ordered due to the quantum fluctuations

and, if present, geometrical frustrations.

Generally, the increase of the dimensionality from one to two reduces quantum fluctua-

tions. For instance, antiferromagnetic chains with S = 1/2 do not order at zero temperature,

however 2D antiferromagnet (AF) on a square lattice possess semi-classical Neel order [1].

Magnetic order can be destroyed by geometrical frustrations, like in an AF on a triangular

lattice [3] or by competing interactions between neighboring and next-neighboring spins. Tun-

ing of competing interactions and quantum fluctuations (for instance by external magnetic

field) can move a system towards quantum criticality. Quantum fluctuations are supported

by three-dimensional magnets as well [4], nevertheless their influence and effects are typically

2 Theoretical introduction and motivation

Figure 1.1: Two bipartite lattices - square, honeycomb and two frustrated - triangular andkagome are shown in (a), (b), (c) and (d), respectively.

much lower compared to the 2D case. In this section, we briefly review the main character-

istics of the 2D S = 1/2 AF on four lattices - two bipartite (i.e. non-frustrated) square and

honeycomb and two frustrated, triangular and kagome (Fig 1.1).

1.1.1 Ground state properties of S = 1/2 Heisenberg antiferromag-

nets

Bipartite lattices

The existence of Neel order in Heisenberg AF (HAF) on a square lattice was proven for the

case of S > 1 [5, 6] and for the anisotropic Ising limit [7, 8]. However there is no strict proof

of the long-range order in case of S = 1/2 HAF. Starting with the work of Anderson [9], the

S = 1/2 HAF on a square lattice (Fig 1.1(a)) was studied for decades. Quantum fluctuations

in the system lead to the renormalization of the order parameter by ∼ 60% compared to the

classical predictions. The Lieb-Mattis theorem [10] postulates that the quantum ground state

is a rotationally invariant singlet for finite bipartite lattices. Low-energy magnetic excitations

called magnons have a dispersion which can be calculated using spin-wave theory. Fig 1.2

shows the result of the first-, second- and third-order spin-wave calculations for the S = 1/2

HAF on a square lattice. The excitation spectra of the semiclassically ordered HAF on a

finite square lattice have an energy gap between a singlet ground state and the first excited

state. This gap vanishes for the Neel ground state in the thermodynamic limit leading to the

formation of the Goldstone mode. Consequently, a finite energy gap for a S = 1/2 HAF on a

square lattice at zero temperature can be the sign for spin anisotropies in the system or the

1.1 Quantum magnetism in two dimensions 3

Figure 1.2: The dispersion of the one-magnon excitation spectra along the high-symmetrydirections for the Heisenberg AF on a square lattice. The black circles represent the result ofseries expansion calculations. The results of first-, second and third-order spin-wave theoryare shown by blue dotted, red dashed and green solid lines, respectively. The figure is takenfrom the publication of W. Zheng et al. [11].

result of a quantum spin liquid ground state that may arise from competing interactions.

Numerical studies of the 2D S = 1/2 HAF on a square lattice using quantum Monte

Carlo, exact diagonalization, coupled cluster as well as series expansion calculations reveal

a quantum renormalization of the one-magnon energy in the entire Brillouin zone. The

one-magnon dispersion along the high-symmetry directions of the Brillouin zone obtained

by series expansion calculations [11] is shown in Fig. 1.2. The main qualitative difference

between the result of this numerical method and the high-order spin-wave theory is a magnon

dispersion along the antiferromagnetic zone boundary. This can be viewed as a pure quantum

effect, because spin-wave theory is based on the assumption that the ground state of S =

1/2 HAF is a classical Neel state. Numerical calculations also predict the existence of a

magnetic continuum at higher energies, near the minima of the zone-boundary dispersion

[11, 12, 13, 14, 15]. The zone boundary dispersion and a magnetic continuum in the excitation

spectra of the 2D S = 1/2 HAF on a square lattice can be a sign of the resonance valence bond

(RVB) ground state, as shown in Fig. 1.3(b). This state can be viewed as a superposition of


local singlets between spins on nearest sites [3].

Kobe and collaborators [16] proposed the ground state energy per bond of the classical

AF on bipartite lattices to be equal to Eclassicalgs /bond = −S2. The ground state energy per

bond and the order parameter was calculated by various techniques. Quantum Monte Carlo

calculations [17, 18] give Egs/bond = −0.334719 and m = 0.3070. Here m is the sublattice

magnetization defined as

m =1

N

∑i

Szi , (1.1)

where N is the total number of spins and the case of m = 1 corresponds to a completely

ferromagnetic alignment of spins. The difference between the calculated value and the clas-

sical expectation underlines the importance of quantum fluctuations in 2D AF on bipartite

lattices.

The reduction of the coordination number, i.e. the number of nearest neighbors, strongly

affects the ground state properties of 2D spin systems. In order to illustrate this effect, we

give the example of the honeycomb lattice (Fig 1.1(b)), where every spin site has only three

nearest neighbors. Several techniques provide evidence that the ground state of this model is

semi-classical Neel order. Such results were obtained theoretically using the quantum Monte

Carlo simulations [19], the coupled-cluster, exact diagonalization of finite systems [20] and the

Schwinger-boson approach [21]. The ground state energy per bond and the order parameter

calculated by Quantum Monte Carlo [19] are equal to Egs/bond = −0.3630 and m = 0.235,

respectively. Those values are lower than it expected from classical approach and even lower

compare to the parameters of HAF on a square lattice. The reduction of magnetic order

parameter indicates enhanced quantum fluctuations in a honeycomb lattice in comparison

with a square lattice due to the lowering of the coordination number.

Frustrated lattices.

In 2D spin systems the effect of quantum fluctuations may be enhanced drastically by geo-

metrical frustrations. S = 1/2 HAF on the triangular lattice is strongly frustrated, however

the number of the nearest-neighbor spins is the highest for the considered 2D systems and


equal to six (Fig. 1.1(c)). Anderson and Fazekas [3, 22] proposed a quantum spin liquid

ground state for S = 1/2 AF on a triangular lattice. However, the signature of the Neel order

was later found using various numerical techniques, like the exact diagonalization of finite

system [23, 24] or quantum Monte Carlo [25]. The translational symmetry is broken due to

the angle of 120 between the spins in different sublattices in the case of the classical ground

state. The low-lying spin excitations in a triangular lattice can be described by spin-wave

theory. The ground state energy per bond and the sublattice magnetization calculated by

Capriotti et al [25] are equal to Egs/bond = −0.1819 and m = 0.205, respectively. This result

implies that quantum fluctuations strongly affect the ground state. The results of quantum

Monte Carlo calculations scaled to the classical values are listed in Tab 1.1 for a range of

different lattices.

Lattice Ecalc0 /bond m/mclass

Square ∼ −0.3347 0.614

Honeycomb ∼ −0.3630 0.47

Triangle ∼ −0.1819 ∼ 0.41

Kagome ∼ −0.2126 ∼ 0

Table 1.1: The summarized data of quantum Monte Carlo calculations of the ground stateenergy per bond and the magnetic order parameter. The order parameter (sublattice mag-netization) is scaled to the classical prediction to give a feeling of amount of the quantumfluctuations in the systems. For kagome lattice presented results obtained by the coupledcluster method.

After the discovery of the semi-classically ordered ground state in S = 1/2 HAF on

triangular lattice, the kagome1 lattice (Fig. 1.2(d)) eventually became the first candidate

for the investigations of a hidden order in 2D [26] or a possible quantum spin liquid state

[27, 28]. Exact diagonalization calculations proposed the quantum spin liquid as ground

state of the kagome lattice. Series expansions calculations performed by Singh and Huse

[29] showed that a valence bond crystal (VBC) with a 36 site unit cell is the ground state

of the S = 1/2 HAF on a kagome lattice (see Fig. 1.3(a)). VBC is characterized by the

formation of local singlets built by an even number of spins with weak correlations [30].

1the name came from Japanese words ”me” meaning the pattern of holes and ”kago” - a basket


Figure 1.3: (a) Ground-state

ordering pattern for the kagome-

lattice Heisenberg model [29].

Dimers with ”strong” and

”weak” bones are shown by

thick blue and thin black lines,

respectively. (b) Cartoon of the

RVB state on a square lattice.

However the experimental evidence for such a ground state

has not yet been found. Hiroi and co-workers [31] per-

formed experimental studies of a novel cuprate volborthite

compound Cu3V2O7(OH)2 · 2H2O using magnetic suscep-

tibility, specific heat, and NMR measurement and did not

find any sign of long-range order or a spin-gapped singlet

ground state down to T = 1.8 K. The ground state en-

ergy per bond calculated using coupled cluster technique

is equal to Egs/bond = −0.2126

We summarize the 2D S = 1/2 HAF on model lattices

that were reviewed in this section as follows:

• S = 1/2 AFs on non-frustrated lattices possess semi-

classical Neel order. An energy gap vanishes in the

thermodynamic limit and the low-lying excitation

evolves into a Goldstone mode.

• Quantum fluctuations reduce the order parameter

and the degree of quantum disorder depends on com-

peting interactions between adjacent spin sites and

on the coordination number.

• The semi-classical Neel order is weak in frustrated

lattices and it is not present in the kagome lattice,

possibly because a quantum spin liquid is the ground

state for this case. The kagome lattice could also

reveal a purely quantum VBC ground state.

1.1.2 Effects of applied field on S = 1/2 Heisenberg antiferromag-

nets

Here we discuss effects of a magnetic field applied to the model spin lattices examined above.

A finite magnetic field orients spins towards the field direction, thus breaking the planar


Figure 1.4: (a) Magnetization calculated of the S = 1/2 HAFM on a square lattice. Theopen circles shows the data obtained by 1/S series expansion calculation, the dashed andsolid line are the results of the first and second order spin-wave theory, respectively. (b)Magnetization of the S = 1/2 HAFM on a triangular lattice in the presence of an externalmagnetic field of strength λ. The results of coupled cluster method (LSUB4, LSUB6 andLSUB8) are compared with exact diagonalization (ED). The plots (a) and (b) are taken from[35] and [40], respectively.

spin alignment, which may be the ground state at zero field. In this sense, the presence

of an external magnetic field can be viewed as a competing interaction, which can affect

quantum fluctuations. Moreover, a magnetic field gives the possibility to tune the ground

state and move the spin systems towards a quantum critical point (see Sec. 1.2). As we already

mentioned, S = 1/2 square-lattice HAF orders at zero temperature. Quantum Monte Carlo

calculations [32] have shown that applied magnetic field induces a transition from Heisenberg

towards XY antiferromagnetic phase and leads to a finite temperature Berezinskii-Kosterlitz-

Thouless transition [33, 34], which is associated with an appearance of 2D topological spin-

vortices.

Fig 1.2(a) shows the magnetization calculated for the S = 1/2 HAFM on a square lattice

as a function of the applied field. The data obtained by 1/S series expansion calculations

[15] are in a good agreement with first- and second-order spin wave theory calculations

performed by Zhitomirsky and Nikuni [35]. The magnetization as the function of applied

field for the classical model is represented by the straight line. The observed deviations seen

in Fig 1.2(a) are caused by quantum effects. Only little is known experimentally about the

effects of application of high magnetic fields on the excitation spectra for 2D AFs [36, 37]


- in contrast to 1D AFs. First predicted by Zhitomirsky and Chernyshev [36] and later by

Luscher and Lauchli [38], the single-magnon excitation in S = 1/2 square-lattice AF overlaps

with a two-magnon continuum leading to instability of the former in high magnetic field,

above Hc ≈ 0.76 Hsat, where Hsat is the saturation value. Experimental investigations of

the excitation spectra at ≈ 0.76 Hsat have so far been impossible, because high exchange

interactions between adjacent spin sites imply the application of unreachable high magnetic

fields in real 2D systems.

A similar smooth dependence of the magnetization on the external magnetic field was

found for the S = 1/2 HAF on a honeycomb lattice [39]. Upon application of magnetic

field, the collinear antiferromagnetic order is canted towards the field direction. Besides the

Goldstone mode associated with unbroken rotational symmetry in the honeycomb plane, a

gapped mode appears in the excitation spectra. Further, the system undergoes a field-induced

phase transition from canted antiferromagnetic towards ferromagnetic state with the spins

aligned along the field direction at Hsat.

The magnetization as function of applied magnetic field for the S = 1/2 HAF on a

triangular lattice is shown in Fig 1.2(b) [40]. These data were obtained by the coupled

cluster method and compared to the result of the exact diagonalization on N = 36 spin sites.

The most pronounced features in the magnetization curves for 2D triangular lattices is the

plateau at m = 1/3 [40, 41]. In the particular case of a triangular lattice, the nature of the

plateau can be understood in the Ising limit: the quantum fluctuations are suppressed and

the state with m = 1/3 corresponds to the classical arrangements of spins pointing up in

two sublattices and down in third one, so called ”up-up-down” state (see Fig. 1.4(b)). The

magnetization plateaus at one-third of the saturation field for the Ising and Heisenberg limits

are qualitatively the same [41]. The appearance of the magnetization equal to m = 1/3 at

the finite range of applied magnetic field is governed by quantum fluctuations [42] which

stabilizes the collinear phase. Calculated by Chubukov and Golosov, the excitation spectra

of the S = 1/2 HAF on a triangular lattice has three resonant modes in magnetic fields below

the saturation value [42].

Numerical calculations [41, 43, 44] predict several features in a magnetization curve of

S = 1/2 HAF on a kagome lattice. Due to the absence of the Neel order, the magnetization

1.2 Quantum phase transitions 9

curve should have a flat region at small applied fields. A second plateau is predicted to occur

at m = 1/3, similar to the case of a triangular lattice [45, 46]. This feature implies that the

system is in the ordered state, analogous to up-up-down state which emerges in a triangular

HAF at one-third of the magnetization [45]. A huge jump in the magnetization curve appears

in S = 1/2 HAF on a kagome lattice close to the saturation field.

Under certain conditions, an applied magnetic field can steer a spin system into a novel

quantum phase. We take a closer look on this case in the next section.

1.2 Quantum phase transitions

A classical phase transition (CPT) involves thermal fluctuations occurring at finite tempera-

tures only. In the thermodynamic limit at T = 0 K, where the thermal energy scale is absent

only the fluctuations associated with the Heisenberg′s uncertainty principle may be present.

In some cases those zero-point fluctuations lead to transitions which, in analogy to CPT, are

called quantum phase transition (QPT) and occur at T = 0 K. In analogy with temperature

driven CPT, every QPT is governed by an external parameter, for instance magnetic field or

pressure.

A system approaches a quantum critical point (QCP) in the thermodynamic limit under

application of the external parameter. Every continuous QPT is characterized by an order

parameter: Being zero in disordered phase an order parameter becomes non-zero while a

system reaches a QCP. Correlations of the order parameter diverges as |ξ| ∝ |t|−ν in the

vicinity of QCP. Here t is a measure of closeness to the QCP and ν is the correlation length

critical exponent. An order parameter fluctuates not only in space but also in time. Close to

the QCP corresponding time correlations tc diverges as tc ∝ |t|−νz, where z is the dynamic

critical exponent.

Critical length and time scales are the only characteristics of the system close to QCP.

Scaling observable variables, all critical exponents depend on ξ and tc exclusively. There-

fore the scaling is universal and depends only on the symmetry of the order parameter.

Consequently, QPTs can be classified by the symmetry of order parameter which forms the

universality class. It means that all observable variables in various QPTs posses a universal


behavior which can be described by a model system with a corresponding symmetry of the

order parameter [47, 48].

Importantly for experimental physics, quantum ordered phases occurring at T = 0 K

survive to a finite temperature range. This makes empirical investigations of QPT possible. A

wide range of QPTs was discovered: from superfluid helium and the cuprate superconductors

which can be tuned from a Mott insulating to a superconducting phase by a carrier doping

[49, 50, 51] to various QPTs in quantum magnets and unconventional metals [52, 53]. The

main role in the phenomena is played by electrons and their collective behavior. Quantum

magnets, i.e. systems with localized electrons in reduced dimensions, belong to the most

important candidates for investigations of novel quantum phases. In the following sections

1.2.1 and 1.2.2 we introduce two specific examples of QPT in low-dimensional magnets.

1.2.1 Bose-Einstein condensation of magnons in quasi-1D systems

Here we show that a field induced QPT from paramagnetic to the 3D XY ordered antifer-

romagnetic phase can be mapped onto Bose-Einstein condensation of magnons (BEC) in

axially symmetric magnets. The starting point is the spin Hamiltonian of the system, which,

for instance, in case of well studied spin-1/2 ladders [54, 55, 56, 57] can be written as

H = Jrung∑rung

S1,rung · S2,rung + Jleg∑i,j

Si · Sj − gµBH∑k

Szk , (1.2)

where the first sum is taken over all rungs in spin ladder, the second sum runs over all legs

and the third term describes the impact of the magnetic field H and its sum runs over all

spins in the system. We assume that the rung exchange interaction, Jrung, is the strongest in

Eq. 1.2, coupling S1 and S2 into dimers. Thus the system effectively consists of interacting

S = 1 particles. Using second order quantization it was shown [55, 58] that Hamiltonian in

Eq. 1.2 can be mapped onto the following form

H =∑i

(Jrung − gµBH)a+i ai +

∑i,j

tija+i aj +

1

2

∑i,j

Uija+i a

+j aiaj, (1.3)


where the operators a+i and ai create and annihilate a boson on dimer i, respectively, tij

describes a hopping between sites i and j and Uij is a repulsion energy. The coupling of

the transverse, Sxi Sxj + Syi S

yj , and longitudinal, Szi S

zj , components in the spin Hamiltonian

in Eq. 1.2 map onto the hopping tij and repulsion Uij terms in the bosonic representation

in Eq. 1.3, respectively. The singlet S = 0 is the ground state of the system described by

the Hamiltonian in Eq. 1.2 and is separated by a finite energy gap ∆ from the exited triplet

Sz = 0,±1. An applied magnetic field induces the Zeeman splitting and lowers the energy

of the dispersive Sz = 1 excitation. When the field energy is equal to the value of the gap,

gµBHc1 = ∆, the excitation Sz = 1 mixes with the ground state and the system undergoes

the QPT from quantum paramagnetic to the 3D antiferromagnetically ordered state. In the

bosonic representation, this process can be viewed as a condensation of magnons carrying

S = 1 and thus obeying Bose-Einstein statistics [59, 60]. In spin space, the field-induced

Bose-Einstein condensation (BEC) of magnons corresponds to the order of the transverse

spin components, perpendicular to the applied field, which spontaneously brakes the O(2)

symmetry of Hamiltonian Eq. 1.2. The list of the respective parameters of Bose gas and

quantum antiferromagnet is given in Tab. 1.2 [61]. The upper critical dimension dc and dy-

namical exponent z of BEC are equal to two. However the magnetically quantum ordered

phase exists at finite temperature region and its dimension is d > 2. Therefore, the experi-

mentally observed field-induced phase transition at low-temperature region corresponding to

the BEC of magnons belongs to the 3D XY universality class with d = 3 and z = 2.

Bose gas Antiferromagnet

Particles Magnons carrying spin-1

Boson number N Spin component Sz

Charge conservation U(1) Rotational invariance O(2)

Condensate wavefunction ψi(r) Transverse magnetic order 〈Sxi + iSyi 〉Chemical potential Magnetic field

Table 1.2: The respective parameters of Bose gas and quantum antiferromagnet. The tableis adopted from the review of T. Giamarchi et al [61].

The first theoretical investigations of a possible BEC of magnons in quasi-one-dimensional


Figure 1.5: The schematic representation of the field driven Bose-Einstein condensation ofmagnons. The figure is taken from [56].

systems was discussed by E. G. Batyev, L. S. Braginskii [62] and I. Affleck [63, 59]. More

than fifteen years later the process of the field induced antiferromagnetic order was observed

experimentally in the gaped S = 1/2 compound TlCuCl3 by A. Oosawa and co-workers

[54] and explained as the BEC of magnons by N. Nikuni et al [55]. The magnetization

measurements [54, 64] have shown that the magnetic subsystem of TlCuCl3 consists of weakly

antiferromagnetically coupled S = 1/2 spin dimers and the first excited state is separated

from the ground state singlet by the energy gap ∆ ∼ 0.7meV. At the quantum critical point,

which corresponds to the critical magnetic field Hc ∼ 5.7T, the energy gap collapses.

The diagram of the field induced QPT from magnetically disordered towards 3D XY an-

tiferromagnetic state is shown in Fig. 1.5. The solid line corresponds to the Zeeman splitting

of the excited triplet state. At the quantum critical point, the triplet mode Sz = +1 reaches

the nonmagnetic ground state Sz = 0 and the system turns into a 3D antiferromagnetic state,

which is proved by the detected Goldstone modes [57].

The isostructural material KCuCl3 with coupled S = 1/2 dimers also undergoes field

induced BEC of magnons. The energy gap is equal to ∆ ∼ 2.6 meV in this compound

[64], implying an application of a magnetic field H ≈ 23 T to be closed. The magnon

condensate was also observed in many spin dimer compounds, as for example NH4CuCl3


Figure 1.6: (a) Nearest and next nearest neighbor exchange interactions are shown by solidand dashed lines, respectively. (b) The J1 − J2 phase diagram [74].

[65, 66], Cs3Cr2Br9 [67], BaCuSi2O6 [68, 69], Cu(NO3)22.5D2O [70], (CH3)2CHNH3CuCl3

[71, 72]. The BEC of magnons is possible in other systems as well, for example in iron based

AFeX3 (A is Cs, Rb and X is Cl, Br) compounds with hexagonal arrangement of spins [73].

1.2.2 Quantum phase transitions in 2D square-lattice antiferro-

magnets

As it was discussed in section 1.1, most of the 2D S = 1/2 HAFs adopts semi-classical long-

range order in the ground state. However the presence of quantum fluctuations can suppress

this order. In the case of non-frustrated lattices, for instance a square lattice, quantum

fluctuations can be enhanced by competing interactions between the nearest neighbor (NN)

and the next-nearest neighbor (NNN) spins. In literature this model is known as ”J1 − J2

model”, where J1 and J2 refer to NN and NNN exchange interaction, respectively, as it is

illustrated in Fig. 1.6(a). A frustrated square lattice with S = 1/2 is a simple example of

the spin system where competing interactions can lead to a completely disordered ground

state, i.e. to a spin liquid state. Fig 1.2(b) [74] shows the summarized diagram of the

ground states as the function of ratio J2/J1. Three main regions in the phase space are


occupied by semi-classical Neel order, ferromagnetic state and the columnar AF (CAF) state

for ∼ 0.4 < J2/J1 < ∞, −∞ < J2/J1 <∼ −0.4 and ∼ −0.7 < J2/J1 <∼ 0.7, respectively.

CAF phase can be viewed as collinear and anticollinear arrangements of magnetic moments on

a nearest sites, which form ”columns” of spins aligned in one direction, as shown schematically

in Fig. 1.6(b) and Fig. 1.7. From the classical point of view, the phase transitions from Neel

order towards CAF and from CAF towards a ferromagnetic order occur at J2/J1 = 0.5 and

J2/J1 = −0.5, respectively (see Tab. 1.3 for details).

Ground state Wave vector Range (J1, J2)

NAF (π, π) J1 > 0, J2 < J1/2

CAF (π, 0), (0, π) |J2| > |J1|/2FM (0, 0) J1 < 0, J2 < −J1/2

Table 1.3: Parameters for classical ground states in the J1 − J2 model.

Numerous theoretical investigations for the J1 − J2 model were performed for the most

interesting and less understood quantum critical regions, which are shown as hatched sectors

in Fig. 1.6(b). The boundaries of the critical phases are not known de facto. The quantum

spin liquid ground state described by the RVB state was proposed by Anderson [75] and

Kivelson et al [76]. Later, based on the 1/S series expansion calculations, Chandra and

Doucot [77] predicted a collapse of the Neel order at the critical ratio J2/J1 ∼ 0.38. The

numerical studies performed by Gelfald, Singh and Huse [78] gave a similar result, which

forecasts a QPT at J2/J1 ' 0.41 and the magnetically disordered state, where spins are

spontaneously dimerized in a columnar pattern for 0.41 < J2/J1 < 0.64 (see Fig. 1.7).

A more complex structure of a quantum disordered state with several QPTs was pro-

posed by Sushkov and colleagues [79]. The result of the calculations yields four quan-

tum critical points for the ratio J2/J1 equal to (J2/J1)1 = 0.34(0.04), (J2/J1)2 ∼ 0.38,

(J2/J1)3 = 0.5(0.02) and (J2/J1)4 ∼ 0.6. Fig. 1.7 shows schematically a summarized phase

diagram for 0 < J2/J1 < 0.7 based on the results of Sushkov et al. A second order phase

transitions from Neel order to Neel state with columnar dimerization is predicted to occur at

(J2/J1)1. When the ratio J2/J1 is increased further, the system undergoes a QPT to dimer-


Figure 1.7: The phase diagram of the S = 1/2 AF on a square lattice as the function ofg = J2/J1 based on the series expansion calculations of Sushkov et al [79].

ized spin liquid and to a columnar dimerized spin liquid with plaquette type modulation at

(J2/J1)2 and (J2/J1)3, respectively. The QPT towards CAF is predicted to be of first order.

The discussion of the possible spin configurations in the quantum disordered phases for the

0.38 < J2/J1 < 0.6 lies beyond the scope of this summary. Nevertheless, it is important to

emphasize that the suggested QPT can occur even for small variations of the J2/J1 param-

eter. Theoretical investigations performed by Siurakshina, Ihle and Hayn [80] suggest the

transition from Neel and CAF states towards the quantum spin liquid phase at J2/J1 = 0.24

and J2/J1 = 0.83, respectively. In contrast to their study, recent calculations based on the

coupled cluster method [81] demonstrate the existence of the paramagnetic ground state for

0.44(1) < J2/J1 < 0.59(1). The discrepancies in the theoretical studies indicate the necessity

of further explorations in this field.

Recently the influence of ferromagnetic NNN interactions on the J1 − J2 model was

studied. Nematic order in the AF on a square lattice with a ferromagnetic J2 was predicted

by numerical calculations of Shannon et al for J2/J1 ∼ −0.5 [82]. The authors found a first

order transition from the saturated ferromagnetic ground state to a gapless spin-liquid state

with bond-nematic order.

However, in contrast to the theoretical studies in this field, the experimental realizations

of J1 − J2 models on a square lattice are quite rare. We will briefly discuss the main layered

vanadium oxide based materials, which have attracted researchers for the last decade (see

Fig. 1.6). The first J1−J2 model was experimentally realized in VoMoO4 and studied by high-


resolution x-ray and neutron powder-diffraction [83]. The ordered magnetic moment in the

semi-classical Neel ground state is equal to m = 0.41(1)µB/V4+, which indicates enhanced

quantum fluctuations due to NNN interactions J2/J1 ∼ 0.2. Another good experimental

realization of the J1 − J2 model is Li2VOXiO4 (X1 = Si,X2 = Ge). These materials were

studied experimentally using specific heat and susceptibility measurements, NMR [84, 85].

It was found that the systems have ratios of the NNN to NN exchange interaction equal

to J2/J1 ∼ 12 and J2/J1 ∼ 4.8 for silicon and germanium components, respectively. The

measurements also revealed the finite interlayer coupling of J2/J⊥ ∼ 0.02 and J2/J⊥ ∼

0.03. The ordered magnetic structure and the ordered magnetic moment in Li2VOSiO4 was

determined by neutron diffraction and resonant x-ray scattering measurements [86]. The

magnetic moment in collinear antiferromagnetic ground state is equal to m = 0.63(3)µB, and

is thus very close to the value expected for the 2D Heisenberg model.

Recently synthesized layered perovskite PbVO3 is the realization of a highly frustrated

J1 − J2 model. The material was studied by measuring the temperature dependence of the

magnetic susceptibility and the specific heat [87, 88]. The results show that the frustration

ratio is equal to J2/J1 ∼ 0.3, indicating its proximity to the critical region which separates a

simple Neel state and Neel state with columnar dimerization (see Fig. 1.6).

The behavior of the 2D S = 1/2 AF in magnetic fields was studied theoretically for the

case J1−J2 model [89, 90]. However, little is known experimentally about the magnetic field

behavior of the 2D S = 1/2 AF, particularly in the presence of NNN interactions.

1.3 Summary 17

1.3 Summary

In conclusion, we reviewed a range of important 2D spin lattices and discussed the presence of

quantum fluctuations due to reduced dimensionality. Zero-point fluctuations in such systems

can lead to purely quantum VBS and spin liquid ground states. An external magnetic

field enhances or suppresses the quantum fluctuations providing the fascinating possibility of

tuning and controlling interactions in the quantum limit. Moreover, it can turn the a spin

system into novel quantum phases, providing unique possibilities for an investigation of novel

aspects of quantum matter.


1.4 Structure of the thesis

The thesis is organized in the following way.

• Chapter 2 is devoted to the experimental technique. Being the prime experimental

method of our work, neutron scattering is discussed here. The instrumentation and

important parts of a sample environments extensively used in our experiments, such as

three-axis spectrometers split-coil magnets and dilution refrigerators, are described in

the same chapter as well.

• Comprehensive study of a weakly-frustrated 2D S = 1/2 AF on a square lattice realized

in the organo-metallic compound Cu(pz)2(ClO4)2 is presented in the Chapter 3. We

determine the field-temperature phase diagram for magnetic fields up to one-third of

the saturation value and investigate the symmetry of the ordered magnetic phase.

The magnetic excitation spectra measured by inelastic neutron scattering in zero and

applied magnetic fields are presented and explained based on a precisely determined spin

Hamiltonian. Moreover we prove experimentally that quantum fluctuations associated

with RVB state exist in the S = 1/2 square lattice AFs and enhanced by a small

additional NNN interaction between spins.

• In Chapter 3 we concentrate on strongly anisotropic quasi-one-dimensional S = 1 chain,

NiCl2 ·4SC(NH2)2, and discuss the investigations of this material using both elastic and

inelastic neutron scattering. Determined ordered antiferromagnetic structure has a

collinear spin alignment with a preserved rotational symmetry. The size of the ordered

magnetic moment is estimated as a function of field-temperature from the neutron

diffraction measurements. The magnetic phase diagram is determined using neutron

diffraction in coordinates field-temperature. We show that the magnetic field applied

parallel to the spin chain closes the energy gap between the ground state and the

excited doublet, which can be described as BEC of magnons. The spin Hamiltonian

is established, based on our neutron spectroscopy experiments performed in a fully

magnetized phase of DTN. The spin dynamics in the antiferromagnetic phase is also

discussed.

Chapter 2

Neutron scattering

2.1 Main properties of the neutron

The existence of a neutral subatomic particle suggested by Lord Ernest Rutherford in the

1920ies was confirmed by the important discovery of the neutron by James Chadwick in

1932 [91, 92, 93]. Later, the possibility of Bragg diffraction of neutrons was demonstrated

[94] and neutron scattering became one of the most important and multifaceted techniques

in condensed matter research. The main properties of the neutron are listed in Tab. 2.1.

Due to the absence of electrical charge, and thus a lack of Coulomb interactions between

charged particles and a neutron, the neutron can penetrate deep into solids and interact

with the bulk. Having a spin of one-half and an associated magnetic moment, neutrons obey

Fermi-Dirac statistics. Therefore, the combination of the chargeless nature and the presence

of a magnetic moment gives an unique opportunity for experimental investigations of bulk

magnetic structures and excitations in condensed matter using neutron scattering.

2.1.1 Neutron sources

Neutron scattering research facilities are based on either a nuclear reactor or a spallation

source. A nuclear reactor uses spontaneous fission of 235U for creating a high-flux of neutrons.

A spallation source produces particles by bombarding a heavy metal target with high-energy

protons. The neutron beam produced in nuclear reactors is continuous, while spallation

20 Neutron scattering

Quantity Value

Mass 1.67492729(28) · 10−27 kgCharge 0

Spin 1/2Magnetic moment −9.6623640 · 10−27 J/T

β decay n→ p+ + e− + νLife time 888(3) s

Table 2.1: Main characteristics of neutron.

sources provide typically a pulsed neutron beam. The most powerful research reactors is the

High Flux Reactor (HFR) at the Institute Laue-Langevin (ILL), France, and provides a flux

of I ∼ 1015 neutrons/(cm2sec) with a thermal power of W ∼ 58.3 MW. Continuous spallation

sources produce usually a flux of one order of magnitude lower compared to nuclear reactors.

The highest value of peak neutron flux equal to I ∼ 3 · 1016 neutrons/(cm2sec) is generated

by a pulsed spallation sources SNS. Another example of the spallation neutron source, ISIS,

produces a peak flux of I ∼ 6 · 1015 neutrons/(cm2sec).

The Swiss Spallation Source (SINQ) located at Paul Scherrer Institute (PSI), Villi-

gen, Switzerland, is based on a 590 MeV Ring Cyclotron, whose accelerator frequency

τ = 50.63 MHz. Such a high frequency provides the possibility to smear out a pulsed

neutron flux and to obtain a continuous final beam.

In order to be useful for scattering experiments, high-energy neutrons obtained after the

spallation process or nuclear reactions are ”cooled down” by passing a moderator block. A

wide distribution of neutron energies results from a use of three main moderators. Containing

liquid deuterium cooled down to T ∼ 20 K, a cold moderator has the maximum transmission

for neutrons with energies E ∼ 5 meV. Providing fluxes of neutrons with energies E ∼

25 meV, thermal moderators are made out of heavy and light water for spallation sources

and nuclear reactor, respectively. Finally hot moderators uses heated up graphite to speed

up thermal neutrons and obtain the energies of order E ∼ 200 meV.

After the moderator block the flux is usually transported to an experimental instrumen-

tation by neutron guides. Acting on neutrons like supermirrors, guides are conventionally

2.1 Main properties of the neutron 21

made out of polished glass coated with special materials, for instance Ni-Ti. Neutron guides

installed at SINQ or SNS include the supermirrors which have been produced using DC-

magnetron sputtering: The mirror consist of a top layer of nickel and a sequence of ∼ 100

layers Ni-N-O/TiVx diffracting neutrons by Bragg reflection. In this way the neutron flux

can be transported into relatively big distances (∼ 100 meters) with small losses. A perfor-

mance of instruments based on neutrons transported by guides, implies usually the use of

cold neutrons and naturally small background scattering (less than 10−1 counts per minute).

In contrast, instruments located in close vicinity to a neutron source (∼ 10 meters) without

a neutron guide, are designed to work in thermal - epithermal regimes and have a higher

background scattering, more than 10−1 counts per minute.

2.1.2 Energy and time scales

The mass of a neutron mn and the velocity v are related to its De Broglie wavelength λ and

the wave-vector k by mnv = hk = h/λ. Therefore, the total energy of a neutron is equal to

E =mnv

2

2=h2|k|2

2mn

. (2.1)

The generally accepted classification of neutron energy ranges is presented in Tab. 2.2. Scat-

tering experiments using cold and thermal neutrons can probe a wide range of properties

in solids, including crystal and magnetic structures, lattice vibrations and anharmonicities,

spin-waves and crystal fields among many others. Therefore, most large scale facilities are

designed to produce a high-flux in cold and thermal regimes. The velocity of thermal neu-

trons is equal to v ∼ 2.2 km/s corresponding to an energy E = 25.3 meV, the temperature

T = 293 K and the wavelength λ = 1.798 A.

A neutron scattering process is shown schematically in Fig. 2.1. The incoming particle

with wave vector ki scatters at the single crystalline sample and changes direction by an

angle of 2Θ to the initial direction. Momentum and energy conservation of the scattering

process leads to the following equations:

hQ = hkf − hki,


Classification Energy

Epithermal (K) ∼ 250− 103 meVHot ∼ 200 meV

Thermal ∼ 25 meVCold 5× 10−2 − 25 meV

Very cold 3× 10−4 − 5 · 10−2 meVUltra cold < 3× 10−4 meV

Table 2.2: Energy classification of neutrons.

hω = Ei − Ef =h2(k2

f − hk2i )

2mn

, (2.2)

where indexes i and f correspond to initial and final (scattered) states, respectively, hQ is

the momentum transferred to the sample. A scattering process is called elastic in the case of

a zero energy transfer and usually it is used to determine a crystal or magnetic structures.

The scattering process with a non-zero energy transfer is called inelastic, and is used to

probe dynamic features of crystal or magnetic structures. Both elastic and inelastic neutron

scattering processes are discussed below. For further details, a number of good textbooks are

recommended [95, 96, 97, 98].

2.2 Elastic neutron scattering

2.2.1 Fermi’s Golden rule

Being the fundamental concept of any scattering process, the cross-section describes an effec-

tive area for collision. In order to set the experimental stage, we have to define the neutron

scattering cross-section. Suppose, neutrons with a certain energy Ei scatter on the sample

into the given direction into the solid angle Ω. In this case the scattering rate is defined

as φ(ki)σ, where φ(ki) is the initial flux of neutrons with the wave vector ki and σ is the

cross-section. Technically, neutron scattering experiments are based on measurements of a

double differential cross-section assigned as a number of scattered neutrons per solid angle

2.2 Elastic neutron scattering 23

Figure 2.1: Scattering process.

dΩf and per energy range from Ef to Ef + dEf . It can be written as the sum of coherent

and incoherent parts:

d2σ

dΩfdEf=

d2σ

dΩfdEf

∣∣∣∣∣coh +d2σ

dΩfdEf

∣∣∣∣∣inc

(2.3)

The first term, coherent, describes collective effects in many body systems, like nuclear or

magnetic Bragg diffraction and inelastic scattering from magnons or phonons. The incoherent

term physically arises from deviations of nucleus from their positions defined by a crystal

symmetry, for instance due to thermal fluctuations or diffusion processes. Our scientific

interest lies in the field of collective effects in solid state and from now on only the coherent

part of the double-differential cross-section will be taken into account. Incoherent scattering

results in a background which we try to eliminate.

In quantum physics, the probability of the transition of the system from one eigenstate

|i〉 to another |f〉 per unit time is defined by the Fermi’s Golden rule. Assuming that U is

the operator which describes the interaction of a neutron with the system and taking into

account the Fermi’s Golden rule, we can write the double-differential cross-section as follows:

d2σ

dΩfdEf=kfki

(mn

2πh2 )2|〈kfλf |U |kiλi〉|2δ(hω + Ei − Ef ), (2.4)

where λi and λf are quantum numbers corresponding to initial and final states, respectively.


2.2.2 Nuclear scattering

Nuclear scattering is the result of the short-range (∼ 1 fm) interaction between a nucleus and

a neutron. Starting from Eq. (2.4), we can write the double differential cross-section which

describes the process as follows:

d2σ

dΩfdEf= N

kfki

∑l

b2l · S(Q, ω), (2.5)

where N is the number of scattered centers in the system and bl is the nuclear scattering

length of the nucleus l which depends on details of the neutron-nuclear interaction potential.

The neutron scattering function, S(Q, ω), describes the contribution of a scattered system

into the coherent cross-section and it is defined as

S(Q, ω) =1

2πhN

∑ll′

∫ +∞

−∞dt〈e−iQ·rl′ (0)e−iQ·rl(t)〉e−iωt, (2.6)

where the angle brackets denote the average over initial states, and the integral is taken over

time. Therefore, based on measurements of S(Q, ω), appropriate microscopic properties of

investigated material can be determined.

Taking into account only elastic neutron scattering on a crystal, the scattering function

Eq. (2.6) can be written as

S(Q, ω) = δ(hω)(2π)3

ν0

∑Kδ(Q−K), (2.7)

where the sum is taken over the reciprocal lattice vectors K. Thus, using Eq. (2.5) and

Eq. (2.7), the elastic cross-section in a crystal medium is given by

dσ

dΩ= N

(2π)3

ν0

∑Kδ(Q−K)|FN(Q)|2, (2.8)

where FN(Q) is an important quantity called nuclear structure factor:

FN(Q) =∑i

bieiQ·die−

12〈(Q·ui)

2〉, (2.9)

2.2 Elastic neutron scattering 25

describing positions di and displacements ui of ions in a unit cell. From Eq. (2.8) we see that

elastic scattering occurs only if the moment transferred to the single crystal is equal to the

reciprocal wave-vector, Q = K. This is the case of Bragg diffraction: satisfying Bragg’s law

nλ = 2d sin(Θ), diffracted peaks appear. Therefore, based on measured neutron diffraction

patterns, i.e. the scattered intensities as a function of the angle Θ, crystal structures of an

investigated material can be analyzed. It is important to note that various structural phase

transitions in solids can be detected by performing diffraction measurements in different

environmental conditions, for instance, temperature or pressure.

2.2.3 Magnetic scattering

Due to the classical magnetic dipole-dipole interactions between neutron and magnetic mo-

ment of electron, neutron scattering also gives the possibility to probe bulk magnetic struc-

tures.

The amplitude for magnetic scattering can be written in the following form:

Im(Q) = (γr0

2)gf(Q), (2.10)

where r0 is the classical electron radius, γ the gyromagnetic ratio, g the Lande splitting factor

and

f(Q) =∫ρs(r)eiQ·rdr

the magnetic form factor that is the Fourier transform of the normalized unpaired spin density

ρs(r). Taking into account magnetic interactions, the differential cross section Eq. (2.4) is

defined as follows:

d2σ

dΩfdEf=kfki

∑i,f

P (λi)|〈λf |∑l

eiQ·rlUsisfl |λi〉|2δ(hω + Ei − Ef ), (2.11)

where Usisfl is the scattering amplitude from spin site si to sf on site l

Usisfl = 〈sf |bl − plS⊥l · σn +BlIl · σn|si〉. (2.12)


Here bl is the nuclear coherent scattering amplitude, σn Pauli spin operator for a neutron,

Bl the spin-dependent nuclear amplitude and Il is the nuclear spin operator. The magnetic

interaction vector S⊥ defined as S⊥ = S − Q(Q · S) means that only spin components

perpendicular to the wave-vector Q will contribute to the magnetic scattering intensity.

Summarizing, we can write the magnetic differential cross section in the following form:

d2σ

dΩfdEf= (γr0)2kf

kie−2W |f(Q)|2

∑α,β

(δα,β −QαQβ

Q2)Sα,β(Q, ω), (2.13)

Sα,β(Q, ω) =∑i,j

eiQ(ri−rj)∑λi,λf

pλi〈λi|Siα|λf〉〈λf |Siβ|λi〉δ(hω + Eλi − Eλf ), (2.14)

where (δα,β − QαQβQ2 ) is the magnetic polarization factor, Sα,β(Q, ω) the magnetic scattering

function and Siα (α = x, y, z) denotes the spin operator of ith ion located at ri. In the case

of elastic scattering on magnetically ordered crystals, (2.13) simplifies to

dσ

dΩf

= Nm(2π)2

υm

∑Gm

e−2W (Gm)δ(Q−Km)|Fm(Km)|2, (2.15)

where the exponent is the Debye-Waller factor. In analogy with the nuclear structure factor

(2.9), the magnetic structure factor is defined as following:

Fm(Km) =∑j

pjS⊥jeiKm·dj . (2.16)

The index m corresponds to the magnetic unit cell, which, generally speaking, can be different

from the nuclear one.

Therefore, using magnetic elastic neutron scattering, magnetic Bragg reflections can be

measured and, generally, the ordered magnetic structure can be determined. Taking into

account the spin dependence of the magnetic amplitude M(Q) = gSµBf(Q), the value of an

ordered magnetic moment can be estimated from a neutron scattering experiment.

2.3 Neutron spectroscopy 27

2.3 Neutron spectroscopy

In addition to the investigation of ordered nuclear and magnetic structures, neutron scattering

also allows the measurement of the dynamics of crystal lattices and magnetic frameworks in

bulk materials. In this section inelastic neutron scattering on nuclear and magnetic structures

are discussed.

2.3.1 Inelastic nuclear scattering

According to Eq. (2.2), a neutron can gain or lose energy while the particle scatters. This

process obeys universal principle of detailed balance

S(−Q,−ω) = e−hω/kBTS(Q, ω), (2.17)

which emphasizes properties of an equilibrium state and it is illustrated schematically in

Fig. 2.2. If neutron loses energy in the scattering process (Fig. 2.2(a)), the scattering system

becomes excited from the ground state λ to a higher-energy state λ′. This transition con-

tributes to the scattering function S(Q, ω). Fig. 2.2(b) shows the reverse process in which

the scattered neutron gains energy and the transition in the scattering system contributes to

S(−Q,−ω). The probability of the scattering system being initially in the low-energy state

λ is higher by e−hω/kBT then the probability of being in the high-energy state λ′.

The fluctuation-dissipation theorem [99] is based on the assumption that in thermal equi-

librium the response of a system on a small perturbation is the same as its response on a

spontaneous fluctuation. Therefore, the measured response of a thermodynamic system can

provide an information about its fluctuation properties. According to this, the relation be-

tween the neutron scattering function and the imaginary part of the dynamic susceptibility

is

S(Q,ω) =χ

′′(Q, ω)

1− e−hω/kBT. (2.18)

The significance of this equation lies in the fact that the measured response of a system

χ′′(Q, ω) can be compared directly to the theoretically predicted value, which often can be


Figure 2.2: Principle of detailed balance.

calculated from first principles.

During inelastic scattering on an ordered atomic system, a neutron creates or destroys a

phonon. In this case the imaginary part of the dynamic susceptibility can be written in the

form

χ′′(Q, ω) =

1

2

(2π)2

υ0

∑G,q

δ(Q− q−G)∑s

1

ωqs|F(Q)|2[δ(ω − ωqs)− δ(ω + ωqs)] (2.19)

where F(Q) is the dynamic structure factor. Based on equation (2.19) phonon dispersion

can be measured by performing inelastic neutron scattering experiments.

2.3.2 Inelastic magnetic scattering

Collective spin excitations are called spin waves. The energy of a spin wave is quantized1

and it has a Q-dependence which can be measured by inelastic neutron scattering. In anal-

ogy with phonons, during inelastic scattering in a magnetically ordered medium, a neutron

creates or destroys a spin wave. Assuming that a propagating spin wave is comprised of

spin displacements from an ordered direction in a ferromagnet, the double differential cross

section Eq. (2.4) can be written for the case of inelastic magnetic scattering in the following

1Quantized spin waves, or magnons, were described in terms of secondary quantization formalism byHolstein and Primakoff [100] who had shown that magnons obey Bose-Einstein statistics.

2.4 Experimental technique 29

manner:

d2σ

dΩfdE= (γr0)2kf

ki

(2π)3

2ν0

S(1 +Q2z)|F (Q)|2e−2W

×∑q,G

[〈nq + 1〉δ(hω − hωq)(Q− q−G) + 〈nq〉δ(hω + hωq)(Q + q−G)], (2.20)

where 〈nq〉 = (exp(hωq/kBT )−1)−1 is the thermal average. Equation (2.20) demonstrates the

possibility to measure the dispersion of spin waves directly using inelastic neutron scattering.

Calculations of Q-dependence of the energy of spin waves is based on a magnetic ordering

geometry and symmetry properties of the system. For a relatively simple case of a two-

dimensional S = 1/2 AF with NN and NNN exchange interactions, the calculation of spin

wave dispersion in linear approximation is given in Appendix A.

2.4 Experimental technique

In the field of inelastic neutron scattering, three-axis spectroscopy (TAS) together with time-

of-flight spectroscopy are the most common used techniques. While the first technique pro-

vides a possibility to measure particular positions in (Q, ω) space point-by-point, rather large

regions of phase space can be explored by the time-of-flight technique. However, the price

paid for a large phase-space probe by time-of-flight spectroscopy is that intensities on a sam-

ple are considerably reduced due to the use of a pulsed neutron beam. Since a major part of

results represented in this manuscript are obtained using TAS, the concept of this method is

describe below.

2.4.1 Three-axis spectroscopy

Fig. 2.3 shows a sketch of a typical three-axis instrument. The performance of the instrument

is based on Bragg’s law. First, passing through a guide, the neutron beam is scattered on

a monochromator crystal in the direction of the sample. Then the beam scattered on the

sample traverses the analyzer and comes to the detector. Monochromators and analyzers

are usually made of single crystals and serve to select and analyze neutrons with a specific


wavelength λ based on Bragg’s diffraction law

nλ = 2d sin(Θ). (2.21)

Here n is an integer number and represents the order of scattering, d is the distance between

planes in a crystal which cause the diffraction of the beam in direction with the angle Θ.

Figure 2.3: Three axis spectrometer. The combina-

tion of monochromator, sample and analyzer axes

provides a wide range of experimental possibilities

for condensed matter research.

Crystals can be bent or turned thus

that an appropriate change of d lead

to Bragg scattering of neutrons with

the selected wavelength λ. Single crys-

tals of pyrolytic graphite (PG), germa-

nium or silicon are commonly used for

monochromating and analysis of neu-

tron beams. In general high order

scattering exists and can be used for

measurements in restricted experimen-

tal conditions, however more often it

complicates experiments. In order to

suppress high order scattering, different

filters can be installed before or after

the sample. It is common to use PG,

beryllium or beryllium-oxide as high or-

der filters. When the neutron beam

with a chosen wavelength according to

Eq. (2.21) hits the sample, the scattered

intensities are distribute in different directions according to equation Eq. (2.4). An angular

dependence of the distribution can be probed by varying scattering angles between incoming

and scattered beams. Rotation of the analyzer defines the direction towards the detector

providing a possibility to obtain information about energy distribution of scattered neutrons.

Measured in this way the scattering function, S(Q, ω), can be used to explore the nuclear or


magnetic structure and dynamic properties.

A substantially easier experimental method is based on the use of the monochromator and

the sample axis only and therefore it is called two-axis technique. The lack of the analyzer

indicates the impossibility to perceive a differences between energies of neutrons scattered

on the sample leading to a higher background compare to the setup with the analyzer.

However, for some measurements this experimental limitation is irrelevant, for instance, the

case of diffraction measurements.

In order to increase a neutron flux on a sample, an incoming beam can be focused by a

bendable monochromator crystal or several crystals mounted on independent holders. Ver-

tical focussing is resulted from the deformed monochromator crystals to a cylinder with its

axis lying in the horizontal plane. At the cost of wave-vector resolution, the gain in flux can

be achieved in this case. By focusing in both directions, vertical and horizontal, total neutron

flux on a sample can be increased by approximately hundred times. Analyzer crystals can be

focused in the same manner. The result of using focused analyzer is identical to making the

distance between analyzer and detector smaller. However an unfavored effect of the latter

option is an increased background due to a larger detector. In contrast, a use of focused

analyzer implicates a lower background.

2.4.2 The experimental resolution function

Heretofore, precisely defined wave-vectors of incoming and scattered beams and the scattering

angle were considered for neutron scattering experiments idealizing the actual situation. In

reality all three-axis spectrometers have a finite resolution due to the following reasons:

1. Any pathway of a neutron beam has finite divergences in directions perpendicular to

the ideal propagation direction.

2. Monochromator and analyzer reflect neutrons with a very small, but finite spread of

wavelengths.

Therefore, detected neutrons are characterized by the incoming, final wave-vectors and the

scattering angle distributed around the average values ki, kf and 2θ. Since those limita-


tions are caused by the experimental technique, the resolution depends only on the specific

instrumental configuration and is independent of actual physics of the studied sample.

As it was shown above (Sec. (2.2), (2.3)), the double-differential neutron scattering cross-

section can be written as

d2σ

dΩfdEf=

kfkiS(Q, ω), (2.22)

with S(Q, ω) defined appropriately for a particular situation. Suppose the spectrometer is

set to the special point in the phase space (Q0, ω0) and three Cartesian coordinates Q⊥,

Q‖ and Qz are defined as parallel to Q0, perpendicular to Q0 in the scattering plane and

perpendicular to the scattering plane of Q0, respectively. In this case the intensity I(Q0, ω0)

detected by the instrument can be calculated using the following four-dimensional integral:

I(Q0, ω0) =∫R(Q−Q0, ω − ω0)S(Q, ω)dω dQ. (2.23)

Relation (2.23) is very general and it shows that measurements performed at the particular

point (Q0, ω0) depend on the scattering function S(Q, ω) convoluted with the instrumental

resolution function R(Q−Q0, ω − ω0), which is defined as

R(Q−Q0, ω − ω0) = R0exp(−1/2∆%M∆%). (2.24)

First derived by Copper and Nathans [101], this equation is based on the assumption that

the distributions of ki and kf are Gaussians. Here M is a four-dimensional function of ki, kf

and 2Θ, depending on all experimental parameters, including all lengths of beam pathways

between different parts of the instrument, vertical and horizontal divergences of the beam,

dimensions of the analyzer, the detector and the mosaicity of monochromator and analyzer.

∆% is a vectorial parameter

∆% = (Q‖ −Q0, Q⊥, Qz,mn

hQ0

(ω − ω0)). (2.25)


The normalization factor R0 used in equation (2.24) is based on all experimental parameters

as the matrix M. The constant amplitude profile for the resolution function is a set of nested

four-dimensional ellipsoids in (Q, ω) space and it can be calculated using equations of the

form ∆%M∆% = Constant. Size and shape of the resolution ellipsoid can be adjusted by

changing the experimental setup, however an increase of the resolution leads to a drop of

detected intensities. Therefore, one of the main experimental issues is to find a compromise

between a good resolution and a reasonable time of the measurement by finding a proper

instrumental configuration of the spectrometer, which will lead to a good signal-background

ratio.

It is worth to mention that for the cases of focused monochromator and analyzer, the way

of calculating a resolution function differs from a flat regime. First described by Cheeser and

Axe [102], R0 and R(Q−Q0, ω − ω0) for the focusing mode were calculated and alter math

was generalized by Popovici [103]. For further details concerning computing the resolution

function we refer to the original publications listed above and to the books written by Hippert

[104] and by Shirane, Shapiro, and Tranquada [97].

In contrast to the resolution analysis of the inelastic neutron data, resolution corrections

for the case of elastic experiments look much simpler. The examination of scattering observed

in a diffraction experiment is conventionally based on the following function:

I(Q0) = C1 + C2Θ0 +mhkl|F|2L(Θ0) · exp(−4 ln 2

(2Θ0 − 2Θhkl

Γ0

)2), (2.26)

where C1 and C2 are the background parameters, Θ0 is the angle corresponding to the

observed Bragg peak, mhkl is the multiplicity of the reflection, L(Θ0) = 1/ sin Θ sin 2Θ is the

Lorentz factor and the function Γ0 describes the lineshape of the peak:

Γ0 =√U tan2 Θ0 + V tan Θ0 +W, (2.27)

where U, V and W are the parameters depend on the geometry of the experiment.

Finally, to perform a proper analysis of data obtained by TAS, an experimentalist must:

1. Determine the instrumental resolution function.


2. Convolute it numerically with a model (or expected) scattering function.

3. Fit the experimental data using resolution-convoluted function.

2.5 Sample environment

As we have shown in Sec. 2.2.3 and Sec. 2.3.2, the neutron scattering technique provides

an essential possibility to study magnetic properties of bulk materials. Magnetic phase

transition are driven by external thermodynamical parameters, for instance, by magnetic

Figure 2.4: The mixing chamber

of dilution refrigerator.

field or temperature. In order to understand a mag-

netic phenomena in its entirety, neutron scattering ex-

periments can be performed in different external envi-

ronments. In this section we briefly describe the main

concepts of split coil magnet and dilution refrigerator,

which were extensively used during the work on this the-

sis.

2.5.1 Split coil magnets

The discovery of type II superconductors made possible

a generation of high magnetic fields by means of super-

conducting magnets. Being superconducting in magnetic

fields, solenoids made out of those materials carry giant

currents producing high magnetic fields. Generally, split

coil magnets are based on niobium-titanium supercon-

ducting wires doped with epoxy, wrapped into a solenoid

and cooled with liquid helium and liquid nitrogen. Two

solenoids are placed on the same axis close to each other.

Therefore, a sample loaded in a small region between the

superconducting solenoids is accessible for incoming neu-

tron beam. To achieve high fields, (Nb, T i)3Sn wires are

2.5 Sample environment 35

used in split coil superconducting magnets as an inner coil. The field generated by a split coil

magnet has a technical limit Blim ≈ 15 T, which was sufficient for our experimental needs.

2.5.2 Dilution refrigerator

The usage of dilution refrigerator was essential for the experimental investigation of quantum

effects in low-dimensional magnets Cu(pz)2(ClO4)2 and NiCl2 ·4SC(NH2)2. Being a side effect

in this experimental work, thermal fluctuations were suppressed by cooling the systems down

to a few tenths of milliKelvin in a dilution refrigerator. The operation of this complex device

is based on a mixture of two helium isotopes, 4He and 3He. At T ≈ 870 mK the mixture

spontaneously separates into 3He rich and diluted phases. The mixing chamber is shown

schematically in the Fig. 2.4. Helium atoms in the diluted phase are attracted to each other

by Van der Waals forces stronger compare to ones in the reached phase. As the result,

3He reached phase is approximately 20% less dense than diluted one. In the process of

phase separation, the system looses energy while transporting 3He atoms from the reached

to the diluted phase. Providing a continuous stream of 3He atoms into the reached phase

and, therefore, a permanent flow of 3He atoms crossing the phase boundary line, dilution

refrigerator cools down to milliKelvin temperature range.


2.6 Summary

In this chapter the essence of neutron scattering and remarkable opportunities, provided by

this experimental technique for condensed matter research, are described. A wide range of

solid state characteristics can be probed by neutron scattering including ordered crystal and

magnetic structures, lattice dynamics and magnetic excitations, phase transitions and critical

magnetic behavior. TAS - a powerful tool for investigations of excitations in bulk solids and

the instrumental resolution function are discussed.

Chapter 3

2D S = 1/2 antiferromagnet on a

square lattice Cu(pz)2(ClO4)2

3.1 Introduction

As it is discussed in Chapter 1, the ground state of the 2D S = 1/2 square lattice Heisen-

berg AF adopts Neel long-range order at zero temperature. Nevertheless, strong quantum

fluctuations arising from geometrical frustration may destroy long-range order in 2D. Even

in the absence of frustration, numerical studies of the 2D S = 1/2 Heisenberg AF on a

square lattice using quantum Monte Carlo, exact diagonalization, coupled cluster as well as

series expansion calculations reveal a quantum renormalization of the one-magnon energy

in the entire Brillouin zone and the existence of a magnetic continuum at higher energies

[12, 13, 11, 14, 15]. In recent years, quantum renormalization effects at zero field have been

studied using neutron scattering in a number of good realizations of S = 1/2 square-lattice

Heisenberg AFs [105, 106, 107, 108, 109, 110].

The addition of antiferromagnetic NNN interactions destabilizes the antiferromagnetic

ground state and increases quantum fluctuations (see Chapter 1): according to the J1 − J2

model [77, 112, 113, 114], where J1 and J2 are the NN and the NNN exchange interactions,

respectively, different ground states are stabilized as a function of J2/J1. A possible spin-

liquid phase appears to be the ground state for 0.38 < J2/J1 < 0.6 and collinear order was

found computationally for J2/J1 > 0.6 (see Sec. 1.3).

38 2D S = 1/2 antiferromagnet on a square lattice Cu(pz)2(ClO4)2

The behavior of the 2D S = 1/2 AF in magnetic fields was studied theoretically for

the case of nearest neighbor interactions [35, 36, 37] as well as for J1 − J2 model [90, 115].

However, little is known experimentally about the magnetic field behavior of the 2D S = 1/2

AF, particularly in the presence of next-nearest neighbor interactions.

In this chapter we present an experimental investigation of the 2D organo-metallic AF

Cu(pz)2(ClO4)2, a good realization of the weakly frustrated quantum AF on a square lattice

with J1 ∼ 1.56 meV. Due to the small energy scale of the dominant exchange interaction,

magnetic fields available for macroscopic measurements and neutron scattering allow the ex-

perimental investigation of this interesting model system for magnetic fields up to about one

third of the saturation field strength. We combine specific heat, neutron diffraction and neu-

tron spectroscopy to determine the spin Hamiltonian and the key magnetic properties of this

model material. Specific heat measurements show that the magnetic properties are nearly

identical for fields applied parallel and perpendicular to the square-lattice plane. This shows

that spin anisotropies are small in contrast to spatial anisotropies, and that it is sufficient

to perform microscopic measurements for just one field direction. Our microscopic neutron

measurements, on the other hand, provide information on the spin Hamiltonian that explain

the nearly identical field-temperature (HT) phase diagrams for the two field directions. Spe-

cific heat and neutron measurements of Cu(pz)2(ClO4)2 thus ideally complement each other.

Moreover, we show that even a small J2/J1 ' 0.02 ratio enhances quantum fluctuations

drastically, leading to a strong magnetic continuum at the antiferromagnetic zone boundary

and the inversion of the zone boundary dispersion in magnetic fields.

3.2 Bulk properties

3.2.1 Crystal structure.

For several decades, copper pyrazine perchlorate, Cu(pz)2(ClO4)2, has been held to be a good

realization of the 2D S = 1/2 AF [116]. Synthesized from a solution of Cu(ClO4)2 · 6H2O,

pyrazine (C4H4N2) and HClO4, single crystals of Cu(pz)2(ClO4)2 have rectangular shape and

layered structure. Typical crystals are shown in Fig.3.1(c) and were grown within a period

of one week. At room temperature, Cu(pz)2(ClO4)2 and its deuterated version crystallizes

3.2 Bulk properties 39

in a monoclinic crystal structure, space group C2/m, with disordered ClO4 (perchlorate)

anions. Below T = 163 K, the material undergoes a structural transition to the C2/c space

group (#15), with lattice parameters a = 14.045(5) A, b = 9.759(3) A, c = 9.800(3) A and

β = 96.491(4), which was determined by X-ray diffraction [117].

The crystal structure is shown in Fig. 3.1(a-b). The Cu2+ ions occupy 4e Wyckoff po-

sitions and pyrazine ligands link magnetic Cu2+ ions into square-lattice planes lying in the

crystallographic bc-plane. The Cu2+-Cu2+ nearest neighbor distances in the bc-plane are all

nearly identical and equal to 6.919(5)A and 6.920(5)A [117]. Tetrahedral perchlorate anions

located between the planes (Fig. 3.1(a)) provide a good spatial isolation of Cu2+ − pz layers

and substantially decrease the interlayer interactions. Thus, perfect square-lattices of copper

ions with a superexchange path mediated by pyrazine molecules are formed in the bc-plane,

as shown in Fig. 3.1(b).

3.2.2 Magnetic susceptibility

Magnetic susceptibility describes the magnetic response of a material to an applied magnetic

field. Generally, the field-induced magnetization is anisotropic and depends on the magnetic

field direction. The susceptibility is thus not a scalar but a tensor of rank two:

χij =Mi

Hj

, (3.1)

where indexes j and i correspond to the directions of an applied field and generated mag-

netization, respectively. We measured the magnetic susceptibility of Cu(pz)2(ClO4)2 using

a commercial superconducting quantum interference device (SQUID) based magnetometer

produced by Quantum Design. A small rectangular single crystal of Cu(pz)2(ClO4)2 with

mass m = 17.75(5) mg was loaded into the magnetometer with the applied field parallel and

perpendicular to the bc-plane. The temperature dependence of the susceptibility measured

from T = 2 K to T = 50 K at µ0H = 1 T is shown in Fig 3.2.

The magnetic susceptibility along the crystallographic a direction is larger than in the

bc-plane. The sharp minimum in temperature dependence of susceptibility around T = 4 K

corresponds to the phase transition from the magnetic disorder towards the Neel long range


Figure 3.1: (Color) (a) Three-dimensional view of the crystal structure of Cu(pz)2(ClO4)2.The Cu2+ ions are shown as big blue spheres and the C, N, Cl and O atoms as small green,cherry, dark-blue and beige spheres, respectively. The D atoms are not shown for simplicity.The yellow lines represent nearest neighbor Cu2+-Cu2+ exchange paths in the square-latticeplane lying in the bc plane. ClO4 tetrahedra provide a good spatial isolation of the square-lattice Cu layers along the a axis. (b) The projection of the crystal structure on the bc planeshows the Cu2+ square-lattice structure. The square lattice are shifted by (0, 0.5, 0) from onesquare-lattice layer to the next. (c) Crystals of Cu(pz)2(ClO4)2 grow naturally and have alayered shape with a well defined crystallographic bc-plane.


antiferromagnetic phase.

Figure 3.2: The magnetic susceptibility χaa, χbb and χcc of Cu(pz)2(ClO4)2 measured asfunction of temperature shown by red, blue and cyan circles, respectively.

The Curie-Weiss law for AFs for T > T N is

χ(T ) =C

T + TN, (3.2)

where C is the Curie constant of the sample and TN is the Neel temperature. The Curie

constant of material is equal to

C =N(−gsµBs)2

kB

, (3.3)

where N is the number of atoms in the sample, gs is Lande g-factor, µB is the Bohr mag-

neton, s is the spin and kB is the Boltzmann constant. Thus, the magnetic susceptibility is

proportional to a squared g-factor. The g-factor values three orthogonal directions are equal

to gb = gc = 2.07(1) and ga = 2.27(1) as determined from the single crystal X-band electron

paramagnetic measurements [118]. The observed difference of 19(1)% in the susceptibility

measured parallel and perpendicular to the bc-plane is thus in the good agreement with the

data obtained from X-band EPR, which predicts the difference of 20.3(7)%. The difference

in the magnetic susceptibility parallel and perpendicular to the Cu2+ planes can thus be

explained by the different gyromagnetic factors. The nearly identical magnetic susceptibility

along the b- and c-direction suggest that spin anisotropies in the copper square lattice plane


are small. The small difference between χbb and χcc may be caused by a small misalignment

of the field away from the bc-plane.

3.2.3 Specific heat

In order to obtain the HT phase diagram of Cu(pz)2(ClO4)2, we measured the specific heat as

a function of temperature for different magnetic field strengths using the Physical Property

Measurement System by Quantum Design. A single crystal of deuterated Cu(pz)2(ClO4)2

with mass m = 13 mg was fixed on a sapphire chip calorimeter with Apiezon-N grease. The

measurements were done using the relaxation technique, which consists of the application

of a heat pulse to a sample and the subsequent tracking the induced temperature change.

The specific heat was obtained in the range from T = 2 K to T = 30 K in magnetic

fields of up to µ0H = 9 T, applied parallel and perpendicular to the copper square-lattice

planes. The measurements were done with the steps of ∆T1 = 0.05 K, ∆T2 = 0.2 K and

∆T3 = 1 K in the temperature ranges T1 = 2 − 6 K, T2 = 6 − 8 K and T3 = 8 − 30 K,

respectively. Care was taken to apply a small heat pulse of 0.1% of the temperature step

∆T and each measurement was repeated three times to increase accuracy. Specific heat of

Apiezon-N grease without Cu(pz)2(ClO4)2 crystal was measured in the entire temperature

range separately and subtracted as a background from the total specific heat of the sample

and grease.

The temperature dependence of the specific heat of Cu(pz)2(ClO4)2 is shown in Fig. 3.3(a,b)

for different magnetic fields applied perpendicular and parallel to the copper square-lattice

plane. At all fields, the temperature dependence of the specific heat reveals a well defined

cusp-like peak, indicating a second order phase transition towards 3D long-range magnetic

order. Previous zero-field studies of Cu(pz)2(ClO4)2 did not show an anomaly in the specific

heat [119]. Most likely, the high accuracy of our measurements played a crucial role in detect-

ing the zero-field anomaly in the specific-heat curve. The small size of the ordering anomaly

is a consequence of the low dimensionality of the magnetism and an ordered magnetic mo-

ment that, due to quantum fluctuations, is considerably smaller than the free-ion value. The

HT phase diagram assembled from the specific heat measurements is shown in Fig. 3.3(c).

The measurements show that the Neel temperature increases with increasing magnetic field,


Figure 3.3: (Color) Specific heat of Cu(pz)2(ClO4)2 as a function of temperature for differentmagnetic fields applied parallel and perpendicular to the bc plane are shown in (a) and (b),respectively. (c) The magnetic phase diagram obtained from the specific heat measurements.

from T N = 4.24(4) K at zero field to T K = 5.59(3) K at µ0H = 9 T.

We also observe an increase of the specific heat with increasing magnetic field in the

paramagnetic phase just above the 3D ordering temperature. We propose that the field

dependence of the specific heat data is a consequence of field-induced anisotropy in the

2D AF. In zero field, a pure 2D Heisenberg AF orders at zero temperature, but quantum

Monte Carlo simulations [32] have shown that the application of an external field induces an

Heisenberg-XY crossover and leads to a finite temperature Berezinskii-Kosterlitz-Thouless

transition TBKT [33, 34]. One consequence of this crossover is the increase of TBKT with

external field for up to H < HSAT/4 and then a gradual decrease of the transition temperature

with increased fields. While the zero-field 3D transition TN in Cu(pz)2(ClO4)2 is driven by the

combination of 3D interaction and intrinsic XY anisotropy, the increase of TN as a function of

field may thus be driven by an increase of the effective anisotropy and the associated increase

of TBKT. Similarly, we propose that the increase of the specific heat above the 3D ordering

temperature is caused by the field-induced strengthening of a XY anisotropy: In the 2D AF on


a square lattice, 2D topological spin-vortices appear above the Berezinskii-Kosterliz-Trousers

(BKT) transition as the preferable thermodynamic configuration. In applied magnetic field

the vortices unbind above the BKT transition, leading to the increase of the specific heat

above the ordering temperature. The anisotropy crossover thus affects the specific heat in a

manner similar to the observed behavior [32].

Remarkably, the HT phase diagrams are identical for fields parallel and perpendicular to

the square-lattice planes. This suggests that the dominant exchange interactions between

nearest copper spins, J1, in bc-plane are close to the isotropic limit in spin space, in contrast

to with the strong spatial two-dimensionality of Cu(pz)2(ClO4)2.

3.3 Neutron diffraction

3.3.1 Experimental details

To determine the ordered magnetic structure of Cu(pz)2(ClO4)2, we performed neutron

diffraction experiments with powdered and single crystal samples. A powder diffraction

experiment was performed using the cold-neutron powder diffractometer DMC at the Paul

Scherrer Institute (PSI), Villigen, Switzerland. A diffraction pattern was measured below and

above the Neel temperature using energy E = 4.637 meV. For the single crystal diffraction

measurements, we used the cold-neutron three-axis spectrometer RITA2 at PSI, Villigen,

Switzerland. The HT phase diagram was also measured in this experiment. A crystal with

dimensions 7×7×1.5 mm and mass of m = 85 mg was wrapped into aluminum foil, fixed with

wires on a sample holder and aligned with its reciprocal (0, k, l) plane in the horizontal scat-

tering plane of the neutron spectrometer. Data were collected at T = 2.3 K and T = 10 K in

magnetic fields up to µ0H = 13.5 T applied nearly perpendicular to the (0, k, l) plane using an

Oxford cryomagnet. Measurements were performed with the pyrolytic graphite (PG) (002)

Bragg reflection as a monochromator. A cooled Be filter was installed before the analyzer

to suppress higher order neutron contamination for the final energy Ei = 5 meV. We also

used an experimental setup without Be filter, which allowed to use the second order neutrons

from the monochromator with Ei = 20 meV, thus allowing to access to reflections at high

wave-vector transfers.

3.3 Neutron diffraction 45

Figure 3.4: The part of powder neutron diffraction patterns measured at T = 1.5 K and atT = 10 K are shown by blue and red lines, respectively.

Figure 3.5: (a) The scattering intensity at Q = (0, 1, 0) as function of rotation angle, mea-sured at T = 2.3 K and at T = 10 K. The inset (b) shows the neutron scattering observedat Q = (0, 0, 1) at the same temperatures.


3.3.2 Magnetic order parameter

Fig. 3.4 shows the low-angle powder neutron diffraction data obtained at T = 2 K and at

T = 10 K. The low-temperature data features a well-defined peak at 2Θ = 24.8 which is

not present at T=10K. This peak corresponds to Bragg diffraction from a magnetic structure

modulated with either Q = (0, 1, 0) or Q = (0, 0, 1). Due to the small difference between b and

c lattice constants (δ ∼ 0.04A), we can not distinguish these two wave-vectors. The magnetic

Bragg powder peaks are weak, due to the small size of the ordered magnetic moment, and

the powder pattern is insufficient for the determination of the ordered magnetic structure.

Therefore it was necessary to perform a single crystal neutron diffraction experiment.

Fig. 3.5 shows two magnetic peaks measured above and below the transition temperature

at Q = (0, 1, 0) and Q = (0, 0, 1) using a single crystal neutron diffraction with final energy

Ef = 20 meV. This data directly demonstrates the presence of magnetic order below TN.

The magnetic Bragg peak widths are limited by the instrumental resolution, confirming that

the magnetic order is long-range.

The field dependence of the magnetic scattering at Q = (0, 1, 0) measured using final

energy Ef = 5 meV reveals an increase of magnetic intensity as a function of field from

zero to µ0H = 13.5 T as is shown in Fig. 3.6. The magnetic scattering was determined

by subtracting the non-magnetic background determined at T = 10 K. The increase of

magnetic diffraction intensity with field is most probably related to a quenching of quantum

fluctuations by the magnetic field, that simultaneously also leads to the observed increase of

the transition temperature TN. This result is in a good agreement with the specific heat data

indicating enhanced XY anisotropy in the applied magnetic field. The intensity measured

at Q = (0, 1, 0) at T = 10 K as the function of applied field did not reveal any magnetic

scattering, showing that magnetic fields do not lead to field-induced antiferromagnetic order

in the paramagnetic phase. This is also evidence for the absence of off-diagonal terms in the

magnetic susceptibility tensor for fields along the a-axis.

The critical magnetic behavior was studied by measuring the peak intensity of the neutron

scattering at the antiferromagnetic wave vector Q = (0, 1, 0) and Q = (0, 3, 0) as function

of temperature in magnetic field up to µ0H = 13.5 T. Typical scans are shown in Fig. 3.7.

The solid line shows that the increase of the antiferromagnetic intensity in the ordered phase


Figure 3.6: The magnetic peak intensity of neutron scattering at Q = (0, 1, 0) as functionof magnetic field measured at T = 2.3 K. The non-magnetic scattering was estimated frommeasurements at T = 10 K and subtracted from the overall peak intensity. The inset showsthe ordered antiferromagnetic moment as a function of field.

Figure 3.7: (Color) (a) The temperature dependence of the neutron scattering peak intensitymeasured at the antiferromagnetic point Q =(0, 1, 0). The data collected at µ0H = 0 T,µ0H = 2 T µ0H = 6 T and µ0H = 13.5 T are shown by circles, squares, triangles anddiamonds, respectively. The red lines are guides to the eye. The inset (b) represents thepeak intensity at Q = (0, 3, 0) as the function of temperature obtained at zero field. TN wasfound to be the same as for Q = (0, 1, 0).


Figure 3.8: (Color) Neutron scattering peak intensity of Q = (0, 1, 0) as a function of tem-perature and magnetic field, applied perpendicular to the square-lattice planes. The resultsobtained by specific heat measurements in magnetic field applied parallel and perpendicu-lar to the copper planes are shown by squares and circles, respectively. The neutron data,measured for magnetic fields perpendicular to the copper planes, are shown by triangles.

close to TN is clearly steeper for high fields. The HT phase diagram compiled from the

temperature scans is shown in Fig. 3.8, confirming the phase diagram obtained from specific

heat measurements.

3.3.3 Ordered magnetic structure

The symmetry of the ordered magnetic phase was studied by single crystal neutron diffraction.

Group theory was used to restrict the search only to magnetic structures that are allowed

by symmetry. The magnetic Bragg peaks at Q = (0, 1, 0) and Q = (0, 3, 0) indicate that the

magnetic structure breaks the C-centering of the chemical lattice and that Cu(pz)2(ClO4)2

adopts an antiferromagnetic structure for T < TN. Symmetry analysis revealed six basis

vectors which belong to four irreducible representations and are listed in Tab. 4.5 (for details

see Appendix A).

The analysis is complicated by the fact that the single crystal probably consists of two

domains with interchanged b- and c-axis, which are nearly identical in length. A twinning


of the single-crystal in this manner is indicated by the observation of both the Q = (0, 2, 3)

and Q = (0, 3, 2) nuclear Bragg peaks with similar intensity, although Q = (0, 3, 2) is not

allowed for a C-centered lattice.

The experimental data are consistent with both Γ2 and Γ4 irreducible representations

listed in Tab. 4.4 and with two basis vectors−→φ 2 and

−→φ 6 (see Appendix A). The ordered

magnetic structure of Cu(pz)2(ClO4)2 can have magnetic moments aligned antiferromag-

netically either along crystallographic b- or c-axis as is shown in Fig. 3.9(a) and Fig. 3.9(b),

respectively. Due to a small number of observed magnetic reflections and the crystallographic

twinning, our experiment cannot distinguish between these two magnetic structures.

The collinear spin arrangement in bc-plane is consistent with the absence of the Dzyaloshinsky-

Moriya interactions between NN. The spatial arrangement of the ordered magnetic moments

in adjacent square-lattice layers is ferro- and antiferromagnetic along ab- and ac-diagonal,

respectively. This is consistent with the chemical structure of Cu(pz)2(ClO4)2, where the

interlayer interaction pathway along ac-diagonal is shorter than the path along ab. The or-

dered magnetic structure of Cu(pz)2(ClO4)2 with the preferable spin direction in bc-plane is

consistent with recent ESR measurements [120] which showed a presence of a small energy

gap δEig ∼ 0.01 meV at the antiferromagnetic zone center, Q = (0, 1, 0). Most probably this

gap is induced by a small Ising-like anisotropy. The resolution of the neutron spectrometer,

which we used to measure the spin gap at Q = (0, 0, 1), was equal to δEn ∼ 0.1 meV and

therefore it did not allow us to observe δEig. Due to a small energy scale of δEig compared to

the dominant exchange interactions between spins in the system, we neglect the presence of a

possible Ising-like anisotropy in further discussions of the spin dynamics in Cu(pz)2(ClO4)2.

The measured Bragg peak intensities were corrected for the instrumental resolution func-

tion. The value of the ordered magnetic moment was obtained from a minimization of

δ = |Rcalc − Rexp|, where Rexp is the measured ratio of the magnetic Bragg peak intensity

to the nuclear Bragg peak intensity, Rcalc = |F(Q)magn|2/|F(Q)nucl|2, F(Q)magn and F(Q)nucl

are the magnetic and nuclear form factors, respectively. The fit was performed for two mag-

netic peaks observed at Q = (0, 1, 0) and Q = (0, 3, 0) and two nuclear peaks measured at

Q = (0, 2, 4) and Q = (0, 0, 6). The value of the ordered magnetic moment in zero field,

which was obtained in this way, is m0 = 0.47(5) µB. The comparison of Rcalc and Rexp for


Figure 3.9: Two possible magnetic structures of Cu(pz)2(ClO4)2 belonging to the Γ2 andΓ4 irreducible representations (see Tab. 4.4), are shown in (a) and (b), respectively. Twoadjacent square-lattice Cu2+ layers, separated by a (0.5,0.5,0) lattice unit translation, aredepicted by open and filled arrows. Cu2+ − Cu2+ interlayer interaction pathway along acdiagonal corresponds to the vertical distance between filled and open symbols in (a) and (b).

two magnetic and two nuclear Bragg peaks is presented in the Tab. 3.1. The calculated value

|F (0,1,0)|2|F (0,2,4)|2

|F (0,1,0)|2|F (0,0,6)|2

|F (0,3,0)|2|F (0,2,4)|2

|F (0,3,0)|2|F (0,0,6)|2

Rexp × 10−4 3.72(7) 5.13(12) 1.89(18) 2.60(25)Rcalc × 10−4 4.35 4.57 2.16 2.26

Table 3.1: The measured and the calculated ratios of squared magnetic to nuclear structurefactors for different Bragg peaks. The calculated values were obtained from a minimizationof δ = |Rcalc − Rexp| and correspond to the ordered magnetic moment m0 = 0.47µB.

of the ordered magnetic moment is smaller than the free-ion magnetic moment, indicating

the presence of strong quantum fluctuations in the magnetic ground state of Cu(pz)2(ClO4)2.

The inset in Fig. 3.6 shows the increase of the ordered antiferromagnetic moment from

m0 = 0.47(5)µB in zero field to m0 = 0.93(5) µB in µ0H = 13.5 T. This is direct evidence for

the suppression of quantum fluctuations by the applied magnetic field due to induced XY

anisotropy as suggested by our specific heat measurements.


3.4 Neutron spectroscopy

3.4.1 Experimental details

We measured the spin dynamics in the antiferromagnetically ordered phase of Cu(pz)2(ClO4)2

in zero and applied magnetic fields using neutron spectroscopy. In total, three experiments

were performed. Two experiments were performed using the cold-neutron three-axis spec-

trometer PANDA at FRM-2, Garching, Germany. One experiment was performed using the

cold-neutron triple-axis spectrometers IN14 at ILL at Grenoble, France.

The zero field studies using PANDA were performed on two single crystals with a total

mass of m = 1 g which were wrapped into aluminum foil, fixed on a sample holder with wires

and co-aligned in an array with a final mosaic spread of 1. The reciprocal (0, k, l) plane of

the sample was aligned in the horizontal scattering plane of the neutron spectrometer. These

measurements were performed in zero magnetic field and at temperature T = 1.42 K using

a 4He cryostat generally referred to as an Orange cryostat. The final energy was either set

to Ef = 4.66 meV or Ef = 2.81 meV using a PG(002) analyzer. Data were collected using a

PG(002) monochromator and a cooled Be filter installed before the analyzer.

The neutron experiments in applied magnetic fields were performed using an array of

deuterated Cu(pz)2(ClO4)2 single crystals with a total mass up to 2g co-aligned with a mosaic

of 0.5. The sample was aligned with its reciprocal (0, k, l) plane in the horizontal scattering

plane. A cryomagnet allowed the application of vertical magnetic fields up to µBH = 14.9 T.

The magnetic field was thus nearly perpendicular to the square-lattice planes. To probe

the ground state as a function of magnetic field, we used a dilution refrigerator, reaching

temperatures of the order of 50 mK in each of the two experiments. The measurements were

performed using a fixed final energy Ef = 4.66 meV and Ef = 2.98 meV for the PANDA

and IN14 experiments, respectively, obtained via the (002) Bragg reflection from a pyrolithic

graphite (PG) monochromator, a focused analyzer and a cooled Be filter before the analyzer.

3.4.2 Spin dynamics in zero field

Constant energy scans were performed near the antiferromagnetic zone centers Q = (0, 0, 1)

and Q = (0,−1, 0) for energy transfer ∆E in the range from ∆E = 0.5 meV to ∆E = 3 meV


Figure 3.10: (Color online) A series of constant energy scans performed along the (0, k,1) and (0, -1, l) directions at different energy transfers ∆E in zero magnetic field and atT = 1.42 K. Please note the changing scale of the vertical axis for the different scans. Thesolid lines correspond to a convolution of two Gaussians with the resolution function.

and are shown in Fig. 3.10. The observed magnetic peaks are resolution limited, indicating

that these magnetic excitations are long-lived magnons associated with a long-range ordered

magnetic structure.

Constant wave-vector scans were performed at the antiferromagnetic zone centers in the

energy transfer range from ∆E = 0 meV to ∆E = 0.7 meV (Fig. 3.11(a), b). These scans

reveal a magnetic mode which is gapped and has a finite energy Ezc = 0.201(8) meV at

the antiferromagnetic zone center. The energy gap at the antiferromagnetic zone center is

attributed to the presence of a small XY anisotropy in the nearest-neighbor two-ion exchange

interactions, because a single-ion anisotropy of type D(Sz)2 is not allowed for S = 1/2.

Constant wave-vector scans away from the antiferromagnetic zone center carried out


at higher energies are shown in Fig. 3.11(c). The energies of the magnetic excitation at

the symmetrically identical antiferromagnetic zone boundary points Qzb1 = (0, 0.5, 1) and

Qzb2 = (0,−0.5, 1) are equal to Ezb1 = 3.629(6) meV and Ezb2 = 3.599(13) meV, respec-

tively. The peaks observed in the constant wave-vector scans at Qzb1 and Qzb2 are resolution

limited. This experimental observation together with the identity of the values Ezb1 and Ezb2

confirms the NN interactions in bc-plane are identical along the square-lattice directions. In

case of different strengths for the NN interactions in the bc-plane, a broadening of the mag-

netic excitations at Qzb1 and Qzb2 would be observed or every energy scan would show two

peaks. The observed one-magnon mode can thus qualitatively be explained by the 2D spin

Hamiltonian with NN exchange interactions:

HNN =∑〈i,j〉Jz

1Szi · Sz

j + Jxy1 (Sx

i · Sxj + Sy

i · Syj ), (3.4)

where 〈i, j〉 indicates the sum over NN in the bc-plane, Jz1 and Jxy

1 are z-, xy-components of

the NN interactions, respectively.

We also studied the spin-wave dynamics along the antiferromagnetic zone boundary by

performing constant wave-vector scans along the Q = (0, 0.5, l) direction from l = 1 to

l = 2. Typical data are shown in Fig. 3.13(a-c) and the observed zone boundary dispersion

is shown in Fig. 3.13(d). The onset of the scattering at Q = (0, 0.5, 1.5) is reduced by

10.7(4)% in energy compared to Q = (0, 0.5, 1). The decrease of the resonant mode energy

at Q = (0, 0.5, 1.5) results from a resonating valence bond quantum fluctuations between NN

spins [13, 110].

In order to subtract a nonmagnetic contribution from the background in the energy scan

at Q = (0, 0.5, 1.5) we performed measurements with the sample turned away from magnetic

scattering. The background-subtracted data are shown in Fig. 3.13(a). The width of the

scattering peak as a function of energy at Q = (0, 0.5, 1.5) is clearly broader than the in-

strumental resolution. This implies the existence of a magnetic continuum scattering in this

region of the antiferromagnetic zone boundary. This non-trivial magnetic continuum and the

dispersion at the zone-boundary result from quantum fluctuations in Cu(pz)2(ClO4)2. The

observed dispersion at the zone boundary is slightly larger than expected from series expan-


Figure 3.11: (Color) The constant Q-scans collected at small energy transfer show the energyof a gapped spin-wave at the antiferromagnetic zone center performed at Q = (0, 0, 1) andat Q = (0,−1, 0) are presented in (a) and (b), respectively. (c) Constant Q-scans performedclose to the antiferromagnetic zone boundary at high energy transfer show the dispersion ofthe spin-wave. The measurements were performed in zero magnetic field and at T = 1.42 K.The solid lines represent the convolution of a Gaussian with the resolution function.

sion calculations and Quantum Monte Carlo simulations for 2D Heisenberg square-lattice

AF with NN interactions [13, 11]. Our collaborators developed series expansion calculations

for the J1 − J2 model [124, 125, 126] and their result shows that the addition of J2 leads

to an enhancement of the zone boundary dispersion (see section 3.4.3 for details). Thus for

Cu(pz)2(ClO4)2 the increased zone-boundary dispersion is attributed to a small AF NNN

interaction of the order of J2 ∼ 0.02− 0.05J1.

We performed the spin-wave calculations in linear approximation based on the following

Hamiltonian:

H = HNN + J2

∑〈i,k〉

Si · Sk, (3.5)

where HNN is given by eq. 3.4, 〈i, k〉 is the sum over NNN in the bc-plane and J2 is the NNN

exchange interaction. The linear spin wave theory (for details see Appendix B) yields two


spin-wave modes with the dispersion

hωq =√A2q −B2

q , (3.6)

where

Aq = 4SJxy1 + S(Jxy1 − Jz1 )(cos(qb) + cos(qc))− 4SJ2 + 4SJ2(cos(qb) · cos(qc)),

Bq = S(Jxy1 + Jz1 )(cos(qb) + cos(qc)). (3.7)

This implies that the exchange anisotropy mostly affects the magnon energy close to the

antiferromagnetic zone center, while the zone boundary energy remains nearly unaffected by

the exchange anisotropy. In the 2D S = 1/2 AF, the energy of a classical (large-S) spin-

wave mode is renormalized due to quantum fluctuations with the best theoretically predicted

renormalization factor Zc = 1.18 [15, 12]. Therefore the energy at the antiferromagnetic zone

boundary is equal to Ezb = 2(ZcJz1 − JR2 ), where JR2 is the renormalized NNN interaction.

The calculated xy-component of NN and NNN exchange interactions are equal to Jxy1 =

1.563(13) meV and J2 ' 0.02Jxy1 (see section 3.4.3), respectively. According to the linear

spin wave theory E2zc = 8Jxy1 (Jxy1 − Jz1 ) and thus Jz1 = 0.9979(2)Jxy1 .

The values of the xy- and z-components of the NN interaction obtained from our neutron

measurements are in a good agreement with the result of magnetic susceptibility measure-

ments [121], which yielded Jxy1 = 1.507(26) meV and Jz1 = 0.9954 Jxy

1 . The small XY

anisotropy indicates that the dominant exchange interaction between nearest copper ions

in the bc-plane in Cu(pz)2(ClO4)2, while spatially very anisotropic, is close to the isotropic

limit in spin space, explaining the strong similarity of the HT phase diagrams measured in

magnetic fields applied parallel and perpendicular to copper square-lattice (Fig. 3.8).

The inelastic-scattering data were fitted with the Gaussian instrumental resolution func-

tion convoluted numerically with the model Hamiltonian (3.5). The result of the fits is shown

by the red lines in Fig. 3.10 and Fig. 3.11, and provides a good description of the observed

spin waves. The color plot of the neutron scattering intensity, which is shown in Fig. 3.12,

summarizes the observed magnetic excitations in both crystallographic directions. The black

lines display the result of the linear spin wave theory, showing that the observed dispersive


Figure 3.12: (Color) Color plot of the scattering intensity, showing the dispersion along theQ = (0,−1, l) and Q = (0, k, 1) directions measured at zero field, presented in the left andright panels, respectively. The color plots were obtained by merging a total of five andthirteen constant wave-vector scans. The solid line represents the dispersion computed fromlinear spin wave theory using Jxy

1 = 1.563 meV, Jz1 = 0.9979Jxy

1 and NNN exchange equal toJ2 = 0.02Jxy

1 as described in the text.

excitation is well characterized by the Hamiltonian (3.5).

The measured spin wave dispersion is similar to that observed in another 2D square-

lattice antiferromagnetic material, namely copper deuteroformate tetradeuterate (CFTD),

where the exchange interaction strength is equal to J = 6.3(3) meV and the energy gap of

E = 0.38(2)meV is present at the antiferromagnetic zone center. However, the energy gap in

CFTD is induced by the presence of small antisymmetric Dzyaloshinsky-Moriya interaction

D = 0.0051(5) meV between NN [105, 106]. Another example with comparable properties is

K2V3O8 with 2D NN exchange strength J = 1.08(3) meV and small energy gap at antiferro-

magnetic point equal to E = 0.072(9) meV [109], which is described by Dzyaloshinsky-Moriya

and easy-axis anisotropies [122]. In contrast, Dzyaloshinsky-Moriya interactions between NN

in Cu(pz)2(ClO4)2 are forbidden by symmetry (see Sec. 4.4) and the energy gap at the anti-


Figure 3.13: (Color) (a) The energy scan at Q = (0, 0.5, 1.5) with the background subtractedas explained in the text. The energy scans performed at wave-vectors Q = (0, 0.5, 1.7) andQ = (0, 0.5, 2) are shown in (b) and (c), respectively. The red curves are fits of a Gaussianfunction convoluted with the resolution function. (d) The antiferromagnetic zone boundarydispersion measured in zero magnetic field and at T = 1.42 K.


ferromagnetic zone center is generated by small XY exchange anisotropy.

3.4.3 Spin dynamics in applied magnetic fields

A color plot of the normalized neutron scattering spectra at the antiferromagnetic zone

boundary is shown in Fig. 3.14 for zero applied field and µ0H = 14.9 T. These data were

measured using the IN14 spectrometer. At zero field, the onset of scattering at reciprocal

wave-vector Q = (0, 0.5, 1.5) is reduced by 11.5(7)% in energy compared to Q = (0, 0.5, 2),

at odds with spin-wave theory. However, this result is consistent with our independent

investigation of Cu(pz)2(ClO4)2 using PANDA spectrometer (see Sec. 3.4.2). The observed

zone-boundary dispersion is larger than expected from Quantum Monte Carlo simulations

and series expansion calculations for the Heisenberg square-lattice AF with nearest-neighbor

interactions, which predict a zone-boundary dispersion of 8% to 10% [13, 11]. Our observation

is opposite to what has been observed in the high-Tc material La2CuO4 where the energy

at Q = (0, 0.5, 1.5) is higher than at Q = (0, 0.5, 2) [111]. The latter has been attributed to

ring-exchanges arising from finite-U/t [123].

The single-magnon energies were computed for different relative strengths J2/J1 for zero

and µ0H = 14.9 T by our collaborators using series expansion calculations. The comparison

of the numerical calculations shown in Fig. 3.15 and our experimental data (see Fig. 3.14)

indicates that the zone-boundary dispersion is increased due to a small AF next-nearest

neighbor interaction of the order of J2 ∼ 0.02− 0.05J1.

According to spin wave theory, the energy at Q = (0, 0.5, 2) is equal to E(0,0.5,2) = 2(J1−

J2) at all fields. Next-nearest neighbor interactions thus lead to a smaller zone-boundary

energy, and thus to a smaller effective nearest-neighbor exchange J . Using J = J1 − J2

and J = 1.53(8) meV determined from susceptibility measurements [119, 127], the field-

induced change of E(0,0.5,2) is obtained from the field dependence of the renormalization factor

Zc = E(0,0.5,2)/2J , yielding Zc = 1.19(2) at zero field in excellent agreement the predicted

value of Zc = 1.18 [13].

Fig. 3.16(a) shows that the spin excitation at Q = (0, 0.5, 1.5) is considerably broader

than experimental resolution, and that neutron scattering extends to about E = 3.9 meV.

This represents clear experimental evidence of a magnetic continuum at Q = (0, 0.5, 1.5)


Figure 3.14: Zone boundary spin dispersion at zero field and µ0H ∼ J . Color plot of thenormalized scattering intensity I(Q, ω) at T = 80 mK, showing the dispersion from Q =(0, 0.5, 1.5) to Q = (0, 0.5, 2) at zero field and µ0H = 14.9 T under otherwise identicalconditions. Both panels present smoothed data obtained by performing six constant Q-scansfrom Q = (0, 0.5, 2) to Q = (0, 0.5, 1.5) with a 0.05 meV energy step.


Figure 3.15: Series expansion calculations of the zone-boundary dispersion. Theoreticalmagnon dispersion for different values of the next-nearest neighbor exchange interaction J2

in zero magnetic field (left panel) and in a µ0H = 14.9 T magnetic field (right panel). Thevalue of the nearest neighbor interaction J has been normalized to fit the experimental data.The normalization constants are shown in the legend [128].

of the square-lattice AF that is not expected from spin-wave theory. The continuum is

clearly stronger than in copper deuteroformate tetradeuterate (CFDT) where next-nearest

neighbor exchange are absent [110], suggesting that next-nearest neighbor exchange enhances

continuum excitations in the 2D S = 1/2 square-lattice AF. The extended continuum in

Cu(pz)2(ClO4)2 is also in contrast to well-defined excitations observed in La2CuO4 near

Q = (0, 0.5, 1.5), suggesting that ring exchange has the opposite effect on the continuum to

next-nearest neighbor interactions. Further, Fig. 3.14 also provides evidence of an energy

gap around Q = (0, 0.5, 2) that separates a main mode and a much weaker continuum above

4.2 meV, as predicted by Ho et al. [14].

Fig. 3.16(c) shows that magnetic fields strongly affects the quantum fluctuations at the

zone boundary: The energy of the magnetic excitation at Q = (0, 0.5, 2) decreases much faster

with field than that at Q = (0, 0.5, 1.5). The zone-boundary dispersion at µ0H = 14.9 T

is inverted from what it was at zero field. The series expansion calculations show that,

with the application of a magnetic field, the inversion of the zone boundary dispersion only

occurs for sufficiently small J2/J1. A five percent J2/J1 no longer shows the reversal of the

dispersion that is seen in our experiments. This implies that even a relatively small next-


Figure 3.16: Field dependence of zone-boundary excitations. Energy scans at Q = (0, 0.5, 1.5)and Q = (0, 0.5, 2) are shown in (a) and (b), respectively. The data was measured at T =80 mK. The black solid line in (b) represents linear spin-wave excitations convoluted with theresolution function. Apart from the continuum region, all the experimentally measured peaksare resolution limited. The field dependent onset of magnetic scattering at Q = (0, 0.5, 1.5)and Q = (0, 0.5, 2) as a function of energy is shown in (c) by red triangles and black circlesrespectively. The dashed red and black lines represent the estimate from the series expansioncalculations with J1 = 1.54 meV and J2 = 0.02J1.


nearest neighbor interaction is effective in enhancing the continuum of excitations, consistent

with the estimate that J2/J1 = 0.02. The quantum renormalization factor Zc decreases

rapidly from Zc = 1.19(2) at zero field and approaches Zc = 0.99(2) at µ0H = 14.9 T, also

consistent with the calculations.

It is known that the wave-vector dependence of the magnetic excitations at the zone

boundary is a result of resonating valence bond quantum fluctuations between nearest-

neighbor spins that reduce the energy of resonant excitations near Q = (0, 0.5, 1.5) below

what is expected from renormalized spin-wave theory [13, 110]. The field-induced reversal

of the zone-boundary dispersion reveals that magnetic fields of the order of H ∼ J strongly

couple to these local quantum fluctuations. This dispersion, that is not expected from spin-

wave theory, demonstrates the presence of local quantum fluctuations in the 2D S = 1/2

square-lattice AF even for fields of the order of H ∼ J . The quantum origin of the disper-

sion reversal is also confirmed by the series expansion calculations. The observed dispersion

suggests that the energy of the magnetic resonance at Q = (0, 0.5, 1.5) is raised by 4.5(7)%

above that of the renormalized spin-wave theory, providing direct evidence of a field-tuned

resonating valence bond fluctuations.

The excitation spectrum at µ0H = 12 T features two well defined magnetic modes of

excitations (Fig. 3.17). The spectrum consists of a Goldstone mode, that indicates unbroken

rotational spin symmetry in the plane perpendicular to the magnetic field, and a gapped

mode. The field dependence of the mass of the gapped mode at the AF zone center, shown

in Fig. 3.18, is linear, which is consistent with the theoretical arguments discussed in [129].

The finite gap at zero field, obtained by the linear fit to the experimental data, implies the

existence of a small XY exchange anisotropy in the system and it is described by a small

anisotropic NN exchange interaction Jz1 = 0.9979(2)Jxy

1 (see sec. 3.4.2). We also analyzed

the spectral weight of the gapped excitation at Q = (0, 0, 1). Our calculations completely

describe the observed field dependence of the peak intensities of the gapped mode at the AF

zone center (Fig. 3.18(c)).

Fig. 3.17 shows the observed dispersion from Q = (0, 1, 0) to Q = (0, 1,−0.5) compared

to linear spin-wave theory. The linear spin-wave theory calculations in a magnetic field are


Figure 3.17: Spin dispersion for µ0H ∼ J . (a) The spin wave dispersion in Cu(pz)2(ClO4)2

measured at µ0H = 12 T and T = 80 mK. The red line represents the spin-wave dispersionwith J1 = 1.54 meV and J2 = 0.02J1. (b,c) Energy scans at Q = (0, 1−0.4), Q = (0, 1−0.3)provide evidence of a mode with a field-induced gap. (d) Constant-energy scattering at1 meV energy transfer provide evidence for the Goldstone mode. The solid line in (b)-(d)corresponds to a convolution of a Gaussian with the resolution function, demonstrating thatthe excitations are resolution limited.


Figure 3.18: Field dependence of the zone-center excitation. Energy scans performed atQ = (0, 0, 1) at µ0H = 2 T, µ0H = 6 T, µ0H = 10 T (a) and the gap energy at the AF pointplotted as the function of field (b). The data was measured at T = 80 mK. The solid linesin (a) are the fits of a Gaussian function convoluted with the resolution function. The filledcircles in (b) represent the experimental data. The solid line in (b) is the linear fit of the gapenergy. Black circles in inset (c) show the measured intensities as function of magnetic fieldand the curve is the scattering intensity calculated using linear spin-wave theory.

performed for the following Hamiltonian:

H = HNN + J2

∑<i,k>

Si · Sk − µ0H∑i

Sai , (3.8)

where HNN is defined in eq. 3.4, 〈i, k〉 next nearest neighbor pairs and Sa is a spin component

perpendicular to the bc plane. The renormalized spin wave theory with Jeff = 1.54(1)meV

and Zc = 1.03(1) for µ0H = 12 T describes qualitatively the observed dispersion, as shown

in Fig 3.17(a), but there are clear differences that may reflect the importance of magnon-

magnon interactions [35, 36]. Possibly, better agreement could be obtained by including

higher-order terms in the spin-wave calculation.

3.5 Summary 65

3.5 Summary

In summary, our comprehensive experimental investigation of Cu(pz)2(ClO4)2 shows that this

material is the first weakly frustrated 2D S = 1/2 AF on a square lattice with an absence of

Dzyaloshinsky-Moriya interaction between NN. We show that the zone-boundary dispersion

and the zone-boundary continuum of the 2D S = 1/2 square-lattice AF are enhanced by

NNN interactions, while the spin-wave energies merely experience a small renormalization.

Magnetic fields of the order of H ∼ J lead to a qualitative change of the quantum fluctu-

ations that suppress the continuum of excitations and renormalize the spin-wave velocity,

but without suppressing the zone-boundary dispersion that arises from non-trivial quantum

fluctuations. In fact, we find that the zone-boundary dispersion is inverted compared to

zero field, providing direct evidence of a field-induced change of resonating valence bond

fluctuations.


Chapter 4

Magnetism in quasi-one-dimensional

S = 1 antiferromagnet NiCl2 · 4SC(NH2)2

4.1 Introduction

Organo-metallic material dichlorotetrakisthiourea-nickel (II) NiCl2 · 4SC(NH2)2, also known

as DTN, consists of Ni2+ ions located in the corners and the centers of the chemical tetragonal

body-centered unit cell [130]. The exchange interaction between neighbor spins S = 1 are

equal to Jc ' 1.64(3) K and Jab ' 0.16(1) K along the crystallographic c-axis and in the

ab-plane, respectively [131]. Because of strongly interacting spins along the tetragonal c-axis,

DTN can be viewed as a system of weakly coupled S = 1 chains.

Due to a presence of a large single-ion XY-anisotropy D ∼ 8 K, a singlet ground state

Sz = 0 at zero magnetic field is separated by an energy gap from the excited doublet Sz = ±1

[131, 132, 133] and the correlations between spins remain short-range in the zero-temperature

limit. At zero temperature, a magnetic field applied along the crystallographic c-axis splits

the excited doublet and mixes its lower member with the ground state at the first critical field,

Hc1, leading to 3D long-range antiferromagnetic order with preserved U(1) symmetry. The

predicted order of the antiferromagnetic phase has transverse spin components, perpendicular

to the applied field and aligned antiparallel between nearest neighbors in the crystallographic

ab-plane. The field-induced QPT from a paramagnetic to a 3D antiferromagnetic phase can

be treated as BEC of magnons in DTN.

68 Magnetism in S = 1 quasi-1D antiferromagnet NiCl2 · 4SC(NH2)2

The field-induced magnetic order and the field-temperature phase diagram of DTN had

been studied previously by several techniques. Magnetization measurements were performed

in magnetic fields up to µ0H = 18 T at different temperatures [132, 133]. The gyromagnetic

Lande factor is equal to g‖ = 2.26, g⊥ = 2.34 parallel and perpendicular to the tetragonal

axis, respectively. The transitions from the quantum paramagnetic to the 3D long-range

antiferromagnetic state and from antiferromagnetic to a fully polarized ferromagnetic state

occur in the zero-temperature limit at the first and the second critical fields, which are

equal to µ0Hc1 = 2.1 T and µ0Hc2 = 12.6 T, respectively [134]. Based on the temperature

dependence of the critical fields, the critical exponent α of this QPT was determined from the

power law Hc(T) − Hc(0) ∼ Tα using specific heat and magnetocaloric effect measurements

by Zapf and collaborators [131]. Using the extrapolation method [69], it was found to be

equal to α = 1.5 as predicted for BEC. The QCP corresponding to Hc1 thus belongs to the

universality class of the dilute Bose gas, with an effective dimensionality d = 2.

The magnetostriction effect was measured in magnetic fields applied parallel (H ‖ c) and

perpendicular (H ⊥ c) to the tetragonal axis of DTN [134, 135]. A field-induced contraction

of the c-axis was observed in magnetic fields Hc1 < H < 6 T for H ‖ c. Higher magnetic

fields, 6 T < H < Hc2, lead to an expansion of the tetragonal axis. This can be understood

as following: The magnetic system increases its antiferromagnetic exchange interaction Jc

by contraction of the c-axis minimizing the magnetic energy. Due to a field-induced canting

of magnetic moments towards the field direction, the NN are increasingly aligned ferromag-

netically above µ0H ∼ 6 T. In order to minimize the magnetic energy associated with the

ferromagnetic coupling, the system thus expands the tetragonal axis in order to reduce the

antiferromagnetic exchange. These magnetostriction measurements also showed that the an-

tiferromagnetic phase between Hc1 and Hc2 shrinks as a function of angle χ between the

magnetic field and the c-axis, collapsing near the critical angle χ = 55. No magnetic order

was observed for magnetic fields applied in the ab-plane.

Systematic studies of the magnetic excitations in DTN using electron spin resonance

(ESR) technique were performed by Zvyagin et al [136, 137]. The observed modes are shown

as a function of applied field in Fig. 4.1. At T = 1.6 K, ESR measurements revealed the

modes A and C, whose energies extrapolate to a low energies towards ∼ 7.7 T. Upon

4.1 Introduction 69

Figure 4.1: The frequency-field dependence of ESR in DTN measured at T = 1.4 K is shownby open circles and at T = 0.45 K are shown by squares and stars (Ref. [137]). Grey circlesrepresents the data obtained at T = 1.6 K. The dashed red and solid cyan lines show resultsof calculations for the simplest axially symmetric two-sublattice antiferromagnetic model andresults of model calculations assuming small Dzyaloshinsky-Moriya interactions, respectively.The figure is taken from the original publication by Zvyagin et al [136].

cooling, these two modes develop into two modes K and L that are separated by a finite

energy. The observation of the gapped mode L and the splitting of mode K in applied

magnetic fields 6 T < H < 10 T indicates the presence of additional interactions that are

not included in the simple axially symmetric two-sublattice antiferromagnetic model. The

authors explained their experimental data by Dzyaloshinsky-Moriya interactions. However,

such the interactions are forbidden by the symmetry of DTN and thus cannot explain the

observed gapped modes.

In this chapter, we present the results of neutron diffraction and neutron spectroscopy

measurements performed on single crystals of DTN. We determined the ordered magnetic

structure in the field-induced AF phase, which is collinear antiferromagnetic order between

nearest-neighbors perpendicular to the applied field. The value of the ordered magnetic

moment as a function of magnetic field and temperature is estimated from neutron diffrac-

tion. We also determine the magnetic phase diagram as a function of magnetic field and

temperature as well as a function of the angle between applied field and the tetragonal axis.

Using neutron spectroscopy, the energy of the excited doublet at the antiferromagnetic

zone center is investigated as a function of magnetic field. We demonstrate that DNT un-


dergoes a BEC of magnons at the first critical field Hc1. In order to determine the spin

Hamiltonian including the strength of the NNN exchange interaction between corner and

center in the chemical unit cell, we measured the one-magnon dispersion in the fully magne-

tized phase of DTN. The excitation spectrum deep in the antiferromagnetic phase is complex

and gives evidence of magnetic interactions not present in the zero-field or high-field limit.

We interpret this as a consequence of a coupling of the magnetic structure to the chemical

lattice that leads to additional interactions in the antiferromagnetic phase.

4.2 Experimental details

The ordered magnetic structure and magnetic field-temperature (HT) phase diagram with

magnetic fields applied parallel to the crystallographic c-axis were studied using the thermal

neutron two-axis diffractometer D23 located at ILL, Grenoble, France. A single crystal of

deuterated DTN with dimensions 5× 5× 7 mm was aligned with its reciprocal (h, k, 0) plane

in the horizontal scattering plane of the neutron diffractometer and loaded into a dilution

refrigerator. Magnetic fields up to µ0H = 12 T were applied along the tetragonal axis. Data

were collected using neutrons with a wave length λ = 1.279 A selected by a pyrolytic graphite

(PG) (002) Bragg reflection. Using a lifting detector which can cover a symmetric angular

range ±30, it was possible to access Bragg reflections located away of the (h, k, 0) plane. In

order to determine the field-induced magnetic order, a total of 124 nuclear and 23 magnetic

Bragg peaks were measured at µ0H = 6 T.

The spin dynamics in the fully magnetized phase of DTN was investigated at the Swiss

Spallation Neutron Source, using the cold-neutron three-axis spectrometer RITA2. The

sample was mounted in the same way as in the D23 experiment and loaded into dilution insert.

To reach the fully magnetized phases, we used a 15 T split-coil magnet and performed the

measurements at µ0H = 13.3 T. The crystal was aligned with its reciprocal (h, k, 0) plane in

the horizontal scattering plane of the spectrometer and the magnetic field was applied along

the c-axis. The dispersion of the low-lying excitation was mapped out through constant

wave-vector scans using neutrons scattered and focused by PG(002) monochromator final

wave length of Ef = 4.7 meV.

4.2 Experimental details 71

Figure 4.2: (a) Three-dimensional view of the

crystal structure of DTN. The nickel and chlo-

rine atoms are shown as green and cherry

spheres, respectively. All other atoms are not

shown for a simplicity. (b) Single crystals of

NiCl2 · 4SC(NH2)2

The dependence of the magnetic phase

diagram on the angle between applied field

and the crystallographic c-axis was mea-

sured using the two-axis neutron diffrac-

tometer E4 at HMI, Berlin, Germany. Four

single crystals of deuterated DTN with di-

mensions 5 × 5 × 10 mm were co-aligned

with a mosaic spread less than 1. The sam-

ple was loaded into a horizontal field mag-

net and aligned with its reciprocal (h, h, l)

plane in the horizontal scattering plane of

the instrument. PG crystals bent verti-

cally were used as the focusing monochro-

mator for the wave length lambda = 2.44 A.

Scattered intensities were observed using a

two-dimensional position-sensitive detector.

The experiment was performed using a di-

lution refrigerator, at temperatures below

T = 0.8 K.

The wave-vector and field dependence of

the spin doublet was studied using FLEX

in magnetic field up to µ0H = 6 T. We used the same sample mount with four co-aligned

crystals, a horizontal field magnet and a dilution refrigerator as in the E4 diffraction ex-

periment. FLEX is a triple-axis spectrometer for cold neutrons located at HMI, Berlin,

Germany. PG(002) reflections were used as monochromator and neutrons with final wave

length of λ = 5.236 A or λ = 4.833 A were detected. A collimation of 60’ located after the

monochromator and a cooled beryllium filter installed before the analyzer were used to sup-

press the beam divergence and higher order neutron contamination. The sample was loaded

into a horizontal field magnet and aligned with its reciprocal (h, h, l) plane in the horizontal

scattering plane of the instrument.


In order to illustrate a field-induced BEC of magnons and measure the magnetic phase

diagram as a function of the angle between an applied field and the crystallographic c-axis,

we had to use a horizontal field magnet. This necessity was imposed by the following: DTN

undergoes a field-induced order only in a field applied close to the tetragonal axis and the

magnetic ordering vector in the antiferromagnetic phase is equal to k = (0.5, 0.5, 0.5) (see

Sec. 4.3.1). This implies that the scattering plane of the neutron spectrometer has to be

located close to the direction of the applied magnetic field. Since scattered neutrons are

detected mainly in the horizontal plane of the instrument, the magnetic field thus has to be

applied in the horizontal direction as well. A disadvantage of any horizontal field magnet

is the limited internal space transparent for neutrons. For this reason, the E4 and FLEX

experiments were accompanied by technical difficulties associated with a limited access to

the reciprocal space of the sample.

4.3 Results and discussions

4.3.1 Ordered magnetic structure

DTN crystallizes in the tetragonal I4 space group (#79) with the lattice parameters a =

9.558 A and c = 8.981 A. Nickel ions occupy 2a Wykoff positions belonging to the corners

and the center of the chemical body-centered tetragonal unit cell, as shown in Fig. 4.2. The

Laue class and the point group of DTN are 4/m and 4, respectively. The space group I4 is

non-centrosymmetric and has a lattice centering vector C = (0.5, 0.5, 0.5). For the body-

centered space group I4, there is a general reflection condition for Bragg peaks Q = (h, k, l),

which is h + k + l = 2n.

We observed magnetic Bragg peaks at Q = (0.5, 0.5, 0.5) ± (h k l), where h, k and l

are even numbers. The magnetic ordering vector is therefore k = (0.5, 0.5, 0.5). A few

of these peaks are shown in Fig. 4.3. The magnetic ordering vector is invariant under two

symmetry operations: the identity (1) and a two-fold rotation around the tetragonal axis

(2c). Therefore, there are two one-dimensional irreducible representations whose characters

are summarized in the character table shown in Tab. 4.1. The decomposition equation for

the magnetic representation is Γmag = 1Γ1 + 2Γ2.

4.3 Results and discussions 73

1 2cΓ1 1 1Γ2 1 -1

Table 4.1: The character table and the irreducible representations obtained from grouptheory analysis for the tetragonal space group I4 (#79) and the magnetic ordering vectork = (0.5, 0.5, 0.5).

Ni1 Ni2Γ1

−→φ 1 (0 0 1) (0 0 -1)

Γ2−→φ 2 (1 0 0) (-1 0 0)−→φ 3 (0 1 0) (0 -1 0)

Table 4.2: Three basis vectors calculated for two nickel positions in primitive unit cell.

The three basis vectors presented in Tab. 4.2 are calculated for the two nickel positions

in a unit cell using the projection operator method acting on a trial vector φα

Ψλαν =

∑gεGk

Dλ∗ν (g)

∑i

δi,giRgφαdet(Rg),

where Ψλαν is the basis vector projected from the λth row of the νth irreducible representation,

Dλ∗ν (g) is the λth row of the matrix representative of the νth irreducible representation for

symmetry operation g, i denotes the atomic position and Rg is the rotational part of the

symmetry operation g. The star of the propagation vector k is formed by two vectors:

• k1 = (0.5, 0.5, 0.5) - symmetry operation (x, y, z)

• k2 = (0.5, −0.5, 0.5) - symmetry operation (-y, x, z)

Magnetic Bragg peaks were measured in a magnetic field µ0H = 6 T parallel to the

c-axis and at temperatures T < 60 mK. Gaussian function describes very well shapes of

observed reflections (see Fig. 4.3) and full widths at half maximum of Gaussians fitted to

the data are almost the same as the resolution limit (FWHM ' 0.57), indicating that the

system is magnetically long-range ordered. Magnetic scattering was detected neither below

µ0H ∼ 2 T nor above T ∼ 1.4 K. We determined the crystal and magnetic structures of the

deuterated single crystal of DTN using the refinement program FullProf. Details of the fit of

the nuclear structure are given in the Appendix C. The list of the observed and calculated


Q |Fobs(Q)|2 |Fcalc(Q)|2(0.5 0.5 0.5) 33.34 (2.68) 9.53

(-0.5 1.5 -0.5) 8.53 (6.79) 5.80(1.5 0.5 -0.5) 8.43 (1.43) 5.79(1.5 -0.5 -0.5) 7.98 (1.18) 5.79(1.5 0.5 0.5) 6.34 (1.11) 7.15(-0.5 1.5 0.5) 7.89 (4.76) 7.15(-1.5 1.5 0.5) 2.67 (1.63) 5.92( 1.5 1.5 0.5) 7.92 (3.22) 5.92(2.5 0.5 -0.5) 3.70 (1.00) 4.58(2.5 -0.5 -0.5) 5.69 (1.62) 4.58(2.5 0.5 0.5) 4.49 (0.89) 5.11(2.5 1.5 -0.5) 17.49 (5.10) 4.16(1.5 5.5 -0.5) 1.39 (0.14) 1.57(-5.5 5.5 0.5) 1.58 (0.70) 0.58(-2.5 1.5 -0.5) 3.27 (1.00) 4.16(2.5 -1.5 -0.5) 3.34 (1.02) 4.16(-2.5 -1.5 -0.5) 4.09 (0.64) 4.16(1.5 5.5 0.5) 1.12 (0.24) 1.54

Table 4.3: The observed and calculated magnetic structure factors for different wave-vectors.The quality of the fit is χ2 = 4.89.

magnetic structure factors is given in Tab. 4.3. Using the previously presented result from

the symmetry analysis, we find that DTN has a collinear arrangement of spins in the ab-

plane at µ0H = 6 T, as shown in Fig. 4.4. The quality of the fit of the magnetic structure

is described by χ2 = 4.89. The value of the fitted ordered magnetic moment in ab-plane

of one nickel ion is equal to mab = 1.941(54) µB. Based on magnetization measurements

[136], the ferromagnetic component of the ordered moment along the c-axis at µ0H = 6 T is

estimated to be mc = 0.8 µB. Hence, the total ordered magnetic moment at µ0H = 6 T is

equal to mtot = 2.1 µB, which is close to the saturation value of the fully ordered moment,

msat = 2.34 µB. The quality of the fit is independent of the momentum direction in the

ab-plane, indicating an unbroken rotational symmetry.

The experimental data were also fitted with a model in which the angle between magnetic

moments located at the corner and the center of the unit cell was fixed to 90. Such a fit

has a worse quality (χ2 = 5.33) than the fit described above and it leads to an unphysical

result which corresponds to a partly disordered magnetic moment at the center of the unit

cell. Therefore we exclude the ”90 model” from the consideration.


Figure 4.3: Magnetic Bragg peaks observed at the wave-vectors Q = (0.5, 0.5, 0.5), Q =(0.5, 0.5, −0.5), Q = (1.5, 0.5, 0.5) and Q = (2.5, 1.5, −0.5) are shown in subplots a,b, c and d, respectively. The measurements were performed in a magnetic field µ0H = 6 Tapplied along the c-axis and at temperatures T < 60 mK.


Figure 4.4: The ordered magnetic structure of DTN. The projections of the magnetic mo-ments located at the corners and the center of the unit cell on the ab-plane are shown byblack and white arrows, respectively. The rotational symmetry in ab-plane is not broken,so the magnetic moments can point in any direction in the plane as long as the magneticstructure remains collinear.

4.3.2 Magnetic phase diagram

To determine the HT phase diagram of DTN, the intensity of the magnetic Bragg reflection

at Q = (0.5, 0.5, 0.5) was measured by neutron diffraction as a function of magnetic field

applied parallel to the tetragonal axis. The experiment was performed at constant temper-

atures from T = 0.1 K to T = 0.8 K with the step of ∆T = 0.1 K. Fig. 4.5 shows the field

scans measured at T = 0.1 K, T = 0.5 K and T = 0.8 K. A fast growth of the antiferro-

magnetic peak intensity above the first critical field is observed at all temperatures. After

reaching the top of the ”dome” located near µ0H = 6 T, the magnetic intensities diminish

slowly and finally disappear at the second critical field, Hc2. The onset field of the antiferro-

magnetic phase increases with increasing temperature, consistent with the thermodynamic

measurements. We also observed a decrease of the transition from the antiferromagnetic to

a fully magnetized state with increasing temperature. Fig. 4.7(a) summarizes the antifer-

romagnetic peak intensity measured at the wave-vector Q = (0.5, 0.5, 0.5) as a function

of temperature and field. The dome-like form of the magnetic phase diagram signifies that

thermal fluctuations destabilize the ordered magnetic moment, thereby shrinking the area of


the antiferromagnetic phase. The HT phase diagram seen with neutrons confirms the phase

diagram obtained by specific heat and magnetocaloric effect measurements [131].

We investigated whether the magnetic structure always has the same symmetry for all

fields. We measured the field dependencies of different magnetic Bragg peak intensities.

Fig. 4.6 shows the scaled peak intensities detected at the wave-vectors Q = (0.5, 0.5, 0.5),

Q = (2.5, 1.5,−0.5) and Q = (1.5, 1.5, 0.5). The close similarity of the scaled data shows

that the relative intensities of theses peaks is field independent, providing evidence that the

magnetic structure in the ordered phase has always the same symmetry.

Based on the magnetic phase diagram measured by neutron diffraction and the mag-

netic structure, we were able estimate the ordered antiferromagnetic moment as a function

of temperature and field, because the magnetic Bragg peak intensity is proportional to the

square of the ordered antiferromagnetic moment. The inset in Fig. 4.5 shows that the an-

tiferromagnetic moment increases from zero below Hc1 to mab = 1.941 µB at µ0H ' 6 T at

T = 0.1 K.

Our experimental investigation is a first microscopic prove of the ordered transverse spin

components of magnetic moments in the antiferromagnetic phase of DTN, perpendicular to

the field direction. Together with a preserved rotational symmetry of spin components in the

ab-plane, this result indicates that QPT at Hc1 can be interpret as BEC of magnons.

The magnetic phase diagram as a function of Ha and Hc (Ha and Hc denotes the component

of the magnetic field along the crystallographic a- and c-axis, respectively) was determined

by measuring the Bragg peak intensity at the wave-vector Q = (−0.5, −0.5, −0.5) as a

function of the angle between the magnetic field and the tetragonal c-axis. The acceptable

area of the diagram was limited by the dark angles of the horizontal field magnet. The

color plot of the neutron scattering peak intensity is shown in Fig 4.8 is the Ha − Hc phase

diagram. The transition from a quantum paramagnetic to antiferromagnetic phase occurs

only if a major component of a magnetic field is applied along the tetragonal axis. This is a

direct consequence of a strong XY anisotropy: A magnetic field applied in the xy-plane acts

on a linear combination of the exited states Sz = ±1 mixing it with a ground state Sz = 0

and thus preventing the system from a magnetic order. Our result independently confirms

the behavior of the first critical field as a function of field angle away from the tetragonal


Figure 4.5: (Color) The neutron scattering intensity measured at the wave-vector Q =(0.5, 0.5, 0.5) as a function of field at Q = (0.5, 0.5, 0.5) at temperatures T = 0.1 K, T = 0.5 Kand T = 0.8 K are shown by blue, green and red circles, respectively. The inset shows theestimated ordered antiferromagnetic moment in ab-plane as a function of field at T = 0.1 K.

Figure 4.6: The scaled neutron scattering peak intensity measured at Q = (0.5, 0.5, 0.5),Q = (2.5, 1.5,−0.5) and Q = (1.5, 1.5, 0.5) are shown by black, yellow and green circles,respectively.


Figure 4.7: (Color) (a) The color plot of the neutron scattering peak intensity measuredat the wave-vector Q = (0.5, 0.5, 0.5) in magnetic fields up to µ0H = 12 T applied alongthe tetragonal axis and temperatures up to T = 0.8 K. (b) The antiferromagnetic momentordered in the crystallographic ab-plane plotted as function of temperature and magneticfield.

axis that was recently determined by the magnetostriction measurements [134].

4.3.3 Bose-Einstein condensation of magnons

The main scientific interest of DTN is a possible field-induced BEC of magnons. We studied

the energy of the magnetic excitations as a function of magnetic field using neutron spec-

troscopy. Due to the restrictions from the horizontal-field magnet, several wave-vectors such

as Q = (0.5, 0.5, 0.5) were not accessible and the constant-Q measurements were performed

at Q = (−1.5, −1.5, 1.5).

Fig 4.9 - 4.10 shows the magnetic excitation spectrum at Q = (−1.5, −1.5, 1.5) as

a function of energy for different magnetic fields strengths. The field was always applied

along the c-axis. The red lines in the energy scans represent a Gaussian convoluted which

was the instrumental resolution function and fitted to the data. The well defined peak at

E = 0.285(5) meV, which is observed at zero field, splits into two peaks in the presence of

a magnetic field. This is direct microscopic evidence that the zero-field magnetic excitations

in DNT are doublet states. The energy of the lower doublet state decreases linearly with

increasing energy, while the energy of the higher doublet state increases with increasing field.


Figure 4.8: (Color) The color plot of the neutron scattering peak intensity measured atQ = (0.5, 0.5, 0.5) at magnetic fields up to H = 5.5T applied in crystallographic a- andc-directions at temperatures below T = 0.1K.

The fitted slope of the decrease of the lower doublet and of the increase of the doublet are

0.139(6) meV/T and 0.132(3) meV/T, which is almost the same.

At Hc1, the energy of the lower doublet is identical to that of the singlet ground state,

leading to a mixing of the lower doublet state with the ground state. As a consequence, the

materials adopts antiferromagnetic long-range order at this field, as observed in our diffraction

experiments. The closing of the energy gap at Hc1 can be described as BEC of magnons,

as it is discussed in Chapter (1). Fig. 4.9(a) shows the summarized field dependence of the

energy of the doublet. Due to the Zeeman effect, the energy of the higher doublet grows

linearly as a function of magnetic field in the antiferromagnetic phase. The fitted slope of

the increase of energy of the higher doublet is equal to 0.221(4) meV/T. This slope is almost

twice larger compare to one observed in the paramagnetic phase. The increase of the slope

in the ordered phase is a direct consequence of the field-induced lowering of the energy of the

linear combination of the lower doublet S = 1 and the Sz = 0 state.

Fig. 4.10 shows the field dependence of the scattering at Q = (−1.5, −1.5, 1.5) in the

ordered phase. A broad peak with a width much larger than the instrumental resolution

is observed between E = 0.35 meV and E = 0.7 meV in the energy scans at µ0H = 4 T


Figure 4.9: (a) The energy of the doublet as the function of magnetic field applied parallel tothe crystallographic c-axis and measured at wave vector Q = (−1.5, −1.5, 1.5). The energyscans performed in magnetic fields µ0H = 0 T, µ0H = 0.5 T and µ0H = 4 T are shown insubplots (b), (c) and (d), respectively.


Figure 4.10: Energy scans performed at Q = (−1.5, −1.5, 1.5) at µ0H = 4 T, µ0H = 5 Tand µ0H = 6 T are shown in (a), (b) and (c), respectively.


and µ0H = 5 T (see Fig. 4.10(a-b)). Upon increase of magnetic field to µ0H = 6 T the

broad scattering evolves into a better defined excitation at E ≈ 0.4 meV as it is shown

in Fig. 4.10(c). A width larger than the instrumental resolution possibly indicates several

energy levels located close to each other. Our observation of a magnetic mode with the

energy E ∼ 0.4 meV at the antiferromagnetic zone center at µ0H ∼ 6 T is consistent

with the result of the ESR measurements [137], however its origin is not well understood

yet. Since Dzyaloshinsky-Moriya interactions are forbidden by symmetry of DTN, a possible

explanation of the observed mode could be given based on a distortion of the crystal lattice

in applied magnetic fields.

4.3.4 Spin dynamics in the fully magnetized phase

It is possible that DTN features NNN neighbor interactions between corner and center cell

spins. In this case, the system may be described by the following spin Hamiltonian [139]:

H =∑r,ν

Jν(κSzrS

zr+eν + SxrS

xr+eν + SyrS

yr+eν )

+Jf∑r,ν′

Sr · Sr+eν′ +D∑r

(Szr )2 − gµBH∑r

Szr , (4.1)

where

eν = a, b, c,

eν′ = c2± a

2± b

2,

κ describes the possibility of a exchange anisotropy and Jf is the exchange interaction be-

tween two tetragonal spin sublattices. The latter interaction was not considered in previous

descriptions of magnetism in DTN.

The parameters of the spin Hamiltonian 4.1 can be determined best in the fully mag-

netized phase of DTN. This is because the spin-waves can be easily calculated for a ferro-

magnetic phase, giving direct access to the exchange and anisotropy parameters. To test the

Hamiltonian in Eq. (4.1), we performed inelastic neutron scattering measurements above the

second critical field, at µ0H = 13.3 T.


Spin-wave calculations predict the following one-magnon dispersion in a fully saturated

phase for the Hamiltonian 4.1 [139]:

EQ = gµ0H −D − 2κ(Jc + 2Jab)− 8Jf + 2(Jc cos(2πl)

+Jab cos(2πk) + Jab cos(2πh)) + 8Jf cos(2πh

2) cos(

2πk

2) cos(

2πl

2). (4.2)

For the Hamiltonian in Eq. (4.1), the second critical field Hc2 is equal to

gµBHc2 = D + 2(1 + κ)(Jc + 2Ja) + 8Jf . (4.3)

Based on the equation 4.2, the interaction between two nickel sublattices, Jf , if present, can

be determined as

Jf =1

16(E(2 0 0) − E(1 0 0)). (4.4)

In a similar way the interaction between spins in ab-plane,

Jab =1

4(E(0.5 1 0) − E(0.5 1.5 0)). (4.5)

We investigated the dispersion of the low-lying magnetic excitation at µ0H = 13.3 T us-

ing the cold neutron three-axis spectrometer RITA2. Typical energy scans are shown in

Fig. 4.11. The red lines are the result of a Gaussian convoluted with the instrumental reso-

lution function, indicating that the observed magnetic excitations are resolution limited and

long-lived.

The dispersion measured along (h, 0, 0) direction, shown in Fig. 4.12(a), implies the

presence of an exchange interaction between the corner and center spins. These measurements

show that the magnon energy at Q = (2, 0, 0) and Q = (1, 0, 0) differ by 0.111(14) meV.

According to Eq. 4.4, we therefore obtain the value Jf = 0.07(1) K. In order to obtain

an independent estimate for Jab, we performed constant Q-scans at Q = (0.5 1 0) and

Q = (0.5 1.5 0), which are shown in Fig. 4.11 (e) and (f), respectively. The magnon energy

changes by δE ∼ 0.05 meV between those two reciprocal vectors. According to Eq. (4.5),


Figure 4.11: Energy scans performed in magnetic field µ0H = 13.3 T at T < 50 mK. Redlines represent a Gaussian convoluted with the instrumental resolution function.


Figure 4.12: Dispersion of the low-lying magnon measured along different reciprocal direc-tions in magnetic field µ0H = 13.3 T at T < 50 mK. The blue lines show the spin-wavedispersion calculated based on the Eq. (4.2) and parameters given in the text.


this corresponds to the exchange interaction in the ab-plane Jab = 0.156(4) K, which is in

excellent agreement with an independent determination of Jab = 0.16 K [131]. Using these

exchange parameters and Eq. (4.2), we find good agreement between the dispersion measured

along (0.5, k, 0) direction and calculated as shown in Fig. 4.12(b).

The exchange interaction anisotropy, κ, and the single-ion XY-anisotropy, D, have the

same wave-vector independent impact on the magnon energy. Therefore those parameters

can not be determined independently from our measurements. We assume that D does not

have a strong field dependence and keep it fixed to the value determined from the zero field

neutron spectroscopy measurements D = 7.7 K [131]. It allows an estimate of the anisotropy

κ of the exchange interactions from the magnon dispersion at µ0H = 13.3 T. The best

agreement between the experimental data and the theoretical spin-wave dispersion is found

for κ = 1.27(1) (see Fig. 4.12). Remarkably, κ > 1 signifies that the exchange interactions

along the crystallographic axes are easy-axis, while the single-ion anisotropy is easy plane.

Due to a magneto-elastic coupling in DTN, it is possible that the single-ion XY-anisotropy

changes in applied fields. In this case our estimate of the exchange interaction anisotropy

will not be precisely correct.

The dispersion calculated using Eq. (4.2) and the parameters D = 7.7 K, Jc = 2.05 K,

Ja = 0.156 K, Jf = 0.07 K and κ = 1.27 is shown in Fig. 4.12(a-d) for different directions

in the reciprocal space, demonstrating excellent agreement with the experimental data. The

second critical field calculated based on the Eq. (4.3) and the obtained parameters is equal to

µ0Hc2 = 12.6 T, which is also in a good agreement with Hc2 measured by specific heat, magne-

tocaloric and magnetization measurements [131, 132]. The proposed spin Hamiltonian (4.1)

with the parameters determined from our neutron spectroscopy experiment provides there-

fore a consistent description of the available macroscopic and spectroscopic data in the fully

magnetized phase of NiCl2 · 4SC(NH2)2.

4.3.5 Spin dynamics at the first critical field

We measured the magnetic excitation spectra at the magnetic field µ0H = 2.1 T applied

parallel to the tetragonal axis and at temperatures T ∼ 50 mK. Constant energy scans

along the (1.5, −1.5, l) reciprocal direction, close to the antiferromagnetic zone center, were


carried out at low-energy transfer from E = 0.1 meV to E = 0.4 meV and are shown in

Fig. 4.13. Most scans show two peaks that shift away from each other in wave-vector with

increasing constant energy transfer.

To follow the excitation further, the constant wave-vector scans away from the antiferro-

magnetic zone center were performed (shown in Fig. 4.14). The energy of the lower-energy

mode increases from E ∼ 0 meV at the zone center to E = 0.724(19) meV at the antiferro-

magnetic zone boundary. Besides the low-lying mode, we observe a high-energy mode whose

energy increases from E = 0.566(3) meV at the zone center to E = 1.27(3) meV at the zone

boundary. The summarized dispersion is shown in Fig. 4.15. The energy scan carried out

at Q = (1.5, −1.5, 1.5) at µ0H = 2.1 T reveals only one peak corresponding to the higher

doublet excitation. This indicates that the lower excitation is mixed with the ground state

at µ0H = 2.1 T.

Long-dashed blued line in Fig. 4.15 is the result of a linear fit of the low-energy data which

demonstrates that µ0H = 2.1 T is possibly larger than the first critical field. In this case the

low-lying doublet forms the Goldstone mode, indicating a preserved rotational symmetry.

This may be a direct confirmation that the QPT at Hc1 is BEC of magnons in DTN [138].

However, the liner fit describes the low-energy data insufficiently. We therefore fitted the data

with a quadratic function, as shown by the short-dashed red line in Fig. 4.15. The quadratic

fit has a slightly better agreement with the experimental data, indicating that µ0H = 2.1 T

may be somewhat above the critical field at which BEC of magnons occurs. In this case the

soften low-energy mode mixes with the ground state at the antiferromagnetic zone center

not yet evolving into the Goldstone mode. The short-dashed line shown in Fig. 4.15 is the

fit of the high-energy excitation shifted down to zero energy at the antiferromagnetic point.

A comparison of the line slopes close to zero energy shows that the low-energy experimental

data can be described rather by a combination of a linear and a quadratic function then by

only one of them. More precise experimental investigations are needed in this critical region

in order to testify an appearance of the Goldstone mode at Hc1.


Figure 4.13: Wave-vector scans performed at low-energy transfer in magnetic field µ0H =2.1 T applied parallel to the tetragonal axis. Red lines are the result of the instrumentalresolution function convoluted numerically with two Gaussians.

4.3.6 Spin dynamics deep in the antiferromagnetic phase

We also studied the spin dynamics in the antiferromagnetic phase of DTN using neutron

spectroscopy. The measurements focused on energies below 1 meV, and thus track the

development of the lower mode of the doublet into the ordered phase. We did not take

measurements of the higher-lying doublet mode.

Constant wave-vector and energy scans were performed along (1.5, −1.5, l) reciprocal

direction at µ0H = 2.1 T and in two directions - (h, h, 1.25) and (0.6, 0.6, l) at µ0H = 6 T.

The choice of these wave-vectors resulted from the use of the horizontal field magnet which

restricted greatly the access to the reciprocal plane of the sample. Background scattering

was determined along the (0.6, 0.6, l) direction by measuring the scattering from the sample

rotated away from any magnetic scattering position.

Fig. 4.16 shows the energy scans at µ0H = 6 T performed along (0.6, 0.6, l) from

l = 1.15 to l = 1.5 after the subtraction of the non-magnetic background. Energy scans along

(h, h, 1.25) from h = 0.5 to h = 1 are shown in Fig. 4.20. Two peaks are observed in both

reciprocal directions, indicating a presence of two magnon modes at most wave-vectors. The


Figure 4.14: Energy scans performed in a magnetic field µ0H = 2.1 T applied parallel tothe tetragonal axis. The result of the numerical convolution of the instrumental resolutionfunction and Gaussian are shown by red lines.


Figure 4.15: Dispersion of two magnetic excitations measured in a field µ0H = 2.1 T appliedparallel to crystallographic c-axis at temperatures T < 60 mK. The results of the linearand quadratic fits of the low-energy data are shown by the long-dashed blue and red lines,respectively. The dispersion of the higher doublet was fitted with a sinusoidal function, asshown by the long-dashed green line. The short-dashed green line shows the fit of higherdoublet shifted down in energy for a comparison.


data was fit to a model containing one or two Gaussians convoluted with the instrumental

resolution function. The result is shown by the red lines in Fig. 4.16, indicating that the

observed peaks are resolution limited. The wave-vector dependence of the energies of two

magnetic modes is shown in Fig. 4.18 for both reciprocal directions.

In contrast to the quantum paramagnetic phase of DTN [131], the measurements in the

antiferromagnetic phase at µ0H = 6 T give clear evidence of a splitting of the lower doublet

mode in the ordered phase into two modes, as shown in Fig. 4.18. One magnetic mode has no

dispersion along the (0.6, 0.6, l) direction with an energy E ∼ 0.4 meV, as shown by the blue

line in Fig. 4.18(a). The energy of the second mode, shown by green the line in Fig. 4.18(a)

drops from E = 0.589(14) meV at Q = (0.6, 0.6, 1.15) to E = 0.100(7) meV close to the

antiferromagnetic zone center, at Q = (0.6, 0.6, 1.5). The intensity of this mode increases

rapidly as the wave-vector changes from Q = (0.6, 0.6, 1.15) towards Q = (0.6, 0.6, 1.5). In

the (h, h, 1.25) direction we detected two dispersionless modes with energies E1 ∼ 0.58 meV

and E1 ∼ 0.42 meV (Fig. 4.18(b)).

Our results show that, while there is a considerable dispersion along the c-axis, there is

no measurable change of the excitation energy in the basal plane, as shown in Fig 4.18. The

splitting of the low-energy spin band may arise from the presence of interpenetrating tetrag-

onal spin sublattices. For purely antiferromagnetic order, the two sublattices are completely

decoupled on a mean field level. In the presence of an external field, the sublattices are mag-

netized starting to interact with each other. This is because of the non-zero magnetization

of the two sublattice that leads to non-zero magnetic fields from one sublattice on the other.

This coupling may lead to a distortion of the crystal lattice and to additional interactions not

present for H < Hc1 or H > Hc2. This interpretation is supported by recent magnetostriction

measurements [134] that provide strong evidence for a coupling between the magnetic order

and the lattice.

A high-resolution wave-vector scans performed close to the antiferromagnetic point at

µ0H = 6 T revealed the low-energy behavior of the magnetic near the antiferromagnetic

ordering wave-vector, as shown in Fig. 4.19. The observed dispersion was fit with a gap and

a quadratic wave-vector dependence. The result plotted as a blue line in Fig. 4.19 indicates

the presence of the energy gap equal to ∆E = 0.135(4) meV. The fit gives a gap energy of the


Figure 4.16: Energy scans performed along the (0.6, 0.6, l) reciprocal direction in a magneticfield µ0H = 6 T applied parallel to the tetragonal axis after the subtraction of the non-magnetic background. The measurements were done at T < 60 mK.


Figure 4.17: Energy scans along the (h, h, 1.25) reciprocal direction performed in a magneticfield µ0H = 6 T applied along tetragonal axis at T < 60 mK.


Figure 4.18: Low-energy magnetic excitation spectra of DTN, measured along the (0.6, 0.6, l)and (h, h, 1.25) reciprocal directions in a magnetic field µ0H = 6 T applied parallel to thecrystallographic c-axis, are shown in (a) and (b), respectively.

dispersive mode comparable to that observed by ESR measurements performed by Zvyagin

and collaborators [137] (see Fig. 4.1), which is equal to ∆E ' 0.103 meV. Our result thus

independently confirms the gap opening at µ0H = 6 T.

The energy gap observed at µ0H = 6 T can be explained by including second order contri-

butions of Jf to the spin Hamiltonian 4.1. Quantum fluctuations renormalize the parameter

Jf , as well as Ja and Jc in the antiferromagnetic phase of DTN. Therefore the values of

exchange interactions measured in the fully magnetized phase may be somewhat different for

the antiferromagnetic state. The spin-wave dispersion measured at µ0H = 6 T (see Fig. 4.18)

is only partly described by the Hamiltonian (Eq. 4.1). In particular, the observed energy

difference between two flat modes along (h, h, 1.25) direction can only be explained by a

rather stronger coupling Jf than detected by our high-field measurements. So, the origin

of the two flat excitations in this direction cannot be explained by the known Hamiltonian.

This indicates that additional magnetic interactions are present that are presently unknown.


Figure 4.19: Low-energy part of the dispersion of low-lying excitation measured at µ0H = 6 Tapplied parallel to the crystallographic c-axis and at T < 60 mK.

4.3.7 Spin dynamics in the antiferromagnetic phase with H 6‖ c

A dispersion of one-magnon excitation was measured at µ0H = 6 T with the field turned

away from the crystallographic c-axis by χ = 20. Energy scans performed in the wave-vector

range from Q = (0.5, 0.5, 1.5) to Q = (0.5, 0.5, 2) are shown in Fig. 4.20. The red lines

are the results of the instrumental resolution function convoluted numerically with Gaus-

sian, suggesting that all observed peaks are resolution limited. The energy of the observed

excitation is plotted as a function of the wave vector in Fig. 4.21. Comparing Fig. 4.19 and

Fig. 4.21, we see that the energy gap at the zone center remains approximately the same for

µ0H = 6 T applied with the angles χ = 0 and χ = 20 to the c-axis. Small bumps close

to E = 0.6 meV in the energy performed at Q = (0.5, 0.5, 1.8), Q = (0.5, 0.5, 1.9) and

Q = (0.5, 0.5, 2) (see Fig. 4.20(f-h)) may possibly be related to a mode observed in field

applied parallel to the tetragonal axis, as shown in Fig. 4.18. Therefore we conclude, that

turning the field away from the tetragonal axis by χ = 20 does not qualitatively change the

dispersion of the low-lying magnetic excitation at µ0H = 6 T. This is mainly caused by a

relatively big field component along the tetragonal axis which is equal to µ0H‖c ' 5.6 T at

χ = 20.


Figure 4.20: Energy scans performed along (0.5, 0.5, l) direction in a magnetic field µ0H =6 T turned away from the crystallographic c-axis by χ = 20 and at temperatures T < 60 mK.


Figure 4.21: One-magnon dispersion measured along (0.5, 0.5, l) direction in magnetic fieldµ0H = 6 T turned away from the crystallographic c-axis by χ = 20 and at temperaturesT < 60 mK.

4.4 Summary

We investigated the magnetic phase diagram and the ordered magnetic structure of the

quantum S = 1 quasi-one-dimensional AF NiCl2 · 4SC(NH2)2 using neutron diffraction. The

ordered magnetic structure in the antiferromagnetic phase of DTN has a collinear antiparallel

spin alignment in crystallographic ab-plane with an unbroken rotational symmetry and the

ordered antiferromagnetic moment equal to mab = 1.922(54) µB. Thus we prove directly the

order of the spin components, perpendicular to an applied magnetic field, in DTN, confirming

the interpretation of the QPT at Hc1 as BEC of magnons.

The phase diagram of DTN was studied as a function of magnetic field and temperature.

Occurring at Hc1 ∼ 2.1 T, the collinear antiferromagnetic order is canted towards the field

direction evolving into a fully magnetized phase above Hc2. A domelike shape of the phase

diagram implies that thermal fluctuations prevent the system from magnetic order. Our

study of the magnetic phase diagram as a function of the angle between an applied magnetic

field and the tetragonal axis, χ, shows the disappearance of the antiferromagnetic order above

χ ∼ 55.

We measured the one-magnon dispersion in the fully magnetized phase, at µ0H = 13.3 T.

Based on the neutron spectroscopy data, the spin Hamiltonian, exchange interactions and

anisotropy parameters were determined precisely. Importantly, the result indicates a presence

4.4 Summary 99

of a small antiferromagnetic exchange interaction between spin sublattices.

Neutron spectroscopy measurements revealed the splitting of the gapped doublet excita-

tion in applied magnetic field. At the first critical field µ0H ∼ 2.1 T the low-lying magnon

mixes with the ground state, which theoretically is described as BEC of magnons.

The study of the dispersion of the low-lying magnon in the antiferromagnetic phase of

DTN reveals the opening of the energy gap at µ0H = 6 T, which we attributed to the presence

of the coupling between tetragonal spin sublattices. However the dispersion of the low-lying

magnon measured in the antiferromagnetic phase is not described completely by the spin

Hamiltonian determined from a high-field measurements and thus remains unexplained.


Outlook

This work has revealed a number of interesting aspects of low-dimensional quantum magnets,

and raised interesting new questions. The outlook research can be summarized as follows:

(A) Weakly-frustrated S = 1/2 AF on a square lattice, Cu(pz)2(ClO4)2:

1. A fascinating issue is an experimental verification of the predicted collapse of one-

magnon excitations in the 2D S = 1/2 AF on a square lattice close to the saturation

magnetic field. The main technical difficulty is connected with the limitation of avail-

able magnetic fields for neutron scattering experiments to approximately 15 Tesla.

Therefore, the NN spin exchange interaction should be equal to J ∼ 0.5 meV at most,

so that the available magnetic fields can reach the saturation-field limit of the S = 1/2

square-lattice antiferromagnet. Good candidates for future research could be materials

with a significantly increased distance between copper ions in a square-lattice plane.

2. Another concern in this topic is related to the enhancement of quantum fluctuations

by an additional antiferromagnetic NNN exchange interaction. Various numerical cal-

culations predict a rich magnetic phase diagram depending on a strength of NNN

interaction. However, due to a lack of materials, only little is known experimentally

about the predicted and possibly exotic phases of a frustrated S = 1/2 AF on a square

lattice.

(B) Quasi-one-dimensional spin-1 system NiCl2·4SC(NH2)2:

3. Our study has shown that the one-magnon dispersion measured in the antiferromagnetic

phase is not well understood. The difficulty of the interpretation of the ordered phase

may be due to a relatively strong coupling of the magnetism to the nuclear lattice,

102 Outlook

which leads to magnetic interactions not present at zero or very high fields. The

investigation of the magnetic symmetry breaking in the ordered phase is therefore

an interesting question. Unpublished data from Los Alamos provides evidence of a

field-induced electric current in the antiferromagnetic phase that could be related to a

change of the electric polarization in this pyroelectric material. This may be similar to

magnetically-induced ferroelectricity in transition metal insulators, with the difference

that the inversion symmetry is already broken in the paramagnetic phase.

4. The BEC of magnons in DTN can be verified independently by measuring the critical

exponent α using neutron scattering. Careful measurements of the magnetic Bragg

peak intensity would be necessary as a function of an applied magnetic field at different

temperatures close to the QCP. Such studies would allow an independent determination

of the critical exponents of the 3D XY antiferromagnetic quantum phase transition.

Appendix A: Symmetry analysis of

Cu(pz)2(ClO4)2

The crystal structure of Cu(pz)2(ClO4)2 belongs to the monoclinic C2/c space group (#15),

whose Laue class and the point group are 2/m. The Cu2+ ions occupy 4e Wyckoff positions

and they are located at r1 = (0 0.7499 0.25), r2 = (0 0.2501 0.75), r3 = (0.5 0.2499 0.25)

and r4 = (0.5 0.7501 0.75). For C2/c space group, reciprocal lattice points are located at

Q = (h, k, l) with k + l = 2n, l = 2n, k = 2n, h + k = 2n. The magnetic Bragg peak was

observed at Q = (0, 1, 0), so the magnetic ordering vector is given as k = (0, 1, 0). The

ordering wave-vector is different from zero due to the C-centering of the lattice.

The subgroup of the ordering wave vector, usually called the little group, consists of four

symmetry operations which belong to four different classes: 1, 2b,1 and mac. Here, 1 is the

identity, 2b is a two-fold rotation around the b-axis, 1 is the inversion and mac is a mirror

plane in the ac plane. Therefore, there are four one-dimensional irreducible representations

whose characters are summarized in the character table given in Tab. 4.4.

1 2b 1 mac

Γ1 1 1 1 1Γ2 1 1 -1 -1Γ3 1 -1 1 -1Γ4 1 -1 -1 1

Table 4.4: The character table and the irreducible representations obtained from grouptheory analysis for monoclinic space group C2/c (#15), the table setting choice is b1) andthe magnetic ordering vector k = (0, 1, 0).

The decomposition equation for the magnetic representation is Γmag = 1Γ1 + 1Γ2 + 2Γ3 +

2Γ4. The six basis vectors presented in Tab. 4.5 are calculated for two Cu2+ positions in

104 Appendix A: Symmetry analysis of Cu(pz)2(ClO4)2

X1 X2

Γ1−→φ 1 (0 1 0) (0 1 0)

Γ2−→φ 2 (0 1 0) (0 -1 0)

Γ3−→φ 3 (1 0 0) (1 0 0)−→φ 4 (0 0 1) (0 0 1)

Γ4−→φ 5 (1 0 0) (-1 0 0)−→φ 6 (0 0 1) (0 0 -1)

Table 4.5: Six basis vectors calculated for two Cu2+ positions in primitive unit cell as ex-plained in Appendix A.

primitive unit cell using the projection operator method acting on a trial vector φα

Ψλαν =

∑gεGk

Dλ∗ν (g)

∑i

δi,giRgφαdet(Rg),

where Ψλαν is the basis vector projected from the λth row of the νth irreducible representation,

Dλ∗ν (g) is the λth row of the matrix representative of the νth irreducible representation for

symmetry operation g, i denotes the atomic position and Rg is the rotational part of the

symmetry operation g.

We now analyze the possibility of Dzyaloshinsky-Moriya (DM) exchange interactions in

Cu(pz)2(ClO4)2. DM interactions are described by a Hamiltonian given by

HDM =∑〈i,j〉

Dij · [Si × Sj],

where Dij is an axial vector. Action of any symmetry operation (including lattice translations)

A on a DM vector Dij must be equal to DA(i)A(j) and Dij = −Dji. We analyze the action

of the inversion symmetry operation 1 on the axial DM vector D12, where i = 1 and j = 2

denotes the NN copper positions r1 = (0 0.7499 0.25) and r2 = (0 0.2501 0.75), respectively.

The result of operation is

1(D12) = (Dx12 Dy

12 Dz12).

The application of the inversion symmetry on the ions positions leads to

1(D12) = D1(1)1(2) = (Dx21 Dy

21 Dz21).

Appendix A: Symmetry analysis of Cu(pz)2(ClO4)2 105

These relations imply (Dx12 Dy

12 Dz12) = (Dx

21 Dy21 Dz

21) which is possible only in case of

D12 = 0. Therefore, DM interactions between NN in Cu(pz)2(ClO4)2 are forbidden by the

crystal symmetry.

106 Appendix A: Symmetry analysis of Cu(pz)2(ClO4)2

Appendix B: The linear spin wave

theory

Assuming that Cu(pz)2(ClO4)2 is in the antiferromagnetic Neel ground state with spins

pointing along z and −z direction, we can make the Holstein-Primakoff transformation of

the spin compounds into bosonic creation and annihilation operators. In linear spin wave

approximation it gives:

Szi = S− a+

i ai, Szj = −S + a+

j aj, Szk = S− a+

k ak,

Sxi =√

2Sai + a+

i

2, Sy

i =√

2Sai − a+

i

2i,

Sxj =√

2Saj + a+

j

2, Sy

j =√

2S−aj + a+

j

2i,

Sxk =√

2Sak + a+

k

2, Sy

k =√

2Sak − a+

k

2i,

where indexes i, j, and k correspond to a reference spin, its nearest and next nearest neighbor

spins, respectively. The quantization axis lies in bc-plane and therefore Jx1 = Jz

1 = J, Jy1 =

J−∆. The spin Hamiltonian written in bosonic operators is:

H =∑〈i,j〉4JS(a+

i ai + a+j aj) + 2S

2J + ∆

2(aiaj + a+

i a+j )+

2S∆

2(aia

+j + a+

i aj)+

∑〈i,k〉

4J2S(aia+k + a+

i ak)− (a+i ai + a+

k ak),

108 Appendix B: The linear spin wave theory

where 〈i, j〉 indicates the sum over NN in the bc-plane and 〈i, k〉 - the sum over NNN in the

bc-plane. After Fourier transformation, the Hamiltonian can be diagonalized using standard

Bogoliubov transformation:

aq = −uqαq + vqβ+q ,

a+q = −uqα

+q + vqβq,

a−q = vqα+q − uqβq,

a+−q = vqαq − uqβ

+q ,

where αq and βq are the bosonic operators and vq, uq are numbers. Finally, after the diago-

nalization we have

Hq = Eg.s. +∑

q

S[2Aqα+q αq + Bq(α+

q α+−q + αqα−q)], (4.6)

and the eigenstates are given by

hωq = (A2q −B2

q )1/2, (4.7)

where

Aq = 4SJ + S∆(cos(qb) + cos(qc))−

4SJ2 + 4SJ2 · cos(qb) · cos(qc)

and

Bq = 2S(J− 1/2∆)(cos(qb) + cos(qc)).

The result is shown schematically in Fig. 4.22. Blue and brown colors represent the low and

high energy of the spin wave, respectively. Having a minimum at the antiferromagnetic zone

center Q = (0, 1, 0), the energy of the one-magnon mode increases rapidly in all reciprocal

directions and approaches the maximum at the zone boundary point Q = (0, 1, 0.5). In this

linear approximation, the energy of the spin wave has a local minima at the antiferromag-

netic zone boundary arising from the presence of the NNN exchange interaction. However

the approximation fails to describe the spectra of Cu(pz)2(ClO4)2 observed by our neutron

Appendix B: The linear spin wave theory 109

Figure 4.22: The result of the linear spin-wave calculation based on the equation 4.7 is plottedschematically in coordinates of L and K.

spectroscopy measurements, due to the presence of strong quantum fluctuations.

110 Appendix B: The linear spin wave theory

Appendix C: The nuclear structure of

NiCl2 · 4SC(NH2)2

Here we show the result of the fit of the nuclear structure of NiCl2 · 4SC(NH2)2 performed

using the FullProf program. The square of nuclear structure factors for different reciprocal

lattice vectors were fitted to the observed intensities after a correction of the Lorentz factor.

The goodness of the final fit is equal to χ2 = 20.63 and the result is shown in Fig. 4.23. A

scaling factor was obtained from the fit of the nuclear structure and was used to estimate

the value of the ordered magnetic moment. The fitted positions of atoms in a unit cell of

NiCl2 · 4SC(NH2)2 are listed in Tab. 4.6.

112 Appendix C: The nuclear structure of NiCl2 · 4SC(NH2)2

Figure 4.23: The result of the calculation of the nuclear structure of DTN. The positions ofopen circles correspond to the calculated and observed nuclear structure factors as ordinateand abscissa, respectively.

Element PositionNi (0.00000 0.00000 -0.08941)

Cl (1) (0.00000 0.00000 -0.58290)Cl (2) (0.00000 0.00000 0.09382)

S (0.03271 0.25634 0.03110)C (0.15359 0.32823 -0.09100)

N (1) (0.19744 0.45836 -0.03954)N (2) (0.20873 0.25899 -0.19580)D (1) (0.15452 0.50868 0.01289)D (2) (0.27245 0.50311 -0.07636)D (3) (0.27929 0.30479 -0.36183)D (4) (0.17566 0.16439 -0.26511)

Table 4.6: Fitted positions of atoms in a unit cell of NiCl2 · 4SC(NH2)2

Appendix C: The nuclear structure of NiCl2 · 4SC(NH2)2 113

|Fmeasured(Q)|2 |Fcalculated(Q)|2 Q1 340.0000 415.2000 ( 3 2 -1)2 236.0000 205.1200 ( 11 0 -1)3 165.1000 131.3900 ( 4 2 0)4 1812.6000 2039.9200 ( 9 1 0)5 281.5000 295.4400 ( 9 0 -1)6 688.0000 629.3700 ( 2 1 -1)7 139.4000 37.6700 ( 8 0 0)8 585.1000 575.2700 ( 10 -2 0)9 1039.6000 843.0800 ( 4 1 -1)10 2670.1001 1922.8300 ( 3 1 0)11 3501.2000 3343.0801 ( 11 -5 0)12 216.2000 214.9800 ( 10 -3 -1)13 38.3000 60.7700 ( 9 -2 -1)14 3431.0000 3886.4099 ( 6 0 0)15 363.9000 329.5900 ( 10 -5 -1)16 158.7000 98.9100 ( 10 -6 0)17 894.5000 864.2600 ( 9 -4 -1)18 5296.1001 5343.7798 ( 4 0 0)19 5319.1001 5343.7798 ( 4 0 0)20 1373.7000 1320.7800 ( 6 -1 -1)21 99.4000 116.3600 ( 9 -5 0)22 1240.1000 1291.8800 ( 8 -4 0)23 2436.5000 1372.2500 ( 2 0 0)24 367.8000 401.9800 ( 6 -2 0)25 58.3000 144.8400 ( 711 0)26 456.0000 499.7400 ( 6 -3 1)27 1479.7000 1659.5601 ( 7 -5 0)28 56.7000 31.2400 ( 4 -2 0)29 2013.1000 2185.8601 ( 6 -6 0)30 997.0000 913.2800 ( 5 -5 0)31 1677.3000 1665.1400 ( 5 -6 -1)32 1738.6000 1522.2800 ( 113 0)33 1305.4000 1588.8700 ( 4 -6 0)34 2958.0000 2747.2800 ( 3 -3 0)35 366.5000 432.5300 ( 3 -4 -1)36 140.9000 127.1900 ( 1 -11 0)37 207.9000 206.3800 ( 0 -11 -1)38 1957.9000 2174.6899 ( 1 -9 0)39 319.6000 298.1200 ( 0 -9 -1)40 723.9000 647.8700 ( 1 -2 -1)41 581.6000 584.5100 ( -2 -10 0)

Table 4.7: The measured and calculated nuclear structure factors for different reciprocallattice vectors.


|Fmeasured(Q)|2 |Fcalculated(Q)|2 Q42 2788.5000 2168.4199 ( 1 -3 0)43 3540.6001 3589.4099 ( -5 -11 0)44 213.6000 216.1100 ( -3 -10 -1)45 889.0000 879.2000 ( -4 -9 -1)46 1407.0000 1387.3700 ( -1 -6 -1)47 773.5000 757.8300 ( -8 -9 -1)48 1294.1000 1323.3700 ( -4 -8 0)49 415.2000 407.0300 ( -2 -6 0)50 1497.8000 1687.5100 ( -5 -7 0)51 1794.5000 1605.5601 ( -9 -7 0)52 74.2000 73.3600 ( -7 -7 0)53 483.4000 444.6200 (-12 -5 -1)54 247.1000 292.1000 ( -5 -6 -1)55 41.8000 31.2700 ( -2 -4 0)56 2036.1000 2184.2300 ( -6 -6 0)57 3419.3999 3398.6599 ( -9 -5 0)58 1623.5000 1647.9900 ( -6 -5 -1)59 1204.6000 1265.8300 ( -8 -5 -1)60 73.5000 76.8800 ( -7 -5 0)61 1270.6000 1278.7800 (-10 -4 0)62 815.4000 806.9200 ( -9 -4 -1)63 1788.6000 1445.8600 (-13 -1 0)64 1303.4000 1554.4000 ( -64 0)65 3020.3000 2742.8301 ( -3 -3 0)66 432.4000 430.1400 ( -4 -3 -1)67 1454.3000 1351.5601 ( -8 -3 -1)68 308.4000 85.2400 ( -1 -1 0)69 202.3000 205.1200 (-11 0 -1)70 1878.6000 2039.9200 ( -9 -1 0)71 30.6000 32.2300 ( -6 -2 0)72 290.0000 295.4400 ( -9 0 1)73 777.1000 629.3700 ( -2 -1 1)74 91.9000 99.1800 (-10 1 1)75 251.4000 219.3400 (-11 4 1)76 579.2000 575.2700 (-10 2 0)77 1087.8000 843.0800 ( -4 -1 -1)78 2657.2000 1922.8300 ( -3 -1 0)79 3466.1001 3343.0801 (-11 5 0)80 234.2000 254.1300 (-11 6 -1)81 212.2000 214.9800 (-10 3 -1)82 64.2000 41.5100 (-11 7 0)

Table 4.8: The measured and calculated nuclear structure factors for different reciprocallattice vectors (continue).

Appendix C: The nuclear structure of NiCl2 · 4SC(NH2)2 115

|Fmeasured(Q)|2 |Fcalculated(Q)|2 Q83 3476.8999 3886.4099 ( -6 0 0)84 885.8000 864.2600 ( -9 4 -1)85 5036.7998 5343.7798 ( -4 0 0)86 5074.5000 5343.7798 ( -4 0 0)87 1400.9000 1320.7800 ( -6 1 -1)88 612.9000 483.2100 ( -9 9 0)89 755.0000 755.9300 ( -9 8 -1)90 1197.6000 1291.8800 ( -8 4 0)91 366.0000 401.9800 ( -6 2 0)92 1516.5000 1659.5601 ( -7 5 0)93 1820.8000 1625.8500 ( -7 9 0)94 55.3000 31.2400 ( -4 2 0)95 123.8000 123.8800 ( -6 9 -1)96 2076.2000 2185.8601 ( -6 6 0)97 3614.7000 3596.3000 ( -5 9 0)98 1659.5000 1665.1400 ( -5 6 -1)99 1244.7000 1287.7600 ( -5 8 -1)100 50.2000 26.0400 ( -2 13 -1)101 1394.0000 1315.5601 ( -4 10 0)102 835.0000 820.2400 ( -4 9 -1)103 1755.7000 1522.2800 ( -1 13 0)104 1349.4000 1588.8700 ( -4 6 0)105 2889.3999 2747.2800 ( -3 3 0)106 374.3000 432.5300 ( -3 4 -1)107 1174.0000 1396.3900 ( -3 8 -1)108 308.6000 308.9500 ( -2 9 -1)109 200.6000 206.3800 ( 0 11 -1)110 265.5000 117.7700 ( -2 5 -1)111 2212.0000 2174.6899 ( -1 9 0)112 26.4000 32.2600 ( -2 6 0)113 317.3000 298.1200 ( 0 9 -1)114 785.9000 647.8700 ( -1 2 -1)115 606.7000 584.5100 ( 2 10 0)116 1132.6000 876.7200 ( -1 4 -1)117 2822.7000 2168.4199 ( -1 3 0)118 3427.3999 3589.4099 ( 5 11 0)119 222.1000 216.1100 ( 3 10 -1)120 404.5000 220.9300 ( 3 9 0)121 702.9000 834.0200 ( 6 9 -1)122 806.1000 757.8300 ( 8 9 -1)123 407.4000 407.0300 ( 2 6 0)124 203.2000 145.6000 ( 7 8 -1)

Table 4.9: The measured and calculated nuclear structure factors for different reciprocallattice vectors (continue).


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Acknowledgments

First of all I am grateful to Dr. Michel Kenzelmann for giving me the opportunity to perform

my PhD work in the Laboratory for Neutron Scattering, PSI & ETH Zurich and in the

Laboratory for Solid State Physics, ETH Zurich. For almost four years, he supervised and

encouraged me, giving advice and constructive criticism. I have benefited a lot from working

with Michel. I am thankful to Prof. Dr. Matthias Troyer for accepting to be my supervisor

and also to Prof. Dr. Joel Mesot for agreeing to be my co-examiner and his support.

I greatly appreciated the help provided by the beamline scientists during the neutron scat-

tering experiments: Peter Link and Astrid Schneidewind FRM-2, Arno Heiss, Eric Ressouche

and Beatrice Grenier from ILL, Jorge Gavilano, Niels Christensen and Christof Niedermayer

from SINQ. I also acknowledge the people from the engineering group of LDM at PSI for

helping with sample environment and technical assistance during experiments at SINQ, espe-

cially Markus Zolliker, Christian Kagi, Dominik Hohl, Walter Latscha and Stephan Fischer.

I thank Dirk Etzdorf and Harald Schneider for technical assistance at PANDA spectrometer

at FRM2, the technical services of ILL involved during the experiment at IN14 at ILL and

Pascal Fouilloux for technical help during the experiment at D23 at ILL.

I am thankful to the chemists from Clark University, especially to Christopher Landee,

Mark Turnbull and Fan Xiao for providing us with single crystal samples of Cu(pz)2(ClO4)2.

I also think A. Paduan-Filho for the growth of particularly large single-crystals of DTN.

This work won’t be complete without theoretical support provided by Rajiv Singh, Tom

Pardini, Cristian D. Batista as well as Vivien Zapf and Marcelo Jaime. I am grateful to them

for a productive collaboration.

Thanks goes to the members of the LNS and LDM, especially to Oksana Zaharko, Kaz-

imierz Conder, Ekaterina Pomjakushina, Volodya Pomjakushin, Denis Cheptiakov and Seva

128 Acknowledgments

Gvasaliya, for useful discussions and support. I am also grateful to Andrey Podlesnyak and

Alex Mirmelstein for all help and assistance in the beginning of my PhD.

Thanks also go to Kruno Prsa, Gelu Marius Rotaru, Roggero Frison, Martin Haag, Johann

Gironnet, Cecile Marcelot, Mohamed Zayed Yasmine Sassa, Gwen Pascua, Martin Mansson,

Yoyo Wang, Marian Stingaciu, Loıc LeDreau and Juan Pablo Urrego-Blanco for the great

time we spent together, and that not only in the office.

I would like to thank my mum Dinara for her unwavering support of my decision to study

physics and my father Lev for his keen endorsement. My warmest thanks are addressed to

Katja Etzel, who not only supported and encouraged me wholeheartedly, but also proofread

some parts of this thesis.

Publications and conferences

2010

• The Two-Dimensional Square-Lattice S=1/2 Antiferromagnet Cu(pz)2(ClO4)2

N. Tsyrulin, F. Xiao, A. Schneidewind, P. Link, H. M. Rønnow, J. Gavilano2, C. P. Landee,

M. M. Turnbull and M. Kenzelmann

Physical Review B 81, 134409 (2010).

2009

• Quantum Effects in a Weakly-Frustrated S=1/2 Two-Dimensional Heisen-

berg Antiferromagnet in an Applied Magnetic Field

N. Tsyrulin, T. Pardini, R. R. P. Singh, F. Xiao, P. Link, A. Schneidewind, A. Hiess,

C. P. Landee, M. M. Turnbull, and M. Kenzelmann

Physical Review Letters, 102, 197201 (2009).

• Quantum Effects in a Weakly-Frustrated S=1/2 Two-Dimensional Square-

Lattice Antiferromagnet in Zero and Applied Magnetic Field

N. Tsyrulin, T. Pardini, R. R. P. Singh, F. Xiao, P. Link, A. Schneidewind, A. Hiess,

C. P. Landee, M. M. Turnbull, and M. Kenzelmann

ILL Annual report.

• Quantum two-dimensional weakly-frustrated S=1/2 antiferromagnet on a

square lattice in zero and applied magnetic fields

Oral presentation at SLS Symposium on Quantum Magnetism, Villigen-PSI, Switzer-

land.

130 Publications and conferences

• Magnetic excitation spectra of two-dimensional weakly-frustrated S=1/2

antiferromagnet on a square lattice Cu(pz)2(ClO4)2

Oral presentation at 2nd FRM2 User Meeting, Garching, Germany.

• Quantum two-dimensional weakly frustrated S=1/2 antiferromagnet on a

square lattice in zero and applied fields

Oral presentation at the International Conference on Magnetism, Karlsruhe, Germany.

• Quantum effects in a weakly-frustrated S=1/2 two-dimensional antiferro-

magnet in an applied magnetic field

Poster presentation at MaNEP Swiss Workshop Meeting, Les Diablerets, Switzerland.

2008

• Quantum effects in S=1/2 two-dimensional Heisenberg antiferromagnet in

applied magnetic field

Poster presentation at XXI Congress and General Assembly of the International Union

of Crystallography, Osaka, Japan (Acta Cryst. A64, C416 (2008).

2007

• Field dependent ordering temperature in copper pyrazine perchlorate,

Cu(pz)2(ClO4)2

C. Landee, F. Xiao, M. Turnbull, N. Tsyrulin, M. Kenzelmann, H. Van Tol

American Physical Society March Meeting.

Curriculum Vitae

Personal data

Name: Nikolay Tsyrulin

Date of birth: 19.11.1983

Place of birth: Kislovodsk/Russian Federation, Russian citizen

Current address

Laboratory for Quantum Magnetism Ecole Polytechnique Federale de Lausanne

PH D3 324, LQM-IPMC-EPFL, Station 3, 1015 Lausanne, Switzerland

Phone: +41 21 69 34310

E-mail: [email protected], [email protected]

Education

March 2006 - January 2010: PhD student in the Laboratory for Neutron Scattering ETHZ

& Paul Scherrer Institute and Laboratory for Solid State Physics ETHZ.

Supervisor: Prof. Dr. M. Troyer and Dr. M. Kenzelmann.

September 2000 - January 2006: Moscow State University named after M. V. Lomonosov,

Russian Federation. Diploma in Physics with specialization in Condensed Matter Physics.

Supervisor: Prof. Dr. A. S. Ilyushin.

1990 - 2000: Secondary education, secondary school 15, Kislovodsk, Russian Federation.

Documents

Nikolay Tsyrulin- Neutron Scattering Studies of Low-Dimensional Quantum Spin Systems