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Surface Science 600 (2006) 1697–1701

Electron transport through a strongly correlated monoatomic chain

M. Krawiec *, T. Kwapinski

Institute of Physics and Nanotechnology Center, M. Curie-Skłodowska University, pl. M. Curie Skłodowskiej 1, 20-031 Lublin, Poland

Available online 8 February 2006

Abstract

We study transport properties of a strongly correlated monoatomic chain coupled to metallic leads. Our system is described by tightbinding Hubbard-like model in the limit of strong on-site electron–electron interactions in the wire. The equation of motion technique inthe slave boson representation has been applied to obtain analytical and numerical results. Calculated linear conductance of the systemshows oscillatory behavior as a function of the wire length. We have also found similar oscillations of the electron charge in the system.Moreover our results show spontaneous spin polarization in the wire. Finally, we compare our results with those for non-interactingchain and discuss their modifications due to the Coulomb interactions in the system.� 2006 Elsevier B.V. All rights reserved.

Keywords: Quantum wire; Conductance oscillations; Electron correlations

1. Introduction

Recently one-dimensional (1D) quantum wires (QW)have attracted much attention due to their potential appli-cations in nanoelectronics [1] and quantum computing [2].The knowledge of the transport properties of such struc-tures is crucial for the design and fabrication of thenanodevices. On the other hand, the quantum wires,although conceptually simple, are very interesting from ascientific point of view as they display extremely rich phe-nomena, very often different from those in two and threedimensions [3,4]. The understanding of the properties ofsuch 1D objects is a major challenge in the field ofnanophysics.

The conductance of the quantum wires has been studiedboth experimentally and theoretically by a number ofauthors (see [5] for a review). The experimental studies re-quire advanced techniques of fabrication of such struc-tures. Those include growing of QW on metallic surfaces[6,7], scanning tunneling microscope techniques [8] ormechanically controlled break junctions [5,9,10]. Those

0039-6028/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.susc.2005.11.053

* Corresponding author. Tel.: +48 81 537 6146; fax: +48 81 537 6191.E-mail addresses: krawiec@kft.umcs.lublin.pl (M. Krawiec), tomasz.

kwapinski@umcs.lublin.pl (T. Kwapinski).

fabrication techniques allowed for revealing of manyphenomena like conductance quantization in units ofG0 = 2e2/h [11], deviations from that (0.7(2e2/h) anomaly)[12], spin–charge separation (Luttinger liquid) [13], oscilla-tions of the conductance as a function of the length of thechain [9,10] or spontaneous spin polarization in QW[12,14].

The purpose of the present paper is twofold. The firstone is to investigate the oscillations of the conductance asa function of the wire length in the case of strong Coulombinteractions. The oscillatory behavior of the conductancemanifests itself as a maximum of the conductance when anumber of the atoms in a wire is odd and minimum whenthe number is even. This effect is known as the even–oddconductance oscillations. Most common examples are theoscillations with a period of two [9,10,15–18] and fouratoms [19]. However, the conductance can oscillate withdifferent (from two and four) periods, depending on theaverage occupation of the wire. Moreover, recently theanalytical formulas for M-atom (M P 2) oscillations havebeen found [20]. However, those analytical formulas re-main valid for non-interacting wire only. In the presenceof strong Coulomb interactions, the even–odd oscillationswith a period of two atoms have been also found [21–26].The M P 2 conductance oscillations have been only

1698 M. Krawiec, T. Kwapinski / Surface Science 600 (2006) 1697–1701

reported for the nearest neighbor Coulomb interactions[27]. Therefore we shall study the oscillations of the con-ductance in the case of strong on-site Coulomb interactionsand see how their period will be modified.

The second purpose is to see if the wire will exhibit anyspontaneous spin polarization in the presence of strongcorrelations. Such spontaneous polarization has been ob-served experimentally [12,14]. But it is well known thatthe ferromagnetism in strictly 1D objects is forbidden dueto the Lieb–Mattis theorem [28].

The paper is organized as follows. In Section 2 we pres-ent a theoretical description of the model wire; in Section 3we show the results of the calculations of the conductance,charge and spin polarization. Finally, in Section 4 we pro-vide some conclusions.

2. Theoretical description

Our system consists of the quantum wire modeled as achain of N atoms coupled to the left L and right R lead de-scribed by the following Hamiltonian in the limit of strongon-site Coulomb interaction (Ui!1) in the slave bosonrepresentation where the real wire electron dir is replacedby the product of the boson bi and the fermion fir operatorsðdir ¼ bþi firÞ [29–31]:

H ¼Xkkr

�kkcþkkrckkr þX

ir

eif þir fir þXhiji;r

tijf þir bibþj fjr

þX

k2LðRÞ;rV LðRÞkcþLðRÞkrbþ1ðNÞf1ðNÞr þ h.c., ð1Þ

where ckkr stands for the electron with the single particleenergy �kk, the wave vector k and the spin r in the leadk = L,R. ei denotes the wire energy level at site i, tij is thehopping integral of the electrons between neighboring wiresites i and j, and VL(R)k is the hybridization matrix elementbetween electrons at site 1(N) and those in the lead L(R).

In the linear response and at the zero temperature theconductance G is proportional to the total transmittanceT, i.e. G ¼ 2e2

h T . In our case the transmittance is given by[20,32]

T N ðEÞ ¼X

r

CLCRjGr1NrðEÞj

2; ð2Þ

where Gr1Nr is the retarded Green function (GF) connecting

the ends of the wire (sites 1 and N) and CL(R) is the elasticrate CLðRÞ ¼ 2p

PkjV LðRÞkj2dðE � �LðRÞkÞ. In calculations we

have assumed constant bare density of states in the leads.Using the equation of motion technique for the retarded

GF with Hamiltonian (1) one can write the general matrixequation for Gr

ijr in the form

bArbGr

r ¼ bN r. ð3ÞDue to the strong on site Coulomb interactions in the

wire Ui, which is assumed to be infinity in our case, theproblem cannot be solved exactly and one has to makeapproximations of the higher order GFs emerging in theequation of motion for the retarded GF Gr

ijr ¼ hhbþ1 fir j

f þjrbjiiE. We have used Hubbard I like approximation [31]according to which the GF hhf þi�rfi�rbþk fkr j f þjrbjiiE isapproximated by hf þi�rfi�rihhbþk fkr j f þjrbjiiE and the otherhigher order GFs are neglected. This approximation is rea-sonably good for not very large values of the hopping t andneglects higher order processes, like for example, the Kondoeffect.

Within the present approximation scheme, bAr in Eq. (3)is N · N tridiagonal symmetric matrix with the elements,

bAr ¼ ðE � eiÞdi;j þ iC2ðdi;1dj;1 þ di;Ndj;N Þ

� t½ð1� ni�rÞdi;iþ1 þ ð1� niþ1�rÞdiþ1;i�; ð4Þ

and bN r is the diagonal matrix of the form,

bN r ¼ ð1� ni�rÞdij; ð5Þwith nir ¼ hf þir firi ¼ � 1

p

RdE ImGr

iirðEÞ being the averageoccupation of the electrons with spin r at site i.

3. Results

In numerical calculations we have assumed all the wiresite energies to be equal (ei = e0) and similarly hopping inte-grals tij = t. All energies are measured with respect to theleads Fermi energy EF = 0 in units of C = CL = CR = 1.Moreover, the occupation nir is calculated self-consistentlyon each wire site.

To find the condition for M-atom conductance oscilla-tions one has to solve the relation: TN = TN+M, where TN

(TN+M) is the transmittance of the wire consisting of N(N + M) atoms, given by Eq. (2). In general, for Ui!1it is not possible to get an analytical expression for theoscillations condition without further assumptions. Notethat for Ui = 0 the problem can be solved exactly and suchcondition can be found [20]. In this case it readscos pl

M

� �¼ EF�e0

2t , where l = 1,2, . . . , M � 1.

In the case of Ui!1, both matrices bAr and bN r dependon the occupation on each wire site ni,r and thus the prob-lem has to be solved numerically. However, if one assumesthat the occupation is the same on each wire site and doesnot depend on spin, i.e., nir = nr = n�r, similar M-atomoscillations condition as for Ui = 0 can be found. In thiscase it reads

cosplM

� �¼ EF � e0

2tð1� nrÞ; ð6Þ

with l = 1,2, . . . , M � 1. Unfortunately, one has to knowthe average occupation nr. The only case of M = 2 canbe solved analytically. As one can see from Eq. (6) the per-iod of two can be obtained for EF � e0 = 0, i.e., when allthe wire single particle energy levels coincide with the Fer-mi energy.

In Fig. 1 the total linear conductance G ¼P

rGr is plot-ted as a function of the wire length N and the energy levele0 for t = 4. As one can see all the figures show similar pat-terns with the regions of large and small conductances.

0

0.5

1

1.5

2GU=0

-6

-4

-2

0

2

4

6

8

ε 0

0

0.5

1

1.5

2Gpara

-6

-4

-2

0

2

4

6

8

ε 0

0

0.5

1

1.5

2Gferro

2 4 6 8 10

N

-6

-4

-2

0

2

4

6

8

ε 0

Fig. 1. The total linear conductance G ¼P

rGr as a function of the wirelength N and the wire energy level e0 for Ui = 0 (top panel), Ui!1 inparamagnetic configuration nir = ni�r (middle panel) and ferromagneticone (bottom panel).

M. Krawiec, T. Kwapinski / Surface Science 600 (2006) 1697–1701 1699

Moreover, for e0 = 0 behavior of the conductance is thesame in all three cases which leads to the conclusion thatcorrelations are not important in this case. It always showseven–odd (M = 2) oscillations. Away from e0 = 0 correla-tions strongly modify the conductance, shifting the maximaof G (for fixed N) towards lower (higher) energies for e0 > 0(e0 < 0). This is due to the modification of the wire hop-ping, which depends now on the occupation nir (see Eq.(4)). This effect also leads to the strong asymmetry forpositive and negative energies. For negative (positive) wireenergies the occupation is large (small), therefore effective

hopping ~t ¼ tð1� ni�rÞ is small (large), thus the conduc-tance decreases (increases). Note that there is no suchasymmetry in the case of Ui = 0. Moreover, different peri-ods of the conductance oscillations can be observed,depending on the position of the wire energy level.

Another important finding is that the wire shows spon-taneous spin polarization. It is well known that in strictly1D wire the spin polarization is prohibited due to theLieb–Mattis theorem [28]. However, this theorem is validfor infinite wire only. In the experimental situation the wireis always connected to the electrodes and this is why thespin polarization is observed experimentally [12,14]. Inter-estingly, it was recently predicted that even an infinite wire,but of the zig–zag shape, can also exhibit the spin polariza-tion [33].

The total conductance in ferromagnetic case is shown inthe bottom panel of Fig. 1. Again, unlike for Ui = 0, theconductance pattern shows strong asymmetry for positiveand negative energies. Note that for negative energies thereare no differences between Gferro (bottom panel) and Gpara

(middle panel). For such energies, iterations alwaysconverge to paramagnetic solution. This can be again ex-plained by effect of the hopping modification. For negativeenergies the wire occupation is large, thus the effective hop-ping ~t ¼ tð1� ni�rÞ is small. One can imagine in this casethat the electrons are more localized and it is more conve-nient for them to spend more time on the same site thanmove to another one. Moreover in the case of the lack ofthe inter-site interactions any collective phenomenon isnot possible. The situation is different for positive energies.In this case the electrons are more mobile, as the effectivehopping is larger due to the small values of the wire occu-pations. Thus they interact with each other via hoppingand it is possible and energetically more favorable to getthe ferromagnetic state. The hopping, which is the correc-tion to the position of the wire energy levels, leads to theeffective splitting of these levels. Thus the conductance isspin polarized in this case. It can be read off from Fig. 2,where the difference between spin up and spin down con-ductance is displayed. It turns out that the strongest differ-ences between G" and G# can be found for intermediatevalues of e0. For energies close to EF the modificationsare weak due to the small values of the hopping while forhigher energies the wire occupation is very small and thusthe difference between spin dependent effective hoppingcan be neglected. Finally it is worthwhile to note that forN = 1 (single atom) there is no ferromagnetic solution asthe ferromagnetism is governed by inter-site hopping inour case. Whether this spontaneous spin polarization is atrue effect or a drawback of the approximation used re-mains an open question, as it is known that the mean fieldlike theories overestimate the role of the magnetism. Theproblem will be further studied.

Corresponding spin polarization n" � n# ðnr ¼P

inir=NÞis shown in Fig. 3. As one can see the spin polarization pat-tern is similar to that of the conductance differences (seeFig. 2).

0

0.1

0.2

0.3

n↑-n↓

2 4 6 8 10

N

0

2

4

6

8

ε 0

Fig. 3. The spin polarization as a function of the wire length N and thewire energy level e0.

0

0.4

0.8

1.2

1.6

2

Gpa

ra

0

0.4

0.8

1.2

1.6

Gpa

ra

0

0.4

0.8

1.2

1.6

2 4 6 8

Gpa

ra

N

0

0.2

0.4

n σ ε0=0

0

0.2

0.4

n σ

ε0=-2.298

ε0=3.135

0

0.2

0.4

2 4 6 8

n σ

N

ε0=-3.1634

ε0=4.71

Fig. 4. The total conductance (left panels) and the occupation (rightpanels) vs. wire length. The left panels show the conductance with differentoscillations periods (2–4) from top to bottom. The positions of the wireenergy levels are indicated in the figure.

-0.8

-0.4

0

0.4

0.8

G↑-G↓

2 4 6 8 10N

0

2

4

6

8

ε 0

Fig. 2. The difference between spin up and spin down conductance as afunction of the wire length N and the wire energy level e0.

1700 M. Krawiec, T. Kwapinski / Surface Science 600 (2006) 1697–1701

In Fig. 4 we show the conductance (left panels) and theoccupation (right panels) as a function of the wire lengthfor a number of the energy levels e0 in the paramagneticconfiguration. The values of e0 have been chosen in sucha way that they lead to the maxima of the conductancefor N = 1, 2 and 3 atom wires. For example, the maximaof the conductance of the N = 2 atom wire corresponds toe0 = �2.298 and 3.135 (see the middle panel of Fig. 1). Inthe same way obtained maxima in Ui = 0 case give theperiods of the conductance oscillations. To find M-atomperiod it is enough to determine the maxima of the con-ductance for N = M � 1 atom wire [20]. As one can readoff from Fig. 4, depending on e0, one gets different periodsof the conductance oscillations. Moreover, except for thespecial case of M = 2, the amplitude of the oscillations de-creases with the wire length. This is a kind of damped

oscillations. No such effect has been observed for Ui = 0[20].

At this point we would like to comment on the otherresults known in the literature. The even–odd (M = 2)oscillations problem was extensively studied within the sec-ond-order perturbation theory in Ui (SOPT) [21,22] and thenumerical renormalization group (NRG) approach [23,24].The results show similar behavior of the conductance forodd number of atoms in a wire: it always reaches the unitarylimit ð2e2

h Þ, independently of Ui. Such behavior is a conse-quence of the Kondo effect. However, in our case the situa-tion is slightly different, as we get M = 2 oscillations in themixed valence regime only (e0 = 0), where the Kondo effectis excluded. Thus our even–odd oscillations are caused bythe resonances associated with the energy level structureof the chain rather than the Kondo effect. On the otherhand, the conductance for even number of atoms in a wireis strongly suppressed, in agreement with SOPT [21,22] andNRG approaches [23,24]. However, NRG calculationsshow that the conductance exponentially depends on Ui,and in the limit of Ui!1 vanishes, contrary to our results,as we get non-zero values of G (see Fig. 4), depending onthe hopping t. The conductance vanishes in the limit of verylarge or very small values of t. Interestingly, when t = C/2,the conductance reaches the unitary limit and shows nooscillations, i.e., is equal to 2e2

h for even and odd N.Fig. 4 (right panels) shows the wire length dependent

occupation which also oscillates with the same period asthe conductance does, except for the special case ofe0 = 0 (M = 2), where the occupation remains constant.

0

0.5

1

1.5

2

Gpa

ra

0

0.1

0.2

0.3

0.4

0.5

-4 -2 0 2 4 6 8

n σ

ε0

0

0.5

1

1.5

2

Gfe

rro

0

0.1

0.2

0.3

0.4

0.5

-4 -2 0 2 4 6 8

n σ ,

n -σ

ε0

Fig. 5. The total conductance in paramagnetic (left top panel) and inferromagnetic (right top panel) configuration and corresponding occupa-tions (bottom panels) vs. e0 of five atom wires.

M. Krawiec, T. Kwapinski / Surface Science 600 (2006) 1697–1701 1701

Moreover, the Coulomb interactions Ui lead to the reduc-tion of the occupation oscillation amplitude. Similar effect,albeit for small Ui, has been found within self-consistentHartree–Fock approximation [34].

In the ferromagnetic case the situation is more complex.For positive values of e0 due to the splitting of the conduc-tance maxima (see Fig. 1) no regular oscillations have beenobserved. On the other hand, for e0 < 0 one gets suchoscillations but in this case the solutions remain alwaysparamagnetic. Fig. 5 shows the comparison of the conduc-tance and the occupations in paramagnetic and ferromag-netic configurations for the wire consisting of five atoms.For negative energies, as discussed before, there are para-magnetic solutions only. For N > 1 and e0 > 0 the ferro-magnetic solutions emerge in certain energy regimes. Inthis case n"5 n# and the resulting conductance peaks aresplit. Interestingly, the spin polarizations occurs only inthe regimes where the occupation has a large slope or thetotal conductance has a maximum. Moreover, the splittingof the conductance leads to the fact that Gferro shows moremaxima than Gpara. A number of maxima of Gferro are re-lated to the wire length (number of atoms—N) and forodd N it gives (3N � 1)/2 maxima, while for even N, oneobserves 3N/2 maxima. In the paramagnetic case it is al-ways equal to a number of atoms N in the wire (comparetop panels of Fig. 5).

Finally we would like to comment on the validity of ourapproach. The present calculations completely neglect theKondo effect which is important at low temperatures anded < 0. We expect some modifications in this regime, as thiseffect leads to the corrections of the conductance of the or-der of e2

h . Thus our results apply for temperatures higherthan the Kondo temperature. On the other hand, we donot expect any qualitative modifications in the mixed va-lence and the empty regimes (ed P 0).

4. Conclusions

In summary we have studied the conductance oscilla-tions of the strongly interacting wire as a function of thewire length. We have found that strong Coulomb interac-tions significantly modify the periods of the oscillationsshowing strong asymmetry for negative and positive wireenergy levels. They also lead to the suppression of the con-ductance with increasing wire length. There are no sucheffects for non-interacting wire. Moreover, strong inter-actions lead to the spontaneous spin polarization for posi-tive wire energies, observed in experiments.

Acknowledgements

This work has been supported by the grant no.1 P03B004 28 of the Polish Committee of Scientific Research.T.K. thanks the Foundation for Polish Science for a finan-cial support.

References

[1] D.R. Bowler, J. Phys. Cond. Matter 16 (2004) R721.[2] A. Bertoni, S. Reggiani, Semicond. Sci. Technol. 19 (2004) S113.[3] J.M. Luttinger, J. Math. Phys. 4 (1963) 1154.[4] F.D.M. Haldane, J. Phys. C: Solid State Phys. 14 (1981) 2585.[5] N. Agraıt et al., Phys. Rep. 377 (2003) 81.[6] M. Jałochowski et al., Surf. Sci. 375 (1997) 203.[7] M. Krawiec et al., Phys. Status Solidi B 242 (2005) 332.[8] A. Yazdani et al., Science 272 (1996) 1921.[9] C.J. Muller et al., Phys. Rev. Lett. 69 (1992) 140.

[10] R.H.M. Smit et al., Phys. Rev. Lett. 87 (2001) 266102.[11] B.J. Wees et al., Phys. Rev. Lett. 60 (1988) 848.[12] K.J. Thomas et al., Phys. Rev. Lett. 77 (1996) 135.[13] O.M. Auslaender et al., Science 308 (2005) 88.[14] B.E. Kane et al., Appl. Phys. Lett. 72 (1998) 3506.[15] H.-S. Sim et al., Phys. Rev. Lett. 87 (2001) 096803.[16] N.D. Lang, Ph. Avouris, Phys. Rev. Lett. 81 (1998) 3515;

N.D. Lang, Ph. Avouris, Phys. Rev. Lett. 84 (2000) 358.[17] E.G. Emberly, G. Kirczenov, Phys. Rev. B 60 (1999) 6028.[18] R. Gutierrez et al., Acta Phys. Pol. 32 (2001) 443.[19] K.S. Thygesen, K.W. Jacobsen, Phys. Rev. Lett. 91 (2003) 146801.[20] T. Kwapinski, J. Phys.: Condens. Matter 17 (2005) 5849.[21] A. Oguri, Phys. Rev. B 59 (1999) 12240;

A. Oguri, Physica B 284–288 (2000) 1932;A. Oguri, Phys. Rev. B 63 (2001) 115305;A. Oguri, J. Phys. Soc. Jpn. 70 (2001) 2666.

[22] Y. Tanaka, A. Oguri, J. Phys. Soc. Jpn. 73 (2004) 163.[23] A. Oguri, A.C. Hewson, J. Phys. Soc. Jpn. 74 (2005) 988.[24] A. Oguri et al., J. Phys. Soc. Jpn. 74 (2005) 1554.[25] R.A. Molina et al., Phys. Rev. B 67 (2003) 235306.[26] V. Meden, U. Schollwock, Phys. Rev. B 67 (2003) 193303.[27] R.A. Molina et al., Europhys. Lett. 67 (2004) 96.[28] E. Lieb, D. Mattis, Phys. Rev. 125 (1962) 164.[29] P. Coleman, Phys. Rev. B 29 (1984) 3035.[30] J.C. Le Guillou, E. Ragoucy, Phys. Rev. B 52 (1995) 2403.[31] M. Krawiec, K.I. Wysokinski, Phys. Rev. B 59 (1999) 9500.[32] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge

University Press, Cambridge, 1995.[33] A.D. Klironomos et al., <cond-mat/0507387>.[34] T. Kostyrko, B.R. Bułka, Phys. Rev. B 67 (2003) 205331.