Electron transport through a strongly correlated monoatomic chain

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Surface Science () 97 7 www.elsevier.com/locate/susc Electron transport troug a strongly correlated monoatomic cain M. Krawiec *, T. Kwapiński Institute of Pysics and anotecnology Center, M.

Curie Skłodowskiej, -3 Lublin, Poland Available online February Abstract We study transport properties of a strongly correlated monoatomic cain coupled to metallic leads.

Te equation of motion tecnique in te slave boson representation as been applied to obtain analytical and numerical results.

1 Surface Science () Electron transport troug a strongly correlated monoatomic cain M. Krawiec *, T. Kwapiński Institute of Pysics and anotecnology Center, M. Curie-Skłodowska University, pl. M. Curie Skłodowskiej, -3 Lublin, Poland Available online February Abstract We study transport properties of a strongly correlated monoatomic cain coupled to metallic leads. Our system is described by tigt binding Hubbard-like model in te limit of strong on-site electron electron interactions in te wire. Te equation of motion tecnique in te slave boson representation as been applied to obtain analytical and numerical results. Calculated linear conductance of te system sows oscillatory beavior as a function of te wire lengt. We ave also found similar oscillations of te electron carge in te system. Moreover our results sow spontaneous spin polarization in te wire. Finally, we compare our results wit tose for non-interacting cain and discuss teir modifications due to te Coulomb interactions in te system. Ó Elsevier B.V. All rigts reserved. Keywords: Quantum wire; Conductance oscillations; Electron correlations. Introduction Recently one-dimensional (D) quantum wires (QW) ave attracted muc attention due to teir potential applications in nanoelectronics [] and quantum computing []. Te knowledge of te transport properties of suc structures is crucial for te design and fabrication of te nanodevices. On te oter and, te quantum wires, altoug conceptually simple, are very interesting from a scientific point of view as tey display extremely ric penomena, very often different from tose in two and tree dimensions [3,]. Te understanding of te properties of suc D objects is a major callenge in te field of nanopysics. Te conductance of te quantum wires as been studied bot experimentally and teoretically by a number of autors (see [5] for a review). Te experimental studies require advanced tecniques of fabrication of suc structures. Tose include growing of QW on metallic surfaces [,7], scanning tunneling microscope tecniques [] or mecanically controlled break junctions [5,9,]. Tose * Corresponding autor. Tel.: ; fax: addresses: krawiec@kft.umcs.lublin.pl (M. Krawiec), tomasz. kwapinski@umcs.lublin.pl (T. Kwapiński). fabrication tecniques allowed for revealing of many penomena like conductance quantization in units of G =e / [], deviations from tat (.7(e /) anomaly) [], spin carge separation (Luttinger liquid) [3], oscillations of te conductance as a function of te lengt of te cain [9,] or spontaneous spin polarization in QW [,]. Te purpose of te present paper is twofold. Te first one is to investigate te oscillations of te conductance as a function of te wire lengt in te case of strong Coulomb interactions. Te oscillatory beavior of te conductance manifests itself as a maximum of te conductance wen a number of te atoms in a wire is odd and minimum wen te number is even. Tis effect is known as te even odd conductance oscillations. Most common examples are te oscillations wit a period of two [9,,5 ] and four atoms [9]. However, te conductance can oscillate wit different (from two and four) periods, depending on te average occupation of te wire. Moreover, recently te analytical formulas for M-atom (M P ) oscillations ave been found []. However, tose analytical formulas remain valid for non-interacting wire only. In te presence of strong Coulomb interactions, te even odd oscillations wit a period of two atoms ave been also found [ ]. Te M P conductance oscillations ave been only 39-/$ - see front matter Ó Elsevier B.V. All rigts reserved. doi:./j.susc.5..53

2 9 M. Krawiec, T. Kwapiński / Surface Science () 97 7 reported for te nearest neigbor Coulomb interactions [7]. Terefore we sall study te oscillations of te conductance in te case of strong on-site Coulomb interactions and see ow teir period will be modified. Te second purpose is to see if te wire will exibit any spontaneous spin polarization in te presence of strong correlations. Suc spontaneous polarization as been observed experimentally [,]. But it is well known tat te ferromagnetism in strictly D objects is forbidden due to te Lieb Mattis teorem []. Te paper is organized as follows. In Section we present a teoretical description of te model wire; in Section 3 we sow te results of te calculations of te conductance, carge and spin polarization. Finally, in Section we provide some conclusions.. Teoretical description Our system consists of te quantum wire modeled as a cain of atoms coupled to te left L and rigt R lead described by te following Hamiltonian in te limit of strong on-site Coulomb interaction (U i!) in te slave boson representation were te real wire electron d ir is replaced by te product of te boson b i and te fermion f ir operators ðd ir ¼ b þ i f ir Þ [9 3]: H ¼ X kk c þ kkr c kkr þ X e i f þ ir f ir þ X t ij f þ ir b ib þ j f jr kkr ir iji;r þ X V LðRÞk c þ LðRÞkr bþ ðþ f ðþr þ.c., klðrþ;r were c kkr stands for te electron wit te single particle energy kk, te wave vector k and te spin r in te lead k =L,R.e i denotes te wire energy level at site i, t ij is te opping integral of te electrons between neigboring wire sites i and j, andv L(R)k is te ybridization matrix element between electrons at site () and tose in te lead L(R). In te linear response and at te zero temperature te conductance G is proportional to te total transmittance T, i.e. G ¼ e T. In our case te transmittance is given by [,3] T ðeþ ¼ X C L C R jg r r ðeþj ; ðþ r were G r r is te retarded Green function (GF) connecting te ends of te wire (sites and ) and C L(R) is te elastic rate C LðRÞ ¼ p P k jv LðRÞkj dðe LðRÞk Þ. In calculations we ave assumed constant bare density of states in te leads. Using te equation of motion tecnique for te retarded GF wit Hamiltonian () one can write te general matrix equation for G r ijr in te form ba rg b r r ¼ b r. ð3þ Due to te strong on site Coulomb interactions in te wire U i, wic is assumed to be infinity in our case, te problem cannot be solved exactly and one as to make approximations of te iger order GFs emerging in te equation of motion for te retarded GF G r ijr ¼bþ f ir j ðþ fjr þb. We ave used Hubbard I like approximation [3] according to wic te GF fi r þ f i rb þ k f kr j fjr þb is approximated by fi r þ f i rib þ k f kr j fjr þb and te oter iger order GFs are neglected. Tis approximation is reasonably good for not very large values of te opping t and neglects iger order processes, like for example, te Kondo effect. Witin te present approximation sceme, ba r in Eq. (3) is tridiagonal symmetric matrix wit te elements, ba r ¼ðE e i Þd i;j þ i C ðd i;d j; þ d i; d j; Þ t½ð n i r Þd i;iþ þð n iþ r Þd iþ;i Š; and b r is te diagonal matrix of te form, b r ¼ð n i r Þd ij ; wit n ir ¼fir þf iri ¼ p occupation of te electrons wit spin r at site i. 3. Results ðþ ð5þ R de ImG r iirðeþ being te average In numerical calculations we ave assumed all te wire site energies to be equal (e i = e ) and similarly opping integrals t ij = t. All energies are measured wit respect to te leads Fermi energy E F = in units of C = C L = C R =. Moreover, te occupation n ir is calculated self-consistently on eac wire site. To find te condition for M-atom conductance oscillations one as to solve te relation: T = T +M, were T (T +M ) is te transmittance of te wire consisting of ( + M) atoms, given by Eq. (). In general, for U i! it is not possible to get an analytical expression for te oscillations condition witout furter assumptions. ote tat for U i = te problem can be solved exactly and suc condition can be found []. In tis case it reads cos pl M ¼ E F e, were l =,,..., M. t In te case of U i!, bot matrices ba r and b r depend on te occupation on eac wire site n i,r and tus te problem as to be solved numerically. However, if one assumes tat te occupation is te same on eac wire site and does not depend on spin, i.e., n ir = n r = n r, similar M-atom oscillations condition as for U i = can be found. In tis case it reads cos pl M ¼ E F e tð n r Þ ; ðþ wit l =,,..., M. Unfortunately, one as to know te average occupation n r. Te only case of M = can be solved analytically. As one can see from Eq. () te period of two can be obtained for E F e =, i.e., wen all te wire single particle energy levels coincide wit te Fermi energy. In Fig. te total linear conductance G ¼ P r G r is plotted as a function of te wire lengt and te energy level e for t =. As one can see all te figures sow similar patterns wit te regions of large and small conductances.

3 M. Krawiec, T. Kwapiński / Surface Science () ε ε ε G U= G ferro Fig.. Te total linear conductance G ¼ P r G r as a function of te wire lengt and te wire energy level e for U i = (top panel), U i!in paramagnetic configuration n ir = n i r (middle panel) and ferromagnetic one (bottom panel). Moreover, for e = beavior of te conductance is te same in all tree cases wic leads to te conclusion tat correlations are not important in tis case. It always sows even odd (M = ) oscillations. Away from e = correlations strongly modify te conductance, sifting te maxima of G (for fixed ) towards lower (iger) energies for e > (e < ). Tis is due to te modification of te wire opping, wic depends now on te occupation n ir (see Eq. ()). Tis effect also leads to te strong asymmetry for positive and negative energies. For negative (positive) wire energies te occupation is large (small), terefore effective opping ~t ¼ tð n i r Þ is small (large), tus te conductance decreases (increases). ote tat tere is no suc asymmetry in te case of U i =. Moreover, different periods of te conductance oscillations can be observed, depending on te position of te wire energy level. Anoter important finding is tat te wire sows spontaneous spin polarization. It is well known tat in strictly D wire te spin polarization is proibited due to te Lieb Mattis teorem []. However, tis teorem is valid for infinite wire only. In te experimental situation te wire is always connected to te electrodes and tis is wy te spin polarization is observed experimentally [,]. Interestingly, it was recently predicted tat even an infinite wire, but of te zig zag sape, can also exibit te spin polarization [33]. Te total conductance in ferromagnetic case is sown in te bottom panel of Fig.. Again, unlike for U i =, te conductance pattern sows strong asymmetry for positive and negative energies. ote tat for negative energies tere are no differences between G ferro (bottom panel) and (middle panel). For suc energies, iterations always converge to paramagnetic solution. Tis can be again explained by effect of te opping modification. For negative energies te wire occupation is large, tus te effective opping ~t ¼ tð n i r Þ is small. One can imagine in tis case tat te electrons are more localized and it is more convenient for tem to spend more time on te same site tan move to anoter one. Moreover in te case of te lack of te inter-site interactions any collective penomenon is not possible. Te situation is different for positive energies. In tis case te electrons are more mobile, as te effective opping is larger due to te small values of te wire occupations. Tus tey interact wit eac oter via opping and it is possible and energetically more favorable to get te ferromagnetic state. Te opping, wic is te correction to te position of te wire energy levels, leads to te effective splitting of tese levels. Tus te conductance is spin polarized in tis case. It can be read off from Fig., were te difference between spin up and spin down conductance is displayed. It turns out tat te strongest differences between G " and G # can be found for intermediate values of e. For energies close to E F te modifications are weak due to te small values of te opping wile for iger energies te wire occupation is very small and tus te difference between spin dependent effective opping can be neglected. Finally it is wortwile to note tat for = (single atom) tere is no ferromagnetic solution as te ferromagnetism is governed by inter-site opping in our case. Weter tis spontaneous spin polarization is a true effect or a drawback of te approximation used remains an open question, as it is known tat te mean field like teories overestimate te role of te magnetism. Te problem will be furter studied. Corresponding spin polarization n " n # ðn r ¼ P i n ir=þ is sown in Fig. 3. As one can see te spin polarization pattern is similar to tat of te conductance differences (see Fig. ).

4 7 M. Krawiec, T. Kwapiński / Surface Science () 97 7 G -G ε = ε ε =-.9 ε = ε =-3.3 Fig.. Te difference between spin up and spin down conductance as a function of te wire lengt and te wire energy level e..... ε =.7 n -n.3 Fig.. Te total conductance (left panels) and te occupation (rigt panels) vs. wire lengt. Te left panels sow te conductance wit different oscillations periods ( ) from top to bottom. Te positions of te wire energy levels are indicated in te figure. ε Fig. 3. Te spin polarization as a function of te wire lengt and te wire energy level e. In Fig. we sow te conductance (left panels) and te occupation (rigt panels) as a function of te wire lengt for a number of te energy levels e in te paramagnetic configuration. Te values of e ave been cosen in suc a way tat tey lead to te maxima of te conductance for =, and 3 atom wires. For example, te maxima of te conductance of te = atom wire corresponds to e =.9 and 3.35 (see te middle panel of Fig. ). In te same way obtained maxima in U i = case give te periods of te conductance oscillations. To find M-atom period it is enoug to determine te maxima of te conductance for = M atom wire []. As one can read off from Fig., depending on e, one gets different periods of te conductance oscillations. Moreover, except for te special case of M =, te amplitude of te oscillations decreases wit te wire lengt. Tis is a kind of damped.. oscillations. o suc effect as been observed for U i = []. At tis point we would like to comment on te oter results known in te literature. Te even odd (M =) oscillations problem was extensively studied witin te second-order perturbation teory in U i (SOPT) [,] and te numerical renormalization group (RG) approac [3,]. Te results sow similar beavior of te conductance for odd number of atoms in a wire: it always reaces te unitary limit ð eþ, independently of U i. Suc beavior is a consequence of te Kondo effect. However, in our case te situation is sligtly different, as we get M = oscillations in te mixed valence regime only (e = ), were te Kondo effect is excluded. Tus our even odd oscillations are caused by te resonances associated wit te energy level structure of te cain rater tan te Kondo effect. On te oter and, te conductance for even number of atoms in a wire is strongly suppressed, in agreement wit SOPT [,] and RG approaces [3,]. However, RG calculations sow tat te conductance exponentially depends on U i, and in te limit of U i!vanises, contrary to our results, as we get non-zero values of G (see Fig. ), depending on te opping t. Te conductance vanises in te limit of very large or very small values of t. Interestingly, wen t = C/, te conductance reaces te unitary limit and sows no oscillations, i.e., is equal to e for even and odd. Fig. (rigt panels) sows te wire lengt dependent occupation wic also oscillates wit te same period as te conductance does, except for te special case of e = (M = ), were te occupation remains constant.

5 M. Krawiec, T. Kwapiński / Surface Science () Conclusions ε Moreover, te Coulomb interactions U i lead to te reduction of te occupation oscillation amplitude. Similar effect, albeit for small U i, as been found witin self-consistent Hartree Fock approximation [3]. In te ferromagnetic case te situation is more complex. For positive values of e due to te splitting of te conductance maxima (see Fig. ) no regular oscillations ave been observed. On te oter and, for e < one gets suc oscillations but in tis case te solutions remain always paramagnetic. Fig. 5 sows te comparison of te conductance and te occupations in paramagnetic and ferromagnetic configurations for te wire consisting of five atoms. For negative energies, as discussed before, tere are paramagnetic solutions only. For > and e > te ferromagnetic solutions emerge in certain energy regimes. In tis case n " 5 n # and te resulting conductance peaks are split. Interestingly, te spin polarizations occurs only in te regimes were te occupation as a large slope or te total conductance as a maximum. Moreover, te splitting of te conductance leads to te fact tat G ferro sows more maxima tan. A number of maxima of G ferro are related to te wire lengt (number of atoms ) and for odd it gives (3 )/ maxima, wile for even, one observes 3/ maxima. In te paramagnetic case it is always equal to a number of atoms in te wire (compare top panels of Fig. 5). Finally we would like to comment on te validity of our approac. Te present calculations completely neglect te Kondo effect wic is important at low temperatures and e d <. We expect some modifications in tis regime, as tis effect leads to te corrections of te conductance of te order of e. Tus our results apply for temperatures iger tan te Kondo temperature. On te oter and, we do not expect any qualitative modifications in te mixed valence and te empty regimes (e d P ). G ferro ε Fig. 5. Te total conductance in paramagnetic (left top panel) and in ferromagnetic (rigt top panel) configuration and corresponding occupations (bottom panels) vs. e of five atom wires., n -σ In summary we ave studied te conductance oscillations of te strongly interacting wire as a function of te wire lengt. We ave found tat strong Coulomb interactions significantly modify te periods of te oscillations sowing strong asymmetry for negative and positive wire energy levels. Tey also lead to te suppression of te conductance wit increasing wire lengt. Tere are no suc effects for non-interacting wire. Moreover, strong interactions lead to te spontaneous spin polarization for positive wire energies, observed in experiments. Acknowledgements Tis work as been supported by te grant no. P3B of te Polis Committee of Scientific Researc. T.K. tanks te Foundation for Polis Science for a financial support. References [] D.R. Bowler, J. Pys. Cond. Matter () R7. [] A. Bertoni, S. Reggiani, Semicond. Sci. Tecnol. 9 () S3. [3] J.M. Luttinger, J. Mat. Pys. (93) 5. [] F.D.M. Haldane, J. Pys. C: Solid State Pys. (9) 55. [5]. Agraït et al., Pys. Rep. 377 (3). [] M. Jałocowski et al., Surf. Sci. 375 (997) 3. [7] M. Krawiec et al., Pys. Status Solidi B (5) 33. [] A. Yazdani et al., Science 7 (99) 9. [9] C.J. Muller et al., Pys. Rev. Lett. 9 (99). [] R.H.M. Smit et al., Pys. Rev. Lett. 7 (). [] B.J. Wees et al., Pys. Rev. Lett. (9). [] K.J. Tomas et al., Pys. Rev. Lett. 77 (99) 35. [3] O.M. Auslaender et al., Science 3 (5). [] B.E. Kane et al., Appl. Pys. Lett. 7 (99) 35. [5] H.-S. Sim et al., Pys. Rev. Lett. 7 () 93. [].D. Lang, P. Avouris, Pys. Rev. Lett. (99) 355;.D. Lang, P. Avouris, Pys. Rev. Lett. () 35. [7] E.G. Emberly, G. Kirczenov, Pys. Rev. B (999). [] R. Gutierrez et al., Acta Pys. Pol. 3 () 3. [9] K.S. Tygesen, K.W. Jacobsen, Pys. Rev. Lett. 9 (3). [] T. Kwapiński, J. Pys.: Condens. Matter 7 (5) 59. [] A. Oguri, Pys. Rev. B 59 (999) ; A. Oguri, Pysica B () 93; A. Oguri, Pys. Rev. B 3 () 535; A. Oguri, J. Pys. Soc. Jpn. 7 (). [] Y. Tanaka, A. Oguri, J. Pys. Soc. Jpn. 73 () 3. [3] A. Oguri, A.C. Hewson, J. Pys. Soc. Jpn. 7 (5) 9. [] A. Oguri et al., J. Pys. Soc. Jpn. 7 (5) 55. [5] R.A. Molina et al., Pys. Rev. B 7 (3) 353. [] V. Meden, U. Scollwöck, Pys. Rev. B 7 (3) [7] R.A. Molina et al., Europys. Lett. 7 () 9. [] E. Lieb, D. Mattis, Pys. Rev. 5 (9). [9] P. Coleman, Pys. Rev. B 9 (9) 335. [3] J.C. Le Guillou, E. Ragoucy, Pys. Rev. B 5 (995) 3. [3] M. Krawiec, K.I. Wysokiński, Pys. Rev. B 59 (999) 95. [3] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, 995. [33] A.D. Klironomos et al., <cond-mat/5737>. [3] T. Kostyrko, B.R. Bułka, Pys. Rev. B 7 (3) 533.

A = h w (1) Error Analysis Physics 141

A = h w (1) Error Analysis Physics 141 Introduction In all brances of pysical science and engineering one deals constantly wit numbers wic results more or less directly from experimental observations. Experimental observations always ave inaccuracies.