The Friedel oscillations in the presence of transport currents
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r The Friedel oscillations in the presence of transport currents
Anna Gorczyca, Maciej M. Ma´ska, and Marcin Mierzejewski
Department of Theoretical Physics, Institute of Physics, University of Silesia, 40-007 Katowice, Poland
We investigate the Friedel oscillations in a nanowire coupled to two macroscopic electrodes ofdifferent potentials. We show that the wave–length of the density oscillations monotonically increaseswith the bias voltage, whereas the amplitude and the spatial decay exponent of the oscillationsremain intact. Using the nonequilibrium Keldysh Green functions, we derive an explicit formulathat describes voltage dependence of the wave–length of the Friedel oscillations.
I. INTRODUCTION
Transport properties of nanosystems, e.g., nanowiresor single molecules, have recently been receiving signifi-cant attention mainly due to their possible application infuture electronic devices.
These properties stronglydiffer from those of macroscopic conductors. The mostimportant obstacle in theoretical investigations of thetransport phenomena originates from the coupling be-tween nanosystem and macroscopic leads. Because ofthis coupling, analysis of the electron correlations is moredifficult than in the equilibrium case.In nanosystems the charge carriers are usually dis-tributed inhomogeneously. There exist several reasonsfor such an inhomogeneity: (i)
First, it may originate from a spatial confinement. In analogy to the case of a quantum well, one may ex-pect that due to a small size of nanosystems, electrons areinhomogeneously distributed. In particular, recent scan-ning tunneling spectroscopy has shown the presence ofthe electronic standing waves at the end of a single–wallcarbon nanotube. (ii) Additionally, in the transport phenomena it mayoriginate from the applied voltage.
In this case thesystem properties are determined by the chemical poten-tials of the left and right electrodes. Different values ofthese potentials may lead to an inhomogeneous distribu-tion as well. (iii)
Similarly to the macroscopic case, inhomogeneouscharge distribution in nanosystems should occur in thepresence of impurities. (iv) Nanowires or molecular wires represent quasi–one–dimensional conductors. Therefore, phenomena typicalfor low dimensional systems, e.g., charge density wavesmay occur as well.
Recently it has been shownthat the charge density waves are strongly modified bythe bias voltage. Apart from the low–voltage regime,they are incommensurate and the corresponding wavevector decreases discontinuously with the increase of thebias voltage.In this paper we focus on the impurity–induced inho-mogeneities. It is known that an impurity in the electrongas produces local changes of the carrier concentration,known as the Friedel oscillations that asymptoticallydecay with the distance from the impurity. The most of recent theoretical investigations of the Friedel oscil-lations concerned the influence of the electronic correla-tions, that is of crucial importance in one–dimensionalsystems. It has been shown that correlationssuppress the decay of the density oscillations.
It isinteresting that these oscillations give information aboutthe impurities as well as the electron–electron interac-tion in Luttinger liquid systems. In macroscopic sys-tems, the Friedel oscillations are closely related to the sin-gularity in the response function for wave–vectors close to2 k F , where k F is the Fermi wave–vector. If the nanosys-tem is isolated (or more generally, is in equilibrium), k F is a well defined quantity. However, in the transport ex-periments the nanosystem is coupled to two macroscopicleads with different Fermi levels and the difference be-tween these Fermi energies increases with the bias volt-age. Therefore, the meaning of k F is ambiguous. Sincethe properties of a nanosystem are determined by thechemical potentials of the left and right electrodes, onemay expect that the Friedel oscillations should depend onthe voltage as well. In this paper we analyze this depen-dence using the formalism of the nonequilibrium KeldyshGreen functions. In particular, we derive an explicit for-mula for the voltage dependence of the wave–length ofthe Friedel oscillations.The paper is organized as follows: In Section II we dis-cuss a microscopic model and details of calculations. Nu-merical results are presented in Section III. Approximateanalytical formulas are derived in Section IV. The lastsection contains a discussion and concluding remarks. II. MODEL AND THE CALCULATIONSSCHEME
We investigate a one–dimensional nanowire with itsends coupled to macroscopic leads. The system underconsideration is described by the Hamiltonian H = H el + H nano + H nano − el , (1)where H el , H nano and H nano − el describe leads, nanowire,and the coupling between the wire and leads, respectively.We assume that electrodes are described by the free elec-tron gas, with a wide energy band: H el = X k ,σ,α ( ε k ,α − µ α ) c † k σα c k σα , (2)where µ α is the chemical potential and α ∈ { L,R } indi-cates the left or right electrode. µ L − µ R = eV , with V being the bias voltage. c † k σα creates an electron withmomentum k and spin σ in the electrode α . The Hamil-tonian of the nanosystem is given by H nano = − t X h ij i σ d † iσ d jσ + U X σ n lσ . (3)Here, d † iσ creates an electron with spin σ at site i of thenanosystem, n iσ = d † iσ d iσ and U is the impurity poten-tial. We have assumed a single impurity localized at site l . The coupling between the nanowire and the leads isgiven by: H nano − el = X k ,i,α,σ (cid:16) g k ,i,α c † k σα d iσ + H . c . (cid:17) . (4)In the following, we assume that the matrix elements g k ,i,α are nonzero only for the edge atoms of thenanowire.The electron distribution has been determined withthe help of the nonequilibrium Keldysh Green functions.Here, we follow the procedure used by Kostyrko andBu lka in Ref. 7. In particular, the local carrier densityis expressed by the lesser Green function, h d † iσ d iσ i = 12 πi Z d ω G
40 50 60 70 80 site n i eV = 0 eV = teV = 2 t FIG. 1: (Color online) Occupation of sites in the vicinity ofthe impurity for a 129–site nanowire. The bias voltage isexplicitly indicated in the legend. For the sake of clarity, thecurves for eV = t and eV = 2 t have been shifted downwardby 0 . .
2, respectively.
III. NUMERICAL RESULTS
We have solved numerically the system of Eqs. (5-9)for nanowires consisting of up to N = 129 lattice sites,with a single impurity in the middle of the wire. The onlynon–vanishing elements of ˆΓ’s have been assumed to befrequency independent h ˆΓ L ( ω ) i = h ˆΓ R ( ω ) i NN = Γ ,where the sites in the chain are enumerated from 1 to N .We have taken the nearest neighbor hopping integral t as an energy unit and assumed the coupling between thenanosystem and the leads as Γ = 0 .
1. The temperatureof both the leads is k B T = 0 .
01. Figure 1 shows the spa-tial distribution of electrons in the nanowire for U = 2and various values of the bias voltage. One can see strongoscillations in the vicinity of the impurity. However, dueto the coupling to the leads the electron distribution vis-ibly differs from the standard Friedel oscillations: n ( x ) = ¯ n + A cos ( Qx + η ) /x δ , (10)where the wave–vector Q = 2 k F and the parameters η and δ depend on the interaction. In our case, the ac-tual value of Q has been obtained from the fast Fouriertransform of the electron distribution n ( x ).Figure 2 shows the voltage dependence of the vector Q . In the equilibrium case ( V = 0) Q = π (with thelattice constant a = 1), so the charge oscillations arecommensurate with the lattice. Since in the half–filledcase k F = π/
2, this wave–vector remains in agreementwith the standard relation Q = 2 k F . However, whenthe bias voltage is switched on, the situation changes eV/t Q / π numerical results2arccos( eV/ t ) FIG. 2: (Color online) The wave–vector Q of the charge os-cillations calculated by means of the fast Fourier transform ofthe numerical solution of Eqs. (5-9) (solid line). The dashedline shows the fit given by Eq. 11. The numerical results havebeen obtained for the same parameters as in Fig. 1 The dis-creteness of Q ( V ) comes from the finite number of the latticesites. and the oscillations are in general no longer commensu-rate. Moreover, one can see a strong dependence of Q on the applied voltage. Q is a monotonically decreasingfunction of voltage and vanishes for a sufficiently large V . Similar situation occurs in the transport phenomenathrough one–dimensional charge density wave systems. Our numerical results indicate that the obtained Q ( V )dependence can be very accurately described by a for-mula eV = 4 t cos( Q/ . (11)The surprising simplicity of Eq. (11) is very suggestive.In the following Section we present an approximate an-alytical approach that explains such a form of Q ( V ). Itis applicable for arbitrary tight–binding Hamiltonian ofnoninteracting electrons and holds true in a wide rangeof the coupling strength Γ . The numerical results pre-sented in Figure 2 allow us to test the applied approxi-mations. IV. ANALYTICAL DISCUSSION
In the equilibrium case, the eigenstates of an isolatedsystems with periodic boundary conditions (pbc) are builtout of plane waves. The Friedel oscillations are relatedto the maximum in the response function defined as aretarded Green function: χ ( Q, ω ) = −hh ˆ ρ ( Q ) | ˆ ρ † ( Q ) ii , (12) calculated for U = 0. Here,ˆ ρ ( Q ) = X i,σ exp( iQR i ) d † iσ d iσ = X k,σ d † k + Qσ d kσ , (13)where the summation is carried out over all momenta k .In the following we demonstrate that this quantity helpsone to explain the dependence Q = Q ( V ) also in thenonequilibrium case.When the nanosystem is connected to macroscopicleads, the pbc become inappropriate since they do notreflect the geometry of the experimental setup. Then,the choice of open boundary conditions (obc) seems tobe more appropriate. For U = 0, the Hamiltonian (3)with obc can be diagonalized with the help of the uni-tary transformation, d † iσ = r N + 1 X k sin( kR i ) d † kσ , (14)where the wave–vectors k take on the following values k = πN + 1 , πN + 1 , . . . , N πN + 1 . (15)The specific form of this transformation accounts for van-ishing of the one–electron wave functions at the edges ofthe nanosystem. In this representation one gets H nano = X kσ ǫ k d † kσ d kσ , (16)where ǫ k = − t cos( k ) . (17)Although, the dispersion relation is exactly the same asfor pbc, the values of k belong to (0 , π ) instead of the 1stBrillouin zone ( − π, π ). One can apply the above trans-formation also to the remaining terms in the Hamiltonian(1) and repeat calculations presented in Sec. II. The re-sulting equations have the same structure as Eqs. (5-9)with the real space variables i replaced by the wave–vectors k . In the new representation, the Hamiltonianmatrix ˆ H is diagonal, however, the matrices ˆΓ α take onmuch more complicated form: h ˆΓ α ( ω ) i kp = 2 N + 1 X ij h ˆΓ α ( ω ) i ij sin( kR i ) sin( pR j ) . (18)In order to analyze the Friedel oscillations we investi-gate the correlation function given by Eq. (12) withˆ ρ ( Q ) = X iσ cos( QR i ) d † iσ d iσ = X kpσ B Q ( k, p ) d † kσ d pσ , (19)where B Q ( k, p ) = 12 (cid:0) δ p,k − Q − δ p,Q − k + δ p,k + Q − δ p, π − ( k + Q ) (cid:1) . (20)Equations of motion allow one to calculate the correlationfunction that in the static limit takes on the form χ ( Q, ω →
0) = X k,p,q,σ B Q ( k, p ) ǫ p − ǫ k (cid:16) B ∗ Q ( p, q ) h d † kσ d qσ i−B ∗ Q ( q, k ) h d † qσ d pσ i (cid:1) + χ ′ . (21)The second term in the above equation, χ ′ , is propor-tional to g hh d † c | d † d ii and will be neglected in the follow-ing analysis. Such simplification is justified only for aweak coupling between the nanosystem and the leads. Inorder to demonstrate the validity of this approximationwe have calculated numerically the resulting correlationfunction for nanowires consisting of 20 and 40 sites (seeFig. 3). The discreteness of the system is clearly visiblefor short nanowires, whereas for large systems the corre-lation function becomes smoother. In the latter case onecan see that χ ( Q, ω →
0) reaches its maximum value for Q given by Eq. (11), what justifies the applied approxi-mation χ ′ ≃ (cid:16) ˆΓ α (cid:17) kp ≃ δ kp Γ ′ (22)Then, all matrices in Eq. (6) become diagonal. In thenext step we assume Γ ′ →
0, what allows one to calculatethe integral over frequencies in Eq. (5). The resultingcorrelation function can be expressed as a sum of twoLindhard functions: χ ( Q, ω →
0) = χ L ( Q ) + χ R ( Q ) , (23)where χ L ( R ) ( Q ) = X k,p,σ |B Q ( k, p ) | f L ( R ) ( ǫ k ) − f L ( R ) ( ǫ p ) ǫ p − ǫ k . (24)The Fermi distribution functions of the left and rightelectrodes read f L ( ǫ k ) = f (cid:18) ǫ k − eV (cid:19) , f R ( ǫ k ) = f (cid:18) ǫ k + eV (cid:19) , (25)with f ( x ) = [exp( x/k B T ) + 1] − . The maximum of theresponse function occurs for such Q , that both the Fermi FIG. 3: (Color online) Correlation function χ = χ ( Q, V ) (Eq.21) determined numerically for 20–site (upper panel) and 40–site (lower panel) chain. functions in the numerator in Eq. (24) vanish simultane-ously. It is easy to check that for both χ L ( Q ) and χ R ( Q )this requirement is equivalent to Eq. (11).At this stage a comment on the approximation givenby Eq. (22) is necessary. It is a crude and generally in-appropriate approximation that strongly affects most ofthe system’s properties. In particular, it would stronglymodify the current–voltage characteristics. However, thecorrelation functions calculated from Eqs. (21) and (23)are almost indistinguishable, what a posteriori justifiesthe use of this approximation for the discussion of chargeinhomogeneities. This surprising result gives some in-sight into the physical mechanism of the charge distribu-tion in nanosystems in the presence of transport currents.This distribution seems to be independent of the detailsof the coupling between the nanosystem and the leadshowever, it is determined by the fact that nanosystemis connected to two macroscopic particle reservoirs withdifferent chemical potentials. V. CONCLUDING REMARKS
Using the nonequilibrium Keldysh Green functions wehave investigated the Friedel oscillations in a nanowirecoupled to two macroscopic electrodes. We have deriveda simple formula for the correlation function that de-termines the wave vector Q of the oscillations. The ap-proximate analytical expression fits the numerical resultsobtained from the Fourier transform of the electron dis-tribution very accurately. Our analysis concerns nanosys-tems described by the tight-binding Hamiltonian with thenearest neighbor hopping. However, it can be straightfor- wardly extended to account for arbitrary hopping matrixelements.The above discussion of the Friedel oscillations focuseson the voltage dependence of the wave–vector Q . Wehave found that the envelope of the charge density os-cillations is almost bias–voltage independent. It meansthat the remaining parameters characterizing the Friedeloscillations, i.e., the amplitude A and the spatial decayexponent δ , are determined predominantly by the inter-nal properties of the nanowire, whereas the wave–lengthof the oscillations depends on the bias–voltage. We be-lieve that investigations of the Friedel oscillations in thetransport phenomena should allow one to get insight intomany important parameters of the experimental setup. Acknowledgments
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