Cornean, Horia Decebal

Jensen, Arne

Moldoveanu, Valeriu

English

We give a short presentation of two recent results. The firrst one is a rigorous proof of the Landauer-Büttiker formula, and the second one concerns resonant quantum transport. The detailed results are in [2]. In the last section we present the results of some numerical computations on a model system.Concerning the literature, then see the starting point of our work, [6]. In [4] a related, but different, problem is studied. See also [5] and the recentwork [1].We give a short presentation of two recent results. The firrst one is a rigorous proof of the Landauer-Büttiker formula, and the second one concerns resonant quantum transport. The detailed results are in [2]. In the last section we present the results of some numerical computations on a model system.Concerning the literature, then see the starting point of our work, [6]. In [4] a related, but different, problem is studied. See also [5] and the recentwork [1]

    Aalborg UniversitetThe Landauer-Büttiker formula and resonant quantum transportCornean, Horia Decebal; Jensen, Arne; Moldoveanu, ValeriuPublication date:2005Document VersionPublisher's PDF, also known as Version of recordLink to publication from Aalborg UniversityCitation for published version (APA):Cornean, H. D., Jensen, A., & Moldoveanu, V. (2005). The Landauer-Büttiker formula and resonant quantumtransport. Aalborg Universitetsforlag. Research Report Series No. R-2005-17General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            ? Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            ? You may not further distribute the material or use it for any profit-making activity or commercial gain            ? You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access tothe work immediately and investigate your claim.Downloaded from vbn.aau.dk on: August 22, 2021AALBORG UNIVERSITY'&$%The Landauer-Büttiker formula andresonant quantum transportbyHoria D. Cornean, Arne Jensen and Valeriu MoldoveanuR-2005-17 Maj 2005Department of Mathematical SciencesAalborg UniversityFredrik Bajers Vej 7G DK - 9220 Aalborg Øst DenmarkPhone: +45 96 35 80 80 Telefax: +45 98 15 81 29URL: http://www.math.aau.dk eISSN 1399–2503 On-line version ISSN 1601–7811The Landauer-Büttiker Formula and ResonantQuantum TransportHoria D. Cornean1, Arne Jensen2, and Valeriu Moldoveanu31 Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej7G, 9220 Aalborg, Denmark. cornean@math.aau.dk2 Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej7G, 9220 Aalborg, Denmark. matarne@math.aau.dk3 National Institute of Materials Physics, P.O.Box MG-7, Magurele, Romania.valim@infim.roWe give a short presentation of two recent results. The first one is a rigorousproof of the Landauer-Büttiker formula, and the second one concerns resonantquantum transport. The detailed results are in [2]. In the last section wepresent the results of some numerical computations on a model system.Concerning the literature, then see the starting point of our work, [6]. In[4] a related, but different, problem is studied. See also [5] and the recentwork [1]1 The Landauer-Büttiker FormulaWe start by introducing the notation and the assumptions. The model usedhere describes a finite sample coupled to a finite number of leads. The leadsmay be finite or semi-infinite. We use a discrete model, i.e. the tight-bindingapproximation. The sample is modeled by a finite set Γ ⊂ Z2. Each lead ismodeled by N = {0, 1, . . . , N} ⊆ N. The case N = N (N = +∞) is thesemi-infinite lead. We assume that we have M ≥ 2 leads. The one-particleHilbert space is thenH = `2(Γ )⊕ `2(N )⊕ · · · ⊕ `2(N )︸ ︷︷ ︸M copies. (1)The Hamiltonian is denoted by H . It is the sum of the following components.For the sample we can take any selfadjoint operator HS on `2(Γ ). In eachlead we take the discrete Laplacian with Dirichlet boundary conditions. Theleads are numbered by α ∈ {1, 2, . . . , M}. ThusHL =M∑α=1HLα , HLα =∑nα∈NtL(|nα〉〈nα + 1|+ |nα〉〈nα − 1|) (2)Functions in `2(N ) are by convention extended to be zero at −1 and N + 1.The parameter tL is the hopping integral. The coupling between the leadsand the sample is described by the tunneling Hamiltonian2 Horia D. Cornean, Arne Jensen, and Valeriu MoldoveanuHT = HLS + HSL, where HLS = τM∑α=1|0α〉〈Sα|, (3)and HSL is the adjoint of HLS. Here |0α〉 denotes the first site on lead α,and |Sα〉 is the contact site on the sample. The parameter τ is the couplingconstant. It is arbitrary in this section, but will be taken small in the nextsection. The total one-particle Hamiltonian is thenH = HS + HL + HT on H. (4)First we consider electronic transport through the system. Initially theleads are finite, all of length N , with N arbitrary. We work exclusively inthe grand canonical ensemble. Thus our system is in contact with a reservoirof energy and particles. We study the linear response of a system of non-interacting Fermions at temperature T and with chemical potential µ. Thesystem is subjected adiabatically to a perturbation, defined as follows.Let χη be a smooth switching function, i.e. 0 ≤ χη(t) ≤ 2, χη(t) = eηt fort ≤ 0, while χη(t) = 1 for t > 1. The time-dependent perturbation is thengiven byV (N, t) = χη(t)M∑α=1VαIα(N).Here Iα(N) =∑Nnα=0|nα〉〈nα| is the identity on the α-copy of `2(N ). Thisperturbation models the adiabatic application of a constant voltage Vα onlead α, which will generate a charge transfer between the leads via the sample.We are interested in deriving the current response of the system due tothe perturbation. In the grand canonical ensemble we need to look at thesecond quantized operators. We omit the details and state the result. Thecurrent at time t = 0 in lead α is given byIα(0) =M∑β=1gαβ(T, µ, η, N)Vβ +O(V2). (5)The gαβ(T, µ, η, N) are the conductance coefficients [3]. It is clear from theabove formula that we work in the linear response regime. Below we are goingto take the limit N →∞, followed by the limit η → 0. The limits have to betaken in this order, since the error term is in fact O(V 2/η2).The next step is to look at the transmittance, which is obtained fromscattering theory, applied to the pair of operators (K, H0), where H0 = HL(N = +∞ case) and K = H0 + HS + HT . Properly formulated this isdone in the two space scattering framework, see [7]. Since the perturbationHS + HT is of finite rank, and since we have explicitly a diagonalization ofthe operator H0, the stationary scattering theory gives an explicit formula forthe scattering matrix, which is an M ×M matrix, depending on the spectralThe Landauer-Büttiker Formula and Resonant Quantum Transport 3parameter λ = 2tL cos(k) of H0. The T -operator is then given by an M ×Mmatrix tαβ(λ), and the transmittance is given byTαβ(λ) = |tαβ(λ)|2. (6)It follows from the explicit formulas that Tαβ(λ) is real analytic on (−2tL, 2tL),and zero outside this interval.With these preparations we can state the main result.Theorem 1. Consider α 6= β, T > 0, µ ∈ (−2tL, 2tL), and η > 0. Assumethat the point spectrum of K (corresponding to the N = +∞ case) is disjointfrom {−2tL, 2tL}. Then taking first the limit N → ∞, and then η → 0, wehavegα,β(T, µ) = limη→0[limN→∞gα,β(T, µ, η, N)]= −12π∫ 2tL−2tL∂fF−D(λ)∂λTαβ(λ)dλ. (7)Here fF−D(λ) = 1/(e(λ−µ)/T + 1) is the Fermi-Dirac function. If we finallytake the limit T → 0, we obtain the Landauer formulagα,β(0+, µ) =12πTαβ(µ). (8)The proof of this main result is quite long and technical. One has to studythe two sides of the equality above. The scattering part (the transmittance)is quite straightforward, using the Feshbach formula. The conductance partis a fairly long chain of arguments, as is the proof of the equality statementin the theorem. We refer to [2] for the details.2 Resonant Transport in a Quantum DotIn the previous section we have allowed the coupling constant τ (see (3)) tobe arbitrarily large. The only assumption was that {−2tL, 2tL} was not inthe point spectrum of K. We now look at the small coupling case, τ → 0.In this case we will assume that the sample Hamiltonian HS does not haveeigenvalues {−2tL, 2tL}. It then follows from a perturbation argument, usingthe Feshbach formula, that the same is true for K, provided τ is sufficientlysmall.Since HS is an operator on the finite dimensional space `2(Γ ), is hasa purely discrete spectrum. We enumerate the eigenvalues in the interval(−2tL, 2tL):σ(HS) ∩ (−2tL, 2tL) = {E1, . . . , EJ}.Let β 6= γ be two different leads. The conductance between these two isnow denoted by Tβ,γ(λ, τ), making the dependence on the coupling constantexplicit, see (6).4 Horia D. Cornean, Arne Jensen, and Valeriu MoldoveanuTheorem 2. Assume that the eigenvalues {E1, . . . , EJ} are nondegenerate,and denote by φ1, . . . φJ the corresponding normalized eigenfunctions. Wethen have the following results:(i) For every λ ∈ (−2tL, 2tL) \ {E1, . . . , EJ} we havelimτ→0Tβ,γ(λ, τ) = 0. (9)(ii) Let λ = Ej . If either 〈Sβ , φj〉 = 0 or 〈Sγ , φj〉 = 0, thenlimτ→0Tβ,γ(Ej , τ) = 0. (10)(iii)Let λ = Ej . If both 〈Sβ , φj〉 6= 0 and 〈Sγ , φj〉 6= 0, then there exist positiveconstants C(Ej), such thatlimτ→0Tβ,γ(Ej , τ) = C(Ej)∣∣∣∣∣〈Sβ , φj〉 · 〈Sγ , φj〉∑Mα=1 |〈Sα, φj〉|2∣∣∣∣∣2. (11)This result can be interpreted as follows. Case (i): If the energy of theincident electron is not close to the eigenvalues of HS, it will not contributeto the current. Case (ii): If the incident energy is close to some eigenvalueof HS , but the eigenfunction is not localized along both contact points Sβand Sγ , again there is no current. Case (iii): In order to have a peak in thecurrent it is necessary for HS to have extended edge states, which couple toseveral leads.3 A Numerical ExampleWe end this contribution with some numerical results on the transportthrough a noninteracting quantum dot described by a discrete lattice contain-ing 20×20 sites and coupled to two leads connected to two opposite corners.The magnetic flux is fixed and measured in arbitrary units, while the lead-dot coupling was set to τ = 0.2. The sample Hamiltonian HS is given by theDirichlet restriction to the above mentioned finite domain ofHS(Vg) =∑(m,n)∈Z2((E0 + Vg)|m, n〉〈m, n|+ t1(e−i Bm2 |m, n〉〈m, n + 1|+ h.c.)+ t2(e−i Bn2 |m, n〉〈m + 1, n|+ h.c.)). (12)Here h.c. means hermitian conjugate, E0 is the reference energy, B is a mag-netic field, from which the magnetic phases appear (the symmetric gauge wasused), while t1 and t2 are hopping integrals between nearest neighbor sites.The constant denoted Vg adds to the on-site energies E0, simulating theso-called ‘plunger gate voltage’ in terms of which the conductance is mea-sured in the physical literature. The variation of Vg has the role to ‘move’ theThe Landauer-Büttiker Formula and Resonant Quantum Transport 5-3.5-3-2.5-2-1.5-1-0.5 0 0  50  100  150  200EnergyEigenvalue labelFig. 1. The dot spectrumdot levels across the fixed Fermi level of the system (recall that the latter isentirely controlled by the semi-infinite leads). Otherwise stated, the eigenval-ues of HS(Vg) equal the ones of HS(Vg = 0) (we denote them by {Ei}), up toa global shift Vg . Using the Landauer-Büttiker formula (8), and the formulas(3.8) and (4.6) in [2], it turns out that the computation of the conductivitybetween the two leads (or equivalently, of T12) reduces to the inversion of aneffective Hamiltonian.Moreover, when Vg is fixed such that there exists an eigenvalue Ei ofHS(Vg = 0) obeying Ei + Vg = EF , the transmittance behavior is describedby (11). Thus one expects to see a series of peaks as Vg is varied. Here theFermi level was fixed to EF = 0.0 and the hopping constants in the latticet1 = 1.01 and t2 = 0.99. Then the resonances appear, whenever Vg = −Ei(since the spectrum of our discrete operator HS(0) is a subset of [−4, 4], thesuitable interval for varying Vg is the same).Before discussing the resonant transport let us analyze the spectrum ofour dot at Vg = 0, in order to emphasize the role of the magnetic field. Werecall that we used Dirichlet boundary conditions (DBC) and the magneticfield appears in the Peierls phases of HS (see (12)). In Figure 1 we plotthe first 200 eigenvalues (this suffices since the spectrum is symmetricallylocated with respect to 0, i.e both Ei and −Ei belong to σ(HS(0))). One no-tices two things. First, there are two narrow energy intervals ([−3.17,−3.16]and [−1.75, 1.65]) covered by many eigenvalues (∼ 33 and 45 respectively).Secondly, the much larger ranges [−3.16,−1.72] and [−1.65,−0.8] containonly 25 and 30 eigenvalues. This particular structure of the spectrum is dueto both the magnetic field and the DBC. The dense regions are reminescencesof the Landau levels of the infinite system while the largely spaced eigenval-ues appear between the Landau levels due to the DBC. As we shall see belowtheir corresponding eigenfunctions are mostly located on the edge of the sam-6 Horia D. Cornean, Arne Jensen, and Valeriu Moldoveanuple. As the energy approaches zero, the distinction between edge and bulkstates is not anymore clear and one can have quite complex topologies foreigenfunctions. We point out that the ‘clarity’, the length, and the number ofedge states regions intercalated in the Landau gaps, increase as the samplegets bigger.Now let us again comment on (11). Here Ei must be replaced by Ei +Vg ,where Ei are eigenvalues of HS(0). Remember that we took µ = 0. Thenumber of leads is M = 2. By inspecting formula (4.6) in [2], one can showthat the constant C(Ei + Vg) will always equal 4 (we have kµ = π/2 andtL = 1). Therefore, each time we fulfill the condition Vg = −Ei, we obtain apeak in the transmittance, which for small τ should be close to4∣∣∣∣∣〈S1, φj〉 · 〈S2, φj〉∑2α=1 |〈Sα, φj〉|2∣∣∣∣∣2≤ 1. (13)We have equality with 1, if and only if |〈S1, φj〉| = |〈S2, φj〉|, and this doesnot depend on the magnitude of these quantities. Therefore, even for weaklycoupled, but completely symmetric eigenfunctions, we can expect to have astrong signal. In fact, in this case the relevant parameter ismin{|〈S1, φj〉||〈S2, φj〉|,|〈S2, φj〉||〈S1, φj〉|}. (14)Now let us investigate how the transmittance behaves, when Vg is varied.Figure 2a shows the peaks corresponding to the first six (negative) eigen-values of HS(Vg = 0). Their amplitude is very small because the associatedeigenvectors are (exponentially) small at the contact sites, and not completelysymmetric (since t1 6= t2). In fact, a few eigenvectors with more symmetrydo generate some small peaks. The spatial localisation of the second and thesixth eigenvector is shown in Figs. 2b,c.The peak aspect changes drastically at lower gate potentials as the Fermilevel encounters levels whose eigenstates have a strong component on thecontact subspace (see Figs. 3b and 3c for the spatial localisation of the 38thand the 49th eigenstate). The transmittance is close to unity in this regime,since the parameter in (14) is also nearly one. This is explained by the factthat t1 and t2 have very close values, and the relative perturbation induced bythe lack of symmetry is much smaller than for the bulk states. One noticesthat the width of the peaks increases as Vg is decreased as well as theirseparation. In Figs. 3b,c we have plotted the 38th eigenfunction, which givesthe first peak on the right of Fig. 3a, and the 49th eigenfunction associatedto the peak around Vg = 3.06.The Landauer-Büttiker Formula and Resonant Quantum Transport 7 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 3.1676  3.1678  3.168  3.1682  3.1684  3.1686  3.1688  3.169TransmittanceGate potential 2 4 6 8 10 12 14 16 18  2 4 6 8 10 12 14 16 18 20 0 0.01 0.02 0.03 0.04 0.05 0.062nd state 2 4 6 8 10 12 14 16 18  2 4 6 8 10 12 14 16 18 20 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.0166th stateFig. 2. Top to bottom: parts a, b, c8 Horia D. Cornean, Arne Jensen, and Valeriu Moldoveanu 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 3  3.02  3.04  3.06  3.08  3.1  3.12  3.14  3.16TransmittanceGate potential 2 4 6 8 10 12 14 16 18  2 4 6 8 10 12 14 16 18 20 0 0.002 0.004 0.006 0.008 0.01 0.01238th state 2 4 6 8 10 12 14 16 18  2 4 6 8 10 12 14 16 18 20 0 0.002 0.004 0.006 0.008 0.01 0.01249th stateFig. 3. Top to bottom: parts a, b, cThe Landauer-Büttiker Formula and Resonant Quantum Transport 9References1. W. Aschbacher, V. Jaksic, Y. Pautrat, C.-A. Pillet Introduction to non-equilibrium quantum statistical mechanics Preprint, 2005.2. H. D. Cornean, A. Jensen, and V. Moldoveanu: A rigorous proof of the Landauer-Büttiker formula. J. Math. Phys. 46, (2005), 042106.3. S. Datta: Electronic transport in mesoscopic systems Cambridge UniversityPress, 1995.4. V. Jaksic, C.-A. Pillet, Comm. Math. Phys. 226 (2002), no. 1, 131–162.5. V. Jaksic, C.-A. Pillet, J. Statist. Phys. 108 (2002), no. 5-6, 787–829.6. V. Moldoveanu, A. Aldea, A. Manolescu, M. Niţă, Phys. Rev. B 63, 045301-045309 (2001).7. D. R. Yafaev, Mathematical Scattering Theory, Amer. Math. Soc. 1992.

The Landauer-Büttiker formula and resonant quantum transport

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The Landauer-Büttiker formula and resonant quantum transport

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