Electron transport in quantum wire superlattices
EElectron transport in quantum wire superlattices
Thomas Grange ∗ Walter Schottky Institut, Technische Universit¨at M¨unchen,Am Coulombwall 3, D-85748, Garching, Germany (Dated: September 12, 2018)Electronic transport is theoretically investigated in laterally confined semiconductor superlatticesusing the formalism of non-equilibrium Green’s functions. Velocity-field characteristics are calcu-lated for nanowire superlattices of varying diameters, from the quantum dot superlattice regimeto the quantum well superlattice regime. Scattering processes due to electron-phonon couplings,phonon anharmonicity, charged impurities, surface and interface roughness and alloy disorder areincluded on a microscopic basis. Elastic scattering mechanisms are treated in a partial coherentway beyond the self-consistent Born approximation. The nature of transport along the superlatticeis shown to depend dramatically on the lateral dimensionality. In the quantum wire regime, theelectron velocity-field characteristics are predicted to deviate strongly from the standard Esaki-Tsuform. The standard peak of negative differential velocity is shifted to lower electric fields, whileadditional current peaks appear due to integer and fractional resonances with optical phonons.
I. INTRODUCTION
Electron transport in superlattices (SLs) has beenwidely studied in the case of 1D periodic arrangement of2D semiconductor layers (Fig. 1a). In contrast, the na-ture of charge transport in superlattices of lower dimen-sionality remains an open question. The nature and effi-ciency of dissipative processes are well known to stronglydepend on the dimensionality, with longest lifetimes andcoherence times in 3D quantum confined structures .Quantum dot (QD) SLs are hence expected to dis-play radically different non-equilibrium transport prop-erties than quantum well (QW) SLs. Potential appli-cations for high performance thermoelectric converters ,photovoltaics , and quantum cascade lasers furthermotivates the fundamental understanding of the non-equilibrium transport properties of SLs with 3D quantumconfinement.Though ballistic transport in a QW SL is ideally apure 1D problem, scattering processes couples all spatialdegrees of freedom (Fig. 1a). In QW SLs, the in-planefree electron motion possesses a continuous dispersion.During the scattering processes, energy transfer occursbetween the axial and lateral motions. Hence the effec-tive energy conservation laws in the motion along the SLaxis can greatly differ from the 3D one of the involvedscattering mechanisms. For example, in QW heterostruc-tures, 3D elastic scattering processes due to static defectsact effectively as 1D inelastic scattering mechanisms withrespect to the motion along the SL. In contrast, in thelimit of purely 1D SL with 0D lateral motion (Fig. 1d),the lateral state is frozen so that the 3D energy conser-vation laws of the scattering processes apply directly tothe 1D motion along the SL.The effect of dissipation on 1D quantum electron trans-port has been investigated in finite quantum region suchas atomic wires and stacked QDs . Quantum dissi-pative transport in QD SLs has been studied in Ref. 14for high electric fields. However full velocity-field char-acteristics of 1D SLs have not been calculated so far. In (a) (b) (c) (d) z y x ρ θ FIG. 1: (Color online). Schematic of superlattices with differ-ent lateral geometries. (a) shows a standard quantum well su-perlattice consisting of 2D ( x , y ) infinite layers of various ma-terials stacked periodically along the z direction. (b),(c) and(d) represent nanowire superlattices of different lateral sizes.Dotted arrows are sketched of classical trajectories of chargecarriers through the various superlattices under an electricfield applied along the z direction. Scattering processes occurdue to various effects (interaction with phonons, structuraldisorder...). It illustrates the transition from transport in-volving 3D scattering processes towards a pure 1D motionwith increasing quantum lateral confinement. Cylindrical co-ordinates ( ρ , θ , z ) used in the calculations are indicated for the(b) nanowire. particular it is still unkown how the Esaki-Tsu negativedifferential velocity (NDV) evolves when the lateral di-mensionality is reduced towards QD SLs.In this article, we present a detailed theory of elec-tron transport in semiconductor SLs laterally confinedin nanowires using the framework of non-equilibriumGreen’s functions (NEGF). The formalism presented hereallows to describe the transport in SLs for any lat-eral dimensionality from the regime of coupled 0D QDs(Fig. 1d) to the one of coupled 2D QWs (Fig. 1a). Forthis purpose, we include in our modeling the scatter-ing mechanisms that are known to be important in pla-nar QW heterostructures as well as the ones specific to a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r quasi-0D systems. Specifically, we consider the mecha-nism of anharmonic polaron decay which has been shownto be dominant in InGaAs self-assembled QDs . Inaddition, as the nature of elastic scattering mechanismsevolves from incoherent to coherent as the dimensional-ity decreases, we include a coherent treatment of elas-tic scattering processes for lateral state-conserving pro-cesses. The electronic transport properties are shownto depend dramatically on the SL lateral dimensionality.With increasing lateral confinement, the drift velocity-voltage characteristics evolve from the Esaki-Tsu form forusual QW SL, dominated by a single NDV effect, towardsricher characteristics where (i) the linear conductivity in-creases, (ii) the standard Esaki-Tsu NDV occurs at muchlower voltages, and (iii) large peaks due to integer andfractional resonances with optical phonons appear.The paper is organized as follows. The theoryof electron transport in laterally confined SLs is de-scribed in section II. The application to the transportin GaAs/AlGaAs nanowire superlattices of varying di-ameters is presented and discussed in section III. II. THEORY
Our aim is to model the non-equilibrium electrontransport in a nanowire SL heterostructure. We first in-troduce the general form of the Hamiltonian. We thenintroduce an electronic basis set, and present the NEGFformalism. We then treat the relevant scattering termsarising from electron-phonon interactions and static dis-order effects. Finally we present the method used forsolving the self-consistent NEGF equations.
A. Hamiltonian of the nanowire superlattice
We consider charge carriers in a nanowire SL in inter-action with phonon modes. The following Hamiltonianis used to model the system:ˆ H = ˆ H + ˆ V e + ˆ V e-ph + ˆ H vib + ˆ V m-fe-e , (1)where ˆ H is the electronic Hamiltonian of a singlecharge carrier in the idealized heterostructure under ahomogeneous electric field (the length gauge will be usedthroughout this paper); ˆ V e is the static electronic poten-tial due to disorder effects such as interface and surfaceroughness, charged impurities and alloy disorder; ˆ V e-ph represents the interaction terms between electrons andphonons; ˆ H vib is the vibrational Hamiltonian of the crys-tal, including its anharmonic part; ˆ V m-fe-e represents themean-field potential arising from the Coulomb interac-tion with other charge carriers. The standard caret no-tation is used to refer to operators. B. Electronic structure and basis states
We consider cylindrical nanowires of diameter D = 2 R having SL heterostructures along their longitudinal axis.We use cylindrical coordinates ( z, ρ, θ ) where z is thenanowire axis (see Fig. 1b). The material compositionand modeling parameters are assumed to depend onlyon the z coordinate. The electronic structure is modeledconsidering a single band within the envelope functionapproximation. We consider a basis set of the form:Ψ α,n ( z, ρ, θ ) = ζ α ( z ) φ n ( ρ, θ ) , (2)where the ζ i ( z ) are defined below and φ n ( ρ, θ ) are eigen-states of the Schr¨odinger equation on a homogeneous disk(i.e. the assumed nanowire cross-section): φ n ( ρ, θ ) = 1 √ πRJ ( m n +1)) ( χ l n m n ) J m n (cid:32) χ l n m n ρR (cid:33) e im n θ (3)where J m is the Bessel function of order m and χ lm is its l th zero. The indexes m n and l n are chosen such thatthey are associated with the n th eigenvalue of the 2Dlateral motion at the z coordinate: E n ( z ) = (cid:126) (cid:0) χ l n m n (cid:1) m ∗ ( z ) R (4)where m ∗ ( z ) is the material-dependent effective mass.For each lateral mode we are then left with a z -dependentelectronic Hamiltonian: (cid:104) n | ˆ H | p (cid:105) = δ n,p (cid:104) ˆ h − eF ˆ z + E n ( z ) (cid:105) , (5)where F is the electric field applied along the z axisand ˆ h = − (cid:126) ∂∂z m ∗ ( z ) ∂∂z + V b ( z ) , (6)with V b ( z ) being the material-dependent band-offset po-tential. The eigenstates of the periodic ˆ h Hamilto-nian satisfy the Bloch theorem and can be classified bywavevector κ z ∈ [ − π/l p , π/l p ] and miniband index ν ˆ h ϕ νκ z ( z ) = ε νκ z ϕ νκ z ( z ) (7)In order to discretize the basis set in the z direc-tion, various choices are possible. A full real-spacediscretization by e.g. finite differences requires alarge basis size in order to quantitatively describe theminiband structure, rendering the NEGF numerical im-plementation challenging. On the other hand, mode-space approaches are also possible, and allow the de-scription of the transport properties with a much smallerbasis . Here, the Hilbert space is reduced by keep-ing only a finite number of minibands, i.e. the low lyingstates of the unbiased structure. An energy cut-off E c isfixed and only the minibands separated from the groundstate by less than E c are kept. Formally we define thefollowing projection operatorˆ P z = ε ν − ε 1. Electron Green’s functions The NEGF formalism is generally introduced in termsof four Green’s functions (GFs) . Here the lesser,upper, retarded and advanced electron GFs are definedrespectively in terms of the following quantities: G <α,β,n ( t , t ) = i (cid:104) ˆ c + β,n ( t )ˆ c α,n ( t ) (cid:105) (15a) G >α,β,n ( t , t ) = − i (cid:104) ˆ c α,n ( t )ˆ c + β,n ( t ) (cid:105) (15b) G Rα,β,n ( t , t ) = − i Θ( t − t ) (cid:104){ ˆ c α,n ( t ) , ˆ c + β,n ( t ) }(cid:105) (15c) G Aα,β,n ( t , t ) = i Θ( t − t ) (cid:104){ ˆ c α,n ( t ) , ˆ c + β,n ( t ) }(cid:105) (15d)where ˆ c + α,n and ˆ c α,n are respectively the creation and an-nihilation operators in the electronic state | Ψ α,n (cid:105) . It isconvenient to introduce the GF operators ˆ G ( t , t ) de-fined here by: (cid:104) α, n | ˆ G ( t , t ) | β, n (cid:48) (cid:105) = δ n,n (cid:48) G α,β,n ( t , t ) , (16)where G refers to any of the four GFs. Since we do notconsider spin-dependent effects, the spin index is not in-dicated here for simplicity and all quantities are implic-itly spin diagonal. The spectral function reads in opera-tor form: ˆ A = i ( ˆ G R − ˆ G A ) = i ( ˆ G > − ˆ G < ) (17)The retarded GF can be expressed from the spectralfunction: ˆ G R ( t , t ) = − i Θ( t − t ) ˆ A ( t , t ) (18)The lesser GF (15a) is an extension of the concept ofdensity matrix for two different times. Its value at equaltimes corresponds to the density matrix:ˆ ρ ( t ) = − i ˆ G < ( t, t ) . (19)Here, off-diagonal terms of the GFs are considered inthe z direction but not in the lateral directions whereonly the diagonal terms of the | φ n (cid:105) eigenmodes are con-sidered. The calculations are thus independent of thebasis choice along the z -axis, allowing full coherent de-scription of transport along the nanowire. In contrast,the use of only diagonal GF terms in the lateral direc-tions does not allow us to describe coherent transporteffects in the lateral directions. This seems a reason-able approximation for thin nanowires where the lateralscattering potentials are weak compared to the lateralquantization energies. On the other hand, in the limitof large diameter nanowire and planar heterostructures,this is equivalent to considering only GF terms which arediagonal in the in-plane momentum, similarly to previ-ous implementations of NEGF in planar quantum wellheterostructures . In this later case, lateral localiza-tion effects might become relevant at low temperatureand/or in presence of strong disorder effects, which re-mains beyond the scope of this work.In steady state, the GFs depend only on the time differ-ence ( t − t ). In this case we will use one-time-dependentGFs ˆ G ( t − t ) = ˆ G ( t , t ). Energy-dependent GFs arethen defined by Fourier-transforming these quantitiesˆ G ( E ) = 1 (cid:126) (cid:90) dt ˆ G ( t ) e iEt/ (cid:126) (20)Moreover, in steady state, the advanced and retardedGFs are linked by ˆ G A ( E ) = ˆ G R ( E ) + , (21)so that the GFs are forming only two independent quan-tities. 2. Phonon Green’s functions The Hamiltonian of the vibration of the crystal readsˆ H vib = ˆ H ph + ˆ V a (22)ˆ H ph = (cid:88) (cid:126) ω q ,p ˆ a + q ,p ˆ a q ,p (23)where ˆ H ph and ˆ V a are respectively the harmonic and an-harmonic parts, and ˆ a + q ,p and ˆ a q ,p are the creation andannihilation operators for a phonon mode of branch p with wavevector q . The phonon GFs are defined by : D < q ,p ( t , t ) = − i (cid:104) ˆ A + q ,p ( t ) ˆ A q ,p ( t ) (cid:105) (24) D > q ,p ( t , t ) = − i (cid:104) ˆ A q ,p ( t ) ˆ A + q ,p ( t ) (cid:105) (25) D R q ,p ( t , t ) = − i Θ( t − t ) (cid:104) [ ˆ A q ( t ) , ˆ A + q ,p ( t )] (cid:105) , (26)with ˆ A q ,p = ˆ a q ,p + ˆ a + − q ,p . (27)In steady-state, we will use the same one-time notation D ( t ) and energy-dependent GFs D ( E ) as for electronicGFs (Eq. 20). When only the harmonic part of the Hamiltonian is taken into account, the lesser and re-tarded equilibrium phonon GFs for a mode of frequency ω read respectively: D < (0) ω ( t ) = − i [( N ω + 1) e iωt + N ω e − iωt ] (28) D R (0) ω ( t ) = i Θ( t )[ e iωt − e − iωt ] , (29)where N ω is the Bose factor at the energy (cid:126) ω . In theenergy domain, it reads: D < (0) ω ( E ) = − πi [( N ω + 1) δ ( E + (cid:126) ω ) + N ω δ ( E − (cid:126) ω )](30) D R (0) ω ( E ) = 1 E + i + − (cid:126) ω − E + i + + (cid:126) ω . (31)Harmonic phonon GFs are usually directly used in elec-tron transport calculations. However, in this work, wewill use instead anharmonic GFs for optical modes whichwill be derived below (Sec. II F). This will allow us toaccount for the mechanism of anharmonic polaron decay. D. Equations of motion The equations of motion in the NEGF formalism canbe expressed in terms of the so-called Dyson and Keldyshrelations; in steady-state, they read respectively :ˆ G R ( E ) = (cid:104) E ˆ I − ˆ H − ˆ V c − ˆΣ R ( E ) (cid:105) − (32)ˆ G < ( E ) = ˆ G R ( E ) ˆΣ < ( E ) ˆ G A ( E ) (33)where ˆ V c is the part of the interaction which is treatedcoherently. Here it reads ˆ V c = ˆ V m-fe-e + ˆ V rand e , where ˆ V m-fe-e is the mean-field Coulomb potential and ˆ V rand e is the partof the electronic disordered potential which is generatedrandomly and treated coherently (see below). Similarlyto the spectral GF, the spectral self-energy is defined asˆΓ = i ( ˆΣ R − ˆΣ A ) = i ( ˆΣ < − ˆΣ > ) , (34)and the retarded self-energy can be expressed as:ˆΣ R ( t ) = − i Θ( t )ˆΓ( t ) = Θ( t )( ˆΣ < ( t ) − ˆΣ > ( t )) . (35)In the energy domain, this relation reads:ˆΣ R ( E ) = − i E ) + 12 H [ˆΓ]( E ) (36)where H denotes the Hilbert tranform H [ˆΓ]( E ) = P (cid:82) d E (cid:48) ˆΓ( E (cid:48) ) /π ( E − E (cid:48) ). E. Self-energies due to electron-phonon interaction We consider the interactions of electrons with longi-tudinal optical (LO), surface optical (SO) and longitudi-nal acoustic (LA) phonon modes laterally confined in thenanowire. Each interaction is of the form:ˆ V e-ph = (cid:88) q ,p ˆ f q ,p ˆ A q ,p . (37)The form factors ˆ f q ,p are given in appendix A for the dif-ferent modes. In the self-consistent Born approximation(SCBA), the self-energies due to electron-phonon inter-action are of the form:ˆΣ < ( t ) = i (cid:88) q ,p ˆ f q ,p ˆ G < ( t ) D < q ,p ( t ) ˆ f q ,p . (38)Within the {| ζ α (cid:105)} basis considered here, the nonvan-ishing coupling terms being diagonal with respect to theaxial wavefunctions, the self-energy readsΣ <αβn ( t ) = i (cid:88) q ,p (cid:88) n (cid:48) f ( n,n (cid:48) ) q ,p ( α ) f ( n (cid:48) ,n ) q ,p ( β ) G <αβn (cid:48) ( t ) D < q ,p ( t ) , (39)where f ( n,n (cid:48) ) q ,p ( α ) = (cid:104) Ψ α,n | ˆ f q ,p | Ψ α,n (cid:48) (cid:105) . (40) 1. Optical phonons For longitudinal optical (LO) and surface optical(SO) phonons, we assume wavevector-independent op-tical phonon GFs. Indeed only long-wavelength opticalphonons with negligible dispersion are effectively coupledto the relevant electronic states. We can then writeΣ < (iO) αβn ( t ) = iD < LO ( t ) (cid:88) n (cid:48) W ( n,n (cid:48) )iO ( α, β ) G <αβn (cid:48) ( t ) , (41)where W ( n,n (cid:48) )iO ( α, β ) = (cid:88) q f ( n,n (cid:48) ) q , iO ( α ) f ( n (cid:48) ,n ) q , iO ( β ) . (42)In the energy domain, it readsΣ < (iO) αβn ( E ) = i (cid:88) n (cid:48) W ( n,n (cid:48) )iO ( α, β ) × (cid:90) dE (cid:48) π D < LO ( E (cid:48) ) G <αβn (cid:48) ( E − E (cid:48) ) . (43)If the phonon GF D < LO is taken harmonic, only quan-tized energy exchanges are allowed. In QDs, it hasbeen shown that energy exchanges strongly differing fromthe optical phonon energy take place due to simulta-neous electron-phonon interaction and anharmonic cou-plings among phonons . In these previous works, the electron-phonon interaction was diagonalized exactly,and the Fermi golden rule was used to calculate scatter-ing rates among polaronic states induced by anharmoniccouplings. Here, instead, the SCBA is used in order totreat the electron–optical-phonon interaction, and anhar-monicity is included in the optical-phonon GFs. As al-ready checked in Ref. 14, the SCBA very well reproducesthe exact polaron formation, especially for temperatures T verifying k b T < E LO where E LO is the optical phononenergy. The calculation of anharmonic GFs is presentedbelow. 2. Acoustic phonons In contrast to optical phonons, the acoustic phononsare assumed to be harmonic and to have a linear dis-persion. The corresponding self-energy can be expressedas: Σ <αβn ( t ) = i (cid:88) n (cid:48) K ( n,n (cid:48) ) α,β ( t ) G <αβn (cid:48) ( t ) , (44)where K ( n,n (cid:48) ) α,β ( t ) = (cid:88) q , LA D < (0) ω q ( t ) f ( n,n (cid:48) ) q , LA ( α ) f ( n (cid:48) ,n ) q , LA ( β ) (45) F. Green’s functions of anharmonic phonons We now calculate the anharmonic phonon GFs in-volved in the electron–optical-phonon self-energy. Weretain only the cubic couplings in the anharmonic terms,and thermal equilibrium population of phonon modes isassumed. As we consider only diagonal terms in thephonon’s GFs, the Dyson equation reads D R q ,p ( E ) = 1 (cid:16) D R (0) q ,p ( E ) (cid:17) − − Π R q ,p ( E ) , (46)where Π R is the phonon retarded self-energy. Thelesser GF is then calculated using the Keldysh relation: D < q ,p ( E ) = D R q ,p ( E )Π < q ,p ( E ) D A q ,p ( E ) . (47)The phonon self-energy is calculated in the Born approx-imation, taking into account cubic anharmonic terms.The phonon lesser self-energy readsΠ < q ,p ( t ) = i (cid:88) q ,p , q ,p | V a ( q , q , q ) | D < (0) q ,p ( t ) D < (0) q ,p ( t ) , (48)in which we have taken the harmonic phonon GFs in theright hand side. The cubic anharmonic coupling V a is ex-pressed in terms of the Gr¨uneisen constant as reported inRef. 15. The phonon spectral and retarded self-energiesare then given respectively byΓ j ( E ) = i (Π Disordered potentials due to randomly distributedcharged impurities, alloy disorder, rough interfaces andrough surfaces break the ideal symmetry of the struc-ture, and couple the dynamics in the various directions.In particular, they couple the 1D motion along the z su-perlattice axis to the the lateral motion. In planar QWheterostructures, as the lateral dispersion forms a con-tinuum, the static disorder produces an incoherent andirreversible evolution within the z -electron basis. In con-trast, in the limit of purely 1D heterostructures, whereonly one lateral state is involved, the evolution is com-pletely reversible, as elastic couplings induce a unitaryevolution within the z -electron basis. A treatment ofelastic scattering beyond the SCBA is thus required inthese low-dimensional structures.In previous studies, the surface roughness in nanowireshas been treated coherently by random generation ofsurfaces . Here, we consider in addition interfaceroughness, impurity scattering and alloy scattering. In-stead of generating microscopic random realizations of allsurfaces, interfaces, charge positions, and alloy atom po-sitions, we develop a simpler approach based on their cor-relations. We believe that it is sufficient for low or mod-erate disorder effects, which are anyway often not knownprecisely from the microscopic point of view. The disor-dered potentials are split into diagonal and off-diagonalparts with respect to the lateral eigenstates:ˆ V diag e = (cid:88) n (cid:104) φ n | ˆ V e | φ n (cid:105) , (52)ˆ V off-diag e = (cid:88) n (cid:54) = m (cid:104) φ n | ˆ V e | φ m (cid:105) . (53)The diagonal component is treated coherently, whilethe off-diagonal part can only be treated incoherentlysince we consider only diagonal elements of the GFs withrespect to the lateral states. 1. Elastic scattering self-energies The off-diagonal elastic coupling component ˆ V off-diag e ,which does not conserve the lateral state, is treatedwithin the SCBA. The corresponding self-energy reads: Σ <αβn ( t ) = i (cid:88) n (cid:48) (cid:54) = n (cid:104) V nn (cid:48) ( α ) V n (cid:48) n ( β ) (cid:105) G <αβn (cid:48) ( t ) , (54)where V nn (cid:48) ( α ) = (cid:104) Ψ α,n | ˆ V e | Ψ α,n (cid:48) (cid:105) . (55) 2. Elastic scattering coherent terms In order to treat coherently the diagonal componentˆ V diag e , we first calculate the following covariance matrix M n ( α, β ) = (cid:104) V nn ( α ) V nn ( β ) (cid:105) . (56)We then generate random potentials with correlationsthat obey this calculated covariance. To this purpose,we need to rotate the covariance matrix in its principalcomponent basis. More precisely, the M n symmetric ma-trix can be diagonalized in M n = R n D n R + n , (57)where D n is a diagonal matrix and R n is a unitary ma-trix. We then generate random diagonal matrices d rand n in the following way: for each diagonal element of index i , we generate a d n ( i, i ) random number obeying a Gaus-sian distribution with a variance D n ( i, i ). The randompotential is then obtained by a back transformation intothe original basis:ˆ V rand e = (cid:88) n | φ n (cid:105) ˆ V rand e,n (cid:104) φ n | , (58)ˆ V rand e,n = R n d rand n R + n . (59)The ˆ V rand potential is then included in V c in the Dysonequation (32) in order to be treated coherently. The allNEGF calculation is then made for different generatedrandom potentials until the distribution of the calculatedobservables reached the desired accuracy. H. Field-periodic boundary conditions We assume the nanowire SL to contain an infinite num-ber of periods of length L . The possible formation ofelectrical field domains is not considered in this work .Instead, the electric field is assumed to have the sameperiodicity as the SL. The field-periodic boundary con-dition for the mean-field component of the electrostaticpotential V m-f reads V m-f ( z + pF L ) = V m-f ( z ) , (60)where p is an arbitrary integer. Note that the externallyapplied electric field F is already included in H . ThePoisson equation reads ∂ V m-f ∂z = − eε (cid:15) s ( ρ e ( z ) + ρ d ( z )) , (61)where ρ e ( z ) and ρ d ( z ) are the charge densities of carriersand dopants respectively. From Eq. 61, this field-periodicboundary assumption implies that the electron probabil-ity distribution ρ e ( z ) is periodic. In the length gaugeused here, the periodicity condition for the GFs reads G α + pM,β + pM,n ( E ) = G α,β,n ( E + pF L ) (62)where M is the number of minibands. Note that innanowires, the SL periodicity can be broken by the in-clusion of the coherent disordered potentials (II G 2). Inorder to investigate such effects, the simulated period canbe increased to several SL periods. In this case, the pe-riod length L in Eqs. 60 and 62 is replaced by L p = m p L where m p is the number of SL periods included in onecomputational period. In order to know the needed num-ber of periods, the integer m p is increased until the cal-culated observables tend towards constant values. I. Electron density and current The expectation values of the various observables canbe derived using the relation between the density matrixand the lesser GF at equal times (Eq. 19). The electronprobability distribution is given by: ρ e ( z ) = − ie (cid:88) α,n (cid:90) d E π G <ααn ( E ) | ζ α ( z ) | , (63)where the factor 2 arises from the spin index. Usingthe expression of the current operator in terms of therestricted operators (Eq. 13), the current reads J z = − e (cid:126) V (cid:88) α,β (cid:104) β | [ˆ h , ˆ z ] | α (cid:105) (cid:88) n (cid:90) d E π G <αβn ( E ) . (64) J. Numerical details The self-consistent problem formed by the Keldyshequation, the Dyson relation, the expression of the var-ious self-energies and the electrostatic Poisson equationis solved iteratively. Starting from an initial guess of theGFs, we calculate iteratively (i) the lesser ˆΣ < and upperˆΣ > self-energies; (ii) the retarded self-energy (Eq. 36);(iii) the coherent part of the interaction, comprising themean-field electrostatic potential and the random poten-tial; (iv) the retarded GF from the Dyson equation; (v)the lesser GFs from the Keldysh relation. This procedureis repeated until a self-consistent solution is reached, with two convergence criteria based on both the lesser GF andthe calculated current. Before each step (i), the GFs arerenormalized in order to fulfill the charge neutrality foreach period. It is checked that this renormalization factorconverges accurately towards unity as the convergence isachieved.The iterative steps (i) and (ii) are performed numer-ically in the time-domain. It is advantageous since theself-energies within the SCBA involves the product inthe time domain. In contrast, energy domain calcula-tions would involve numerically expensive convolutions,as a broad continuum of anharmonic optical phonons isconsidered. On the contrary, the iterative steps (iv) and(v) are more easily computed in the energy domain, sothat Fourier transforms of the GFs and the self-energiesare used respectively in between steps (v)-(i) and (ii)-(iv). Though adaptive energy grids have been shown tobe useful in order to resolve GF peaks with a reducednumber of energy points , a homogeneous energy grid isused here in order to make use of a fast Fourier transformalgorithm.A maximum coherence length l c is set in the calcula-tions, so that we consider only the GF terms G α,β,n for | ζ α − ζ β | ≤ l c . Due to the field-periodic boundary con-ditions, the GFs G α,β,n are calculated for α belongingto a given period, while β varies away from α within theset coherence length l c . In the simulation of the nanowireSL, we check that l c is taken large enough by increasing ituntil convergence of the calculated current. In the regimeof QW SL, we have verified that there is no difference inthe calculated observables considering one or several pe-riods, provided the doping densities remain low enoughto prevent the formation of field domain instabilities. III. RESULTS AND DISCUSSIONSA. Transport in nanowire SLs: transition from QWto QD SLs As an application of the theory presented above, wepresent calculations of the electronic transport in aGaAs/Al . Ga . As nanowire SL. One period consists in5 nm GaAs (well) and 5 nm Al . Ga . As (barrier). TheSL is assumed to be homogeneously n-doped with a con-centration of 10 cm − . Assumed values for surface andinterface roughnesses are given in appendix B. The fun-damental subband has a width of 9 . E c is taken around100 meV). Fig. 2 shows the drift velocity as a functionof the nanowire diameter for two different electric fieldsalong the SL. For large nanowire diameters, the currentdensity tends toward a constant value. This is interpretedas the 2D regime for the lateral motion. In contrast, forsmaller diameters, the quantum wire regime is reached,and the current density varies with the lateral confine- Drift velocity (103 m.s-1) N a n o w i r e d i a m e t e r ( n m ) E - E ( m e V ) FIG. 2: (Color online). Drift velocity in a nanowire superlat-tice as a function of the nanowire diameter for various electricfields. The superlattice period consists of 5 nm GaAs followedby 5 nm AlGaAs. The temperature is set to 300 K. 01 02 05 01 0 01 5 0 N a n o w i r e d i a m e t e r 1 5 n m 3 0 n m 5 0 n m 1 5 0 n m Spectral function (a.u.) N a n o w i r e d i a m e t e r 1 5 n m 3 0 n m 5 0 n m 1 5 0 n m Electron density (a.u.) E n e r g y ( m e V ) FIG. 3: (Color online). Spectral function A lat ( E ) (upperpanel) and electron density n lat ( E ) (lower panel) for variousnanowire diameters (see text for definitions). The electricfield is 20 kV/cm and the temperature is 300 K. ment. In order to confirm this interpretation, we plot inFig. 3 the quantities A lat ( E ) = 1 s (cid:88) n A α,α,n ( E ) (65) n lat ( E ) = − s Im (cid:34)(cid:88) n G <α,α,n ( E ) (cid:35) , (66)which represent respectively the spectral function and theelectron density summed over the lateral states; s = πR is the nanowire cross-section and α is the single basisstate per period considered. For a nanowire diameter of E l e c t r i c f i e l d ( k V / c m ) Drift velocity (m.s-1) V o l t a g e d r o p p e r p e r i o d ( m V ) N a n o w i r e d i a m e t e r 1 5 0 n m 5 0 n m 3 0 n m 1 5 n m Drift velocity (103 m.s-1) V o l t a g e d r o p p e r p e r i o d ( m V ) 01 0 02 0 0 FIG. 4: (Color online). Drift velocity–voltage characteristicsat 300K for various diameters of the nanowire superlattice. 150 nm, the smooth shape of the spectral function indi-cates that the lateral quantization effects are overcomeby broadening effects. Its flat shape is characteristic of a2D subband. On the other hand, for nanowire diametersbelow 17 nm, the current displays relatively small varia-tions in Fig. 2. This corresponds to a quasi-0D regime forthe lateral motion, where only the ground lateral state isoccupied, as shown in Fig. 3 for a nanowire diameter of15 nm. In between these two limit regimes, large varia-tions of the current densities are calculated in Fig. 2, cor-responding to an intermediate dimensional regime whereseveral but still distinct lateral states are involved, as de-picted in Fig. 3 for nanowire diameters of 30 and 50 nm.Fig. 4 shows the calculated velocity-voltage character-istics for various nanowire diameters. For a nanowire di-ameter of 150 nm, where the 2D lateral regime is reachedin good approximation, the standard Esaki-Tsu behavioris observed: negative differential velocity (NDV) is ex-pected to occur when the carriers are excited beyond aninflection point in the miniband dispersion . For smallerdiameters, the characteristics deviate from this simplestandard behavior. The most remarkable difference is theappearance of a large peak when the voltage drop per pe-riod matches the LO-phonon energy ( (cid:126) ω LO =37 meV inGaAs). In addition, smaller resonances become visibleat (cid:126) ω LO / (cid:126) ω LO / 3. The interpretation is the fol-lowing: in the 2D regime, the lateral motion acts as acontinuous energy reservoir, which renders the 3D en-ergy conservation laws of the scattering mechanisms notdirectly visible on the SL transport properties. In par-ticular, though LO phonons provide the most efficientinelastic scattering processes, their quasi-monochromaticnature is not evidenced on the velocity–voltage charac-teristics. In contrast, in the limit of 1D SLs with 0Dlateral motion, the energy is exchanged directly betweenthe quantized SL levels (i.e. the Wannier-Stark states)and the phonons. Hence resonances in the transport oc-cur when two consecutive Wannier-Stark levels are sepa-rated by the LO-phonon energy (cid:126) ω LO . The smaller res-onances around (cid:126) ω LO / (cid:126) ω LO / , where the NDV peak is predicted to occur for edF τ / (cid:126) = 1 ( τ being the scattering time). Beyond thiscritical point, the Esaki-Tsu model predicts that elec-trons are accelerated beyond an inflection point in theenergy–momentum dispersion relation and hence expe-rience a negative effective mass. This edF = (cid:126) /τ crit-ical point can also be interpreted as the transition be-tween the domain of validity of miniband transport andWannier-Stark hopping models . In both pictures, theoverall reduction of scattering processes with decreasingdimensionality explains qualitatively the observed shiftof the NDV peak towards lower voltages. B. Role and nature of elastic scattering processes In order to analyze the role and the nature of elas-tic scattering processes, simulations with different treat-ments of elastic scattering are shown in Fig. 5. First, wediscuss the comparison between full calculations with theone neglecting elastic scattering. These two calculationsnotably differ for thick NWs. In contrast, for thin NWs,the relative difference in the calculated transport is verysmall, indicating that elastic scattering only plays a mi-nor role. This strong reduction of elastic scattering pro-cesses with decreasing dimensionality is consistent withthe intuitive expectation that elastic mechanisms are sup-pressed by the discretization of energy levels.Now we compare the calculations using the standardSCBA, the “selective SCBA” defined above and the fullcalculation. In the QW limit the standard SCBA, the“selective SCBA” and the full calculation are found togive the same results. In contrast, in the quantum wireregime, the standard SCBA totally differs from the twoother models. The reason is that the standard SCBAfails when the fluctuations of the energy levels becomelarger than their linewidths. In contrast, in the presentfull calculation, the main coupling elements, which are di-agonal with respect to the lateral eigenstates, are treatedcoherently. If we now compare the full calculation to theselective SCBA calculation, we can see that the selec-tive SCBA provides an excellent approximation to thefull calculation, down to 15 nm thin nanowires. Our in-terpretation is that the typical energy fluctuations of thelevels stay smaller than the miniband width and are too Drift velocity (103 m.s-1) D = 1 5 n m F u l l c a l c u l a t i o n N o e l a s t i c s c a t t e r i n g S t a n d a r d S C B A S e l e c t i v e S C B A V o l t a g e d r o p p e r p e r i o d ( m V ) D = 3 0 n m F u l l c a l c u l a t i o n N o e l a s t i c s c a t t e r i n g S t a n d a r d S C B A S e l e c t i v e S C B A Drift velocity (103 m.s-1) V o l t a g e d r o p p e r p e r i o d ( m V ) D = 5 0 n m F u l l c a l c u l a t i o n N o e l a s t i c s c a t t e r i n g S t a n d a r d S C B A S e l e c t i v e S C B A Drift velocity (103 m.s-1)-2) V o l t a g e d r o p p e r p e r i o d ( m V ) D = 1 5 0 n m F u l l c a l c u l a t i o n N o e l a s t i c s c a t t e r i n g S t a n d a r d S C B A S e l e c t i v e S C B A Drift velocity (103 m.s-1) V o l t a g e d r o p p e r p e r i o d ( m V ) FIG. 5: (Color online). Drift velocity–voltage characteristicsat 300K for different treatments of the scattering mechanisms. D = 3 0 n m A n h a r m o n i c p h o n o n s H a r m o n i c p h o n o n s Drift velocity (103 m.s-1) V o l t a g e d r o p p e r p e r i o d ( m V ) D = 5 0 n m A n h a r m o n i c p h o n o n s H a r m o n i c p h o n o n s Drift velocity (103 m.s-1) V o l t a g e d r o p p e r p e r i o d ( m V ) D = 1 5 0 n m A n h a r m o n i c p h o n o n s H a r m o n i c p h o n o n s Drift velocity (103 m.s-1) V o l t a g e d r o p p e r p e r i o d ( m V ) FIG. 6: (Color online). Drift velocity–voltage characteristicsat 300K with and without the inclusion of phonon anhar-monicity. weak to induce localization effects. C. Role of phonon anharmonicity In Fig. 6 we show a comparison of the calculationswith and without the inclusion of phonon anharmonicityfor various NW diameters. While the inclusion of anhar- monicity is found to be almost irrelevant for thick NWsin the QW SL limit, striking differences are observed forthin NWs. Indeed the direct LO-phonon emission is grad-ually suppressed with decreasing dimensionality towards0D systems, while the mechanism of anharmonic polarondecay becomes dominant. For very thin NW diameterscorresponding to the pure 1D SL limit, note that the sim-ulation without phonon anharmonicity does not convergedue to the strong reduction of the linewidths below ournumerical possibilities in energy resolution. Hence ourcalculations show a clear transition in the dominant en-ergy dissipation pathway as the lateral dimensionality isreduced, from direct LO-phonon emission to polaron an-harmonicity. IV. CONCLUSION Within the NEGF formalism, we have developed a the-ory of electronic transport in nanowire SLs. This modelallows the calculation of transport in a wide range ofNW SL thickness, for lateral dimensionality ranging from0D to 2D, i.e. from QD SL to QW SL. We have stud-ied how the transport evolves with this change of lat-eral dimensionality. While velocity-voltage characteris-tics are dominated by the standard Esaki-Tsu NDV inQW SLs, electron-phonon resonances appear when thelateral quantum regime is reached. In addition, the elec-tron mobility increases and the NDV peaks are stronglyshifted to lower electric fields. This is accompanied byimportant changes in the dominant scattering mecha-nisms. In QW SLs, the transport is mainly controlledby LO-phonon scattering and elastic scattering processes.In contrast, when the lateral dimensionality is reduced,these two mechanisms are progressively suppressed, andanharmonic polaron decay becomes dominant. V. ACKNOWLEDGMENTS This work has been supported by the Alexander vonHumboldt foundation and the Austrian Science FundFWF through Project No. F25-P14 (SFB IR-ON).S. Birner, M. Branstetter, C. Deutsch, P. Greck, G.Koblm¨uller, M. Krall, T. Kubis, S. Rotter, K. Unter-rainer, and P. Vogl are gratefully acknowledged for fruit-ful discussions. Appendix A: Electron-phonon interactions incylindrical nanowires In this appendix we give the form factor of thecouplings of electrons with longitudinal-optical (LO),surface-optical (SO), and longitudinal-acoustical (LA)phonon modes confined in a cylindrical nanowire. Weconsider the full confinement of phonons by the nanowiresurfaces but neglect the effect of the SL interfaces.1 1. Interaction with longitudinal optical modes The polar interaction between electrons and opticalphonons confined in a cylindrical nanowire is treatedwithin the dielectric continuum model . The zone-center LO phonons involved in the couplings with elec-trons are assumed to be monochromatic with frequency ω LO . The interaction with the longitudinal optical (LO)modes of the nanowire readsˆ H e-LO = (cid:88) m,l,q z f LO m,l,q z ( ρ, θ, z ) ˆ A LO m,l,q z , (A1)where ˆ A LO m,l,q z = ˆ a + m,l,q z + ˆ a − m,l, − q z (A2)is the sum of phonon creation and annihilation operatorswith opposite momentum. The form factor reads f LO m,l,q z ( ρ, θ, z ) = C LO m,l,q z J m (cid:18) χ lm ρR (cid:19) e imφ e iq z z , (A3)where | C LO m,l,q z | = e (cid:126) ω LO πε LJ m +1 ( χ lm )( χ l m + R q z ) (cid:18) ε ∞ − ε s (cid:19) . (A4) ε is the vacuum permittivity; ε s and ε ∞ are respectivelythe static and high frequency relative permittivities. 2. Interaction with surface optical modes The interaction with the surface optical modes (SO)reads ˆ H e-SO = (cid:88) m,l,q z C SO m,l,q z I m ( q z ρ ) e imθ e iq z z ˆ A SO m,l,q z , (A5)where | C SO m,l,q z | = e (cid:126) ω m,l,q z πε LI m ( q z R )( I m − ( q z R ) + I m +1 ( q z R )) × q z R (cid:18) (cid:15) ∞ − (cid:15) − (cid:15) s − (cid:15) (cid:19) (A6)and (cid:15) = − I m ( q z R )(( K m − ( q z R ) + K m +1 ( q z R )) K m ( q z R )(( I m − ( q z R ) + I m +1 ( q z R )) (cid:15) ext (A7)where (cid:15) ext is the relative permittivity outside thenanowire. ω m,l,q z = (cid:18) (cid:15) − (cid:15) ∞ (cid:15) ∞ − (cid:15) (cid:19) ω (A8) 3. Interaction with acoustic phonons We consider the deformation potential interaction withacoustic phonons confined laterally in the nanowire.In principle, the boundary condition couple longitudi-nal acoustic (LA) and transverse acoustic (TA) modes.Coupled LA-TA modes interacting with electrons havebeen considered by Yu et al . However, there, onlyphonon modes with axial symmetry have been consid-ered. We expect this approximation to be valid inthe thin nanowire limit but to completely fail for largenanowires where LA phonons do not necessarily prop-agate parallel to the nanowire axis. Here, instead, weneglect the coupling between LA and TA modes (i.e. be-tween compressive and shear modes) at the surface of thenanowire, and consider the interaction of electrons withall the LA modes. This treatment is exact in the bulklimit and is expected to provide a reasonable approxima-tion for nanowires of moderate size, as far as the NWdiameter is large compared to the lattice constant. Webelieve such assumption for the boundary conditions doesnot change significantly the results for the electron trans-port properties, as (i) the nanowire thicknesses studiedhere are relatively large compared to the atomic scaleand (ii) the efficiency of the acoustic phonon scatteringmechanism is always much weaker than other scatteringmechanisms. The displacement fields for purely compres-sive phonon modes can be written as u = ∇ φ (A9)We consider the free boundary condition, so that ∇ u = 0at the surface and φ can be written as φ = (cid:88) m,l,q z C m,l,q z J m (cid:18) χ lm ρR (cid:19) e imφ e iq z z (A10)The phonon quantization readsˆ u = (cid:88) m,l,q z u m,l,q z ˆ A LA m,l,q z , (A11)with the normalization condition: (cid:90) d r | u m,l,q z ( r ) | = (cid:126) µω m,l,q z , (A12) µ is the material density and ω m,l,q z the phonon fre-quency.The integration using the form of Eq. A10 gives: | C m,l,q z | = (cid:126) πLµω ( χ l m + R q z ) J m +1 ( χ lm ) (A13)The Hamiltonian describing the electron–LA-phononinteraction is taken of the formˆ H e-LA = D ∇ . ˆ u, (A14)2where D is the deformation potential of the band of in-terest. Here it gives:ˆ H e-LA = D ( χ l m /R + q z ) ˆ φ, (A15)and finally:ˆ H e-LA = (cid:88) m,l,q z f LA m,l,q z ( ρ, θ, z ) ˆ A LA m,l,q z , (A16)where the form factor reads: f LA m,l,q z ( ρ, θ, z ) = C LA m,l,q z J m (cid:18) χ lm ρR (cid:19) e imφ e iq z z , (A17) | C LA m,l,q z | = D (cid:126) (cid:112) χ l m + R q z πRLµc s J m +1 ( χ lm ) , (A18)where c s the sound celerity. The dispersion relation reads ω m,l,q z = c s (cid:114) χ l m R + q z . (A19) Appendix B: Elastic scattering mechanisms Imperfections and intrinsic atomic disorder break thestructure symmetry and the crystal periodicity, thus re-sponsible for elastic scattering processes. We considerbelow the effects of surface roughness of the nanowires,interface roughness of the superlattice heterostructures,randomly located charged impurities and alloy disorder. 1. Interface roughness We note z i the mean z -coordinate of the i th interface.The deviation from this average position at the in-planecoordinate r is denoted by δz i ( r ). The interface rough-ness can be statistically characterized by (i) the distri-bution of the interface position along the heterostructuregrowth axis, and (ii) its in-plane autocorrelation. Allinterfaces are assumed to obey the same statistics. Wenote P ( z ) the distribution of the δz i interface deviationand σ its root mean square. We note C ( r ) the in-planeautocorrelation form factor so that: (cid:104) δz i ( r ) δz i ( r ) (cid:105) = σ C ( | r − r | ) (B1)We note ∆ V the heterostructure band offset. The cou-pling correlations involved in the calculation of elasticscattering processes read: (cid:104) V (ir) nn (cid:48) ( α ) V (ir) n (cid:48) n ( β ) (cid:105) = ∆ V Y nn (cid:48) (cid:88) i W ( i ) αβ , (B2) where the lateral form factor reads Y nn (cid:48) = (cid:90) d r d r φ n ( r ) φ n (cid:48) ( r ) C ( | r − r | ) , (B3)and the axial form factor for the i th interface reads W ( i ) αβ = (cid:90) z i +1 z i − d z i P ( z i − z i ) w ( i ) α ( z i ) w ( i ) β ( z i ) , (B4)with w ( i ) α ( z i ) = (cid:90) [ z i ,z i ] d z | ζ α ( z ) | . (B5)In the numerical study, P ( z ) is taken as a Gaussian dis-tribution with a standard deviation σ = 0 . 15 nm and theautocorrelation C ( r ) decreases as an exponential e − r/λ with a correlation length λ = 8 nm. 2. Surface roughness We consider small fluctuations of the surface around itsideal cylindrical position. The surface of the nanowiresare considered as infinite barriers for the electrons. Mo-tivated by the fact that a homogeneous change δR in thenanowire diameter would induce a change ∂ E n ∂R δR in the n -th lateral state energy, we make the following assump-tion for the surface roughness couplings : (cid:104) n | V (sr) ( θ, z ) | n (cid:48) (cid:105) = (cid:114) ∂ E n ∂R ∂ E n (cid:48) ∂R δR θ,z (B6)We assume an exponential autocorrelation of the form (cid:104) δR ( θ , z ) δR ( θ , z ) (cid:105) = δR e −| z − z | /L sc × e − R | arg[ e i ( θ − θ ] | /L sc (B7)in which we have factorized the axial and azimuthal cor-relation terms in order to simplify the calculations. Fi-nally we find that the correlations of the coupling termsare given by (cid:104) V (sr) nn (cid:48) ( α ) V (sr) n (cid:48) n ( β ) (cid:105) = 4 δR R E n E n (cid:48) Y (sr) nn (cid:48) W (sr) αβ , (B8)where the lateral form factor reads Y (sr) nn (cid:48) = (cid:90) π − π d θ π e i ( m n − m n (cid:48) ) θ e − R | θ | /L sc = L sc πR (cid:2) − e − πR/L sc (cid:3) if m n = m n (cid:48) R [ − e − πR/Lsc ( − mn − mn (cid:48) ] πL sc [( m n − m n (cid:48) ) +( R/L sc ) ] if m n (cid:54) = m n (cid:48) , (B9)and the axial form factor reads W (sr) αβ = (cid:90) d z (cid:90) d z (cid:48) | ζ α ( z ) | | ζ β ( z (cid:48) ) | e −| z − z (cid:48) | /L sc . (B10)In the numerical study, the standard deviation is takenas δR = 0 . 15 nm and the surface correlation length istaken as L sc = 8 nm.3 3. Charged impurities The random location of charged impurities is also asource of elastic scattering. For simplicity of the calcu-lation and for clear comparison with the 2D limit, weassume that the filling factor of the nanowires is unity.The Coulomb potential created by ionized impurities atthe position r d reads V (c) r d ( r ) = e e − ( | r − r d | ) /L s πε ε s | r − r d | (B11)where L s is the screening length. Following Nelanderand Wacker , L s is calculated within the Debye screen-ing model for a 3D homogeneous electron gas with thesame average electron density and at the lattice temper-ature. After performing an in-plane Fourier transformthis potential reads : V (c) ρ d ,z d ( ρ, z ) = (cid:90) d k (cid:107) π e i k (cid:107) .ρ e (cid:15) ε s (cid:15) s k s e − k s | z − z d | (B12)with k s = (cid:112) k (cid:107) + 1 /L s . The coupling correlation termsare obtained by averaging over the different possible po-sition of the charged impurities: (cid:104) V (c) nn (cid:48) ( α ) V (c) n (cid:48) n ( β ) (cid:105) z d ,ρ d = (cid:90) dz d (cid:90) d ρ d N d ( z d ) ×(cid:104) Ψ α,n | V (c) z d ,ρ d | Ψ α,n (cid:48) (cid:105)(cid:104) Ψ β,n (cid:48) | V (c) z d ,ρ d | Ψ β,n (cid:105) . (B13)After some algebra we find: (cid:104) V (c) nn (cid:48) ( α ) V (c) n (cid:48) n ( β ) (cid:105) = πe ε ε s (cid:90) + ∞ k (cid:107) dk (cid:107) k s F k (cid:107) ( n, n (cid:48) )Λ k (cid:107) ( α, β ) , (B14)where F k are Hankel transform coefficients: F k ( n, n (cid:48) ) = (cid:90) R dρ ρ φ n ( ρ ) φ n (cid:48) ( ρ ) J m n,n (cid:48) ( kρ ) , (B15)with m n,n (cid:48) = | m n − m (cid:48) n | , and the axial form factor reads:Λ k ( α, β ) = (cid:90) dz (cid:90) dz | ζ α ( z ) | | ζ β ( z ) | λ k ( z , z ) , (B16) with λ k ( z , z ) = (cid:90) dzN d ( z ) e − k ( | z − z | + | z − z | ) . (B17) 4. Alloy disorder We consider a ternary alloy of the form A x B − x C . Weassume a random and uncorrelated distribution of theatoms A and B . We assume that, at the atomic scale,the carriers experience local band offset potentials V AC or V BC with probability x and 1 − x , respectively .Here these potentials are taken as the one of the binarycompounds. The mean potential reads V ( z ) = xV AC + (1 − x ) V BC , (B18)while the alloy scattering potential is defined by:∆ V = V AC − V BC (B19)The variance of the local band offset is then: (cid:104) ( V − V ) (cid:105) = x ( V AC − V ) + (1 − x )( V BC − V ) = x (1 − x )∆ V (B20)We denote V the volume occupied by a binary pair ofatoms ( V = a / a being thelattice constant). 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