Magnetotunnelling in resonant tunnelling structures with spin-orbit interaction
Goran Isić, Dragan Indjin, Vitomir Milanović, Jelena Radovanović, Zoran Ikonić, Paul Harrison
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Magnetotunnelling in resonant tunnelling structures withspin-orbit interaction
Goran Isi´c
School of Electronic and Electrical Engineering,University of Leeds, LS2 9JT, UK andInstitute of Physics, University of Belgrade,Pregrevica 118, 11080 Belgrade, Serbia
Dragan Indjin, Zoran Ikoni´c, and Paul Harrison
School of Electronic and Electrical Engineering, University of Leeds, LS2 9JT, UK
Vitomir Milanovi´c and Jelena Radovanovi´c
School of Electrical Engineering, University of Belgrade,Bulevar kralja Aleksandra 73, 11120 Belgrade, Serbia (Dated: November 7, 2018)
Abstract
Magnetotunnelling spectroscopy of resonant tunnelling structures provides information on thenature of the two-dimensional electron gas in the well. We describe a model based on nonequilib-rium Green’s functions that allows for a comprehensive study of the density of states, tunnellingcurrents and current spin polarization. The investigated effects include the electron-phonon in-teraction, interface roughness scattering, Zeeman effect and the Rashba spin-orbit interaction. Aqualitative agreement with experimental data is found regarding the satellite peaks. The spin po-larization is predicted to be larger than ten percent for magnetic fields above 2 Tesla and having astructure even at the satellite peaks. The Rashba effect is confirmed to be observable as a beatingpattern in the density of states but found to be too small to affect the tunnelling current. . INTRODUCTION Resonant tunnelling was first observed in scattering of electrons by noble gases where it isknown as the Ramsauer effect . After the invention of quantum mechanics, the appearanceof the scattering minimum at a particular electron energy was explained by coupling witha quasi-bound state of the gas atom. Following the pioneering work of Esaki and Tsu inthe seventies, the same effect was found in resonant tunnelling structures (RTSs) where therole of electrostatic potential barriers is played by thin semiconductor layers.RTS is the basic nanoelectronic device that exploits the quantum nature of electrons. Asthe electron transport through a RTS is a combination of coherent tunnelling and phase-breaking collisions, RTSs are the ideal testing ground for quantum transport theories. Inthe envisioned high-performance electronic applications, one aims to maximize the maincurrent peak where coherent transport dominates while minimizing the satellite peaks thatoccur due to scattering. Resonances appear in scattering as well because it is enhancedfor the localized quasi-bound states, the most pronounced feature being the longitudinaloptical (LO) phonon satellite peak . While leading to performance degradation and powerdissipation, the scattering mechanisms in RTSs are interesting in themselves for allowingthe study of various aspects of the twodimensional electron gas (2DEG) physics via cur-rent spectroscopy. When a perpendicular magnetic field is applied, the 2DEG density ofstates (DOS) collapses into discrete Landau levels (LLs) thus pronouncing the resonant fea-tures in scattering. For this reason, magnetotunnelling experiments in RTSs have beenestablished as an important tool in investigating the 2DEG physics.The electron-phonon interaction within the 2DEG of III-V RTSs leads to a formationof magnetopolarons manifested as a characteristic anticrossing of LLs which has beenobserved in magnetotunnelling experiments . Another important phenomenon in InAs-based RTSs is the spin-orbit interaction (SOI). Due to potential applications in semicon-ductor spintronics , the zero-magnetic field spin splitting in a 2DEG has been receivingconsiderable attention since the field effect spin transistor was proposed by Datta and Das .At smaller magnetic fields SOI is known to significantly affect the 2DEG LLs leading to abeating pattern in the DOS. This causes a beating pattern in magnetoresistivity which isoften used to characterize SOI in a 2DEG .Here we report on a theoretical study of magnetotunnelling with LO phonon and interface2oughness (IR) scattering taking a particular care of the carrier spin. The spin splitting isdescribed as a combination of SOI and the Zeeman effect, both of which are known to bepronounced in InAs quantum wells. A detailed description is given of a nonequilibriumGreen’s function (NEGF) model with scattering treated within the self-consistent firstBorn approximation (FBA). The LO phonon and IR correlation functions are approximatedwith delta functions, the suitability of which is discussed in Appendices A and B. Themodel is applied to a symmetrical InAs-GaAs double barrier RTS and shown to yield I-Vcurves that qualitatively match the experimental data. An estimate of the spin polarizationis found, and the role of SOI in spin-dependent transport through RTSs discussed. II. MODEL DESCRIPTION
In the first part we discuss the electronic states of the RTS in absence of scattering.Eigenstates of the lateral and spin degrees of freedom found here are used in the second partto formulate the transport model with scattering.
A. Effective Hamiltonian and the Landau levels
The RTS is assumed to be grown along the [100] direction denoted as the z -axis. Itcomprises of: (1) a highly doped InAs emitter contact, (2) the 3nm wide left barrier formedby a GaAs layer, (3) the 6nm InAs well, (4) the 3nm wide right barrier made of GaAs and(5) a highly doped InAs collector contact. We adopt a one-band envelope function approxi-mation (EFA) in which the electronic states of a heterostructure are determined by theireffective mass m ∗ and the effective potential V ( z ) comprising the conduction band edgevariation and the externally applied voltage. A generalized multiband EFA allows for manyparameters and is applicable for modelling a wide range of semiconductor heterostructures .The one-band approximation used here is often adopted for single-valley conduction bandheterostructure states. This is because up to few hundred milielectron volts above theconduction band minimum, the states are of a dominantly s-like character so both energydispersion and matrix elements can be sufficiently well estimated using effective parame-ters. The use of more complicated models such as the multiband EFA or, especially withNEGF calculations, multiband empirical tight-binding models is justified when a more3 IG. 1. The symmetric InAs-GaAs RTS under bias. The barrier and well widths are 3 and 6nm,respectively. The conduction band edge with slope due to the external bias is shown with the full(red) line. The shading represents the LDOS in logarithmic scale. The pale series of peaks in theemitter (left) and collector (right) region represent Van Hove singularities at LL subband bottomswhile the intense peaks in the well are the quasi-bound LLs. Up to around 700meV only the lowest z -subband is present, the second z -subband states appearing as broadened double-node peaks. quantitative model is aimed for or even mandatory if effects like inter-valley scattering orinter-band tunnelling are studied.The conduction band effective potential V ( z ) of the investigated RTS is depicted bythe thick solid line in Fig. (1). The externally applied collector to emitter voltage U ce isassumed to be positive so that electrons have a tendency to flow towards the collector whoseband edge is taken as a reference energy. The EFA Hamiltonian of the biased RTS with anexternal magnetic field applied parallel to the growth direction reads H = π m ∗ + V ( z ) + 12 g ∗ µ B Bσ z + α R ~ ( π x σ y − π y σ x ) . (1)The first term is the kinetic energy, π = p + e A being the kinetic momentum of the electronand m ∗ its conduction band effective mass. The third term represents the Zeeman effectdetermined by the effective g -factor, g ∗ , and σ x,y,z are the Pauli spin matrices .The fourth term is the Rashba spin-orbit interaction (RSOI) Hamiltonian that accountsfor SOI due to the heterostructure inversion asymmetry . The Dresselhaus SOI (DSOI)4aused by the bulk inversion asymmetry is neglected. This is believed to hold in InAsquantum wells, considering the theoretical arguments in Ref. 40 and experimental findingsin Refs. 19 and 41 (for InAs-GaSb quantum wells) and Refs. 42 and 43 (for InGaAs-InAlAs quantum wells). The RSOI in a heterostructure is a combined effect of the externallyapplied potential and the spatial variation of the band edge. In asymmetric heterostructuresthe latter lifts the spin degeneracy even in absence of an external electric field E ext . Forsymmetric structures the Rashba constant α R can be taken to be proportional to E ext α R = CE ext , C = C E + C I , (2)where the direct contribution of the electric field is evaluated as C E = e ~ m ∗ (2 E g + ∆ SO )∆ SO E g ( E g + ∆ SO )(3 E g + 2∆ SO ) . (3)The interface contribution C I depends on details of the heterostructure potential. We havefound that in the range of external fields of interest (around the main and satellite currentpeaks) taking into account only the first term gives α R in the range 1 − ≈ − eVm whichis close to values for a InAlAs-InGaAs quantum well reported in that vary in the range2 − ≈ − eVm. For this reason, unless it is varied as a parameter, α R is evaluated fromEqs. (2) and (3) with E g (InAs) = 356meV, ∆ SO (InAs) = 380meV while C I is set to zero.Thus, our model has three main effective parameters, m ∗ , g ∗ and α R . The energy scaleover which these parameters change considerably is set by the bandgap. In InAs the effectivemass increases approximately linearly from the zone center value m ∗ = 0 . m to aroundtwice that at E = 0 . g -factor predicted by the k · π modelis g ∗ ( E ) = g ∗ (0) E g ( E g + ∆ SO )∆ SO (cid:18) E g + E − E g + ∆ SO + E (cid:19) , (4)with the zone center value g ∗ (0) given by g ∗ (0) = (cid:18) − m m ∗ (0) (cid:19) SO SO + 3 E g . (5) g ∗ ( E ) calculated from the above equations with InAs parameters is a monotonously increas-ing function starting at g ∗ (0) = − . − . g -factor is also known to depend on the electron-electron interaction, thus a range of valueshas been reported in InAs quantum wells . In Ref. 45 | g ∗ | of a 2DEG in a 4nm wide InAsquantum well has been found to be around 6. The situation with α R is less clear since it5 IG. 2. A scheme of the E ns levels and their relation with the ordinary ( α R = 0) LLs for negative g . (a) Shows the arrangement of the levels at high magnetic fieds. (b) is the dispersion E ns ( B ) forsmall and moderate magnetic fields. For clarity, (b) is drawn with exaggerated α R . vanishes in bulk while a value in a heterostructure is highly dependent on both the con-stituents and electrostatic effects. It will be seen below that the mechanisms involved inelectron transport through the investigated RTS have an energy scale of few tens of milielec-tron volts, determined by the chemical potential of the contacts and the InAs LO phononenergy. Consequently, keeping in mind the range of meaningful values, it makes sense tofix the effective parameters which is done by setting m ∗ (InAs) = 0 . m , g ∗ = − α R as explained above.In absence of RSOI ( α R = 0), the eigenkets describing the lateral ρ = ( x, y ) and spindegrees of freedom are the spin split LLs | nσ i with energies E nσ = (cid:18) n + 12 (cid:19) ~ ω C ± ∆ B , n = 0 , , ..., ∆ B = g ∗ µ B B, ω C = eBm ∗ , (6)where the plus is to be taken for spin up, σ = ↑ , and the minus for spin down along the z axis, σ = ↓ . Adopting the Landau gauge, the corresponding wavefunctions are h ρ ↑ | kn ↑i = h ρ ↓ | kn ↓i = e ikx √ L x Φ n (cid:16) yL − kL (cid:17) , h ρ ↑ | kn ↓i = h ρ ↓ | kn ↑i = 0 . (7)6n this equation k and L x denote the quasi-momentum and system length along the x axis,while L = p ~ /eB is the magnetic length. The quantization of the lateral degrees of freedomis manifested in the local density of states (LDOS) shown in Fig. 1 where it leads to a seriesof Van Hove singularities (corresponding to the bottom of each LL subband) in the emitterand collector and to a series of quasi-discrete levels in the well.For nonzero RSOI the LLs given by Eqs. (6) and (7) are mixed into eigenstates labelledas | kns i where the allowed values of the pair of quantum numbers n and s are ns = 0+ , − , , − , , .... (8)The corresponding energies are E ns = n ~ ω C + s s(cid:18) ~ ω C − ∆ B (cid:19) + nR , R = √ α R /L. (9)The | kns i states can be expressed in terms of | knσ i as | kns i = A ↑ ns | k ( n − ↑i + A ↓ ns | kn ↓i , (10)with coefficients A σns given by A ↑ n + = √ nR p nR + ( E n + − E n − ↑ ) , A ↓ n + = i ( E n + − E n − ↑ ) p nR + ( E n + − E n − ↑ ) ,A ↑ n − = i ( E n ↓ − E n − ) p nR + ( E n ↓ − E n − ) , A ↓ n − = √ nR p nR + ( E n ↓ − E n − ) . (11)In evaluating Eq. (10) for for ns = 0+, | k ( n − ↑i is to be taken as the zero ket. Thearrangement of the E nσ and E ns energy levels for a negative effective g -factor is depictedin Fig. 2. By analyzing Eqs. (9) and (10), several observations can be made regarding theeffect of RSOI on LLs : (1) the | i ket is identical (apart from the irrelevant phase factor)to | ↓i and they have the same energy, E = E ↑ , as depicted by the dots connecting thesetwo levels in Fig. 2; (2) the two closely spaced spin-split levels E n ↑ and E n ↓ become E ( n +1) − and E n + , respectively; (3) the Zeeman spin splitting, ∆ B , becomes ∆ E n = E n + − E ( n +1) − which increases with n and is greater than ∆ B for all n . B. The quantum transport model
Initially, magnetotunnelling in RTS has been studied within the frame of theLandauer-B¨uttiker formalism neglecting the phase-breaking scattering altogether or us-ing ad hoc schemes to describe it . In Refs. 52–54 the Matsubara technique suitable7or a rigorous treatment of many-body effects at finite temperatures is used to evaluatethe RTS transmission probability in the presence of LO phonon scattering. The combinedelectrostatic charging and LO phonon scattering effects have been studied in Ref. 56 byevaluating a conveniently devised equation of motion for the electron field operators. Whilethe degree of complexity and ability to account for various physical phenomena of the mod-els used in these studies vary, their common difficulty is a lack of a consistent descriptionof the non-equilibrium state of the biased RTS. It is for this reason that the NEGF modelthat simultaneously accounts for the quantum mechanical details and the non-equilibriumstatistics is the principal method in studying the RTS. Existing NEGF studies of RTSmagnetotunnelling are scarce and include the study of scattering on LO phonons and theeffect of shallow impurity states .The quantities used in the NEGF method are the Green’s functions G γ and self-energiesΣ γ both of which can be of three kinds: lesser ( γ = < ), greater ( γ = > ) and retarded( γ = R ). In explaining the model we need to jump from the mixed, | kns i| z i , to the real-space, | ρ σ i| z i = | r σ i , representation and vice versa, so to improve the paper readability westart by giving a step-by-step explanation of how the equation for determining G R is arrivedat. The retarded Green’s function G R is defined in the time domain as G Rs s ( k n z , k n z ; t , t ) = − i ~ θ ( t − t ) (cid:10)(cid:8) Ψ s ( k n z ; t ) , Ψ † ( k n z ; t ) (cid:9)(cid:11) . (12)Here t , are the time coordinates, θ ( t ) is the Heaviside step function, h ... i is the ensembleaverage and { ... } the fermion anticommutator. Ψ s ( knz, t ) and Ψ † s ( knz, t ) are the destructionand creation operators for the | kns i state at coordinate z and time t . The spin coordinate( s = ± ) is arbitrarily written as a subscript. As the problem under consideration is station-ary, all the quantities depend only on the time difference t − t and we switch to the energydomain, e.g. G Rs s ( k n z , k n z ; E ) = Z d ( t − t ) e iE ( t − t ) / ~ G Rs s ( k n z , k n z ; t − t ) . (13)Since the Hamiltonian (1) allows for decoupling the lateral and the z degrees of freedom while | kns i are eigenstates, G R is diagonal in kns in absence of scattering (any perturbation to H ).The onset of a scattering mechanism generally mixes the | kns i states so off-diagonal termsmay appear. However, depending on the scattering model details, the coupling between8ny given | k n s i and all the other | kns i states induced by the perturbation can oftenbe properly described by a diagonal term of the self-energy Σ R . All the Green’s functionsand self-energies considered here are diagonal in k and n but not in s , so the full notationfrom Eq. (13) will be abbreviated by G Rs s ( knz z ; E ). This (quasi-) decoupling of the kn coordinates will be clarified later after the scatterer correlation functions are introduced.The retarded Green’s function is found by fixing kn and inverting the matrix G Rs s ( knz z ; E ) = (cid:0) E − H s s ( knz z ) − Σ Rs s ( knz z ; E ) (cid:1) − , (14)where the z degree of freedom is represented by a tight-binding model with the nearest-neighbour hopping energy t = ~ / m ∗ a and care taken to ensure Hermiticity around theheterointerfaces where the effective mass changes abruptly . It is equivalent to replacingthe derivatives along z by finite-differences defined on a set of equidistant points with a being the point-to-point distance. Therefore, if the z axis interval enclosing the RTS isrepresented by N mesh points, Eq. (14) is a 2 N × N matrix equation, the doubling beingdue to spin.The self-energies in (14) are a sum of four terms that are calculated self-consistentlyΣ γ = Σ γ E + Σ γ C + Σ γ IR + Σ γ LO . (15)The emitter (Σ γ E ) and collector (Σ γ C ) self-energies describe the interaction of electrons withinthe RTS with the exterior of the device, as explained in Ref. 26. The interface roughness(Σ γ IR ) and LO phonon (Σ γ IR ) self-energies are assumed zero in the first iteration.The lesser ( λ = < ) and greater ( λ = > ) Green’s functions are found from the kineticequation G λs s ( knz z ; E ) = X s ,s Z z C z E dz Z z C z E dz G Rs s ( knz z ; E )Σ λs s ( knz z ; E ) G As s ( knz z ; E ) , (16)where G A is the advanced Green’s function (in the energy domain, it is the complex conjugateof G R ). In subsequent iterations FBA is used to evaluate the interface roughness ( φ = IR)and phonon ( φ = LO) lesser and greater self-energiesΣ λφ,σσ ( r r ; E ) = 12 π Z dE ′ ~ G λσσ ( r r ; E − E ′ ) D λφ ( r r ; E ′ ) , (17)where D λφ ( r r ; E ) denotes the scatterer correlation function assumed to be independent onthe electron spin (we neglect the scattering via SOI). For IR scattering the lesser ( D < ) and9reater ( D > ) correlation functions are equal D IR ( r r ; E ) = 2 π ~ δ ( E ) δ ( r − r ) X z I U ( z I ) δ ( z − z I ) , (18)where the summation is done over the heterointerfaces located at z = z I . The LO-phononcorrelation functions are taken to be D ≷ LO ( r , r ; ω ) = 2 π ~ δ ( E ∓ E LO ) U δ ( r − r ) , (19)where E LO = 29meV is the LO phonon energy in InAs. The delta-like form of correla-tion functions given by Eqs. (18) and (19) is used because it considerably simplifies thecalculations by decoupling the equations for states with different kn coordinates. Toclarify the approximation being made, note that the IR correlation function used in theliterature is usually a Gaussian function with the deviation set by the roughness laterallength Λ (typically few nanometers), while the Fr¨ohlich interaction with LO phonons gives adependence proportional to / | r − r | . Roughly, the approximation consists in assumingthat the self-energy at a given point in space is determined solely by the electronic prop-erties at that same point while the more realistic models allow for a contribution from allthe nearby points. A quantitative discussion of scatterer correlation functions aimed at de-termining meaningful values for scattering strengths U IR and U LO is given in Appendices Aand B.The mixed representation of the lesser and greater self-energies is then found asΣ ≷ IR s s ( kn, z , z ; ω ) = δ ( z − z ) X I U ( z I ) δ ( z − z I ) X σ G ≷ σσ ( ρ z , ρ z ; E ) A σ ∗ ns A σns , (20)for IR scattering andΣ ≷ LO s s ( n, z , z ; ω ) = δ ( z − z ) U X σ G ≷ σσ ( ρ z , ρ z ; E ∓ E LO ) A σ ∗ ns A σns , (21)for LO phonons. The real-space representation of Green’s functions in the above equationsis given by G γσσ ( ρ z , ρ z ; ω ) = 12 πL X ns s A σns A σ ∗ ns G γs s ( n, z , z ; ω ) , σ = ↑ , ↓ . (22)The retarded self-energy is related to the lesser and greater self-energies byΣ R ( E ) = 12 [Σ > ( E ) − Σ < ( E )] + i π PV Z dE ′ Σ > ( E ′ ) − Σ < ( E ′ ) E − E ′ , (23)10here PV denotes the Cauchy principal value. The left term is the anti-Hermitian part thatcauses the broadening of energy levels. The right term is Hermitian and corresponds to thescattering-induced energy shift (the energy correction found by time-independent perturba-tion theory ). In studying the electron transport through a RTS, the energy shift (of theorder of a milielectron volt) due to various interactions is not very significant especially sincethe exact quasi-bound state energies are hard to estimate anyway. On the other hand, thelevel broadening leads to qualitative differences and its exact value is crucial in estimatingthe degree of spin polarization. Therefore, in order to avoid the difficulties in the numericalevaluation of the Cauchy principal value, the Hermitian part is neglected, as often done inNEGF calculations .Once the values of Σ R are found, the calculations are repeated to obtain a self-consistentsolution, as in Ref. 30. III. RESULTS AND DISCUSSION
Magnetotunnelling through a RTS is governed by the coupling of the emitter and collectorelectron baths via the discrete states in the well, the coupling being a combined effect ofcoherent tunnelling through the barriers and incoherent transport assisted by scattering onthe interface roughness and LO phonons. Consequently, the starting point of our discussionis the (quasi)discrete LL spectrum in the well, which is described by the spin-polarizedDOS in the RTS. After clarifying the structure of this spectrum, we move on to studythe properties of the vertical transport and discuss the features in the RTS I-V curves.Throughout this section low temperature ( T = 4 . µ E = µ C = 20meV) are assumed. A. Properties of the RTS DOS
In absence of RSOI, the dispersion of LLs, E nσ ( B ), is represented by the usual Landaufan diagram. The presence of RSOI leads to a nonlinear dispersion E ns ( B ) given by Eq.(9) and plotted for first few LLs in Fig. 2. By investigating Eq. (9) it is seen that thearrangement of E ns levels shown schematically in Fig. 2 is reached at B large enough that ~ ω C ≫ √ nR . The large spacing between the doublet E n + and E ( n − − and any other level11mplies a simple DOS structure consisting of a series of double peaks (separated by thespin-splitting energy ∆ E n = E n + − E ( n − − ) arranged at a distance ~ ω C apart. However, arich structure is seen in DOS at smaller magnetic fields.Considering that ~ ω C is proportional to B and that R is proportional to √ B , at B → E n − levels lie below all the E n + levels ( E being the lowest amongst them). Withincreasing B , each E n − level approaches the corresponding E ( n +1)+ level crossing all the E n + levels below it. This produces a characteristic beating pattern in the DOS for any fixed B . While the exact energy spacing ∆ E between the two lowest nodes is easily calculatedfrom the E ns ( B ) dispersion, to understand its order of magnitude and dependence on B andthe RSOI parameter α R we note that the crossing occurs when the energy difference ~ ω C between two successive LLs is compensated by √ nR . Therefore n ∼ ( ~ ω C /R ) and∆ E ∼ n ~ ω C = ( ~ ω C ) /R , (24)where the tilde denotes the order of magnitude. As an example, for α R = 0 . × − eVm, B = 0 .
5T (giving ~ ω C ≈ . E ∼ . E ) σ = Z z C z E dz LDOS σ ( E, z ) , (25)with LDOS σ ( E, z ) = i π (cid:0) G Rσσ ( ρ z, ρ z ; E ) − G Aσσ ( ρ z, ρ z ; E ) (cid:1) , (26)and the total DOS is a sum of its spin polarized components, DOS( E ) = DOS ↑ ( E ) +DOS ↓ ( E ).There are two conditions that need to be satisfied for observing the crossing of the fewlowest LLs: (1) R ∼ ~ ω C and (2) ~ ω C > Γ, where Γ denotes the LL broadening. The lattercondition ensures that B is large enough for a discrete spectrum to appear. In a RTS, Γ isnonzero even for noninteracting electrons due to the finite probability of electron escapingfrom the well via tunnelling through barriers. For the structure shown in Fig. 1, Γ = 0 . .
5T ( ~ ω C ≈ . U ce = 270mV), from Eq. (3) we find α R = 10 − eVm, giving R ≈ . IG. 3. Spin-polarized DOS of the RTS biased with U ce = 270mV showing the dispersion E ns ( B ).To observe the LLs in this range of magnetic fields, the LLs must be very narrow. Here theballistic limit is taken while the value of α R is exaggerated to five times the value predicted by Eq.3 (approximately 4 . × − eVm). The plotted quantity is log [1 + DOS σ ( E ) /D ]. of the RTS for external bias U ce = 270mV (main peak of the current) calculated in theballistic limit and with α R set to five times the value given by Eq. (3) so that the crossingof LLs is clearly seen. The 2DEG spin-polarized DOS at zero magnetic field is given by D = m ∗ / π ~ . The energy is always measured relative to the collector band edge, sothat the emitter band edge is U e = eU ce . If the RTS had a DOS as shown in Fig. 3 it wouldmean that the RSOI could be investigated by studying the structure of the RTS I-V curves.However, considering that the realistic value of Γ is few meV and that the effective mass inan InAs quantum well is likely to be higher than 0 . m , we conclude that it is not realisticto expect that the RSOI effects would affect the shape of the I-V curves.When scattering is included with strengths U and U as found in Appendices A and B,the broadening for magnetic fields around 1T becomes Γ ≈ . E ns dispersion impossible. However, RSOI can be experi-mentally observed via the beating pattern in Shubnikov-de Haas (SdH) oscillations . Itis a consequence of the beating pattern seen in the DOS. This is illustrated in Fig. 4. Thebeating pattern survives even with significant scattering effects. The distance between the13 IG. 4. DOS of RTS biased with U ce = 270mV and B = 0 . E LO above the emitter conductionband bottom. In SdH measurements this effect is not observed since the LO phonon emission atthe Fermi level is blocked by the Pauli exclusion principle. The blue curve is the case with higherIR scattering, showing that the beating pattern exists even when the LL broadening is larger than ~ ω C . nodes is found to be around 100meV, in good agreement with the estimate given by Eq.(24).The effect of LO-phonon scattering becomes apparent if the DOS is observed on a largerscale. Figure 5 shows the (a) ballistic and (b) LO-phonon scattering DOS of the biasedRTS. Compared to Fig. 5 (a), additional energy levels are seen in (b). These are the phononreplicas of the LLs displaced from them by multiples of E LO . The problem of electron-LOphonon interaction in a 2DEG, being of fundamental significance, has received a considerableattention . Polaron effects have been studied using cyclotron resonance in a quantumwell, or by investigating the I-V curve features in magnetotunnelling in perpendicular andin-plane magnetic fields.The DOS shown in Fig. 5 (b) demonstrates that the current model captures some of themain features of the electron (magnetopolaron) DOS in a (quasi)2DEG: (1) appearance of the14 IG. 5. The evolution of DOS with B for B >
1. The RTS is biased with U ce = 270mV. (a)Ballistic limit where the LL broadening is entirely due to coherent coupling with emitter andcollector states. (b) DOS when the LO phonon scattering is included with U = 4 U . The redline shows the emmiter Fermi level. The Pauli blocking of the first replica of the lowest LL levelbelow 4T (when it is below the red line) is not complete because the system is not in equilibriumso the lowest LL is only partially occupied. The plotted quantity is log [1 + DOS ( E ) /D ]. phonon replicas, (2) the anticrossing of LLs and their replicas at magnetophonon resonancesand (3) a considerable broadening of LLs only around the magnetophonon resonance. Effectsrelated to the real part of the self-energy, see Eq. (23), such as the polaron shift and effectivemass renormalization, are not included in this model. The LO phonon scattering strength U determines the height of the DOS corresponding to phonon replicas. It also affects theanticrossing energy. A quantitative connection between the LO-phonon self-energy and U is not simple to establish, owing to the fact that the self-energy is found as a self-consistentsolution, but a rough estimate is that the self-energy scales as the square root of U . Thesignificance of the phonon replicas in the RTS is that they provide an additional channel forelectrons tunnelling through the RTS. For example, the lowest spin-split replicas seen in Fig.5 (b) correspond to the lowest lying | i and | −i LLs. Once their energies are aligned withelectrons in the emitter having the same lateral coordinates, ns = 0+ and ns = 1 − , resonantintra-LL (preserving the LL coordinates) tunnelling with LO phonon emission occurs.The effect of IR scattering on the DOS is simpler and consists of broadening all thestates (both the LLs and their replicas). In contrast to the phonon scattering, it is known15 IG. 6. The effect of IR on spin-polarized DOS, DOS ↑ is shown in full lines and DOS ↓ in dashedlines. The bias voltage is U ce = 270mV, while the magnetic field is B = 5T. to depend only on the DOS and not on the occupation of the levels . In Appendix A weargue that if a delta model is used for the roughness correlation, the scattering strength atinterface z I should be taken as U IR = V ∆ π Λ . (27)In this approach only the product of ∆ IR and Λ is significant, so we fix ∆ IR = 0 . − and yield a broadening of more than 15meV. Since the typical levelbroadening in a high quality RTS is expected to be several meV, smaller values of Λ havebeen used in calculations. In Fig. 6 a comparison is made of the spin-polarized DOS inthe biased RTS at B = 5T for the ballistic (Λ = 0), Λ = 1nm and Λ = 2nm cases. TheIR scattering increases the width of the LLs to around 4meV for Λ = 1nm and 8meV forΛ = 2nm. The LL profile is found to significantly deviate from the Lorentzian profile whichcan be explained by the discrete spectrum (a Lorentzian profile is obtained if the retardedself-energy is approximately constant around the given level which is a characteristic of acontinuous spectra). 16 . Terminal currents The paramagnetic current flowing through the RTS is obtained by evaluating the traceof the current operator I op ( E ). Following Ref. 26, instead of I op , we evaluate K p ( E ) = − e π ~ (cid:2) Σ
( E ) − Σ >p ( E ) G < ( E ) (cid:3) , p = E , C , LO (28)which has the same trace as I op ( E ). The symbol p denotes the system with which the currentis exchanged . For calculating the I-V curves, the emitter (E) and collector (C) currentsare needed. The exchange of particles with the phonon bath is described by the p = LOterm. Utilizing the decoupling of kn coordinates, K p ( E ) in the mixed representation reads K p,s s ( knz , z ; E ) = − e π ~ X s Z dz (cid:2) Σ
s s ( knz , z ; E ) − Σ >p,s s ( knz , z ; E ) G / cm and 50A / cm , respectively. The red lines represent the spin-up, while the black linesare the spin-down currents. also seen in Fig. 5 (b) where the second replica of | k −i clearly has a lower DOS than the firstreplica. While the results indicate that second and third order processes make a differencein calculations, it should be noted that the reason that the two- and three-phonon peaksin and are more intense then the lower order peaks is the sequential emission of onephonon at a time. This is possible, even though the magnetophonon resonance E LO = ~ ω C is not exactly achieved, due to the significant level broadening caused by IR scattering.The effect of varying the magnetic field on the total currents is shown in Fig. 9. There isvery little difference between the B = 1T and B = 0T curves, the latter not being shown inFig. 9 (see, for example, Fig. 4 in Ref. 30). The is because the broadening of quasiboundstates is larger than ~ ω C = 4 . B = 2T, the emitter chemical potential is around20meV so two LL subbands are occupied, leading to the structure in the main peak (theonset of the coherent | k −i current is around U ce = 265mV), and the satellite peaks - thetwo peaks in the 310 − | k −i and | k −i . At B = 3T,21 IG. 10. Spin polarization P for various B . The curves are vertically displaced by 0 . µ = 21 . / × ~ ω C = 21 . | k −i emitter subband participates in the currentbut cannot be observed in the I-V curve. From B = 4T and above, only the lowest LLemitter subband is occupied. The peaks seen in the I-V curves occur when the alignmentbetween either a higher LL or a phonon replica with the emitter Fermi sea is achieved. Theevolution of the peaks towards higher values of U ec reflects the evolution of the DOS with B , as discussed on the example of the B = 5T case.The spin polarization P of the current, defined as P = i C , ↑ − i C , ↓ i C , ↑ + i C , ↓ , (32)with the magnetic field varied as a parameter is plotted in Fig. 10. The highest valueof P is always reached at the main peak but significant values occur at satellite peaks aswell. To explain this, we argue that the value of P is determined by two factors: (1) themagnetization of the emitter Fermi sea which becomes significant once only the lowest LLsubband is populated and (2) the spin splitting of quasi-bound LLs in the well. Obviously,the magnetization is also determined by the spin splitting (but that of the emitter states).However, the distinction is important because (1) and (2) represent two mechanisms leading22o nonzero P : the former influences the electron supply while the latter determines how wellwill the RTS itself filter the electrons based on their spin. P is higher at the main peakbecause most of the current is coherent and the quasi-bound LLs involved are narrower.However, the fact that the structure in P reflects so clearly the scattering processes meansthat (2) is significant even at the satellite peaks. IV. SUMMARY
A comprehensive model for quantum transport in RTSs in a perpendicular magneticfield has been described. Effects like the formation of magnetopolarons and IR induced LLbroadening are accurately reflected in the evolution of the I-V curves with changing themagnetic field. Using effective parameters that realistically reflect the InAs bandstructure,a beating pattern in the DOS has been obtained even with large IR scattering inducedbroadening, which is in agreement with experimental findings that the RSOI constant canbe estimated from SdH measurements. It has been found that the I-V curves are notsensitive to RSOI, and that the spin polarization is dominated by the Zeeman effect. Thespin polarization of the current is found to be significant for magnetic fields of few Teslaeven at satellite peaks, suggesting that its measurement might yield additional informationin magnetospectroscopic studies.
ACKNOWLEDGMENTS
Authors acknowledge the support of NATO Collaborative Linkage Grant (Reference No.CBP.EAP.CLG 983316). G.I. is grateful for the support from ORSAS (UK), Universityof Leeds, the School of Electronic and Electrical Engineering, and the Serbian Ministry ofScience Project No. OI171005. V.M. and J.R. are grateful for support from the SerbianMinistry of Science Project No. III45010.
Appendix A: Correlation function for IR scattering
The quantity used to describe interface roughness is the interface fluctuation ∆ IR ( r ),i.e. the length by which the interface is displaced from the average value. By definition,23he average (over r ) value of ∆ IR ( r ) is zero. Typical values of ∆ IR are of the order ofone monolayer, i.e. few angstroms (see e.g. Ref. 68), therefore the z -dependence of thescattering potential caused by the roughness, V IR ( r ), can be taken as a delta function and∆ IR considered to be a function of only the lateral coordinates ∆ IR ( ρ ) V IR ( r ) = δ ( z − z I ) V b ∆ IR ( ρ ) , (A1)where V b represents the conduction band discontinuity at the heterointerface.If the interaction between electrons and a scatterer is described by the Hamiltonian H ′ = Z d r Ψ † ( r ; t ) V ( r ; t )Ψ( r ; t ) , (A2)where V ( r ; t ) can be a static external potential (in which case t is redundant) or involvedynamic degrees of freedom as in case of phonons, the scatterer correlation functions aredefined as D > ( r , r ; t , t ) = h V ( r ; t ) V ( r ; t ) i , D < ( r , r ; t , t ) = h V ( r ; t ) V ( r ; t ) i . (A3)The brackets h ... i imply the statistical average which is the configurational average in caseof the IR scattering or the ensemble average in case of phonons.For IR scattering the interaction potential V IR ( r ) is static so the lesser and greater cor-relation functions are equal. After transforming to the energy domain, we find D IR ( r , r ; E ) = 2 π ~ δ ( E ) δ ( z − z I ) δ ( z − z ) V h ∆ IR ( ρ )∆ IR ( ρ ) i . (A4)The quantity h ∆ IR ( ρ )∆ IR ( ρ ) i is called the IR autocorrelation function in the literatureand usually assumed to be of the Gaussian form h ∆ IR ( ρ )∆ IR ( ρ ) i av = ∆ e − ( ρ − ρ ) / Λ , (A5)where ∆ is referred to as the roughness height and Λ as the roughness lateral length. Thevalues reported in Ref. 68 are ∆ = 3 − A and Λ = 50 − A . In problems where Λ is smallcompared to other relevant lengths, the Gaussian may be approximated by a delta function,so D ( r r ; E ) = 2 π ~ δ ( E ) δ ( z − z I ) δ ( z − z ) δ ( ρ − ρ ) V ∆ π Λ . (A6)When a magnetic field is perpendicular to the interface the magnetic length L = p ~ /eB is the relevant quantity. As it becomes smaller than 10nm for B > . Appendix B: LO phonon correlation function
For polar coupling to bulk LO phonons (Fr¨olich interaction), the term V ( r ; t ) in Eq. (A2)is given by V ( r ; t ) = X q M q e − i qr A q ( t ) , M q = 1 q s e E LO ε (cid:18) ε ( ∞ ) − ε (0) (cid:19) , (B1)where A q ( t ) = a q ( t ) + a † q ( t ) is the phonon operator while a q ( t ) and a † q ( t ) are the phonondestruction and creation operators, respectively. Assuming that the phonons are in a ther-modynamic equilibrium and neglecting the LO phonon dispersion, E q ≡ E LO , we find h A q ( t ) A q ( t ) i = δ q , − q (cid:8) e − iE LO ( t − t ) / ~ [ n B ( E LO ) + 1] + e iE LO ( t − t ) / ~ n B ( E LO ) (cid:9) . (B2)At low temperatures such that k B T ≪ E LO , the Bose-Einstein occupation factor is verysmall, n B ( E LO ) ≪
1, so it can be set to zero (only spontaneous phonon emission is con-sidered). Upon switching to the energy domain and summing up over q , the LO phononcorrelation functions are found as D ≷ LO ( r , r ; ω ) = δ ( ω ∓ ω LO ) C LO | r − r | , C LO = e ~ ω LO ε (cid:18) ε ( ∞ ) − ε (0) (cid:19) . (B3)In the main text, the correlation functions are approximated by D ≷ δ LO ( r , r ; ω ) = 2 πδ ( ω ∓ ω LO ) U δ ( r − r ) . (B4)In contrast to the case of IR scattering where the scatterer correlation is a short rangeGaussian function, approximating the long range 1 /r correlation by a delta function mayappear bizarre. The first issue to be resolved when the 1 /r correlation is used is findingthe relevant length scale. In extended systems, such as the bulk, the scale is set by thescreening length. As the problem of electron-phonon interaction in a RTS is analogous tothe problem of electron-phonon interaction in a quantum well, we may consider the latter,leaving screening aside, and conclude that for our purposes the scale is set by the quantum25 IG. 11. LO phonon scattering strength U as a function of the magnetic field, calculatedaccording to Eq. (B9) and expressed in units of U . well width L z and the magnetic length L . But since L z is the extension of the entireinvestigated system, approximating D ≷ LO by D ≷ δ LO is obviously flawed in a fundamental way.However, we will now show that the scattering strength U LO can be chosen so that D ≷ δ LO gives scattering rates for electronic states in a quantum well that are close to scattering ratesobtained using D ≷ LO .We consider an infinitely deep quantum well of width L z in a perpendicular magneticfield. k , n and the space coordinates are the same as in the main text. As RSOI and theZeeman effect are irrelevant, we neglect them. The eigenkets are | kna i where a denotes thequantum number for motion along the z axis. As the aim is only to compare the two models,it is sufficient to consider FBA without self-consistency. The Green’s functions below are,therefore, the non-interacting Green’s functions with only diagonal elements in kna , labeledby G ≷ ( na ; E ). The self-energies for the delta model are evaluated straightforwardly asΣ ≷ δ LO ( k n a , k n a ; E ) = δ k k δ n n U πL X n a G ≷ ( n a ; E ∓ E LO ) Z L z dz |h z | a i| |h z | a i| . (B5)The self-energies for D ≷ LO are not diagonal in kn , but for the first order effect we need to26onsider only the diagonal elements labelled by Σ ≷ LO ( kna ; E )Σ ≷ LO ( kna ; E ) = X n a G ≷ ( n a ; E ∓ E LO ) Ω(2 π ) Z d q M q | F ( q z , a , a ) | | H ( q k , n , n ) | , (B6)where F ( q z , a , a ) = Z dz h a | z i e − iq z z h z | a i , (B7)and | H ( q k , n , n ) | = e − ξ / n ! n ! (cid:18) ξ (cid:19) n − n (cid:12)(cid:12)(cid:12)(cid:12) L n − nn (cid:18) ξ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , n ≥ n, ξ = q k L. (B8)In these equations q is the phonon wavevector with q k and q z being its in-plane and per-pendicular components while L mn is the generalized Laguerre polynomial. For more detailson evaluating the electron-LO phonon matrix elements and scattering rates in a 2DEGsee Refs. 71–73. Assuming that only the lowest lying a = 1 subband in an infinitely deepquantum well is populated (as is the case with the RTS investigated in the main text) with h z | i = p /L z sin( πz/L z ), the self-energies given by Eqs. (B5) and (B6) are seen to beequal if U = L L z e E LO π ε (cid:18) ε ( ∞ ) − ε (0) (cid:19) Q ( n , n , B ) , (B9) Q ( n , n , B ) = Z d q | F ( q z , , | | H ( q k , n , n ) | q . (B10)The value of U given by Eq. (B9) ensures that the scattering rate from LL n to thelower lying n due to LO phonon emission obtained assuming the delta correlation D ≷ δ LO isthe same as the one obtained from the exact D ≷ LO . The dependence of U on B and LLsinvolved in scattering is analyzed by noting that only n , n pairs which differ in energy by E LO are relevant. Therefore, we fix the lower LL index n in Eq. (B9) and numericallyintegrate Q ( n , n , B ) to obtain the scattering strength as a function of B , while choosingthe upper LL index n so that n = n + [ E LO / ~ ω C ], where the square brackets [ u ] denotethe integer closest to u . The calculated U for L z = 6nm are shown in Fig. 11 in units of U = 1meV nm /D . These results show that the scattering strengths that should be usedto accurately describe the strength of various inter-LL transitions in the investigated rangeof magnetic fields are well within the order of magnitude with each other. Consequently,a D ≷ δ LO correlation with a fixed (i.e. independent of n , n and B ) value of U may be27xpected to give predictions similar to those of a D ≷ LO model. D. Bohm,
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