Quantum confinement, energy spectra and backscattering of Dirac fermions in quantum wire in magnetic field
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Quantum confinement, energy spectra and backscattering of Dirac fermions inquantum wire in magnetic field
V.V. Enaldiev
1, 2, ∗ and V.A. Volkov
1, 2, † V.A. Kotelnikov Institute of Radio-engineering and Electronics of RussianAcademy of Sciences, Mokhovaya st. 11-7, Moscow, 125009 Russia Moscow Institute of Physics and Technology, 141700,Institutskii per. 9, Dolgoprudny, Moscow Region, Russia (Dated: June 8, 2018)We address the problems of an energy spectrum and backscattering of massive Dirac fermionsconfined in a cylindrical quantum wire. The Dirac fermions are described by the 3D Dirac equationsupplemented by time-reversal-invariant boundary conditions at a surface of the wire. Even in zeromagnetic field, spectra quantum-confined and surface states substantially depend on a boundaryparameter a . At the wire surface with a > a <
0) the surface states form 1D massive subbandsinside (outside) the bulk gap. The longitudinal magnetic field transforms the energy spectra. In thelimit of the thick wires and the weak magnetic fields, the 1D massless surface subbands arise at half-integer number of magnetic flux quanta passing through the wire cross section. We reveal conditionswhen backscattering of the surface Dirac fermions by a non-magnetic impurity is suppressed. Inaddition, we calculate a conductance formed by the massless surface Dirac fermions in the magneticfield in collisional and ballistic regimes.
PACS numbers: 73.21.Hb,73.20.-r,73.25.+i
I. INTRODUCTION
For the last few years the concept of topological insula-tors has attracted great attention to a study of masslesssurface (or edge) Dirac fermions (DFs) in crystals. Theseexcitations can be realized in semiconductor structures oftwo types: 1) at a flat surface of the 3D topological insu-lators (TIs), like Bi Se , Bi T e or at an edge of the2D TIs in
CdHgT e quantum wells; 2) at a flat surfaceof narrow-gap semiconductors , like bismuth, bismuth-based solid alloys, lead chalcogenides. In a continuousmodel both systems are described by multicomponentenvelope functions. However, it is assumed that the sur-face states (SSs) in the two types of systems arise dueto different reasons. Emergence of the SSs in the TIs iscaused by changing of the topological Z indices acrossa crystal–vacuum interface . The topological indicesare solely defined by the Bloch Hamiltonian . Becausethey are robust to moderate perturbations of the BlochHamiltonian and can change only by closing of the gap,the SSs are topologically protected. For the systems ofthe second type, the SSs arise due to an abrupt cutoffof a crystal potential, as in Refs.[7,8]. In latter case thetopological arguments are not relevant. These SSs aresometimes called the Tamm states or the Shockley states,or the Tamm–Schockley states.3D TIs ( Bi Se -type) are described in terms of theenvelope functions by a modified 3D Dirac Hamil-tonian with momentum depending mass term m ( p ) = M − M p : H T I = (cid:18) m ( p ) σ A σp A σp − m ( p ) σ (cid:19) , (1)where p = ( p x , p y , p z ) is the 3D momentum operator, A, M, M are the model parameters, σ = ( σ x , σ y , σ z ) is the vector of the Pauli matrices, σ is an identity matrix2 ×
2. To derive the surface state spectrum of H T I , oneneeds to specify boundary conditions (BCs) for the en-velope functions. The Hamiltonian H T I has the secondorder in the momentum operators, therefore it is requiredfour restrictions for the envelope functions at the surface.Since the envelope functions are only a smoothly varyingfactor of a real wave function, the problem of the properBCs for them is not trivial. One of the most widespreadand the simplest BCs are the open BCs . TheseBCs guarantee appearing of the SSs in the bulk gap andagree with a spectrum of the tight–binding calculationsfor the (111) surface . However, the issue of correct BCsfor other surface termination is poorly studied.In the systems of the second type, the envelope func-tions of the massive Dirac fermions obey the 3D Diracequation with a constant mass term : (cid:18) mc σ c σp c σp − mc σ (cid:19) (cid:18) Ψ c Ψ v (cid:19) = E (cid:18) Ψ c Ψ v (cid:19) , (2)where m is an effective mass, c is an effective ”speed oflight”. In eq. (2) two-component spinors Ψ c , Ψ v are theenvelope functions corresponding to the conduction (c)and valence (v) bands. In this case we need only two re-strictions for components of the spinors at the surface, asthe Dirac Hamiltonian is of the first order in the momen-tum operators. One can derive the BC for the spinorsfrom the Hermiticity and time-reversal symmetry of theDirac Hamiltonian in a region with the surface S :( σ Ψ v − ia σn Ψ c ) r ∈ S = 0 , (3)where n = n ( S ) is an inner normal to the surface S , a is the real phenomenological parameter, which charac-terizes both the microscopic surface structure and bulkband structure.In a halfspace z ≥ (see Fig.1) E = − sv | p || | + E , (4)where v = 2 a c/ (1 + a ) is the SSs speed, p || = ( p x , p y , E = mc (1 − a ) / (1 + a ) is theenergy of the Dirac point counted from the middle of thebulk gap, here and below an effective chirality s = ± σ z ⊗ ( σ , [ n , p ]). We willcall these states the Tamm–Dirac (TD) states. One canclassify the surface properties depending on a sign ofthe boundary parameter a . At a ≥ E is in the bulk gap, fig.1a, that is thespectrum of the TD states is similar with the spectrumof the SSs of TI. At a < a = 1, the spectrum of the TD states possessesthe particle–antiparticle symmetry with E = 0. Itshould be noted, that such a case is the most popular inthe theory of TIs. Below, we will show that deviationfrom this symmetry ( a = 1 , E = 0) has importantconsequences for backscattering between the TD statesin a quantum wire.One can try to clarify the physical meaning ofthe parameter a in a model of sharp ”inverseheterocontact” (see also Ref.18). The model is de-scribed by the 3D Dirac equation (2) with abruptlyvarying mass and work function (it is not includedin (2)) across the heterocontact. The surface with of a > . Deviation of a from unity accountsfor particle–antiparticle asymmetry of the interface.Thus, the massless 2D SSs arise both at a flat sur-face of the TI with the Hamiltonian (1) and at a flatsurface of crystal, which is described by the 3D Diracequation (2) with the BCs (3). Note that H T I (1) re-duces to the Dirac Hamiltonian in the limit M = 0.Hence, it is reasonable to correlate results for the SSsof the two types of systems. Spin polarization of theTD states is h s D i = [ sE / mc ](sin ϕ, − cos ϕ, s D = σ ⊗ σ / , p || = | p || | (cos ϕ, sin ϕ, a = 1), the TDstates are not spin polarized ( h s D i = 0). The topologicalSSs at the (111) surface of Bi Se has spin polariza-tion h s T I i = ( − sin ϕ, cos ϕ, s/
2. Recently it has beenshown, that the Dirac point energy and the spin polar-ization of the topological SSs depend on crystallographicorientation of the surface , as Bi Se is an anisotropiccrystal. Consequently, more appropriate BCs for H T I (1), that are differed from the open BCs, can lead to ad-ditional dependence of spin polarization and the Diracpoint energy on the surface orientation.Consider now an effect of a longitudinal magnetic fieldon the spectrum of the DFs, confined in a quantum wire.From the one hand, a diagonalization of H T I (1) in aquantum wire in the magnetic field is a sophisticated problem. To the best of our knowledge, the problem isnot solved. From the other hand, the BCs (3) allow tosolve the 3D Dirac equation (2) and to find analytically aspectrum of the surface DFs in the magnetic field, takingthe quantum confinement into account.In the present paper we study in frames of latter ap-proach properties of the TD states in a cylindrical quan-tum wire allowing two possible classes of surfaces (i.e.different signs of a ).Recently, Aharonov–Bohm magnetoresistance oscilla-tions have been observed in nanowires of Bi Se and Bi T e . Period of the oscillations corresponded topassing of the one flux quantum Φ through the nanowirecross section area. It was suggested that the effect wascaused by the topological SSs. Spectrum of these SSs inthe quantum wire consists of 1D surface subbands andhas periodic dependence on a magnetic flux Φ throughthe wire. Due to the Berry phase π , massless surfacesubbands emerge at half-integer values of Φ / Φ . InRefs.[22,23] a SS conductance along a TI quantum wirewas numerically calculated in two models. First, theconductance was calculated in the model of an effectivesurface Hamiltonian for different disorder strength andvarious doping regimes. At the low doping (when theFermi level is in the vicinity of the Dirac point) the con-ductance is formed by the perfectly transmitted mode. Itapproaches e /h at half-integer values of Φ / Φ for everydisorder strength, because of suppression of backscatter-ing in this model. At the high doping the conductance os-cillates with Φ period only at weak disorder. At strongdisorder oscillations of the magnetoconductance disap-pear in this regime. Second, in the Fu–Kane–Mele lat-tice model small gap arose in 1D subband spectrumat half-integer values of Φ / Φ , due to time-reversal sym-metry breaking in the magnetic field. This results in adeviation of the wire conductance from e /h .In our paper we focus on the solution of the 3D Diracequation supplemented by the BCs (3). One of goalsconstitutes in a qualitative comparison between resultsobtained in frames of (1) and (2)-(3). Refs.[21,22,23]present the results of the TI Hamiltonian diagonaliza-tion. In Sec. II we calculate the energy spectrum of theDirac fermions, as surface as well as quantum–confinedones, confined in the cylindrical quantum wire. Besides,we find conditions, when backscattering of the surfaceDFs by a scalar potential is suppressed in zero magneticfield. In Sec. III we consider influence of the magneticfield on the spectrum of the states in the wire. It is shownthat the massless surface DFs emerge periodically on themagnetic flux through the quantum wire. Finally, we ex-plicitly determine factors that break down suppression ofbackscattering of the surface massless DFs in the mag-netic field and calculate the conductance in this regime. E FIG. 1. Energy spectrum of the Tamm–Dirac states (bold solid lines) of the 3D Dirac equation (2) in a halfspace z ≥ p || = ( p x , p y ,
0) is an in-plane momentum. a) For the surface with the boundaryparameter 0 ≥ a ≤
1, the TD states lie inside of the bulk gap, on a cone with the Dirac point E and end points at p e = 2 | a | mc/ (1 − a ); b) For the surface with − < a < ~ s of the TD states at the cone surface, corresponding to s = ± < a ≤
1. Grey color corresponds to thebulk states continuum.
II. DIRAC FERMIONS IN CYLINDRICALQUANTUM WIRE WITHOUT MAGNETICFIELD
Consider a cylindrical quantum wire with a radius R .Choose the cylindrical axis z along the wire axis. The en-velope functions of DFs Ψ c , Ψ v obey the 3D Dirac equa-tion (2) and satisfy BCs (3) with constant a at the cylin-drical surface. From now on we set ~ = c = 1 everywhere,except where it is needed. Cylindrical symmetry of the3D Dirac equation (2) and BCs (3) implies conservationof longitudinal momentum k z and total angular momen-tum J z = σ ⊗ j z , with eigenvalues j = ± / , ± / , . . . ,where j z = σ ( − i∂ θ ) + σ z /
2. Hence, one can find Ψ c asfollows: Ψ c = (cid:18) ψ c ( r ) e i ( j − / θ ψ c ( r ) e i ( j +1 / θ (cid:19) e ik z z . (5)By use of the Dirac equation (2), it is convenient to ex-press Ψ v via Ψ c . The radial wave functions ψ c ( r ) , ψ c ( r )obey the Bessel equation (cid:16) − ∂ ∂r − ∂r∂r + ( j ∓ / r (cid:17) ψ c ,c = (cid:0) E − m − k z (cid:1) ψ c ,c , (6)and BCs i (cid:16) ∂ r − j − / R − a ( E + m ) (cid:17) k z − k z i (cid:16) ∂ r + j +1 / R − a ( E + m ) (cid:17) (cid:18) ψ c ( r ) ψ c ( r ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = R = 0 . (7)Therefore ψ c ,c ( r ) = C , J j ∓ / ( kr ), where k = p E − m − k z , J j ∓ / ( kr ) – the Bessel functionsof the first kind, C , C – arbitrary constants. The BCs(7) impose a relationship for the constants C , C and give a dispersion equation k (cid:20) J j − / ( kR ) J j +1 / ( kR ) − J j +1 / ( kR ) J j − / ( kR ) (cid:21) == (cid:18) a − a (cid:19) E + m (cid:18) a + 1 a (cid:19) . (8)Imagine values of k correspond to the TD states, realvalues of k describe quantum–confined states of bulkDFs. Because of a symmetry property of the disper-sion equation (8) a → /a , E → − E , it is enoughto consider the case | a | ≤
1. Besides, the disper-sion equation does not depend on j sign as followsfrom the property of the Bessel function of integerindex J − j ∓ / ( x ) = ( − j ± / J j ± / ( x ). Consequentlyall 1D subbands have double degeneracy due to the j sign. Result of qualitative graphic solution of the dis-persion equation (8) is shown on Fig.(2). The energyof quantum–confined states at | k z | → ∞ aspires to E = ± q m + k z + γ j ± / ,n /R , where γ j ± / ,n is the n-th zero of the Bessel function J j ± / . At a > v ~ | j | /R in the thick wireslimit ( | a | mcR/ ~ ≫ Bi Se -typequantum wires . The energy spectrum of the TD stateshas asymptotes E = − sv ~ s k z + j R + E , (9)when their decay length κ − is much smaller than ra-dius R ( κR ≫ j ). The four radial components of an E g E E g < a < 1 a) b) k R z k R z a <-1< E j [ ] c * h / R E j [ ] hc * / R FIG. 2. Energy spectrum E j ( k z ) of states in a quantum wire without magnetic field for two classes of the surface: a) at0 < a ≤
1, b) at − < a < E g = 2 mc is the bulk gap, R is a wire radius. Grey color fills regions of quantum–confinedstates. Solid bold lines emphasize the spectrum of the TD states, associated with different values of total angular momentum j .All subbands have double degeneracy due to the sign of j . Dash lines show boundaries between TD states and quantum–confinedones. eigenstate | j, k z i are ψ c ( r ) = CJ j − / ( kr ) ,ψ c ( r ) = C ik z (cid:16) k − a ( E + m ) e J (cid:17) J j +1 / ( kr ) ,ψ v ( r ) = C k z (cid:16) E − m − a k ( E + m ) e J (cid:17) J j − / ( kr ) ,ψ v ( r ) = Cia e JJ j +1 / ( kr ) , (10)where e J = J j − / ( kR ) /J j +1 / ( kR ), here and be-low C is a normalization factor. While time re-versal symmetry prohibits scattering between Kramerspartners, ( k z , j ) → ( − k z , − j ), an act of backscattering,( k z , j ) → ( − k z , j ), is allowed.We now show that the backscattering amplitude is sup-pressed in the limit κR ≫ j . The wave function of theTD state (10) in this limit is | j, k z , s i ≈ C e ( j − / θ − isk z p k z + j /R e ( j +1 / θ sa k z p k z + j /R e ( j − / θ ia e ( j +1 / θ e κ ( r − R )+ ik z z √ πκr . (11)Therefore the amplitude of backscattering off a scalarpotential V ( r ) is h− k z , j, s | V | s, j, k z i ≈ π (cid:16) k z R j (cid:17) Z π e V ( R, θ, k z ) dθ, (12)where e V ( r, θ, k z ) = R + ∞−∞ exp ( i k z z ) V ( r, θ, z ) dz/ π isthe Fourier transform of V ( r ) along z axis. At inte-grating R R e V ( r, θ, k z ) e κ ( r − R ) dr , we assume that themain value of the integral comes from the vicinity of R . This holds true when e V ( r, θ, k z ) declines slower than e κ ( r − R ) . Otherwise the radial integral has exponentialsmallness in κR , due to exponential decay of the TD wavefunction. From the Eq.(12) one can see that backscatter-ing is suppressed at j /k z R ≪ III. DIRAC FERMIONS IN CYLINDRICALQUANTUM WIRE IN LONGITUDINALMAGNETIC FIELD
In magnetic field along the wire axis B = (0 , , B )we make the Peierls substitution p → p + e A inthe 3D Dirac equation (2), − e is the electron charge.For a vector–potential we choose the cylindrical gauge A = ( − By/ , Bx/ , B >
0. The radial components ψ c , ψ c ofthe spinor Ψ c obey the following Eqs. (cid:18) − ∂ ∂r − ∂r∂r + ( j ∓ / r + j ± / λ + r λ (cid:19) ψ c ,c =( E − m − k z ) ψ c ,c , (13)where λ = 1 /eB is the magnetic length squared. Nor-malizable solutions of the radial equations (13) are ex-pressed in terms of the Kummer function M ( α, β, ξ ) .We are interested in the TD states in the bulk gap. While B >
0, these states have negative total angular momen-tum j ≤ − /
2. Their radial components are ψ c ( r ) = Cg ( r ) M (cid:16) − λ k , − j + , r λ (cid:17) ,ψ c ( r ) = iCk z r (cid:16) j − ) − a R ( E + m ) f M (cid:17) × g ( r ) M (cid:16) − λ k , − j + , r λ (cid:17) ,ψ v ( r ) = Ck z (cid:16) E − m − a Rk f M j − / (cid:17) × g ( r ) M (cid:16) − λ k , − j + , r λ (cid:17) ,ψ v ( r ) = iCa Rr f M g ( r ) M (cid:16) − λ k , − j + , r λ (cid:17) , (14)where g ( r ) = e − r λ ( √ λ/r ) j − / and f M = M (1 − λ k / , − j + 3 / , R / λ ) M ( − λ k / , − j + 1 / , R / λ ) . Substituting the functions (14) in the BCs (3) wederive the dispersion equation in magnetic field for j ≤ − / h j − / − a R ( E + m ) f M i × (cid:20) R k j − /
2) + a R ( E + m ) f M (cid:21) + k z R = 0 . (15)Numerical solution of this equation (15) yields an en-ergy spectrum E ( j, k z ), see Fig.3. Since in zero magneticfield the TD states is in the bulk band gap at a > κR ≫ j , κR ≫ | j | · Φ / Φ ), the dispersion law of the TD states is E = sv ~ s k z + ( j + Φ / Φ ) R + E , (16)where Φ = πBR is the magnetic flux through the wirecross section.While the magnetic field is weak, the massless TDsubbands periodically emerge at half-integer values of j ≡ − Φ / Φ with a period of Φ . With further increaseof the magnetic field, the period is of weakly dependenceon B . This effect originates from non-zero decay lengthof the TD states. To study properties of backscatteringbetween the massless TD states by a scalar potential V ,we employ the asymptotic expansion of the radial com-ponents (14) in the limit κR ≫ j , κR ≫ | j | · Φ / Φ : ψ c ( r ) = C (cid:20) − j ( j − − j − ) r λ κr (cid:21) (cid:16) √ λr (cid:17) / e κ ( r − R ) ,ψ c ( r ) = Cs | k z | ik z (cid:20) − j ( j +1) − j − ) r λ κr (cid:21) (cid:16) √ λr (cid:17) / e κ ( r − R ) ,ψ v ( r ) = Csa | k z | k z (cid:20) − j ( j − − j − ) r λ κr (cid:21) (cid:16) √ λr (cid:17) / e κ ( r − R ) ,ψ v ( r ) = Cia (cid:20) − j ( j +1) − j − ) r λ κr (cid:21) (cid:16) √ λr (cid:17) / e κ ( r − R ) . (17) Calculation of the backscattering amplitude in abovelimit results in h j , − k z , s | V | j , k z , s i ≈ π E mc ∗ j κR Z π e V ( R , θ, k z ) dθ. (18)Although the amplitude does not vanish for arbitrary val-ues of a , but it is zero if particle–antiparticle symmetry( a = 1, E = 0) is preserved.To illustrate properties of backscattering processes, wecalculate a conductance of the cylinder quantum wirewith a length L formed by the massless TD subbands indifferent regimes. In the collisional regime the conduc-tance is expressed in terms of the conductivity as follows:Σ = σ/L . The conductivity σ is calculated by means ofthe classic Boltzmann equation in the τ –approximation.For simplicity we set random potential of impurity cen-ters like this V ( r ) = V P Ni δ ( z − z i ), z i is a site of the i -th center along the wire axis, V is power of the impu-rity center, N is a number of the centers. Therefore, acorrection f ( k z ) to the equilibrium Fermi–Dirac distri-bution f ( E ) is f ( k z ) = ev z τ ( E ) ∂f ( E ) ∂E F, (19)where v z = sv , F is electric field along z axis, τ ( E ) isrelaxation time. In considered approximation1 τ ( E ) = 2 L ~ v h| V − k z k z | i , (20)where h| V − k z k z | i = nV E (2 j ) /L ( m c ∗ )( κ R ) isthe backscattering amplitude (18) averaged over the im-purity center sites, n = N/L is a concentration of theimpurity centers. Using of the density current formula j = − e R + ∞−∞ f ( k z ) v z dk z / π permits us to derive the con-ductance at low temperature:Σ = 2 e h vτ F L , (21)where τ F ≡ τ ( E F ). As the massless TD subbandsemerge at half-integer values of Φ / Φ in the weak mag-netic fields, consequently the wire conductance will os-cillate in B and reach peaks at the same fluxes. How-ever a peak magnitude is decreased with an increase ofΦ / Φ in this regime. It follows from the relationship j ≡ − Φ / Φ and Eqs.(21), (20). Another effect of a mag-netic field increase is deterioration of strictly B -periodicoscillations of the conductance. In the ballistic regime( vτ F ≫ L ), the conductance is obeyed the Landauer for-mula Σ = e /h . In such a case magnitude of the peakdoes not depend on B . These periodic or quasiperiodicoscillations of the conductance is a manifestation of theAharonov–Bohm effect. IV. DISCUSSION AND SUMMARY
So, we studied the energy spectra of the massive Diracfermions, confined in a cylindrical quantum wire. Micro- -25 -20 -15 -10 -5 03002001000-100-200 E [meV] jj a) k z [ ] E [meV] 0 b) E E g E FIG. 3. Energy spectrum of DFs in a cylinder quantum wire in a longitudinal magnetic field: a) E j ( k z = 0) as a function oftotal angular momentum j ; b) E j ( k z ) as a function of the k z for three values of the angular momentum j = − . , − . , − . j = j = − . R = 50nm, a = 0 . B = 6 . E g = 2 mc = 0 . c = 10 m/s. scopic properties of the wire surface are characterized bya real phenomenological parameter a . The latter con-trols the boundary conditions for the 3D Dirac equation.There are two classes of the surfaces in dependence onthe a sign. The DFs spectra in the TD states are deter-mined by the surface class and magnetic flux Φ passingthrough the wire cross section. At zero magnetic fieldthe spectrum of the TD states consists of 1D subbandsindexed by the total angular momentum. For one of thesurface class ( a > Bi Se quantum wire . In the longitu-dinal magnetic field, the massless TD subbands emergein the wire of the same surface class. In the limit ofsmall decay length of the TD states ( κR ≫ j ) and inthe weak magnetic field ( κR ≫ | j | · Φ / Φ ), the mass-less TD subbands arise at half-integer values of Φ / Φ .In this limit, the backscattering amplitude between themassless TD states, Eq.(18), is a product of two smallparameters, Φ / Φ κR and (1 − a ). The former definesthe smallness of TD decay length, and the latter is a measure of particle-antiparticle asymmetry. At strongermagnetic field the emergence of the massless TD sub-bands has quasiperiodic character.In the case of particle–antiparticle symmetry ( a → σ tends to infinity as(1 − a ) − . It should be emphasized that the masslessTD states are not protected, generally speaking, frombackscattering if a = 1. The theory of topological SSsin the TI quantum wires does not describe this result.The Φ-periodic emergence of the massless TD states canlead to the Aharonov–Bohm effect in the wire resistance.This result is in qualitative agreement with the TI-wirecase. ACKNOWLEDGMENTS
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