Low-temperature linear transport of two-dimensional massive Dirac fermions in silicene: residual conductivity and spin/valley Hall effects
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Low-temperature linear transport of two-dimensional massive Dirac fermions insilicene: residual conductivity and spin/valley Hall effects
Yuan Yao, S. Y. Liu, ∗ and X. L. Lei Department of Physics and Astronomy, Shanghai Jiaotong University,800 Dongchuan Road, Shanghai 200240, China andCollaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
Considering finite-temperature screened electron-impurity scattering, we present a kinetic equa-tion approach to investigate transport properties of two-dimensional massive fermions in silicene. Wefind that the longitudinal conductivity is always nonvanishing when chemical potential lies withinthe energy gap. This residual conductivity arises from interband correlation and strongly dependson strength of electron-impurity scattering. We also clarify that the electron-impurity interactionmakes substantial contributions to the spin- and valley-Hall conductivities, which, however, are al-most independent of impurity density. The dependencies of longitudinal conductivity as well as ofspin- and valley-Hall conductivities on chemical potential, on temperature, and on gap energy areanalyzed.
PACS numbers: 73.50.Bk,73.25.+i,72.80.Vp,72.10.-d
I. INTRODUCTION
In the last decade, the electronic properties ingraphene, a single-atom-thick two-dimensional (2D) layerof carbon atoms in a hexagonal honeycombed lattice,have been extensively studied both theoretically andexperimentally.
In this system, the low-energy carriersnear two nodal (”Dirac”) points in the Brillouin zone pos-sess linear energy spectrum and they behave just as mass-less, two-dimensional, relativistic Dirac fermions.
The high mobility as well as a long mean free path atroom temperature, makes graphene a promising candi-date for future electronic applications.However, the application of graphene in spintronics isquite limited: due to the weak spin-orbit coupling (SOC),the energy gap in graphene is very small. Recently,graphene’s silicon analog, silicene, has been proposed and synthesized.
It also has Dirac cones which aresimilar to those of graphene, but its energy gap due tointrinsic SOC at the Dirac points may reach the valueabout 2∆ SO = 1 . ∼ . SO is the characteristicenergy of this SOC). Besides, buckled lattice struc-ture with two different sublattice planes separated by avertical distance enables us to break the inversion sym-metry in silicene by applying an external perpendicularelectric field. This makes another possibility to controlthe energy gap in silicene (this energy gap is denoted by2∆ z ). Tunning the values of ∆ SO and ∆ z , a phasetransition from a quantum spin-Hall state ( | ∆ z | < ∆ SO )to a trivial insulating state ( | ∆ z | > ∆ SO ) is expected tobe observed. It was reported that, such a tran-sition from the topological insulator (TI) to the trivialband insulating (BI) state produces a quenching of thequantum spin Hall effect (SHE) and an onset of an anal-ogous quantum valley Hall effect (VHE). In Refs. 29 and32, the intrinsic SHE and VHE induced by ac electric fieldalso have been examined. However, up to now, the effectof electron-impurity interaction on these phenomena has not been analyzed yet. The previous studies on spin-Halleffect in conventional 2D systems with Rashba and/orDresselhaus spin-orbit couplings indicated that thecontribution of electron-impurity scattering to spin-Hallconductivity is quite important. It even can lead to thevanishing of total spin-Hall conductivity in conventional2D electron gas with Rashba SOC.
Hence, to inves-tigate the SHE and VHE in silicene, extrinsic mechanismassociated with electron-impurity interaction is expectedto play a substantial role.In previous studies on transport in graphene, one ofthe most intriguing phenomenon is the residual conduc-tivity observed in carrier-density dependence of conduc-tivity: the conductivity reduces to a finite value of or-der of e /h when chemical potential tends to zero. Much theoretical effort has been devoted to quantita-tively explain this minimum longitudinal conductivity.It is generally accepted that the origin of residualconductivity is associated with the formation of electron-hole puddles in graphene, induced by randomly dis-tributed charge impurity when the global average car-rier density is low. However, in clean samples, such asin suspended graphene samples, where the charged im-purities are removed upon annealing and puddle forma-tion should be suppressed, the residual conductivity stillcan be observed and hence the mechanisms essen-tially independent of disorders are required. In previ-ous studies, two such mechanisms were proposed. Oneis the interband correlation: notwithstanding the vanish-ing of equilibrium electron density, dc electric field ex-cites an electron from the valence to conduction band,resulting in a nonvanishing conductivity.
The otherone is associated with the low-energy states at the edgesof samples. It was demonstrated that this edge-statemechanism makes sizable contribution to the subgap con-ductance in bilayer graphene even for highly imperfectedges. ? Although the residual conductivity in gappedsystems, such as in bilayer graphene, ? –67 has been ex-tensively studied, but it is still unclear in silicene up tonow.In present paper, considering interband coherence, wepresent a pseudohelicity-basis kinetic equation approachto investigate the effects of electron-impurity interactionon longitudinal conductivity as well as on the spin- andvalley-Hall conductivities in both the TI and BI states ofsilicene. We find that, in addition to the intrinsic SHEand VHE, electron-impurity interaction makes substan-tial contributions to spin- and valley-Hall conductivities(SHC and VHC) although these contributions are prac-tically independent of impurity density. We also clarifythat the longitudinal residual conductivity can reach thevalue of order of e /h at low temperature if the scat-tering is relatively weak. The dependencies of residualconductivity and of spin- and valley-Hall conductivitieson temperature, on band gap and on chemical potentialare analyzed by considering finite-temperature screenedelectron-impurity scattering.The paper is organized as follows. In section II, wepresent the kinetic equation in pseudo-helicity basis. Thenumerical results are shown in Section III. Finally, weconclude our results in section IV and append the resultswithin relaxation time approximation in Appendix. II. THEORETICAL FORMULATION
Consider a 2D massive Dirac fermion with momentum k ≡ ( k x , k y ) and electric charge − e in buckled silicene.Its motion near the K or K ′ Dirac node can be describedby a Hamiltonian of the form [ λ ησ = ∆ z − ησ ∆ SO ]ˇ h ησ ( k ) = v F ( ηk x ˆ τ x + k y ˆ τ y ) + λ ησ ˆ τ z , (1)with η = ± K and K ′ , and σ = ± τ i ( i = x, y, z ) representthe Pauli matrices, v F ≈ . × m/s is the Fermi veloc-ity of Dirac fermion in silicene and the characteristic en-ergy of intrinsic SOC is chosen to be ∆ SO ≈ . ∆ z comes from a hybridization of p z orbitals with σ orbitals of silicon atoms and its value can be tunnedby applying external electric field along z -direction. Hamiltonian (1) can be diagonalized analytically: theeigen wavefunction Ψ ησµ k ( r ) ( µ = ± ) can be written asΨ ησµ k ( r ) = ψ ησµ ( k )e i k · r with ψ ησµ ( k ) given by ψ ησµ ( k ) = 1 p g ησ ; k ( g ησ ; k − µλ ησ ) (cid:18) ηµv F k e − iηϕ k g ησ ; k − µλ ησ (cid:19) , (2)and the corresponding eigenvalue takes the form ε ησµ ( k ) = µg ησ ; k with g ησ ; k ≡ q v F k + λ ησ . Here, k and ϕ k are the magnitude and angle of momentum k ,respectively.It is useful to introduce a unitary transformation U k ≡ [ ψ + ( k ) , ψ − ( k )], which corresponds to a change from thepseudospin basis to the pseudohelicity basis. By meansof this transformation, Hamiltonian (1) is diagonalizedas ˆ h ησ ( k ) = U + k ˇ h ησ ( k ) U k = diag[ ε ησ ;+ ( k ) , ε ησ ; − ( k )]. To drive the system out of equilibrium, an electric field E is assumed to apply in the x − y plane. In pseudospinbasis, this field can be described by a scalar potential, V = e E · r . The portion of observed electric currentcontributed from electrons with spin index σ in η val-ley is determined by J ησ ( T ) = − e P k Tr[ˇ j ησ k ˇ ρ ησ ( k , T )]with ˇ ρ ησ ( k , T ) as the pseudospin-basis distribution func-tion. ˇ j ησ k is the single-particle current in pseudospinbasis, ˇ j ησ k ≡ − e ∇ k ˇ h ησ ( k ), which has vanishing diago-nal elements: ˇ j ησ ; x = ev F η ˆ τ x and ˇ j ησ ; y = ev F ˆ τ y . Bymeans of the unitary transformation U k , J ησ ( T ) can bedetermined in pseudo-helicity basis via J ησ ( T ) = − e X k Tr[ˆ j ησ k ˆ ρ ησ ( k , T )] (3)with ˆ j ησ k = U + k ˇ j ησ k U k and ˆ ρ ησ ( k , T ) = U + k ˇ ρ ησ ( k , T ) U k being the pseudohelicity-basis single-particle current op-erator and distribution function, respectively. Eq. (3) canbe explicitly written as J ησ ( T ) = J (1) ησ ( T ) + J (2) ησ ( T ) + J (3) ησ ( T ) with J (1) ησ ( T ) = − v F e X k k g ησ ; k [ˆ ρ ησ ;++ ( k , T ) − ˆ ρ ησ ; −− ( k , T )] , (4) J (2) ησ ( T ) = − v F e X k k λ ησ kg ησ ; k Re[ˆ ρ ησ ;+ − ( k , T )] , (5)and J (3) ησ ( T ) = − v F e X k η [ k × n ]Im[ˆ ρ ησ ;+ − ( k , T )] /k. (6)Here, n is the unit vector along z -axis and ˆ ρ ησ ; µν ( k , T )[ µ, ν = ± ] are elements of matrix function ˆ ρ ησ ( k , T ).To evaluate the current, one has to determine the car-rier distribution function. In pseudospin basis it obeysthe kinetic equation of the form (cid:20) ∂∂T − e E · ∇ k (cid:21) ˇ ρ ησ ( k , T ) + i [ˇ h ησ ( k ) , ˇ ρ ησ ( k , T )] = − ˇ I, (7)with ˇ I as the collision term. Applying the unitary trans-formation, the kinetic equation for pseudohelicity-basisdistribution, ˆ ρ ησ ( k , T ), takes the form (for the sake ofbrevity, the arguments of distribution function, k and T ,will hereafter be omitted) (cid:20) ∂∂T − e E · ∇ k (cid:21) ˆ ρ ησ + e E · [ˆ ρ ησ , U + k ∇ k U k ]+ i [ˆ h ησ ( k ) , ˆ ρ ησ ] = − ˆ I. (8)Here, ˆ I is a scattering term determined byˆ I = Z dω π [ ˆΣ r ( k , ω ) ˆG < ( k , ω ) + ˆΣ < ( k , ω ) ˆG a ( k , ω ) − ˆG r ( k , ω ) ˆΣ < ( k , ω ) − ˆG < ( k , ω ) ˆΣ a ( k , ω )] , (9)with ˆG r,a,< ( k , ω ) and ˆΣ r,a,< ( k , ω ) as the retarded, ad-vanced and ”lesser” pseudohelicity-basis Green’s func-tions and self-energies, respectively. Note that in Eqs. (8)and (9), the electron-impurity scattering is embedded in ˆΣ r,a,< ( k , ω ).Without loss of generality, we further assume that theelectric field is applied along the x -axis. Thus, the kineticequation, Eq. (8), can be explicitly written as ( µ = ± ) (cid:18) ∂∂T − e E · ∇ k (cid:19) ρ ησ ; µµ + µeEv F λ ησ cos ϕ k g ησ ; k Re ρ ησ ;+ − + η sin ϕ k g ησ ; k Im ρ ησ ;+ − ! = − ˆ I µµ , (10)and (cid:18) ∂∂T − e E · ∇ k (cid:19) ρ ησ ;+ − + eEv F ( − g ησ ; k [ λ ησ cos ϕ k + ηig ησ ; k sin ϕ k ]( ρ ησ ;++ − ρ ησ ; −− ) − iηλ ησ sin ϕ k v F kg ησ ; k ρ ησ ;+ − (cid:27) + 2 ig ησ ; k ρ ησ ;+ − = − ˆ I + − . (11)A simplest approach to the complicated collision termˆ I is the relaxation time approximation (RTA). Usingit, the kinetic equation in steady-state linear-responseregime can be solved analytically. The corresponding re-sults are presented in Appendix. Here, in order to inves-tigate th effects of long-range electron-impurity scatter-ing on longitudinal conductivity as well as on spin- andvalley-Hall conductivities, we employ a two-band gener-alized Kadanoff-Baym ansatz (GKBA) to simplify thecollision term ˆ I . This ansatz, which expresses the lesser Green’s function through the Wigner distributionfunction, has been proven sufficiently accurate to analyzetransport and optical properties in semiconductors. Further, we consider electron-impurity scattering in theBoltzmann limit, where the effect of electric field on ˆG r,a is ignored and ˆΣ r,a and ˆΣ < are considered in the self-consistent Born approximation. After complicated butstraightforward calculation, ˆ I can be explicitly writtenasˆ I µµ = n i X q | V ( k − q ) | π D µµ g ησ ; k g ησ ; q (cid:8)(cid:0) C ϕ k − ϕ q kqv F + λ ησ + g ησ ; k g ησ ; q (cid:1) [ˆ ρ ησ ; µµ ( k ) − ˆ ρ ησ ; µµ ( q )]+ µλ ησ v F ( q C ϕ k − ϕ q − k )Re[ˆ ρ ησ ;+ − ( k )] − µλ ησ v F ( k C ϕ k − ϕ q − q )Re[ˆ ρ ησ ;+ − ( q )] + µη S ϕ k − ϕ q v F kg ησ ; q Im[ˆ ρ ησ ;+ − ( q )] } (cid:9) ;(12)ˆ I + − = n i X q ,µ | V ( k − q ) | π D µµ g ησ ; k g ησ ; q (cid:8) µ (cid:2) λ ησ v F (cid:0) q C ϕ k − ϕ q − k (cid:1) + ηi S ϕ k − ϕ q v F qg ησ ; k (cid:3) [ˆ ρ ησ ; µµ ( k ) − ˆ ρ ησ ; µµ ( q )]+ (cid:0) g ησ ; k g ησ ; q − λ ησ − C ϕ k − ϕ q kqv F (cid:1) (cid:2) ˆ ρ ησ ;+ − ( k ) + C ϕ k − ϕ q ˆ ρ ησ ; − + ( q ) (cid:3) − kqv F S ϕ k − ϕ q ˆ ρ ησ ; − + ( q ) o . (13)Here, D µµ ≡ δ ( ε ησµ ( k ) − ε ησµ ( q )), S ϕ k − ϕ q ≡ sin( ϕ k − ϕ q ), C ϕ k − ϕ q ≡ cos( ϕ k − ϕ q ), n i is the impuritydensity, and V ( k − q ) is the electron-impurity scatteringpotential.In present paper, we restrict our consideration to thesteady-state linear response regime. In connection withthis, the distribution function can be expressed as a sumof two terms: ˆ ρ ησ ≈ ˆ ρ (0) ησ + ˆ ρ (1) ησ , where ˆ ρ (0) ησ and ˆ ρ (1) ησ ,respectively, are the unperturbed part and the linear-electric-field part of ˆ ρ ησ . Thus, the kinetic equation for ˆ ρ (1) ησ ( k ) can be written as − e E · ∇ k ˆ ρ (0) ησ ; µµ = − ˆ I (1) µµ , (14)and − eEv F g ησ ; k [ λ ησ cos ϕ k + ηig ησ ; k sin ϕ k ](ˆ ρ (0) ησ ;++ − ˆ ρ (0) ησ ; −− )+ 2 ig ησ ; k ˆ ρ (1) ησ ;+ − = − ˆ I (1)+ − , (15)with ˆ I (1) as the linear-electric-field part of the collisionterm.Further, for convenience, we introduce electron andhole distribution functions: f e; ησ ( k ) ≡ ˆ ρ ησ ;++ ( k ) and f h; ησ ( k ) ≡ − ρ ησ ; −− ( k ). Their unperturbed partstake the forms f (0)e; ησ ( k ) = n F ( ε + ( k )) and f (0)h; ησ = 1 − n F ( ε − ( k )), respectively [ n F ( ε ) = [exp[( ε − µ ) / ( k T )] +1] − is the Fermi-Dirac distribution function, µ is the chemical potential, and T is the lattice tempera-ture]. The linear-electric-field parts of f e; ησ , f h; ησ andˆ ρ ησ ;+ − ( k ) can be obtained from Eqs. (14) and (15) byexpanding them in terms of Fourier series of momentumangle: A (1) ( k ) = A (1 ,c ) ( k ) cos ϕ + A (1 ,s ) ( k ) sin ϕ with A representing f e; ησ , f h; ησ , or ˆ ρ ησ ;+ − ( k ). The coefficientsof expansion are determined by [ i = e , h] eE ddk f (0) i ; ησ ( k ) = Γ ( k ) f (1 ,c ) i ; ησ ( k ) − Γ ( k )Re[ˆ ρ (1 ,c ) ησ ;+ − ( k )] − η Γ ( k )Im[ˆ ρ (1 ,s ) ησ ;+ − ( k )] , (16)0 = Γ ( k ) f (1 ,s ) i ; ησ ( k ) − Γ ( k )Re[ˆ ρ (1 ,s ) ησ ;+ − ( k )] + η Γ ( k )Im[ˆ ρ (1 ,c ) ησ ;+ − ( k )] , (17) eE v F λ ησ g ησ ; k X i =e , h f (0) i ; ησ ( k ) − = −
12 Γ ( k ) X i =e , h f (1 ,c ) i ; ησ ( k ) + Γ ( k )Re[ˆ ρ (1 ,c ) ησ ;+ − ( k )] − g ησ ; k Im[ˆ ρ (1 ,c ) ησ ;+ − ( k )] , (18)0 = −
12 Γ ( k ) X i =e , h f (1 ,s ) i ; ησ ( k ) + Γ ( k )Re[ˆ ρ (1 ,s ) ησ ;+ − ( k )] − g ησ ; k Im[ˆ ρ (1 ,s ) ησ ;+ − ( k )] , (19) ηeEv F g ησ ; k X i =e , h f (0) i ; ησ ( k ) − = − η ( k ) X i =e , h f (1 ,c ) i ; ησ ( k ) + Γ ( k )Im[ˆ ρ (1 ,s ) ησ ;+ − ( k )] + 2 g ησ ; k Re[ˆ ρ (1 ,s ) ησ ;+ − ( k )] , (20)and 0 = η ( k ) X i =e , h f (1 ,s ) i ; ησ ( k ) + Γ ( k )Im[ˆ ρ (1 ,c ) ησ ;+ − ( k )] + 2 g ησ ; k Re[ˆ ρ (1 ,c ) ησ ;+ − ( k )] . (21)In these equations, the scattering rates Γ i ( k ) ( i =1 , , ,
4) take the formsΓ i ( k ) = n i X q | V ( k − q ) | πδ ( g ησ ; k − g ησ ; q ) B i ( k , q ) , (22)where B ( k , q ) ≡ (1 − C ϕ k − ϕ q )[1 + ( λ ησ + v F kq C ϕ k − ϕ q ) / ( g ησ ; k g ησ ; q )], B ( k , q ) ≡ (1 − C ϕ k − ϕ q ) λ ησ v F q/ ( g ησ ; k g ησ ; q ), B ( k , q ) ≡S ϕ k − ϕ q v F k/g ησ ; q , and B ( k , q ) ≡ [1 − λ ησ / ( g ησ ; k g ησ ; q )](1 − C ϕ k − ϕ q ).After the coefficients of Fourier series are determined,the charge current contributed from electrons with spin σ in η valley, J ησ , can be obtained via J ησ ; x = − v F e X k v F k g ησ ; k X i =e , h f (1 ,c ) i ; ησ ( k ) + λ ησ g ησ ; k Re[ˆ ρ (1 ,c ) ησ ;+ − ( k )] + η Im[ˆ ρ (1 ,s ) ησ ;+ − ( k )] , (23)and J ησ ; y = − v F e X k v F k g ησ ; k X i =e , h f (1 ,s ) i ; ησ ( k ) + λ ησ g ησ ; k Re[ˆ ρ (1 ,s ) ησ ;+ − ( k )] − η Im[ˆ ρ (1 ,c ) ησ ;+ − ( k )] . (24)From Eq. (23) it is clear that not only the nonequilibrium distribution of electrons or holes makes nonvanishing con-tribution to J ησ ; x , the interband correlation process in-duced by dc electric field also contributes to the longi-tudinal current. Due to this interband coherence, zero-temperature J ησ ; x remains finite when chemical potentiallies within the energy gap. Another interesting propertyof J ησ ; x is the symmetric relation: J ++; x = J −− ; x and J + − ; x = J − +; x , i.e. the total current of electrons withboth spins in K node and that in K ′ node are the same.From Eq. (24) we see that the transverse current J ησ ; y is nonvanishing although the total Hall current J y = P η,σ J ησ ; y is zero. The nonvanishing of J ησ ; y not only di-rectly comes from the interband coherence, i.e. fromnonvanishing of Re[ˆ ρ (1 ,s ) ησ ;+ − ( k )] and Im[ˆ ρ (1 ,c ) ησ ;+ − ( k )], it isalso associated with the nonvanishing of f (1 ,s ) i ; ησ ( k ) aris-ing from an impurity-mediated process: due to electron-impurity scattering, the coherence process between con-duction and valence bands causes electrons or holes mov-ing transversely [this can be seen from Eq. (17)]. Notethat in the formalism of transport within relaxation-timeapproximation presented in Appendix, f (1 ,s ) i ; ησ ( k ) is com-pletely absent.It is interesting to analyze the impurity-density de-pendencies of J ησ ; x and J ησ ; y in the limit n i → n i → f (1 ,s ) i ; ησ ,Re ρ (1 ,s ) ησ , and Im ρ (1 ,c ) ησ are essentially independent of n i ,while the dependencies of f (1 ,c ) i ; ησ , Re ρ (1 ,c ) ησ , and Im ρ (1 ,s ) ησ on n i are quite complicated: the Laurent series of each ofthese functions in the case n i → n i and another one is inverselyproportional to n i . Consequently, when n i → J ησ ; y is independent of n i but J ησ ; x contains two components, J ησ ; x → Cn i + D/n i . III. NUMERICAL RESULTS
Further, considering finite-temperature screenedelectron-impurity scattering, we present a numericalcalculation to investigate the longitudinal transport andspin- and valley-Hall effects. Whence linear system ofEqs. (16)-(21) is solved, the currents J ησ ; x and J ησ ; y aredetermined. Thus, we can obtain the total longitudinal,spin- and valley-Hall conductivities, σ xx , σ ( s ) yx , and σ ( v ) yx ,defined as σ xx = J x /E , σ ( s ) yx = P σ,η ; i =1 , , σJ ( i ) ησ ; y / ( − eE )and σ ( v ) yx = P σ,η ; i =1 , , ηJ ( i ) ησ ; y / ( − eE ), respectively. In cal-culation, a screened scattering potential due to chargedimpurities is considered: V ( q ) = e / [2 κǫ ( q + q s ) ǫ ( q )]with κ as the dielectric constant of substrate. Here,to overcome the divergence of scattering potential inthe limit of q → µ lies within energy gap,a formal cut-off parameter Λ ≡ v F q s is introduced. ǫ ( q ) = 1 − v ( q )Π( q, T, µ ) is the static dielectric func-tion, v ( q ) = e / (2 ǫ q ) is the 2D Coulomb potential,and Π( q, T, µ ) is the static polarization function for chemical potential µ at finite temperature T . At zerotemperature Π( q, T = 0 , µ ) takes the form Π( q, T = 0 , µ ) = − µ πv F X σ,η [ F ( q ) θ ( | λ ησ | − µ )+ G ( q ) θ ( µ − | λ ησ | )] , with F ( q ) and G ( q ), respectively, taking the forms [ k ησF = q µ − λ ησ ] F ( q ) = | λ ησ | µ + v F q − λ ησ v F qµ arcsin s v F q v F q + 4 λ ησ ! (25)and G ( q ) = 1 − θ ( q − k ησF ) " p q − k ησF ) q − v F q − λ ησ v F qµ arctan p v F q − k ησF ) v F µ ! . The finite-temperature static polarization function is de-termined viaΠ( q, T, µ ) = 14 T Z ∞−∞ dε Π( q, T = 0 , ε )cosh [( µ − ǫ ) / (2 T )] . (26) A. Longitudinal conductivity
In Fig. 1, we plot the dependencies of longitudinal con-ductivity on chemical potential at various temperature.We see that when chemical potential decreases but stilllies above the bottom of conduction band, the conduc-tivity decreases rapidly. This is due to the fact that thedensity of carriers excited by temperature exponentiallydecreases. However, when µ decreases to near the bot-tom of conduction band and further drops into the energygap [see Fig. 1(b)], the behaviors of σ xx versus µ at rela-tively low temperature and at high temperature are quitedifferent. At low temperature, the conductivity first de-creases and then increases with a decrease of chemicalpotential. When chemical potential is close to the centerof the energy gap, the conductivity finally reaches a con-stant value of order of e /h , forming a plateau. However,when temperature increases, the width of this plateaubecomes shorter. It even disappears at relatively hightemperature k T > || ∆ z | − ∆ SO | : in this case, the con-ductivity first decreases and then increases when µ de-creases. Note that if the chemical potential µ ascendsfrom the bottom of conduction band and goes close tothe value | ∆ z | + ∆ SO , the second conduction band is nolonger empty: gradual change of σ xx with µ is brokenand ”dog-leg” shaped connection can be observed near µ ≈ ∆ z + ∆ SO = 3 . i.e. at T = 0 .
1, 0 .
5, 1 . σ xx versus µ depend on the strength of electron-impurity -1 σ xx ( e / h ) T = Λ =2.0meV n i =10 m -2 µ (meV)e-i scattering: (a) ∆ z =2meV ∆ SO =3.9meV σ xx ( e / h ) T = µ (meV) (b) σ xx ( e / h ) n i = m -2 µ (meV) FIG. 1. (Color online) (a) Dependencies of longitudinalconductivity on chemical potential at various temperatures.From top to bottom, the lattice temperatures are T = 10 . .
0, 6 .
0, 4 .
0, 1 .
0, 0 .
5, and 0 . SO =3.9 meV,∆ z = 2 . . n i = 1 × − m − and cut-off parameter is chosen to beΛ = 2 meV. In inset of (b) σ xx versus µ at lattice tempera-ture T = 0 .
001 K are plotted for various impurity densities: n i = 1 × , 5 × , and 1 × m − . scattering. To show this, in the inset of Fig. 1(b) weplot the dependencies of conductivity on chemical poten-tial for various impurity densities at temperature T =0 .
001 K. It can be seen that the values of residual con-ductivity increase when the strength of electron-impurityascends. Note that our approach is valid only for rela-tively clean samples ( i.e. for small n i ). If impurity den-sity further increases, one has to consider the collisionalbroadening effect induced by electron-impurity interac-tion on transport, which is beyond the scope of presentstudy.When a chemical potential increases across the bot-tom of conduction band, the metal-insulator transition n i =1x10 m -2 Λ =2.0meV1.85 ∆ SO =3.9meV ∆ z =2meV2.01.9 1.5 1.0 0.5 σ xx ( e / h ) T (K) µ =0meV FIG. 2. (Color online) Dependencies of longitudinal conduc-tivity on temperature for various chemical potentials: µ =0,0.5, 1.0, 1.5, 1.85, 1.9, and 2.0 meV. Other parameters are thesame as in Fig. 1. ∆ SO =3.9meV T = 0.1K 2.0 4.0 6.0 10.0 σ xx ( e / h ) Λ =2.0meV n i =1x10 m -2 µ =0meV ∆ z (meV) FIG. 3. (Color online) σ xx versus ∆ z at various lattice tem-peratures: T =0.1, 2, 4, 6, and 10 K. Other parameters are thesame as in Fig. 1. (MIT) is expected to be observed. To demonstrate this,in Fig. 2 we plot the dependencies of longitudinal conduc-tivity on temperature for various chemical potentials. Wefind that when temperature increases, the conductivityfirst decreases from the residual-conductivity value andthen increases when µ lies within the energy gap, whileit monotonically increases in the case µ > || ∆ z | − ∆ SO | .Note that the observed descent of σ xx with an ascent of T for µ < || ∆ z | − ∆ SO | is associated with temperaturedependence of screening of electron-impurity scattering.In previous studies on MIT in conventional 2D electronsystems, finite-temperature screening plays a key role inmechanism proposed by Das Sarma and Hwang. InFig. 2, we also can see that when µ lies within the energy σ s yx ( e / π ) Λ =2.0meV n i =1x10 m -2 Intrinsic andextrinsic SHE(b) ∆ SO =3.9meV µ =0meV T = 0.1K 2.0 4.0 6.0 10.0(a) ∆ z (meV) σ s yx ( e / π ) Intrinsic SHE n i = 1x10 m -2
5 10 Τ =0.1K σ s yx ( e / π ) Λ =2.0meVIntrinsic andextrinsic SHE(c) ∆ z (meV) µ =0meV ∆ SO =3.9meV FIG. 4. (Color online) Intrinsic (a) as well as intrinsic andextrinsic (b) spin-Hall conductivities versus ∆ z at various lat-tice temperatures: T =0.1, 2, 4, 6, and 10 K. (c) σ ( s ) xy versus∆ z at T = 0 . n i = 1 × ,5 × , and 1 × m − . Other parameters in (a)-(c) arethe same as in Fig. 1. gap and it is far from the bottom of conduction band (inour case, µ =0, 0.5, 1.0, 1.5 meV), a plateau is formedin σ xx versus T when temperature increases from zero.This implies that for such chemical potentials and latticetemperatures, the interband correlation makes dominantcontribution to total conductivity. When chemical po-tential further ascends and is close to the bottom of con-duction bands, the plateau becomes smaller and finallydisappears.In silicene, due to the specific buckled structure, ∆ z can be tunned by applying an electric field perpendicularto the plane of atoms. In Fig. 3, we plot the dependen-cies of longitudinal conductivity on ∆ z at various latticetemperatures in the case µ = 0. We can see that when∆ z goes close to ∆ SO from both the left and right sides,( i.e. the energy gap decreases), conductivity monoton-ically increases at relatively low temperature ( T = 0 . T = 4, 6, and 10 K). -2-1012-10 -5 0 5 10-1.0-0.50.00.51.0-10 -5 0 5 10-2-1012 ∆ SO =3.9meV σ v yx ( e / h ) Λ =2meV n i =1x10 m -2 µ =0meV Intrinsic and extrinsic VHE(b) T = 0.1K 2.0 4.0 6.0 10.0(a) ∆ z (meV) σ v yx ( e / h ) Intrinsic VHE Τ =0.1 K n i = 1x10 m -2
5 10 ∆ SO =3.9meV σ v yx ( e / h ) Λ =2meV µ =0meVIntrinsic and extrinsic VHE(c) ∆ z (meV) FIG. 5. (Color online) Intrinsic (a) as well as intrinsic andextrinsic (b) valley-Hall conductivities versus ∆ z at variouslattice temperatures. (c) σ ( v ) xy versus ∆ z at T = 0 . B. Spin- and Valley-Hall effects
When | ∆ z | changes across ∆ SO , the topological phasetransition may occur and the spin- and valley-Hall con-ductivities are expected to change abruptly. In Figs. 4and 5, we plot the dependencies of SHC and VHC on∆ z at various temperatures. From Figs. 4(a) and 5(a),it is clear that at low temperature, the intrinsic SHC isnonvanishing only when | ∆ z | < ∆ SO , while the intrinsicVHC is nonvanishing only in the case | ∆ z | > ∆ SO andits sign changes with the change of sign of ∆ z . Theirnonvanishing values are universal: σ ( s ) yx = e/ (2 π ) for | ∆ z | < ∆ SO ; | σ ( v ) yx | = e/h for | ∆ z | > ∆ SO . When tem-perature increases, the sharp changes of σ ( s ) yx and σ ( v ) yx near | ∆ z | = ∆ SO are smeared out. From Figs. 4(b)and 5(b), we see that the electron-impurity scatteringmakes additional contribution to the total σ ( s ) yx and σ ( v ) yx :the plateau values of σ ( s ) yx and of σ ( v ) yx , respectively, are σ ( s ) yx ≈ . e/ (2 π ) (for | ∆ z | < ∆ SO ) and | σ ( v ) yx | ≈ . e/h (for | ∆ z | > ∆ SO ). At the same time, differing from theintrinsic SHC and VHC, σ ( s ) yx no longer vanishes when -0.3-0.2-0.10.0 (b)(a) ∆ SO =3.9meV ∆ z =2meV2.2 T (K) σ S yx ( e / π ) µ =0meV n i =1x10 m -2 Λ =2.0meV3.0 1.53.52.2 2.52.0 σ V yx ( e / h ) T (K) µ =4.0meV FIG. 6. (Color online) Temperature dependencies of totalspin- (a) and valley-Hall (b) conductivities for various chemi-cal potentials. In (a), from top to bottom, the chemical poten-tials are µ = 0, 0 .
5, 1 .
0, 1 .
5, 1 .
85, 1 .
9, 2 .
0, 2 .
1, and 2 . | ∆ z | > ∆ SO and σ ( v ) yx is also finite in the case | ∆ z | < ∆ SO at relatively low temperature: when | ∆ z | increases fromthe case | ∆ z | < ∆ SO to the case | ∆ z | > ∆ SO , σ ( s ) yx first in-creases from the plateau value, quickly falls to a negativevalue and then increases towards zero. Such peak-dipstructure also can be observed in σ ( v ) yx versus ∆ z near | ∆ z | = ∆ SO . When temperature ascends, the peak-dipfeatures in ∆ z -dependencies of σ ( s ) yx and σ ( v ) yx are gradu-ally smeared out and finally disappear, but the drasticchange of values near phase-transition point still can beobserved.Obviously, the plateau values of spin- and valley-Hallconductivity are no longer universal in the presence ofelectron-impurity scattering. In Figs. 4(c) and 5(c), weplot the total spin- and valley-Hall conductivities versus∆ z for various impurity density. It is clear that althoughthe electron-impurity scattering makes substantial con-tribution to spin- and valley-Hall conductivities, the val- ues of total σ ( s ) yx and σ ( v ) yx slightly depend on the impuritydensity. This can be seen from the fact that in the limitof n i → f (1 ,s ) ησ , Re ρ (1 ,s ) ησ , and Im ρ (1 ,c ) ησ is essentially in-dependent of impurity density.It is interesting to analyze the temperature depen-dencies of total spin- and valley-Hall conductivities forvarious chemical potentials, which are plotted in Fig. 6.When temperature increases, the spin-Hall conductivitydecreases in the case | µ | > || ∆ z |− ∆ SO | while it increasesfor | µ | < || ∆ z | − ∆ SO | , exhibiting a crossover behaviordue to ”metal-insulator transition”. In contrast to this,the valley-Hall conductivity monotonically increases withan increase of temperature for µ > || ∆ z | − ∆ SO | . In thecase | µ | increases across the bottom of conduction band,it rapidly reduces towards zero. IV. CONCLUSIONS
Linear longitudinal transport as well as spin- andvalley-Hall effects in silicene have been investigated bymeans of a pseudohelicity-basis kinetic equation ap-proach. A numerical calculation was carried out by con-sidering finite-temperature-screened electron-impurityscattering. We found that, when the density of equi-librium carriers vanishes, the low-temperature longitu-dinal conductivity still is nonvanishing. This residualconductivity strongly depends on the electron-impurityscattering and leads to a plateau in the dependence ofconductivity on chemical potential at relatively low tem-perature. We also clarified that the electron-impurityinteraction makes substantial contribution to the totalspin- and valley-Hall conductivities, although the valuesof these condictivities are almost independent of impuritydensity. The temperature dependencies of longitudinalconductivity and of spin- and valley-Hall conductivitiesfor various chemical potentials were also carried out. Wefound that the changes of σ xx and σ ( s ) yx with temperatureare quite different for chemical potentials lying below andabove the bottom of conduction band, and they exhibitcrossover behaviors due to metal-insulator transition. ACKNOWLEDGMENTS
This work was supported by the project of NationalKey Basic Research Program of China (973 Program)(Grant No. 2012CB927403) and National Natural Sci-ence Foundation of China (Grant Nos. 11274227 and91121021).
Appendix: Steady-state linear transport withinrelaxation time approximation
In the relaxation time approximation, the scatteringterm can be written as ˆ I µν = Γ µν ( ρ ησ ; µν − ρ (0) ησ ; µν ) withΓ µν as the formal parameters of scattering rates. The el-ements of unperturbed distribution function, ρ (0) ησ ; µν , takethe forms: ˆ ρ (0) ησ ; µµ = n F ( ε ησµ ( k )) and ˆ ρ (0) ησ ; µν = 0 if µ = ν .From Eqs. (14) and (15) it follows that ˆ ρ (1) ησ ;+ − takesthe formˆ ρ (1) ησ ;+ − ( k ) = eEv F ( λ ησ cos ϕ k + ηig ησ ; k sin ϕ k )2 g ησ ; k (Γ + − + 2 ig ησ ; k ) × [ˆ ρ (0) ησ ;++ ( k ) − ˆ ρ (0) ησ ; −− ( k )] , and ˆ ρ (1) ησ ; µµ ( k ) are given byˆ ρ (1) ησ ; µµ ( k ) = eE cos ϕ k Γ µµ ddk ˆ ρ (0) ησ ; µµ ( k ) . (A.1)Hence, the components of longitudinal current J ησ ; x , J ( i ) ησ ; x ( i = 1 , , J (1) ησ ; x = − v F e E X k ,µ k µµ g ησ ; k ∂ ˆ ρ (0) ησ ; µµ ( k ) ∂ε ησµ ( k ) , (A.2) J (2) ησ ; x = − e Eλ ησ v F Γ + − X k ˆ ρ (0) ησ ;++ ( k ) − ˆ ρ (0) ησ ; −− ( k )2 g ησ ; k (Γ − + 4 g ησ ; k ) , (A.3)and J (3) ησ ; x = − e Ev F Γ + − X k [ˆ ρ (0) ησ ;++ ( k ) − ˆ ρ (0) ησ ; −− ( k )]2 g ησ ; k [Γ − + 4 g ησ ; k ] . (A.4)Obviously, J (1) ησ ; y = 0 but J (2) ησ ; y and J (3) ησ ; y are finite dueto nonvanishing of ˆ ρ ησ ;+ − : J (2) ησ ; y = J (3) ησ ; y = − ηe Eλ ησ v F X k ˆ ρ (0) ησ ;++ ( k ) − ˆ ρ (0) ησ ; −− ( k ) g ησ ; k (Γ − + 4 g ησ ; k ) . (A.5)From Eqs. (A.3) and (A.4) it is clear that the interbandcorrelation makes nonvanishing contribution to longitu-dinal current, which is proportional to Γ + − , i.e. the scat-tering rate. This is significantly different from the contri-bution to current from the nonequilibrium electrons andholes [Eq. (A.2)], which is proportional to the relaxationtime, i.e. , 1 / Γ µµ .Another interesting result obtained from Eqs. (A.3)and (A.4) is that J x and J x remain nonvanishing evenwhen the equilibrium densities of electrons and holes arezero. To carry out this residual conductivity, we con-sider the case in which the lattice temperature tendsto zero and the chemical potential lies within the en-ergy gap: | µ | < || ∆ z | − ∆ SO | . Obviously, J (1) ησ ; x van-ishes since there is no electron near the Fermi surface: ∂ ˆ ρ (0) ησ ; µµ ( k ) ∂ε µk = − δ ( ε ησµ ( k ) − µ ) →
0. However, J (2) ησ ; x and J (3) ησ ; x are nonvanishing and take finite values. Using thefact that ˆ ρ (0) ησ ;++ ( k ) − ˆ ρ (0) ησ ; −− ( k ) = − T = 0, we obtain the conductivity associated with J (2) ησ ; x and J (3) ησ ; x ( σ (2) xx ≡ P η,σ J (2) ησ ; x /E , σ (3) xx ≡ P η,σ J (3) ησ ; x /E ) σ (2) xx (cid:12)(cid:12)(cid:12) T =0 = X η,σ v F λ ησ Γ + − e π Z ∞ kdkg ησ ; k [Γ − + 4 g ησ ; k ]= X ησ e | λ ησ | π Γ − {| λ ησ | [2 arctan(2 | λ ησ | / Γ + − ) − π ]+Γ + − } , (A.6)and σ (3) xx (cid:12)(cid:12)(cid:12) T =0 = X η,σ v F Γ + − e π Z ∞ kdkg ησ ; k [Γ − + 4 g ησ ; k ]= e π X η,σ [ π − | λ ησ | / Γ + − )] . (A.7)From Eqs. (A.6) and (A.7) we clarify that the resid-ual conductivity depends on the factor Γ + − /λ ησ . Inthe case of small Γ + − /λ ησ for any η and σ , wehave σ (2) xx (cid:12)(cid:12)(cid:12) T =0 ≈ P η,σ e Γ + − / (48 π | λ ησ | ) and σ (3) xx (cid:12)(cid:12)(cid:12) T =0 ≈ P η,σ e Γ + − / (16 π | λ ησ | ). Thus, the total residual conduc-tivity takes the form σ xx | T =0 = σ (2) xx (cid:12)(cid:12)(cid:12) T =0 + σ (3) xx (cid:12)(cid:12)(cid:12) T =0 = P η,σ e Γ + − / (12 π | λ ησ | ), which depends on scattering andon λ ησ . In the opposite limit, i.e. in the case ofsmall λ ησ / Γ + − for any η and σ , σ (2) xx (cid:12)(cid:12)(cid:12) T =0 ≈ σ (3) xx (cid:12)(cid:12)(cid:12) T =0 ≈ e / π/ e /h ) reduces to a universalvalue. Note that in the previous investigation on trans-port of zero-gap Dirac fermions, various values of con-ductivity have been obtained. It is interesting to analyze the spin- and valley-Hall conductivities within relaxation time approxima-tion. From (A.5) we find that, when µ lies within the en-ergy gap, zero-temperature σ ( s ) yx and σ ( v ) yx take the forms σ ( s ) yx (cid:12)(cid:12)(cid:12) T =0 = − e X η,σ ησ λ ησ π Γ + − [ π − | λ ησ | / Γ + − )](A.8)and σ ( v ) yx (cid:12)(cid:12)(cid:12) T =0 = − e X η,σ λ ησ π Γ + − [ π − | λ ησ | / Γ + − )](A.9)respectively. In the clean limit Γ + − / | λ ησ | →
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