The evaluation of non-topological components in Berry phase and momentum relaxation time in a gapped 3D topological insulator
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r The evaluation of non-topological components in Berry phase and momentumrelaxation time in a gapped 3D topological insulator
Parijat Sengupta , ∗ Gerhard Klimeck , and Enrico Bellotti Department of Electrical and Computer Engineering, Boston University, Boston, MA 02215 Dept of Electrical and Computer Engineering, Purdue University Network for Computational Nanotechnology, Purdue University, West Lafayette, IN, 47907
The zero gap surface states of a 3D-topological insulator host Dirac fermions with spin lockedto the momentum. The gap-less Dirac fermions exhibit electronic behaviour different from thosepredicted in conventional materials. While calculations based on a simple linear dispersion canaccount for observed experimental patterns, a more accurate match is obtained by including higherorder −→ k terms in the Hamiltonian. In this work, in presence of a time reversal symmetry breakingexternal magnetic field and higher order warping term, alteration to the topologically ordained Berryphase of (2 n +1) π , momentum relaxation time, and the magneto-conductivity tensors is established. I. Introduction
Topological insulators (TI) such as Bi Se and Bi Te are characterized by surface states which possess distinctcharacteristics, for instance, a non-trivial Berry phase of(2 n + 1) π , mass-less Dirac fermions, and a spin helicalstructure under ideal conditions. Time reversal sym-metry(TRS) breaking perturbations destroy the topolog-ical nature of the surface states and introduce additionalcontributions that have been experimentally establishedprimarily through ARPES data. Further, the linear dis-persion, at points away from the Dirac cone is altered byhigher order k -terms that in turn modify the topologi-cally ordained behaviour. An external magnetic fieldwhich breaks TRS and higher order k -terms that warpthe band structure is considered in this work to 1) com-pute deviations to the Berry phase from the well-known(2 n + 1) π and 2) evaluate the momentum relaxation timefor electrons scattered on the surface of a 3D-topologicalinsulator.The paper is outlined as follows: In Section II, an ex-pression for the Berry phase using dispersion and wavefunctions obtained from the warped and a gapped TIHamiltonian is calculated which explicitly shows a non-topological component followed by a determination of themomentum relaxation time within a Boltzmann equationframework. The expressions derived in this section arenumerically evaluated in Section III. A brief summaryof results and their potential implications concludes thepaper. II. Theory and Model
The two-dimensional Dirac Hamiltonian is H surf.states = ~ v f ( σ x k y − σ y k x ) (1)where v f denotes Fermi -velocity and σ i ; i = x, y are thePauli matrices. The two-dimensional Dirac Hamiltonianis in principle sufficient to probe the surface states; how-ever, ARPES studies of the Fermi -surface at energies sig-nificantly far away from the Dirac point reveals a snow-flake like hexagram structure that is markedly dif-ferent from a simple circular Fermi surface observed byapplication of Eq. 1. This departure from experimentis reconciled by noting that the simple two-dimensional Dirac Hamiltonian fails to account for the underlyingcrystal symmetries. The deformation of the Fermi sur-face can be theoretically reproduced if higher order k terms are incorporated in the Hamiltonian. Since thetwo-dimensional Dirac Hamiltonian must comply withthe C v point-group and time reversal symmetry, thenext higher order terms that must be added are cubicin k . The modified Hamiltonian therefore must looklike H ( k ) = ǫ ( k )+ ~ v f ( σ x k y − σ y k x )+ λ ~ (cid:0) k + k − (cid:1) σ z (2) ǫ ( k ) introduces the particle-hole anisotropy and the cu-bic terms denote warping. Using Eq. 2, and ignoringparticle-hole anisotropy, the surface state spectrum is ǫ ± ( k ) = ǫ ( k ) ± q ~ v f k + λ ~ k cos (3 θ ) (3)where θ = tan − (cid:18) k y k x (cid:19) . The spectrum contains the low-est order correction to the perfect helicity of the Diraccone predicted in Eq. 1. The cos (3 θ ) term possesses thesymmetry of the C v point group and the Hamiltonian isevidently time reversal symmetric. A. Berry phase of gapped Dirac fermions
The Berry phase of the gapped surface spectrum of a3D topological insulator was derived in Ref. 15. A finiteband gap can be induced either through the proximityeffect of a ferromagnet or an s -wave superconductor. The final expression for Berry phase is given as γ η = π ± ∆ pro q ∆ pro + ( ~ v f k ) (4)where ∆ pro is the band gap split on account of proximityeffects and η = ± | Ψ ( r ; R ) i )of the warped Hamiltonian inserted in the Schr¨odingerequation H ( R ) | Ψ ( r ; R ) i = E n ( R ) | Ψ ( r ; R ) i can be writ-ten asΨ η = 1 √ (cid:18) λ η ( k ) exp ( − iθ ) ηλ − η ( k ) (cid:19) (5a)where λ η ( k ) = vuuut ± ∆ ′ pro q ∆ ′ pro + ( ~ v f k ) (5b)∆ ′ pro = ∆ pro + λ ~ (cid:0) k + k − (cid:1) σ z (5c)The warping term effectively augments the band gapsplitting originally introduced in the topological insu-lator. The complete Berry phase in k -space such that R = k can be written as γ n ( C ) = i I C h Ψ ( r ; R ) |∇ R | Ψ ( r ; R ) i dR (6)Expanding the Berry connection A η = h Ψ ( r ; R ) |∇ R | Ψ ( r ; R ) i in Eq. 6 using the wavefunctions given in Eq. 5a yields A η = i (cid:0) λ ∗ η exp ( iθ ) ηλ ∗− η (cid:1) (cid:18) ( ∂ k λ η − iλ η ∂ k θ ) exp ( − iθ ) η∂ k λ − η (cid:19) (7)Simplifying the above expression, A η ( k ) = i (cid:0) λ ∗ η ∂ k λ η − i | λ η | ∂ k θ + λ ∗− η ∂ k λ − η (cid:1) = 12 | λ η ( k ) | ∂ k θ (8)The last expression has been condensed by noting that ∂ k λ − η evaluates exactly as ∂ k λ η with sign reversed,therefore taken together they are equal to zero. ∂ k λ η is worked out. ∂ k λ η ( k ) = ∂ k vuut ± ∆ pro q ∆ pro + ( ~ v f k ) = η λ η ( k ) ∂ k q ∆ pro + ( ~ v f k ) (9)The final expression for Berry connection in matrix no-tation is therefore A η = 12 ± ∆ pro q ∆ ′ pro + ( ~ v f k ) k (cid:18) − k y k x (cid:19) (10)In deriving Eq. 10, the angular derivatives ∂ k x θ ( k ) = − k y k and ∂ k y θ ( k ) = k x k were used. Inserting Eq. 10 inEq. 6 and integrating over a closed path in k -space gives γ η = I C dk x · A η ( k ) + I C dk y · A η ( k ) (11)Since the energy contour under the influence of warp-ing is no longer a circle but has a dependence on ( k, θ ), the usual relations k x = kcos ( θ ) and k y = ksin ( θ ) aremodified to k x = k ( θ ) cos ( θ ) and k y = k ( θ ) sin ( θ ). Theangular derivatives are therefore dk x dθ = dkdθ cosθ − ksinθ (12a)and dk y dθ = dkdθ sinθ + kcosθ (12b)The Berry phase with warping using Eq. 11 and substi-tuting for angular derivatives from Eq. 12 gives γ η = 12 I dk x ± ∆ pro q ∆ ′ pro + ( ~ v f k ) − ksinθk + 12 I dk y ± ∆ pro q ∆ ′ pro + ( ~ v f k ) kcosθk γ η = 12 Z π dθ ± ∆ + λ ~ k cos (3 θ ) q (∆ + λ ~ k cos (3 θ )) + ( ~ v f k ) (13)An analytic evaluation of the integral is not possible anda numerical solution is presented in Section III. The in-tegral, as evident, consists of the topological contribu-tion of π and the non-topological part shown in Eq. 14.The warping term effectively increases the proximity in-duced band gap and the non-topological component ofthe Berry phase. γ non − top = ∆ + λ ~ k cos (3 θ ) q (∆ + λ ~ k cos (3 θ )) + ( ~ v f k ) (14)The Berry phase induced fictitious magnetic field bycomputing ▽ × A η and setting the warping term to zerois −→ B fic = ∓ ~ v f ∆ pro (cid:18)q ∆ pro + ( ~ v f k ) (cid:19) (15)The additional fictitious magnetic field generates a veloc-ity given as v a = dkdt × −→ B fic . This velocity is transverseto the electric field and gives rise to the intrinsic Hallcurrent. Writing dkdt = − e ~ E , the corresponding Hallcurrent is j Hall = − e X σ Z BZ dk π f ( k ) v a (16a)= − e X σ −→ E × Z BZ dk π f ( k ) −→ B fic (16b)The Hall conductivity which is σ xy = ∂∂E y ( j Hall,x )and evaluating the curl such that ∂∂E y ( E × B ) = B fic,z gives σ xy = e ~ Z BZ dk π f ( k ) B fic,z (16c)where f ( k ) is the electron distribution of the given band.The Berry supported anomalous velocity in Eq. 16cgives rise to a orbital magnetization( −→ B fic ) dependentHall effect which flows transverse to the electric field.The anomalous velocity ceases to exist for a zero-gapsystem as can be easily derived by setting ∆ to zero inEq. 10 and evaluating the curl of the Berry vector po-tential. In presence of a proximity induced band gap,further augmented by warping, the anomalous velocity isfinite, a consequence of which is the intrinsic anomalousHall conductivity. This anomalous Hall conductivity istherefore band gap and warping dependent. B. Momentum relaxation time
The reciprocal of the relaxation time by solving theBoltzmann equation is τ = β Z d k ′ π δ ( ε k − ε k ) | χ kk ′ | (1 − cosθ ) (17)where χ kk ′ = |h Ψ ′ | Ψ i| ζ (cid:16) s, s ′ (cid:17) . The additional spin-scattering factor ζ (cid:16) s, s ′ (cid:17) takes in to account the helicalspin structure of the TI surface states. If the externalmagnetic field is sufficiently large, such that the spins arealigned parallel to it, ζ (cid:16) s, s ′ (cid:17) can be set to unity. β is aconstant determined from nature of the scattering source.The wave vector scattering matrix can be computed asfollows T ( k, k ′ ) = | (cid:0) λ + ( k ) exp ( iθ ) λ − ( k ) (cid:1) (cid:18) λ + ( k ) λ − ( k ) (cid:19) | (18a)where λ ± is given by Eq. 5b. Expanding the inner prod-uct gives T ( k, k ′ ) = 14 h(cid:0) λ cosθ + λ − (cid:1) + λ sinθ i (18b)Simplifying Eq. 18b, one obtains T ( k, k ′ ) = h + ( ~ v f k ) (1 + cosθ ) i (cid:16) ∆ + ( ~ v f k ) (cid:17) (18c)In deriving Eq. 18c, it is assumed that scattering takesplace between two equi-energetic states such that | k | = | k ′ | . Further, the states are assumed to be of positivehelicity (conduction band for a topological insulator) and exp ( iθ ) is set to unity for the initial wave function( θ = 0).The relaxation time can now be evaluated by multiplying Eq. 18c by the factor (1 − cosθ ) and integrating overall angles1 τ = β Z π T ( k, k ′ ) (1 − cosθ ) dθ = πβ ~ v f k ) + 4∆ (cid:16) ∆ + ( ~ v f k ) (cid:17) (19)Setting ∆ to zero for a pristine 3D-topological insulator,the relaxation time reduces to 1 τ = πβ τ t τ nt = ( ~ v f k ) + 4∆ ( ~ v f k ) + ∆ = 1 + 4 κ κ (20)where τ t ( τ nt ) denotes the individual relaxation time fora topological(trivial) insulator and κ = ∆ ( ~ v f k ) . If κ ≫ warp = ∆ + ~ λcos θk .Using ∆ warp , the scattering time expression in Eq. 19changes to1 τ = α Z π (cid:0) ∆ + λ ~ cos θk (cid:1) (1 − cosθ ) + ( ~ v f k ) sin θ n (∆ + λ ~ cos θk ) + ( ~ v f k ) o dθ (21)The integral in Eq. 21 is numerically evaluated in SectionIII.The scattering time expression derived in Eq. 21 canbe inserted in a Boltzmann equation and solved underthe relaxation time approximation to determine the re-sponse of the two-dimensional electronic system throughthe magneto-conductivity tensors σ xx and σ xy . Themagneto-conductivity tensors are evaluated with an ex-ternal magnetic field directed along the z -axis and anelectric field that is confined to the x-y plane. The elec-tric current in terms of Boltzmann transport equationper spin is therefore J = e π Z d kv k δf k (22)where δf k is the deviation from the Fermi-Dirac distri-bution, v k = v f ( cosθ, sinθ ) and θ = tan − (cid:18) k y k x (cid:19) . v f isthe Fermi -velocity. The Boltzmann equation assumingno spatial variation is ∂f ( r, k, t ) ∂t + dkdt · ∂f ( r, k, t ) ∂k = 0 (23a)where f ( r, k, t ) is the distribution function written as asum of the equilibrium distribution and deviation underan electric and magnetic field f ( r, k, t ) = f ( r, k, t ) + δf ( r, k, t ). Since dkdt = − q ( E + v × B ), Eq. 23a can bewritten as − δfτ − q (cid:18) E · v k ∂f∂ε + ( v × B ) · ∂ ( δf ) ∂k (cid:19) = 0 (23b)In writing Eq. 23b, ∂f ( r, k, t ) ∂t is approximated as − δfτ using the relaxation time approximation. E is the electricfield on surface of the TI and τ is carrier relaxation time.Solving Eq. 23b and inserting δf in Eq. 22, yields themagneto-conductivity tensors σ xx = e | ε f | π ~ τ ω c τ (24a) σ xy = e | ε f | π ~ ω c τ ω c τ (24b)The ratio of the longitudinal and transverse conductivitytensors for a pristine and gapped topological insulator istherefore σ ntxx σ txx = τ nt (cid:0) ω c τ t (cid:1) τ t (1 + ω c τ nt ) (25a)and σ ntxy σ txy = τ nt (cid:0) ω c τ t (cid:1) τ t (1 + ω c τ nt ) (25b)where ω c = ev f B/ε is the cyclotron frequency for Diracfermions at a certain energy ε . III. Results
The presence of a non-topological component in agapped 3D-TI (Fig. 1) is numerically evaluated using ex-pressions derived in Sec II. The surface state of the TIis assumed to possess a finite band gap brought aboutmost commonly through the proximity effect by interfac-ing with a ferromagnet that has an out-of-plane magne-tization component.
A. Numerical evaluation of Berry phase withwarping
The combined influence of warping and a finite bandgap on the Berry phase is computed numerically us-ing Eq. 13. The warping term ~ λ (meV˚A ) and theband gap ∆ are varied while | k | is held constant at 0.031 / ˚ A . The accumulated Berry phase for a surface elec-tron with positive spin helicity in presence of proximityinduced magnetic field and higher order warping termsis shown in Fig. 2. As the strength of the warping termand band gap increases, the deviation from the topolog-ically determined value of π is more pronounced. Thenon-topological contribution arising purely on account ofwarping in absence of a band gap can be computed by FIG. 1: Dispersion of a 20.0 nm thick Bi Se topological in-sulator slab around the Dirac point (Fig. 1a) at 0.02 eV. Thedispersion has a finite band gap (Fig. 1b) when a ferromag-net is coated on the surfaces. The ferromagnet is assumed topossess an out of plane( z -directed) magnetization axis. Themagnetization strength is an exchange energy set to 30.0 meV setting ∆ to zero in Eq. 14. Further, if the warping con-tribution is chosen such that it is equal in magnitude tothe surface state energy because of the linear Hamilto-nian, the Berry phase expression changes to γ η = 12 Z π dθ (cid:18) ± √ (cid:19) (26)which integrates to 1 . π for a positive helicity elec-tron. This condition can be fulfilled when ~ v f k = λ ~ k cos (3 θ ). Setting cos (3 θ ) to 0.5, the warping term to 200 ev˚A , and v f = 5 × m/s, gives k = 0.18 ˚A.This roughly corresponds to an energy equal to 0.4 evfor the surface states. This in a way also determines acut-off energy beyond which the warped Hamiltonian isthe dominant component and outweighs the linear con-tribution.A significant manifestation of the increase in Berryphase can be seen as a way to enhance the Berry cur-vature (Eq. 15) which is the geometric analog of a realmagnetic field. Figure 2 shows the dependence of the di-mensionless ratio of the Berry curvature to square of themagnetic length (cid:18) l b = ~ eB (cid:19) as a function of an exter-nal magnetic field. The magnetic length is approximatedas 25.0 nm / √ B . The band gap split is calculated usingthe Zeeman splitting ( gµ B B ), where the g -factor is setto 20 for electrons on surface of a TI. The Fermi -velocityfor surface electrons in Bi Te is 5 × m/s and the k -vector is 0.3 1/˚A. The Berry curvature in Eq. 15 doesnot include the warping term, which has been ignored towrite a more compact expression. B. Momentum relaxation time
The ratio of momentum relaxation times for the caseof an ungapped topological insulator to a gapped sam-ple is roughly four times if κ = ∆ ( ~ v f k ) in Eq. 20 issignificantly larger than unity. The gap (∆ = gµ B B ) iscomputed by assuming an externally impressed magneticfield. Figure 4 shows the relaxation time ratio τ t /τ nt as FIG. 2: The Berry phase ( γ ) magnitude plotted as function ofthe warping strength. γ increases with greater warping thusmagnifying the non-topological contribution to the standardvalue of π . The overall phase is shown for two values of theband gap indicated on the sub-plots. As warping increases,the band gap contribution diminishes and the curve flattensout.FIG. 3: The ratio of Berry phase induced orbital magneticfield (Berry curvature) to magnetic length is plotted againstan externally applied magnetic field. The Berry curvature isaugmented as the magnetic field strength increases. Warpingeffects are ignored in this calculation. a function of the warping strength. The correspondingratio for a magnetic field equal to 50 T without warpingis also indicated as a constant. For relatively small valuesof the magnetic field, the linearly dispersing surface-stateenergy ~ v f k dominates the Zeeman splitting term. Forinstance, with B = 50 T along the z -axis, and a g -factorequal to 20, the Zeeman-induced band gap is roughly FIG. 4: The relaxation time is computed for various warpingstrengths. At zero warping strength, the relaxation times for afinite band gap topological insulator coincide. The relaxationtime is significantly altered due to a higher warping of thebands. k -point = 0.3 1/˚A is 6.2 eV. The surface energy thereforemasks the Zeeman split and brings the ratio of scatteringtimes for a topological and trivial insulator close to unity.For k -points sufficiently close to the Dirac point, whichis a low energy case, for example, at k = 5e-3 1/˚A, theratio changes to 1.71. The overall validity of a surfacestate energy contribution limited to a linear Hamiltonianholds only when the k -point is chosen close to the chargeneutral Dirac point at Γ. The ratio of scattering times forpoints in momentum-space farther away from Γ is morecorrectly represented through a numerical evaluation ofEq. 21. The warping term usually dominates the bandgap split induced by the magnetic field and a markeddeviation in the ratio of scattering times from the non-warped case is seen in Fig. 4. A higher warping strengthsignificantly shifts the ratio curve away from values com-parable to unity for cases where the topological insulatoris pristine or the higher-order k terms have limited con-tribution.The ratio of longitudinal and transverse conductivitytensors without warping is evaluated by using Eq. 25aand Eq. 25b under a sufficiently strong magnetic field of10 T at 0.2 eV. The cyclotron frequency is 1.25 GHz;the ratio σ ntxx /σ ntxy by approximating 1 + ω c τ nt as ω c τ nt is τ t /τ nt . The transverse conductivity under these con-ditions is almost close to unity. As mentioned above,at points in proximity to Γ, such as k = 5e-3 1/˚A, thelongitudinal conductivity tensor ratio is 1.71. The lon-gitudinal conductivity tensor therefore exhibits the samebehaviour as the scattering times noted above. IV. Conclusion
The influence of time reversal symmetry(TRS) break-ing magnetic field that opens a band gap at the surfaceand higher order terms defined by a warping of the eigenenergy spectrum substantially alter the topological Berryphase of (2 n + 1) π and any phenomenon that dependson the relaxation rate of the surface carriers. The Berryphase from the topologically determined value of π getsan additional component that depends on the band gapsplit(TRS breaking) and the warping strength. For agiven band gap, the Berry phase increases with the warp-ing strength. The altered Berry phase for a finite gaptopological insulator described by a warped Hamiltonianalso gives rise to anomalous Hall velocity through orbitalmagentization. The relaxation time within the Boltz-mann formalism (assuming spins aligned to the externalmagnetic field) is also warping dependent and at highervalues offsets the magnetic field splitting. More realis-tic cases of time relaxation expression can be derived for charged impurities with screening and for the Kondo-effect which involves spin-flip scattering by magneticimpurities. The spin scattering factor which has beenchosen as unity to simplify calculations must be set as ascattering angle dependent quantity in a future work. Acknowledgments
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