Electron scattering by step like defect in TI nanofilm

8/10/2019 Electron scattering by step like defect in TI nanofilm

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PHYSICAL REVIEW B 87, 075318 (2013)

Electron scattering by a steplike defect in topological insulator nanofilms

Thakshila M. Herath,1 Prabath Hewageegana,2 and Vadym Apalkov1

1 Department of Physics and Astronomy, Georgia State University, Atlanta, Georgia 30303, USA2 Department of Physics, University of Kelaniya, Kelaniya 11600, Sri Lanka

(Received 3 December 2012; published 21 February 2013)

We study the properties of a steplike defect on the surface of ultrathin topological insulator nanofilms. Wecalculate the reflectance of an electron from such a defect for different parameters of the nanofilm and different

parameters of the defect. We show that an electron incident on a steplike defect not only produces reflected and

transmitted waves but also generates the modes, which are localized at the steplike defect. Such modes result in

an enhancement of electron density at the defect by ≈60%. The magnitude of the enhancement depends on the

parameters of the nanofilm and the height of the step and is the largest in the case of total electron reflection.

DOI: 10.1103/PhysRevB.87.075318 PACS number(s): 73.20.−r, 72.10.Fk, 73.63.−b, 85.75.−d

I. INTRODUCTION

Topological insulators (TIs),1–3 a new class of materialsdiscovered recently, have very unique topological properties.

While the bulk of topological insulator shows a full gap, thesurface states are gapless and are topologically4–8 protected

by the time reversal symmetry. The topological insulatorswere first proposed theoretically for HgTe quantum wells9 and

then observed experimentally.10 Recently a new type of TIs,

three-dimensional TIs, has been also discovered.11–17 The 3DTIs were predicted theoretically and observed experimentally

in Bix Sb1−x , Bi2Te3, Sb2Te3, and Bi2Se3 materials. Theunique features of 3D TIs are gapless surface states with

low-energy dispersion, which is similar to the dispersion lawof a massless Dirac fermion. Such relativistic dispersion law

has been observed experimentally by angle-resolved photoe-mission spectroscopy,13,14,16–18 oscillations in the local density

of states,19 magnetoconductivity measurements,20 quantumoscillations of magnetization,21 Aharonov-Bohm oscillations

in TI nanoribbons,22 Landau level spectroscopy,23,24 andquantum Hall effect.25 In the bulk, the TIs have a gap, the

largest value of which is realized in Bi2Se3 TI with the bulk band gap 0.3 eV.

The growing interest in TI systems is related to the uniquedispersion law of their surface states. The surface states in3D TIs are gapless and have a relativistic massless energydispersion lawwithchiral spin texture;i.e., thedirection of spinis correlated with the direction of momentum. The low-energydispersion is described by the Dirac cones. In general the TIcan have an odd number of Dirac cones, but in experimentallyrealized 3D TI systems only one Dirac cone in the surface

dispersion exists. Such Dirac cone is described by an effectiveHamiltonian of relativistic type.

The relativistic massless dispersion law is also realized inanother system, graphene.26 Although the low-energy disper-sions in TI and graphene are similar, there is a fundamentaldifference between these systems. Namely, the TI has only oneDirac cone (or odd number of Dirac cones), while graphenehas two Dirac cones, corresponding to two valleys. Thestates of each Dirac cone in graphene are chiral, but chiralitycorresponds to pseudospin, not the real spin, while the states of TI arespin chiral, where thereal electron spin is correlated withits momentum. Thus, each state in graphene has double spindegeneracy, whereas the states in TI have no spin degeneracy.

Another fundamental difference between graphene andTI is that the surface states in graphene are purely two-dimensional, whereas the surface states of TI are three-dimensional with finite spatial extension into the bulk of

TI. The finite spatial width of the surface states of TIbrings additional features to TI systems; for example, fora TI nanofilm of small thickness, the surface states at twoboundaries of the nanofilm are coupled due to the 3D natureof TI surface states.27–29 Such coupling of the states of two TIsurfaces opens a gap in otherwise gapless surface relativisticdispersion, resulting in the energy dispersion similar to thedispersion of narrow-gap semiconductors. The value of thegap depends on the thickness of TI nanofilm. By changingthe thickness of the nanofilm one can control the gap in thedispersion law and introduce the defects into the system. Thesimplest defectis thesteplikedefect, which divides two regionsof TI nanofilm with different thicknesses. Such type of defect

with the height of a step ∼30.5 ˚A was introduced on a surfaceof Bi2Te3 to study experimentally the oscillations of the local

density of states introduced by the step defect. The steplikedefect was also considered theoretically in Refs. 30 and 31within the 2D model of surface states of TI. The defect wasintroduced into the 2D effective Hamiltonian of the surfacestates as a δ-function potential.

Here we consider the properties of a one-dimensionalsteplike defect on a surface of TI nanofilm. An electron,propagating along the surface of TI nanofilm, can be scatteredby the defect, which results in generation of reflected andtransmitted electron waves. We study how electron reflectanceand transmittance depend on the parameters of the defect.Although some properties of the electron reflection can beexplained within a picture of reflection from a steplikebarrier, there are properties which are specific to the TIsystem. Namely, the scattering of an electron by a steplikedefect generates not only reflected and transmitted electronwaves, but also localized (evanescent) modes. Such modes arelocalized at the defect, which results in increase of electrondensity at the defect.

The paper is organized as the following. In Sec. II, weintroduce an effective Hamiltonian of TI nanofilm and describethe procedure of calculating the amplitudes of reflected,transmitted, and localized modes. In Sec. III, we present theresults of thecalculations. In Sec. IV we present the concludingdiscussion of the obtained results.

075318-11098-0121/2013/87(7)/075318(7) ©2013 American Physical Society

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HERATH, HEWAGEEGANA, AND APALKOV PHYSICAL REVIEW B 87, 075318 (2013)

II. MODEL AND MAIN EQUATIONS

We consider a TI nanofilm with steplike singularity, whichmeans that the TI nanofilm consists of two regions withdifferent thicknesses; see Fig. 1(a). We assume that the linearsteplike defect is described by an equation x = 0 with axisy parallel to the steplike defect and axis z perpendicular to

the nanofilm; see Fig. 1(a). In two regions, determined by thedefect, the thickness of the nanofilm is L1 at x < 0 (region 1)and L2 at x > 0 (region 2). Below we consider both L1 < L2

and L1 > L2 cases.We study the reflection (scattering) of an electron by

the steplike defect. The electron, propagating in region 1 inthe positive direction of axis x, is incident on the defectand is partially reflected (transmitted) by the defect. Thespecific feature of the TI system is that the scattering of the incident electron by the defect produces not only thepropagating (reflected and transmitted) electron waves but alsothe evanescent (localized) waves as schematically illustratedin Fig. 1(b).

To study the scattering of an electron by the steplike defect,we employ the effective low energy model of the TI nanofilm,which is described by a 4 × 4 matrix Hamiltonian with theform32

H =H+ 0

0 H−

= D1

π 2

L2 + D2

p2

h2 + A2

h

σ x py − σ y px

+

B1

π 2

L2 + B2

p2

h2

τ z ⊗ σ z, (1)

where σ i (i = x,y,z) are Pauli matrices corresponding to spindegrees of freedom, and τ z is the Pauli matrix correspondingto electron pseudospin, which describes the block diagonalHamiltonian matrix with H+ (for τ z = 1) and H− (for

τ z = −1). Here pi = −h∂/∂ri (i = x,y ) is the electron 2D

L L

Reflected Transmitted

Incident

(a)

(b)

localized modes x 0

Region 1 Region 2

FIG. 1. (a) Schematic illustration of a steplike defect on a surface

of topological insulator nanofilm. Thesteplikedefect at x = 0 divides

TI nanofilm into two regions: region 1 with nanofilm thickness L1

and region 2 with nanofilm thickness L2. (b) Schematic illustration

of electron reflection from the steplike defect. Incident, reflected, and

transmitted propagating waves are shown by arrows. The localized

modes, generated by electron reflection, are also shown. The modes

are localized at the defect.

momentum, L is the thickness of the nanofilm, and A2, B1,B2, D1, D2 are parameters of the 3D model of TI. Belowwe consider Ti2Se3 TI, for which these parameters take thefollowing values:33 A2 = 4.1 eV A, B1 = 10 eV A2, B2 =56.6 eV A2, D1 = 1.3 eV A2, D2 = 19.6 eV A2. The thirdterm in Hamiltonian (1), which is determined by constant A2,describes the spin-orbitinteraction in the effective Hamiltonian

of the TI.Since the electron dynamics for two components of pseu-

dospin are uncoupled, we consider scattering of an electronwith only one component of pseudospin; for example, τ z = 1.For such component of pseudospin the scattering is describedby a 2 × 2 Hamiltonian, where the two components of thewave function correspond to two spin components, σ z = ±1.The scattering amplitudes for pseudospin component τ z = −1are the same as for τ z = 1.

Within each region (1 or 2) the nanofilm thickness isconstant. Then the electron wave functions corresponding toHamiltonian (1) have the plane wave form ∝ ei(kx x+ky y),where (kx,ky ) is the electron wave vector. The corresponding

energy spectrum, which is given by an expression

E±(k,L)=

D1

π 2

L2 + D2k2

±

B1

π 2

L2+B2k2

2

+ A22k2,

(2)

has a gap g(L) = 2B1π 2

L2 , determined by the thickness of the

nanofilm. Here k =√

k2x + k2

y . For a given wave vector (kx ,ky )there are two energy bands, E+(k) and E−(k), which can beidentified as “conduction” and “valence” bands of TI nanofilm.The two-component wave function, corresponding to the wavevector (kx,ky ) and energy E±(kx ,ky ), has the form

kx ,ky ,L = 1 A2

2|k|2 + (F 1 + N 1k2 − E±)2

× −A2(ikx + ky )

F 1 + N 1k2 − E±(k,L)

ei(kx x+ky y), (3)

where we introduced the notations F 1 = (D1 + B1)π 2/L2 andN 1 = (D2 + B2).

For the reflection (elastic electron scattering) problem, weneed to identify the reflected and transmitted states for a givenincident electron wave. Such states are determined by thecondition that the energy of the reflected (transmitted) electronis equal to the energy of the incident electron.

A specific feature of the TI system is the existence of fourdifferent wave vectors k which have the same energy. Namely,at a given energy E there are four solutions of an equationE = E±(k). These solutions are given by the expressions

k1,±(E,L) = ±

β +

β2 + 4αγ

2α, (4)

k2,±(E,L) = ±

β −

β2 + 4αγ

2α, (5)

where α = B22 − D2

2 , β = (B1B2 − D1D2) 2π 2

L2 + 2ED2 +A2

2, γ = E(E − D12π 2

L2 ) + (D21 − B2

1 ) π 4

L4 . If energy E is notin the energy gap of the TI nanofilm, then wave vectors k1,

±075318-2



ELECTRON SCATTERING BY A STEPLIKE DEFECT IN . . . PHYSICAL REVIEW B 87, 075318 (2013)

are real and correspond to two propagating waves, whereasthe wave vectors k2,± are purely imaginary and describedecaying and growing modes. If energy E is in the gap,i.e., E−(k = 0) < E < E+(k = 0), then all solutions, k1,± andk2,±, are imaginary, which means that there are no in-gappropagating modes and all the modes are either decaying orgrowing modes.

The existence of 4 different wave vectors with the sameenergy introduces an additional features in the electronscattering by a defect in the TI nanofilm. For a steplike defectthe problem of scattering (reflection) is formulated as follows.An electron with wave vector (kx > 0,ky ) and energy E±(k) ispropagating in the positive direction of axis x and is incidenton the steplike defect. The direction of electron propagation ischaracterized by the angle of incidence, π/2 ϕ 0, definedas tan ϕ = ky /kx . The scattering of an electron by the defectresults in generation of two reflected and two transmittedpropagating or evanescent waves; see Fig. 1(b). Such wavesaredetermined by the condition that the energy of the reflected ortransmitted wave is equal to the energy of the incident electron

and the y component of the wave vector, ky , of the incidentelectron is equal to the y component of the transmitted andreflected waves. Here for the reflected (transmitted) waves weneed to choose the waves, which propagate or decay in thenegative (positive) direction of axis x .

The corresponding wave functions are described by Eq. (3)and are characterized by the following parameters:

(1) The incident wave is kx ,ky ,L1, where k = k1,+(E,L1)

and E is the energy of incident electron.(2) The reflected waves are −kx ,ky ,L1

(propagating wave)

and −iλ,ky ,L1 (evanescent wave), where λ > 0 and −λ2 +

k2y = [k2,+(E,L1)]2.

(3) Transmitted waves are kx ,ky ,L2 (propagating

or evanescent wave) and iλ ,ky ,L2 (evanescent wave),

where (kx)2 + k2y = [k1,+(E,L2)]2, λ > 0, and−(λ)2 + k2

y =[k2,+(E,L2)]2.

Then the electron wave function is

0 =

kx ,ky ,L1 + r1

−kx ,ky ,L1 + r2

−iλ,ky ,L1, x < 0,

t 1 kx ,ky ,L2 + t 2 iλ,ky ,L2

, x > 0, (6)

where r1 and r2 are amplitudes of the reflected waves and t 1and t 2 are amplitudes of the transmitted waves. The amplitudesr1, r2, t 1, and t 2 are found from the boundary condition, i.e.,continuity of the wave function, 0, and its derivative, ∂x

0, atx = 0, which is equivalent to continuity of thecurrent (electronflux). Then the reflection and transmission coefficients are R

=|r 21 | and T = 1 − R, respectively. Since the evanescent modes

do not contribute to electron current, there is no contribution of the evanescent modes into the reflectance and transmittance.

III. RESULTS AND DISCUSSION

A. Reflectance

Since the evanescent waves produce zero current, only thepropagating waves contribute to the electron reflectance. Thus,if the energy of the incident electron, which is in the region1 of the nanofilm with thickness L1, is in the energy gapof the TI nanofilm in region 2 with thickness L2, then thereare no propagating transmitted waves and the transmittance

0 20 40 60 80 100 0

0.2

0.4

0.6

0.8

1.0

k0.005Å

L 40 Å 0˚

0.02

0.01

0.04

0.1

R e

f l e c t a n c e

L2(Å)

FIG. 2. (Color online) Electron reflectance is shown as a function

of the thickness L2 (see Fig. 1) of nanofilm in region 2 for different

values of electron energy. The thickness of TI nanofilm in region 1 is

L1 = 40 A and the angle of incidence is ϕ = 0. The numbers next to

the lines are the wave vectors of the incident electron in units of A−1

.

in this case is 0; i.e., the reflectance is exactly 1. Such total

internal reflection can be realized only if g(L1) = 2B1π 2

L21

<

g(L2) = 2B1π 2

L22

; i.e., L1 > L2.

If the energy of the incident electron is not in the gap of the TI nanofilm in region 2 then there is a finite reflectanceof the incident electron. In Fig. 2 the reflectance is shown asa function of thickness L2 [see Fig. 1(a)] for different wavevectors k of the incident electron, i.e., for different energies of the incident electron. The thickness of the nanofilm in region1 is fixed and is equal to L1 = 40 A. The reflectance is zero atL2 = L1 = 40 A, which corresponds to the homogeneous TInanofilm with constant thickness. At L2

=L1 the reflectance

is not zero, but it shows different behaviors for L2 > L1 andL2 < L1. At L2 > L1 the reflectance monotonically increaseswith L2, reaching some asymptotic values at large L2, whichare around 20%–40% depending on the electron energy.

At L2 < L1, within a narrow interval of thickness L2,the reflectance shows very strong dependence on L2. Ateach value of electron energy (or wave vector), there is

a critical thickness L(cr)2 so that at L2 L

(cr)2 an electron

experiences total internal reflection with reflectance equal to1. This corresponds to the condition that the electron energyE is in the energy gap of TI nanofilm in region 2. The

critical thickness is determined by an equation E+(0,L(cr)2 ) =

E, which gives L(cr)

2 = π 2(D1 +

B1

)/E. Near the criticalthickness the reflectance shows very strong dependence on L2

with sharp increase from zero to one within narrow interval of thicknesses L2.

The reflectance as a function of electron wave vector(energy) is shown in Fig. 3 for different values of L2.At L2 > L1 = 40 A, the reflectance is large within narrowinterval of wave vectors near k = 0 and becomes 1 at k = 0.At L2 < L1, there is a range of wave vectors, which satisfythe condition of total reflection; i.e., the energy E is in thegap of nanofilm of thickness L2. The total reflection occursat small wave vectors (small energies), i.e., at k < k(cr)(L2).The critical wave vector is determined by the condition that theelectron energy is exactly at the edge of the “conduction” band

075318-3




0 0.02 0.04 0.06 0.08 0.100

0.2

0.4

0.6

0.8

1.0 L = 10 Å

L = 40 Å

20 = 0°30

50 100

k(Å-1)

R e f l e c t a n c e


of the wave vector k of the incident electron for different thicknesses

L2 of TI nanofilm in region 2. The thickness of TI nanofilm in region

1 is L1 = 40 A and the angle of incidence is ϕ = 0. The numbers

next to the lines are the values of the thickness L2.

dispersion for the nanofilm of region 2. Thus, the critical wavevector satisfies an equation E = E+(k(cr),L2). The dependenceof the reflectance on the wave vector (energy) is strong near thecritical value, k ≈ k(cr). Within a good accuracy, one can saythat the reflectance is 1 within a finite range of wave vectors,k < k(cr)(L2), and zero at k > k(cr)(L2).

This property can be used to design an electron filter toselect electrons within a given range of energies. Namely, if the electron beam with broad energy distribution is incidenton a steplike singularity with the known values of L1 andL2 < L1, then only the electrons with k < k(cr)(L2) will betotally reflected.

In Fig. 4 we show the dependence of the reflectance on the

angle of incidence, ϕ , for different wave vectors (energies) of the incident electron and different thicknesses, L2, of the TInanofilm in region 2. The total internal reflection, which can berealized only at L2 < L1, occurs at ϕ > ϕ(cr). The critical angleis determined by condition E = E+(k sin ϕ(cr),L2); i.e., kx inregion 2 is zero at the critical angle and the y component of the wave vector is k sin ϕ(cr). Comparing small [Fig. 4(a)] andlarge electron energies [Fig. 4(b)], one can see that the criticalangle ϕ(cr) increases with electron energy. The critical anglealso increases with the thickness L2. If the thickness L2 of the nanofilm in region 2 satisfies an inequality E+(0,L2) E,then even at zero angle of incidence the electron experiencestotal reflection. In this case for all angles there is total internalreflection of the incident electron [see L

2 =10 and 20 A

in Fig. 4(a) and L2 = 10 A in Fig. 4(b)]. Near the criticalangle ϕ ≈ ϕ(cr) (at L2 < L1), the reflectance shows strongdependence on the angle ϕ with sharp increase from zeroto 1 within small interval of angles. Comparing the resultsfor L2 = 100 A in Figs. 4(a) and 4(b) one can concludethat the dependence of the reflectance on the angle becomesstronger with increasing electron energy. At L2 > L1, the totalinternal reflection cannot be realized and the reflectance showsmonotonic dependence on the angle ϕ .

The above data on the reflectance from the steplike defectshow that the total reflection can be realized only at L2 < L1.At such values of nanofilm thicknessL2, by varying parametersof the system, e.g., angle of incidence or electron energy, one

0

0.2

0.4

0.6

0.8

L =10Å,20Å

k = 0.02 Å

L = 40 Å

30

100 50

0.2 0.4 0.8 1.0 1.2 1.4

0.2

0.4

0.6

0.8

L =10Å

20 30

100 50

(a)

(rad )

(b)1.0

0

k = 0.1Å

L = 40 Å

0.6

1.0

R e f l e c t a n c e


of angle of incidence ϕ for different values of the nanofilm thickness

L2 in region 2. The thickness of the nanofilm in region 1 is L1 =40 A and the wave vector of the incident electron is (a) k = 0.02 A−1

and (b) k = 0.1 A−1. The numbers next to the lines are the values of

the thickness L2. In panel (a), the electron reflectance is exactly one

at L2 = 10 A and 20 A.

can change the reflectance from small value (almost zero) to1 within a narrow interval of parameters. Thus, the reflectanceshows strong dependence on the parameters of the system neartheir critical values.

B. Evanescent modes

The scattering of an electron by a steplike defect generateslocalized evanescent modes. Such modes are shown schemat-ically in Fig. 1(b). The evanescent modes are localized atthe steplike defect and exponentially decay in both directionsaway from the defect (in both negative and positive directionsof axis x); see Fig. 1(b). The evanescent modes do not directlycontribute to electron reflectance, but theyincrease the electrondensity near the steplike defect.

The amplitudes of the evanescent modes are stronglycorrelated with electron reflectance. The amplitudes of theevanescent modes are large only when the electron experiencestotal reflection; otherwise the amplitudes are relatively small.Typical spatial distributions of electron density in the evanes-cent modes are shown in Fig. 5 for different values of electronwave vector (electron energy) and different thicknesses L2 of nanofilm in region 2. Here ρ1 and ρ2 in Fig. 5 correspondto two components of the electron wave function in the TInanofilm; see Eq. (3). The two components of the wavefunction correspond to two directions of electron spin, i.e.,spin-up and spin-down. The figure illustrates two cases: weak reflection [Figs. 5(a) and 5(c)] and strong total reflection[Figs. 5(b) and 5(d)].

Forweak reflection, theamplitudes of theevanescent modesare small. The corresponding increase of the local electrondensity near the steplike defect is less than 1%. The wave

075318-4




x (Å)

L1 = 100 Å

L2 = 20 Å

k = 0.05 Å-1

(a)

(c) (d)

( x )(×10-3)

1

3

5

10 20-10-20

x (Å)10 20-10-20

x )(×10-3)

0.1

0.3

0.5

(b) x )(×10-3)

x )(×10-3)

10

10

10

50

20

20

20

30

100

150

-20 -10

-10-20

L1 = 100 Å

L2 = 20 Å

k = 0.1 Å-1

L1 = 100 Å

L2 = 10 Å

k = 0.1 Å-1

L1 = 100 Å

L2 = 10 Å

k = 0.05 Å-1

x (Å)

x (Å)

ρ ( ρ

( ρ

( ρ

ρ

ρ ρ

ρ ρ

ρ ρ

ρ

FIG. 5. (Color online) Spatial distribution of electron density in

the evanescent (localized) modes is shown for different parameters

of the nanofilm and different electron energies (wave vectors). Twodensities ρ1 and ρ2 correspond to two components of the wave

function. The parameters of the system (thicknesses L1 and L2 and

electron wave vector k) are shown in corresponding panels.

function of the evanescence modes is almost a continuousfunction of the position; see Figs. 5(a) and 5(c), where thedensity of the evanescent mode is continuous at x = 0.

For total reflection [see Figs. 5(b) and 5(d)], the typicalamplitude of the evanescent mode is large, which results instrong increase, by up to 60%, of local electron density nearthe defect. The amplitudes of the evanescent modes are largein region 1 only, whereas they are small in region 2. The

localization length of the evanescent modes is around 10 A.The wave function, which describes the localized evanescentmode, is discontinuous at x = 0. Note that the continuityof the electron wave function is satisfied for the total wavefunction, which includes both propagating and evanescentmodes. Figures 5(b) and 5(d) also show that with decreasingelectron energy, i.e., when the energy becomes farther awayfrom the edge of the “conduction” band, the amplitude of theevanescent mode increases.

Under the condition of total reflection, the density ρ1 (orρ2) of the evanescent modes is the largest in the region 1,whereas the density in region 2 is almost an order of magnitudesmaller. Below we describe the evanescent modes in terms of their densities at the steplike defect in region 1, i.e., at pointx = 0−.

In Fig. 6, two components of electron density of evanescentmodes at x = 0− are shown as the functions of nanofilmthickness, L2, in region 2. The electron density in evanescentmode is large for small thickness L2 and monotonicallydecreases with L2. In both Figs. 6(a) and 6(b) there aresingularities in the dependence of electron density on thicknessL2. These singularities occur at L2 ≈ 19 A in Fig. 6(a) andat L2 ≈ 11 A in Fig. 6(b) and correspond to critical points of transition from total reflection at small L2 to weak reflection atlarge L2;seeFig. 2. Thus, thedata shown in Fig. 6 illustrate thatlarge amplitudes of evanescent modes are realized only in thecase of total electron reflection. The farther the system is from

Ρ

Ρ

k 0.05 Å-1

L

k 0.1 Å

0 5 10 15 20 25 30

0.2

100 5 15 20 25 30

0.4

0.6

0.8

0.1

0.2

0.3

0.4

Ρ

Ρ

(a)

(b)

Ρ

Ρ

FIG. 6. (Color online) The electron density in the evanescent

mode near the steplike defect in region 1, i.e., at x = 0−, is shown

as a function of the nanofilm thickness L2 in region 2. Two densities

ρ1 and ρ2 correspond to two components of the wave function. The

nanofilm thickness in region 1 is L1 = 40 A and the wave vector

of the incident electron is (a) k = 0.05 A−1 and (b) k = 0.1 A−1.

Singularities in the graphs at L2 ≈ 18 A [panel (a)] and L2 = 12 A

[panel (b)] correspond to transition from total electron reflection at

small L2 to small reflection at large L2.

the critical point, i.e., the transition point from total reflection

to small reflection, the larger the amplitudes of evanescentmodes are.

In Fig. 7 the densities of evanescent modes, ρ1(0−) andρ2(0−), are shown as the functions of the angle of incidence,ϕ. The system is in the regime of total reflection. The densitiesof evanescent modes have nonmonotonic dependence on theangle ϕ with maximum values at some finite values of ϕ; e.g.,

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.05

0.10

0.15

(rad )

L

L

k

Ρ Ρ

Ρ Ρ

FIG. 7. (Color online) The electron density in the evanescent

mode near the steplike defect in region 1, i.e., at x = 0−, is shown

as a function of the angle of incidence. Two densities ρ1 and ρ2

correspond to two components of the wave function. The nanofilm

thickness in region 1 is L1 = 100 A and in region 2 is L2 = 10 A.

The wave vector of the incident electron is k

=0.05 A−1.

075318-5




ρ2 has maximum at ϕ ≈ 18◦. At ϕ = 90◦, i.e., an electronis propagating parallel to the defect, the reflectance is 1 (seeFig. 4) and no evanescent mode is generated at the defect.

The above results are shown for pseudospin τ z = 1. Inthis case the densities ρ1 and ρ2 correspond to up anddown directions of electron spin, respectively. For pseudospinτ z

= −1, the results for electron reflectance and transmittance

and the intensity of evanescent modes will be exactly the sameas for τ z = 1 with the only difference that for τ z = −1 thedensities ρ1 and ρ2 correspond to down and up directions of electron spin.

The decaying states at the surface of 3D TI near thelinear defect have been discussed in Refs. 34–36, where thedecaying modes are due to warping terms in the effective TIHamiltonian.

IV. CONCLUDING DISCUSSION

The scattering of an electron propagating on a surface of TI nanofilm is determined by the specific energy dispersion

law of the TI system. Namely, the electron dispersion in theTI nanofilm has a gap, the magnitude of which depends on thethickness of the TI nanofilm, g ∝ L−2. The gapped structureof electron energy dispersion introduces two types of electronstates on a surface of TI nanofilm. If electron energy is inthe energy gap, then there are only decaying (evanescent) andgrowing electron modes in the nanofilm, whereas if electronenergy is not in the energy gap, then there are both propagatingand evanescent (and growing) modes. Coexistence of bothpropagating and evanescent modes is a specific feature of the TI nanofilm. Due to such coexistence of different typesof modes, the scattering of an electron by a steplike defectproduces not only reflected and transmitted electron waves but

also modes which are localized at the defect.The amplitude of the reflected electron wave stronglydepends on the position of electron energy relative to the

energy gap in region 2 of the nanofilm, i.e., in the transmittedregion. If electron energy is in the energy gap of region 2, thenthe reflectance is exactly 1, which corresponds to total internalreflection. If the electron energy is not in the energy gap,then the reflectance is small and the electron is almost totallytransmitted to region 2. The critical points dividing these tworegimes are defined by the condition that the electron energy

is at the band edge of energy dispersion in region 2. Near thecritical points, the dependence of the electron reflectance onthe parameters of the system, such as thickness of the nanofilmor electron energy, is strong. The reflectance changes from 1(total reflection) to zero (small reflection) within a narrowinterval of parameters.

The amplitudes of the generated evanescent modes, whichare localized at the steplike defect, are large in the case of totalinternal reflection and small if the reflectance is small. Thus,if an electron with a given energy E is incident on a region of the TI nanofilm which does not support electron propagatingwaves at energy E, then there is a strong enhancement of local electron density near the boundary of such region. The

electron density near the boundary (defect) increases by up to60%. Such strong increase of the local electron density canbe observable in a many-electron system, when the scatteringof many electrons by the defect is studied. The interelectroninteractions combined with the local increase of the electrondensity near the defect can introduce additional scatteringpotential near the steplike defect. Such additional potential canmodify the scattering properties of the many-electron system.

In the analysis of electron scattering from a steplike defectwe assumed that the width w of the step is infinitely small. Fora finite width of the step, the above results are valid as long asthe width is much smaller than the electron wavelength; i.e.,kw 1. For example, for k = 0.05 A−1 the limitations on the

width of the step is w 20 A. For larger values of w 20 A,the scattering amplitude and formation of evanescent modesare expected to be suppressed.

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Electron scattering by step like defect in TI nanofilm

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