Geodesic scattering by surface deformations of a topological insulator.

Geodesic scattering by surface deformations of a topological insulator.

Dahlhaus, J.P.; Hou, C-Y.; Akhmerov, A.R.; Beenakker, C.W.J.

Citation

Dahlhaus, J. P., Hou, C. -Y., Akhmerov, A. R., & Beenakker, C. W. J. (2010). Geodesic scattering by surface deformations of a topological insulator. Physical Review B, 82(8), 085312. doi:10.1103/PhysRevB.82.085312

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Geodesic scattering by surface deformations of a topological insulator

J. P. Dahlhaus, C.-Y. Hou, A. R. Akhmerov, and C. W. J. Beenakker Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

共Received 8 July 2010; published 12 August 2010兲

We consider the classical ballistic dynamics of massless electrons on the conducting surface of a three- dimensional topological insulator, influenced by random variations in the surface height. By solving the geodesic equation and the Boltzmann equation in the limit of shallow deformations, we obtain the scattering cross section and the conductivity ␴, for arbitrary anisotropic dispersion relation. At large surface electron densities n this geodesic scattering mechanism共with␴⬀

冑

n兲 is more effective at limiting the surface conduc- tivity than electrostatic potential scattering.

DOI:10.1103/PhysRevB.82.085312 PACS number共s兲: 73.23.Ad, 73.25.⫹i, 73.50.Bk

I. INTRODUCTION

Topological insulators such as Bi₂Se₃form a new class of materials, characterized by an insulating bulk and a conduct- ing surface.^1,2The surface states are massless Dirac fermions with spin tied to momentum by spin-orbit coupling. Time- reversal symmetry prohibits backscattering and prevents dis- order from localizing the surface states. The surface conduc- tivity can therefore be unusually large, offering potential applications for electronics. The limitations on the conduc- tivity of Dirac fermions imposed by random potential fluc- tuations are well understood 共mostly from extensive studies of graphene³兲. Here we study an altogether different nonelec- trostatic scattering mechanism, originating from random sur- face deformations.

The epitaxial growth of Bi₂Se₃films is known to produce random variations in the height profile z =␨共x,y兲 of the surface.⁴ These surface deformations correspond to terraces of additional layers of the material 共of typical height H

= 2 nm and width W = 10 nm兲. Since the Dirac fermions are bound to the surface, they are forced to follow its geometry.

Like photons in curved space time, the electrons follow the geodesic or shortest path between two points, although here the curvature is purely spatial.⁵ 共The metric tensor of the surface does not couple space to time.兲 The geodesic motion around deformations constitutes a scattering mechanism that by its very nature is energy independent and which therefore is qualitatively different from potential scattering.

Our problem has no direct analog in the context of graphene. Ripples of a graphene sheet do scatter the elec- trons but this is not geodesic scattering: ripples in graphene are described by gauge fields and scalar potentials in a flat space.³ Space curvature effects may appear around conical defects 共pentagon and heptagon rings兲 but these are rare in graphene.⁶ An early study of geodesic scattering in con- densed matter that we have found in the literature is by Dug- aev and Petrov⁷ with possible applications to intercalated layered crystals. The present work goes beyond their analysis by including the effects of an anisotropic dispersion relation, which is a major complication but relevant for topological insulators.

The paper is organized as follows. In Sec.II we investi- gate the classical motion of the surface electrons in the pres- ence of surface deformations. The geodesic equation is

solved in the regime H/WⰆ1 of shallow deformations, to obtain the differential scattering cross sectionS. In Sec. III we use the linearized Boltzmann integral equation to com- pute the conductivity tensor ␴^fromS. This is a notoriously difficult problem for an anisotropic dispersion relation.⁸ In the regime H/WⰆ1 we are able to find a closed-form solu- tion, by converting the integral equation into a differential equation. Results are given in Sec. IV. In Sec. Vwe discuss the experimental signatures that distinguish geodesic scatter- ing from potential scattering.

II. GEODESIC SCATTERING A. Geodesic motion

We consider the surface of a topological insulator in the x-y plane, deformed by a locally varying height z =␨共x,y兲.

The dispersion relation of a locally flat surface is an elliptical hyperboloid

E =

冑

v_x²p_x²+v_y²p_y²+v_z²p_z²+⑀^2, 共2.1兲 where we have taken the x , y , z axes as the principal axes of the elliptical cone. In general, all three velocity components vx,vy,vz may be different. For an isotropic dispersion rela- tion in the x-y plane we have in-plane velocitiesvx=vy=vF

but the out-of-plane velocityvzmay still differ.

We have included a mass term ⑀^{in Eq.} 共2.1兲 in order to have a nonzero Lagrangian

L =

兺

_i ^{x˙ipi− E = −⑀}

冑

^{1 −}

^兺

i

共x˙i/vi兲² 共2.2兲

with x˙i= dxi/dt=⳵^E/⳵^piand i = x , y , z. In the final equation of motion ⑀will drop out.

The constraint that the motion follows the surface implies z˙ =共⳵␨/⳵x兲x˙+共⳵␨/⳵y兲y˙, which can be used to eliminate z˙

from the Lagrangian. The result can be written in the form L = −⑀

冑

1 −v_x⁻²g_␮␯x˙^␮x˙^␯ 共2.3兲 with g_␮␯ the metric tensor 共made dimensionless by pulling out a factor v_x²兲. Summation over repeated indices

␮,␯= 1 , 2 = x , y is implied and upper or lower indices distin- guish contravariant or covariant vectors.

Explicitly, we find

gxx= 1 +共⳵␨/⳵x兲²v_xz², 共2.4a兲

gyy=v_xy² +共⳵␨/⳵y兲²v_xz², 共2.4b兲

gxy= gyx=共⳵␨/⳵x兲共⳵␨/⳵y兲vxz2 , 共2.4c兲 where we have abbreviated v_ij=v_i/vj. The inverse of the tensor g_␮␯, denoted by g^␮␯, has elements

g^xx= D⁻¹关1 + 共⳵␨/⳵y兲²v_yz²兴, 共2.5a兲

g^yy= D⁻¹关vyx

2 +共⳵␨/⳵x兲²v_yz²兴, 共2.5b兲

g^xy= g^yx= − D⁻¹共⳵␨/⳵x兲共⳵␨/⳵y兲vyz

2, 共2.5c兲

D = 1 +共⳵␨/⳵x兲²v_xz² +共⳵␨/⳵y兲²v_yz² . 共2.5d兲 The Euler-Lagrange equation ⳵^L/⳵^x␮=共d/dt兲⳵^L/⳵^x˙␮ gives the inhomogeneous geodesic equation^9,10

x¨^␭+⌫_␮␯^␭ x˙^␮x˙^␯= x˙^␭1 L

dL

dt. 共2.6兲

The coefficients⌫_␮␯^␭ are the Christoffel symbols

⌫_␮␯^␭ ⬅g^␭␦

2

冉

⳵^{⳵x␯g␦␮+}

⳵

⳵^{x␮g␦␯−}

⳵

⳵^x␦g␮␯

冊

^{. 共2.7兲}

The nonzero right-hand side in Eq. 共2.6兲 may be elimi- nated by a reparametrization of time, from t to ␶ ^{such that} d␶/dt=−L共t兲/⑀. We thus arrive at the homogeneous geodesic equation

d²x^␭ d␶^{2 +⌫␮␯␭}

dx^␮ d␶

dx^␯

d␶ ^{= 0. 共2.8兲}

Since ⑀does not appear in this equation of motion, it holds also in the limit of massless electrons.

B. Scattering angle

We consider the scattering from a surface deformation

␨共x,y兲 of characteristic width W and height H large com- pared to the Fermi wavelength␭F. The scattering may then be described by the classical equation of motion, which is the geodesic equation共2.8兲.

An electron with wave vector k incident on the deforma- tion with impact parameter b at an angle␪kwith the x axis is scattered by an angle␪共␪k, b兲, resulting in a differential scat- tering cross section S共␪k,␪兲=兩db/d␪兩. Multiple trajectories may lead to the same scattering angle so that␪共␪k, b兲 cannot be inverted. Then the function has to be split into several invertible branches i and the cross section becomes S共␪k,␪兲=兺i兩dbi共␪k,␪兲/d␪兩.

These quantities may be calculated by numerically solv- ing the geodesic equation. Analytical progress is possible in the physically relevant regime H/WⰆ1 of shallow deforma- tions. As shown in second section of the Appendix, the scat- tering angle is then given by

␪共␪k,b兲 = −

冕

_−⬁^{⬁ ⌫˜xxy共x˜,b兲dx˜. 共2.9兲}

Here⌫˜_␮␯^␭ 共x˜,y˜兲 is obtained from ⌫_␮␯^␭ 共x,y兲 by a rotation of the coordinate axes over an angle ␪k共so that the electron is in- cident parallel to the x˜ axis兲. To leading order in H/W and b/W the scattering angle scales as␪⁼O共H²b/W³兲.

One simple example is the case of a Gaussian deforma- tion

␨共x,y兲 = H exp关− 共x²+ y²兲/2W²兴, 共2.10兲 which yields 共see third section of the Appendix兲

␪共␪k,b兲 = −

冑

␲ 2

H²v_yz

W³ be^−b2/W2⫻ 共cos²␪k+v_yx² sin²␪k兲 共2.11兲 in the shallow deformation limit. The geometry is depicted in Fig. 1. We will use this example throughout the paper to illustrate our general results.

III. CALCULATION OF THE CONDUCTIVITY A. Linearized Boltzmann equation

We investigate how geodesic scattering influences the sur- face conductivity ␴of the topological insulator. We assume

␴Ⰷe²/h so that we may use a semiclassical Boltzmann equation approach. In the presence of an external electric field E, the occupation fk= f₀共Ek兲+gk of the electron states deviates to first order in E according to the linearized Bolt- zmann equation

⳵^f0

⳵Ek

ev_k· E =

兺

k⬘

Q共k,k

⬘

兲共gk− g_k⬘兲. 共3.1兲 Here,vk=⳵Ek/ប⳵k is the velocity and Q共k,k

⬘

兲 the scattering rate from k to k

⬘

关equal to Q共k

⬘

, k兲 because of detailed bal- ance兴. The sum over k

⬘

runs over all states of the 共d-dimensional兲 momentum space. In the continuum limit, 兺k→V兰dk/共2␲兲^d, where V is the d-dimensional volume 共d=2 in our case兲. Spin degrees of freedom do not contribute to the sum since the helical surface states have definite spin FIG. 1. Geodesic trajectory of an electron deflected by a circu- larly symmetric deformation 共characteristic width W兲. The impact parameter b, incident angle␪k, and scattering angle␪ are indicated.

The gray scale background shows the height profile of the Gaussian deformation in Eq.共2.10兲.

DAHLHAUS et al. PHYSICAL REVIEW B 82, 085312共2010兲

085312-2

direction. Particle conservation leads to the normalization condition

兺

k

gk= 0. 共3.2兲

The electric field can be eliminated from Eq. 共3.1兲 by means of the vector mean-free path⌳k, defined by^8,11

gk= ⳵^f0

⳵Ek

eE ·⌳k, 共3.3兲

兺

k⬘

Q共k,k

⬘

^{兲共⌳k−⌳k}_⬘^{兲 = vk. 共3.4兲} For elastic scattering, Q共k,k

⬘

^兲=␦共Ek−Ek⬘兲q共k,k

⬘

^{兲. Using} dk = dk_⬜dS_F= dEkdS_F/兩បvk兩, with dSF a Fermi surface ele- ment, Eq. 共3.4兲 can be rewritten in terms of the density of states N共E_F兲 at the Fermi energy

N共EF兲 = 共2␲兲^−d

冖

^{dSF兩បvk兩−1. 共3.5兲}

The integral养dSFextends over the Fermi surface. The result is

VN共EF兲具q共k,k

⬘

兲共⌳k−⌳k⬘兲典k⬘=v_k 共3.6兲 with 具¯典k denoting the weighted average over the Fermi surface

具f共k兲典k=

冖

^{dSFf共k兲兩បvk兩−1}

冖

^{dSF兩បvk兩−1 . 共3.7兲}

The normalization condition in Eq.共3.2兲 becomes 具⌳k典k= 0.

At zero temperature, the conductivity tensor is given by

␴^=e2

V

兺

_k ^{␦共Ek−EF兲vk丢⌳k= e2N共EF兲具vk丢⌳k典k.}

共3.8兲 The direct product 丢 indicates the dyadic tensor with ele- ments关vk兴i关⌳k兴j. Substitution of Eq.共3.6兲 for vkand the use of q共k,k

⬘

^兲=q共k

⬘

, k兲 shows that␴ is a symmetric tensor.

For a low density N of scatterers, the scattering rate q共k,k

⬘

兲 can be related to the differential cross section S of a single scatterer 共averaged over all scatterers兲. In the two- dimensional case of interest here, the relation is

N兩vk兩S共␪k,␪k⬘兲d␪k⬘= q共k,k

⬘

兲 V 共2␲兲²

dS_F

⬘

兩បvk⬘兩, 共3.9兲 where ␪k is the angle betweenvk and the x axis. The Eq.

共3.6兲 which determines the vector mean-free path then takes the form

N兩vk兩

冕

0 2␲

d␪k⬘S共␪k,␪k⬘兲共⌳k−⌳k⬘兲 = vk. 共3.10兲

For the solution of this equation共and the interpretation of the results兲, it is convenient to follow Ziman^8,12and define an anisotropic relaxation time␶共k兲 by

1

␶共k兲= VN共EF兲具共1 − vˆk·vˆk⬘兲q共k,k

⬘

^兲典k_⬘^{. 共3.11兲} Using Eq.共3.9兲 this can be rewritten as

1

␶共k兲=N兩vk兩

冕

0 2␲

d␪k⬘S共␪k,␪k⬘兲关1 − cos共␪k⬘−␪k兲兴.

共3.12兲

B. Isotropic dispersion relation

For isotropic dispersion relations共when Ek depends only on 兩k兩 so that the velocity v=vFkˆ is aligned with the wave vector兲, the linearized Boltzmann equation can be solved exactly.⁸This applies, for example, to surfaces perpendicular to the 关111兴 direction of Bi2Se₃. We consider this simplest case first.

Since the deformations do not have a preferred orientation and the dispersion is isotropic, the average scattering cross section S共␪k,␪k⬘兲 only depends on the scattering angle

␪⁼␪k−␪k⬘, independently of the incident direction. The so- lution to Eq. 共3.6兲 is then ⌳k=␶^vk with a relaxation time ␶ given by

1

␶^=NvF

冕

0 2␲

d␪S共␪兲共1 − cos␪兲. 共3.13兲

Substitution into Eq. 共3.8兲 leads to a scalar conductivity ␴ given by the Drude formula

␴^{= e2}N共EF兲vF 2␶

d=e² h

EF

ប

␶

2. 共3.14兲

In the second equality we inserted the density of states N共EF兲=EF/共2␲ប²v_F²兲 of a Dirac cone with a circular cross section.

The regime H/WⰆ1 of shallow surface deformations is characterized by predominantly forward scattering 共兩␪兩Ⰶ1兲.

Then the relaxation time in Eq.共3.13兲 is given by the second moment of the scattering angle

1

␶⁼ 1

2NvF

冕

^{d␪S共␪兲␪2. 共3.15兲}

We substitute the relation S共␪兲=具兩d␪共b兲/db兩⁻¹典, where 具¯典 indicates an average over the共randomly oriented兲 scatterers.

The integration over scattering angles␪becomes an integra- tion over impact parameters b

1

␶⁼ 1

2NvF

冓 ^冕

^{db␪2共b兲}

冔

^{. 共3.16兲}

From Eq.共2.9兲 we infer the scaling 1/␶⬀W⫻共H/W兲⁴of the relaxation rate with the characteristic height and width of the surface deformations.共The additional factor of W comes from the integral over b.兲 This scaling was first obtained by

Dugaev and Petrov.⁷Eq.共3.14兲 then gives the scaling of the conductivity

␴^{= constant}⫻e² h

EF

ប 1 NvF

W³

H⁴. 共3.17兲

C. Anisotropic dispersion relation

We now turn to the case of an anisotropic dispersion re- lation. There is then, in general, no closed-form solution of the linearized Boltzmann equation.¹³ One widely used ap- proximation for the conductivity, due to Ziman,¹² has the form

␴Ziman= e²N共EF兲具vk丢vk␶共k兲典k 共3.18兲 with␶共k兲 the anisotropic relaxation time in Eq. 共3.11兲. As we will show in the following, this is a poor approximation for our problem but fortunately it is not needed: In the relevant limit H/WⰆ1 of scattering from shallow surface deforma- tions an exact solution becomes possible. For shallow defor- mations forward scattering dominates,兩␪兩=兩␪k−␪k⬘兩Ⰶ1. This allows for an expansion of⌳k⬘around␪k, which reduces the integral equation共3.6兲 to a differential equation.

With the notation

Mp共␾兲 =

冕

0 2␲

d␪S共␾,␾+␪兲␪^p 共3.19兲

the expansion to second order of Eq.共3.10兲 can be written as

M₁共␾兲 d

d␾^␭共␾兲 +1

2M₂共␾兲 d²

d␾^2␭共␾兲 = − 1

Ne^i␾. 共3.20兲 We introduced a complex variable ␭=⌳x+ i⌳y to combine the two components of the vector mean-free path. Denoting the radius of curvature of the Fermi surface by

␬共␾兲=dSF/d␾, the normalization condition in Eq.共3.2兲 be- comes

冕

0 2␲

d␾␬共␾兲

v共␾兲␭共␾兲 = 0. 共3.21兲 Once we have the solution of Eq. 共3.20兲, the conductivity tensor elements follow from

␴xx⫾␴yy=e² hRe

冕

0

2␲d␾

2␲^e⫿i␾␬共␾兲␭共␾兲, 共3.22a兲

␴xy=␴yx=e² h 1 2Im

冕

0

2␲d␾ 2␲^e

i␾␬共␾兲␭共␾兲. 共3.22b兲

A further simplification is possible if the average scatter- ing angle vanishes, M₁共␾兲=0. Then the second moment M₂共␾兲 of the scattering angle is, within the forward scatter- ing approximation, directly related to the anisotropic relax- ation time

1

␶共␾兲=1

2Nv共␾兲M2共␾兲. 共3.23兲 Equation 共3.20兲 can now be solved in terms of the Fourier transforms

ᐉn=

冕

0 2␲d␾

2␲^{e−in␾v共}␾兲␶共␾兲, 共3.24a兲

␬n=

冕

0 2␲d␾

2␲^e−in␾␬共␾兲, 共3.24b兲

␭n=

冕

0 2␲d␾

2␲^{e−in␾␭共}␾兲 共3.24c兲 resulting in

␭n= ᐉn−1

n² + constant⫻␦n,0. 共3.25兲 The normalization constant can be determined from Eq.

共3.21兲.

Inserting the solution into Eq. 共3.22兲 we obtain the con- ductivity

␴xx⫾␴yy=e²

hRe_n=−⬁

兺

^{⬁ ᐉn−1}_n^{␬2−n⫾1, 共3.26a兲}

␴xy=␴yx=e² h

1

2Im_n=−⬁

兺

^{⬁ ᐉn−1}_n^{␬2−n−1. 共3.26b兲}

For simplicity we have assumed an inversion symmetric Fermi surface, for which ␬_⫾1= 0 so that the normalization constant in Eq.共3.25兲 does not contribute to the conductivity.

In the case of an isotropic Fermi surface, only the Fourier components l₀=vF␶ ^and ␬0= kF are nonzero. From Eq.

共3.26兲, we then find␴xy= 0 =␴yx,␴xx=␴yy=共e²/2h兲kFvF␶^{, in} agreement with Eq. 共3.14兲.

Comparing with the Ziman approximation in Eq. 共3.18兲 for the conductivity in terms of the anisotropic relaxation time, we see that it can be written in the same form, Eq.

共3.26兲, but without the factor 1/n². It therefore deviates strongly from our forward-scattering limit, except in the case of an isotropic Fermi surface共when only n=1 contributes兲.

IV. RESULTS

A. Isotropic dispersion relation

In the shallow deformation limit the conductivity is given by Eq.共3.17兲, up to a numerical prefactor of order unity. We have calculated this prefactor for Gaussian deformations of the form 共2.10兲, randomly distributed over the surface. We assume that the deformations are shallow, H/WⰆ1. For sim- plicity, we also take the same parameters H and W for each deformation. From Eqs.共2.11兲, 共3.14兲, and 共3.16兲 we obtain the result

DAHLHAUS et al. PHYSICAL REVIEW B 82, 085312共2010兲

085312-4

␴⁼¹⁶

冑

2

␲

冑

_␲បv^EF^FN W³ 共HvF/vz兲⁴

e²

h. 共4.1兲

The factorvF/vzis there to allow for an out-of-plane velocity vzthat is different from the in-plane velocityvx=vy=vF. The result in Eq.共4.1兲 confirms the scaling behavior in Eq. 共3.17兲 and gives the numerical prefactor.

To relax the assumption H/WⰆ1 of shallow deforma- tions, we solved the geodesic equation共2.8兲 numerically for the Gaussian case. The corresponding Christoffel symbols were taken from Eq.共A2兲 with vx=vy=vF. Using the scatter- ing angle␪共b兲 that we obtained from the numerics, we cal- culated the conductivity following from Eqs. 共3.13兲 and 共3.14兲.

As shown in Fig.2, the numerical results deviate from the scaling in Eq. 共4.1兲 only for relatively large ratios H/Wⲏ0.5. The deviations are oscillatory, due to electron trajectories that circle around the deformation as depicted in the inset共b兲 of Fig.2. Inset共a兲 shows generic trajectories for electrons scattering off a shallow Gaussian deformation. No- tice the focusing of trajectories as an analog of gravitational lensing.

B. Anisotropic dispersion relation

As an example of an anisotropic dispersion relation, we consider elliptic equienergy contours Ek=ប共vx

2k_x²+v_y²k_y²兲^1/2 with principal axes x and y. As in the previous section, we investigate shallow Gaussian surface deformations. These have zero average scattering angle, M₁共␾兲=0, and second moment

M₂共␾兲 =1

C共sin²␾^+vyx2 cos²␾兲². 共4.2兲

The coefficient C is given by

C =16

冑

2

␲

冑

␲ W³

H⁴v_y⁴/vz4. 共4.3兲 From Eq.共A12兲 we deduce that Eq. 共4.2兲 actually holds more generally for any circularly symmetric deformation, the only difference being in the expression for C.

Using Eqs.共3.23兲 and 共3.24a兲 one obtains the Fourier co- efficients

ᐉ_⫾n= C

N

冉

^{1 −1 +vvyx}yx

冊

^{兩n兩/2共1 + 兩n兩vv}yx^yx+vyx2兲

3 共4.4兲

for n even, and zero for n odd. The elliptic dispersion rela- tion leads to

␬共␾兲 = EF

បvx

vyx

共sin²␾+v_yx² cos²␾兲^3/2. 共4.5兲 The Fourier coefficients␬nare also nonzero only for n even.

共Since their expressions are rather lengthy, we do not list them here.兲

From Eq.共3.26兲 we find that the off-diagonal components of the conductivity tensor vanish while the diagonal compo- nents are given by

␴

再

^yyxx

冎

^=eh2n

^兺

ⱖ1

1

2n²共ᐉn+1⫾ ᐉn−1兲共␬n+1⫾␬n−1兲. 共4.6兲 The series converges rapidly.

The ratio ␴xx/␴yy depends only on the anisotropy vyx=vy/vx. It is plotted in Fig. 3. For comparison, we also show the Ziman approximation ␴Ziman 关obtained from the forward-scattering limit in Eq.共4.6兲 without the 1/n²factor兴.

As expected, it deviates substantially upon increasing the anisotropy 共notice the logarithmic scale兲.

V. COMPARISON WITH POTENTIAL SCATTERING A. Carrier density dependence

The energy independence of the mean-free pathᐉ=vF␶^is the hallmark of geodesic scattering. It implies the square root dependence ␴⬀

冑

n of the conductivity on the surface electron density n. This follows from Eq. 共3.17兲 with EF=បvF

冑

4␲n for an isotropic Dirac cone, or more generally from the scaling ␴⬀SF for a noncircular Fermi surface 共of area SF⬀

冑

_n兲.

FIG. 2. 共Color online兲 Surface conductivity of a topological in- sulator as a function of the height H of randomly positioned Gaussian deformations 共width W=10 nm, density N=0.1 W⁻²兲.

We took an isotropic dispersion relation, with in-plane velocities v_x=v_y=v_F= 5⫻10⁵ m/s, and a smaller out-of-plane velocity v_z=v_F/3. The Fermi energy is fixed at EF= 150 meV. As discussed in Sec.V, these are realistic parameter values for the关111兴 surface of Bi₂Se₃. Dots represent numerical results whereas the line shows the shallow deformation limit in Eq.共4.1兲.

FIG. 3. The solid line shows the ratio of conductivities␴xx/␴yy

as a function of anisotropy v_y/vx, calculated from Eq. 共4.6兲. The dashed line corresponds to the Ziman approximation.

As discussed in the context of graphene,^3,14 electrostatic potential scattering typically gives a faster increase in the conductivity with increasing carrier density. Coulomb scat- tering from charged impurities and resonant scattering from short-range impurities both give a linear increase␴⬀n 共up to logarithmic factors兲. Scattering from a potential landscape with a Gaussian correlator gives an even more rapid increase

␴⬀n^3/2. Geodesic scattering, with ␴⬀n^1/2, would therefore form the dominant conduction-limiting scattering mechanism at high carrier densities.

For a quantitative comparison of geodesic and potential scattering, we consider the 关111兴 surface of Bi2Se₃ with Gaussian deformations given by Eq.共4.1兲. We take isotropic in-plane velocities vx=vy=vF= 5⫻10⁵ m/s and a smaller out-of-plane velocity v_z=v_F/3.^15,16 We adopt the following numerical parameters for the deformations from an experi- mental image:⁴ characteristic width W = 10 nm and height H = 2 nm, covering 40% of the surface area so N=10¹¹ cm⁻². The carrier density dependence of the con- ductivity for geodesic scattering, following from Eq.共4.1兲, is plotted in Fig.4 共solid curve兲.

To compare the geodesic scattering to typical potential scatterers, we also show the corresponding results for scat- tering from charged impurities共dashed兲 and Gaussian poten- tial fluctuations 共dotted兲 in Fig. 4. For charged impurities 共charge Q=e兲 we considered the unscreened Coulomb poten- tial U共r兲=共Qe/4␲⑀0⑀r兲兩r兩⁻¹, as the extreme case of a long- ranged potential. We took ⑀r= 80 as a typical value for the dielectric constant and kept the other parameter values as before. The semiclassical conductivity is then given by^3,17

␴^=e2 h

n Nc

2␲ប²v_F²

u₀² , u₀= Qe 4⑀0⑀r

. 共5.1兲

For Fig. 4 we used Nc= 2.5⫻10¹¹ cm⁻² as the density of impurities.

For a potential landscape with Gaussian correlator共range

␰, dimensionless strength U₀兲

具U共r兲U共r

⬘

兲典 =U₀共បvF兲²

2␲␰^{2 exp}

冉

^{−兩r − r}2␰²

^⬘

^兩2

冊

^共5.2兲

the conductivity takes the functional form¹⁸

␴^=e2 h

4␲ⁿ␰^2e4␲n␰2

U₀I₁共4␲ⁿ␰²兲. 共5.3兲 共The function I1 is a Bessel function.兲 For Fig. 4 we took U₀= 0.1 and ␰= W = 10 nm. The parameter values used in Fig.4are only for the purpose of illustration but the point to make is that geodesic scattering dominates over potential scattering for large carrier densities.

B. Anisotropy dependence of conductivity

In the case of an anisotropic共elliptical兲 dispersion relation the conductivity will be direction dependent. This situation arises for example if the surface of Bi₂Se₃is not in the关111兴 direction. Geodesic scattering implies a certain universality for the directionality dependence of the conductivity, if we may assume that the surface deformations are shallow 共H/WⰆ1兲 and without a preferential orientation 共circularly symmetric on average兲. The ratio␴xx/␴yyis then only a func- tion ofvy/vx, independent of other parameters共such as elec- tron density or density and height of the deformations兲. This universal function is plotted in Fig.3共solid curve兲.

In Fig. 5 we compare this result for geodesic scattering with corresponding results for potential scattering. Three typical impurity potentials are considered, of different range:

long-ranged unscreened Coulomb potentials, medium-ranged Gaussian potential fluctuations, and short-ranged potentials.

The conductivities are obtained following the general ap- proach of Ref. 19, by first computing the transition rates in Born approximation and then solving numerically the linear- ized Boltzmann equation. We took the same material param- eter values as in the previous section.

The unscreened Coulomb potential gives a ratio ␴xx/␴yy

which depends only on vy/vx 共dashed line兲. For Gaussian potential fluctuations, the ratio␴xx/␴yyis a function of both v_y/vx and n. It is plotted as a dotted line in Fig. 5 for n␰^{2= 1.}共If␰= W = 10 nm this corresponds to the carrier den- sity n = 10¹² cm⁻².兲 In the same figure we also plot 共dot- dashed line兲 the limit ␰→0 共at fixed n兲 of a short-ranged potential.

FIG. 4. Conductivity as a function of carrier density. The influ- ence of three different sources of scattering is shown: surface de- formations 共solid line兲, unscreened Coulomb impurities 共dashed line兲, and Gaussian correlated potential fluctuations 共dotted line兲.

The parameters used for the plot are given in the text.

FIG. 5. Ratios of conductivities along the two main axes of the dispersion relation are shown as a function of anisotropyv_y/vx. The influence of four different sources of scattering is shown: surface deformations 共solid line兲, unscreened Coulomb impurities 共dashed line兲, Gaussian potential fluctuations 共dotted line兲, and short-ranged potentials 共dot-dashed line兲. The parameters used for the plot are given in the text.

DAHLHAUS et al. PHYSICAL REVIEW B 82, 085312共2010兲

085312-6

From the double-logarithmic plot in Fig. 5 one can see that there is an approximate power-law dependence,

␴xx/␴yy⬀共vy/vx兲^−p, over at least one decade. The exponent is p⬇3.3 for geodesic scattering, while p=2 for short-range potential scattering. Scattering from long-ranged Coulomb impurities or from medium-ranged Gaussian potential fluc- tuations gives p⬍2.

Anisotropic charge transport in the presence of un- screened Coulomb impurities for an elliptic dispersion rela- tion was also discussed in the context of strained graphene.²⁰ There it was argued that␴xx/␴yy⬀共vy/vx兲⁻²on the basis of a power-counting argument. Our numerical solution of the Boltzmann equation gives a smaller exponent p⬇1.3 in that case.

To conclude, charge transport dominated by surface de- formations has a much stronger anisotropy dependence than that governed by impurity potentials. This highly anisotropic transport behavior is a distinct characteristic of geodesic scattering.

ACKNOWLEDGMENTS

We thank F. Hassler for valuable discussions. This re- search was supported by the Dutch Science Foundation NWO/FOM and by an ERC Advanced Investigator Grant.

APPENDIX: CALCULATION OF THE SCATTERING CROSS SECTION

1. Christoffel symbols in rotated basis

In order to calculate the scattering angle in the geometry of Fig.1, it is convenient to rotate the coordinate axis in the x-y plane such that the electron is incident parallel to the x axis. Under the linear transformation from x , y to

˜ = x cosx ␪k+ y sin␪k, y˜ = −x sin␪k+ y cos␪k, the Christoffel symbol⌫_␮␯^␭ transforms to

⌫˜_␮␯^␭ 共x˜,y˜兲 = ⳵^˜x␭

⳵^{x␭⬘⌫␮⬘␯⬘}

␭⬘ 共x,y兲⳵^x␮⬘

⳵^˜x␮

⳵^x␯⬘

⳵^{˜x␯ . 共A1兲} Using the expressions共2.7兲 for metric tensor and Christoffel symbols, we arrive at

⌫˜_␮␯^x = D⁻¹ ⳵²␨

⳵^˜x␮⳵^˜x␯

⫻

冋

^{vxz2 ⳵␨⳵˜x−共vxz2 −vyz2兲sin␪k}

^冉

^{⳵␨⳵˜xsin␪k+⳵␨⳵˜ycos␪k}

^冊 册

^,

共A2a兲

⌫˜_␮␯^y = D⁻¹ ⳵²␨

⳵^˜x␮⳵^˜x␯

⫻

冋

^{vyz2 ⳵␨⳵˜y−共vxz2 −vyz2兲sin␪k}

^冉

^{⳵␨⳵˜xcos␪k− ⳵␨⳵˜ysin␪k}

^冊 册

^.

共A2b兲 The factor D from Eq.共2.5d兲, written in terms of the rotated coordinates, reads

D = 1 +v_xz²

冉

^⳵␨⳵^˜xcos␪k−⳵␨

⳵^˜ysin␪k

冊

²

+v_yz²

冉

^⳵␨⳵^˜xsin␪k+⳵␨

⳵^˜ycos␪k

冊

^{2. 共A3兲}

The Christoffel symbols in Eq. 共A2兲 appear in the geodesic equation for the rotated coordinates

d²˜x^␭ d␶^{2 +⌫˜␮␯␭}

dx˜^␮ d␶

dx˜^␯

d␶ ^{= 0. 共A4兲}

2. Geodesic equation for shallow deformation The geodesic equation 共A4兲 can be considerably simpli- fied in the shallow deformation limit H/WⰆ1. Let us con- sider a particle incident on a deformation along the x˜ direc- tion from −⬁ with impact parameter b and velocity

v = v_xv_y共vy2cos²␪k+v_x²sin²␪k兲^−1/2. 共A5兲 Since the derivative dy˜/d␶is smaller than dx˜/d␶by a factor 共H/W兲², we can drop this derivative from the geodesic equa- tion. The result is

d²˜x d␶^2+⌫˜xx

x

冉

^dxd^˜␶

冊

^{2= 0, 共A6a兲}

d²˜y d␶^2+⌫˜xx

y

冉

^dxd^˜␶

冊

^{2= 0. 共A6b兲}

Furthermore, since dx˜/d␶=v关1+O共H/W兲²兴, we can write d/d␶^=vd/dx˜. This leads to

d²˜y dx˜²= −⌫˜xx

y. 共A7兲

The scattering angle ␪Ⰶ1 is obtained from

␪^{= lim}˜x→⬁dy˜/dx˜ hence

␪共␪k,b兲 = −

冕

_−⬁^{⬁兩⌫˜xxydx˜兩˜y→b. 共A8兲}

Inserting Eq. 共A2b兲 into Eq. 共A8兲 and noting that D = 1 +O共H/W兲², we obtain the scattering angle to leading order in H/W

␪共␪k,b兲 = −

冕

_−⬁^{⬁ dx˜}

冋 ^冉

^{␣⳵⳵␨˜y−␥⳵⳵␨˜x}

^冊

^{⳵⳵˜2x␨2}

册

^˜y→b

. 共A9兲

We abbreviated

␣^=vyz2 cos²␪k+v_xz² sin²␪k, 共A10a兲

␥⁼共vxz

2 −v_yz²兲sin␪kcos␪k. 共A10b兲

3. Circularly symmetric deformation

For a circularly symmetric height profile ␨共x,y兲, depen- dent only on r =

冑

x²+ y²=

冑

˜x²+ y˜², the term proportional to␥

in Eq. 共A9兲 vanishes 共because it is an integral over an odd function of x˜兲. The expression for the scattering angle thus simplifies further to

␪共␪k,b兲 = −␣

冕

_−⬁^{⬁ dx}

冋

^{⳵␨⳵y⳵⳵2x␨2}

册

^y→b

. 共A11兲

For the Gaussian deformation in Eq. 共2.10兲 we obtain the scattering angle in Eq.共2.11兲 given in the main text.

The entire dependence of the scattering angle ␪ ^{on the} angle of incidence ␪k is contained in the prefactor ␣. This implies that the moments Mp=兰db␪^pof the scattering angle depend on the angle of incidence as

Mp共␪k兲 = cp␣^{p= c}pv_xz^p共sin²␪k+v_yx² cos²␪k兲^p 共A12兲 with cp a coefficient independent of␪k.

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