The persistent charge and spin currents in topological insulator Bi2Se3nanowiresWen-Kai Lou, Fang Cheng, and Jun Li Citation: J. Appl. Phys. 110, 093714 (2011); doi: 10.1063/1.3658853 View online: http://dx.doi.org/10.1063/1.3658853 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v110/i9 Published by the American Institute of Physics. Related ArticlesTuning the conductivity of vanadium dioxide films on silicon by swift heavy ion irradiation AIP Advances 1, 032168 (2011) Determination of Rashba and Dresselhaus spin-orbit fields J. Appl. Phys. 110, 064306 (2011) Magnetic-flux-induced persistent currents in nonlinear mesoscopic rings J. Appl. Phys. 109, 07E139 (2011) Persistent currents in ballistic normal-metal rings Low Temp. Phys. 36, 982 (2010) Persistent currents, flux quantization, and magnetomotive forces in normal metals and superconductors (ReviewArticle) Low Temp. Phys. 36, 841 (2010) Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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The persistent charge and spin currents in topological insulator Bi2Se3

nanowires

Wen-Kai Lou,1,a) Fang Cheng,1 and Jun Li1,2

1SKLSM, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, China2Department of Physics, Semiconductor Photonics Research Center, Xiamen University, Xiamen 361005,China

(Received 26 May 2011; accepted 28 September 2011; published online 8 November 2011)

We investigate theoretically the surface states of three-dimensional topological insulator cylinder

nanowires analytically and numerically. In contrast to the conventional semiconductor cylinder

nanowires, these surface states exhibit unique massless Dirac dispersion and interesting

transport properties. We find that the persistent charge current and persistent spin current,

i.e., the Aharonov-Bohm oscillation, can be induced by the driven magnetic flux. The amplitude

of persistent charge current shows an oscillating behavior with increasing the electron density.VC 2011 American Institute of Physics. [doi:10.1063/1.3658853]

I. INTRODUCTION

Spin-orbit interaction (SOI) is a relativistic effect which

could lead to many exotic physical phenomena.1 In conven-

tional semiconductors with spacial inversion asymmetry, SOI

is relatively weak and causes the spin splitting and spin relax-

ation.2 In some special narrow bandgap materials with heavy

atoms, such as HgTe and Bi2Se3, SOI is so strong that it can

cause a band inversion between the conduction band and val-

ance band and therefore leads to the topological phase with an

energy gap in the bulk and metallic surface or edge states at

its boundary.3–5 Recently, these materials with topological

phases, named as topological insulators (TIs), have attracted

intensive attentions both in theories and experiments.6–8 The

predicted two-dimensional (2D) topological phase in HgTequantum wells was shown to exhibit quantum spin Hall

effect,3,9 and the surface states in three-dimensional (3D) TIs,

e.g., Bi2Se3, Bi2Te3, and Sb2Te3, were confirmed by angle-

resolved photoemission spectroscopy.10,11 These surface and

edge states are protected by the time reversal symmetry and

robust against disorder effects, e.g., crystal defects and

nonmagnetic impurities, leading to dissipationless electron

transport in the absence of high magnetic fields.

BixSb1�x, an alloy with a complex structure of surface

states, was first verified to be a 3D TI.12,13 Soon after, Bi2X3

(X¼ Se, Te) were predicted to be 3D TIs according to the

first-principle calculations14 and confirmed in later experi-

ments.10,11 The TIs were predicted to possess helical surface

states with a single Dirac cone in the bulk bandgap (around

0.3 eV). These surface states can lead to novel magnetic

properties.15 Many exotic physical phenomena could emerge

in low dimensional Bi2X3 nanostructures as a consequence of

the quantum confinement. For example, ultrathin Bi2X3 film

as well as Bi2X3 nanowires are expected to exhibit topologi-

cally nontrivial surface states, which serve as a new platform

for studying the novel transport property. In addition, the

bulk states can be tuned by the chemical potential more

effectively compared to the bulk samples, because the contri-

bution from the bulk states is suppressed in these thin films

and nanostructures.

Searching for new TIs and probing novel transport

properties are the central issues in this rapid growing field.

However, the properties of nanostructures made of TIs are

relatively unexplored. It is interesting to ask what happens

when the size of TIs decreases from bulk to the nanometer

scale. Bi2X3 nanowires,16–19 which hold large surface-to-

volume ratios, therefore can manifest much more surface

states effects than bulk materials. In this work, we investi-

gate theoretically the electron surface states, the persistent

charge current, and persistent spin current, i.e., the

Aharonov-Bohm effect in a Bi2Se3 cylindrical nanowire.

This paper is organized as follows. The theoretical

model is presented in Sec. II, the numerical results and dis-

cussions are given in Sec. III. In Sec. IV, we give a brief

conclusion.

II. THEORY

A. Model and Hamiltonian

We consider a Bi2Se3 cylindrical nanowire shown sche-

matically in Fig. 1. Note that Bi2Te3 nanowire exhibits simi-

lar properties. The low-energy spectrum of 3D TIs can be

well described by the k � p model14 with the band edge basis

jP1þ�; "i, jP2�þ; "i, jP1þ�; #i, and jP2�þ; #i. Here, P denotes

the p-like orbital states of the atoms and 1(2) stands for Bi(X¼ Se, Te) atoms. The superscript 6 depicts the parity of

the wave function. The subscript 6 stands for the Pz orbital

state couples with pxþ ipy or px� ipy, and :(;) is the spin-up

(spin-down) state, respectively. The Hamiltonian reads

H0 ¼ e kð Þ

þ

M kð Þ B kzð Þkz 0 A kk� �

k�B kzð Þkz �M kð Þ AðkkÞk� 0

0 A kk� �

kþ M kð Þ �B kzð Þkz

A kk� �

kþ 0 �B kzð Þkz �M kð Þ

0BB@

1CCA;

(1)a)Author to whom correspondence should be addressed. Electronic mail:

wklou@semi.ac.cn.

0021-8979/2011/110(9)/093714/8/$30.00 VC 2011 American Institute of Physics110, 093714-1

JOURNAL OF APPLIED PHYSICS 110, 093714 (2011)

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where e kð Þ ¼ C0 þ C1k2z þ C2k2

k , M kð Þ ¼ M0 þM1k2z

þM2k2k, A kk

� �¼A0þA2k2

k, B kzð Þ¼B0þB2k2z , and k2

k ¼ k2x

þk2y . This Hamiltonian possesses the following symmetries:

(i) a threefold spatial rotation symmetry (R3) around z axis,

(ii) a twofold spatial rotation symmetry (R2) around x axis,

(iii) the spatial reversal symmetry I, and (iv) time reversal

symmetry T. Choosing e(k)¼ 0 will lead to an additional

particle-hole symmetry. The off-diagonal term A(k||) k6 in

the Hamiltonian (1) couples opposite spins of different

atoms. Utilizing the Dirac matrices Ci i¼ 1 � � �5ð Þ, the 4� 4

Hamiltonian can be rewritten as

H0 ¼ e kð ÞIþM kð ÞC5 þ B kzð ÞkzC3 þ A kk� �

kxC1 þ kyC2

� �;

(2)

where I is the 4� 4 identity matrix. The five Dirac matrices

are C1 ¼ rx � sx, C2 ¼ ry � sx, C3 ¼ rz � sx, C4 ¼ r0 � sy,

and C5 ¼ r0 � sz. r0, s0 are the 2� 2 identity matrices and

rx,y,z (sx,y,z) are the Pauli matrices acting on the spin (parity)

space. The Dirac matrices satisfy {Ci, Cj}¼ 2dij. The sub-

band dispersions and the corresponding eigenstates are

obtained from the Schrodinger equation H0(k)|w(k)i¼E0(k)|w(k)i. The eigenstates and eigenenergies can be

obtained numerically by expanding the wave function in

terms of the first kind of the cylindrical Bessel basis20

wj;kz¼X

n

bL;kz;n;"AL;nJL kLnq

� �eiLu

cL;kz;n;"AL;nJL kLnq

� �eiLu

bL;kz;n;#ALþ1;nJLþ1 kLþ1n q

� �ei Lþ1ð Þu

cL;kz;n;#ALþ1;nJLþ1 kLþ1n q

� �ei Lþ1ð Þu

0BB@

1CCAeikz�z; (3)

where L¼ 0, 61, 62. is the quantum number of the azi-

muthal orbital angular momentum and j¼ L 6 1/2 is the

quantum number of the z-component of the total angular mo-

mentum. kLn is the n-th zero point of the first kind of the cyl-

inder Bessel functions JL(x), kz is the wavevector along the

wire direction, i.e., the z axis, AL;n ¼ 1=ffiffiffipp

RJLþ1 kLn

� �is the

normalization constant, and R is the radius of the cylindrical

nanowire. The total angular momentum operator is defined

by J ¼ Lþ S, where S is the angular momentum that charac-

terizes the zone-center Bloch functions |S, Szi and L is the

angular momentum of the envelope part of the wave func-

tion.21 Since the Hamiltonian is rotational invariant about

the z axis, the z-component of the total angular momentum is

a good quantum number, i.e., Jz; H0

� �¼ 0. Therefore, the

summation in Eq. (3) runs only over n.

If a magnetic field is applied along the wire direction,

the Hamiltonian of the system becomes H kð Þ ¼ H0 kð Þ þ Hz.

The Zeeman term Hz takes the form

Hz ¼lB

2

g1zBz 0 0 0

0 g2zBz 0 0

0 0 �g1zBz 0

0 0 0 �g2zBz

0BB@

1CCA; (4)

where lB is the Bohr magneton and g1z,2z are the longitudinal

effective Lande g-factors. The Schrodinger equation

becomes H kð Þjw kð Þi ¼ E kð Þjw kð Þi. In the presence of mag-

netic field, one should replace �hk by the canonical momen-

tum �hK ¼ �hkþ eA, where we adopt the symmetry gauge

for magnetic vector A ¼ B2�y; x; 0ð Þ. Utilizing the property

of the cylindrical Bessel function K6JL kLnq

� �eiLu

¼ 6i kLnJL61 kL

nq� �

þ ðeBq=2�hÞJL kLnq

� �� �ei L61ð Þu with K6

¼Kx 6 iKy, we can calculate the matrix elements of the

Hamiltonian analytically. In this paper, we take the unit of

magnetic field as U0/(pR2), where U0¼ h/(2e) is the mag-

netic flux quantum.

B. Surface states of the nanowire

The massless Dirac Hamiltonian on a curved surface

can be expressed by22

H ¼ ð~n1 �~r cos hþ~n2 �~r sin hÞð~n1 �~pÞþ ð~n2 �~r cos h�~n1 �~r sin hÞð~n2 �~pÞ;

(5)

where h is the angle between spin and momentum and ~n1;2

are the orthogonal directions of the plane. For the surface

states of the Bi2Se3 system, electron spins are locked perpen-

dicularly to their orbital momentums. Adopting the cylindri-

cal coordinate (er, eu, ez), the surface state Hamiltonian of

Bi2Se3 nanowire can be expressed simply by

Hsurf ¼ vF~nr � ~r�~pð Þr¼ vF rupz � purz

� �, where ~nr is the

unit vector perpendicular to the surface of the nanowire

and vF is the Fermi velocity. The momentum operators

are pz ¼ �i�h@=@z and pu ¼ �h=Rð Þ �i@=@uþHð Þ, where

the dimensionless flux parameter H comes from the

external magnetic field. It is interesting to notice that the

Fermi velocities in the axial and tangent direction are

anisotropy; in the axial direction, the Fermi velocity

is vFu ¼ A0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� C2=M2ð Þ2

q� nm=�h � 0:6� 106m=s, and in

FIG. 1. (Color online) Sketch of a cylindrical Bi2X3 nanowire. The blue

sphere is the electron and the blue arrow indicates spin orientation of the

electron. R is the radius of the nanowire.

093714-2 Lou, Cheng, and Li J. Appl. Phys. 110, 093714 (2011)

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the tangent direction, the Fermi velocity is vFz

¼ B0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� C2=M2ð Þ2

q� nm=�h � 0:3� 106m=s. Taking this

anisotropy into consideration, we can rewrite the surface

Hamiltonian as

Hsurf ¼�vF

upu �ivFz e�iupz

ivFz eiupz vF

upu

� �: (6)

The surface states satisfy the Schodinger equation Hsurf ws

¼ Ews, in which the surfaces states wave function ws can be

written as

ws ¼ veijueikzz; (7)

where j ¼ � � � ;� 32þH;� 1

2þH; 1

2þH; 3

2þH; � � � and v

¼ e�iu2n1; e

iu2n2

� �T. The secular equation becomes

�vFu

R j� 12

� ��h �ivF

z �hkz

ivFz �hkz

vFu

R jþ 12

� ��h

!n1

n2

� �¼ E

n1

n2

� �: (8)

The energy dispersion can be obtained analytically

E6 ¼vFu�h

2R6�h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivF

z

� �2k2

z þj2 vF

u

2

R2

vuut; (9)

where 6 refers to the conduction and valence bands, respec-

tively. Because of the spatial reversal symmetry, the states

|kz, ji and |�kz, ji are degenerate. And v can be expressed by

vþ ¼h jð Þ sin h

2þ h �jð Þ cos h

2

� �e�iu

2

h jð Þ cos h2

eic þ h �jð Þ sin h2

eic� �

eiu2

!; (10a)

v� ¼h jð Þ cos h

2þ h �jð Þ sin h

2

� �e�iu

2

�h jð Þ sin h2

eic � h �jð Þ cos h2

eic� �

eiu2

!; (10b)

where h(x) is the Heaviside step function, c ¼ p2

(when

kz� 0) or � p2

(when kz< 0), and tanh ¼ vFz kz

.jvF

/

.R

h i��� ���.Note that this effective angular momentum j includes the

contribution from the external magnetic field.

III. NUMERICAL RESULTS AND DISCUSSIONS

A. Energy spectrum of Bi2Se3 nanowire

In this section, we present the numerical results for the

cylindrical Bi2Se3 nanowire. In our calculations, the material

parameters are taken from Ref. 14 and shown in Table I.

Fig. 2 shows the energy spectrum for a Bi2Se3 nanowire

with R¼ 60 nm, which is calculated from the four-band k � p

model (see Eq. (1)). The surface states display a perfect lin-

ear dispersion and a mini-gap (�7 meV) between the up- and

down-branches of the surface state subbands. This mini-gap

comes from the quantization of the orbital angular momen-

tum. The spacing of the adjacent surface state subbands is

determined by the circumference of the nanowire cross sec-

tion. The energy spectrum is twofold degenerate because of

the time and spatial reversal symmetry. In addition, under

the SU(2) operation, the spin of electron gets a p phase when

it cycles the nanowire. Therefore, the quantum number jmust be half of an odd integer.23

From the subband dispersions of the surface states of the

nanowire which are shown in Fig. 2, we can obtain the den-

sity of state (DOS). For one-dimensional system, the DOS

per unit length is given by D Eð Þ ¼P

i gi dkiz Eð Þ=dE

�� ��=p,

where i runs over all the subbands and gi¼ 2 counts the

Kramer’s degeneracy of each subband. Fig. 3 shows the den-

sity of the surface states. The many spikes in the DOS come

TABLE I. The parameters used in our calculation. U, P, and V are short for

the words: unit, parameters, and values, respectively.

U P V P V P V P V

mev C0 �6.8 M0 280.0 A0 410.0 B0 220.0

mev � nm C1 13.0 M1 �100.0

Mev � nm2 C2 196.0 M2 �566.0 A2 0.0 B2 0.0

FIG. 2. (Color online) The energy spectrum of a TI nanowire with

R¼ 60 nm. The black dashed lines and the green solid lines denote the bulk

and surface states, respectively.

FIG. 3. The DOS of the surface states of Bi2Se3 TI nanowire with the radius

R¼ 60 nm.

093714-3 Lou, Cheng, and Li J. Appl. Phys. 110, 093714 (2011)

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from the one-dimensional feature of the surface state sub-

bands. The spikes of the DOS correspond to the bottom of

the quantized surface subbands.

In Fig. 4(a), we plot the energies of the surface and bulk

states as a function of the angular momentum quantum num-

ber m at kz¼ 0. The spectra of spin-down and spin-up surface

states show perfect linear dispersion against m. Because

e(k)= 0 in the k � p Hamiltonian (see Eq. (1)) in our calcula-

tion, the energy spectra exhibit particle-hole asymmetry for

bulk states and the surface states at large in-plane momen-

tum. The density distribution of the conduction band and va-

lence band bulk states are localized at the center of the

nanowire (see Figs. 4(b) and 4(c), respectively). In contrast

to the bulk states, the surface states show interesting ring-

like density distribution in the vicinity of the surface of the

nanowire (see Figs. 4(d) and 4(e)). When this massless Dirac

fermion is confined in a nanowire, its lowest energy modes

should be the whispering gallery mode, similar to a photon

confined in a cylinder cavity.23 This gives us an intuitive pic-

ture to understand the origin of these exotic ring-like quan-

tum states in a nanowire. We also plot the energy spectrum

at kz¼ 0.1/nm (kz= 0) in Fig. 5. The density distributions

for electron of the bulk and surface states at kz¼ 0.1/nm (see

Figs. 5(b)–5(e)) are similar to that at kz¼ 0. Compared to

Fig. 4, the difference in Fig. 5 is that the mini-gap becomes

much larger and the dispersion near the Dirac point shows a

hyperlinear behavior in Fig. 5(a). The large gap is induced

by the mass term at kz= 0 (see Eq. (9)). In order to under-

stand the behavior of the surface states, we compare the nu-

merical results with the analytical results obtained by

solving the massless Dirac equation on the curved surface

and plot the energy-spacing of the surface states against the

size of the nanowire in Fig. 6. One can see a very good

agreement between the numerical results and analytical solu-

tions. Interestingly, the energies of the ring-like surface

states display a perfect linear dependence on the inverse of

circumference of the nanowire, i.e., 1/R. This behavior is

very different from that of the bulk states, which shows a

linear dependence on 1/R2. This difference indicates that

the bulk states are quantized in the cross section of the

nanowire, while the surface states are quantized along the

circumference.

B. The spin states and spin orientations

Using the basis P1þ�;12

�� �, P2�þ;

12

�� �, P1þ�;

�12

�� �, and

P2�þ;�12

�� �, the spin operator can be expressed by S ¼ �h

2r,

where the vector of spin matrix r ¼ ðr1; r2; r3Þ,

FIG. 4. (Color online) (a) The energy spectrum of the Bi2Se3 TI nanowire as

a function of the angular momentum quantum number m with R¼ 60 nm

and kz¼ 0. The spin-down and spin-up surface states are denoted by the red

and blue squares, respectively. The black squares represent the bulk conduc-

tion and valence band states. (b) and (c) The density distributions of the bulk

conduction and valence band states (marked by the black arrows in (a)). (d)

and (e) are similar to (b) and (c), but for the surface states (marked by the

red and blue arrows in (a)).

FIG. 5. (Color online) (a) The energy spectrum of a TI nanowire as a func-

tion of the angular momentum quantum number m with R¼ 60 nm and

kz¼ 0.1/nm. The surface states of conduction band and valence band are

denoted by the red and blue squares, respectively. (b) and (c) The density

distributions of the bulk conduction and valence band states (marked by the

black arrows in (a)). (d) and (e) are similar to (b) and (c), but for the surface

states (marked by the red and blue arrows in (a)).

FIG. 6. (Color online) The energies spacing between up and down branches

of the surface states of TI wires at kz¼ 0 as a function of the radius R. The

solid lines are the numerical results calculated from the four band k � pmodel. The red dashed lines are obtained from the analytical solution of sur-

face Hamiltonian (6). DEj¼ 2jA0/R, where A0 is the four band Hamiltonian

parameter, j¼61/2, 63/2,. is total azimuthal angular quantum number.

093714-4 Lou, Cheng, and Li J. Appl. Phys. 110, 093714 (2011)

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r1 ¼ rx � s0; r2 ¼ ry � s0, and r3 ¼ rz � s0:rx;y;z is the

Pauli matrix and s0 is the identical matrix. In these cylindri-

cal coordinates, the components of the expectation value of

spin operator can be expressed as

Sq ¼�h

2u1u3e�iu þ u3u1eiu þ u2u4e�iu þ u4u2eiu� �

;

(11a)

Su ¼ �i�h

2u1u3e�iu � u3u1eiu þ u2u4e�iu � u4u2eiu� �

;

(11b)

Sz ¼�h

2/1/1 þ /2/2 � /3/3 � /4/4

� �; (11c)

where U1,2,3,4 is the four components envelope function of

wj;kz¼ /1;/2;/3;/4ð Þ>. The expected spin orientation can

be easily obtained by hSi ¼ hwj;kzjSjwj;kz

i. Sq and Su are uindependent,24 which can be seen from Eqs. (3) and (11).

We find that the spins of the surface states lie in the surface

plane of the nanowire and Sq¼ 0. From the analytical solu-

tion of the surface Hamiltonian, we obtain the average spin

orientation rih i ¼ ws6 rij jws

6

�(see Table II).

From Table II, we can find that:

1. For the up (down) branch, the states jkz, j, si and j�kz, �j,�si form Kramer degenerate pairs. The system possesses

both the time and spatial reversal symmetry. Therefore,

the states jkz, j, si and jkz, �j, �si are degenerate.

2. hrqi¼ 0 shows that the spin orientation is always tangen-

tial along the nanowire surface due to the spin-momentum

locking ~r �~p ¼ 0.

3. When kz¼ 0 (thus h¼ 0), the electron spins point along

the nanowire axis. When |kz| is very large and the small |j|number (thus h�p/2), the spins lie in the surface plane

perpendicular to the nanowire axis and point along the

tangent direction of the cross section.

The spatial distributions of the spin orientation for the

electron surface states with zero and large momenta,

obtained from the numerical solution of the four-band k � pmodel, are shown in Fig. 7. The spins point to the axis of the

nanowire when kz¼ 0. This feature agrees with the conclu-

sion obtained from the surface Hamiltonian (see Eq. (6) and

Table II). Note that the surface states with a moderate posi-

tive j number are purely spin-down states (see Fig. 7(a)) for

the up branch of energy spectrum (see Fig. 4(a)), while the

surface states with a moderate negative j number are purely

spin-up states (see Fig. 7(b)). While in the cases of the large

kzj j � jj jvF/p= 2vF

z R� �

and the small |j| number, the spins lie

in the plane perpendicular to the nanowire axis. The spin ori-

entation depends on the direction of the electron orbital

motion. For instance, for an electron with a positive kz of the

up branch of the surface band, its spin points along the tan-

gent direction anti-clockwisely (see Fig. 7(c)), while for an

electron with a negative kz, the electron spin points in an op-

posite direction (see Fig. 7(d)). For the down branch, the

spin orientation is the opposite to that of the up branch.

In order to understand the behavior of spin orientation

further, we draw the expected value of the spin orientation of

the surface states at kz¼ 0 and kz= 0 in Figs. 8(a) and 8(b),

respectively. In Fig. 8(a), when jjj is not very large, as

TABLE II. The spin orientation calculated by the analytical surface Hamil-

tonian (6), where tanh ¼ vFz kz= vF

uj=Rh i��� ���.

wsi j kz ru

�rzh i rq

�~r �~p

wsþ j> 0 kz� 0 sin h �cos h 0 0

kz< 0 �sin hj< 0 kz� 0 sin h cos h

kz< 0 � sin hws� j> 0 kz� 0 �sin h cos h

kz< 0 sin hj< 0 kz� 0 �sin h �cos h

kz< 0 sin h

FIG. 7. (Color online) The spatial distribution of spin orientation for differ-

ent surface states with: (a) kz¼ 0, j> 0; (b) kz¼ 0, j< 0; (c)

kz ¼ jj jvFup= 2vF

z R� �

0 and small |j| number; (d) kz ¼ � jj jvFup= 2vF

z R� �

and

small |j| number.

FIG. 8. (Color online) The spin orientation for different surface states at (a)

kz¼ 0; (b) kz¼ 0.1/nm. The surface states of the up and down branches of

surface state are shown in red and blue symbols, respectively.

093714-5 Lou, Cheng, and Li J. Appl. Phys. 110, 093714 (2011)

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depicted in Figs. 7(a) and 7(b), the spin of surface states

stays the same value and points to the same direction. For

the large jjj case, due to the strong coupling between the sur-

face states and the bulk states, all large positive j 0 states

are fully spin up, and all large negative j� 0 states are fully

spin down. In Fig. 8(b), when jjj is very small, as illustrated

in Figs. 7(c) and 7(d), the spin lies in the plane perpendicular

to the axis of nanowire. The spin of surface states with a

large kz and a moderate jjj number will deviate this perpen-

dicular plane. The numerical results shown in Figs. 7 and 8

agree well with the analytical results in the Table II.

C. The magneto energy spectrum and persistentcurrents

Due to the ring-like distribution of TI nanowire surface

states, the Aharonov-Bohm oscillations of the persistent

charge current (CC) and spin current (SC) induced by the

magnetic flux can be found in this Bi2Se3 TI nanowire. In

Fig. 9, we show the magnetic energy spectrum Bi2Se3 nano-

wire at (a) kz¼ 0 and (b) kz¼ 0.1/nm, respectively. The mag-

netic level spectra En(U) of this TI nanowire surfaces states

are periodic functions of magnetic flux U with a period 2U0,

i.e., En(U)¼En(Uþ 2U0), where U0¼ h/(2e) (see Fig. 9). In

Fig. 9(a), the energy spectrum shows a linear dependence of

magnetic flux. It is interesting to notice that the surface

bands cross at the odd numbers of U0, the mass term in

Eq. (9) disappears at certain total azimuth angular momen-

tum quantum number j, and the mini-gaps between the up

and down branches of the surface bands at kz¼ 0 (see Fig. 2)

are closed. The closed mini-gap can be opened again with

increasing the magnetic fields further. In Fig. 9(b), the

energy spectrum displays a nonlinear dependence on the

magnetic flux due to the mass term in Eq. (9), and the energy

gap always exists under any value of the magnetic fields.

The persistent current is determined by the magnetic flux

U crossing the ring and the boundary conditions of the single-

particle wave function. Electrons in the state Ej(k, U) carry a

current Ij ¼ �Ð kFj

0 dEj k;Uð Þ�

dU� �

dk. The Fermi wave vector

kFj can be obtained by N ¼Ð EF

Ej kz¼0ð Þ DdE from the energy dis-

persion, where N is the total number of electrons, D ¼P

j Dj

is the DOS per unit length with Dj ¼ j@Ej=@kj� ��1

=p. In the

presence of an external magnetic field, due to the breaking of

time reversal symmetry, the noninteger flux U lifts the degen-

eracy. The total current in the system, given by the summation

of all occupied states Ej(k, U) up to the Fermi energy is,

Icc ¼ �X

j

ðkFj

0

@Ej k;Uð Þ@U

dk; (12a)

Isc ¼ � �h

2e

Xj

ðkFj

0

@Ej k;Uð Þ@U

rh idk; (12b)

where Icc (Isc) is the total persistent CC (SC). The persistent

CC of the surface states shows an oscillating dependence on

the magnetic flux, as depicted in Fig. 10. With increasing the

magnetic flux, the persistent CC oscillates with a period

T¼U0 rather than 2U0 (see the down panel of Fig. 10). This

period is half of the oscillating period of the energy spec-

trum, because the Fermi energy oscillating with a period

T¼U0 as a function of the magnetic flux (see the up panel of

Fig. 10). In the absence of an external magnetic field, the

spin-up electrons rotate reverse to the spin-down electrons

around the surface of nanowire, and therefore the currents

cancel each other. The total persistent CC is exactly zero.

The external magnetic field has two impacts on the motion

of electrons of surface states on the nanowire. First, the

external magnetic field increases the electron rotating clock-

wisely and slows down the velocities of the electrons of

FIG. 9. (Color online) The magneto energy spectra. (a) The energy of the

surface states under different magnetic field of a TI wire with radius

R¼ 60 nm and kz¼ 0; (b) the same as (a), but with kz¼ 0.1/nm. The red and

black lines denote the up and down branches of the surface bands,

respectively.

FIG. 10. (Color online) The Fermi energy (the upper panel) and the persis-

tent CC (the lower panel) as a function of the magnetic flux, respectively.

The electron density is n¼ 5� 106/cm.

093714-6 Lou, Cheng, and Li J. Appl. Phys. 110, 093714 (2011)

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rotation anticlockwisely. Second, the magnetic field lifts the

degeneracy and causes the DOS of j ji which is different

from that of j�ji. Thus there is a persistent CC for the sur-

face states of TIs nanowire under a magnetic flux. Similarly,

the negative magnetic flux would induce an opposite persis-

tent CC. The oscillating persistent CC could induce magnet

moment oscillates as well. And the oscillation of the magnet

moment, known as the Aharonov-Bohm effect, can be

detected by the nano-SQUID technique. Therefore, the exis-

tence of surface states of Bi2Se3 nanowire can be detected

utilizing this Aharonov-Bohm effect.

It is interesting to note that the maximum amplitude also

shows an oscillating behavior with increasing the electron

density n (see the red line in Fig. 11). Because @Ej(k, U)/@Uhas the opposite signs for the two nearest subbands (see

Fig. 9), the persistent CCs for these nearest subbands are

opposite. Therefore, the summation in Eq. (13) may be de-

structive for even number of occupation levels, which causes

a trough in Fig. 11, and constructive for odd number of occu-

pation levels, which leads to a crest in Fig. 11. Consequently,

the maximum amplitude shows an oscillating behavior with

increasing the electron density n. Since the Fermi energy

increases monotonically with increasing the electron density

n (see the blue line in Fig. 11), the populations of states j jiwith a higher electron density are larger than that of states

j ji with a lower electron density. Therefore, the maximum

amplitude of persistent CC increases with increasing the

electron density (see the red line in Fig. 11).

Since electrons carry spins as well as charges, their

motion gives rise to a spin current besides the charge cur-

rent.25,26 The spin current, different from the charge current,

is actually a tensor. The persistent SC coming from the sur-

face states shows an oscillating behavior verse the magnetic

flux, as shown in Fig. 12. Because the sign of hrzi is opposite

for |ji and |�ji states, there is a pure large persistent SCðIscuzÞ

even without magnetic flux (U¼ 0). With increasing of mag-

netic fields, the persistent SCðIscuzÞ oscillates with increasing

magnetic flux with a period U0. For the states with a fixed jnumber but with different signs of kz (kz> 0 and kz< 0), hruihas opposite signs (see Table II), which leads to Isc

uu ! 0:

IV. CONCLUSION

In summary, we investigate theoretically the energy

spectrum, spin orientation, and persistent charge and spin

currents of the surface states of topological insulator Bi2Se3

nanowires. We find that the surface states show a perfect lin-

ear dispersion and helical feature, i.e., the spin-momentum

locking. The electron spin is perpendicular to its momentum

and points to the tangent direction of the nanowire surface.

The density distributions of the surface states are localized

near the boundary and show a ring-like behavior. We find

that the persistent CC (SC) oscillates with magnetic fields.

We propose the surface states can be detected from the

Aharonov-Bohm effect. Our results suggest that topological

insulator Bi2Se3 nanowires afford promising candidates for

future spintronic devices application.

ACKNOWLEDGMENTS

This work was supported by the NSFC Grant Nos.

11104232 and 11004017.

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