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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:
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.
1R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electronand Hole Systems (Springer-Verlag, Berlin, 2003).
2W. Yang and K. Chang, Phys. Rev. B 74, 193314 (2006); ibid, 73, 113303
(2006).3B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science 314, 1757 (2006).4C. L. Kane and E. J. Mele, Science 314, 1692 (2006).5W. Yang, K. Chang, and S. C. Zhang, Phys. Rev. Lett. 100, 056602
(2008); C. Brune, A. Roth, E. G. Novik, M. Konig, H. Buhmann, E. M.
Hankiewicz, W. Hanke, J. Sinova, and L. W. Molenkamp, Nat. Phys. 6,
448 (2010).6X. L. Qi and S. C. Zhang, Phys. Today 63(1), 33 (2010).7M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).8J. E. Moore, Nature 464, 194 (2010).9M. Konig, S. Wiedmann, C. Brune, A. Roth, H. Buhmann, L. W.
Molenkamp, X. L. Qi, and S. C. Zhang, Science 318, 766 (2007).10Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer,
Y. S. Hor, R. J. Cava, and M. Z. Hasan, Nat. Phys. 5, 398 (2009).
FIG. 11. (Color online) The maximum amplitude of persistent charge cur-
rent (CCM) and Fermi energy (EF�E0) as a function of the electron density.
E0 is a zero-point energy.
FIG. 12. The persistent SC as a function of the magnetic flux at electron
density n¼ 5� 106/cm.
093714-7 Lou, Cheng, and Li J. Appl. Phys. 110, 093714 (2011)
Downloaded 30 Jan 2012 to 14.139.220.33. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
11Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J.
Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain,
and Z.-X. Shen, Science 325, 178 (2009).12D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z.
Hansan, Nature 452, 970 (2008).13D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F.
Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava, and M. Z. Hasan,
Science 323, 919 (2009).14H. J. Zhang, C. X. Liu, X. Dai, Z. Fang, and S. C. Zhang, Nat. Phys. 5,
438 (2009).15J. J. Zhu, D. X. Yao, S. C. Zhang, and K. Chang, Phys. Rev. Lett. 106,
097201 (2011).16H. Peng, K. Lai, D. Sheng, S. Meister, Y. Chen, X.-L. Qi, S.-C. Zhang,
Z.-X. Shen, and Y. Cui, Nature Mater. 9, 225 (2009).
17D. Kong, J. C. Randel, H. Peng, J. J. Cha, S. Meister, K. Lai, Y. Chen,
Z.-X. Shen, H. C. Manoharan, and Y. Cui, Nano Lett. 10, 329 (2010).18J. J. Cha, J. R. Williams, D. Kong, S. Meister, H. Peng, A. J. Bestwick,
P. Gallagher, D. Goldhaber-Gordon, and Y. Cui, Nano Lett. 10, 1076 (2010).19S. S. Hong, W. Kundhikanjana, J. J. Cha, K. Lai, D. Kong, S. Meister,
M. A. Kelly, Z.-X. Shen, and Y. Cui, Nano Lett. 10, 3118 (2010).20P. C. Sercel and K. J. Vahala, Phys. Rev. B 44, 5681 (1991).21J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955).22Y. Zhang, Y. Ran, and A. Vishwanath, Phys. Rev. B 79, 245331 (2009).23K. Chang and W. K. Lou, Phys. Rev. Lett. 106, 206802 (2011).24R. Egger, A. Zazunov, and A. L. Yeyati, Phys. Rev. Lett. 105, 136403
(2010).25L. Wendler, V. M. Fomin, and A. A. Krokhin, Phys. Rev. B 50, 4642 (1994).26J. S. Sheng and K. Chang, Phys. Rev. B 74, 235315 (2006).
093714-8 Lou, Cheng, and Li J. Appl. Phys. 110, 093714 (2011)
Downloaded 30 Jan 2012 to 14.139.220.33. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions