Solid State Communications 188 (2014) 45–48
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Solid State Communications
journal homepage: www.elsevier.com/locate/ssc
Transport in a quantum spin Hall bar: Effect of in-plane magnetic ﬁeld
Fang Cheng a,b,n, L.Z. Lin b, D. Zhang b
a
b
Department of Physics and Electronic Science, Changsha University of Science and Technology, Changsha 410004, China
SKLSM, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, China
art ic l e i nf o
a b s t r a c t
Article history:
Received 30 May 2013
Received in revised form
18 September 2013
Accepted 25 February 2014
by S. Tarucha
Available online 6 March 2014
We demonstrate theoretically that edge transport in quantum spin Hall bar can be controlled by in-plane
magnetic ﬁelds. The in-plane magnetic ﬁeld couples the opposite spin orientation helical edge states at
the opposite edges, and induces the gaps in the energy spectrum. The hybridized electron wave
functions ψ ↑ ðx; ky Þ of the edge states can be destroyed with increasing the in-plane magnetic ﬁelds.
When the Fermi surface is located within this energy gap induced by the in-plane magnetic ﬁeld, one
can expect that the conductance of the edge states becomes e2/h. By tuning the magnetic ﬁeld and Fermi
energy, the edge channels can be transited from opaque to transparent. This switching behavior offers us
an efﬁcient way to control the topological edge state transport.
& 2014 Elsevier Ltd. All rights reserved.
Keywords:
A. Topological insulator
D. Electronic states
D. In-plane magnetic ﬁeld
D. Quantum transport
1. Introduction
Topological insulators (TIs), a strong spin–orbit coupling system,
exhibit rich and fascinating physics, which have been investigated
intensively both theoretically and experimentally [1–4]. The twodimensional (2D) TIs have been realized in HgTe quantum wells
(QWs) and InAs/GaSb QWs [5,6] by tuning the thickness of the QWs
or electric ﬁeld [7–11]. HgTe is a narrow gap semiconductor with
very strong spin–orbit interaction (SOI) [12]. Strong SOI inverts the
band structure of HgTe, leading to a topological insulating phase. In
this phase, HgTe possesses an insulating in the bulk with a gap
separating the valence and conduction bands but with gapless
helical edge states that are topologically protected by the timereversal symmetry [13,14]. The existence of the helical edge states in
2D TIs was proved by recent experiments [15,16]. The helical feature
and suppressed backscattering render edge states an attractive
platform for high mobility charge- and spin-transport devices.
Since the topological edge states in 2D TI are protected by timereversal symmetry and robust against to backscattering, control of
the edge states, e.g., switch on/off, is a challenging issue from the
viewpoint of basic physics and potential device application.
Recently, there have been a few proposals to control the edge
state transport using a quantum point contact [17–20]. These
electrical means can control the transport, magnetic properties
and even quantum phase transition, and provide us an efﬁcient
way to control spin transport [11,20–22]. It is natural to ask if there
is any other method to control the edge state transport?
In this letter, we study the effect of in-plane magnetic ﬁeld on the
transport property of a quantum spin Hall (QSH) bar. The magnetic
ﬁeld can lead to a large Zeeman term because of the large g factor of
HgTe material. The Zeeman term couples spin-up and spin-down
electron and holes, and induces the gaps in the energy spectrum.
Electrons with the opposite spin orientation at the opposite edges
couple together due to in-plane magnetic ﬁelds. And the density
distributions of hybridized electron wave functions ψ ↑ ðx; ky Þ become
more localized in the center of the QSH bar, indicating destroy of the
edge state. When the Fermi surface is located within this energy gap
induced by the in-plane magnetic ﬁeld, one can expect that there is
only an edge state and the conductance of the edge states becomes
one-half of the conductance quantum 2e2/h. The in-plane magnetic
ﬁeld can control the coupling between the edge states at opposite
edges and between the topological edge state and the bulk state.
Tuning the in-plane magnetic ﬁeld, one can switch-on/off the edge
channel in the ﬁnite width QSH bar system when the Fermi energy is
the gap. This feature provides us an efﬁcient means to control the edge
state transport in QSH bars.
2. Theoretical model
n
Corresponding author at: Department of Physics and Electronic Science,
Changsha University of Science and Technology, Changsha 410004, China.
Tel.: þ 86 15807314598.
E-mail address: [email protected] (F. Cheng).
http://dx.doi.org/10.1016/j.ssc.2014.02.028
0038-1098 & 2014 Elsevier Ltd. All rights reserved.
The total Hamiltonian for the system in the presence of an
external in-plane magnetic ﬁeld is
H ¼ H0 þ HZ ;
ð1Þ
46
F. Cheng et al. / Solid State Communications 188 (2014) 45–48
where the ﬁrst term is the single-particle Hamiltonian of electron
in HgTe QWs and the second term HZ is the Zeeman effect. The
electron transport in the quasi-one-dimensional (Q1D) QSH bar is
along the longitudinal y-direction. The four-band single-particle
Hamiltonian H reads as
0
1
ϵk þ MðkÞ
Ak g e μB B
0
B
C
ϵk MðkÞ
0
g h μB B C
B Ak þ
C;
H¼B
ð2Þ
B g μB B
0
ϵk þ MðkÞ
Ak þ C
e
@
A
0
g h μB B
Ak ϵk MðkÞ
where k ¼ ðkx ; ky Þ is the in-plane momentum of electrons,
2
2
ϵk ¼ C þ Vðx; yÞ Dðkx þ ky Þ with Vðx; yÞ being the conﬁnement
2
2
potential, MðkÞ ¼ M Bðkx þky Þ, k 7 ¼ kx 7 iky , A, B, C, D, and M
are the parameters describing the band structure of the HgTe/CdTe
QW. g e=h denotes electron or hole g factor, respectively. μB is the
Bohr magneton. B is the external transversal x-direction magnetic
ﬁeld.
The transport property of a Q1D QSH bar can be obtained by
discretizing the Q1D system into a series of in-plane stripes along
the transport direction. Assuming a hard-wall in-plane conﬁning
potential, the traveling-wave-like or evanescent-wave-like eigenstates of the Schrödinger equation Hψ ¼ Eψ in a given region λ can
be written as the form
λ
ψ λ ðx; yÞ ¼ expðiky yÞ∑χ λn φn ðxÞ;
ð3Þ
n
where
rﬃﬃﬃﬃﬃﬃ
2
nπx
sin
φn ðxÞ ¼
W
W
ε (meV)
in which W is the width of the lead, and the subband index n¼1,2,…
N with N being the number of the basis function which is chosen to
ensure the convergence of the energies of the low subbands near the
Dirac point. fχ λn g ðλ ¼ L; RÞ are the expanded coefﬁcients. The longλ
itudinal wave vector ky and the eigenvector χ λn (n¼ 1,2,3,…)
L
n
R
By using scattering matrix theory, we can calculate the coefﬁcients
r m , t m in the left and right leads. Thus we can obtain the total
conductance from the Landauer–Büttiker formula
υRm 2
t j ;
L m
m;n υn
RM
G ¼ G0 ∑
3. Numerical results and discussions
In the case of a QSH bar in the absence of in-plane magnetic
ﬁeld, the ﬁnite size effect induces the overlap of the wave
functions of the edge states localized at the opposite edges, and
can open a minigap in the energy spectrum of the edge states at
ky ¼ 0 (see Fig. 1(a)). The in-plane magnetic ﬁeld couples the edge
states at the opposite edges (see Eq. (1)), and induces a mass term
for massless Dirac electrons in the edge states. Therefore the gaps
in the energy spectrum increase with increasing the magnetic
ﬁelds.
0
8
8
4
4
0
−0.02
0
ky (nm−1)
0.02
ð5Þ
where G0 ¼ e2/h is the conductance unit, RM denotes the summation
over all right-moving modes in the left and right leads, tm is the
transmission coefﬁcient where the electron incidents from the
subband n in the left lead to be scattered into the subband m in
the right lead, and υλm ¼ 〈^υ λm 〉 ¼ 〈∂H=∂ky 〉 are the group velocity of the
electron in the subband m in the leads along the QSH bar, i.e., the yaxis direction.
4
0.02
ð4Þ
mn
4
0
mn
ψ R ¼ ∑ t m χ Rm;n eikm y φn ðxÞ:
8
−0.02
L
ψ L ¼ eikI y ∑χ LI;n φn ðxÞ þ ∑ r m χ Lm;n e ikm y φn ðxÞ;
8
0
ε (meV)
are determined from the generalized eigenvalue problem [20].
Assuming an electron injected from a given energy with wave vector
kLI in the left lead, the wave functions in the left lead and the right
lead can be written as
0
−0.02
−0.02
0
0.02
0
0.02
ky (nm−1)
Fig. 1. The energy spectra with width W ¼ 200 nm under different in-plane magnetic ﬁelds (a) B¼ 0, (b) B ¼0.5 T (c) B¼ 1 T, and (d) B¼ 2 T. The parameters used in the
calculation are A ¼ 364.5 meV, B ¼ 686 meV nm2, C¼ 0, D ¼ 512 meV nm2, M¼ 10 meV.
F. Cheng et al. / Solid State Communications 188 (2014) 45–48
We plot the density distributions of edge states for a QSH bar
with the width W¼200 nm for the different strengths of the
in-plane magnetic ﬁelds. In the case of a QSH bar in the absence
of in-plane magnetic ﬁelds, the two states of ψ ↑ ðx; 7 ky Þ (and
ψ ↓ ðx; 7 ky Þ) are degenerate but localized at the opposite edges. The
overlap between these two edge states, the ﬁnite size effect, can
open a minigap (see Fig. 1(a)). Electrons with the opposite spin
orientation at the opposite edges couple together due to in-plane
magnetic ﬁelds, i.e., the off-diagonal elements in the Hamiltonian
(see Eq. (1)). The wave functions of two spin directions are
hybridized by the in-plane magnetic ﬁeld. Therefore the densities
of the hybridized wave functions ψ ↑=↓ ðx; ky Þ (and ψ ↑=↓ ðx; ky Þ) are
symmetrically distributed at the two sides. The in-plane magnetic
ﬁeld can control the coupling between the topological edge state
and the bulk states. Fig. 2(b)–(d) shows clearly that the density
distributions of the hybridized wave functions ψ ↑ ðx; ky Þ become
more localized in the center of the QSH bar with the increase of inplane magnetic ﬁeld B, indicating destroy of the edge state.
The variation of the conductance as a function of the Fermi
energy is shown in Fig. 3 for the different strengths of the in-plane
magnetic ﬁelds. In the absence of an in-plane magnetic ﬁeld, the
conductance in a QSH bar shows a perfect plateau 2e2/h in the bulk
gap because of the two 1D spin-resolved conducting channels at the
edges, and displays a minigap near the Dirac point (see black curve
in Fig. 3). The width of the gap in the conductance plateau is
determined by the width of the QSH bar, i.e., the overlap between
the wave functions of the edge states. The minigap of the conductance plateau can be changed signiﬁcantly by applying an
in-plane magnetic ﬁeld. The Zeeman term couples spin-up and
spin-down electron and holes (see Eq. (1)), and induces a mass term
for massless Dirac electrons in the edge states. Therefore the gaps in
the plateau increase with increasing the magnetic ﬁelds. There is an
odd number of the conductance quantum e2/h in the spectrum.
When the Fermi surface is located within this energy gap induced by
the in-plane magnetic ﬁeld, the conductance becomes e2/h, which
indicates that the conductance plateau comes from hybridized
ψ ↓ ðx; ky Þ topological edge states. The topological edge states of the
hybridized electron wave functions ψ ↑ ðx; ky Þ are destroyed because
of the in-plane magnetic ﬁeld. More interestingly, one can also
47
switch on/off this edge current by changing slightly the in-plane
magnetic ﬁelds when the Fermi energy locates at the minigap.
The magnetic ﬁeld dependence of the QSH edge current also
shows the interesting switching on/off feature of edge current (see
Fig. 4(a)). Increasing the strength of the in-plane magnetic ﬁeld,
a crossover from the transmitting case to the opaque case occurs.
Clearly, the crossover takes place at different magnetic ﬁelds for
different Fermi energies. It means that the crossover can also be
controlled by tuning the Fermi energy for a ﬁxed magnetic ﬁeld.
Fig. 4(b) shows the conductance as a function of the width of the
QSH bar for a ﬁxed Fermi energy and three different strengths of
the in-plane magnetic ﬁeld. From this ﬁgure, we can see that the
transition from the zero conductance to the plateau G0 or 2G0
depends not only on the width of the system, but also on the
strength of the in-plane magnetic ﬁeld. In the vicinity of the gap
where the tunneling is forbidden, the critical width of the system
depends sensitively on the in-plane magnetic ﬁeld. Increasing the
width of the QSH bar or decreasing the strength of the in-plane
magnetic ﬁeld will lead to weakening of the inter-edge coupling.
Both the ﬁnite size and the in-plane magnetic ﬁeld modify the
Fig. 3. (Color online) The Fermi energy dependence of the conductance G, where
the black line is for B¼ 0, the red line for B¼ 0.5 T and the blue line for B¼ 1 T.
Fig. 2. (Color online) The density distribution of the edge states for a QSH bar with width W¼ 200 nm under different in-plane magnetic ﬁelds: (a) B ¼0, (b) B¼ 0.5 T,
(c) B¼ 1 T, and (d) B ¼2 T. The red solid line corresponds to wave function ψ ↓ ðx; ky Þ ðψ ↑ ðx; ky ÞÞ, and black dashed line to ψ ↑ ðx; ky Þ ðψ ↓ ðx; ky ÞÞ at ky ¼ 0:01 nm 1 .
48
F. Cheng et al. / Solid State Communications 188 (2014) 45–48
Fig. 4. (Color online) The conductance as a function of (a) the strength of the in-plane magnetic ﬁeld for two different Fermi energies EF ¼ 6 meV (the black solid line), 7 meV
(the red dashed line). The width of the system is a ﬁxed value W¼ 200 nm. (b) The width of the system for a ﬁxed EF ¼ 6 meV and different strengths of magnetic ﬁeld B¼ 0
(the black solid line), 0.5 T (the red dashed line), and 1 T (the blue dashed-dot line).
overlap of the wave functions of the edges states localized at the
opposite edges. Therefore the critical width for the blocking of
the edge channels can be tuned by in-plane magnetic ﬁelds. The
crossover from the zero conductance to the plateau G0 or 2G0
occurs very sharply, indicating a perfect switching effect.
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4. Conclusions
In summary, we demonstrate theoretically that the topological
edge state transport through a QSH bar can be manipulated using
in-plane magnetic ﬁelds. Electron transport through the edge
states can be transited from opaque to transparent and from the
plateau G0 to the plateau 2G0 by tuning the Fermi energy cross the
gap in the QSH bar system. This feature offers us an efﬁcient way
to control the topological edge state transport, and paves a way to
construct the edge state electronic device.
Acknowledgments
This work is partly supported by the NSFC Grant nos. 11004017,
11104263, 11274108, Hunan Provincial Natural Science Foundation
of China no. 13JJ2026, Scientiﬁc Research Fund of Hunan Provincial
Education Department 12B010, Foundation for University Key
Teacher by the Ministry of Education, Science and Technology
Innovative Research Team in Higher Educational Institutions of
Hunan Province and the construct program of the key discipline in
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