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Contact Effects in thin 3D-Topological Insulators: How does the current flow?

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Abstract The effect of different contact configurations (semi-infinite extended-channel, normal metal and ferromagnetic metal) on quantum transport through thin Bi 2 Se 3 three-dimensional (3D) topological insulator (TI) slab (channel) has been investigated through Non-Equilibrium Green Function. The issue of contact dependent current flow and distribution across quintuple layers of 3D-TI has been addressed in this work and applied to expound the explanation for recent experimental work on electrical detection of spin-momentum locking on topological surface for long channel device. A theoretical model is propounded to develop a microscopic understanding of transport in 3D-TI in which contact type and magnetization concur with helical surface states of the TI channel to manifest seemingly counter-intuitive current distribution across layers. The quantum transport calculations for short channel devices with magnetic source and drain contacts postulate negative surface current for anti-phase magnetization whose axis is transverse to both current and quintuple layers. For in-phase magnetization at the two terminals, it is shown that observations can change fundamentally to result in anomalous current distribution. Such results are explained to stem from the confinement of 3D-TI between ferromagnetic contacts along the transport direction. A simple mechanism to validate topological insulators via quantum transport experiments has also been suggested.

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### OPEN

SUBJECT AREAS:

ELECTRONIC AND
SPINTRONIC DEVICES

TOPOLOGICAL INSULATORS

Received
20 November 2014

Accepted
23 February 2015

Published

30 March 2015

Correspondence and

requests for materials

should be addressed to

[G.G. (a0089293@](mailto:a0089293@nus.edu.sg)

[nus.edu.sg) or (G.L.](mailto:a0089293@nus.edu.sg)

[(elelg@nus.edu.sg)](mailto:elelg@nus.edu.sg)

## Contact Effects in thin 3D-Topological Insulators: How does the current flow?

Gaurav Gupta, Mansoor Bin Abdul Jalil & Gengchiau Liang

Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576.

The effect of different contact configurations (semi-infinite extended-channel, normal metal and
ferromagnetic metal) on quantum transport through thin Bi2Se3 three-dimensional (3D) topological
insulator (TI) slab (channel) has been investigated through Non-Equilibrium Green Function. The issue of
contact dependent current flow and distribution across quintuple layers of 3D-TI has been addressed in this
work and applied to expound the explanation for recent experimental work on electrical detection of
spin-momentum locking on topological surface for long channel device. A theoretical model is propounded
to develop a microscopic understanding of transport in 3D-TI in which contact type and magnetization
concur with helical surface states of the TI channel to manifest seemingly counter-intuitive current
distribution across layers. The quantum transport calculations for short channel devices with magnetic
source and drain contacts postulate negative surface current for anti-phase magnetization whose axis is
transverse to both current and quintuple layers. For in-phase magnetization at the two terminals, it is shown
that observations can change fundamentally to result in anomalous current distribution. Such results are
explained to stem from the confinement of 3D-TI between ferromagnetic contacts along the transport
direction. A simple mechanism to validate topological insulators via quantum transport experiments has
also been suggested.

he peculiar suppression of backscattering <sup>1</sup> of helical fermions in odd-number <sup>2</sup> of gapless Dirac <sup>3</sup> surface
bands of three-dimensional (3D) topological insulators (TI) <sup>2,4</sup>, accompanied with an insulating bulk, has
recently drawn tremendous interest for Very-Large Scale Integration (VLSI) interconnect <sup>5</sup>, spintronic <sup>6,7</sup> and
# T
quantum computing <sup>8</sup> applications. Time-reversal symmetry (TRS) renders this protection to surface states
against scattering from non-magnetic impurities and vacancies because in these states the spin of an electron
is locked to its momentum vector, which results in a distinct momentum-space spin-texture, and therefore
without spin-flip mechanism (e.g. doping with magnetic impurities) or breaking of TRS (e.g. application of a
perpendicular magnetic field) an electron cannot be backscattered. The spin-texture which is left (right) handed
for top-surface (bottom-surface) conduction band (see Fig. 1(a)) and vice-versa for valence bands has recently
been shown to engender spin-polarized surface current <sup>9,10</sup> with an average polarization transverse <sup>11</sup> to nonequilibrium transport direction. Kramer’s degeneracy theorem <sup>12,13</sup> posits that these surface bands with antagonistic spin-texture on opposite surfaces are degenerate. If 3D-TI is thin enough, but thicker than 5 quintuple layers <sup>9</sup>

(QL) (1 QL, 0.943 nm) to eschew crossover to two-dimensional limit <sup>14</sup>, the wavefunction overlap between
opposite surfaces then enables an electron to be backscattered from forward moving state of one surface to
backward moving state of opposite one. However, for very large thickness a sizeable fraction of current flows
through the bulk layers which weakens the signature of topological surface in electrical transport <sup>9</sup> . Thin 3D-TI
(thickness, 10 nm <sup>15</sup> ) with weak inter-surface coupling but surface dominant transport is therefore especially
interesting and important <sup>16</sup> . We show in this work that this thickness regime may also provide a simple way to
validate spin-momentum locking on topological surfaces through quantum transport experiments, in comparison to more complicated Berry-phase matching <sup>17</sup> and optical (Spin-ARPES <sup>18</sup> and Circular Dichroism <sup>19</sup> )
experiments.

Furthermore, for the observation of the topological properties the device should be electronically operated (i) at
small channel bias with Fermi-level (EF) close to the Dirac-point, to avoid the dominance of trivial bulk bands in
observed results; and (ii) at low temperature to circumvent phonon scattering <sup>20</sup> . Physically, defect-free or compensated <sup>21</sup> 3D-TI of short channel must be used because strong spin-charge coupling <sup>22,23</sup> in 3D-TI would limit the
spin-relaxation length <sup>24</sup> to the mean-free path. Small channel length becomes especially important for the study of
novel magneto-electric effects that may arise in 3D-TI due to ferromagnetic contacts <sup>23,25</sup> . Most experiments have,
howbeit, been performed on very long channel devices <sup>9,26</sup> because placing multiple probes for measurements that
can accurately characterize the carrier transport become very challenging for small channel lengths. Therefore,

finally evolves into more complicated results predicted in this work
for short-channel devices with both metallic source and drain contacts where quantum effects may manifest exotic results and provide
entirely new ways of identifying a topological insulator via electrical
transport experiments.

The paper is organized as follows. Subsequent to this introduction,
methods section describes the Hamiltonian and NEGF formalism
implemented in this work for appraising different contacts. Results
and Discussion section then presents our results on the effect of contacts on transport and more specifically on current distribution in 3DTI. Finally, we conclude our findings with suggestions for experiments.

Methods
Bi2Se327 is chosen as a representative 3D-TI in this study owing to its large bulk
bandgap of,0.3 eV, which is largest among extant 3D-TIs, and hence, provides large
energy window for transport through topological surface bands. The parameters for
device Hamiltonian in eq. (1) are chosen to fit the experimental dispersions, and its
validation is elaborated in our previous work <sup>20,28</sup> . Each infinite quintuple layer (x-y
plane; see Fig. 1(b)) of Bi2Se3 is described in pz spin-orbital basis by k.p model by Hp

2

6664

3

ð1Þ
7775,

Hp~

k <sup>2</sup> =m1 dzk <sup>2</sup> =m2 ivkx{vky 0
dzk <sup>2</sup> =m2 k <sup>2</sup> =m1 0 {ivkxzvky
{ivkx{vky 0 k <sup>2</sup> =m1 dzk <sup>2</sup> =m2
0 ivkxzvky dzk <sup>2</sup> =m2 k <sup>2</sup> =m1

Figure 1 | (a) Spin-texture in momentum space for conduction bands for
top and bottom surface. Blue arrows are spin vectors, red circle is energy
contour and black arrows denote momentum vectors. (b) Atomic
structure of Bi2Se3 (3D-TI). 5 atomic layers constitute one quintuple layer
(QL, 0.943 nm). (c) Device Structure for quantum transport modeling
through Bi2Se3 slab. The axis are given over the device along with
convention for polar representation of Magnetization vector (M) in this
work. h is polar angle and [ is azimuthal angle. Slab dimensions of 10 QL
thickness – periodic width (uncoupled mode space) 265 nm channel
length 210 nm contact on both side of channel, temperature of 0 K, a
potential drop of 35 mV across channel (VDS) and Fermi-level (Ef) 5
0.075 eV has been considered as a representative case throughout this work
unless specified otherwise. (d) Illustration of the potential distribution
through the energy bands along the transport direction. mS (mD) is
electrochemical potential at source (drain) end.

despite of recent emphasis on the importance of short channel for
investigating topological insulators, study of quantum transport and
current distribution through such a device is completely lacking. For
short-channel device the issue is further exacerbated by the confinement along transport direction. We show in this work that confinement especially due to ferromagnetic contacts can radically influence
the experimental investigation of transport in 3D-TI. In this work,
therefore, we carry out a systematic and comprehensive study of
quantum transport and current distribution in thin 3D-TI based
on Bi2Se3 for different contact configurations, which not only enables
us to explain recent experimental work on electrical detection of
spin-momentum locking on topological surface for long channel
device, but also allows us to make new predictions for short-channel
TI device like negative surface current and an exotic current distribution which may not be expected from conventional understanding
of spin-texture for surface states of 3D-TI. Therefore, in this work,
first a simple system with ‘Ferromagnetic Metal source-TI ChannelExtended TI Drain’ (FM-TI-exTI) is comprehensively examined to
understand the recently published experimental data <sup>9</sup> (long channel
device). We then present a more microscopic analysis for different
combinations of three different contact-types i.e. ferromagnetic
metal (FM) normal metal (NM) and extended TI (exTI) which

where v 5 2.5 eV A <sup>˚</sup> is the Fermi velocity, m1 5 0.125 eV <sup>21</sup> A <sup>˚</sup> <sup>22</sup> and m2 5
20.04 eV <sup>21</sup> A <sup>˚</sup> <sup>22</sup> are the orbital masses and the parameter d 5 20.22 eV is introduced
to generate a gap. In-plane wavevector k is computed as k <sup>2</sup> 5 kx2 1 ky2 where kx and ky

are wavevectors along x-axis and y-axis respectively. The hopping between adjacent
layers is described by tight-binding parameter tz 5 0.35 eV in hopping matrix T as per
eq. (2).

Green Function : ½G <sup>r</sup> �~½(Ezig)I{H0{U0{SS{SD{Sph� <sup>{1</sup> ð4Þ

Correlation Function : ½G <sup>n</sup> �~½G <sup>r</sup> �½S <sup>in</sup> �½G <sup>rH</sup>      - ð5Þ

-
Spectral Function : ½A�~i ½G <sup>r</sup> �{½G <sup>rH</sup>       - ð6Þ

I tt i F ti ½S <sup>in</sup>      - ½C �f z½C �f ð7Þ

0 0 0 0

tz 0 0 0

0 0 0 0

0 0 tz 0

3

ð2Þ
7775,

T~

2

6664

For modeling the transport in the x-direction (see Fig. 1(c)), kx wave-vector in eq. (1)
is discretized via finite difference method (FDM) by substituting kx by 2i h/hx.
Transverse direction (y-axis) is modeled in uncoupled mode space (k-space (eigenmodes))for an infinitely wide channel (periodic condition) with no potential variation along the transverse direction <sup>29–31</sup>, to compute with reasonable computational
resources <sup>32</sup> . Recursive-Green function (RGF) <sup>33</sup> and Non-Equilibrium Green function
(NEGF) algorithm are employed to describe the transport through a defect-free 3DTI slab. To model the effect of infinitely or semi-infinitely long channel, computationally one or both of the terminal contacts are treated as if made of the same material
as the channel <sup>34,35</sup> This is done by applying open boundary condition at the contact
for reflection less propagation of the plane wave and the contact self-energy is
computed from the self-consistent solution of the surface green function <sup>36</sup> . This type
of contact is referred as exTI (extended TI or extended channel) to distinguish it from
other contact configurations. From the experiment’s point of view, this refers to the
scenario where terminals are far away from the surface probes to affect the measurements i.e. when voltage probes (VPs) (measurement probes) are sufficiently far
from the current-injection probes (CPs) for very long channel device (typically in mm
range). The measurement characterizes only the region between VPs, and captures
the transport only in the TI material. Semi-infinite 3D-TI contacts are more appropriate to model such a system. On the other hand, modeling metallic contacts induces
the effect of confinement and a hard wall boundary condition. When metallic (normal or ferromagnetic) contacts are applied at both ends, the system is referred to be a
short-channel device. For such systems, the confinement and the type of contact is
expected to influence the readings. Acoustic phonons scattering <sup>20</sup> is modeled as selfenergy in the channel and converged self-consistently. The key equations are summarized as follows:

Current : Ip?q~ <sup><u>q</u></sup>

h

ð

dE Trace(i½HpqG <sup>n</sup>

<sup>n</sup> qp <sup>{Gn</sup> pq

<sup>n</sup> pq <sup>Hqp�)</sup> ð3Þ

Level Broadening : ½CS=D�~i½SS=D{SHS=D <sup>�</sup> ð8Þ

h i

tHS=D

- �h

tS=D g <sup>S</sup> S

ð9Þ

Contact Self -Energy : ½SS=D�~ tS=D

h i

g <sup>S</sup> S=D

Phonons Self -Energy : ½Sph�~Dac½G <sup>r</sup>           - ð10Þ

Transmission : TðEÞ~Trace(CSG <sup>r</sup> CDG <sup>rH</sup> ) ð11Þ

where ‘p’ and ‘q’ are two successive y-z surfaces between which the current is evaluated
and Hpq is hopping matrix from p to q planes. H0 is the device Hamiltonian and U0 is
potential distribution in device. Dac is the deformation potential for modeling acoustic
phonons. tS/D is the coupling between the channel and the contacts, and g <sup>S</sup> S/D is the
surface Green’s function of the contact (source/drain). For the semi-infinite metallic
contacts, g <sup>S</sup> S/D is phenomenologically calculated <sup>37</sup> assuming constant density of states
(DOSM38) (,0.008 eV21) of metal and a contact-coupling coefficient (a), in range of 0

to 1, that describes the quality of contact with channel. The contact self-energy for
metallic contacts therefore becomes

Self -Energy (Metal) : ½SS=D�~{iD

~{ipa <sup>2</sup> trace(tS=DtHS=D <sup>)DOSM½ISide�</sup>

ð12Þ

where, ISide is identity matrix of size, number of layer 3 basis size, between metallic
contact and TI channel. For accurate modeling, an ab-initio computation is required to
determine the hopping energy between Bi2Se3 and specific contact material.
Nonetheless, as we show in the discussion section, the conclusions are is not sensitive to
the exact value.

For ferromagnetic metal contact, D was magnetized by modifying the coupling for
spin-up (down) by a factor of 1 1 (2)CM/100, where CM is contact magnetization (in
percent). Then depending on the magnetization vector, the unitary transformation <sup>39</sup>

was applied on D. Here, we would emphasize that coupling parameters for metallic
contacts describe the transport only phenomenologically <sup>37</sup> to help us to understand
the underlying physics and neglects more complex effects like exchange interaction at
the interface. The charge correction from Poisson equation has also been neglected
because of the low-field condition <sup>20</sup> (near-equilibrium transport).

Results and Discussion
Firstly, we investigate a FM-TI-exTI system. Figure 2(a) shows the
effect of quality of FM source contact-coupling (a) with the channel
on the drain current and its spin-polarization (SP). ‘a’ is a material
dependent parameter as each material (even NM) will have different
coupling strength with the TI. With the improvement in the quality
of the contact-coupling, metallic reservoirs can more easily inject
electrons into the channel. This enhances the current through all
layers. On the basis of Fig. 2(a), it is stressed that absolute value of
a does not affect the conclusions qualitatively because the results only
get scaled with the actual value of current. Therefore, for the subsequent discussion, we select the value of 0.6 for a i.e. a moderate
quality contact. It is also observed that despite of electron injection
from 100% (2y) magnetized source (MS), drain current is only
,34.7% spin-polarized because of non-zero current through subsurface layers. We would also like to note that this phenomenon
cannot be captured in the usual 2 3 2 Dirac-model used for modeling
topological surface which normally overestimates this effect <sup>40</sup> . Since,
only the normal mode (ky 5 0) for electron energy very close to
Dirac-point has complete 2y spin orientation (see Fig. 1(a)), for
higher energies the spin-vector starts to go out-of-plane <sup>41</sup> and for
larger transverse modes (ky ? 0) spin-vector has 6x-spin component, together they result in low SP. Furthermore, SP is independent of
actual magnitude of current injection in this system because contact
is uniformly coupled to all layers of the slab and hence, scales the
current through each layer proportionally. A real contact, which
probably can be simulated via ab-initio models, may slightly differ
in this aspect because the contact coupling is material dependent and
spin-injection efficiency into the channel depends on the contactchannel interface.

Next, Fig. 2(b) illustrates that SP scales directly with degree of
source magnetization (MS), where 0% MS corresponds to Normal
Metal. Therefore, a contact material with larger magnetization at
source end should result in higher SP at the drain end A small

Figure 2 | FM-TI-exTI system. Spin-Polarization (SP) at Drain end and
current through slab over a range of (a) contact-coupling strength (a) for
100% ‘2y’ source magnetization (MS); (b) ‘2y’ MS for a 5 0.6. From here
on only a 5 0.6 is considered for moderate quality of contact. Current
distribution across slab layers for (c) 100% and (d) 40% MS along 6y axis.
Note that current chiefly flows through layer with conducive spin-texture
(see Fig. 1(a)). Inset shows the symmetric current distribution for 0% MS.
SP at drain end and current for 100% MS along 6y axis over a range of
(e) Fermi-level (Ef) (f) slab thickness.

decrease in current is observed with increasing MS. It is due to mismatch between magnetized contact and TI channel, which increases
with increasing MS because of different coupling with 6y spins for
asymmetric injection. It is captured in the contact self-energy. Since,
MS and a are material dependent parameter, although the qualitative
effect is same, the observed magnitudes of drain current and SP will
depend on the choice of FM (e.g. Fe or Co <sup>9</sup> ).

Figure 2(c, d) next present a clear evidence of spin-momentum
locking on the topological surface. For 2y spin injection, current
flows chiefly on the top-surface and on bottom-surface for 1y electron spin in conformance with spin-texture (see Fig. 1(a)). In
Fig. 2(c) (100% MS), note that there is a finite current through other
layers, even on opposite surface. This can chiefly be attributed to two
reasons. Firstly, the current in the channel is not 100% spin-polarized
and transport has modes other than ky 5 0. Secondly, topological
behavior of the system forces the current to flow on the surfaces.
Only very near to the source contact current is verified to be least on
the opposite surface. Therefore, there is a small transition region after
source-contact where current redistributes across layers, as elaborated later in the discussion, with conservation in y-z plane.
Accordingly, going from full (100%) magnetization to weaker source
magnetization of 40% in Fig. 2(d) the current becomes less antisymmetric about middle layers. Subsequently, the effect of Fermilevel is shown in Fig. 2(e). For ballistic operation at 0 K in surface
metallic bands of TI, the current through the device increases with
increase in Fermi-level (Ef) and channel-bias (VDS). For the operation in surface bands the increase in surface current is much more
compared to sub-surface layers for both Ef and VDS, unlike the effect
of scaling contact coupling a Higher current through surface results

Figure 3 | FM-TI-exTI system. (a) Experimentally observed trend for
signature of spin-momentum locking on Bi2Se3, taken from Fig. 5(c) of
Ref. 9 (Adapted by permission from Macmillan Publishers Ltd: [Nature
Nanotechnology] (9), copyright (2014)). (b) Computed spin-polarization
at drain end across temperature (without phonons) for 1y (100%) MS
contact. 2y Magnetization produces same values but with opposite
polarity. (c, d) Current distribution across slab layers for ballistic transport
at 0 K ( | SPDrain |, 34.7%) and 300 K ( | SPDrain |, 0.43%) and with
acoustic scattering at 300 K ( | SPDrain |, 0.32%) for 100% source
magnetization (MS) along (a) ‘1y’ and (b) ‘2y’ axis.

in higher SP at the drain end (see Fig. 5(a) of Ref. 9) but of opposite
polarity for 6y spin injection. Note that if the Ef is eventually moved
close to or in the bulk bands, then inspite of higher current, SP may
reduce because of unpolarized bulk bands. Next, as illustrated in
Fig. 2(f), the increase in slab thickness nevertheless reduces the fraction of current flowing through surface layer inspite of increase in
total current due to current flow through more layers. This results in
decreasing spin-polarization at the drain end with increasing slab
thickness (see Fig. 5(b) of Ref. 9).

Figure 5 | (a) Transmission distribution at equilibrium (VDS 5 0 V) for
normal metal contacts at both ends of the TI channel. Note that the
confinement along the transport direction quantizes the bands which
cannot be captured in energy-dispersion along transport direction
(x-axis). Current distribution across slab layers for (b) three different
combinations with normal metal contact, (c) with anti-phase
ferromagnetic contacts, and (d) in-phase ferromagnetic contacts.

The analysis of effect of temperature reproduces the experimental
data in Fig. 3(a, b) which showed that the signature of spinmomentum locking rapidly erodes with increasing temperature.
Evaluation at room-temperature is thereafter illustrated in Fig. 3
(c, d) for both 6y spin injection. Although the ballistic current
through surface layer increases from 0 K to 300 K, the spread in
Fermi-distribution drives the electron energies into higher energy
states which are unpolarized <sup>20</sup> . This causes the degradation of the
SP at drain end from 34.7% to mere 0.43%. Both current and SP
further degrade when electron-phonon interactions are further taken
into account. Besides phonon scattering, note that at higher tempera

Figure 4 | Current distribution along transport direction for (2y) 100% MD FM Drain for normal mode (ky 5 0). Source contact-type is stated over
each column Operating condition is as follows: (a c) E 5 m f 5 0 f 5 1; (d f) E 5 m f 5 1 f 5 0 Refer text for symbols

tures the thermal effect on the magnetization of the ferromagnetic
leads in an experiment may also result in the suppression of the spin
polarization. However, since the ferromagnetic leads are quite large
(60–80 mm) there will be minimal superparamagnetic effect. Hence,
significant thermal effect on its magnetization would occur only close
to its Curie temperature, which at above 1000 K for iron or cobalt is
much higher than the temperature considered in our simulation.
It would thus not be critical to the main conclusions drawn in
this work, which focuses more on near zero Kelvin operation.
Consequently, it is important to execute such experiments at low
temperature (see Fig. 5 (c, d) of Ref. 9 where measurement signal
was lost by 125 K) in order to suppress the phonon scattering and get
reasonably spin-polarized current.

We have construed the recent experimental results for the electrical transport that observed spin-polarization resulting from spinmomentum locking on the surfaces of 3D-TI. This, however, pertains
to a long-channel device and as stated earlier there could be more
exciting phenomena at smaller channel lengths where physics at or
due to contacts becomes more significant. Before discussing our
observations for such devices, we briefly digress to investigate the
effect of contacts more closely to develop a better understanding of
underlying mechanism. In the subsequent discussion, fS (fD) is
Fermi-distribution at source (drain), E is the energy grid for electron
injection for energies from mD to mS for positive channel bias VDS. At
0 K, fS(E 5 mS) 5 0.5 otherwise it is 1 whereas fD(E 5 mD) 5 0.5
otherwise it is 0 on the energy grid. Therefore, drain contact injects
charge (in-scattering) into the channel (see eq. (7)) only for E 5 mD.

Consider the simplest case of only one energy point (see Fig. 4) for
normal mode (ky 5 0) electron transport for (2y) 100% MD FM
drain contact, because for normal mode spin-vector exactly aligns
with in-plane contact magnetization, and three configurations of
source contact i.e. exTI (extended-TI) and (6y) 100% MS. Firstly,
to focus on the drain we set E 5 mD, fS(E) 5 0 (note that it is normally
1 for transport at 0 K), fD(E) 5 1 in Fig. 4(a–c). It is observed that for
exTI source, there is roughly no top layer current, whereas for 6y MS
there is significant positive current through the top layer. For exTI
configuration, there is a negative current (current direction opposite
to voltage bias) in bottom layer in Fig. 4(a) because for injection from
drain i.e. in 2kx direction the bottom surface has the appropriate
spin-momentum state (see Fig. 1(a)). A similar isolated injection
from source (2y MS) results in entire current on top surface (not
shown). For 1y FM source, magnitude of top and bottom layer
current is same but it shows the opposite polarity. For 2y FM source,
although magnitude of current simultaneously increases through
both layers, there is more current through bottom layer. This indicates that magnetic contacts are causing some reflection in the current. For 2y source, since source-contact has same phase as drain,
the transmission through contact is higher and hence, there is weaker
reflection compared to 1y FM source configuration. Close examination reveals that even at the exTI source-contact there is a very small
reflection current. Note that in Fig. 4(a–c) the injected spin from the
drain is constant and only the reflection gets modified with sourcecontact type. Next, to focus on the source, we set E 5 mS, fS(E) 5 1,
fD(E) 5 0 (note that it is normally 0.5 for transport at 0 K) in
Fig. 4(d–f) for injection from the source contact. Before examining
the current distribution, we comment on two subtle observations
that are evident from the plots: (a) the cyclic trend (oscillations)
for surface layer current along transport direction due to the
reflection from the drain contact; (b) the wavelength of oscillation
is a function of electron energy (compare the two rows of sub-plots in
Fig. 4) but nearly independent of channel length (verified by simulating from devices of various lengths from 30 nm to 120 nm), with
lower energy electrons having longer wavelength as expected from
De Broglie’s relation. The amplitude of the oscillation is a function of
degree of in-plane magnetization along y-axis and the percentage of
magnetization (strength) (verified by simulating for entire range of

polar and azimuthal angles). Now by appraising current distribution
in Fig. 4(d–f), it is observed that for 1y FM source positive current
flows through the bottom layer (see from spin-texture in Fig. 1(a)
that forward mode of bottom layer supports 1y spin) and negative
reflection current through the top layer. The magnitude of current is
same although of opposite polarity. Here, we would note that this is
equal only for ky 5 0 mode (the simplest case under consideration).
For 2y FM source, positive current flows through the top layer and
negative reflection current through the bottom layer. The magnitude
of top current is greater than that of bottom (measured via cursor in
matlab, this may not be clearly visible from the figure) again because
of better phase matching at drain end which gives relatively weaker
reflection current and higher transmission. Nevertheless, for exTI
source, it is observed that there is positive current through bottom
layer and close to zero current on top layer. Spin polarization shows
that this current has 1y spin. This is most counter-intuitive because
it would be expected from spin-texture in Fig. 1(a) that 2y drain
spin-polarization matches with the forward moving state of top surface, therefore suggesting that current should flow through the top
layer. Even a single energy level model with spin (2 3 2 matrix) <sup>39</sup> with
‘bup’ and ‘bdown’ coupling for up and down spin respectively would
imply that we have source coupling to both 1y and 2y spins (exTI
source) whereas for the drain the coupling should only be finite for
2y spin, and therefore 2y spin should flow through top layer.
Conversely, our calculation results indicate otherwise. This can be
explained on the basis of transmission and reflection concept as
follows. Consider that source injects ‘I1’ mA/mm of 2y current on
the top layer and ‘I0’ mA/mm of 1y current on the bottom layer (in
forward) direction. Because drain is 2y magnetized, entire current
on bottom layer is reflected through the top layer i.e. top layer has a
reflection current of ‘2I0’ mA/mm. The current injected in the top
layer, sees same phase (magnetization direction) in drain and has
non-zero transmission (lower reflection). Therefore, ‘2(I1 2 I2)’ mA/mm
current is reflected through the bottom layer. As a result, net current on
the top layer is ‘I1 2 I0’ and on bottom it is ‘I0 2 I1 1 I2’. Since,
TI contact injects roughly equal ‘I0’ and ‘I1’, we have nearly zero
current on the top layer and ‘1I2’ mA/mm on the bottom layer. In
general:

-
T(zy)IBottom{Surf ð13Þ

Top Layer Current : T({y)ITop{Surf {R(zy) T(zy)IBottom{Surf

-
Bottom Layer Current : T(zy)IBottom{Surf {R({y) T({y)ITop{Surf ð14Þ

where, T (R) is transmission (reflection), in range of 0 to 1, of a
particular spin from source to channel and ITop(Bottom)_Surf is surface
current for TI system with exTI contacts (approximately without any
reflection because some reflection will still be there caused by mode
mismatch in non-equilibrium condition). Here, T and R are function
of (i) strength of contact magnetization (for instance 40% or 100%),
(ii) azimuthal angle, (iii) polar angles of each contact (i.e. type of
magnetization), (iv) electron injection energy Ei, and (v) ky mode.
For (iv) and (v) the TI spin-texture and thus ITop(Bottom)_Surf depend
on them. Also note that although eq. (13–14) only consider the top
and bottom layer, the surface current can be affected by transmission
and reflection for all sub-surface layers and therefore the expression
is just for an empirical understanding of the underlying physics and
lays the foundation for subsequent discussion. The exact modeling of
eq. (13–14) is beyond the scope of current work.

We now transition to short channel devices for which neither of
the contacts can be modeled as exTI. Figure 5(a) illustrates transmission spectrum over finely discretized grid for energy and transverse
modes (ky) for device in equilibrium condition. Unlike infinite channel length (contacts modeled as semi-infinite extended channel), the
confinement along the transport direction (x-axis) quantizes the
bands, and is therefore highly subjective to channel length, which
cannot be captured in energy-dispersion along transport direction
because energy dispersion calculation presumes plane wave pro

pagation along the corresponding wave-vector axis. This is, however,
revealed clearly in transmission spectrum simulated over all relevant
transverse and energy modes. The current depends on all five criterions discussed above for eq. (13–14) for both contacts. The transverse
mode (ky) which contributes most for a given energy point on grid
will depend on the choice of that E point. For high energies it will in
principle be further from ky 5 0 centre-point on ky grid, unless it is
through sub-band edge at ky 5 0, as illustrated from distribution of
red-yellow spots in transmission spectrum. Furthermore, this can
also be understood in terms of various cross-sectional energy contours from the conical frustum of energy-dispersion curve of surface
conduction band and analyzing the permissible kx and ky modes in
non-equilibrium. Note that at equilibrium it should be exactly at the
intersection of chosen ky mode and energy-point on the sub-band
illustrated in Fig. 5(a). Furthermore, as advanced later in discussion
for Fig. 5(d), the distribution of dominant ky modes leads to some
non-trivial counter-intuitive results because the overall result is governed by spin-momentum locking at these dominant modes instead
of the normal mode (ky 5 0) which was discussed above. Fig. 5(b)
illustrates the effect of normal metal contact. If both contacts are
normal-metal (red squares) indicating that we have symmetrical
distribution of current about the middle layer, the most of which
flows on surface layers as expected from topological properties of the
3D-TI. This distribution is same as that obtained by modeling both
contacts as exTI, only the magnitude is different because of the
difference in magnitude of contact coupling. For (1y) 100% FM
source contact (blue circles) positive current flows through the bottom surface, in accordance with spin-texture, however, there is
strong reflection current in top layer because backward moving state
of top layer has 1y spin-vector locked to 2kx momentum on an
average. Here we stress that net current along y-z plane is conserved
throughout transport direction and total current is indeed positive.
For NM source with (1y) 100% FM drain contacts (black triangles),
the positive current flows through top layer and negative reflection
through the bottom. This case is exactly opposite of the one discussed
for Fig. 4(d) and so is the current distribution across layers. Fig. 5(c)
then considers both contacts to be ferromagnetic but anti-phase with
each other. Similarly, the current distribution is as expected from the
spin-texture for each surface but with larger magnitudes because the
forward moving state of one surface exactly matches with the backward moving state of opposite surface. The same phase of ferromagnetic contacts is considered in Fig. 5(d) (see supplementary for the
effect of slab thickness on current distribution for anti-phase and inphase contact configuration). 6x or 6z spin-injection can be equally
resolved along 6y -axis and hence, we get the same current distribution as for NM or exTI source and drain contacts. Therefore,
magnetization orientations along 6x or 6z for both or either contact
is not expected to show any spin-momentum locking signature in
transport experiments (see Fig. 3(e, f) of Ref. 9). For 6y in-phase
magnetization, which should see small (high) reflection (transmission) at drain, we have the counter-intuitive results as pointed out
earlier. If normal mode had dominated the transport, then as evident
from Fig. 4(c, f), the largest fraction of positive current would have
flown through the top layer and a negative current through the
bottom surface. However, the spin-dependent transmission and
reflection at ky modes away from ky 5 0 point dominates the transport resulting in more current through top (bottom) surface for 1y
(2y) magnetization and a sizeable positive current through respective opposite surfaces.

To get deeper physical insight into this phenomenon by developing step-by-step understanding, Fig. 6 illustrates the energy
resolved information for important quantum transport parameters
for three sample contact configuration (NM contacts, FM contacts
in in-phase and anti-phase 100% magnetization along y-axis) as
follows: current (I) (Fig. 6(a)) as a function of slab layers (spatial
distribution along z axis) Transmission (TE) (Fig 6(b c)) and

spin-polarization of the current (SP) for top (Fig. 6(d) and bottom
surface (Fig. 6(e)) as a function of ky modes. Firstly, current distribution in Fig. 6(a) affirms the topological behaviour of the system by
illustrating that the current is mainly flowing through top and bottom surface layers but through certain energy-eigenmodes. As illustrated by the red-yellow color in Fig. 6(a1), for NM contacts,
the current flows equally through both surface layers, and subsurface layers (light-blue color) have lower current contributions.
Subsequently, Fig. 6(a2) and (a3) illustrate energy resolved current
flow for FM contacts, where red and blue color, respectively, show
strong positive and negative values. The respective energy integrated
values result in trends observed in Fig. 5. Specifically, notice the
energy dependent selection of layers in Fig. 6(a2) which corresponds
to the exotic behavior observed in Fig. 5(d) for in-phase (2y) FM
contacts. To demystify this observation, we now break down the
transport parameters of eq. (3–11). Next, the transmission in
Fig. 6(b) shows that the metallic source and drain contacts confine
the channel to induce the discretization of bands and certain energy
modes have higher contribution (see peaks) to the transport. We
note that this band-quantization is not caused by the confinement
along z-axis. Only z-axis termination (periodic width with semiinfinite contacts) results in Dirac-surface bands whose transmission
and DOS are linear function of energy as shown in Refs. 20, 28.
Figure 6(b) is further decomposed with respect to ky modes in
Fig. 6(c). By comparing Fig. 6(c) with peaks of 6(b) we can determine the dominant ky modes (within square box) at dominant
energy-modes. Fig. 6(c) also shows that the transmission is a function of contacts’ (source and drain together) magnetization (compare color-bars and golden spots among three systems within the
square box). This analysis which explains current distribution along
energy-axis in Fig. 6(a) is insufficient to explain surface layer selection for different contact configuration. Therefore, Fig. 6(d, e) next
examine the spin-polarization of current at each energy and kymode through top and bottom surface layer, with dominant modes
enclosed in square box. For NM contacts in Fig. 6(d1) and (e1)
observe that 2y (1y) spins are flowing through top (bottom) surface as expected from spin-texture, but the values integrated over
entire y-z plane would result in zero polarization for the net current.
For 2y FM source in Fig. 6(d2, d3, e2 and e3), the spin-polarization
of current flowing through each surface is strongly influenced by
drain’s magnetization and can in fact be of opposite polarity for
certain ky modes (positive (red) in Fig. 6(d2 and e2) while nearly
zeros but negative (light bluish-green) for Fig. 6(d3 and e3)). In
addition to FM contacts, this is probably due to combined
effect of confinement induced band-quantization and the nonequilibrium condition which breaks the symmetry along xdirection. The y-spin component is opposite in adjacent quadrants
in k-space for a given ky and therefore, spins for projections of Dirac
bands on ky-axis in non-equilibrium, which may be larger for certain
modes than others, may result in either of the polarities. Although a
further study, for instance ab-initio modeling with FM contacts,
might be needed to give the detailed understanding of this kind of
phenomenon, our investigation in this work consequently reveals
that the net solution for spin-polarization at least depends on magnetization axis of source and drain, sub-band quantization, ky
and Ei, and hence strongly influences the Transmission (T) and
Reflection (R) coefficients in eq. (13–14). Therefore, from microscopic dissemination of quantum information, we note that for ferromagnetic contacts the current distribution is highly subjective to
the choice of Fermi-level, bias (determines electron injection-energy)
and contact magnetization, which may result in exotic observations
for current distribution across layers of 3D-TI. Next, we observe that
negative top surface current can be driven for certain configurations
of metallic contacts (see Fig. 5(b, c)) for short channel devices.
Therefore, the experimental observation of negative resistance for
surface transport on thin slabs of potential candidates for 3D TI

Figure 6 | Energy resolved microscopic examination of current (a), Transmission (b), ky-resolved Transmission spectrum (c), spin-polarization of
current through top (d) and bottom surface (e) layers for three sample cases (as stated over each column (1)–(3)) taken from Fig. 5. Zoom-in plots are
illustrated instead of showing full-information to emphasize on the key features. Square box are drawn to draw attention to dominant modes (see text).
Specifically, observe the confinement induced band quantization which limits the current through certain dominant energy and ky modes in (c). Also note
that the current through surface layers is symmetrical about middle of the slab for NM contacts, but is spin-dependent for FM contacts. The net current
integrated over all layers is always positive in above cases. Spin-polarization of certain dominant energy and ky modes in (d) and (e) decide the selection of
topological surface for transport illustrated in (a). Note that NM contacts strictly comply with spin-texture diagram in Fig. 1. In contrast, for FM contacts
the net solution atleast depends on magnetization axis of source and drain, sub-band quantization, transverse mode and injection energy.

can serve as a simple way to verify spin-momentum locking and
hence validate the existence of topological insulator.

Finally, we would like to note that presence of helical surface state
is necessary but not the sufficient condition for detecting a TI because
the count of Dirac cones must also be unraveled. More specifically,
Z2 class <sup>2</sup> of TI has odd number of Dirac cones whereas another class
called topological crystalline insulator (TCI) <sup>42</sup> has even number <sup>43,44</sup>

of Dirac cones with same helicity for a given surface. Since only
Bi2Se3 has been studied in this work, which belongs to Z2 class, we
could be right in suggesting our method for the electrical detection of
the topological insulators for this system only. Hence, as part of
future work we would suggest to further mature this methodology
for conclusively predicting topological insulators via electrical
detection

Conclusion
In summary, we address the issue of influence of different contact
types and configurations on current flow through thin Bi2Se3 3D-TI.
Specifically, we examined the current distribution across layers as it
flows through the device for the extended-channel, normal metal
and ferromagnetic metal contacts. Our model is shown to explain
the recent experimental work <sup>9</sup> on electrical detection of spinmomentum locking on topological surface. For experimental observations, it is suggested to perform the experiment on thin samples at
low temperatures with magnetic contacts of high-polarization. We
also show that for short channel devices the spin-dependent transmission and reflection at the contacts can result in observations
seemingly counter-intuitive from simple spin-texture understanding
of the topological surfaces. For the anti-phase magnetization between
terminal contacts, along the y-axis, negative surface current can be
generated which may provide a simple mechanism to validate topological insulators via quantum transport experiments by observing
negative surface resistance. Recently various optical <sup>45,46</sup>, electrical <sup>47</sup>

and magnetic <sup>48</sup> techniques have been established for selectively probing edge and surface transport of materials and may help in validating the predictions made in this work. Furthermore, for the in-phase
configuration, it is shown that confinement and hence induced
quantization of energy and momentum modes may result in current
distribution antagonistic to trend expected from general understanding of spin-texture. The comprehensive understanding of transport
in 3D-TI with ferromagnetic contacts should expedite the development of novel spintronic devices based on topological insulators.

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Acknowledgments
We thank Dr. B.T. Jonker for allowing us to reuse their experimental figure. We thank Dr.
Hsin Lin and Dr. Arun Bansil for useful discussions. The work at the National University of
Singapore was supported by MOE under Grant No. R263000689112 and
MOE2013-T2-2-125. Computations were performed on Landauer Cluster of
Computational Nanoelectronics and Nanodevice LAB and Graphene Research Centre
(GRC) Cluster at NUS. M.B.A.J. acknowledges the Singapore National Research
Foundation for support under NRF Award No. NRF-CRP9-2011-01.

Author contributions

G.G. performed the computations and wrote the manuscript. G.G., M.B.A.J. and G.L.
critically analyzed the data. All authors reviewed the manuscript. G.L. also managed the
overall execution of the project

Additional information
[Supplementary information accompanies this paper at http://www.nature.com/](http://www.nature.com/scientificreports)
[scientificreports](http://www.nature.com/scientificreports)

Competing financial interests: The authors declare no competing financial interests.

How to cite this article: Gupta, G., Jalil, M.B.A. & Liang, G. Contact Effects in thin
3D-Topological Insulators: How does the current flow? Sci. Rep. 5, 9479; DOI:10.1038/
srep09479 (2015).

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