We study the transport properties of the Dirac fermions with Fermi velocity
$v_F$ on the surface of a topological insulator across a ferromagnetic strip
providing an exchange field ${\mathcal J}$ over a region of width $d$. We show
that the conductance of such a junction changes from oscillatory to a
monotonically decreasing function of $d$ beyond a critical ${\mathcal J}$. This
leads to the possible realization of a magnetic switch using these junctions.
We also study the conductance of these Dirac fermions across a potential
barrier of width $d$ and potential $V_0$ in the presence of such a
ferromagnetic strip and show that beyond a critical ${\mathcal J}$, the
criteria of conductance maxima changes from $\chi= e V_0 d/\hbar v_F = n \pi$
to $\chi= (n+1/2)\pi$ for integer $n$. We point out that these novel phenomena
have no analogs in graphene and suggest experiments which can probe them.Comment: v1 4 pages 5 fig

Mondal, S.

Sen, D.

Sengupta, K.

Shankar, R.

English

arXiv

We study the transport properties of the Dirac fermions with a Fermi   velocity v(F) on the surface of a topological insulator across a ferromagnetic strip providing an exchange field J over a region of width d. We show that the conductance of such a junction, in the clean limit and at low temperature, changes from oscillatory to a monotonically decreasing function of d beyond a critical J. This leads to the possible realization of a magnetic switch using these junctions. We also study the conductance of these Dirac fermions across a potential barrier of width d and potential V-0 in the presence of such a ferromagnetic strip and show that beyond a critical J, the criteria of conductance maxima changes from chi = eV(0)d/(h) over barv(F) = n pi to chi = (n + 1/2)pi for integer n. We point out that these novel phenomena have no analogs in graphene and suggest experiments which can probe them

Mondal, S

Sen, D

Sengupta, K

Shankar, R

Open Access Repository of IISc Research Publications

Tuning the Conductance of Dirac Fermions on the Surface of a  Topological Insulator

S. Mondal

D. Sen

K. Sengupta

R. Shankar

Crossref

Tuning the Conductance of Dirac Fermions on the Surface of a Topological Insulator

We study the transport properties of the Dirac fermions with a Fermi velocity vF on the surface of a topological insulator across a ferromagnetic strip providing an exchange field J over a region of width d. We show that the conductance of such a junction, in the clean limit and at low temperature, changes from oscillatory to a monotonically decreasing function of d beyond a critical J. This leads to the possible realization of a magnetic switch using these junctions. We also study the conductance of these Dirac fermions across a potential barrier of width d and potential V0 in the presence of such a ferromagnetic strip and show that beyond a critical J, the criteria of conductance maxima changes from X=eV0d/&#x045B;vF=n&#960; to X=(n+&#189;)&#960; for integer n. We point out that these novel phenomena have no analogs in graphene and suggest experiments which can probe them

Publications of the IAS Fellows

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Tuning the conductance of Dirac fermions on the surface of a topological insulator
S. Mondal,1 D. Sen,2 K. Sengupta,1 and R. Shankar3
1Theoretical Physics Division, Indian Association for the Cultivation of Sciences, Kolkata 700 032, India
2 Center for High Energy Physics, Indian Institute of Science, Bangalore 560 012, India
3 The Institute of Mathematical Sciences, C.I.T Campus, Chennai 600 113, India
(Dated: August 14, 2009)
We study the transport properties of the Dirac fermions with Fermi velocity vF on the surface of a
topological insulator across a ferromagnetic strip providing an exchange field J over a region of width
d. We show that the conductance of such a junction changes from oscillatory to a monotonically
decreasing function of d beyond a critical J . This leads to the possible realization of a magnetic
switch using these junctions. We also study the conductance of these Dirac fermions across a
potential barrier of width d and potential V0 in the presence of such a ferromagnetic strip and show
that beyond a critical J , the criteria of conductance maxima changes from χ = eV0d/~vF = npi to
χ = (n+1/2)pi for integer n. We point out that these novel phenomena have no analogs in graphene
and suggest experiments which can probe them.
PACS numbers: 71.10.Pm, 73.20.-r
Topological insulators in both two- and three- dimen-
sions (2D and 3D) have attracted a lot of theoretical and
experimental attention in recent years [1, 2, 3, 4]. It has
been shown in Ref. [4] that such 3D insulators can be
completely characterized by four integers ν0 and ν1,2,3.
The former specifies the class of topological insulators to
be strong (ν0 = 1) or weak (ν0 = 0), while the latter in-
tegers characterize the time-reversal invariant momenta
of the system given by ~M0 = (ν1~b1, ν2~b2, ν3~b3)/2, where
~b1,2,3 are the reciprocal lattice vectors. The topological
features of strong topological insulators (STI) are robust
against the presence of time-reversal invariant perturba-
tions such as disorder or lattice imperfections. It has been
theoretically predicted [1, 4] and experimentally verified
[2] that the surface of a STI has an odd number of Dirac
cones whose positions are determined by the projection
of ~M0 on to the surface Brillouin zone. The position
and number of these cones depend on both the nature of
the surface concerned and the integers ν1,2,3. For several
compounds such as HgTe and Bi2Se3, specific surfaces
with a single Dirac cone near the Γ point of the 2D Bril-
louin zone have been found [2, 5]. Such a Dirac cone is
described by the Hamiltonian
H =
∫
dkxdky
(2π)2
ψ†(~k) (~vF~σ · ~k − µI) ψ(~k), (1)
where ~σ(I) denotes the Pauli (identity) matrices in spin
space, ψ = (ψ↑, ψ↓)
T is the annihilation operator for the
Dirac spinor, vF is the Fermi velocity, and µ is the chemi-
cal potential [6]. Recently, several novel features of these
surface Dirac electrons such as the existence of Majorana
fermions in the presence of a magnet-superconductor in-
terface on the surface [6, 7, 8], generation of a time-
reversal symmetric px + ipy superconducting state via
proximity to a s-wave superconductor [6], anomalous
magnetoresistance of ferromagnet-ferromagnet junctions
[9] and novel spin textures with chiral properties [10] have
been studied in detail.
FIG. 1: Proposed experimental setups: a) Left panel: The
ferromagnetic film extends over region II of width d provid-
ing an exchange field in this region. b) Right panel: The
film extends over region III while the region II has a barrier
characterized by a voltage V0. V and I denote the bias volt-
age and current across the junction respectively. See text for
details.
In this letter, we study the transport properties of
these surface Dirac fermions in two experimentally re-
alizable situations shown in Fig. 1. The first study con-
cerns their transport across a region with a width d with
a proximity-induced exchange field J arising from the
magnetization ~m = m0yˆ of a proximate ferromagnetic
film as shown in the left panel of Fig. 1. We demon-
strate that the tunneling conductance G of these Dirac
fermions through such a junction can either be an oscilla-
tory or a monotonically decaying function of the junction
width d. One can interpolate between these two quali-
tatively different behaviors of G by changing m0 (and
thus J ) by an applied in-plane magnetic field leading to
the possible use of this junction as a magnetic switch.
The second study concerns the transport properties of
Dirac fermions across a barrier characterized by a width
d and a potential V0 in region II with a magnetic film
proximate to region III as shown in the right panel of
Fig. 1. We note that it is well known from the context
of Dirac fermions in graphene [11] that such a junction,
2in the absence of the induced magnetization, exhibits
transmission resonances with maxima of transmission at
χ = eV0d/~vF = nπ, where n is an integer. Here we
show that beyond a critical strength of m0, the maxima
of the transmission shifts to χ = (n + 1/2)π. Upon fur-
ther increasing m0, one can reach a regime where the
conductance across the junctions vanishes. We stress
that the properties of Dirac fermions elucidated in both
these studies are a consequence of their spinor structure
in physical spin space, and thus have no analogs for ei-
ther conventional Schro¨dinger electrons in 2D or Dirac
electrons in graphene [12].
We begin with an analysis of the junction shown in the
left panel of Fig. 1. The Dirac fermions in region I and
III are described by the Hamiltonian in Eq. (1). Con-
sequently, the wave functions of these fermions moving
along ±x in these regions for a fixed transverse momen-
tum ky and energy ǫ can be written as
ψ±i = (1,±e±iα) ei(±kxx+kyy)/
√
2, (2)
where i takes values I and III, α = arcsin(~vF ky/|ǫ+ µ|)
and kx(ǫ) =
√
[(ǫ+ µ)/~vF ]2 − k2y. In region II, the
presence of the ferromagnetic strip with a magnetiza-
tion ~m0 = m0yˆ leads to the additional term Hinduced =∫
dxdy J θ(x)θ(d − x)ψ†(~x)σy ψ(~x), where J ∼ m0 is
the exchange field due to the presence of the strip [9],
and θ(x) denotes the Heaviside step function. Note
that Hinduced may be thought as a vector potential
term arising due to a fictitious magnetic field ~Bf =
(J /evF )[δ(x) − δ(d− x)]zˆ. This analogy shows that our
choice of the in-pane magnetization along yˆ is completely
general; all gauge invariant quantities such as transmis-
sion are independent of x-component of ~m0 in the present
geometry. For a given m0, the precise magnitude of J
depends on several factors such as the exchange coupling
of the film and can be tuned, for soft ferromagnetic films,
by an externally applied field [9]. The wave function for
the Dirac fermions in region II moving along ±x in the
presence of such an exchange field is given by
ψ±II = (1,±e±iβ) ei(±k
′
x
x+kyy)/
√
2, (3)
where β = arcsin(~vF (ky +M)/|ǫ+ µ|), M = J /(~vF ),
and k
′
x(ǫ) =
√
[(ǫ+ µ)/~vF ]2 − (ky +M)2. Note that
beyond a critical Mc = ±2|ǫ + µ|/(~vF ) (and hence a
critical Jc = ±2|ǫ+ µ|), k′x becomes imaginary for all ky
leading to spatially decaying modes in region II.
Let us now consider an electron incident on region II
from the left with a transverse momentum ky and energy
ǫ. Taking into account reflection and transmission pro-
cesses at x = 0 and x = d, the wave function of the elec-
tron can be written as ψI = ψ
+
I +rψ
−
I , ψ
II = pψ+II+qψ
−
II ,
and ψIII = tψ
+
III . Here r and t are the reflection and
transmission amplitudes and p (q) denotes the amplitude
of right (left) moving electrons in region II. Matching
FIG. 2: Plot of tunneling conductance G/G0 for a fixed V
and µ as a function of the effective width z = d|eV + µ|/~vF
for ~vFM/|eV + µ| = 0.3 (green dotted line), 0.7 (blue solid
line), 1.3 (black dash-dotted line) and 2.1 (red dashed line).
The value of the critical M is given by ~vFM/|eV + µ| = 2.
See text for details.
boundary conditions on ψI and ψII at x = 0 and ψII
and ψIII at x = d leads to
1 + r = p+ q, eiα − re−iα = peiβ − qe−iβ ,
teikxd = peik
′
x
d + qe−ik
′
x
d,
tei(kxd+α) = pei(k
′
x
d+β) − qe−i(k
′
x
d+β). (4)
Solving for t from Eq. (4), one finally obtains the con-
ductance G = dI/dV = (G0/2)
∫ pi/2
−pi/2
T cos(α)dα. Here
G0 = ρ(eV )we
2/(π~2vF ), ρ(eV ) = |(µ+eV )|/[2π(~vF )2]
is the density of states (DOS) of the Dirac fermions and
is a constant for µ≫ eV , w is the sample width, and the
transmission T = |t|2 is given by
T = cos2(α) cos2(β)/[cos2(k
′
xd) cos
2(α) cos2(β)
+ sin2(k
′
xd)(1− sin(α) sin(β))2]. (5)
Eq. (5) and the expression for G represent one of the
main results of this work. We note that for a given α, T
has an oscillatory (monotonically decaying) dependence
on d provided k
′
x is real (imaginary). Since k
′
x depends,
for a given α, on M , we find that one can switch from an
oscillatory to a monotonically decaying d dependence of
transmission in a given channel (labeled by ky or equiva-
lently α) by turning on a magnetic field which controlsm0
and hence M . Also since −1 ≤ sin(α) ≤ 1, we find that
beyond a critical M = Mc, the transmission in all chan-
nels exhibits a monotonically decaying dependence on d.
Consequently, for a thick enough junction one can tune G
at fixed V and µ from a finite value to nearly zero by tun-
ing M (i.e., m0) through Mc. Thus such a junction may
3FIG. 3: Plot G/G0 versus eV/µ for several representatives
values ~vFM/µ ranging from 3 (left-most black solid curve)
to 5 (right-most magenta dash-double dotted line) in steps of
0.5 The effective junction width z0 = 5 for all plots. The inset
shows a plot of eVc/µ versus ~vFM/µ. See text for details.
be used as a magnetic switch. These qualitatively differ-
ent behaviors of the junction conductance G forM below
and above Mc is demonstrated in Fig. 2 by plotting G as
a function of effective barrier width z = d|eV + µ|/~vF
for several representative values of ~vFM/|eV +µ|. Since
T and hence G depends on M through the dimensionless
parameter ~vFM/|eV + µ|, this effect can also be ob-
served by varying the applied voltage V for a fixed µ, d,
andM . In that case, for a reasonably large dimensionless
barrier thickness z0 = dµ/~vF , G/G0 becomes finite only
beyond a critical voltage |eVc + µ| = ~vFM/2 as shown
in Fig. 3 for several representative values of z0. This crit-
ical voltage Vc can be determined numerically by finding
the lowest voltage for which G/G0 exhibits a monotonic
decay as a function of z0. The plot of eVc/µ as a function
of ~vFM/µ, shown in inset of Fig. 3, demonstrates the
expected linear relationship between Vc and M . We note
such a magnetic field or applied bias voltage dependence
of the junction conductance necessitates that the Dirac
electrons represents spinors in physical spin space and is
therefore impossible to achieve in graphene [11].
Next, we analyze the junction shown in the right panel
of Fig. 1 where the region III below a ferromagnetic film
is separated from region I by a potential barrier in re-
gion II. Such a barrier can be applied by changing the
chemical region in region II either by a gate voltage V0
or via doping [5]. In the rest of this work, we will analyze
the problem in the thin barrier limit for which V0 → ∞
and d → 0, keeping the dimensionless barrier strength
χ = eV0d/(~vF ) finite. The wave function of the Dirac
fermions moving along ±x with a fixed momentum ky
FIG. 4: Plot of tunneling conductance G1/G0 versus the ef-
fective barrier strength χ and ~vFM for fixed applied voltage
V and chemical potential µ. G1 vanishes for |M | ≥ Mc =
2|eV + µ|/~vF .
and energy ǫ in this region is given by
ψ
′ ±
II = (1,±e±iγ) ei(±k
′′
x
x+kyy)/
√
2, (6)
where γ = arcsin(~vFky/|ǫ + eV0 + µ|) and k′′x(ǫ) =√
[(ǫ+ eV0 + µ)/~vF ]2 − k2y. The wave functions in re-
gion I and III are given by Eqs. (2) and (3) respectively:
ψ
′
I = ψI and ψ
′
III = ψII . Note that one can have a
propagating solution in region III only if |M | ≤ |Mc|.
The transmission problem for such a junction can be
solved by an procedure similar to the one outlined above
for the magnetic strip problem. For an electron ap-
proaching the barrier region from the left, we write down
forms of the wave function in the three regions I, II
and III: ψ
′
I = ψ
+
I + r1ψ
−
I , ψ
′
II = p1ψ
′ +
II + q1ψ
′ −
II , and
ψ
′
III = t1ψ
+
II . As outlined earlier, one can then match
boundary conditions at x = 0 and x = d, and obtain the
transmission coefficient T1 = |t1|2k′x/kx as
T1 = 2 cos(β) cos(α)/[1 + cos(β − α)
− cos2(χ){cos(β − α)− cos(β + α)}]. (7)
Note that in the absence of the ferromagnetic film over
region III, β = α, and T1 → T 01 = cos2(α)/[1 −
cos2(χ) sin2(α)]. The expression for T 01 , reproduced
here for the special case of M = 0, is well known
from analogous studies in the context of graphene, and
it exhibits both Klein paradox (T 01 = 1 for α = 0)
and transmission resonance (T 01 = 1 for χ = nπ)
[12]. When M 6= 0, we find that the transmission
for normal incidence (ky = 0) does become indepen-
dent of the barrier strength, but its magnitude deviates
from unity: T normal1 = 2
√
1− (~vFM/|eV + µ|)2/(1 +√
1− (~vFM/|eV + µ|)2). The value of T normal1 de-
creases monotonically from 1 for M = 0 to 0 for |M | =
|eV + µ|/(~vF ) and can thus be tuned by changing M
(or V ) for a fixed V (or M) and µ.
4FIG. 5: Top left panel: Plot of G1/G0 versus χ for
~vFM/|eV + µ| = 0.1 (black solid line), 0.7075 (red dashed
line), and 1.4 (blue dash-dotted line) for fixed V and µ. Top
right panel: Plot of G1/G0 versus χ at M = M
∗ showing
the period halving. Bottom left panel: Plot of χmax versus
~vFM/|eV + µ| showing the shift of conductance maxima.
Bottom right panel: Plot of ∆G1/G0 versus ~vFM/|eV + µ|
which crosses 0 for M < Mc at M = M
∗. The dotted line is
a guide to the eye.
The conductance of such a junction is given by G1 =
(G0/2)
∫ α2
−α1
T1 cos(α)dα, where α1,2 are determined from
the solution of cos(β) = 0 for a givenM (Eq. (3)). A plot
of G1 as a function of ~vFM/|eV + µ| and χ (for a fixed
eV and µ) is shown in Fig. 4. We find that the amplitude
of G1 decreases monotonically as a function of |M | reach-
ing 0 atM = Mc beyond which there are no propagating
modes in region III. Also, as we increase M , the conduc-
tance maxima shifts from χ = nπ to χ = (n + 1/2)π
beyond a fixed value of M∗(V ) ≃ ±c0|eV + µ|/(~vF ) as
shown in top left panel of Fig. 5. Numerically, we find
c0 = 0.7075. At M = M
∗, G1(χ = nπ) = G1(χ =
(n+1/2)π), leading to a period halving of G1(χ) from π
to π/2 . This is shown in top right panel of Fig. 5 where
G1(M = M
∗) is plotted as a function of χ. We note that
near M∗, the amplitude of oscillation of G1 as a function
of χ becomes very small so that G1 is almost indepen-
dent of χ. In the bottom left panel of Fig. 5, we plot
χ = χmax (the value of χ at which the first conductance
maxima occurs) as a function of ~vFM/|eV + µ| which
clearly demonstrates the shift. This is further highlighted
by plotting ∆G1 = G1(χ = 0)−G1(χ = π/2) as a func-
tion of ~vFM/|eV + µ| in the bottom right panel of Fig.
5. For M < Mc, ∆G1 crosses zero at M = M
∗ indicat-
ing the position of the above-mentioned period halving.
Thus we conclude that the position of the conductance
maxima depends crucially on ~vFM/|eV +µ| and can be
tuned by changing either M or V .
The experimental verification of our results would in-
volve preparation of junctions by depositing ferromag-
netic films on the surface of a topological insulator. For
the geometry shown in the left panel of Fig. 1, we propose
measurement of G as a function of m0 whose magnitude
and direction can be tuned by an externally applied in-
plane magnetic field for soft ferromagnetic films [9]. We
predict that depending on m0, G should demonstrate ei-
ther a monotonically decreasing or an oscillatory behav-
ior as a function of d. Another, probably more experi-
mentally convenient, way to realize this effect would be
to measure Vc of a junction of width d for several M and
confirm that Vc varies linearly with M with a slope of
~vF /(2e), provided µ and d remain fixed. For the geom-
etry depicted in the right panel of Fig. 1, one would, in
addition, need to create a barrier by tuning the chemical
potential of an intermediate thin region of the sample as
done earlier for graphene [11]. Here we propose measure-
ment of G1 as a function of V0 (or equivalently χ) for
several representative values of m0 and a fixed V . We
predict that the maxima of the tunneling conductance
would shift from χ = nπ to χ = (n + 1/2)π beyond a
critical m0 for a fixed V , or equivalently, below a critical
V , for a fixed m0.
In conclusion, we have studied the transport of Dirac
fermions on the surface of a topological insulator in the
presence of proximate ferromagnetic films in two experi-
mentally realizable geometries. Our study unravels novel
features of the junction conductances which have no ana-
log in either graphene or 2D Schro¨dinger electrons and
can be verified in realistic experimental setups.
[1] B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science
314, 1757 (2006).
[2] M. Koenig et al., Science 318, 766 (2007); D. Hsieh et
al., Nature 452, 970 (2008);
[3] C.L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801
(1995); ibid, Phys. Rev. Lett. 95, 146802 (2006).
[4] L. Fu, C.L. Kane, and E. J. Mele, Phys. Rev. Lett. 98,
106803 (2007); R. Roy, arXiv:cond-mat/0607531 (unpub-
lished); J. E. Moore and L. Balents, Phys. Rev. B 75,
121306 (2007).
[5] Y. Xia et al., Nature Phys. 5, 398 (2009); ibid,
arXiv:0907.3089 (unpublished).
[6] L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407
(2008).
[7] A. R. Akhmerov, J. Nilsson, and C. W. J. Beenakker,
Phys. Rev. Lett. 102, 216404 (2009).
[8] Y. Tanaka, T. Yokoyama, and N. Nagaosa,
arXiv:0907.2088 (unpublished).
[9] T. Yokoyama, Y. Tanaka, and N. Nagaosa,
arXiv:0907.2810 (unpublished).
[10] D. Hsieh et al., Science 323 919 (2009); D. Hsieh et al.,
arXiv:0904.1260 (unpublished).
[11] A. Castro Neto et al., Rev. Mod. Phys. 81, 109 (2009);
C. W. J. Beenakker, Rev. Mod. Phys. 80, 1337 (2008);
A. K. Geim, Science 324, 1530 (2009).
[12] M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Na-
ture Phys. 2, 620 (2006); C. W. J. Beenakker, Phys. Rev.
Lett. 97, 067007 (2006); S. Bhattacharjee and K. Sen-
gupta, Phys. Rev. Lett. 97, 217001 (2006).


Tuning the conductance of Dirac fermions on the surface of a topological insulator

Indian Academy of Sciences

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Tuning the conductance of Dirac fermions on the surface of a topological insulator
S. Mondal,1 D. Sen,2 K. Sengupta,1 and R. Shankar3
1Theoretical Physics Division, Indian Association for the Cultivation of Sciences, Kolkata 700 032, India
2 Center for High Energy Physics, Indian Institute of Science, Bangalore 560 012, India
3 The Institute of Mathematical Sciences, C.I.T Campus, Chennai 600 113, India
(Dated: August 14, 2009)
We study the transport properties of the Dirac fermions with Fermi velocity vF on the surface of a
topological insulator across a ferromagnetic strip providing an exchange field J over a region of width
d. We show that the conductance of such a junction changes from oscillatory to a monotonically
decreasing function of d beyond a critical J . This leads to the possible realization of a magnetic
switch using these junctions. We also study the conductance of these Dirac fermions across a
potential barrier of width d and potential V0 in the presence of such a ferromagnetic strip and show
that beyond a critical J , the criteria of conductance maxima changes from χ = eV0d/~vF = nπ to
χ = (n+1/2)π for integer n. We point out that these novel phenomena have no analogs in graphene
and suggest experiments which can probe them.
PACS numbers: 71.10.Pm, 73.20.-r
Topological insulators in both two- and three- dimen-
sions (2D and 3D) have attracted a lot of theoretical and
experimental attention in recent years [1, 2, 3, 4]. It has
been shown in Ref. [4] that such 3D insulators can be
completely characterized by four integers ν0 and ν1,2,3.
The former specifies the class of topological insulators to
be strong (ν0 = 1) or weak (ν0 = 0), while the latter in-
tegers characterize the time-reversal invariant momenta
of the system given by ~M0 = (ν1~b1, ν2~b2, ν3~b3)/2, where
~b1,2,3 are the reciprocal lattice vectors. The topological
features of strong topological insulators (STI) are robust
against the presence of time-reversal invariant perturba-
tions such as disorder or lattice imperfections. It has been
theoretically predicted [1, 4] and experimentally verified
[2] that the surface of a STI has an odd number of Dirac
cones whose positions are determined by the projection
of ~M0 on to the surface Brillouin zone. The position
and number of these cones depend on both the nature of
the surface concerned and the integers ν1,2,3. For several
compounds such as HgTe and Bi2Se3, specific surfaces
with a single Dirac cone near the Γ point of the 2D Bril-
louin zone have been found [2, 5]. Such a Dirac cone is
described by the Hamiltonian
H =
∫
dkxdky
(2π)2
ψ†(~k) (~vF~σ · ~k − µI) ψ(~k), (1)
where ~σ(I) denotes the Pauli (identity) matrices in spin
space, ψ = (ψ↑, ψ↓)
T is the annihilation operator for the
Dirac spinor, vF is the Fermi velocity, and µ is the chemi-
cal potential [6]. Recently, several novel features of these
surface Dirac electrons such as the existence of Majorana
fermions in the presence of a magnet-superconductor in-
terface on the surface [6, 7, 8], generation of a time-
reversal symmetric px + ipy superconducting state via
proximity to a s-wave superconductor [6], anomalous
magnetoresistance of ferromagnet-ferromagnet junctions
[9] and novel spin textures with chiral properties [10] have
been studied in detail.
FIG. 1: Proposed experimental setups: a) Left panel: The
ferromagnetic film extends over region II of width d provid-
ing an exchange field in this region. b) Right panel: The
film extends over region III while the region II has a barrier
characterized by a voltage V0. V and I denote the bias volt-
age and current across the junction respectively. See text for
details.
In this letter, we study the transport properties of
these surface Dirac fermions in two experimentally re-
alizable situations shown in Fig. 1. The first study con-
cerns their transport across a region with a width d with
a proximity-induced exchange field J arising from the
magnetization ~m = m0ŷ of a proximate ferromagnetic
film as shown in the left panel of Fig. 1. We demon-
strate that the tunneling conductance G of these Dirac
fermions through such a junction can either be an oscilla-
tory or a monotonically decaying function of the junction
width d. One can interpolate between these two quali-
tatively different behaviors of G by changing m0 (and
thus J ) by an applied in-plane magnetic field leading to
the possible use of this junction as a magnetic switch.
The second study concerns the transport properties of
Dirac fermions across a barrier characterized by a width
d and a potential V0 in region II with a magnetic film
proximate to region III as shown in the right panel of
Fig. 1. We note that it is well known from the context
of Dirac fermions in graphene [11] that such a junction,
2
in the absence of the induced magnetization, exhibits
transmission resonances with maxima of transmission at
χ = eV0d/~vF = nπ, where n is an integer. Here we
show that beyond a critical strength of m0, the maxima
of the transmission shifts to χ = (n + 1/2)π. Upon fur-
ther increasing m0, one can reach a regime where the
conductance across the junctions vanishes. We stress
that the properties of Dirac fermions elucidated in both
these studies are a consequence of their spinor structure
in physical spin space, and thus have no analogs for ei-
ther conventional Schrödinger electrons in 2D or Dirac
electrons in graphene [12].
We begin with an analysis of the junction shown in the
left panel of Fig. 1. The Dirac fermions in region I and
III are described by the Hamiltonian in Eq. (1). Con-
sequently, the wave functions of these fermions moving
along ±x in these regions for a fixed transverse momen-
tum ky and energy ǫ can be written as
ψ±i = (1,±e±iα) ei(±kxx+kyy)/
√
2, (2)
where i takes values I and III, α = arcsin(~vF ky/|ǫ+ µ|)
and kx(ǫ) =
√
[(ǫ+ µ)/~vF ]2 − k2y. In region II, the
presence of the ferromagnetic strip with a magnetiza-
tion ~m0 = m0ŷ leads to the additional term Hinduced =
∫
dxdy J θ(x)θ(d − x)ψ†(~x)σy ψ(~x), where J ∼ m0 is
the exchange field due to the presence of the strip [9],
and θ(x) denotes the Heaviside step function. Note
that Hinduced may be thought as a vector potential
term arising due to a fictitious magnetic field ~Bf =
(J /evF )[δ(x) − δ(d− x)]ẑ. This analogy shows that our
choice of the in-pane magnetization along ŷ is completely
general; all gauge invariant quantities such as transmis-
sion are independent of x-component of ~m0 in the present
geometry. For a given m0, the precise magnitude of J
depends on several factors such as the exchange coupling
of the film and can be tuned, for soft ferromagnetic films,
by an externally applied field [9]. The wave function for
the Dirac fermions in region II moving along ±x in the
presence of such an exchange field is given by
ψ±II = (1,±e±iβ) ei(±k
′
x
x+kyy)/
√
2, (3)
where β = arcsin(~vF (ky +M)/|ǫ+ µ|), M = J /(~vF ),
and k
′
x(ǫ) =
√
[(ǫ+ µ)/~vF ]2 − (ky +M)2. Note that
beyond a critical Mc = ±2|ǫ + µ|/(~vF ) (and hence a
critical Jc = ±2|ǫ+ µ|), k
′
x becomes imaginary for all ky
leading to spatially decaying modes in region II.
Let us now consider an electron incident on region II
from the left with a transverse momentum ky and energy
ǫ. Taking into account reflection and transmission pro-
cesses at x = 0 and x = d, the wave function of the elec-
tron can be written as ψI = ψ
+
I +rψ
−
I , ψ
II = pψ+II+qψ
−
II ,
and ψIII = tψ
+
III . Here r and t are the reflection and
transmission amplitudes and p (q) denotes the amplitude
of right (left) moving electrons in region II. Matching
FIG. 2: Plot of tunneling conductance G/G0 for a fixed V
and µ as a function of the effective width z = d|eV + µ|/~vF
for ~vF M/|eV + µ| = 0.3 (green dotted line), 0.7 (blue solid
line), 1.3 (black dash-dotted line) and 2.1 (red dashed line).
The value of the critical M is given by ~vF M/|eV + µ| = 2.
See text for details.
boundary conditions on ψI and ψII at x = 0 and ψII
and ψIII at x = d leads to
1 + r = p+ q, eiα − re−iα = peiβ − qe−iβ ,
teikxd = peik
′
x
d + qe−ik
′
x
d,
tei(kxd+α) = pei(k
′
x
d+β) − qe−i(k
′
x
d+β). (4)
Solving for t from Eq. (4), one finally obtains the con-
ductance G = dI/dV = (G0/2)
∫ π/2
−π/2
T cos(α)dα. Here
G0 = ρ(eV )we
2/(π~2vF ), ρ(eV ) = |(µ+eV )|/[2π(~vF )2]
is the density of states (DOS) of the Dirac fermions and
is a constant for µ≫ eV , w is the sample width, and the
transmission T = |t|2 is given by
T = cos2(α) cos2(β)/[cos2(k
′
xd) cos
2(α) cos2(β)
+ sin2(k
′
xd)(1 − sin(α) sin(β))2]. (5)
Eq. (5) and the expression for G represent one of the
main results of this work. We note that for a given α, T
has an oscillatory (monotonically decaying) dependence
on d provided k
′
x is real (imaginary). Since k
′
x depends,
for a given α, on M , we find that one can switch from an
oscillatory to a monotonically decaying d dependence of
transmission in a given channel (labeled by ky or equiva-
lently α) by turning on a magnetic field which controlsm0
and hence M . Also since −1 ≤ sin(α) ≤ 1, we find that
beyond a critical M = Mc, the transmission in all chan-
nels exhibits a monotonically decaying dependence on d.
Consequently, for a thick enough junction one can tune G
at fixed V and µ from a finite value to nearly zero by tun-
ing M (i.e., m0) through Mc. Thus such a junction may
3
FIG. 3: Plot G/G0 versus eV/µ for several representatives
values ~vF M/µ ranging from 3 (left-most black solid curve)
to 5 (right-most magenta dash-double dotted line) in steps of
0.5 The effective junction width z0 = 5 for all plots. The inset
shows a plot of eVc/µ versus ~vF M/µ. See text for details.
be used as a magnetic switch. These qualitatively differ-
ent behaviors of the junction conductance G for M below
and above Mc is demonstrated in Fig. 2 by plotting G as
a function of effective barrier width z = d|eV + µ|/~vF
for several representative values of ~vFM/|eV +µ|. Since
T and hence G depends on M through the dimensionless
parameter ~vFM/|eV + µ|, this effect can also be ob-
served by varying the applied voltage V for a fixed µ, d,
and M . In that case, for a reasonably large dimensionless
barrier thickness z0 = dµ/~vF , G/G0 becomes finite only
beyond a critical voltage |eVc + µ| = ~vFM/2 as shown
in Fig. 3 for several representative values of z0. This crit-
ical voltage Vc can be determined numerically by finding
the lowest voltage for which G/G0 exhibits a monotonic
decay as a function of z0. The plot of eVc/µ as a function
of ~vFM/µ, shown in inset of Fig. 3, demonstrates the
expected linear relationship between Vc and M . We note
such a magnetic field or applied bias voltage dependence
of the junction conductance necessitates that the Dirac
electrons represents spinors in physical spin space and is
therefore impossible to achieve in graphene [11].
Next, we analyze the junction shown in the right panel
of Fig. 1 where the region III below a ferromagnetic film
is separated from region I by a potential barrier in re-
gion II. Such a barrier can be applied by changing the
chemical region in region II either by a gate voltage V0
or via doping [5]. In the rest of this work, we will analyze
the problem in the thin barrier limit for which V0 → ∞
and d → 0, keeping the dimensionless barrier strength
χ = eV0d/(~vF ) finite. The wave function of the Dirac
fermions moving along ±x with a fixed momentum ky
FIG. 4: Plot of tunneling conductance G1/G0 versus the ef-
fective barrier strength χ and ~vF M for fixed applied voltage
V and chemical potential µ. G1 vanishes for |M | ≥ Mc =
2|eV + µ|/~vF .
and energy ǫ in this region is given by
ψ
′ ±
II = (1,±e±iγ) ei(±k
′′
x
x+kyy)/
√
2, (6)
where γ = arcsin(~vFky/|ǫ + eV0 + µ|) and k
′′
x(ǫ) =
√
[(ǫ+ eV0 + µ)/~vF ]2 − k2y. The wave functions in re-
gion I and III are given by Eqs. (2) and (3) respectively:
ψ
′
I = ψI and ψ
′
III = ψII . Note that one can have a
propagating solution in region III only if |M | ≤ |Mc|.
The transmission problem for such a junction can be
solved by an procedure similar to the one outlined above
for the magnetic strip problem. For an electron ap-
proaching the barrier region from the left, we write down
forms of the wave function in the three regions I, II
and III: ψ
′
I = ψ
+
I + r1ψ
−
I , ψ
′
II = p1ψ
′ +
II + q1ψ
′ −
II , and
ψ
′
III = t1ψ
+
II . As outlined earlier, one can then match
boundary conditions at x = 0 and x = d, and obtain the
transmission coefficient T1 = |t1|2k
′
x/kx as
T1 = 2 cos(β) cos(α)/[1 + cos(β − α)
− cos2(χ){cos(β − α) − cos(β + α)}]. (7)
Note that in the absence of the ferromagnetic film over
region III, β = α, and T1 → T 01 = cos2(α)/[1 −
cos2(χ) sin2(α)]. The expression for T 01 , reproduced
here for the special case of M = 0, is well known
from analogous studies in the context of graphene, and
it exhibits both Klein paradox (T 01 = 1 for α = 0)
and transmission resonance (T 01 = 1 for χ = nπ)
[12]. When M 6= 0, we find that the transmission
for normal incidence (ky = 0) does become indepen-
dent of the barrier strength, but its magnitude deviates
from unity: T normal1 = 2
√
1 − (~vFM/|eV + µ|)2/(1 +
√
1 − (~vFM/|eV + µ|)2). The value of T normal1 de-
creases monotonically from 1 for M = 0 to 0 for |M | =
|eV + µ|/(~vF ) and can thus be tuned by changing M
(or V ) for a fixed V (or M) and µ.
4
FIG. 5: Top left panel: Plot of G1/G0 versus χ for
~vF M/|eV + µ| = 0.1 (black solid line), 0.7075 (red dashed
line), and 1.4 (blue dash-dotted line) for fixed V and µ. Top
right panel: Plot of G1/G0 versus χ at M = M
∗ showing
the period halving. Bottom left panel: Plot of χmax versus
~vF M/|eV + µ| showing the shift of conductance maxima.
Bottom right panel: Plot of ∆G1/G0 versus ~vF M/|eV + µ|
which crosses 0 for M < Mc at M = M
∗. The dotted line is
a guide to the eye.
The conductance of such a junction is given by G1 =
(G0/2)
∫ α2
−α1
T1 cos(α)dα, where α1,2 are determined from
the solution of cos(β) = 0 for a given M (Eq. (3)). A plot
of G1 as a function of ~vFM/|eV + µ| and χ (for a fixed
eV and µ) is shown in Fig. 4. We find that the amplitude
of G1 decreases monotonically as a function of |M | reach-
ing 0 at M = Mc beyond which there are no propagating
modes in region III. Also, as we increase M , the conduc-
tance maxima shifts from χ = nπ to χ = (n + 1/2)π
beyond a fixed value of M∗(V ) ≃ ±c0|eV + µ|/(~vF ) as
shown in top left panel of Fig. 5. Numerically, we find
c0 = 0.7075. At M = M
∗, G1(χ = nπ) = G1(χ =
(n+ 1/2)π), leading to a period halving of G1(χ) from π
to π/2 . This is shown in top right panel of Fig. 5 where
G1(M = M
∗) is plotted as a function of χ. We note that
near M∗, the amplitude of oscillation of G1 as a function
of χ becomes very small so that G1 is almost indepen-
dent of χ. In the bottom left panel of Fig. 5, we plot
χ = χmax (the value of χ at which the first conductance
maxima occurs) as a function of ~vFM/|eV + µ| which
clearly demonstrates the shift. This is further highlighted
by plotting ∆G1 = G1(χ = 0) −G1(χ = π/2) as a func-
tion of ~vFM/|eV + µ| in the bottom right panel of Fig.
5. For M < Mc, ∆G1 crosses zero at M = M
∗ indicat-
ing the position of the above-mentioned period halving.
Thus we conclude that the position of the conductance
maxima depends crucially on ~vFM/|eV +µ| and can be
tuned by changing either M or V .
The experimental verification of our results would in-
volve preparation of junctions by depositing ferromag-
netic films on the surface of a topological insulator. For
the geometry shown in the left panel of Fig. 1, we propose
measurement of G as a function of m0 whose magnitude
and direction can be tuned by an externally applied in-
plane magnetic field for soft ferromagnetic films [9]. We
predict that depending on m0, G should demonstrate ei-
ther a monotonically decreasing or an oscillatory behav-
ior as a function of d. Another, probably more experi-
mentally convenient, way to realize this effect would be
to measure Vc of a junction of width d for several M and
confirm that Vc varies linearly with M with a slope of
~vF /(2e), provided µ and d remain fixed. For the geom-
etry depicted in the right panel of Fig. 1, one would, in
addition, need to create a barrier by tuning the chemical
potential of an intermediate thin region of the sample as
done earlier for graphene [11]. Here we propose measure-
ment of G1 as a function of V0 (or equivalently χ) for
several representative values of m0 and a fixed V . We
predict that the maxima of the tunneling conductance
would shift from χ = nπ to χ = (n + 1/2)π beyond a
critical m0 for a fixed V , or equivalently, below a critical
V , for a fixed m0.
In conclusion, we have studied the transport of Dirac
fermions on the surface of a topological insulator in the
presence of proximate ferromagnetic films in two experi-
mentally realizable geometries. Our study unravels novel
features of the junction conductances which have no ana-
log in either graphene or 2D Schrödinger electrons and
can be verified in realistic experimental setups.
[1] B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science
314, 1757 (2006).
[2] M. Koenig et al., Science 318, 766 (2007); D. Hsieh et
al., Nature 452, 970 (2008);
[3] C.L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801
(1995); ibid, Phys. Rev. Lett. 95, 146802 (2006).
[4] L. Fu, C.L. Kane, and E. J. Mele, Phys. Rev. Lett. 98,
106803 (2007); R. Roy, arXiv:cond-mat/0607531 (unpub-
lished); J. E. Moore and L. Balents, Phys. Rev. B 75,
121306 (2007).
[5] Y. Xia et al., Nature Phys. 5, 398 (2009); ibid,
arXiv:0907.3089 (unpublished).
[6] L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407
(2008).
[7] A. R. Akhmerov, J. Nilsson, and C. W. J. Beenakker,
Phys. Rev. Lett. 102, 216404 (2009).
[8] Y. Tanaka, T. Yokoyama, and N. Nagaosa,
arXiv:0907.2088 (unpublished).
[9] T. Yokoyama, Y. Tanaka, and N. Nagaosa,
arXiv:0907.2810 (unpublished).
[10] D. Hsieh et al., Science 323 919 (2009); D. Hsieh et al.,
arXiv:0904.1260 (unpublished).
[11] A. Castro Neto et al., Rev. Mod. Phys. 81, 109 (2009);
C. W. J. Beenakker, Rev. Mod. Phys. 80, 1337 (2008);
A. K. Geim, Science 324, 1530 (2009).
[12] M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Na-
ture Phys. 2, 620 (2006); C. W. J. Beenakker, Phys. Rev.
Lett. 97, 067007 (2006); S. Bhattacharjee and K. Sen-
gupta, Phys. Rev. Lett. 97, 217001 (2006).


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Tuning the conductance of Dirac fermions on the surface of a topological insulator

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