Anomalous Hall effect and persistent vall ey
currents in graphene p-n junctions
MASSACHUSETTS
INSTIUTE
OF TECHNOLOGY
by
AUG 15 201
Polnop Samutpraphoot
LUBRARIES
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Bachelor of Science in Physics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2014
@ Massachusetts
Institute of Technology 2014. All rights reserved.
Signature redacted
A uthor .........
I
f
.......................
Department of Physics
May 9, 2014
Signature redacted,
Certified by.....................
Professor Leonid S. Levitov
Department of Physics
Thesis Supervisor
Signature redacted
Accepted by ......
t
Professor Nergis Mavalvala
Department of Physics
Senior Thesis Coordinator
2
Anomalous Hall effect and persistent valley currents in
graphene p-n junctions
by
Polnop Samutpraphoot
Submitted to the Department of Physics
on May 9, 2014, in partial fulfillment of the
requirements for the degree of
Bachelor of Science in Physics
Abstract
Dirac particles can exhibit Hall-like transport induced by Berry's gauge field in the
absence of magnetic field. We develop a detailed picture of this unusual effect for
charge carriers in graphene nanostructures. The Hall effect is nonzero in each valley
but is of opposite signs in different valleys, giving rise to charge-neutral valley currents.
Our analysis reveals that p-n junctions in graphene support persistent valley currents
that remain nonzero in the system ground state (in thermodynamic equilibrium). The
valley currents can be controlled via the bias and gate voltages, enabling a variety of
potentially useful valley transport phenomena.
Thesis Supervisor: Professor Leonid S. Levitov
Title: Department of Physics
3
4
Acknowledgments
I am indebted to Professor Leonid Levitov for his introduction to this thesis topic.
The problem is mathematically manageable, physically elegant, yet experimentally
within reach. I would also like to thank him and Justin Song for taking their time
to guide and support me both in science and in academic life. It has been a great
pleasure collaborating with both of them, without whom the completion of this thesis
would have been impossible.
I am grateful to the Department of Physics at MIT and its people, for making
my MIT experience a memorable one. I would first like to thank Professor Patrick
Lee for his academic advice and suggestion to explore research in the Condensed
Matter Theory Group.
I would also like to thank everyone in Professor Vladan
Vuletic's group, where I conducted my first research project in physics, the Junior
Lab staff, who guided me in learning what it meant to do science, the administrative
staff members who are always happy to provide help, and friends in the department
who are always enthusiastic when it comes to discussing physics. Special thanks to
Thiparat Chotibut of Harvard Physics for encouraging me to explore what Boston
has to offer intellectually and culturally.
The research activity leading up to this thesis was supported by MIT's Undergraduate Research Opportunities Program (UROP).
5
6
Contents
1
Introduction
11
15
2 The quasiclassical dynamics
2.1
Quasiclassical equations of motion .....
....................
2.2
Calculation of the Berry's curvature . . . . . . . . . . . . . . . . . . .
19
2.3
Quasiclassical trajectories
20
. . . . . . . . . . . . . . . . . . . . . . . .
3 The Liouville flow picture
25
3.1
Transport equations
. . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.2
The anomalous Hall effect . . . . . . . . . . . . . . . . . . . . . . . .
27
4 Persistent valley currents
5
15
31
4.1
Current spatial distribution
. . . . . . . . . . . . . . . . . . . . . . .
4.2
Valley currents vs. charge currents
. . . . . . . . . . . . . . . . . . .
33
4.3
The effect of bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
Conclusion
31
37
7
8
List of Figures
1-1
a) Schematic of a p-n junction in graphene layer placed on a hexagonalboron-nitride (h-BN) layer. Gate potentials
U induce inhomogeneous
carrier (electron or hole) doping, creating regions of opposite polarity.
The zoomed-in lattice structure (inset) shows broken inversion symmetry induced by h-BN, which leads to gap opening shown in b).
2-1
. .
12
The gauge field (in arbitrary units) and Berry's curvature (shaded
background) for the positive energy eigenstate.
. . . . . . . . . . . .
20
2-2 Position space trajectories for massive Dirac particles moving in a constant electric field with transverse drift given by the anomalous Hall
velocity. The solutions shown are for the case of py = 0 and etot = 0,
as given by Eq.(2.17).
4-1
. . . . . . . . . . . . . . . . . . . . . . . . . ..
23
Band diagram for fermion occupancy and the position dependence of
Fermi surface. The gapped region -xO < x < xo is surrounded by two
doped regions: the p-doped region on the left, and the n-doped region
on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-2
Particle current distribution shows the quantized Hall effect behavior
within the gapped region and the roll-off behavior outside this region.
4-3
32
33
In the case of forward bias, Fermi surface shifts in a way that the
gapped region (pF=0) is broadened. Here EF1 is the Fermi level of the
p-doped region, and EF2 of the n-doped region. Note that we have omit
the depiction of the carriers shown in Fig. 4-1.
9
. . . . . . . . . . . .
35
10
Chapter 1
Introduction
Gauge fields, when coupled to particle motion, generate nonconservative velocitydependent forces that lead to peculiar particle dynamics. The best known example of
such forces is the canonical Lorentz force acting on charges moving in magnetic field.
Another example, which received a lot of attention recently, is the momentum-space
analog of Lorentz force which arises due to Berry's phase and Berry's curvature (also
known as Berry's vorticity) [1-6]. This force, which enters the equations of motion
through a combination x + Q x p where Q is Berry's curvature in momentum space,
is known in the literature as anomalous velocity. Anomalous velocity manifests itself
in topological transport properties such as anomalous Hall effect (AHE) which arises
in the absence of magnetic field [3].
Berry's phase manifests itself in numerous physical systems [5, 7-11] including
graphene [12-14]. Here we will focus on graphene nanostructures where transport
measurements give direct access to the physics originating from Berry's phase. The
bandstructure of graphene mimics the dispersion relation of massless Dirac particles
in two dimensions [14]. There are two distinct regions, called valleys K and K', per
Brilluoin zone that exhibit this dispersion relation described as H = vu- p, where p is
the lattice momentum measured with respect to the Dirac points and o- is the vector
of Pauli matrices. These Dirac bands are known to possess
7r Berry's phase, which
is the main ingredient of the transport properties we will study [15,16]. In addition,
weak inter-valley scattering allows long-range propagation of a current associated
11
with a single valley [17].
Stacking a layer of graphene on top of a layer of h-BN induces a gap at the Dirac
points [18-20]. As depicted in Fig. 1-1, the distinct lattice sites of h-BN breaks the
inversion symmetry of graphene, thus lifting the degeneracy at each Dirac point. The
3
,
bands can be described with the Dirac Hamiltonian with a mass term H = vo--p+A-
where A is the energy gap . As we will show, each band possesses non-zero Berry's
curvature, which manifests into a topological current.
a)
h-BN
Figure 1-1: a) Schematic of a p-n junction in graphene layer placed on a hexagonalboron-nitride (h-BN) layer. Gate potentials U induce inhomogeneous carrier (electron or hole) doping, creating regions of opposite polarity. The zoomed-in lattice
structure (inset) shows broken inversion symmetry induced by h-BN, which leads to
gap opening shown in b).
Transport phenomena occur when inhomogeneity is introduced in physical systems. We are particularly interested in transport induced by inhomogeneity in p-n
junctions. A graphene p-n junction can be constructed by placing G/h-BN onto a
split gate to create the doped regions, as illustrated in Fig. 1-1. Between the doped
regions we expect that the built-in electric field E to form and drive the particles in
the transverse direction, creating an anomalous Hall current
12
j.
We can also apply a
bias voltage (not presented) by introducing extra connectors at both ends.
We present an investigation of the total valley generated in this system starting
from analyzing the dynamics of the Dirac quasiparticle in a uniform built-in field in
two dimensions. The dynamics is governed by a 2 x 2 Dirac Hamiltonian with a mass
term,
H
71)
eEx
(1.1)
which provides a minimal model for particles in valley K. Here p = pi t ip 2 , and
P2
switches sign for valley K', following the convention in Ref. [14].
In Chapter
2, we calculate the Berry's curvature accrued by particles in the eigenstates of this
Hamiltonian and explore possible trajectories of quasiparticles driven by anomalous
velocity. We then develop a toolbox for analyzing many particle dynamics and derive
the anomalous Hall effect in Chapter 3 and apply these tools to calculate valley
current distribution in a p-n junction in Chapter 4.
13
14
Chapter 2
The quasiclassical dynamics
2.1
Quasiclassical equations of motion
Here we derive equations of motion from the least action principle by adding Berry's
gauge field to the canonical action. This approach affords a general framework in
which Berry's phases of different types can be easily incorporated. As we will see,
it can be used to treat the anomalous Hall effect for a generic Berry's gauge field,
providing a direct connection to the magnetic vector potential, Lorentz force and the
conventional Hall effect.
The validity of our approach can be understood within the general framework
of the adiabatic approximation. This approximation, originating from the adiabatic
theorem in Quantum Mechanics, guarantees decoupling of different eigenstate manifolds for the dynamics with the characteristic frequencies small compared to the
energy level spacing. For massive Dirac particles, this condition can be written as
hw < A.
Alternatively, it can be stated as a condition for spatial length scales:
r ~ v/w >
= hv/A, where
is the "Compton wavelength" for massive Dirac
particles. As an example, applied to the p-n junction described by the Hamiltonian
(1.1), the adiabaticity condition translates into a smooth inhomogeneity condition
xo = A/eE > , giving eE < A 2 /hv.
The adiabatic Hamiltonian is given by an instantaneous eigenvalue obtained for
15
an adiabatically evolving eigenstate,
Had(x) = Ea(x) + ef(r),
Hluc,(x)) = Ea lu(x)),
(2.1)
where the states are parameterized by particle coordinate and momentum. This is
expressed by a four-component vector x,, (M = 1, 2,3,4) which labels points in phase
space, x = (X 1 , X 2 , x 3 , X 4 )
=
(ri, r2 ,P1,P2 ). Here
#(r)
is electrostatic potential which
can describe the built-in electric field arising due to charge inhomogeneity and also
an external electric field which drives current in the system. Berry's connection is
defined by
A =
ih
(na(x)IOx.ua(x)) + c.c.,
(2.2)
where c.c. stands for complex conjugate. The adiabatic phase corresponding to this
definition of Berry's connection is given by AO =
f Adx,,.
We note that, while the framework developed is completely general and applicable for the Hamiltonian and gauge field with position and momentum dependence,
we focus on the problem described by the Hamiltonian in Eq.(1.1) in the following
discussion. The adiabatic eigenvalues are of the form E (x) = + /A 2 + v 2 p2 , with
the phase-space coordinates xu = (r, p) taken on particle trajectory.
The action, written in a position-momentum symmetric form, reads:
S
=
J(7r + A")dx, - Had(X)dt,
(2.3)
where 7r" = (p1, P2, 0, 0). This form of 7r" generates the canonical action f p'dri. The
part f A~dx, accounts for coupling to Berry's gauge field. Since in general all four
components of A" are nonzero, the part f A~dx, describes coupling in both real space
and momentum space.
The dynamics governed by the action, Eq.(2.3), can be cast in the Hamiltonian
form by introducing the canonical structure in the phase space, described by the
16
1-form w = (7r + All)dx, and the 2-form
dw = Q1 dx, A dx,
(2.4)
The skew-syrmmetric 4 x 4 matrix Q,, encodes Berry's curvature position and momentum dependence in general. Evaluating dw, we express Q/" through Berry's curvature
tensor:
A" (2.5)
v= Al" + F"",
F" -
.
The 4 x 4 matrix A is of a canonical symplectic form
A =
(2.6)
1
0
where 1 is a 2 x 2 unit block matrix.
Equations of motion can be obtained by taking variation with respect to x1, and
setting JxS to zero, giving
OHaA
-
ax,
-
ax
,
MA"
A
a X,.
These equations can be written in a more compact form by using the Berry's curvature
tensor and the matrix QA' defined above in Eq.(2.4):
'X=H
* (2.7)
To avoid confusion, we note that FlV is distinct from the conventional electromagnetic
field tensor in that it involves extra momentum components, v = 3,4, but no timelike components.
An applied perpendicular magnetic field would contribute in a
conventional way to the spatial components of F" which enter Ql", p, v = 1, 2.
However, in contrast to the electromagnetic field tensor, an electric field would enter
through the right hand side of Eq.(2.7) via VxH.
Interestingly, the additional terms Fl"Lb appearing due to the Berry's connection
17
do not result in any work done on the system. This can be verified most easily
by multiplying Eq.(2.7) by t. and observing that the left hand side is zero by the
antisymmetry of both tensors, giving jHad = z'j Oaca
=
0.
As a sanity check, we can show that these equations are reduced to the standard
Lorentz force equations for magnetic vector potential, which is of a 'space-like' form
(i.e. it depends only on spatial coordinates and has A 3 = A 4 = 0). After dropping A 3
and A 4 we have the least action principle for particle trajectory in the phase space,
q(t), p(t), describing particle action as
S =
dt (pi~i - H(p,q) + Aiji).
(2.8)
Here the vector potential A describes the magnetic field B = V x A and, to simplify
the discussion, we set electric charge value to unity. Taking variation with respect
to pi and setting 6 pS to zero, we find 4i = OH/api. Similarly, taking variation with
respect to qi and setting 6 qS to zero, we find
.
. aH (9A,- .
aAi
p= + 0 q -Aq
aqi
oqi
aq,-
It is straightforward to verify that the last two terms in this equation is of the form
identical to the Lorentz force,
=-VH + 1 x B.
This form can be established most easily using the double cross product formula,
a x (b x c) = b(a -c) - c(a -b).
We finally note that the action used in the above derivation is distinct from that
used in the canonical approach. The latter is the action given in terms of the canonical
momentum variables, S = f pdq - Hdt, whereas here we are using an action, Eq. (2.8),
written in terms of the kinetic momentum.
18
2.2
Calculation of the Berry's curvature
Here we derive Berry's gauge fields for the Dirac Hamiltonian in Eq. (1.1).
adiabatic limit TA >
In the
h, instantaneous eigenstates are evaluated with parameters
fixed at a given value,
ju+) = a
A
,e,
= V/A 2 + v 2 Ip 2
(2.9)
)
v(p1 + ip 2
with the normalization constant a
eigenenergies are c
,p
1
=
= +c,. A 1 ,2 = i (u
lA u
2 = p2 + p2.
The associated
) &1,2 A vanish regardless of the form
of A. Only two out of four components are nonzero:
A3 =
hp 2
2c,(cp
_
A)'
A=
A4 = -
hp1
2cp(cp t A)'
.(2.10)
The non-vanishing components of F" must therefore be expressed through OA 3 ,4
(p = 1...4).
The 4 x 4 tensor Fl" is conveniently analyzed by splitting it into 2 x 2 blocks for 'allposition' and 'all-momentum' components F,, and Fpp and the 'mixed' components
Frp and Fpr. The part Frr vanishes since A 1 ,2 = 0, and Fp and Fpr vanish since A 3 ,4
are position independent. Evaluating the all-momentum part Fpp, we find
43
= wp,
WP = T
3/2,
2 (A 2 + v21p12) 3 / 2
(2.11)
(
F 34 = -F
which is the canonical Berry's curvature. The quantity wp gives rise to the anomalous velocity (see Eq. (3.23) in Ref. [5] and discussion below). Its integral over the
momentum space as A -+ 0 gives +7r,
in agreement with the total Berry's phase
calculated for a single layer of graphene [15,16].
In the absence of spatial dependence in the Dirac mass, Fpp is the only nonzero
part of F". If an external magnetic field is applied, it contributes to F"' by modifying
Frr as discussed above.
19
P2 (v/A)
pi(v/A)
Figure 2-1:
The gauge field (in arbitrary units) and Berry's curvature (shaded
background) for the positive energy eigenstate.
2.3
Quasiclassical trajectories
We will study the dynamics of a single particle whose Hamiltonian is given by Eq. (1.1)
with position-independent A in an external electric field E and zero magnetic field.
In this simple case, the equations of motion Eq.(2.7) take the form
1 = eE,
i- = VPep + WP X 1,
(2.12)
where we treat Berry's curvature as a vector normal to the 2D plane, Wp 11z. We
choose the coordinate axes such that x = (x, y, px, py), where the x-direction aligns
20
with the electric field. Note that the problem possesses translational symmetry in the
y-direction.
We first inspect the conservation of energy by integrating the x-component of
Eqs.(2.12),
= Vsei =
+V
2
Px/cp, using the momentum components p'i = eE and
15x = 0 for the change of variables and obtain
x =k
dp
eE
V/
+v 2 p2 + xo,
k\A2
VP
2
+v 2p 2
eE
(2.13)
which can be rewritten as
\/A2 + v 2 p2 -
Ctot =
eEx,
(2.14)
where we have now included the negative kinetic energy solution. Energy conservation
ensures that the anomalous Hall velocity term does no work, in agreement with our
discussion in the previous section.
For visualization of the dynamics, we examine trajectories that follow the equations of motion.
By inspecting the momentum components, we immediately see
that the trajectories in the momentum space are straight lines in the direction of
px with translational symmetry in the direction of py. The position space solutions,
however, requires extra steps to solve for. The solution x(t) to the x-component of
Eqs. (2.12) follows the classical relativistic dynamics and is hyperbolic with time. The
y-component,
2
=
k
y +
wpeE,
(2.15)
is modified by the anomalous Hall velocity term. We thus expect an additional drift
in the direction transverse to the electric field.
Macroscopically, the group velocity term v 2 py/Ep in Eq.(2.15) is averaged out for
both Dirac bands because of the odd parity in py. For this reason, we are interested
in the trajectory with py
=
0 in particular as it can be easily solved analytically while
still capturing the important features. We also choose the reference of the electrostatic
potential such that Etot = 0. Dividing the two spatial components of Eqs. (2.12) and
21
using
6+ =
+
p=
v 2p 2 +
2
and wp
=
-hv 2 A/2E
given by Eq. (2.9) and Eq.
(2.11) yields
dy
dx
where xO = A/eE and (
hAeE
_
2p.,(V2p2 + A2)
=
2 X2(X2
_
X0)1/2)1
(2.16)
hv/A. We can integrate Eq. (2.16) using hyperbolic
substitution and obtain
2(y - yo))
2
(
)2
(2.17)
The arbitrary constant of integration yo leads to continuous translational symmetry
in the y-direction.
The trajectories are plotted in Fig. 2-2. We infer from Eq.(2.14) that the positive energy solutions are on the right hand side and drift upward and vice versa.
Although these trajectories show single particle dynamics, we can qualitatively visualize macroscopic dynamics by noticing that as one moves further away from the
center, the transverse drift becomes smaller. We should therefore expect that the
macroscopic current contains roll-off behavior at both wings.
In the classical regime, particles approaching from afar do not leak into the gapped
region -x < xo < x. The transverse drift in the y direction is related to breaking of
the time-reversal symmetry due to the anomalous velocity term w, x p. As we will
show, the current in this region takes a quantized value as appropriate for the Hall
effect a gapped system.
22
2xo
y
x
EE
Figure 2-2: Position space trajectories for massive Dirac particles moving in a constant
electric field with transverse drift given by the anomalous Hall velocity. The solutions
shown are for the case of py - 0 and ctot -0, as given by Eq. 2.17).
23
24
Chapter 3
The Liouville flow picture
3.1
Transport equations
The framework developed above can be extended to investigate the macroscopic behavior of the valley current in the p-n junction system.
Many-particle transport
in our system can be described by Boltzmann-type transport equations written for
the probability distribution in the four-dimensional phase space, parameterized by
X/ = (r, p). Particle distribution in phase space can be characterized by the probability dP = n(x, t)dV. In the absence of collisions, time development of n(x, t) is
governed by the Liouville flow equation.
(&t + zi',) n(x, t) = 0,
(3.1)
where the velocity is given as a function of xP by the quasiclassical expression,
1
=
(Q')pvOvHad,
(3.2)
see Eq.(2.7). The 4 x 4 matrix Q- in general has both r and p dependence.
It can be seen from Eq.(3.1) that any distribution given by a function of Had is a
25
steady state of the flow. This is the case, in particular, for the Fermi function
1
no(r, p) = e(Had(r,p)4)
1
'
=
kBT,
(3-3)
describing electron system in equilibrium.
Importantly, Liouville's flow equation can be transformed to the form of a continuity equation. Introducing current density jI' = i/n(x,, t), we obtain the continuity
equation in a canonical form,
Otn + 1,j4 = 0.
(3.4)
Spontaneous transitions due to particle collisions can be accounted for by a collision term
Otn + ojjl' = I[n].
(3.5)
For example, scattering by disorder is described by
I[n] = E wp,, (n(r, p', t) - n(r, p, t))
(3.6)
P'
where wPP, is the transition rate for elastic scattering between momentum states p
and p'. Below, we will use a simple form for wp,,, which ignores the skew scattering
and the side jump effects. The impact of these effects will be analyzed elsewhere.
It is straightforward to verify that the Liouville's flow associated with the dynamics
given in Eq.(2.7), possesses conservation laws identical to those found for Eq.(2.7).
In particular, energy conservation holds:
d
J
dVHan =
-
J
dVHad (O9tj) = 0,
(3.7)
where f dV... stands for the integration over the entire phase space.
Transport equations, Eq.(3.5), acquire a conventional form in the absence of
Berry's gauge field.
With the phase-space velocity given by i =
26
H = v, p =
ap
-- = -eE, Eqs.(3.5),(3.6) yield
(Pt+
vV, + eEVp) n(r, p, t) = I[n(r, p, t)],
(3.8)
which is the canonical form of the Boltzmann kinetic equation. In contrast, since
for systems subject to a generic Berry's gauge field, the equations of motion take a
more general form, Eq.(2.7), the transport equation must be used. This gives rise to
unconventional transport phenomena analyzed in the next section.
3.2
The anomalous Hall effect
We are now in a position to incorporate the anomalous Hall effect into the transport
current density over momenta and the two Dirac bands (indicated by
=
ein(p) = E e (v(p)
-
e2 W, x E) n(p).
)
equation. The spatial current density can be obtained by summing the phase space
(3.9)
Combining Eq.(3.5) with the - and p given in Eq.(2.12), we can write transport
equation as
[Ot + eE - VP + (v(p) - WP x eE) - Vr] n = I[n].
(3.10)
Here E, which is p-independent, commutes with V,, and can thus be placed either
before or after V,. Similarly, for position-independent A, both v(p) and wp commute
with V, and therefore can be placed either before or after V,.
Taking into account that spatially uniform and time-independent field, while driving the p dependence out of equilibrium, leaves the distribution spatially uniform and
time-independent, we see that the distribution function obeys
eEVpn = I[n].
Solving Eq.(3.11) at first order in E, we find n(p)
27
(3.11)
=
no(p) + Jnr(p), where no(p) is the
equilibrium distribution and the perturbation 6n(p) satisfies
eEVPno(p)
--yn(p),
=
-Y= (WP,(1- cos9P,P/)),
(3.12)
where we take into account that I1no] = 0 in equilibrium. Here y is the transport
scattering rate (angular brackets denote averaging over angles).
Substituting this result in the expression for current, Eq.(3.9), and noting that
EP v(p)no(p) = 0 as there can be no drift without a drive (E = 0), we find the
relation between the applied field and induced current:
=
- e 2 V(P) (EVp)no(p) - e2(wp x E)no(p)
where E,,, ... stands for E
f
d2
.
(3.13)
-.. Here the first term represents the conven-
tional longitudinal ohmic current, whereas the second term describes the anomalous
Hall current, which is transverse to the electric field. Since Vpno(p) is peaked near
the Fermi surface, the first term receives contribution only from a narrow band of
states near the Fermi level. In contrast, the second term depends on the contribution
of states both near the Fermi level and deep beneath it.
The above expression for the Hall current, which is proportional to Berry's curvature integrated over all p states under the Fermi level, e, < EF, in both Dirac bands.
Expressing wp as a curl of Berry's gauge field and using Stokes theorem, we can relate
the flux of Q, through the Fermi surface to the net Berry's phase accrued by tracing
the Fermi surface. This gives an expression for the Hall conductivity of the form
UH=
Ai(p)dp 2 ,
___
(27r)2h
i
=
17,2,
(3.14)
P=PF
This quantity UH depends on Berry's connection only on the Fermi surface, regardless of the behavior inside the band. It thus encodes geometric properties. Performing
28
this integral, or its equivalent area integral, leads to a familiar expression
e2A
PFH
2h EPF
6
PF
=
2 ,
2+V2p
(3.15)
identical to that found in Ref. [1-3, 5]. We reiterate that the analysis above does
not account for the effects that may significantly alter the Hall conductivity such as
skew scattering and side jumps in the position space accompanying scattering in the
momentum space.
29
30
Chapter 4
Persistent valley currents
4.1
Current spatial distribution
The Hall conductivity given by Eq. (3.15) is position dependent since the built-in
electric field E creates potential gradient. Fig. 4-1 shows particle distribution in the
ground state, described by the Fermi function, Eq.(3.3) for T = 0. The potential
gradient divides the space into three regions, p-doped, n-doped, and undoped (or
gapped), the boundaries of which are -xO and xO as shown.
In the region where x > xO, the potential shifts the bands in a way that the Fermi
surface sits on the positive Dirac band (conduction band). The radius PF of the Fermi
surface is position-dependent and is given by
A 2 + v 2p 2 = eEjx
=
6
PF.
In this
section, we are interested in particle current for simplicity. We now drop a factor
of e from the electric current density in Eq.(3.9).
The uncharged current density
j(x) = uHE/e becomes
j(X > XO)
e A
2hcPF
E=
1 'A
-2h Ix|
(4.1)
The roll-off is caused by the energy dependence ~ 1/cF of the anomalous velocity.
When summed over the two dimensional momentum, this gives j
1/cPF
1/X.
By symmetry, we expect the same roll-off in the magnitude of current density for
x < -xO.
In this region, the Fermi surface lies on the negative value of energy, causing
31
WC
x
XO
-...
I-
+XO
+~
S
S
S
.%
S00
I
Figure 4-1: Band diagram for fermion occupancy and the position dependence of
Fermi surface. The gapped region -xO < x < xo is surrounded by two doped regions:
the p-doped region on the left, and the n-doped region on the right.
shortage in negative energy carriers, which can also be viewed as Dirac antiparticles or
holes. Since the negative energy carriers flow opposite to the positive ones according to
Fig. 2-2, the shortage is equivalent to particle current flowing into the same direction
as the current in x > xO. We can also deduce this directly by inspecting the anomalous
Hall velocity term wPeE in Eq.(2.15) as both the charges and the Berry's curvatures
of Dirac bands of opposite signs of energy are opposite. In the central region, -xo <
x < xo, we have PF = 0, and thus j(-xo < x < xO) =
2h
E as in the conventional
anomalous Hall effect.
The distribution of the anomalous Hall current over the entire space is depicted
in Fig. 4-2. The roll-off behavior of the wings qualitatively agree with the transverse
velocity distribution of the trajectories shown in Fig. 2-2. In both pictures, the roll-off
behavior is observed to be outside the middle strip of the same width 2xO = 2A/eE.
32
44
A
A
4
k
A
+
A
tt i t t t t
t t t t t t
tt t
tt t t tt
t t t
A 44
A
A
44
A
+
A
+t
A
iii
+
A
A
A
A
A
44+
A
A
-
i
-
i
4
t
It
+
4
t
A
E
urn
i
t t tt
t
t t t t
t t t t t
t
A
+
4
AA
it
i tt ttt t tt I
t t t t t
t
t
t
44At
4
Figure 4-2: Particle current distribution shows the quantized Hall effect behavior
within the gapped region and the roll-off behavior outside this region.
4.2
Valley currents vs. charge currents
In the previous section, we have seen that the charge transport of the system is trivial
as electrons and holes move in the same direction, giving rise to zero charge current.
However, the valley transport properties are non-trivial. We recall that in hexagonal
lattices there are two distinct Dirac points per Brilluoin zone, label K and K'.
The analysis in the previous sections leading up to this point is done in one valley,
which we choose to be K. Given that the two valleys are related by time-reversal
symmetry, the extension to the K' valley is easily done by reflection the Hamiltonian
in Eq.(1.1) across the pi axis, that is
P2 (
-P2,
or equivalently p
(-
PT [14].
This switches the sign of wp in Eq.(2.11) and therefore the direction of the anomalous
Hall velocity.
The opposite directions of the transverse currents for the two valleys give rise to
33
the valley Hall effect [6]. The valley current, defined as j,,
=
j
-
jK,, where j is
the current density derived in the previous section and s labels the spin, is non-zero.
Summing over the valleys and spins, the magnitude of the total valley current derwity
is therefore
j, =
4j(x), where j(x) is given in the previous section. This is the valley
Hall effect described in [6]. It can be treated to be a direct analogue of the spin Hall
effect as the weak inter-valley scattering allows us to treat different valleys as separate
species as we do with spins.
4.3
The effect of bias
The result in the previous section can be conveniently extended to the case where
there is external bias voltage V, causing the Fermi level to shift as shown in Fig. 4-3.
We observe that the central region where PF is widened for forward bias, and vice
versa by an additional length of V/E. We label the bounds as tx, = t(A+eV)/2eE.
Regardless of the functional form of the Fermi surface, as long as it lies within the
gap the middle region
out of the edges
lxi
lxi > x1
< xi, the current density is uniform. The roll-off behavior
remains
-
1/x.
As the current in the middle of the junction stays the same, the larger area of
constant current results in a larger total current.
At eV > A, the width x 1 is
directly proportional to the bias voltage V, so this the total valley current. This linear
relationship between the valley current and the bias voltage allows us to manipulate
the valley degree of freedom and opens up possibilities to construct new devices in
valleytronics [21].
34
-Xi
x
.I s
.m~ ..
,.......
EF1
8
F2
+x1
Figure 4-3: In the case of forward bias, Fermi surface shifts in a way that the gapped
region (pF=O) is broadened. Here cFi is the Fermi level of the p-doped region, and
cF2 of the n-doped region. Note that we have omit the depiction of the carriers shown
in Fig. 4-1.
35
36
Chapter 5
Conclusion
Our analysis predicts persistent currents in graphene p-n junctions, which arise from
Berry's gauge field. We have studied these currents with the help of quasiclassical
equations of motion for massive Dirac particles and used Liouville transport equations
to evaluate the spatial distribution of these currents. We have demonstrated that the
charge current is zero, whereas the net valley current is nonzero. The current peaks
in the gapped region and decays as 1/x outside this region. The width of the gapped
region increases (decreases) under forward (reverse) bias, allowing control over the
total current.
transport.
Graphene p-n junctions therefore afford a platform to study valley
The freedom to induce and manipulate valley currents opens door to
potentially useful valleytronics applications.
37
38
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