Journal of Electromagnetic Analysis and Applications, 2012, 4, 426-431
http://dx.doi.org/10.4236/jemaa.2012.410059 Published Online October 2012 (http://www.SciRP.org/journal/jemaa)
1
Chiral Current in a Graphene Battery
Hector Torres-Silva1, Diego Torres Cabezas2
1Escuela de Ingeniería Eléctrica Electrónica, Universidad de Tarapacá, Arica, Chile; 2Modernización y Gobierno Electrónico, Mi-
nisterio Secretaría General de la Presidencia, Santiago, Chile.
Email: htorres@uta.cl
Received August 2nd, 2012; revised September 5th, 2012; accepted September 15th, 2012
ABSTRACT
We review the formulation of graphene’s massless Dirac equation, under the chiral electromagnetism approach, hope-
fully demystifying the material’s unusual chiral, relativistic, effective theory. In Dirac’s theory, many authors replace
the speed of light by the Fermi velocity, in this paper we deduce that in graphene the Fermi velocity is obtained from
the connection between the electromagnetic chirality and the fine structure constant when the electric wave E is quasi
parallel to the magnetic wave H. With this approach we can consider the properties of electric circuits involving gra-
phene or Weyl semimetals. The existence of the induced chiral magnetic current in a graphene subjected to magnetic
field causes an interesting and unusual behavior of such circuits. We discuss an explicit example of a circuit involving
the current generation in a “chiral battery”. The special properties of this circuit may be utilized for creating “chiral
electronic” devices.
Keywords: Chiral Dirac; Battery; Grapheme; Weyl Semimetal
1. Introduction
Recently, the 2D and 3D materials with linearly dispers-
ing excitations [1] have attracted significant attention.
The existence of these “chiral” excitations stems from
the point touchings of conduction and valence bands. The
corresponding dynamics is described by the Hamiltonian
ˆ
F
H
vk
 ; where
F
v is the Fermi velocity of the
quasi-particle, k is the momentum in the first Brillouin
zone, and ˆ
are the Pauli matrices. This Hamiltonian
describes massless particles with positive or negative
(depending on the sign) chiralities, e.g. neutrinos, and the
corresponding wave equation is known as the Weyl
equation—hence the name Weyl semimetal [1]. Weyl
semimetals are closely related to 2D graphene [2], and to
the topological insulators [3]—3D materials with a gap-
ped bulk and a surface supporting chiral excitations.
Specific realizations of Weyl semimetals have been pro-
posed, including a multilayer structure composed of identi-
cal thin films of a magnetically doped 3D topological
insulator, separated by ordinary-insulator spacer layers
[4].
Weyl semimetals provide a unique opportunity to
study the macroscopic behavior of systems composed by
chiral fermions. In particular, they allow [5] to study, in a
condensed matter system, the chiral magnetic effect ex-
pected [6-11], and closely related phenomena [12-16].
The effects of the anomaly on the transport in Weyl
semimetals, including the chiral magnetic effect, have
recently been investigated in [17-20].
In this paper we argue that the existence of chiral
magnetic current in graphene subjected to magnetic field
can cause an interesting, and potentially useful for prac-
tical applications, behavior of circuits such a chiral bat-
tery [9].
The material called graphene which is a single layer of
atoms arranged in honeycomb lattice could let electronics
to process information and produce radio transmission 10
times better than silicon based devices.
From the point of view of its electronic properties,
graphene is a two-dimensional zero-gap semiconductor
with the cone energy spectrum, and its low-energy qua-
siparticles are formally described by the Dirac-like Ham-
iltonian [21,22]. 0ˆ
F
Hiv
 where F
61
10 msv
is the Fermi velocity and
,
ˆ
x
y

are the Pauli
matrices. The fact that charge carriers in graphene are
described by the Dirac-like Equation (1), rather than the
usual Schrödinger equation, can be seen as a conse-
quence of graphene’s crystal structure, which consists of
two equivalent carbon sublattices [21,22]. Quantum me-
chanical hopping between the sublattices leads to the
formation of two cosine-like energy bands, and their in-
tersection near the edges of the Brillouin zone yields the
conical energy spectrum. As a result, quasiparticles in
graphene exhibit the linear dispersion relation
GF
EEkv
, as if they were massless relativistic par-
ticles with momentum k (for example, photons) but the
Copyright © 2012 SciRes. JEMAA
Chiral Current in a Graphene Battery 427
role of the speed of light is played here by the Fermi ve-
locity 300
F
vc. Owing to the linear spectrum, it is
expected that graphene’s quasiparticles will behave dif-
ferently from those in conventional metals and semicon-
ductors where the energy spectrum can be approximated
by a parabolic (free-electron-like) dispersion relation.
The low-energy excitations of this system are then de-
scribed by the massless two-dimensional Weyl-Dirac
equation and their energy dispersion relation F
vk
is that of relativistic massless fermions with particle-hole
symmetry. The maths is simple but the principles are
deep. We will review the formulation of graphene’s
massless Dirac Hamiltonian, under the chiral electro-
magnetism approach, like a meta-material media [23-29],
hopefully demystifying the material’s unusual chiral,
relativistic, effective theory. These results are derived of
the Chiral Electrodynamics with T as the chiral parame-
ter and0
kc
[27,30-32]. The chiral vector poten-
tial c
A
can be expressed as

22
20
22 22
0
20
11
cc c
o
kT
AA A
kT kT

 

As 222
x
yz
kkkk, if sin
x
kk
, 0
y
k
, and
cos
z
kk
, we have the matrix: (see the formula below)
The dispersion relation of the transversal wave is


2
2222 42222
000
14sincoskkTkkkT

 

0
00
1kk kkT
 .
That is

1
00 0
11kk kTkT
c
 
If we put then

0
1
F
vc kT F
vc
which is the
fine structure constant. Here we have that c
A
EH, that
is the electric field is quasi parallel to the magnetic
wave
E
H
. The novel result here is that in our chiral theory
we do not make
F
cvbut we obtain
F
v
0

as
if0 or if
0. In this section we have given an approach of
the chiral electromagnetism applied to grapheme. In Sec-
tion 2, we discuss the two component equations of Dirac
electron in grapheme. Section 3 describes a graphene bat-
tery device.

0
kT 0kT1vc
0
F
kT

1
F
vc kT
2. Two Component Equations of Dirac
Electron in Graphene
The usual choice of an orthogonal set of four plane-wave
solutions of the free-particle Dirac equation does not lend
itself readily to direct and complete physical interpreta-
tion except in low energy approximation. A different
choice of solutions can be made which yields a direct
physical interpretation at all energies. Besides the separa-
tion of positive and negative energy states there is a fur-
ther separation of states for which the spin is respectively
parallel or anti-parallel to the direction of the momentum
vector. This can be obtained from the Maxwell’s equation
without charges and current in the wave EH configu-
ration, so EiH
, orEicB
[30-32]. Here we con-
siderer a bidimensional graphene system so the Dirac’s
four-component equation for the “relativistic” electron is:
ˆD
iH
t

, (1)
where with
,t
EH

ˆ
E

E
, ˆ
H
H
so
we can write
2
ˆˆ
D
F
Hv mv
F
αp, (2)
0,1,2
0
k
k
k
k


 ,3,
(3)
0
0z
I
I



(4)
and I is the two-by-two identity matrix and k
and
are written in chiral or Weyl representation, the Fermi
velocity
F
v is deduced from the chiral electrodynamics
with
0
1
FkTvc , where T is the chiral parameter in
a metamaterial condition. This result is capital to our
approach because we find a contact point between the
graphene system and optical metamaterial making
1vc kT
0

F, no making
F
cv as other authors do
it. This Hamiltonian commutes with the momentum vector
. In order to resolve this degeneracy we seek a dy-
namical variable which commutes with both H and
p
p
.
Such a variable is ˆp
, where ˆ
is the matrix Pauli.
The eigenfunctions of the commuting variables and
p
ˆp
are simultaneous:

22
ˆpp

, (5)
Thus for a simultaneous eigenstate of and
pˆp
,
the value of ˆp
will be +p or –p, corresponding to
states for which the spin is parallel or antiparallel, re-
spectively, to the momentum vector like a graphene sys-
tem.


2222 2
00 0
222222
0000
22222
000
12cos0
2cos 12sin
02sin1
x
y
z
kkTk jkkTA
jk kTkkTkjk kTA
A
jkkTkk Tk


 



 





 

0
Copyright © 2012 SciRes. JEMAA
Chiral Current in a Graphene Battery
428
A simultaneous eigenfunction of H and p will have the
form of a plane wave

exp,1, 2,3, 4
jj
uiprEtj



, (6)
where the
j
are the four components of the state func-
tion and
j
u four numbers to be determined. Then E can
have either of the two values.

1
2
24 22
FF
Emvv
 p. (7)
We now demand that
j
be also an eigenfunction of
ˆp
belonging to one of the eigenvalues , say,
where , The eigenvalue equation is
pE
p
Ep
ˆpp
 

E, (8)
Since can be given either of the two values
W
and , the two values , we have found for given p
four linearly independent plane wave solutions. It is easily
verified that they are mutually orthogonal.
pEp
The physical interpretation of the solutions is now
clear. Each solution represents a homogeneous beam of
particles of definite momentum p, of definite energy,
either
, and with the spin polarized either parallel or
anti-parallel to the direction of propagation. From here
we can obtain the well known equation for graphene
0ˆˆ
FF
H
ivv p
 
B
. The sign ± correspond to
the different chiralities of the Weyl fermions. When we
apply a magnetic field 0 along z, we obtain

00
ˆ
Fe
H
vp cA
, where 0
A
is the vector poten-
tial corresponding to 0. The chiral anomaly occurs
when the current density in terms of right and left-handed
spinors is nonzero.
B

††
ee
EE HH
j
 


and the total chiral
current is
3
d
cj
I
x
(9)
In Section 3, we extend this study to 3-D graphene
with linearly dispersing excitations envolving an interes-
ting application of the Chiral Magnetic Effect—a re-
chargeable battery which stores chirality—the chiral bat-
tery.
3. Chiral Current in a Graphene Battery
Device
Chirality in graphene is not related to the usual spin
states considered above but instead refers to the sublat-
tice states. If we have some finite amount of this material,
it can be used as a battery. The battery can be charged
using the axial anomaly by placing it in parallel electric
and magnetic fields. The charging time will be deter-
mined by the axial anomaly. The battery stores energy,
since the Fermi-levels of right- and left-handed modes
differ.
In the absence of electric and magnetic fields, chirality
is conserved, so the battery does not discharge. If the
battery is connected to a circuit element with resistance R
and we apply a magnetic field to the battery in the right
direction, a current J will be induced due to the Chiral
Magnetic Effect. The behavior of this current as a func-
tion of the applied magnetic field can be obtained. The
current will cause a potential difference total over
the circuit element. As a result, the same potential dif-
ference will also exist over the battery. Hence an electric
field will arise parallel to the magnetic field. In this case
the axial anomaly operates again to decrease the chirality.
Hence a slow rate of discharge will be determined by the
axial anomaly as well.
VIR
Following the argument by Nielsen and Ninomiya, [17]
let us imagine that there is an electric field 0 applied
in the same direction as the magnetic field 0. This 0
will lead to transfer of particles between the two Weyl
nodes of graphene through the zero node, which has a
definite chirality. The slow rate of particles between the
left () and right (+) node per unit volume is given by
E
B E
2
00
22
de
d2π
NN EB
tc

Since the Weyl nodes have an energy difference W
,
this particle transfer process has an associated power per
unit volume
d2
d
W
NN
Pt


So that 0
PJE
and
J
can be obtained. Consider
first a cylindrical sample of graphene inside a solenoid
that provides an external, constant magnetic field of
strength B along the longitudinal direction of cylindrical
geometry, say , see Figure 1. The top and bottom of
the graphene material are touching metallic plates that
can conduct electric currents owing through the sample.
These two metallic plates are then connected to an out-
side circuit which is characterized by a resistance R. The
cross section area of the graphene sample is and the
z
2
πr
B
0
R
r
l
Figure 1. The chiral battery: Graphene system (shown in
grey) connected to the circuit with resistance R in an exter-
nal magnetic field B0.
Copyright © 2012 SciRes. JEMAA
Chiral Current in a Graphene Battery 429
longitudinal length is d. The induced chiral magnetic
current density along , is (Equation (9))
z
2
2
e
4π
c
J
B
c
(9)
is the energy separation between the Weyl nodes.
If 1 meV

J
, and the magnetic field 0, we
have , an easily measurable current [9].
1 TB
2
/cm0.1 A
c
In the case of a global equilibrium all the contribution
to Equation (9) from different valleys cancel each other.
In the presence of 0 parallel to 0, an imbalance of
electron populations is created so there is a finite current
and it responds only on the component of the electric
field parallel to .
B E
0
Once an external magnetic field is applied, the energy
stored in the difference of the chemical potentials of left-
and right-handed fermions can be released by generating
the current (
B
c
I
), hence the name chiral battery [9].
Note that according to (9) a Weyl semimetal is a kind
of battery that provides a definite amount of current,
contrary to conventional batteries that support a definite
voltage. The total anomaly-induced current through the
sample is c
2
π
c
I
rJ. If the entire current is total t
II
,
then there is a voltage drop along the resistance R given
by total . Since the same amount of voltage drop
should also occur along the graphene sample, there is an
electric field along direction with a magnitude
VIR
z
0d
t
EV IR d (10)
Note the negative sign of E. This electric field gives
rise to a normal current through conductivity
2
2
0total
π
π
d
j
j
rR R
total
I
rEII
R
 
; (11)
where 2
d
π
j
Rr
is the intrinsic resistance of the Weyl semimetal sample.
The total current t
I
should be the sum of
j
I
and c
I
,
that is determined self-consistently as
1
c
tjc
j
I
III RR

(12)
This is the equation governing the performance of the
chiral battery.
Let us now see how the energy discharge works for the
chiral battery. From the total current (12) through the
resistance R and the normal current
j
I
. In through
j
R,
the energy discharge rate should be
2
2
22 2
22
d
d1
4π
tjj
j
R
RIRIAB
tRR
c

 

2
e
(13)
using (11) and (12). This should match the reduction of
internal energy of the Weyl semimetal sample. In the
presence of both electric field E as in (10) and the mag-
netic field B, the charge density of ith Weyl point
changes is ()
,1c

22
00
22
dee
d4π4π
iii
ww
tkk
t

E
BEBEB (14)
Here as the fast time electromagnetic field is
ww
ic
E
B, there is not contribution of this term, so the
slow time average is
2
0
2
de
dd
4π
iit
tkI
B
t
 (15)
The total volume of the sample is Ad, so that the slow
total rate of increase of i’th charge is
2
total 0
2
de
d4π
ii
Qk
A
IRB
t (16)
from which the rate of internal energy change is
2
2
22
int
total 0
22
de
d1
4π
c
j
R
J
AI RAB
tRR
c

 

(17)
Using the expression (12) for I, which indeed agrees
precisely with (13), the time-dependence of the chiral
battery performance relies on the detailed equation of
state between i
and (chiral chemical potential).
i
Let us estimate the amount of energy E stored in the
chiral battery per unit volume. It is equal to the Helm-
holtz free energy, which is the energy that can be used to
do work. The free energy is the difference between the
thermodynamic potential with a chiral charge density and
without and is easily founded. Following [9], we then
obtain that the energy density is
μ
437
d7.010 enm1.010 Jcm
d
FF
vv
Ev
vol cc
 3
The typical distance between the lattice sites in a crys-
tal is of order 0.1 nm. Suppose we can store 1 unit of
chirality per lattice site, i.e. an excess of 100 right-
handed fermions over left-handed fermions per nm3. In
typical materials like graphene 2
10
F
vc
, so the typi-
cal storage capacity of the chiral battery is of order
53
10 J cm30 Whcm3
. This is comparable or better
than conventional batteries whose energy density is typi-
cally 10 - 100 Wh/Kg; note besides that the current in our
case is spin-polarized and so may be used for spintronic
applications.
The circuit discussed above represents only a couple
of examples from a vast array of devices that one can
envision. We hope that the chiral electronics based on
graphene circuits can serve as a new way to explore the
macroscopic dynamics induced by the chiral anomaly,
and perhaps open a path towards new electronic devices.
4. Conclusions
In this paper, we reviewed the formulation of graphene’s
Copyright © 2012 SciRes. JEMAA
Chiral Current in a Graphene Battery
430
massless Dirac equation, under the chiral electromagne-
tism approach. In Dirac’s theory, many authors replace
the speed of light by the Fermi velocity, in this paper we
showed that in graphene the Fermi velocity is obtained
from the connection between the electromagnetic chiral-
ity and the fine structure constant F
vc
when the
electric wave E is quasiparallel to the magnetic wave H.
With this approach we considered the properties of electric
circuits involving graphene. The existence of the induced
chiral magnetic current in a graphene subjected to mag-
netic field causes an interesting and unusual behavior of
such circuits. We discuss an explicit example of a circuit
involving the chiral current in a “graphene battery”. The
special properties of this circuit may be utilized for cre-
ating other “chiral electronic” devices.
5. Acknowledgements
I wish to thank my colleague Ricardo Ovalle (Escuela de
Ingeniería Electrica Electrónica, EIEE), for many useful
discussions on batteries.
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