Electronic transport through a graphene-based ferromagnetic/normal/ferromagnetic junction.

Abstract

Electronic transport in a graphene-based ferromagnetic/normal/ferromagnetic junction is investigated by means of the Landauer-Büttiker formalism and the nonequilibrium Green function technique. For the zigzag edge case, the results show that the conductance is always larger than e(2)/h for the parallel configuration of lead magnetizations, but for the antiparallel configuration the conductance becomes zero because of the band-selective rule. Therefore, a magnetoresistance (MR) plateau emerges with the value 100% when the Fermi energy is located around the Dirac point. In addition, choosing narrower graphene ribbons can yield wider 100% MR plateaus and the length change of the central graphene region does not affect the 100% MR plateaus. Although the disorder will reduce the MR plateau, the plateau value can still be kept about 50% even in a large disorder strength case. In addition, when the magnetizations of the left and right leads have a relative angle, the conductance changes as a cosine function of the angle. What is more, for the armchair edge case, the MR is usually small. So, it is more favorable to fabricate a graphene-based spin valve device by using a zigzag edge graphene ribbon.

Full text

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arXiv:1002.3665v1 [cond-mat.mes-hall] 19 Feb 2010
Electronic transport through a graphene-based
ferromagnetic/normal/ferromagnetic junction
Jiang-chai Chen1, Shu-guang Cheng2, Shun-Qing Shen3, and
Qing-feng Sun1,∗
1Beijing National Lab for Condensed Matter Physics and Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, China
2Department of Physics, Northwest University, Xi’an 710069, China
3Department of Physics, and Center of Theoretical and Computational Physics, The
University of Hong Kong, Hong Kong, China
E-mail: sunqf@aphy.iphy.ac.cn
Abstract.
junction is investigated by means of Landauer-B¨ uttiker formulism and the nonequilib-
rium Green’s function technique. For the zigzag edge case, the results show that the
conductance is always larger than e2/h for the parallel configuration of lead magneti-
zations, but for the antiparallel configuration the conductance becomes zero because of
the band-selective rule. So a magnetoresistance (MR) plateau emerges with the value
100% when the Fermi energy is located around the Dirac point. Besides, choosing nar-
rower graphene ribbons can obtain the wider 100% MR plateaus and the length change
of the central graphene region does not affect the 100% MR plateaus. Although the
disorder will reduce the MR plateau, the plateau value can be still kept about 50%
even in a large disorder strength case. In addition, when the magnetizations of the left
and right leads have a relative angle, the conductance changes as a cosine function of
the angle. What is more, for the armchair edge case, the MR is usually small. So, it is
more favorable to fabricate the graphene-based spin valve device by using the zigzag
edge graphene ribbon.
Electronic transport in a graphene-basedferromagnetic/normal/ferromagnetic

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Electronic transport through a graphene-based ferromagnetic/normal/ferromagnetic junction2
1. Introduction
Graphene, a two-dimensional single-layer crystal of carbon atoms arraying in a
honeycomb lattice, is a novel and exciting material [1].
of a graphene has a linear dispersion relation, and the dynamics of the charge carriers
obeys the massless Dirac-like equation [2].
the graphene has many peculiar properties, such as that the Hall plateaus occur at
half-integer multiples of ge2/h with the spin and valley degeneracy g = 4, and the
conductivity at the zero magnetic field has a non-zero minimal value [3]. For the neutral
graphene, its Fermi level is located at the Dirac points, the corners of the hexagonal first
Brillouin zone. In experiment, a gate voltage can be used to tune the Fermi level, which
can be above or below the Dirac points [4]. In addition, graphene also exhibits many
excellent transport characteristics: the high mobility [5], long spin relaxation length [6],
and stable behaviors under ambient conditions. At room temperature, its mobilities can
be above 104cm2/V · s, implying that the mean free path can be as long as a few hundred
nanometers. Because of weak spin-orbit coupling [7] and low hyperfine interaction [8], its
spin relaxation length can reach the order of the micrometer at room temperature. Thus,
graphene may be an excellent candidate for microelectronic applications, in particular
for spintronic applications [9].
In general, the charge carriers in graphene are not spin-polarized. For spintronic
applications, people try to inject the spin-polarized current or to induce the spin-
polarized carriers in the graphene. Recently, many works are focusing on this issue. For
example, Haugen et al [10] suggested that the spin-polarized carriers can be realized
by growing the graphene on a ferromagnetic (FM) insulator (e.g. EuO). Owing to the
magnetic proximity effect, an exchange split between the spin-up and spin-down carriers
in the graphene is induced, and then the carriers are spin-polarized. Based on the first-
principles calculations, Son et al [11] predicted that the zigzag graphene nanoribbon
becomes a half metal when an in-plane transverse electric field is applied. Also, Lin
et al [12] demonstrated that electron-electron correlation in armchair graphene nano-
ribbon can generate the flat-band ferromagnetism. On the experiment side, a large
spin injection into the graphene has been realized by connecting the graphene to an
FM electrode [6, 13, 14]. Furthermore, several groups [14, 15] have performed nonlocal
magnetoresistance(MR) measurements, in which a net spin current is brought between
the injector and detector.
The well-known application in spintronics is the spin-valve effect [16], in which
the resistance of devices can be changed by manipulating the relative orientation of
the magnetizations. Motivated by the spintronic application with the novel material,
the spin-polarized transport through graphene are currently attracting a great deal of
attentions [17, 18, 19, 20, 21, 22, 23]. Using the tight-binding model, Brey and Fertig
[20] studied the MR of the FM/graphene/FM junction in the limit of infinite width.
They found that the MR is rather small since the conductivity is weakly dependent on
the relative magnetization orientations of FM leads. Ding et al [21] investigated the
The low energy excitation
Because of the unique band structures,

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Electronic transport through a graphene-based ferromagnetic/normal/ferromagnetic junction3
similar device with two FM leads by a continuous model. The results showed that the
MR versus the bias exhibits a cusp around the zero bias in absence of external magnetic
field and oscillating behavior at the high magnetic field. Based on the first-principles
calculation, Kim and Kim [22] predicted that a graphene-based spin-valve device could
have a high MR.
In this paper, we study the conductance and MR of a graphene-based spin valve.
The device consists of a graphene nanoribbon coupling to two FM leads, as shown in
figure 1(a). Here the width of the device is finite in the order of 10 nanometers, i.e.
the size effect is considered. In recent experiments [24], a few 10-nanometer or sub-
10-nanometer graphene nanoribbons have already been fabricated successfully. For a
finite-width graphene nanoribbon, the wave vectors along the confined direction are
discrete, and the transverse subbands emerge. The characteristics of the subbands are
strongly dependent on the chirality of the graphene-nanoribbon edge [25], e.g. the zigzag
edge or armchair edge. So the conductance and MR should be strongly dependent on
the boundary condition of graphene nanoribbon. In addition, we consider that two FM
leads also have the hexagonal lattice structure as the graphene. In other words, the
leads are graphene-based FM or called the FM graphene. On the experiment, the FM
electrode (e.g. the cobalt electrode) usually overlaps on the graphene through a thin
oxide layer [6, 14]. Because of the magnetic proximity effect and the Zeeman effect, an
exchange split in the graphene underneath the oxide layer is induced, then its carriers are
spin-polarized. So the graphene covered with the FM electrode has the magnetization.
Then the spin-polarized charge carriers, driven by a bias, travel from one FM graphene
through the central normal graphene to another FM graphene.
In the tight-binding model the Landauer-B¨ uttiker formula and non-equilibrium
Green’s function method are applied to calculate the conductance and MR. For the
zigzag graphene ribbon case, we found that when the graphene nanoribbon is narrow
enough and its separation △ between the first subband and the zeroth subband is larger
than the exchange split energy M, the conductance for the antiparallel magnetization
configuration can almost be zero in a quite large energy window around the Dirac point.
But for the parallel configuration the conductance is always larger than e2/h, so the MR
exhibits a plateau with the value 100%. As the width of the nanoribbon increases, the
sub-band separation △ gradually decreases, and the MR can keep at the value 100% at
the beginning of △ > M, then decreases when △ < M. In the presence of the disorder,
the MR slightly drops, but even in quite large disorder it can keep the value 50%, which
is still much larger than 10%, the lower limit of the giant magnetoresistance effect. What
is more, for the armchair graphene nanoribbon case, the MR is always small regardless
of the width of nanoribbon. By comparing two cases of graphene ribbons with different
edges, it is more reasonable to apply zigzag graphene ribbons to the spin-valve devices.
The rest of the paper is organized as follows: In Section 2, we describe the model
and the details of the calculations. In Sections 3 and 4 we will give the numerical results
of the conductance and MR for the zigzag and armchair edge cases, respectively. Finally,
a brief conclusion is presented in Section 5.

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Electronic transport through a graphene-based ferromagnetic/normal/ferromagnetic junction4
?
?
?
?
?
?
?
?
?
?
?
?
?
Figure 1.
junction. (b) and (c): the conductance G vs. the Fermi energy EF for the different
magnetization M at the parallel configuration (θ = 0). The size of the central graphene
is N = 50 and L = 10.
(Color online)(a).The schematic of a graphene-based FM/normal/FM
2. Model and formulations
We consider a graphene-based spin valve (as shown in figure 1(a)) which consists of a
central normal graphene strip and two FM graphene ribbons. The total Hamiltonian
H of the device can be divided into four terms, H = HC+ HL+ HR+ HT, where
HC describes the central normal graphene region, HL and HR are the Hamiltonian
of the left and right FM-graphene leads, respectively, and HT = HTL+ HTR is the
coupling Hamiltonian of central region to the left and right leads. In the tight-binding
approaximation[26], the Hamiltonians HC, HL, HR, and HT can be written as
HC
=
?
?
−
i∈C
ǫCa†
iσIai−
?
<ij>(i,j∈C)
(ta†
iσIaj+ h.c.),
Hα=L,R=
i∈α
a†
i(ǫασI+ σ · Mα)ai
?
<ij>(i,j∈α)
(ta†
iσIaj+ h.c.),

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Electronic transport through a graphene-based ferromagnetic/normal/ferromagnetic junction5
HT
= −
?
<ij>(i∈C,j∈L,R)
(ta†
iσIaj+ h.c.),
where s =↑,↓ represents the spin of electrons; a†
with spin s on site i, and ai =
σ = (σx,σy,σz) are the Pauli matrices and σI is a 2 × 2 unit matrix. ǫC, ǫL, and
ǫRare the on-site energies (i.e. the energy of the Dirac point) in the center region, left
and right FM leads, respectively, which can be tuned by the gate voltage. The size of the
center graphene region is described by the width N and length L. In figure 1(a), a zigzag
edge graphene nanoribbon with N = 4 and L = 9 is shown. The terms including the
factor t in Hamiltonian describe the nearest neighbor hopping with the hopping energy
t. MLand MRare the magnetizations of the left and right FM leads. Here we allow
that the magnetizations MLand MRcan be along arbitrary direction. Without loss of
generality we assume that the magnetization MLof left lead is along the z-axis, then
ML= ML(0,0,1), and the magnetization MRof the right lead is along the direction
(θ,ϕ) with MR= MR(sinθcosϕ,sinθsinϕ,cosθ).
Before performing the calculation, we take a unitary transformation with ˜ ai= Uai
for all site i in the right FM lead, where the unitary matrix U is
is(ais) creates (annihilates) an electron
; <ij> stands for a nearest-neighbor pair.
?ai↑
ai↓
?
U =
?
cos(θ/2)
eiϕsin(θ/2) −cos(θ/2)
e−iϕsin(θ/2)
?
.
Under this unitary transformation, the Hamiltonians HC, HL, and HTL remain
unchanged, and HRand HTRvary into
HR =
?
−
i∈R
˜ a†
i(ǫRσI+ σ · M′
R)˜ ai
?
?
<ij>(i,j∈R)
(t˜ a†
iσI˜ aj+ h.c.),
HTR= −
<ij>(i∈C,j∈R)
(ta†
iU˜ aj+ h.c.),
where M′
right FM lead is along the direction of M′
spin space.
The current flowing through the device can be calculated from the Landauer-
B¨ uttiker formula [27]
R= MR(0,0,1). After the unitary transformation, the z-axis of the spin in the
R, and the Hamiltonian HRis diagonal in the
I = (e/h)
?
dǫTLR(ǫ)[fL(ǫ) − fR(ǫ)],
where fL/R(ǫ) = 1/{exp[(ǫ − µL/R)/kBT] + 1} is the Dirac-Fermi distribution function
of the left and right FM leads and TLR(ǫ) = Tr[ΓLGrΓRGa] is the transmission
coefficient, with the line-width function Γα(ǫ) = i[Σr
functions Gr(ǫ) = [Ga(ǫ)]†= 1/[ǫ − HC− Σr
function coupling to the leads, which has to be calculated numerically by solving the
surface Green’s function of the leads [28, 29]. After solving the current I, the linear
conductance G can be obtained straightforwardly, G = limV →0dI/dV , with the bias
α(ǫ) − Σa
L/Ris the retarded self-energy
α(ǫ)] and the Green’s
L− Σr
R]. Σr