Manipulating the fully spin-polarized edge currents in graphene ribbon

0.4 MB·English

by  Lei Xu

Manipulating the fully spin-polarized edge currents in graphene ribbon
Lei Xu,1 Jin An,1,∗ and Chang-De Gong2,1,†
1National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
2Center for Statistical and Theoretical Condensed Matter Physics,
and Department of Physics, Zhejiang Normal University, Jinhua 321004, China
(Dated: January 18, 2011)
Electron fully spin-polarized edge states in graphene emerged at the interfaces of a nonuniform
magneticfieldarestudiednumerically inatight-bindingmodel,with boththeorbitalandZeeman-
1 splitting effects of magnetic field considered. We show that the fully spin-polarized currents can
1 be manipulated by a gate voltage. In order to make use of the fully spin-polarized currents in the
0 spin related transport, a three-terminal experiment is designed and expected to export the fully
2 spin-polarized currents. This may haveimportant applications in spin based nanodevices.

n
a PACSnumbers: 72.25.-b,73.20.-r,71.70.Di
J

5
I. INTRODUCTION z (a)
1 Magnetic strip graphene layer

y
]
i Recently, spin-dependent transport phenomena have x
c
s received intensive studies because of their potential ap- L
- plications in spintronics devices. Understanding how to
l (b) 0.02 (c)
mtr goenneeoraftteheankdeympaoniinptuslaftoer tthhee sdpeivne-lpooplmareiznetdofcutrhreenstpsinis- Bz mveacgtonre tpico tfeineltdial Ay
based devices. Rashba spin-orbit interaction, which can B2 B2L E0.00
t. becontrolledbyan(oreffective)electricfield,providesan B1 B1L
a
m efficient way to achieve a spin-polarized current, for ex- -0.02
ample in graphene ribbon,1 doped semiconductors2 and 0 L x L+l 2L+l 0.36 0.38 0.40ky/ 0.74 0.76 0.78
-
d 2D electron system.3 In most of these applications, the
n inducedchargecurrentsareonlypartiallypolarized,4but FIG. 1: (Color online) (a) Schematic illustration of the sys-
o a fully spin-polarized (FSP) current is much more diffi- tem: an array of ferromagnetic strips with different magneti-
c zations is placed on the top of a rectangular graphene layer.
cult to achieve experimentally. Up to now, FSP cur-
[ Inthetwo-stripcase,(b)themagneticfield(solidline)andits
rent is found to be realized only in very limited materi-
gaugepotential(dottedline)asfunctionsofx,and(c)thecor-
1 als, for example, in the half-metallic states of the ferro-
responding energy spectrum for a zigzag-edge graphene rib-
2v magnetic metals5,6 and ferromagnetic semiconductors,7 bonwithL=213nm,l=2.13nm,B1 =2TandB2 =−3T.
where the electronic valance band is partially filled
7
for one spin component, whereas completely empty for
9
2 the other. From material viewpoint, two-dimensional
. (2D) graphene, as a candidate material of new genera-
1
tion electronics8–10 for its high mobility and low carrier
0
density, is expected to have great application in spin- states at the interfaces, especially, in graphene.
1
1 tronics devices.11–14 How to realize a FSP current in
: graphene is then quite meaningful for future graphene-
v In this paper, we study monolayer graphene in the
based electronic devices. In graphene nanoribbon with
i presence of a stepwise NMF, which can be realized ex-
X zigzag edges, it has been predicted from a fist-principle
perimentally in several ways,21–23 to exhibit systemati-
r calculation15 and numerical simulations16,17 that with
a cally how to manipulate the FSP currents generated at
the application of a transverse electric field, a half-
theinterfacesoftheNMF.Thesituationisschematically
metallic state can be achieved, which leads naturally to
shown in Fig. 1(a). For the uniform magnetic field case,
FSPedgecurrentspropagatingalongsampleboundaries.
the FSP state is found to be absent, and the transport
On the other hand, for a 2D electron system, a suffi- in graphene can be described by the so-called counter-
cient strong magnetic field will polarize the spins of all propagating chiral edge states,24 when the chemical po-
electrons in the same direction as the field, but only a tentialµisbetweenthespingapduetoZeemansplitting.
much smaller magnetic field is needed to achieve a FSP However, for the stepwise NMF case, several spin gaps
current.18 The cost is that a nonuniform magnetic field are opened and the FSP edge states may emerge at the
(NMF) must be applied to the system. Actually, it has magnetic interfaces when µ is between the gaps. In the
been predicted that a partially spin-polarized current followingwewilladdresstheapproachtomanipulatethis
will appear at the interfaces of inhomogeneous magnetic novel FSP edge currents. Furthermore, an experimental
fields,19,20 but much less attention are taken on the FSP setup is designed to export the edge currents.
2
0.004
II. MODEL
(a3) (b3)
(a2) (b2)
(a1) (b1)
Webeginourdiscussionbythetight-bindingmodelon E0.000 (a) Case I 2 1 (b) CaseII 2 1
a honeycomb lattice in the presence of a perpendicular
magnetic field B=(0,0,B (x)), which is given by,
z
-0.004
0.32 0.36 0.72 0.76 0.36 0.38 0.74 0.76 0.78
ky/ ky/
H0 =−t X[eiRijA·dlcˆ†icˆj+H.c.]−XhZ(x)(ni↑−ni↓), (a1) (b1)
<ij> i 2T 2T
(1)
where the first term describes electrons hopping be- 3T -3T
tween the nearest neighbors, suffering an additional
phase causedby the orbitaleffect of magnetic field, with (a2) (b2)
cˆ† = (cˆ† ,cˆ† ) electron creation operators, whereas the 2T 2T
i i↑ i↓
second one the Zeeman splitting term. Here h (x) =
Z
1gµ B (x) with g as the Land´e factor. The magnetic 3T -3T
2 B z
fieldB (x) is assumedtochangeonlyinx direction,and
z (a3) (b3)
the stepwise one which is focused on in this paper, has
2T 2T
the following form,
3T -3T
B 0<x<L,
 1
Bz(x)=B1+ B2−lB1(x−L) L<x<L+l, FIG.2: (Coloronline)Toppanels: Electronenergyspectrum
B2 L+l<x<2L+l. of graphene ribbon under a stepwise magnetic field with one
 (2) interface, with L=213 nm (1000 zigzag chains) for (a) Case
For the gauge potential A, the Landau gauge A = I with B1 = 2 T and B2 = 3 T, (b) Case II with B1 = 2 T
(0,A (x),0)isadopted,withA (x)= xB (x′)dx′given and B2 = −3 T. Here the transition length for the varia-
y y R0 z tion of magnetic field long x direction is l = 2.13 nm (10
as:
zigzagchains). Theredandblacklinesdenotebandsofspin-
up and spin-down states, respectively. The interface edge
B x 0<x<L, states are labelled with thick lines. The Zeeman energy is
 1
Ay(x)=B1x+ B22−lB1(x2−L2) L<x<L+l, cwhiothseBnt=ob2eTo.nTehteenbtlhacokfdthaeshlaedrgleisntesLLgivspeatchiengreopfrtehseenrteagtiiovne
B x− B2−B1l L+l <x<2L+l.
 2 2 (3) cpoorsrietisopnosndoifngthfeoucrh-etmerimcainlaploctoenntfiigaulsraµt.ionOstihnerHpaallnmelse:asuTrhee-
Forsimplicity,inthe followingdiscussion,energyismea- ment. Thespatial distribution of theedge currentsisshown,
sured in unit of the hopping integral t. wheretheredandblacklineswitharrowsrepresentthespin-
up and spin-down electron edge currents, respectively. The
thicklineswitharrow at theinterfacesin (b3)representthat
each current is carried by two edge states.
III. RESULTS AND DISCUSSIONS
A. Generation and manipulation of the FSP edge
currents
To analyze all possible edge states, we choose open
(periodic)boundaryconditioninthex(y)direction,and
numericallydiagonalizetheHamiltonianH onarectan- Let’s focus our discussion on the regime where the
0
gularsample underaNMF toobtainthe electronenergy chemical potential µ is between the zeroth LLs of the
spectrum. Inoneinterfacecase,thespectrumisshownin two spin species. To make things more explicit, we show
Fig.1(c),whichissymmetricwithrespecttozeroenergy in Figs. 2(a) and 2(b) the energy spectrum around zero
duetotheparticle-holesymmetry(which,strictlyspeak- energy. The splitting of the zeroth LL and formation of
ing, should be combined with the time-reversal symme- two spin gaps ∆1 and ∆2 can be clearly seen. When µ
try)preservedeveninaNMF.Thespectrumiscomposed lies within the interval |µ| < ∆1/2 or |µ| > ∆2/2, there
oftwoseriesofLandaulevels(LLs),includingallthebulk existonly sampleboundaryedge states,andno interface
states of the system, and branches of edge states, which edge states atall. However,whenµ lies betweenthe two
play a dominant role in quantum transport in the pres- spin gaps ∆1/2 < |µ| < ∆2/2, interface edge states are
ence of magnetic field. The edge states can be classified generated.
into the normal ones located at the sample boundaries, For 0 < µ < ∆ /2, edge currents are propagating
1
and the interface ones located at the interface where the in opposite directions for opposite spin polarizations at
magnetic field changes abruptly. both sample boundaries [see Fig. 2(a1)], which is similar
3
0.005 (a4) (b4) (c4) (d4)
(a3) (b3) (c3) (d3)
(a2) (b2) (c2) (d2)
E0.000 (a) Case III 3 2 1(a1) (b1)(b) Case IV 3 2 1 (c) Case V 1 2 (3c1) (d(d1)) Case VI 3 2 1
-0.005
0.280.320.36 0.680.720.76 0.25 0.30 0.350.65 0.70 0.75 0.36 0.40 0.76 0.80 0.32 0.36 0.68 0.72 0.76
ky/ ky/ ky/ ky/
(a1) (b1) (c1) (d1)
2T 3T 2T 2T
3T 2T -4T 4T
4T 4T 3T -3T
(a2) (b2) (c2) (d2)
2T 3T 2T 2T
3T 2T -4T 4T
4T 4T 3T -3T
(a3) (b3) (c3) (d3)
2T 3T 2T 2T
3T 2T -4T 4T
4T 4T 3T -3T
(a4) (b4) (c4) (d4)
2T 3T 2T 2T
3T 2T -4T 4T
4T 4T 3T -3T
FIG. 3: (Color online) Similar to that of Figs. 2, but for four representative configurations of two interfaces (a) Case III with
B1 = 2 T, B2 = 3 T and B3 = 4 T, (b) Case IV with B1 = 3 T, B2 = 2 T and B3 = 4 T, (c) Case V with B1 = 2 T,
B2 =−4 T and B3 =3 T, and (d) Case VI with B1 =2 T, B2 =4 T and B3 =−3T.
to the quantum spin Hall state in graphene by consid- graphenesample,andbychanginggraduallythevoltage,
ering intrinsic SO interaction,4 and also similar to the one can vary the carrier density in the sample, so the
statepredictedinRef.24inauniformmagneticfield. For chemical potential µ gradually. We take Case I as an
µ >∆ /2, however, the edge currents at sample bound- example. Whenµlieswithintheminimumspingap|µ|<
2
ariesbecomenormalbecausetheypropagateinthesame ∆ /2,theinterfaceedgecurrentchannelisclosed. As|µ|
1
direction for both spin polarizations [see Fig. 2(a3)]. In- increases, the interface channel with a FSP edge current
terestingly,whenthechemicalpotentialµlieswithinthe is opened if µ lies within the two gaps ∆ /2 < |µ| <
1
gap∆ /2<µ<∆ /2,theinterfacechannelsareopened ∆ /2. Finally the channel is closed again in the interval
1 2 2
and two spin-downedge states with the same flowingdi- |µ| > ∆ /2. This implies that one can control the FSP
2
rection emerge at the interface, and certainly, they are edge currents electrically.
FSP [Fig. 2(a2)]. Meanwhile, the edge current at one
The spin gap due to Zeeman splitting is estimated as
sample boundary becomes normal whereas that at the
∼ 3 K at B = 2 T, whereas the corresponding first
other boundary is still left novel. For case where the
LL E is estimated to be about 500K. Furthermore, it
1
magnetic fields in the two sides of the interface are in
was argued that the spin gap can be enhanced to a few
opposite directions, similar conclusion can be made ex-
hundred kelvin for a realistic magnetic field due to elec-
cept that when µ > ∆2/2, no FSP state is present but tron exchange interaction.24 So the FSP current is ex-
the interface current channel is not closed and there ex-
pected to be observed experimentally in relatively large
ist a normal edge current at the interface instead [see
range of temperatures. To visualize the effect of the
Figs. 2(b1)-2(b3)].
Zeeman splitting clearly and not changing the physics,
From the viewpoint of application, one can use the relatively large spin gaps have been chosen for calculat-
properties mentioned above to manipulate the FSP edge ing the energy spectrum in Fig. 2. We also note that
currents. Imaging placing a voltage gate on the top of the localization character along x direction for the edge
4
0.004 0.4
statesis welldefinedbecausethe magnetic lengthhere is (a) 0.008 (b)
lB = ~c/eB =18 nm, much less than the width of the 0.2
p
sample L (=213 nm). 0.004- 2/2 E - 1/2
WenextconsiderthecaseofNMFwithtwointerfaces. E0.000 " ! 0.0
Therearetwelveintotaldifferentconfigurationsforpat-
-0.2
tern of magnetic field, but among them, only four( Case
IwIhIi-chCaisseshIoVw)nairne Freigp.re3s.enTtahtriveee aspnidnogfagprseaatreinfoterrmesetd, -0.0040.32 0.36ky/ 0.72 0.76 -0.4 - 2/2 - 1/2 E 1/2 2/2
in each case, so similar to Fig. 2, the states between the
FIG. 4: (Color online) (a) Electron energy spectrum of
zero LLs can be generally classified into four regimes,
grapheneinthepresenceofRashbaSOinteractionforCaseI.
according to the position of the chemical potential µ.
Case III is very special. For this interesting case, if ParametersarethesameasthatinFig.2(a)andλ=∆1/50.
(b) Thespin tilt angle θ as a function of E for different edge
generally, there are more interfaces with the magnetic
states. The inset in (b) is an enlarged view of the tilt an-
field in each region in the same direction and increasing gle within the spin gap −∆2/2 < µ < −∆1/2. Open and
monotonically, the FSP interface edge channels will be filled triangles denote the interface edge states, while open
opened one by one from the low-field interface to high- and filled circles denotethe sample boundary edge states.
fieldone,where the oldone closingandnew one opening
happen simultaneously. For Case IV, when µ increases
gradually from 0 < µ < ∆1/2 to µ > ∆3/2, the two continuously with ky. Therefore, theoretically speaking,
interface channels are opened simultaneously, after that the FSP interface current is still preserved at T = 0 K
one channel is closed and finally the other one is closed. [Fig. 4 (a)], except that the edge current is now polar-
For Case V and Case VI, although the FSP interface ized along other direction instead which slight deviates
channels are opened and closed in different orders,these from z axis by an angle θ(k ). In the spin gap region
F
channels are not closed in the end but become normal ∆ /2 < |µ| < ∆ /2, for the parameters shown in Fig.4,
1 2
ones instead [Figs. 3(c4) and 3(d4)]. thetiltinganglefortheinterfaceedgestatesisestimated
Based on the above discussion, we come to a general to be |θ| < 0.0074. It is very small and changes nearly
conclusionthat for a NMF with many interfaces,so long linearly with energywith the slope ∆θ/∆E ∼1.3 in this
as the magnetic length is much less than the width of gap [see the inset in Fig. 4 (b)]. Physically, the slope
each uniform region, there always exist FSP edge states ∆θ/∆E isbelievedtobe nearlyproportionaltoλ,which
eitheratalltheinterfacesoratseveralofthem,whenthe is rather small in real situations (For a perpendicular
chemical potential µ lies within the gap ∆min/2<|µ|< electricfieldEz ∼50V/300nm,4,27 λ∼0.07K.25,26). At
∆max/2. All the calculations above are performed for finite temperatures, in order to obtain a almost FSP in-
the zigzag graphene ribbon. For the armchair graphene terfacecurrent,thethermodynamicalenergyk T should
B
ribbon, similar edge states and so similar conclusion can bemuchlessthanthe spingap(∆ −∆ )/2,whichises-
2 1
be obtained. timated to be about 30 K for B = 2 T.24 Therefore, in
many situations achievable in experiments, the spin-flip
processes induced by the Rashba SO interaction can be
B. Rashba spin-orbit coupling effect negligible, and an almost FSP current can be realized.
We now discuss the effect of the spin-flip processes
caused by spin-orbit(SO) interaction. To estimate the C. Exporting the FSP edge currents
influenceoftheSOinteractiononthepolarizededgecur-
rent,weonlyconsiderherethe RashbaSOinteractionas
So far, we have shown explicitly how to generate and
an example, whose Hamiltonian can be written as
manipulate the FSP edge currents, but another associ-
HR =λ X[ieiRijA·dlcˆ†i(σ×dij)·ˆzcˆj +H.c.], (4) asttiellduannsdolvinetde,rewshtiinchg iqsuehsotwiontowehxipchortnatthueraFlSlyPacruisrersenist
<ij>
to serve as a current source for a spin related device.
whereλis the RashbaSOcouplingstrengthandσ is the This is not a trivial question, because to export the cur-
Paulimatrix. d isavectorpointingfromjtoitsnearest rent, one has to destroy the interface in some extent
ij
neighbor i, and ˆz is a unit vector in z direction. Figure to introduce another lead, but the FSP current is com-
4 (a) shows the spectrum at low energies with Rashba posed of edge states, which will disappear when the in-
SO interaction, which is found to be almost unchanged terface is destroyed or removed away. For this purpose,
comparedwiththecasewithoutSOcoupling[Fig.2(a)]. we design an experimental setup shown schematically In
However, in contrast to the case without SO interaction Fig. 5(a), which is expected to export FSP current to
where electrons are polarized along z axis, the spin po- the possible spin related device connected with the arm-
larization of the edge states in the presence of the SO chair graphene sample by a narrowgraphene strip. Here
interaction can be described by S = (sinθ,0,cosθ) with the armchair graphene ribbon is adopted, because for
θ the tilting angle deviated from the z axis, changing thezigzaggrapheneribbon,theexistenceoftheflatedge
5
(a) (c) where ΣR =[ΣA ]† is the self-energy due to lead p and
z y Nanodevice z y 3 the elempe,nσts of tph,σe matrix ΣRp,σ are given by
x x
B B
1 3B/2 2 1 3B/2 2 ΣRp,σ(i,j)=−Xχm(i)χm(j)eikma, (7)
m
0.2
(b) Lead 3
(d)
0.1 with the function χ (i) = 2 sin mπi describing
y m qN+1 N+1
E0.0 ssppiinn--duopwn " ! x Lead 2 the transverse profile of mode m in lead p and km the
-0.1 wavevectordeterminedbytheequationµ=−2coskm−
2cos mπ . HereN isthenumberofsitesconnectedwith
-0.2 Lead 1 N+1
2600 26n40 2680 the leads and m = 1,2,··· ,N. The retarded Green’s
function of the graphene sample GR is written as
FIG.5: (Coloronline)(a)Schematicdiagramofourproposed
experimental setup for application in spintronics device. (b) GR =[EI−H −ΣR]−1, ΣR = ΣR (8)
Thecorrespondingeigenvaluescalculatedonthesamplein(a) 0 X p,σ
withthemagneticfluxperhexagonφ=2π/80and1.5×2π/80 p,σ
(B ∼ 1000 T). The sample size is 40×120 consisting of 40 and GA = [GR]†. Here, to make the calculation simple,
armchairchainswith120sitesineachchain,whilethenarrow
thethreeleadsareassumedtobedescribedbyanearest-
graphene strip size is 4×40. The Zeeman energy is taken as
neighbortight-bindingmodelonasquarelatticewiththe
hZ =0.1 and hZ =0.15 for the two regions respectively. (c)
same hopping integral as graphene.
Thecorrespondingthree-terminaldevicewiththesamplesize
as(b). (d)Geometry of thegraphenesampleconnected with The transmissioncoefficients Tpσq are plotted in Fig. 5.
threeleadsforcalculatingtransmissioncoefficients. Toreduce For spin-up electrons, T↑ and T↑ show quasi-periodic
12 21
the scattering between the sample and the leads, there exist oscillation and the maxima of them are the quantized
a transition region with length 1.14 nm, where the magnetic
value1. Thisoscillatingbehaviorissimilartothatfound
fieldismadetobelinearly decreased tozerowhen approach- in graphene-based ferromagnetic double junctions,31and
ing leads.
can be attributed to electrons’ multiple reflections be-
tweenterminal1and2. Astheenergyincreasesgradually
bandswillcauseferromagneticinstability andleadtolo- from0to0.2,bothT↑ andT↑ areapproachingto1,sug-
12 21
calferromagneticpolarizationsatsample edges,28 giving gestingoneleft-movingspin-upedgestateononesample
rise to spin-flip processes. boundaryandoneright-movingoneontheother,sharing
To testify this idea, we first calculate the eigenvalues thesimilarpicturewithFig.2(a). However,mostimpor-
of this particular system for an open boundary condi- tantly,wefindthatwithintheerrorrange,T↑ =T↑ =0
13 31
tion. We find that LLs are still present even in the andT↑ =T↑ =0inthewholeenergyrange0<E <0.2
23 32
condition that the interface is partially destroyed by the weareinterestedin. Thisiscorrespondingtotheabsence
introduced lead. However, states for both spin species of spin-up edge states at the interface.
exist between the LLs ∆1/2 < |µ| < ∆2/2 [Fig. 5(b)], For spin-down electrons, when 0 < µ < ∆1/2 and
leaving the question whether spin-up electrons in this µ > ∆ /2, T↓ ≈ T↓, the situation is similar to that
regime contribute to the current flowing through the de- 2 12 21
for spin-up electrons. However,when ∆ /2<µ<∆ /2,
1 2
vice still unaddressed. Therefore, in order to confirm T↓ andT↓ showstronglyfluctuations,anddeviatefrom
thatthecurrentflowingthroughthedeviceisFSP,were- 12 21
the quantized value 2 [Fig. 6(a)], which should be taken
fer to the three-terminalsetup asillustratedinFig.5(c),
intheidealcaseshowninFig.2(a). Thismaybeascribed
wheretherectangulargraphenesampleisconnectedwith
to the bonds breaking atthe interface of terminal 3,and
three ideal semi-infinite leads. The detail of these con-
lattice mismatching at the interfaces of terminal 1and 2
nectionsbetweengraphenesampleandtheleadsisshown
which results in electrons scatteringfrom edges states to
in Fig. 5(d). In this situation, four bonds across the
the nearby zeroth LL [see Fig. 5(b)]. In the same range,
interface are broken, so that lead 3 made of the nar-
the fact that T↓ > T↓ > 0 and T↓ > T↓ > 0 indi-
row graphene strip can be connected with the sample. 13 31 32 23
catesthatboththe left- andright-movingcurrentsexist,
This strip is then upright so as to avoid the influence
of magnetic field. According to the Fisher-Lee relation, but the net current is left moving. The finiteness of T2↓3
the electron transmission coefficient from lead q to lead also implies that a small fraction of spin-down electrons
incoming from terminal 3 have to be scattered to sam-
p for multi-terminal Landauer-Bu¨ttiker formula is given
by29,30 ple boundaries via bulk states, since there is no current
channel at the interface from terminal 3 to 2. Similarly,
Tpσq =Tr[ΓσpGRΓσqGA], (5) the finiteness of T3↓1 > 0 means that the right-moving
ThescatteringratematrixΓσ duetothecouplingtolead spin-down electrons propagating along sample bound-
p arieshavesomeprobability to be scatteredto terminal3
p has a form
viabulkstates. Theseprocessesareschematicallyshown
Γσ(i,j)=i[ΣR (i,j)−ΣA (i,j)], (6) in Fig. 6(g). We remark that although these scattering
p p,σ p,σ
6
2.0 (a) (b) processes will cause resistance, spin is conserved since
T1.5 TT1221 TT1221 tthoetrheeisconnocslupsinio-nflitphpartotcheesseelsecctornosnidceurrerdenhterfleo.wTinhgisoluetadosf
1.0 terminal 3 has spin-down polarization, and so is FSP.
1.000 1.000
0.5 Inactualapplication,onecanapplytheelectricalvolt-
0.998 0.996 age to the three terminals as V = V = V and V = 0,
1.0 1 2 3
(c) (d) then the FSP (spin-down) electrons will be transported
T13 T13 from lead 3 to leads 1 and 2. In this situation, the FSP
T0.5 T31 T31 currentflowing out ofterminal 3, I3 =(T3↓1+T3↓2)Ve2/h
is estimated as 0.85Ve2/h ∼ 0.95Ve2/h [see Fig. 6(h)].
0.0 Indeed, the width of the strip is d = 0.87 nm and the
magneticlengthisaboutl =0.8nm. Thefactofd∼l
B B
1.0
(e) (f) guaranteesthattherearesufficientelectronsthroughthe
T T strip. Therefore, the graphene sample with the designed
23 23
0.5 T32 T32 setup under a NMF can be viewed as the FSP current
T source for a spin related nanodevice.
0.0 IV. SUMMARY
0.0 0.1 0.2 0.0 0.1 0.2
E E
1.0 Insummary,wehavenumericallyinvestigatedtheFSP
(h) edgecurrentsattheinterfaceingrapheneunderanonuni-
(g)
1 3 2 2Ve/h)0.5 I3 froernmtsmisargenaelitziecdfibelyd.aTgahteemvoalntiapguel,aatinodniotfstphoeteFnStiPalcaupr--
I (3 plication in spintronic devices is also discussed. An ex-
0.0
perimental setup is designed and is expected to export
0.0 0.1 0.2 the FSP current, which is confirmed numerically by a
E
three-terminal transmission calculation.
FIG.6: (Color online) (a)-(f)Thetransmission coefficients as
functionsoftheFermienergy. Thecoefficientsin(a),(c)and
(e) are for spin-down electrons, while the others for spin-up
electrons. The dash lines at E = 0.1 and 0.15 indicate the
Acknowledgments
positionsoftwospingaps. (g)Schematicdiagram ofelectron
current flow when ∆1/2 < µ < ∆2/2 (i.e., 0.1 < µ < 0.15).
(h)Thecurrentfollowingoutofterminal3,I3,whenterminal The author J. An thanks X. G. Wan, H. H. Wen, J.
voltages are chosen as V1 = V2 = V and V3 = 0. The inset X. Li, Q. H. Wang for useful discussions. This work
in (a) is an enlarged view of T1↓2 and T2↓1 in the gap 0<µ< was supported by NSFC Projects No. 10504009, and
∆1/2,and theinset in(b)isan enlarged viewof T1↑2 and T2↑1 No. 10874073,and973ProjectsNo. 2006CB921802,No.
in theenergy range 0<E <0.2. 2006CB601002,and No. 2009CB929504.
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