Spin-Polarized Electrons in Bilayer Graphene Flakes

1.5 MB·English

by  P. A. Orellana

Spin-Polarized Electrons in Bilayer Graphene Flakes
P.A. Orellana,1,a) L. Rosales,2 L. Chico,3 and M. Pacheco2
1)Departamento de F´ısica, Universidad Cat´olica del Norte, Casilla 1280, Antofagasta,
Chile
2)Departamento de F´ısica, Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso,
Chile
3)Departamento de Teor´ıa y Simulaci´on de Materiales, Instituto de Ciencia de Materiales de Madrid, CSIC,
28049 Cantoblanco, Spain
(Dated: 22 January 2013)
3
1 Weshowthatabilayergrapheneflakedepositedaboveaferromagneticinsulatorcanbehaveasaspin-filtering
0 device. The ferromagnetic material induces exchange splitting in the graphene flake, and due to the Fano
2
antiresonancesoccurringin the transmissionof the grapheneflake as a function of flake length and energy,it

n is possible to obtain a net spin current. This happens when an antiresonance for one spin channel coincides
a with a maximum transmission for the opposite spin. We propose these structures as a means to obtain
J
spin-polarized currents and spin filters in graphene-based systems.

1
2
PACS numbers: 73.63.-b,72.50.Vp,75.25.-b

Keywords: spin-polarized system, graphene bilayer, transport properties
]
l
l
a
!"
h I. INTRODUCTION
-
s
e In the last years, there has been much interest in ex-
m #
ploring the unique properties of nanostructures for spin-
. tronicdevices,whichutilizethespindegreeoffreedomof
t
a theelectronasthebasisoftheiroperation1. Tothisend,
m novel ways of generating and detecting spin-polarized
- currents have been explored. For instance, Song et al.2
d FIG. 1. Schematic view of a graphene flake deposited onto
describedhowaspinfiltermightbeachievedinopensys-
n a graphene nanoribbon. This system can be considered as
temsbyexploitingtheFanoresonancesoccurringintheir
o a bilayer flake with monolayer ribbon contacts. The bilayer
c transmission characteristics. The Fano effect3,4 arises portion ofthesystem isplaced overaferromagnetic material
[ whenquantuminterferencetakesplacebetweentwocom- (blue).

petingpathways,oneconnectingwithacontinuumofen-
1
ergy states and the other with discrete states. In open
v
4 systemsinwhichthespindegeneracyofcarriershasbeen focused on graphene applications12. Recently, it has
7 lifted, this effect might be used as an effective means to been experimentally demonstrated the growth of Eu
9 generate spin polarization of transmitted carriers. The on graphene, thus bringing closer the possibility of the
4 idea is to tune the system so that a transmission reso- proposed scheme to induce spin polarized currents in
1. nance for one spin channel coincides with an antireso- graphene-basedsystems.13
0 nancefor the oppositespin. Inthis way,aspinpolarized In this work we study the transport properties of a
3 current arises. ferromagneticbilayergrapheneflake,showingthatitcan
1 Previous works have shown that graphene bilayer behaveasa spinfilter. We considerthatthe flakeis con-
v: flakesexhibitFanoantiresonancesinthetransmission5,6. tacted by monolayer nanoribbons, and the contacts are
i We propose to exploit these antiresonances to produce connectedto the same layerof the flake,as shownin fig-
X
spin polarized currents in a graphene-based system by ure 1. Such configuration can be achieved by placing a
r putting the graphene flake in contact with a magnetic monolayer flake onto a nanoribbon. It is also possible
a
insulator, such as EuO7,8. Exchange splitting is in- to form a bilayer flake by overlapping two semi-infinite
duced in the graphene flake due to the magnetic prox- nanoribbons5,14; however, we concentrate here in the
imity effect9–11. This produces an opposite energy shift bottom-bottom geometry, given that the changes due to
in the spin-up and down antiresonances in the con- theconfigurationofthecontactsmerelyproduceavaria-
ductance, yielding a spin polarization of the current. tioninthepositionofthetransmissionantiresonances5,6.
The feasibility ofspin-polarizedcurrentsin graphenede- We employ a one-orbital tight-binding model which we
vices is important for the development of all-graphene solveanalyticallywithin the single-modeapproximation.
electronics, which is one of the goals in the research This solution agrees perfectly with the numerical result
obtained by a Green function method in the one-mode
energy range.
Our main results are the following:
a)Electronicmail: [email protected] (i) We obtain an analytical solution for the spin-
2
2
lations. Just above the gap, which diminishes with the
ribbon width, the behaviour is as reported here.
With respect to the bilayer flake, we focus in one type
1.5 of stacking, the AA, taking into account only one hop-
ping parameter γ between atoms placed directly on top
1
of each other. We assume that γ = 0.1γ . Note that,
1 0
although the AB stacking is more stable for graphite,
T 1
the directorAAstackinghasbeenexperimentallyfound
in few-layer graphene17. Moreover, the antiresonances
which give rise to the energy windows where spin po-
0.5 larization takes place in these systems are present both
in AA- or in AB-stacked bilayer graphene flakes, so the
physics of the problem can be studied in any of the
two geometries. Our choice of the AA stacking is mo-
0
0 0.2 0.4 0.6 0.8 1 tivated by that fact that, as we show in this paper, the
Energy(γ ) AA-stacked system can be easily mapped into a one-
0
dimensionalchainwithtwohoppings,yieldinganalytical
expressionsofthetransmissionthatperfectlyfitthecon-
FIG. 2. Transmission versus energy for spin-up carriers of
thearmchairbilayerflakeofwidthW =5andlength L=30 ductanceobtainednumericallyinthesinglemoderegime.
(black thin line) and the coupled-chain model of equivalent Within this description, we measure the length of the
length,n=120(bluethickline). Thelatterisshownonlyfor flakeintranslationalunitcells, whichcorrespondsto the
theone-mode energy intervalof thearmchair ribbon system. number of 4-atom units along the flake. For the width
weusethestandardnotation,givingthenumberofdimer
chains across the flake.
dependent transmission through bilayer graphene flakes. The Hamiltonian of the complete system can be writ-
The comparison of this analytical result to the numer- tenasasumoftheleftandrightleadpartsplusacentral
ically computed transmission, obtained by a recursive part, which constitutes the bilayer flake:
Green function method, is excellent in the one-mode en-
ergy range.
(ii)Theanalyticalexpressionsforthetransmissionallows H=H +H +H (1)
L R F
us to explore thoroughly the parameter space, locating
the most advantageous system sizes to obtain a net spin with
current.
(iii) We have found that the maximum spin polarization HL,R =−γ0 c†i,σcj,σ, (2)
is obtainedwhen sharpantiresonancesareproducedin a hiX,ji,σ
plateau with maximum transmission. These correspond
to quasi-localized states in the bilayer graphene flake. where c† (c ) stands for the creation (annihilation)
i,σ i,σ
Thus, the tuning of the flake length is important to ob- operator of an electron with spin σ on site i, and the
tain a net spin current. sum hi,ji is over nearest neighbors. The Hamiltonian of
the central conductor, i.e., the bilayer graphene flake, is
given by
II. MODEL
H =−γ c† c −γ (c† c +h.c.)
As we focus on the electronic properties close to the F 0 i,σ,α j,σ,α 1 i,σ,u i,σ,d
Fermienergy,weemployaπ-orbitaltight-bindingmodel, hi,Xji,σ,α Xi,σ
which gives an excellent description of the low-energy + σ∆ c† c (3)
ex i,σ,α i,σ,α
propertiesofgraphenesystems. Weconsiderthein-plane iX,σ,α
nearest-neighbor interaction given by a single hopping
parameter γ , which is approximately 3 eV, concentrat- In this latter expression α runs over u, d, i.e., the up-
0
ing on metallic armchair nanoribbons, which can play per and lower flake, respectively, the spin index σ takes
the role of contacts in this system. We would like to the values ±1,and∆ is the exchangesplitting induced
ex
note that, in principle, all armchair nanoribbons show bythe ferromagneticinsulator. Forgraphenesystemson
a gap due to edge reconstruction, as evidenced by first- EuOwe takethe value ∆ =0.001γ 7,8. For the sakeof
ex 0
principles calculations15. A change of the hopping pa- simplicityintheanalyticalapproach,weassumethatthe
rameter at the edges can reproduce this effect within sameferromagnetictermisincludedinbothlayersofthe
the one-orbital approximation15,16; however, we choose flake. However,wehavenumericallycomputedbothpos-
all hoppings to be equal in order to solve analytically sibilities(i.e.,spininteractioninoneorintwolayers)and
the problem and then compare to the numerical calcu- found the main difference is that the energy shift of the
3
antiresonances is twice when the ferromagnetic coupling tem is uniform with a high symmetry, one can take ad-
affects the two layers. vantage of this and employ an analytical approach to
The transport properties of this structure can be ob- solve the problem in the single-mode approximation.
tained numerically by computing the Green function The tight-binding equations of motion for the ampli-
of the system, which yields the conductance in the tudeofprobabilitytofindanelectroninsitei(j)ofatom
Landauer-Kubo approximation18. However, as this sys- A(B) in a bilayer ribbon flake are given by
(E−σ∆ )ψA,α =γ (ψB,α +ψB,α +ψB,α )+γ ψA,α¯
ex j,m,σ 0 j,m,σ j−1,m−1,σ j−1,m+1,σ 1 j,m,σ
(E−σ∆ )ψB,α =γ (ψA,α +ψA,α +ψA,α )+γ ψB,α¯ . (4)
ex j,m,σ 0 j,m,σ j+1,m−1,σ j+1,m+1,σ 1 j,m,σ
Assumingthatthesolutionsinthey-directionareacom- (4)arereducedtotheequationsofmotionoftwocoupled
binationofplanewavesoftheforme±imq,theexpressions dimer chains (see Appendix), which eventually lead to a
spin-dependent transmission given by
(4− E2)G2
γ2 σ
T (E)= 0 , (5)
σ [(G2 −F2−1) E −2F ]2+(G −F +1)2(1− E2)
σ σ 2γ0 σ σ σ 4γ02
where G , F are given by
σ σ
1 E E
Gσ = 2(cid:20)Un−1(cid:18) σ2,+(cid:19)+Un−1(cid:18) σ2,−(cid:19)(cid:21) 0.20 T
0
0.18 0.050
U Eσ,+ U Eσ,− 0.10
1 n−1 2 n−1 2 0.16 0.15
Fσ = 2 Un(cid:16)(cid:16)Eσ2,+(cid:17)(cid:17) + Un(cid:16)(cid:16)Eσ2,−(cid:17)(cid:17). (6) () 0.14 0000....22330505
y 0.12 0.40
IntheaboveU (x)aretheChebyshevpolynomialsofthe g 0.45
n r 0.50
second kind and Eσ,± =E−σ∆ex∓γ1. ne 0.10 00..5650
In order to quantify the spin-dependent transport, it E 0.08 0.65
0.70
isnecessarytoevaluatethedegreeofspinpolarizationof 0.06 0.75
0.80
the electric current. There are several ways to do this, 0.85
but following Ref.2 we introduce the weighted spin po- 0.04 00..9905
larization as: 0.02 1.0
0.00
|T −T |
↑ ↓ 20 40 60 80 100 120 140 160 180 200
P = T . (7)
σ σ
|T +T |
↑ ↓ n
Notice that this definition takes into account not only
the relativefractionofoneofthe spins, butalsothe con- FIG. 3. Contour plot of the transmission versus energy and
coupled-chainlengthnwithoutferromagneticgate(∆ =0).
tribution of those spins to the electric current. In other ex
words, we will require that not only the first term of the
right-handsideof(7)tobeoforderofunity,butalsothe
transmission probability Tσ(E) should be significant. tion of the structure, as done previously for similar sys-
tems without spin polarization5,19. Given that the shift
produced by the effective exchange is small, we just
III. RESULTS present the results for the spin-up channel. As the arm-
chair ribbon length is measured in units comprised of
We begin by comparing the analytical results for fouratoms,theequivalencebetweentheone-dimensional
the one-dimensional chain to the conductance of the chain and the armchair ribbon length is n=4N. Fig. 2
graphene nanoribbon system obtained numerically em- shows the transmission for the n = 120 one-dimensional
ployingadecimationtechniquetoobtaintheGreenfunc- chain model along with the transmission for the arm-
4
0.5
11
n=40 up
00..44 n=40 up
00..88 down
down
00..66 0.3
T σ
P
00..44 00..22
00..22 0.1
00
00 0.05 0.1 0.15 00..22 00
0.072 0.074 0.076 0.078 0.08
11
11
n=60
00..88 n=60
00..88
00..66
T σ 00..66
P
00..44
00..44
00..22
00..22
00
00 0.05 0.1 0.15 00..22 00
0.048 0.05 0.052 0.054 0.056
11
11
n=120
00..88
00..88 n=120
00..66
T σ 00..66
P
00..44
00..44
00..22
00..22
00
00 0.05 0.1 0.15 00..22 00
γ ) 0.022 0.024 0.026 0.028 0.03
Energy( γ )
0 Energy(
0
FIG. 4. (Color online) Spin-dependent transmission for the
FIG. 5. Spin polarization versus energy for different flake
graphene flakes of length n = 40,60 and 120 with ∆ =
ex lengths, with ∆ =0.001γ .
0.001γ . Thesolid black line is for spin up,and thesolid red ex 0
0
line represents thespin down.
functionofenergyandflakesizefor∆ =0. Withinthis
ex
chair nanoribbon system of width W =5 (two complete single-mode approximation,our results presentan excel-
hexagons) and bilayer flake length N = 30. The energy lent agreement to those obtained within the continuum
interval is chosen to display the results for the armchair Dirac model for graphene5.
system with two modes contributing to the total trans- Fromthisfigure3itisclearthatforflakelengthn=30
mission. Asexpected,theresultsfortheone-dimensional there is a minimum in the transmission, and that the
chaintransmissionandtheoneobtainedforthearmchair sharpest antiresonances occur for n = 60, slightly above
nanoribbon system perfectly coincide in the whole one- 120andsoon,wherespinpolarizationwillbemaximized.
mode energy window. Thus, in the following, we focus The n = 40 has a noticeable oscillation of the transmis-
ontheanalyticresultsinthesingle-modeapproximation. sion,butdoes notreachthe maximumvalue of1. To see
Theanalyticalsolutionoftheone-dimensionalcoupled more clearly this effect, we choose three lengths for the
chain allows us for exploring thoroughly the dependence flake,namely, 40,60, and 120,and plot the transmission
of the transmission coefficient on the flake length n as versus energy for the system placed over a ferromagnet,
a function of energy. As the exchange energy is very as described above. Figure 4 shows the transmission for
small, producing a shift of a few meV in the antireso- spin-up and down electrons in these three cases. We see
nances or any other features of the conductance, it is thatthen=40casehasasmootheroscillationthatdoes
sufficienttostudythetransmissionforzeroexchangeen- not rise to transmission 1; however, both n = 60 and
ergy in order to identify the situations more convenient n = 120 reach the maximum value of the conductance
for spin filtering. Figure 3 presents a contour plot of the and have very sharp antiresonances,as anticipated from
transmissionforthe one-dimensionalcoupledchainsas a the contour plot.
5
allow us to propose this spin filter device.
P
down
0.20
IV. SUMMARY
0.18 0
0.10
0.16 We have studied the transport properties of a bilayer
0.20
)00.14 0.30 graphene flake placed in close proximity to a ferromag-
( netic insulator. This produces an effective Zeeman split-
y 0.12 0.40
g tingthatcanbeexploitedtoobtainaspinfilteringeffect.
r 0.50
e 0.10 We have shown that the tight-binding model for the fer-
n 0.60
E 0.08 romagnetic flake can be mapped into a one-dimensional
0.70
chain that can be analytically solved in the single mode
0.06 0.80
approximation. Theanalyticalsolutionallowsforathor-
0.90
0.04 ough exploration of the system parameters and the sub-
1.0
sequentidentificationofthemostconvenientsystemsizes
0.02
for a maximum spin polarization.
0.00
40 60 80 100 120 140 160 180 200
n Appendix A: Derivation of the transmission for the coupled
FIG.6. (Coloronline)Contourplotofweightedspindownpo- linear chains
larization for the bilayer graphene flake with ∆ =0.001γ ,
ex 0
as a function of theFermi energy and flakelength. In this Appendix we give in more detail the procedure
toderivethespin-polarizedtransmission(5)analytically.
We use the plane-wave form of the solutions
For these three flake lengths, we compute the spin po- ψA,α =Aq,αe±imq
larization as indicated in equation (7). Even for n=40, j,m,σ j,σ
where the transmission oscillates smoothly and the min- ψB,α =Bq,αe±imq, (A1)
j,m,σ j,σ
imum values for opposite spins are barely distinguish-
which, substituted in the equations of motion (4), yield
able, there is a non-negligible value of the polarization,
depicted in the top panel of figure 5. However, there is (E−ǫj,σ)Aqj,,σα =γ0(Bjq,,σα+2cosqBjq−,α1,σ)+γ1Aqj,,σα¯
not appreciable spin filtering, given that the two peaks (E−ǫ )Bq,α =γ (Aq,α+2cosqAq,α )+γ Bq,α¯ (A2)
j,σ j,σ 0 j,σ j+1,σ 1 j,σ
overlap in energy. For the other flake lengths, namely,
Werestrictourcalculationtothecaseofmetallicarm-
n=60 and n=120, the antiresonances in the transmis-
chair nanoribbons and lower energies, close to the Dirac
sion are very sharp and well-separated in energy. This
point (q = π/3). For this case, (A2) can be mapped
yields a maximum polarization, shown in the center and
into the equations for two linear chains by replacing
bottom panels of figure 5.
Aq,α =fα , Bq,α =fα ,
Theanalyticalsolutionallowsustoexplorethoroughly j,σ l,σ j,σ l+1,σ
thesystemsizesandenergyranges,withtheaimtoiden- (E−σ∆ )fu =γ (fu +fu )+γ fd
ex l,σ 0 l+1,σ l−1,σ 1 l,σ
tify the optimal cases for spin filtering. Figure 6 is a
(E−σ∆ )fd =γ (fd +fd )+γ fu (A3)
contour plot of the weighted spin down polarization as ex l,σ 0 l+1,σ l−1,σ 1 l,σ
a function of energy and bilayer flake length. As the Employing the symmetric and antisymmetric combi-
spin-up and down antiresonances are very close in en- nation of the solutions, f± = fu±fd, the equations of
ergy,thisplotissufficienttoidentifythecaseswithmax- motion become
imalweightedpolarization. Themaximaofthe polariza- (E−σ∆ −γ )f+ =γ (f+ +f+ )
ex 1 l,σ 0 l+1,σ l−1,σ
tion contour plot follow lines that correspond to quasi-
localized states that give rise to sharp antiresonances. (E−σ∆ex+γ1)fl−,σ =γ0(fl−+1,σ+fl−−1,σ). (A4)
It is enlightening to compare this contour plot to the In order to obtain the transmission through the sys-
one depicting the transmission (figure 3), where we can tem we consideropen boundary conditions. Plane waves
observe vertical wide bands of maximum transmission incomingfromtheleftwithamplitudeunityarereflected
(in red) that shrink for some values of energy. The lines fromthe scatteringregion(the bilayerflake)with ampli-
drawn by the polarization maxima go precisely through tude r , and are transmitted to the right contact with
σ
the points where the maximal transmission regions nar- amplitude tσ:
row dramatically into a marked minimum. fd =eikl+r e−ikl, l<0
l,σ σ
In the literature, it is has been showed that the Fano
fd =t eikl, l>N. (A5)
effect is robustunder the severalconditions, suchas,the l,σ σ
presence of impurities, electron-electron interaction, ex- This leads to a system of equations on the reflection
ternal fields, etc.4. In this sense, we believe that our and transmission amplitudes. The transmission is given
model captures the essence of this phenomenon, which by T (E)=|t |2.
σ σ
6
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