Generation of pure bulk valley current in graphene

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by  Yongjin Jiang

Generation of pure bulk valley current in graphene
Yongjin Jiang1,∗ Tony Low2, Kai Chang3, Mikhail I. Katsnelson4, and Francisco Guinea5
1 Department of Physics, ZheJiang Normal University, Zhejiang 321004, People’s Republic of China
2 IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA
3 SKLSM, Institute of Semiconductors, Chinese Academy of Sciences,
P.O. Box 912, Beijing 100083, Peoples Republic of China
4 Radboud University Nijmegen, Institute for Molecules and Materials,
Heyendaalseweg 135, 6525AJ Nijmegen, The Netherlands
5 Instituto de Ciencia de Materiales de Madrid. CSIC. Sor Juana In´es de la Cruz 3. 28049 Madrid, Spain
(Dated: September 11, 2012)
2
Thegenerationofvalleycurrentisafundamentalgoalingraphenevalleytronicsbutnopractical
1
waysofitsrealizationareknownyet. Weproposeaworkableschemeforthegenerationofbulkvalley
0
currentinagraphenemechanicalresonatorthroughadiabaticcyclicdeformationsofthestrainsand
2
chemical potential in the suspended region. The accompanied strain gauge fields can break the

p spatial mirror symmetry of the problem within each of the two inequivalent valleys, leading to
e a finite valley current due to quantum pumping. An all-electrical measurement configuration is
S
designed to detect the novel state with pure bulk valley currents.

9
PACSnumbers: 72.80.Vp,85.85.+j,73.63.-b

]
l
l
a Apart from pseudospin (chirality), charge carriers in z
h Lead L Lead R
graphene are also characterized by the valley index y y
- x y 0
s (sometimes called isospin) originated from the existence
x
me oftwoconical(Dirac)pointsperBrillouinzone[1]. Aval- graphene q x
ley polarized state requires the absence of time reversal 0
t. symmetry, as the two valleys are related by this symme- gate
a
m try. (a) (b)
- Motivated by the growing field of spintronics[2], it
d FIG.1: Illustrationanddefinitionsofthe(a)graphenebased
was proposed that the manipulation with the valley
n NEMdeviceand(b)crystallographiccoordinatesystemsused
o index may open a new way to transmit information in the paper.
c through graphene, and different manipulation schemes
[ were proposed[3–10]. After initial enthusiastic attitude,

interest in “valleytronics” declined somehow, as it was graphene with the Dirac Hamiltonian,
1
v soon realized that a valley polarized current will be de-
0 graded by intervalley scattering induced by atomic scale H±((cid:126)k,A(cid:126))=(cid:126)vf[±(kx∓Ax)σx+(ky∓Ay)σy]+µ1l2, (1)
7 disorder[1], making it difficult the maintenance of val-
7 where +(−) denotes K(K(cid:48)) valley index, v is the Fermi
ley polarized states. In addition, a number of proposals f
1 velocity and µ is the chemical potential. A(cid:126)((cid:126)r) is an ef-
werebasedonthespatialseparationofvalleycurrentsat
.
9 fective gauge field describing the modifications to the
zigzag edges[3], which requires well-defined edge orienta-
0 hopping amplitudes induced by the strain fields u ((cid:126)r)
tion and at the same time free of short-range scattering. ij
2
[16, 17], and has opposite signs at the two valleys as re-
1
We present a new scheme to induce valley polarized quired by time-reversal symmetry. We described the de-
:
v currents in graphene which avoids some of the pitfalls of formation of the suspended region, i.e. −L < y < L,
Xi previous proposals. The breaking of time reversal sym- with a simple uniaxial strain given by uy2y = u an2d
metry is achieved by means of time dependent fields, u = u = 0. It will be shown below that for arbi-
r xx xy
a instead of a magnetic field. The induction of valley trary crystallographic orientation θ (see Fig.1b), A(cid:126)((cid:126)r) in
polarization by a.c. fields has been proven[11] in MoS2 the suspended region is given by the expression
where optical radiation was used in order to excite val-
ley polarized charge carriers. As in this experiment, A(cid:126)((cid:126)r)= βκu(-cos3θ,sin3θ), (2)
the scheme discussed below generates valley currents a
throughout the entire system. However, the a.c. driv-
where β ≈ 2 is the electron Gruneisen parameter and
ing force in our proposal is due to mechanical vibrations
κ ≈ 1 is a parameter related to graphene’s elastic prop-
ofananoelectrical-mechanicalsystem(NEMS[12–15]),as 3
erty as described in [16]. a ≈ 1.4˚A is the interatomic
illustrated in Fig.1a.
distance. Adiabatic cyclic variations of the internal pa-
We employ a continuum-medium description of rameters, such as deformations in the strains (u) and
2
chemical potential (µ) in the suspended region, over a magnetic vector potential reads:
work cycle can constitute a scheme for adiabatic quan-
tum pumping[18, 19] (for the general theory of quantum βκ
A = (u −u ),
pumping, see Ref. [20, 21]). For the charge pumping x a xx yy
one needs to break the inversion symmetry, e.g., mak- 2βκ
A = − u . (4)
ing the right and left leads different (e.g., by different y a xy
doping)[21]. Here we will demonstrate that for the case
For arbitrary orientation i.e. θ (cid:54)= 0, Eq.(1) and Eq.(4)
ofsymmetricleadsthevalleycurrentthroughthesystem
have to be recasted in the new coordinate frame (x,y).
is, in general, nonzero. In this case, the pumping cur-
The two coordinate system are related by
rent through each channel will be shown to follow the
generalrelation(withperiodicboundaryconditionalong
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
x x cosθ sinθ
the transversal x direction), =R 0 , R= . (5)
y y −sinθ cosθ
0
Ipump,K(k ,θ)=−Ipump,K(cid:48)(−k ,θ), (3)
L/R x L/R x The wave vector(cid:126)k transforms in the same way as(cid:126)r such
where the subscript L/R refer to left/right leads(I is that (cid:126)k ·(cid:126)r is a rotational invariant quantity. The strain
definedpostivewhenthecurrentflowsoutofthedeLv/iRce). field is defined as, uij = 21(∂∂uxji + ∂∂xuji), which is a sym-
This significant relation results from an intrinsic sym- metric tensor of rank two. Hereafter we use the sub-
mety of graphene NEMs, as we will show later. Eq.(3) script/superscript “0” to denote physical quantities in
represents the central result of our paper. Below, we the original frame (x ,y ). Thus, we have (cid:126)k = R−1(cid:126)k
0 0 0
present the derivations leading to Eq.(3) and discuss the and
physical consequences that follows, such as the genera-
tion of pure bulk valley current and its possible experi- u0 −u0 = cos2θ(u −u )−2sin2θu ,
xx yy xx yy xy
mental detection. 1
We start by considering the case θ = 0, where the u0xy = 2sin2θ(uxx−uyy)+cos2θuxy. (6)
trench is directed along the zigzag direction (see the
(x ,y )coordinatesystemshowninFig.1b). Inthiscase, By using the new coordinates in the Dirac Hamiltonian,
0 0
the Hamiltonian has the form of Eq.(1) and the pseudo- we can transform it to the rotated frame:
(cid:18) µ/(cid:126)v e∓iθ[(±k −ik )∓(±A −iA )](cid:19)
H±((cid:126)k,A(cid:126))=(cid:126)v f x y x y , (7)
f e±iθ[(±k +ik )∓(±A +iA )]) µ/(cid:126)v
x y x y f
wherewehavedefinedthepseudo-magneticfieldA(cid:126)inthe the derived Hamiltonian given by Eq.(7) is physically
rotated frame as[22]: equivalent to Eq.(1). Suppose that between −L <
2
y < L, the system has uniaxial strain u = u and
(cid:18)A (cid:19) (cid:18) βκ(u −u )(cid:19) 2 yy
x =R(3θ) a xx yy . (8) uxx =uxy =0, hence, Eq.(8) is reduced to
Ay −2aβκuxy
(cid:26)0, |y|> L
Eqs.(7) and (8) constitute the continuum-medium de- (Ax,Ay)((cid:126)r)= (βκu)(−cos3θ,sin3θ), |y|≤ L2 (9)
scription of stained-graphene in an arbitrarily rotated a 2
frame. Here follows two comments about this general
withtheexpressiongiveninEq.(2)forthesuspendedre-
form. First, if we perform the gauge transformation of
gion. In the following, assuming a particular geometry
the wave function on B sublattice ψ±K −→ ψ±Ke±iθ,
B B (the width W(cid:29)L), the x direction is treated as trans-
the Dirac Hamiltonian can be made explicitly invariant
lationally invariant. From Eq.(7) and Eq.(9), one can
(i.e. Eq.(1)) under rotation. Thus, we can simply drop
easily see that
the factor e±iθ in Eq.(7) in subsequent discussion. Sec-
ond, from the definition for pseudo-magnetic field, it is H+(k ,k ,A(cid:126),θ)=H−(−k ,k ,A(cid:126),−θ). (10)
x y x y
obvious that the form is of 2π/3 periodic in θ, which
reflects the trigonal symmetry of the underlying honey- We can call such combined symmetry as the crystalline-
comb lattice. angle-combined mirror symmetry in the continuum-
Next, we describe the quantum pumping problem medium model. It turns out that such combined sym-
based on graphene NEMs[19, 23]. As discussed above, metry has a significant consequence on the relation of
3
pumping currents in each valley, i.e., Eq.(3), as will be Symmetries related with t and r . Next, we discuss
d d
elaborated further. symmetry properties of the scattering amplitudes. First,
Duetothementionedsymmetry,wemayfocusonlyon we note that ϕ (d) = ϕ (−d), ϕ (d) = −ϕ (−d) and
G G L L
the K Dirac cone, whose Hamiltonian has the following they are independent of the sign of θ. Then, we can
form for the different regions (i.e., i = L,R,G denotes obtain the relations satisfied by C ’s: C (d)=−C (−d),
i 1 1
y <−L, y > L, |y|≤ L, respectively): C (−d)∗ = C (d). Based on these relations and the
2 2 2 2,3 2,3
odd parity of A (θ), we arrive at:
(cid:18) (cid:15) /(cid:126)v (k˜ −ik˜ ) (cid:19) y
H+((cid:126)k,A(cid:126)) = (cid:126)v i f x y i , (11)
i f (k˜ +ik˜ ) (cid:15) /(cid:126)v t (k ,θ) = t (k ,−θ),
x y i i f d x −d x
r (k ,θ) = r (k ,−θ). (16)
where we have defined (k˜ ,k˜ ) = (k −A ,k −A ) . d x d x
x y i x x y y i
In this paper, we consider mainly the symmetric case Symmetry of pumped valley-dependent current. Ac-
(cid:15)L =(cid:15)R (cid:54)=(cid:15)G. cordingtotheadiabaticpumpingtheory[20]andthesym-
The eigenenergies and eigenstates of the Hamiltonian metry relations satisfied by r and t , we obtain the fol-
d d
in Eq.(11) read lowing relation for the pumping current Ipump,K(k ,θ)
L x
(cid:113) for K valley:
E ((cid:126)k) = (cid:15) +s(cid:126)v k˜2+k˜2,
i i f x y
(cid:32) 1 (cid:33) ILpump,K(kx,θ)−IRpump,K(kx,−θ)
ψi((cid:126)k) = ei(kxx+kyiy) (cid:126)vf(k˜x+ik˜y)i , (12) = ieω (cid:90) 2π/ωds[dr1(θ)r (θ)∗− dr−1(θ)r (θ)∗], (17),
Es−(cid:15)i 4π2 ds 1 ds −1
0
where s = ±1 refers to electron/hole band, respectively.
where we’ve used s as time symbol to avoid confusion
Due to translational invariance in x direction, k is the
x with transmission coefficient. Now, using the symme-
same in all three regions. We consider now the equilib-
try relations satisfied by C ’s and the fact that the com-
riumsituationwhereallthreeregionscanbedescribedby i
acommonchemicalpotentialµ. Obviously,k˜ isalways mon complex factors for C2 and C3, i.e., ieiϕL, is in-
(cid:113) xi dependent of time(see Eq.(19) for typical time depen-
real since kx is real. Then, k˜yi = ± kf2i −k˜x2i, where dence of pumping parameters for graphene NEM), we
k = µ−(cid:15)i and the ± sign is selected to give the correct can simply prove that the integrand in Eq.(17) is zero.
sifgin in(cid:126)vtfhe group velocity, depending whether it is an By further taking into account the current conserva-
incident, transmitted, or reflected waves. Note that k˜y tion condition( ILpump,K(kx,−θ) + IRpump,K(kx,−θ) =
can be purely imaginary representing evanescent waves 0) and the symmetry relation guaranteed by Eq.(10)
in the central region. (i.e., Ipump,K(cid:48)(−k ,θ) = Ipump,K(k ,−θ)), we arrive at
L x L x
It is straightforward to calculate the scattering matrix Eq.(3), the relation for pumped valley-dependent cur-
for our device. Without loss of generality, we can focus rent, which is the central result of this paper.
on the case with the electron doping (µ−(cid:15) >0) in the Next, we discuss some direct consequences of Eq.(3).
L
leads. The scattering coefficients are calculated to be Byintegratingoutthetransversalwavevectork ,wecan
x
get the following relation,
C (d)+C (d)
rd = e−ikyLL 2 C1(d)3 , Ipump,K(θ)=−Ipump,K(cid:48)(θ). (18)
L/R L/R
td = e−ikyLL−4sinϕLCsi(ndϕ)GeiAyLd, (13) Eq.(18)meansthatthetotalpumpingcurrentatagiven
1
lead is opposite for different valleys. Thus, the total
where r and t are the reflection and transmition coef-
d d chargecurrentisexactlyzero. Thissituationisanalogous
ficients with d = 1(−1) corresponding to the cases with
tothepurespincurrentgenerationinspintronics[24,25],
incidentwavesfromy =−∞(leftlead)andy =∞(right
thus we call this effect pure valley current generation.
lead), respectively; ϕ and ϕ are defined through
L G In summary, we have shown that the application of an
alternating back gate voltage to graphene NEMs can in-
k ±ik d k˜ ±ik˜
e±iϕL = x yL ,e±iϕG = xG yG. (14) duce a pure valley current via adiabatic pumping. In
k k
fL fG adiabatic pumping theory, there are two necessary con-
ditions for finite charge pumping effect, i.e., time rever-
The C (d) in Eq.(13) are defined as
i
sal symmetry breaking and mirror symmetry breaking
C (d) = 4isin(k˜ Ld)(1−cosϕ cosϕ ) (15) of the the left/right leads[21]. The above derivations
1 yG L G
− 4cos(k˜ Ld)sinϕ sinϕ , show that the valley pumping effect can be realized in a
yG L G seemingly symmetrictwo-terminalgeometrywithidenti-
C (d) = −2i(1+e2iϕL)sin(k˜ Ld),
2 yG cal leads. However, the mirror symmetry of the system
C (d) = 2ieiϕL[sin(k˜ Ld+ϕ )+sin(k˜ Ld−ϕ )]. (i.e. y→−y) has actually been broken for each valley
3 yG G yG G
4
L~15nm
KQc0.10 L~100nm
e
cl
y0.05
C
r
e
p
e 0.00
g
r
a
Ch-0.05
d FIG. 3: (Color online) Pictorial sketch of the pumping gen-
e
p-0.10 eration of pure valley current and an all-electrical detection
m /p3 2/p3 p
u scheme. The Hall voltage difference V12 = V1 − V2 and
P -4 -3 -2 -1 0 1 2 3 4 V34 = V3 −V4 across the NEMs is predicted to bear oppo-
Crystal Orientation q site sign due to the flow of pure valley current.
FIG. 2: (Color online) The pumped charge per cycle of the
type x axis (θ = π/2+nπ/3,n ∈ Z) while it is zero at
K valley,denotedasQK. Thevalleycurrentcanbeobtained
c
by eωQK/2π. The width is fixed to be 5000a (≈ 700nm), zigzag-type x axis (θ = nπ/3,n ∈ Z). The periodicity
c
calculated for two different length as indicated. We fixed the 2π/3 with θ is easily seen. The valley current can be
phase difference of the driving parameters φ=π/2 and take defined as Ipump,v(θ)=Ipump,K(θ)−Ipump,K(cid:48)(θ), which
|A(cid:126) |/k ≈0.002,whichamountstoastrainu≈1.75×10−4. issimplytwicethevalueofIpump,K(θ). Fortypicalreso-
av f
nance frequency of ω ≈ 10MHz[14] to 0.16GHz [15], we
arrive at numerical estimates Ipump,v ≈0.1−10pA/µm,
when θ (cid:54)= nπ. Such symmetry breaking is embodied
3 a quantity measurable in experiment.
in the continuum theory through a nonzero A compo-
y The possibility of pure bulk valley current in this sim-
nent. For comparison, we have Ipump,K/K(cid:48)(θ = 0) = 0 ple pumping scenario looks very promising. The prob-
L
when Ay = 0, which can be infered from the relation lem is how to probe the valley current. Here we propose
Ipump,K/K(cid:48)(k ,θ)=−Ipump,K/K(cid:48)(k ,−θ)implicitinthe an all-electrical measurement as shown in Fig.3. From
L x L x
discussion below Eq.(17). As is worthy to recapitu- Eq.(3), we can infer that the charge current pertaining
late, the symmetry property intrinsic to the graphene to carriers from the valley K not only has opposite lon-
NEM(i.e., thecrystalline-angle-combined mirror symme- gitudinal component with respect to the charge carriers
try) can impose strong restriction of pumping currents from the valley K(cid:48), but also their transversal velocities
on different valleys. For more quantitative understand- are opposite. As pictorially shown in Fig.3, we expect
ing, we present some numerical results of Ipump,K(θ) (in charges accumulating on opposite edges on the two sides
L
terms of the pumped charge per cycle ) for the K valley oftheNEMs. Basedonthisobservation, wepredictthat
in Fig.2 using some typical experimental parameters, as the resultant Hall voltage on the left lead has opposite
discussed below. signwiththatontherightlead,i.e.,Sign(V12/V34)=−1,
As stated above, the strain (u) and Dirac potential which is the characteristic feature of the bulk valley cur-
((cid:15) ) in the suspended region are modulated by the ac rent flow.
G
backgatevoltage. Nearresonance,theydiffersbyaphase Theabovediscussioncanbemademoreclearifwecon-
differenceφwithatypicaltimedependencegivenby(with sider the valley-dependent pumping Hall current, which
conventional time symbol t)[19]: canbecalculatedasILH/aRll,K(θ)=(cid:80)kxILpu/Rmp,K(kx,θ)kkxy.
From Eq.(3) and the particle conservation law, we can
(cid:15)G = (cid:15)G0[1+δ(cid:15)cos(ωt)]12, obtain:
A(cid:126) = A(cid:126) [1+δAcos(ωt+φ)]2. (19)
0 IHall,K(θ)=IHall,K(cid:48)(θ)=−IHall,K(θ). (20)
L/R L/R R/L
Assuming typical numbers for the static part (cid:15) =
G0
(cid:15) = 0.3eV, A(cid:126) = 0.02k (cos3θ,−sin3θ), and the os- Such Hall current on different leads can result in the op-
L 0 f
cillating amplitude δ(cid:15) = δA = 0.2 , we calculated the posite Hall voltage due to the charge accumulation near
pumpedchargepercycleQK(θ)forK valley,asshownin edges. Thus the existence of valley polarized currents
c
Fig.2. By definition, the pumping current Ipump,K(θ) = possibly can be detected in the leads by means of nonlo-
L
eωQK(θ)/2π. Our calculation indicates a nearly linear cal multiterminal measurements[26].
c
scaling(unshown) of the pumping effect on length L of Ourdiscussionaboveisrestrictedtothecasewithsym-
the NEMs, which is similar to results in[19]. As explic- metrical leads. It is straightforward to extend our study
itly shown, the maximum pumping effect is reached for to the more general case with differently doped leads.
the crystallographic angles corresponding to armchair- In the general situation, the current is not purely valley
5
current,i.e.,Ipump,K(θ)(cid:54)=−Ipump,K(cid:48)(θ),thusthecharge [5] J. M. Pereira, F. M. Peeters, R. N. Costa Filho and G.
L/R L/R
pumping current is finite. An example is again given by A. Farias, J. Phys. Condens. Matter 21, 045301 (2009).
the case θ = 0, where Ipump,K(0) = Ipump,K(cid:48)(0)[19] and [6] Z.Wu,F.Zhai,F.M.Peeters,H.Q.Xu,andK.Chang,
L/R L/R Phys. Rev. Lett. 106, 176802 (2011).
the valley current is zero. [7] D.Xiao,W.YaoandQ.Niu,Phys.Rev.Lett.99,236809
To conclude, we have shown that through pumping (2007).
induced by mechanical vibrations bulk valley polarized [8] J.L.Garcia-Pomar,A.Cortija,andM.Nieto-Vesperinas,
currents can be generated in graphene leads connecting Phys. Rev. Lett. 100, 236801 (2010).
[9] H. Schomerus, Phys. Rev. B 82, 165409 (2010).
the graphene resonator with trench directed at a gen-
[10] T. Low and F. Guinea, Nano Lett. 10, 3551 (2010).
eral crystallographic angle. We have demonstrated that
[11] D.Xiao,G.-B.Liu,W.Feng,X.Xu,andW.Yang,Phys.
the generated current is purely valley current (with zero
Rev. Lett. 108, 196802 (2012).
net charge pumping current) in the setup with the same [12] J.S.Bunch,A.M.vanderZande,S.S.Verbridge,I.W.
doping rate in the graphene leads. Together with an all- Frank,D.M.Tanenbaum,J.M.Parpia,H.G.Craighead
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direction of exploiting the valley degree of freedom, thus [13] D. Garcia-Sanchez, A. M. van der Zande, B. L. A. San
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[14] C. Chen, S. Rosenblatt, K. I. Bolotin, W. Kalb, P. Kim,
YJ and KC acknowledges the support from the Na-
I.Kymissis,H.L.Stormer,T.F.HeinzandJ.Hone,Nat.
tionalNaturalScienceFoundationofChina(undergrant
Nanotech. 4, 861 (2009).
No.11004174(YJ) and No. 10934007(KC)). TL also ac- [15] A. Eichler, J. Moser, J. Chaste, M. Zdrojek, I. Wilson-
knowledges partial support from NRI-INDEX. MIK ac- Rae and A. Bachtold, Nat. Nanotech. 6, 339 (2011).
knowledges a financial support from FOM, The Nether- [16] H. Suzuura and T. Ando, Phys. Rev B. 65, 235412
lands. FG acknowledges financial support from Spanish (2002).
[17] M. A. H. Vozmediano, M. I. Katsnelson and F. Guinea,
MICINN(grantsFIS2008-00124, FIS2011-23713, CON-
Phys. Rep. 496, 109 (2010).
SOLIDER CSD2007-00010), and ERC, grant 290846.
[18] E. Prada, P. San-Jose, and H. Schomerus, Phys. Rev. B
80, 245414 (2009).
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