Single-parameter pumping in graphene

Pablo San-Jose,1 Elsa Prada,2 Sigmund Kohler,2 and Henning Schomerus3

1Instituto de Estructura de la Materia (IEM-CSIC), Serrano 123, 28006 Madrid, Spain2Instituto de Ciencia de Materiales de Madrid (ICMM-CSIC), Cantoblanco, 28049 Madrid, Spain

3Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom(Dated: October 12, 2011)

We propose a quantum pump mechanism based on the particular properties of graphene, namelychirality and bipolarity. The underlying physics is the excitation of evanescent modes entering apotential barrier from one lead, while those from the other lead do not reach the driving region.This induces a large nonequilibrium current with electrons stemming from a broad range of ener-gies, in contrast to the narrow resonances that govern the corresponding effect in semiconductorheterostructures. Moreover, the pump mechanism in graphene turns out to be robust, with a sim-ple parameter dependence, which is beneficial for applications. Numerical results from a Floquetscattering formalism are complemented with analytical solutions for small to moderate driving.

PACS numbers: 05.60.Gg, 73.40.Gk, 72.80.Vp, 72.40.+w

I. INTRODUCTION

Ratchets and pumps are devices in which spatio-temporal symmetry breaking turns an ac force withoutnet bias into directed motion.1,2 If the time-dependenceenters via only one parameter, the pump current van-ishes in the adiabatic limit.3 Therefore, single-parameterpumping requires non-equilibrium conditions enforced bydriving beyond the adiabatic limit. This distinguishessingle-parameter pumps from devices that operate withoriented work cycles, like turnstiles4 or sluices.5 Quantumpumps can be implemented with quantum dots in a two-dimensional electron gas (2DEG) driven by microwaves,6

surface-acoustic waves,7 ac gate voltages8–11 or non-equilibrium noise.12 Here we propose a single-parameterpump based on the particular properties of graphene andshow that these display a broad-band response, in con-trast to 2DEG-based ratchets or pumps for which isolatedresonances govern the effect.6,12,13

Owing to the chiral and gapless nature of charge car-riers in graphene, an electron hitting a potential stepin this material may propagate forwards as a hole withopposite momentum.14,15 Due to this Klein tunneling itis difficult to confine electrons by electrostatic potentials.This phenomenon even extends to evanescent modes, i.e.,modes that decay exponentially as a function of the bar-rier penetration. In graphene, electrons populating suchmodes can tunnel a large distance.16 This impedimentto electrostatic confinement constitutes a drawback formany switching and sensing applications.

An inspection of present quantum pump designs sug-gests that graphene pumps would be negatively affectedby this issue as well. For example, pumps have been real-ized with electrostatically defined double quantum dotsin a 2DEG.6,12 In these experiments, a driving field in-duces dipole transitions of an electron from a metastablestate below the Fermi energy in the, say, left dot to ametastable state in the right dot. Subsequently the elec-tron will leave to the right lead, and an electron fromthe left lead will fill the empty state in the left dot. The

emerging pump current therefore requires resonant con-ditions with a pair of energetically well-defined states,i.e., well isolated states with long life times, which donot exist in graphene.

Nevertheless, the alternative, graphene-specific mech-anism identified here allows realizing highly efficientpumping, which may indeed outperform conventional de-vices. The large pump current emerges in the bipolarregime around the Dirac point. This is due to a scatteringprocess where a whole continuum of evanescent modes ispromoted into unidirectionally propagating states, whichcouple well to the leads because of chirality. Since thismechanism does not rely on intricate resonance condi-tions, but invokes graphene’s intrinsic features of bipolar-

L L

W

UHtL=U×cosHΩtL

HaL

HbL HcLGraphene 2DEG

qk

¶ÑΩ

FIG. 1. Sketch of a single-parameter graphene pump con-nected to two leads. The left half of the graphene ribbon isexposed to an ac gate voltage. Dispersion of (b) grapheneand (c) a 2DEG, with longitudinal and transverse momentak and q, including the branches of evanescent modes. Photonabsorption (and in the case of graphene also emission) mayexcite an evanescent mode to propagating.

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103.

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v2 [

cond

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.mes

-hal

l] 1

1 O

ct 2

011

2

ity and chirality, the pump current is robust and displaysa simple parameter dependence curve, as is desirable forelectronic sensor applications.

This paper is organized as follows: In the next sectionwe describe our proposal for a single-parameter pumpand give typical values for length and energies scales.In Sec. III we introduce the Floquet scattering theorythat applies to non-equilibrium pumping in graphene.We first present the general formalism and then con-sider the weak driving regime, where we derive analyticalexpressions for the one-photon transmission probability.In Sec. IV we present numerical results for the pumpedcurrent comparing the performances of graphene and a2DEG pump. Moreover, we compare the numerics withthe semiclassical approximation derived in the precedingsection and find good agreement. The specific pump-ing mechanism by excitation of evanescent modes is ex-plained in Sec. IV A. Finally, we conclude in Sec. V.

II. DESCRIPTION OF THE SYSTEM

As a single-parameter pump setup, sketched in Fig. 1a,we consider a graphene ribbon of length 2L and width Wattached to two metallic electron reservoirs, with its lefthalf driven by a time-dependent gate. The driving byvariation of only one parameter excludes the emergenceof an adiabatic pump current.3,17 However, consideringthat spatial symmetry is broken by the placement of thegate, a finite dc current arises in non-equilibrium condi-tions, which can be achieved by non-adiabatic driving.In order to assess the importance of graphene’s chiralityand bipolarity in the non-adiabatic context, we contrastour results with those found for the corresponding setupof a 2DEG in a semiconductor heterostructure.

We assume that the sample sizes are smaller than themean free path and that W L, so that the effectscoming from the boundaries, electron-electron interac-tions and disorder play a minor role. Besides, we re-strict ourselves to the low energy regime such that theDirac approximation remains valid. Then, our quasi one-dimensional model contains three natural energy scales,namely the driving frequency ω, the driving amplitude U ,and the energy associated with the length L of the de-

vice. Additionally, the leads’ Fermi momentum ~k(∞)F

becomes a relevant scale in a 2DEG, since the con-tact resistance depends on its value. For graphene withFermi velocity vF ≈ 106m/s and L = 5µm, the lat-ter is EGL = ~vF /L ≈ 0.13meV. For a 2DEG withthe same geometry and effective mass m∗ = 0.067me

(GaAs/AlGaAs), it is roughly four orders of magnitudesmaller, ENL = ~2/2m∗L2 ≈ 0.02µeV. Driving beyondthe adiabatic limit requires frequencies ω & EL/~. Elec-trons can then absorb or emit photons, and after a shorttransient period, these excitations will establish a non-equilibrium population of the electronic states.

III. FLOQUET SCATTERING THEORY INGRAPHENE

A quantitative description of nonadiabatic chargetransport is provided by Floquet scattering theory. Sub-sequently, we show that this theory provides the proba-

bility T(n)LR for an electron to be scattered from the left

to the right lead under the absorption or emission of nphotons, and with it we can calculate the dc current.18 Inorder to focus on the graphene-specific features, we willrestrict ourselves to the zero temperature limit so that allelectronic states below the Fermi energy EF are initiallyoccupied.

A. General formalism

When scattered at a periodically time-dependent po-tential, an electron with initial energy ε may absorb oremit |n| quanta of the driving field (n < 0 corresponds toemission), such that its final energy is ε + n~ω. This isembodied in the decomposition of the transmission prob-ability from the left to the right reservoir

TLR(ε) =

∞∑n=−∞

T(n)LR (ε). (1)

The corresponding time-averaged dc current is given bythe generalized Landauer formula18,19

I =ge

h

∫dε∑n

[T

(n)LR (ε)fL(ε)− T (n)

RL (ε)fR(ε)], (2)

where e is the electron charge and f(ε) is the Fermi-Diracdistribution. Spin and valley degeneracy of grapheneis responsible for the prefactor g = 4, while g = 2accounts for the spin in a 2DEG. The application ofEq. (2) to graphene (or any other two-dimensional ma-terial) requires extending the summation to transversemomenta ~q. For a driving field that breaks reflec-tion symmetry, one generally finds TRL(ε) 6= TLR(ε).Then, even when the leads are in equilibrium such thatfL(ε) = fR(ε) ≡ f(ε), a net current may flow, and apump current emerges,

I =ge

h

∫dεf(ε)∆T (ε), (3)

where ∆T = TLR − TRL.For the computation of the transmission probabilities

we adopt the Floquet scattering formalism of Ref. 20 andconsider electrons in two dimensions under the influenceof a time-dependent potential described by the Hamilto-nian

H(t) = H0(x) + U(x) cos(ωt). (4)

Here, H0 comprises the kinetic energy and the staticpotential V (x), while U(x) is the profile of the time-dependent potential with frequency ω. Since the po-tential is y-independent, the transverse momentum is

3

VL VR

HLL HlL HrL HRL

UHtL

ÑΩ

n=-2

n=-1

n=0

n=1

n=2

FIG. 2. Quasi-one-dimensional scattering potential. Profileof the pump, modeled as a barrier for which the left half(region l) experiences an oscillatory gate voltage. Regions Land R correspond to highly doped leads. Dotted lines denotesidebands to which the electron energy changes by absorptionand emission of photons from the driving field.

conserved and the problem becomes effectively one-dimensional. We assume that both the static potentialV (x) and the driving profile U(x) are piecewise constant,and that U(x) = 0 outside the scattering region; seeFig. 2. For graphene, H0 = ~vFk · σ + V (x) with thewavevector k = ±kex + qey. The free solution with en-ergy E reads

ϕ±E =e±ikx√2|E|k/~

(|E|/~vF±k + iq

), (5)

where k is positive and fulfills the dispersion relationE2 = ~2v2F (k2 + q2), while ± is the sign of the corre-sponding current. The normalization has been chosensuch that propagating waves have unit longitudinal cur-rent, vF (ϕ±E)†σxϕ

±E = ±1. This is convenient since then

the coefficients of a superposition become probability am-plitudes. For evanescent solutions with imaginary longi-tudinal wavenumber k = iκ, the current vanishes.

According to the Floquet theorem, the Schrodingerequation with a time-periodic Hamiltonian H(t) = H(t+2π/ω) possesses a complete set of solutions with struc-ture ψ = e−iεt/~φ(t), where the Floquet state φ(t) =φ(t+2π/ω) obeys the time-periodicity of the Hamiltonianand ε is the quasienergy. Here we are looking for Floquetscattering states, i.e., solutions of the Schrodinger equa-tion that (i) are of Floquet structure and (ii) have anincoming plane wave as boundary condition. For clarity,we derive here only the transmission from left to right—for the opposite direction, it follows by simple re-labeling.

Condition (i) is equivalent to employing for the wave-function in any of the four regions ` = L, l, r, R theansatz ψ`(x, t) = e−iεt/~φ`(x, t). The time-periodic partsφ`(x, t) still have to be determined, while the quasienergyε turns out to equal the energy of the incoming wave. In-serting ψ` into the Schrodinger equation of region ` yieldsfor φ`(x, t) a partial differential equation which we solve

by a separation ansatz. The resulting solutions

φ±n,`(x, t) = e−inωt−i(U`/~ω) sin(ωt)ϕ±ε+n~ω−V`(x) (6)

=

∞∑n′=−∞

Jn′−n (U`/~ω) e−in′ωtϕ±ε+n~ω−V`

(x),

comply with the requirement of time-periodicity providedthat the separation parameter n is of integer value. Theseparation parameter labels all possible solutions and de-termines the time-averaged energy ε + n~ω. The Besselfunction of the first kind Jn stems from the relationexp[−iz sin(ωt)] =

∑n Jn(z) exp(−inωt). Note that in

the present case, U` is non-zero only in the driving re-gion ` = l. The ansatz (6) has also been used to studyphoto-assisted tunneling in graphene.21,22

Condition (ii) means that in region L, the Floquet stateconsists of an incoming plane wave and a reflected part,φL = ϕ+

ε−VL+∑n rnφ

−n,L, while in region R, we have only

an outgoing state, φR =∑n tnφ

+n,R. With the normal-

ization chosen for the free solutions (5), the coefficients ofthe latter superposition relate to the left-to-right trans-mission probability under absorption or emission of nquanta according to

T(n)LR (ε) = |tn(ε)|2. (7)

In the scattering regions l and r, the Floquet solutionmust be a superposition of the states (6), i.e., φ` =∑n t

(`)n φ+n,` +

∑n r

(`)n φ−n,`. The coefficients t

(`)n and r

(`)n

follow from the requirement that the scattering stateshave to be continuous at any time. These matching con-ditions can be written as a set of linear equations, withthe incoming wave appearing as inhomogenity. In matrixnotation it reads∑

n′

Mn,n′ · (rn′ , t(l)n′ , r(l)n′ , t

(r)n′ , r

(r)n′ , tn′) = δn,0ϕ

+L,in, (8)

where ϕ+L,in ≡ ϕ

+ε−VL

denotes the incoming wave writtenas a 6-dimensional vector. The 6× 6 matrices Mn,n′ areconstructed from the wavefunctions (6) evaluated at theinterfaces. They can be written efficiently as

Mn,n′ = M(ε− n′~ω)∑`

Jn′−n (U`/~ω)P`, (9)

where M(ε) contains the wave matching condition at en-ergy ε in the absence of driving. The matrix P` is aprojector to region `, constructed such that MP` con-tains only wavefunctions from region `. In particular,Pl = diag(0, 1, 1, 0, 0, 0) projects onto the driving region.Equation (8) fully determines the transmission and reflec-tion amplitudes tn and rn. The tight-binding version18 ofthis method is suitable for studying ac driving of smallercarbon-based conductors, such as ribbons with a size ofonly a few lattice constants23 or nanotubes.24,25 Via time-dependent density functional theory, it can be generalizedto the presence of interactions.26

4

For electrons in a 2DEG with effective mass m∗, thestatic Hamiltonian reads H0 = ~2k2/2m∗+V (x). Its freesolutions are scalar fields, but since also their derivativesmust be continuous, it is convenient to write them inspinor notation,

ϕ±E =e±ikx√~k/m∗

(1±ik

), (10)

with the dispersion relation E = ~2k2/2m∗. The normal-ization is such that the first vector component has unitcurrent, 1

m∗ (ϕ±E,1)†(−i~∂x)ϕ±E,1 = ±1. With these ingre-dients, the Floquet scattering formalism can be directlyapplied to the 2DEG case.

B. Weak driving limit

The set of linear equations (8) can be solved analyt-ically in the limit of small driving amplitudes or largefrequencies, such that p ≡ (U/2~ω)2 1. We aim atfinding analytical expressions for the one-photon trans-mission probabilities T (±1) in terms of the static trans-mission for Klein tunneling at energy ε with respect tothe top of a high barrier15 of length L,

T (ε, q) =k2ε

k2ε + q2 sin2(kεL), (11)

where kε = [(ε/~vF )2 − q2]1/2. The semiclassical limitL → ∞ of the above relation is obtained by inte-gration over fast oscillations, which yields T (ε, q) ≈√

1− (~vF q/ε)2. In order to relate these expressions toour formalism, we extract from the solution of Eq. (8) inthe undriven limit, U = 0, the transmission amplitude

t0 by multiplication with the vector p†R = (0, 0, 0, 0, 0, 1)and obtain

T (ε) = |p†RM−1(ε)ϕ+

L,in|2. (12)

Next we simplify Eq. (8) using for the Bessel func-tions the approximations J0(U/~ω) = 1, J±1(U/~ω) =±U/2~ω, while Jn(U/~ω) = O|n| (U/~ω) for U/~ω 1.Thus, to first order in U/~ω, only the Floquet indices n =0,±1 remain. The resulting 18 × 18 matrix (Mn,n′) canbe inverted via the approximation [A + (U/2~ω)B]−1 =A−1− (U/2~ω)A−1BA−1, which provides t±1 and, thus,

T(±1)LR = p

∣∣∣p†RM−1(ε∓ ~ω)M(ε)PlM−1(ε)ϕ+

L,in

∣∣∣2 . (13)

The calculation of the elastic transmission T(0)LR up to

leading order in p is more tedious, but fortunately notrequired, because our Hamiltonian obeys time-reversal

symmetry. Therefore, T(0)LR(ε) = T

(0)RL(ε) which implies

that the elastic channel does not contribute to the pumpcurrent.18

Our numerical calculations will reveal that the relevantcontributions to the pump current stem from modes that,

in the absence of driving, are evanescent in the barrierregion. For such modes, Eq. (13) can be connected toEq. (12) in a simple way, because the modified sourceterm in Eq. (13) has an invariant forward amplitude,M(ε)PlM

−1(ε)ϕ+L,in = ϕ+

L,in + α−ϕ−L,in. Although α− isnot necessarily zero, it does not affect the transmission,since it merely redefines the reflection amplitude r0. Asa result, the one-photon transmission, Eq. (13), becomes,besides a prefactor p, identical to the static transmission(12) in the evanescent region,

T(±1)LR (ε) = p T (ε± ~ω, q). (14)

For the evanescent modes entering from the right, thesame reasoning lets us conclude that the waves decayexponentially and do not reach the driven region l. Con-

sequently, PlM(−1)(ε)ϕ−R,in ≈ 0, and T

(±1)RL (ε) p can

be neglected. Then the net transmission appearing inEq. (3) becomes

∆T = p [T (ε+ ~ω, q) + T (ε− ~ω, q)] +O(e−2κL). (15)

The magnitude of the correction reflects the fact thatwe have neglected the exponentally small transmission ofthe evanescent modes which decays with the imaginarywavenumber iκ = i[q2−(ε/~vF )2]1/2. Since 0 ≤ T ≤ 1, itfollows that the maximal net transmission for evanescentwaves is 2p.

IV. NON-ADIABATIC PUMP CURRENT

In the relevant weak driving regime U ~ω, theabsorption or emission probabilities are of the orderp ≡ (U/2~ω)2 1 [see Eq. (14)], so that ∆T ∼ p. More-over, for short and wide systems (width W L), thetotal current takes the form of an integral over modesq. Hence, the current (3) at a given Fermi energy EF(measured from the Dirac point of the barrier) can beexpressed as

I =ge

h

(U

2~ω

)2

W

∫ EF

−∞dε

∫ ∞−∞

dq∆T

p. (16)

We compute ∆T numerically by wave-matching in Flo-quet space [see Eq. (8)]. Using typical parameters L =5µm, W/L = 4, U = 40µeV, ~ω = 2meV (around500GHz), we obtain the results shown in Fig. 3. Aslong as p 1, the structure of the transmission forgraphene depends only on the product Lω, unlike for the2DEG. Graphene develops a far larger pump current thana 2DEG, saturating to an L- and ω-independent maximalvalue Imax ≈ 3nA for |EF | ≥ ~ω, while the smaller pumpcurrent in the 2DEG case already saturates at |EF | & 0.The semiclassical approximation described in Sec. III B,valid for ~ω EL, yields

IG ≈e

~(U/2)2

EGW×

(2− |EF |

~ω

)EF

~ω , |EF | < ~ω±1, |EF | > ~ω

, (17)

5

2DEG

-1 0 1 2 3 4 50

0.1

0.2

-2 -1 0 1 2

0

0.05

0.1

EL

¶ EF

I@n

AD

-4 -2 0 2 4-3

-2

-1

0

1

2

3

EF ÑΩ

I@n

AD

Graphene

2DEG

FIG. 3. Pump current I for graphene and 2DEG. Dashedlines mark the semiclassical result, which closely matches thenumerical ones. The upper inset is a blowup of the 2DEGcase. The lower inset depicts the differential current demon-strating that the main contribution to the pump current arisesfrom an energy range ~ω around the Dirac point, populatedby modes that are evanescent in the barrier region.

where the energy EW is the analogue of EL with L re-placed by the width W . The corresponding calculationfor a 2DEG provides the result

IN ≈e

~(U/2)2

2k(∞)F WENW

×

0, EF < −~ω(1 + EF

~ω)2, −~ω < EF < 0

1, EF > 0

.

These approximations are plotted as dashed curves inFig. 3. Unlike for graphene, the pump current in the

2DEG depends on the leads’ large Fermi momentum k(∞)F

(around 12 nm−1 for gold electrodes), which has been as-sumed equal for both leads. The relative pump perfor-

mance ν ≡ ImaxG /Imax

N = ~k(∞)F /m∗vF , assuming gold

electrodes and GaAs/AlGaAs 2DEGs, is ν ≈ 20, i.e., thepump current in the graphene device is much larger thanin the 2DEG device.

A. Direction-dependent excitation of evanescentmodes

In the following we show that the large and robustpump current of the graphene device stems from a mech-anism whereby the ac field promotes evanescent modesfrom the left lead with probability p into propagatingmodes, unlike evanescent modes from the right lead,which couple poorly to the driving region. In order tosupport this picture, we consider the differential responsedI/dEF , shown in the lower inset of Fig. 3. It indicatesthat the main contribution to the current stems from thebipolar regime |ε| < ~ω in the case of graphene, or fromthe gap boundary region −~ω < ε < 0 in the case of the2DEG. In the absence of driving, these energy ranges arepopulated by electronic modes that become evanescent

2DEG

Graphene3

2

1

00 0.25 0.75 10.5

(c)

-1 12DEG

0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

q/qw

2-2 Graphene

0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

q/qw

q/qw

DT

/pe

/Ñw

(a) (b)

FIG. 4. Scaled net transmission ∆T/p, where p = (U/2~ω)2,for (a) the graphene and (b) the 2DEG pump as a func-tion of transverse wave number q and the initial energy εfor weak driving U ~ω at photon energy ~ω = 2meV ∼500GHz. The dashed lines separate regions with propagat-ing and evanescent modes under the barrier. (c) Cut at zeroenergy revealing the behavior in the evanescent region. Thedashed line marks the static transmission 2T for grapheneat energy ~ω, while the dotted curve is its semiclassical ap-proximation. The transverse wave number q is scaled byqω, which is defined by qωL = ~ω/EG

L for graphene, and(qωL)2 = ~ω/EN

L for the 2DEG.

in the barrier region. Whether modes are evanescent orpropagating furthermore depends on their transverse mo-mentum q. Their response to driving is encoded in thefunction ∆T/p, which represents the differential responsefor a given mode with transverse momentum q and totalenergy ε [see Eq. (16)].

Figure 4 depicts ∆T/p for graphene and the 2DEG.In the 2DEG case, at each energy only a discrete set ofresonant modes contributes to the pump effect. Theseresonances correspond, up to a shift of ±~ω in energy, toquasibound levels in the static system, that form becauseof the velocity mismatch at the interface with the metal-lic leads, which results in strong confinement. The con-tact resistance increases with increasing Fermi momen-

tum k(∞)F L, and the resonances become increasingly nar-

row, resulting in a suppressed response of the 2DEG. Bycontrast, in graphene a broad range of modes contributesat all energies, and the response is particularly strong in

6

>

FIG. 5. Evanescent mode pump mechanism. Evanescentmodes penetrating the barrier from the left absorb or emitphotons such that they become propagating. The correspond-ing modes from the right lead decay before reaching the regionwith the ac gating. Filled-in arrows indicate strong popula-tion; non-filled arrows mark negligible scattering channels.

a diamond-shaped region within the bipolar regime (redsquare in Fig. 4), in agreement with our earlier obser-vation in Fig. 3. Graphene’s response does not exhibitsharp resonances due to the fact that, in the static case,carrier chirality prohibits their confinement, so that carri-ers in the graphene pump remain strongly coupled to theleads, even when they are driven far out of equilibrium.Instead, graphene’s response in Fig. 4(a) has features as-sociated with the cone ε = ~vF q (dashed lines) and itsreplicas shifted by multiples of ~ω. For the weak drivingconsidered here, only the two replicas at ε = ~vF q ± ~ωare visible. The first cone divides the modes enteringthe scattering region into two categories, propagatingand evanescent: any incoming carrier in mode q willbecome evanescent under the barrier if its energy ful-fills |ε| < ~vF q, and will remain propagating otherwise.The other two cones determine whether carriers popu-late evanescent or propagating modes after absorption oremission of a single photon. Analogously, in the 2DEGthe threshold between propagating and evanescent modesis found at ε = ~2q2/2m∗, and under photon emission orabsorption shifts to ε = ~2q2/2m∗ ± ~ω.

In both systems, it can now be seen that the mainmechanism of charge transfer is established by a processwhereby an evanescent mode coming from the left leadmay get transmitted to the right by absorbing or emittinga photon, as long as in this process it jumps to an existingpropagating mode that may travel into the right lead.An evanescent coming from the right, in contrast, firstencounters the static region not covered by the gate, andso gets reflected with a high probability (see Fig. 5). Thisspatial asymmetry rectifies the carrier flux such that anet transport from left to right emerges.

In the 2DEG, when photon absorption is at resonancewith a quasibound state, this process via evanescentmodes yields a maximal value ∆T/p ∼ 1, but becausethe resonances are narrow the integrated current remainssmall. In graphene, on the other hand, all evanescentmodes with qL < ~ω/EGL exhibit a response ∆T/p . 2,close to twice the optimal value of resonant evanescent

modes in a 2DEG, but without the need to satisfy anyresonance condition. The factor two comes from the twopossible transitions to a propagating mode, by absorp-tion but also by emission of a photon, i.e., it is directlyrelated to bipolarity. Resonant conditions are not re-quired because Klein tunneling in the propagating modesand macroscopic tunneling in evanescent modes keep thecontacts always open; this is directly related to chirality.In combination, these two graphene-specific effects max-imize the differential pump response around the Diracpoint, and thus determine the robust characteristic fea-tures of the total pump current in Fig. 3.

V. CONCLUSIONS

With this work, we have put forward a pump mech-anism that makes use of the evanescent modes under apotential barrier in graphene. If an ac gate voltage actsupon the, say, left half of the barrier, electrons penetrat-ing the barrier from the left lead are excited into prop-agating modes and, thus, will be scattered to the rightlead. By contrast, electrons from the opposite side willreach the driving region only with exponentially smallprobability. This breaking of spatio-temporal symme-tries induces surprisingly large non-adiabatic pump cur-rents in the range of several nA, significantly larger thanwhat has been observed so far with semiconductors. Themain reason for this efficiency is that a whole range ofevanescent modes is excited, while in the correspondingsetup with a 2DEG, only isolated resonances contribute.Despite the large pump currents, the main effect stemsfrom single-photon absorption, which allows one to ob-tain analytical results. Moreover, owing to current con-servation, ratchets which are built by several pumps inseries, exhibit identical operation characteristics (repeti-tion of the element is therefore not desirable for practicalimplementations).

A further important observation is that the resultingpump current changes smoothly, symmetrically, and al-most monotonically with the distance of the Fermi en-ergy to the Dirac point. This behaviour is a direct con-sequence of bipolarity and, thus, particular to graphenepumps. As an application, it allows steering the dc cur-rent into a direction of choice by simply shifting the bar-rier height via a local dc gate voltage across the Diracpoint. This effect can also be used to detect a static elec-tric field by measuring the current in response to a smalloscillating probe electric field, or detecting the amplitudeof such an oscillating field when the static field is fixedto a moderately large value, beyond which the responseflattens out.

ACKNOWLEDGMENTS

We acknowledge support by the CSIC JAE-Doc pro-gram and the Spanish Ministry of Science and Innovation

7

through grants FIS2008-00124/FIS (P.S.-J), FIS2009- 08744 (E.P.) and MAT2008-02626 (S.K.).

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