Quantum pumping in graphene with a perpendicular magnetic field

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Quantum pumping in graphene with a perpendicular magnetic field

Rakesh P. Tiwaria兲 and M. Blaauboer

Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands

共Received 10 September 2010; accepted 29 November 2010; published online 17 December 2010兲 We consider quantum pumping of Dirac fermions in a monolayer of graphene in the presence of a perpendicular magnetic field in the central pumping region. The two external pump parameters are electrical voltages applied to the graphene sheet on either side of the pumping region. We analyze this pump within scattering matrix formalism and calculate both pumped charge and spin currents. The predicted charge currents are of the order of 1000 nA, which is readily observable using current technology. © 2010 American Institute of Physics.关doi:10.1063/1.3528457兴

By periodically changing the confining potential of a quantum system, quantum pumping allows a net dc of carriers to flow in the absence of an applied bias.1–3 The possibility of generating electrical currents in solid-state nanostructures using this phenomenon has attracted consid-erable attention.4The same phenomenon allows us to adia-batically generate spin-polarized currents in solid-state nanostructures,5which is of great interest in the field of spin-tronics. Usually the pumped currents are small 共⬃100 pA兲 and difficult to identify; as such, a small signal may be ob-scured by rectification. Indeed, quantum pumps realized so far are commonly believed to have demonstrated rectified currents.6–8In this article we propose a quantum pump real-ized in graphene in the quantum Hall regime. We predict large pumped currents共⬃1000 nA兲 at modest values of gate voltages and magnetic fields, which increase the hope for unambiguous demonstration of quantum pumping.

Ever since the synthesis of high-quality graphene,9there has been tremendous interest in the properties of this single-layer form of carbon.10 Partly as a result of its electronic structure,9,11–13 graphene exhibits many unusual electronic and magnetic transport properties, such as a unique type of quantum Hall effect,9,14 ballistic conduction by massless Dirac fermions,9,14 a size-dependent band gap,15 and large magnetoresistance.13,16–18 Recently some quantum pumps based on graphene were proposed. Zhu and Chen19 consid-ered a monolayer charge pump with two gate voltages as pumping parameters. Prada et al.20 considered a similar charge pump and found that the current is pumped through evanescent modes in the pumping region. In our proposed pump, both charge and spin currents are pumped through traveling modes and the magnitudes are much larger 共⬃1000 nA兲, making it more desirable for applications.

Figure 1 shows a schematic of the device we consider. On the left and right ends of the graphene sheet two external voltages V₁共t兲=V₁+␦V₁sin共␻t兲 and V₂共t兲=V₂+␦V₂sin共␻t +␾兲 are applied. An external field B0zˆ共perpendicular to the

graphene sheet兲 is applied in the central part of the sheet of length L, which forms the pumping region. We consider short and wide pumps关width共W兲Ⰷlength共L兲兴 for which the microscopic details of the edges become insignificant and the pumped current is obtained by integrating over the transverse modes. Also, we assume that the dimensions of the pump are

much larger than the carbon-carbon lattice constant. In that case the intervalley tunneling is suppressed and it is suffi-cient to consider just one valley.21

We use the scattering matrix formalism for calculating the pumped current.3 We divide the system into three re-gions. In region 1 共x⬍−L/2兲 and region 3 共x⬎L/2兲 gate voltages V1共t兲 and V2共t兲 are applied, respectively. In these

regions both up-spin and down-spin carriers obey the Dirac equation of motion described by the Hamiltonian H =vF␴· p + eVj, where j = 1 for region 1 and j = 2 for region 3,

p is the momentum operator, vF is the Fermi velocity, and

␴=共␴x,␴y,␴z兲 represents the three Pauli matrices. In region 2, we describe the effect of the magnetic field B0and include

the effect of Zeeman splitting. We choose the Landau gauge, such that the vector potential is A共x兲=共0,B0x , 0兲. Then in

region 2 the Hamiltonian for up-spin and down-spin carriers is H_↑=vF␴·关p+eA共x兲兴+gⴱ␮BB0/2 and H↓=vF␴·关p

+ eA共x兲兴−gⴱ␮BB0/2, respectively. Here gⴱ is the spin

g-factor 共assumed to be 2兲共Ref. 22兲 of the carriers and␮B represents the Bohr magneton. To simplify the notation we introduce dimensionless units: lB=

冑

ប/共eB0兲, r→lBr, q →q/lB, kx→kx/lB, E→E0E, and E0=បvF/lB.បkxrepresents the momentum along the x-axis.

In the adiabatic regime and for 共bi兲linear response the spin-dependent pumped current through each mode into the left lead is given by3

a兲_{Electronic mail: rapat82@gmail.com.}

L

−L/2

L/2

V (t)

0 B

2 W

1 x

y

V (t)

FIG. 1.共Color online兲 Schematic of the proposed quantum pump. An exter-nal field B0zˆ is applied in the central pumping region. The gate voltages

V1共t兲 and V2共t兲 are the two pumping parameters.

APPLIED PHYSICS LETTERS 97, 243112共2010兲

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Ip␴= ␻e sin␾␦V1␦V2 2␲ Im

冋

⳵r␴ⴱ ⳵V₁ ⳵r␴ ⳵V₂+ ⳵t␴ⴱ ⳵V₁ ⳵t␴ ⳵V₂

册

. 共1兲

Here r␴ and t␴,␴=↑ or ↓, represent the coefficients for re-flection and transmission of carriers with spin␴into the left reservoir 共a similar expression can be derived for pumping into the right reservoir兲. The pumped charge current is I_pcharge= I_p↑+ I_p↓and spin current is I_pspin= I_p↑− I_p↓. To calculate r↑ and t↑, we write down the wave functions in the three regions corresponding to the respective Hamiltonians, and then de-mand that the wave function is continuous across the bound-aries at x = −L/2 and x=L/2. Due to the translational invari-ance of the geometry in the transverse direction, the momentum along the y-axis is conserved and is denoted as បq. In region 1 the wave function is given by

⌿1= eiqy

冉

eikx共x+L/2兲_{+ r}↑_e−ikx共x+L/2兲

eikx共x+L/2兲_ei␾1_{+ r}↑_e−ikx共x+L/2兲_e−i␾1

冊

, 共2兲

where tan␾1⬅q/kx and the energy of these excitations is given by E =共eV1+បvF

冑

kx2+ q2兲/E0. Similarly the wave

func-tion in region 3 is given by ⌿3= t↑eiqy

冉

eik_x⬘共x−L/2兲 eikx⬘共x−L/2兲_ei␾2

冊

, 共3兲

where tan␾2⬅q/kx

⬘

and the energy of these excitations is given by E =共eV2+បvF

冑

kx

⬘

2+ q2兲/E0. In the central region the

wave function can be written as a linear combination of We-ber functions,23 ⌿2= eiqy

冢

C1Dp−1共z兲 + C2Dp−1共− z兲 i

冑

2 E 关C1Dp共z兲 − C2Dp共− z兲兴

冣

, 共4兲

where z =

冑

2共x+q兲 and p =关E−共gⴱ␮BB0/2E0兲兴2/2.

Setting ␣₃= D_p−1关

冑

2共q−L/2兲兴, ␣₄= D_p−1关

冑

2共−q+L/2兲兴,

␥1=

冑

2/EDp关

冑

2共q−L/2兲兴, ␥2=

冑

2/EDp关

冑

2共−q+L/2兲兴, ␻1= Dp−1关

冑

2共q+L/2兲兴, ␻2= Dp−1关

冑

2共−q−L/2兲兴, ␩1

=

冑

2/EDp关

冑

2共q+L/2兲兴, and ␩2=

冑

2/EDp关

冑

2共−q−L/2兲兴, we find t↑=2i共␩2␻1+␩1␻2兲cos␾1 f+g+− f−g− , 共5兲 r↑= f −_h+_{− f}+_h− f+g+− f−g−, 共6兲 where f+= i␩2+␻2ei␾2, f−= i␩1−␻1ei␾2, g+= i␥1+␣3e−i␾1,

g−= i␥2−␣4e−i␾1, h+= i␥2+␣4ei␾1, and h−= i␥1−␣3ei␾1.

Simi-lar expressions are found for r↓ and t↓, the only difference being that p =关E+共gⴱ␮BB0/2E0兲兴2/2 instead of p=关E

−共gⴱ␮BB0/2E0兲兴2/2.

Now we present the numerical results for the pumped charge and spin currents obtained using the above expres-sions for the reflection and transmission coefficients. The current is calculated using Eq. 共1兲 and we show the total pumped currents which are obtained by integrating over all the modes, i.e., I_p,Tcharge=兰_−␲/2␲/2 cos共␾1兲Ip

charge

d␾1 and Ip,T

spin

=兰₋␲/2_␲/2cos共␾₁兲Ipspind␾1. We choose W = 5 ␮m, L = 0.5 ␮m,

V₁= V₂= 0.1 V, ␦V₁=␦V₂= 0.1 mV, vF= 106 m/s, and ␻/共2␲兲=5 GHz.19,22

The phase difference between the two external voltages ␾is chosen to be ␲/2 so as to maximize

the pumped current. First we calculate the pumped current for fixed energy E =␮= 100.1 meV. ␮ represents the Fermi energy in the two leads and the single particle state energy in the central pumping region. All these energies are taken to be equal in order to eliminate the possibility of generating cur-rent by applied external bias and secure energy-conserved tunneling. The temperature is set to zero in all our calcula-tions and we ignore electron relaxation processes. ␮is cho-sen to be 100.1 meV in order to include the contribution of the lowest lying excitations for V1= V2= 0.1 V and ␦V1

=␦V2= 0.1 mV.

The upper panel of Fig. 2 shows the pumped charge current as we change the magnetic field for the above men-tioned parameters. The magnetic field is varied from 0.01 to 0.5 T. As expected, the pumped currents are very sensitive to the magnetic field. The pumped current is large when the magnetic length lB is comparable to the dimensions of the pump. For B₀= 0.1 T, lB⬃100 nm and we see large values of the pumped currents 共⬃1000 nA兲. As we increase the magnetic field, the magnetic length decreases and also the total pumped current decreases. The lower panel of Fig. 2 shows the calculated spin current for the same parameters. In general the pumped spin current is smaller than the pumped charge current. We believe that the enhancement of the pumped current in the proposed pump, compared to previ-ously studied pumps without magnetic field,19,20 is due to interference arising from the magnetic length and the dimen-sions of the pump.

To compare the calculated pumped currents with recti-fied currents we calculate the conductance as a function of magnetic field for the same parameters 共see Fig. 3兲. The conductance for up-spin carriers is given by G = G0兰−␲/2␲/2 cos共␾1兲t↑ⴱt↑d␾1 with G0= 2e2kFW/共␲h兲, where kF is the Fermi wave-vector. The conductance for down-spin carriers共not shown兲 is almost identical. Peaks in the conduc-tance correspond to a large jump in pumped currents, and the pumped charge current changes sign around that peak. This can be used to distinguish between any rectification current and the pumped current.

Finally we comment on the experimental realization of this pump. Fabricating metallic gates on graphene nanostruc-tures with widths ranging from 10 to 100 nm and lengths of

0.1 0.2 0.3 0.4 0.5 B₀ (T) -8000 -6000 -4000 -2000 0 2000 I p,T ch arge (nA) 0.1 0.2 0.3 0.4 0.5 B₀ (T) 0 1000 2000 3000 4000 5000 6000 I p,T spin (nA)

FIG. 2. Upper panel: pumped charge current as a function of B₀. Lower panel: pumped spin current as a function of B0.

243112-2 R. P. Tiwari and M. Blaauboer Appl. Phys. Lett. 97, 243112共2010兲

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1 – 2 ␮m can be achieved using current technologies.11,15 Perhaps the most difficult part will be to apply local mag-netic fields of strengths up to 0.5 T in the central pumping region although efforts have already been made in this direc-tion, for example, using nanomagnets.24 The latter can in principle be embedded under the central pumping region of the graphene pump. It should be noted that Eq.共1兲represents the current pumped throughout one whole pumping cycle with no external bias applied. In practice because of the phase difference ␾ between V₁共t兲 and V₂共t兲共chosen to be

␲/2兲, there will be a small bias across the pump. The pump cycle can be chosen in a symmetric way such that the bias reverses after half of the period and any rectification current over the whole cycle is canceled.

To summarize, we have proposed a quantum pump based on graphene in a perpendicular magnetic field. The pumped current is carried by traveling modes and has a large magni-tude 共⬃1000 nA兲. We find that the pumped current has a characteristic dependence on B0. Experimental verification of

the pump properties would provide a much needed demon-stration of the phenomenon of quantum pumping.

This research was supported by the Dutch Science Foun-dation NWO/FOM.

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0.1 0.2 0.3 0.4 0.5 B₀(T) 0 0.5 1 1.5 2 G/G 0

FIG. 3. Conductance G/G0as a function of B0.

243112-3 R. P. Tiwari and M. Blaauboer Appl. Phys. Lett. 97, 243112共2010兲