Numerical study of parametric pumping current in mesoscopic systems in the presence of magnetic field

0.28 MB·

by  Fuming Xu

Numerical study ofparametric pumping current inmesoscopicsystems inthe presence ofmagnetic
field
Fuming Xu1, Yanxia Xing1,2, and Jian Wang1∗
1Department of Physics and the Center of Theoretical and Computational Physics,
The University of Hong Kong, Hong Kong, China.
2Department of Physics,
Beijing Institute of Technology, Beijing 100081, China.
7
1 Wenumericallystudytheparametricpumpedcurrentwhenmagneticfieldisappliedbothintheadiabaticand
0 non-adiabaticregimes. Inparticular, weinvestigatethenatureofpumpedcurrentforsystemswithresonance
2 aswellasanti-resonance. Itisfoundthatintheadiabaticregime,thepumpedcurrentchangessignacrossthe
sharpresonancewithlonglifetimewhilethenon-adiabaticpumpedcurrentatfinitefrequencydoesnot. When
n
a thelifetimeofresonantlevelisshort,thebehaviorsofadiabaticandnon-adiabaticpumpedcurrentaresimilar
J withsignchanges.Ourresultsshowthatattheenergywherecompletetransmissionoccurstheadiabaticpumped
currentiszerowhilenon-adiabaticpumpedcurrentisnon-zero.Differentfromtheresonantcase,bothadiabatic
6
andnon-adiabaticpumpedcurrentarezeroatanti-resonancewithcompletereflection. Wealsoinvestigatethe
] pumped current when the other system parameters such as magnetic field, pumped frequency, and pumping
l potentials. Interestingbehaviorsarerevealed. Finally,westudythesymmetryrelationofpumpedcurrentfor
l
a several systemswithdifferent spatial symmetryupon reversal of magneticfield. Different fromtheprevious
h theoreticalprediction,wefindthatasystemwithgeneralinversionsymmetrycanpumpoutafinitecurrentin
- theadiabaticregime.Atsmallmagneticfield,thepumpedcurrenthasanapproximaterelationI(B)≈I(−B)
s
e bothinadiabaticandnon-adiabaticregimes.
m
. PACSnumbers:72.10.-d,72.10.Bg,73.23.-b,73.40.Gk
t
a
m
I. INTRODUCTION field. Theconclusionwasconfirmedbytheoreticalworksus-
- ing Floquet scattering matrix method28–30. Later it was nu-
d
n Theideaofparametricelectronpumpwasfirstaddressedby merically suggested31 that the pumped spin current also has
o Thouless1,whichisamechanismthatatzerobiasadccurrent certainspatialsymmetries.
c
[ ispumpedoutbyperiodicallyvaryingtwoormoresystempa- Inthispaper,weaimtonumericallyinvestigatethepumped
rameters.Overtheyears,therehasbeenintensiveresearchin- currentinthepresenceofmagneticfield. Bothadiabaticand
1 terestconcentratedonparametricelectronpump.2–6 Electron non-adiabaticpumpedcurrentarecalculated.Wefocusonthe
v pumphasbeenrealizedon quantumdotsetup7 consistingof natureofpumpedcurrentforthemesoscopicsystemswithres-
9
1 AlGaAs/GaAs heterojunction8. Low dimensionalnanostruc- onance with complete transmission and anti-resonance with
5 tures, such as carbon nanotubes(CNT)9,10 and graphene11,12 complete reflection. We find that the behaviors of adiabatic
1 were also proposedas potentialcandidates. Investigationon and non-adiabatic pumped current are very different. In the
0 electronpumpalsotriggerstheproposalofspinpump13–15,in non-adiabatic regime, the pumped current is nonzero at res-
1. whichaspincurrentisinducedbyvariousmeans. onance while it is zero at anti-resonance. However, the adi-
0 At low pumping frequencylimit, the variation of the sys- abatic pumped current is always zero regardless of types of
7 temisrelativelyslowthantheprocessofenergyrelaxation16. resonance. Sincethereisnoexternaldrivingforce,thedirec-
1 Hencethesystemisnearlyinequilibriumandwecoulddeal tion of current depend only on the system parameters. Our
:
v with the adiabatic pump by equilibrium methods. On the numericalresultsshowthatthe adiabaticpumpedcurrentre-
i otherhand,non-adiabaticpumpreferstothecasethatpump- verses it sign at the resonance or anti-resonance. For non-
X
ing process is operated at a finite frequency. In the non- adiabatic pumped current, the sign reversal depends on the
r adiabaticregime,non-equilibriumtransporttheoryshouldbe lifetimeoftheresonantstates.Thenon-adiabaticpumpedcur-
a
employed. Theoreticalmethodsadoptedintheresearchfield rent change the sign near the resonant point only when the
includes conventional scattering matrix theory17–19, Floquet lifetimeisshort. Wealsostudythepumpedcurrentasafunc-
scattering matrix20,21 and non-equilibrium Green’s function tion of magnetic field. We find that as the system enters the
(NEGF) method22,23, as well as other methodologiesto both quantumHallregimewithincreasingmagneticfieldstrength,
adiabatic24andnon-adiabatic25electronpumps. thepumpingcurrentvanishes. SinceinquantumHallregime
Theelectronpumpisaphasecoherentphenomenon,since electron wave functionappearsas edge state, it will circum-
the cyclic variation of system parameters affects the phase vent the confining potentials shown in Fig.1. Pumping po-
of wave function with respect to its initial value26. As a re- tentialsoverlappinginspacewithconfiningpotentialspresent
sult, itis verysensitiveto the externalmagneticfield. Inthe nomodulationontheelectronwavefunctionduringthevari-
experimental work of Switkes et.al8, at adiabatic limit, the ation period. Hence there is zero pumped current in the
pumped current of a open quantum dot system with certain QuantumHallregime. We also examinethepumpedcurrent
spatialsymmetryisshowedinvariantuponreversalmagnetic and its relation with other system parameterssuch as pump-
2
a) IUD d) GUD II. THEORETICALFORMALISMANDMETHODOLOGY
V V V V
1 y 2 1 1 Weconsideraquantumdotsystemconsistingofacoherent
scatteringregionandtwoidealleadswhichconnectthedotto
V1 x V2 V2 V2 electronreservoirs. Thewholesystem isplacedin x-yplane
andamagneticfieldisapplied. ThesingleelectronHamilto-
b) ILR e) GLR
nianofthescatteringregionissimply
V V
1 1
(p+eA/c)2
H = +V(x,y,t)
2m∗
V V V V
2 2 1 2
whereAisthevectorpotentialofthemagneticfield. Herethe
c) IIV f) GIV
magneticfieldischosentobealongz-directionwithB=(0,0,
V V V B).Thevectorpotentialhasonlyx-componentintheLandau
1 2 2
gauge,A=(-By,0,0).
V2 V1 V1 H =H0+Vp
where
∂ e ∂
FIG.1:Sketchofthespatialreflectionsymmetries:(a)instantaneous H0 =(−i~ − By)2+(−i~ )2+V0(x,y)
∂x c ∂y
up-down(IUD);(b)instantaneousleft-right(ILR);(c)instantaneous
inversion (IIV); (d) general up-down (GUD); (e) general left-right and V is a time-dependent pumping potential given by
p
(GLR);(f)generalinversion(GIV).Shadowrectangularindicatesfor V (x,y,t)= V (x,y)cos(ωt+φ ).
p j j j
thepumpingregionanddarkgrayblocksstandforpotentialbarriers Forthe adiaPbaticelectronpump,theaveragecurrentflow-
definingthespatialsymmetryofthesystem.Thepumpingpotentials
ingthroughleadαduetotheslowvariationofsystemparam-
arerightontopoftheseconfiningpotentials.
eterV inoneperiodisgivenby17
j
1 τ dQ (t)
α
I = dt
α
ing frequency and pumping potential amplitude. Finally we τ Z0 dt
qω τ dN dV
alsoinvestigatethesymmetrypropertiesofthepumpedelec- = dt α j (1)
tron current of systems with certain spatial symmetry in the 2π Z0 Xj dVj dt
presence of magnetic field by the Green’s function method.
SixspatialsymmetriesstudiedinRef.29 areconsidered,both whereτ = 2π/ω isthevariationperiodofparameterVj and
at the adiabatic and non-adiabaticcases, which are instanta- ωthecorrespondingfrequency.Forsimplicitywetakeω =1
neousup-down(IUD),left-right(ILR),andinversionsymme- intheadiabaticcase. α=LorRlabelsthelead. Theso-called
tries (IIV) and the corresponding non-instantaneous/general emissivitydNα/dVj isconventionallydefinedintermsofthe
up-down(GUD),left-right(GLR),andinversionsymmetries scatteringmatrixSαβ as17,27
(GIV),respectively(see Fig.1). Theelectronpumpis driven
dN dE ∂S
α αβ ∗
by periodicalmodulationof potentialswhichshare the same = (−∂ f) Im S (2)
spatial coordinates with the confining potentials which pre- dVj Z 2π E Xβ ∂Vj αβ
servereflectionsymmetryofthesystem. Mostofournumeri-
In the language of Green’s function, the above equation is
calresultsagreewiththeconclusionsfromFloquetscattering
equivalenttothefollowingform23
theory28,29, exceptforthegeneralinversionsymmetry(GIV)
( setup f in Fig.1 ). In contrast with the theoretical predic- τ dE dV
tionthattheadiabaticpumpedcurrentIad ≈0forthisspatial Iα =q dt (−∂Ef)Tr ΓαGr pGa (3)
Z0 Z 2π (cid:20) dt (cid:21)
symmetry, our numericalcalculation shows that the pumped
currentis finite and furtherinvestigationrevealsthatthere is wheretheinstantaneousretardedGreen’sfunctionGr inreal
anapproximatesymmetryrelationofthecurrentassetupeat spaceisdefinedas
small magnetic field, which is the experimentalsetup8. The Gr(E,t)=(E−H(t)−Σr)−1 (4)
conclusion suggests that the general left-right (GLR) spatial
symmetryhasa ratherstrongimpactonthe pumpedcurrent, whereΣr istheself-energyduetotheleads.
whichleadsto thequiteaccuraterelationI(B) = I(−B) in Whereas for non-adiabatic pump at finite frequency, the
the experimentalfinding. The result also holds for the non- pumpedcurrentup to the secondorder in pumpingpotential
adiabaticcase. isderivedas22
Our paper is organizedas follows. In the next section we dE
will describe the numericalmethod first followed by the nu- Iα =−iq Z 8πTr[ΓαGr0Vj((f −f−)(Gr0−−
mericalresultsanddiscussionsinsectionIII.Finally,conclu- jkX=1,2
sionisgiveninsectionIV. Ga0−)eiφkj +(f −f+)(Gr0+−Ga0+)e−iφkj)VkGa0] (5)
3
whereΓ isthelinewidthfunctionofleadαdefinedasΓ = From the Landauer-Bu¨ttiker formula, the electric current
α α
i[Σrα −Σaα]; f = f(E) and f± = f(E ±ω) are the Fermi alongthelefttorightregionisgivenby
distributionfunctions;φ =ϕ −ϕ isthephasedifference
kj j k
between the two pumping potentials. Here G0r = G0r(E) 2e
andG0r± =G0r±(E±ω)aretheretardedGreen’sfunctions IR←L = h Z dETR←L(E)f(E)
where there is no pumping potentials. In the following sec-
tion,wewilluseEqs. (3)and(5)tocarryonnumericalinves- while IL←R is defined in a similar way. Then the pumped
currentthroughtheleftleadisdefinedas29,30
tigationsandallthenumericalworkaredoneatthezerotem-
perature.Inthecalculationweconsiderasquarequantumdot
withsize0.7µm×0.7µm,ofthesameorderasintheexper- IL =IL→R−IR→L =0 (6)
imentalsetup.8 Twoopenleadswith thesame widthconnect
Theconclusionholdsforanyparticularmoment,whichmeans
thedottotheelectronreservoirs.Thequantumdotisthendis-
cretizedintoa40×40mesh. Hoppingenergyt=~2/2m∗a2 thattherewillbenopumpedcurrentatall. Henceinthefol-
setstheenergyscalewithathelatticespacingandm∗ theef- lowingwewillnotdiscussthecaseofsetupc.
First we examine the relation between the adiabatically
fectivemassofelectronsin thequantumdot. Dimensionsof
otherrelevantquantitiesarethenfixedwithrespecttot. pumped current Iad and phase difference φ12 of the pump-
ing potentialscalculated from Eq.(3). A sinusoidalbehavior
is observed at a relatively small pumping amplitude V =0.5
p
forallsetupsinFig.1. ThesinusoidalformofIad(φ12)repre-
sents a generic property of adiabatic electron pump at small
III. NUMERICALRESULTSANDDISCUSSION
V . Driven by the cyclic variation of two time-dependent
p
system parameters, the pumped current is directly related to
Inthissection,numericalresultswillbepresented. Totest the area enclosed by the parametersin parametric space. At
ournumericalmethod,wefirststudythereflectionsymmetry smallpumpingamplitudes,theleadingorderofIadispropor-
ofpumpedcurrentoninverseofmagneticfield.Otherproper- tionaltothephasedifferencebetweenthepumpingpotentials,
tiesofthecurrentwillbediscussedinthesecondsub-section. Iad ∝ Vpsinφ1217. However, the relation doesn’t hold for
To check the symmetryof the pumpedcurrentwe assume largepumpingamplitude.Todemonstratethiswehavecalcu-
V0 = j=1,2Vj(x,y)inournumericalcalculation. InFig.1, latedcurrentpumpingthroughthe setupwith symmetryILR
we haPveschematically plotted6 setupswith differentspatial atalargepotentialVp=1.6. AsshowninFig.2thesinusoidal
symmetriesofinterest. Insetupsa,b,andc,thespatialsym- relation is clearly destroyed. Except for this difference aris-
metries are kept at any moment during the pumping period. ing from the pumping amplitude Vp, there is a general anti-
Hencewelabeltheminstantaneousup-down(IUD),instanta- symmetryrelationbetweenthepumpedcurrentandthephase
neousleft-right(ILR),andinstantaneousinversion(IIV)sym- differenceφ12forallsetups:I(φ12)=−I(−φ12). Naturally,
metries,respectively. Ontheotherhand,symmetriesarebro- I(φ12 = nπ) = 0. Thisisunderstandablesincetwosimulta-
kenduringthewholepumpingcycleexceptwhenφ = nπ neouslyvaryingparametersenclosealineratherthananarea
jk
insetupd,e,andf.Theyarecorrespondinglylabeledasgen- in the parametric space, the pumped current vanishes. This
eral up-down (GUD), general left-right (GLR), and general result, however, does not hold for non-adiabatic case where
inversion (GIV) symmetries. All potential profiles locate at thefrequencygivesadditionaldimensionofparametricspace.
theboundaryofthe pumpingregion,i.e., thefirstand/orlast Another result of interest is that, in contrast to the theoreti-
layerinthediscretelattice(seethedarkgrayregion). calpredictionIad ≈ 0forthesetupofGIVsymmetry29, the
pumped current from the setup of GIV is finite and has the
sameorderofmagnitudeasthatofGLRsymmetry.
Fig.3 plots the pumped current versus magnetic field
A. Symmetryofpumpedcurrent strength B at phase difference φ12 = π/2, where the mag-
nitude of Iad(φ12) is maximized. In the upper panel (a)
of Fig.3, we see that current is either even or odd func-
Before the presenting numerical results, we would like to
tion of magnetic field strength B for symmetries IUD, ILR,
point out that for setup c with instantaneous inversion sym-
and GUD, I(B) = ±I(−B) which agree with the theoreti-
metry (IIV), the theoretical predictions28,29 and our numeri- cal predictions.28,29 In panel (b), it is clear that the pumped
cal calculations give the same result: the pumped current is
current is invariant upon the reversal of magnetic field for
exact zero at both adiabatic and non-adiabatic cases that is GLR.28,29 ThesystemwithGIVsymmetryshowsanapprox-
independentof B and φ12. The phenomenacan be straight- imation relation I(B) ≈ I(−B) only at small B, which is
forwardly understood by Floquet scattering matrix theory29.
similar to that of GLR symmetry. This does not agree with
InIIVsetup,itisobviousthatthetransmissioncoefficientof thetheoreticalprediction.29Tofurtherinvestigatetherelation
an electron traveling from the left lead to the right TR←L is I(B)≈I(−B)westudiedthreeintermediatesetupsbetween
alwaysequaltothatofanelectronmovingintheoppositedi-
GIVandGLR.Inpanels(e)and(f)ofFig.1,thelengthofthe
rection,TL←R,i.e., pumpedpotentialprofileisfixedas20bothforV1 andV2 in
the system with 40×40 mesh. Then we shift down the po-
TR←L =TL←R tentialprofileV2 oftheGIVsymmetry5latticespacingeach
4

0.02
a)
0.010 ILR 0.001 IUD-0.004
ILR
0.005 Vp = 1.6 GUD
0.000
0.01
-0.005
d 0.000
-0.010 a L
I

V=1
0
d 0.00
aIL -0.001 Vp=0.5
E=0.62
F
IUD V=1 -0.009 φ =π/2
-0.01 ILR V0=0.5 b) 12
GUD p
GLR EF=0.62 -0.012
GIV B=0.001
-0.02-π -π/2 φ0 π/2 π ad -L0.015 GIV
12 I down 5
-0.018 down 10
down 15
FIG.2:Theadiabaticallypumpedcurrentasafunctionofphasedif- GLR
ferenceφ12fordifferentspatialsymmetriesofthepumpingsystemat -0.021
pumpingamplitudeVp =0.5.Inset:pumpedcurrentofsystemwith -0.0010 -0.0005 0.0000 0.0005 0.0010
symmetryILRatVp = 1.6. Othersystemparameters: EF = 0.62, B
B=0.001,V0=1.
FIG. 3: Panel (a): The adiabatically pumped current versus mag-
netic field strength B for system symmetries a(IUD), b(ILR), and
time. After 4 shifts, the system changes from GIV to GLR d(GUD).CurveforIUDisoffsetby-0.004foracompactillustration.
Panel(b):IadvsBforspatialsymmetryGLR,GIV,andthreeinter-
symmetry and in this process three intermediate systems are
generated.Numericalresultsshowninpanel(b)ofFig.3sug- mediatesetups. Calculationparameters: EF = 0.62, φ12 = π/2,
gestthatallthesesetupshavetherelationI(B) ≈ I(−B)at V0=1,Vp =0.5.
smallmagneticfields,althoughtherearenospatialreflection
symmetriesin these systems. The closer system to the GLR
symmetry,thelargerthemagneticfieldforthisrelation.These
resultssuggestthatthegeneralleft-rightsymmetryisarather
strongspatialsymmetryandevenaroughsetup(cyan-down- (a) of Fig.4 thatthe pumpedcurrentobeysan antisymmetric
triangle curve in panel (b)) can lead to an accurate invariant relation with phase difference: Inad(φ12) = −Inad(−φ12).
relation of pumpedcurrent, at least for small magneticfield. Inad(φ12 = nπ) = 0 is a natural result of this antisymme-
Recall the experimental result8 of an adiabatic pump where try relation. Combining with the result from the adiabatic
the experimentalsetup can nothave precise GLR symmetry, case (Fig.2), we see that this antisymmetryrelation between
butanaccuraterelationI(B)=I(−B)wasstillachieved.In pumpedcurrentandphasedifferenceφ12 isageneralfeature
addition, the amplitudeof pumpedcurrentin this setup with of the GIV symmetry. Besides, from panel (b) of Fig.4 we
GLRsymmetryisrelativelyhigh,comparedwithothersym- foundthat,Inad forGIVsystematafixedphaseφ12 = π/2
metries. shows Inad(B) ≈ Inad(−B) at small magnetic field. This
approximate symmetry relation can not be obtained theoret-
Now we turn our attention to the non-adiabatic electron
ically. Our results confirm the theoretical predictionson the
pump with finite pumping frequency. The numerical re-
parityofpumpedcurrentonreversalofmagneticfieldforse-
sults are presented in Fig.4. One of the major differences
between adiabatic and non-adiabatic pump is that a non- tupsIUD andILR29, whicharerespectivelyI(B) = I(−B)
adiabaticpumpcanoperatewithonlyonesystemparameter, and I(B) = −I(−B) (see panel (b)). However, it doesn’t
hold for GUD and GLR in panel (c). In this case, one can
sincethefinite pumpingfrequencysuppliesoneextradegree
offreedomanditcouldactasanotherpumpingparameter. In only get the relations I(B,φ) = I(−B,−φ) for GUD and
the experiment8 it was found that I(φ12 = 0) 6= 0. Later I(B,φ) = −I(−B,−φ) for GLR29. When the two pump-
a theoretical work22 attributed this phenomenon as a conse- ing potentials operate in phase or out of phase (φ12 = nπ),
quenceofphoton-assistedprocessesanditisanonlineartrans- they reduce to a simple version: I(B) = I(−B) for GUD
port feature of non-adiabatic electron pump. In our numer- andI(B) = −I(−B)forGLR,whicharethesameforIUD
ical results, we also found that Inad(φ12 = nπ) 6= 0 is a andILRatφ12 = nπ. Itis worthmentioningthatthesetwo
general property of the pumped current, except for systems relationsareincontrarytotheadiabaticcasewhereφ12 6=nπ.
with spatial symmetries IIV or GIV. Although the pumping
frequency ω can play the role of a variation parameter, the We collect the results and summarize these conclusions
pumped current in the system with symmetry IIV is always drawnfrombothadiabaticandnonadiabaticpumpsandthere
zero. For the setup with GIV symmetry, we see from panel areshowninTable.Iindetail.
5
adiabaticpump nonadiabaticpump
φ12 =nπ φ12 6=nπ φ12 =nπ φ12 6=nπ
IUD I =0⋄ I(B)=I(−B)⋄ I(B)=I(−B)⋄ I(B)=I(−B)⋄
ILR I =0⋄ I(B)=−I(−B)⋄ I(B)=−I(−B)⋄ I(B)=−I(−B)⋄
IIV I =0⋄ I =0⋄ I =0⋄ I =0⋄
GUD I =0⋄ I(B)=−I(−B)⋄ I(B)=I(−B)⋄ I(B,φ)=I(−B,−φ)⋄
GLR I =0⋄ I(B)=I(−B)⋄ I(B)=−I(−B)⋄ I(B,φ)=−I(−B,−φ)⋄
I =0⋄ I ≈0⊲ I =0⋄ I(B,φ)=−I(B,−φ)⋄
GIV ∗ ∗
I(B)≈I(−B) I(B)≈I(−B)
TABLEI:Symmetryofthepumpedcurrentsoninversionofthemagneticfieldforbothadiabaticandnonadiabaticelectronpumps.
⋄standsforthetheoreticalpredictionfromRef.29confirmedbyournumericalcalculation.
⊲representstheoreticalrelationwithoutnumericalsustainment.
∗correspondstoournewfindingincontrasttothetheoreticalprediction.
B. Transportpropertiesofthepumpedcurrent
0.2 a) IUD
ILR
In the last section we have concentrated on the symmetry
GIV/10
0.1 of the pumped currentwith magnetic field B and phase dif-
nad L0.0 sfeyrsetenmcepφa1r2amasettehresvoanritahbelepsu.mNpoewdcwuerrsetnutd.yNtuhmeeefrficeacltcoaflocuthlae-r
I
-0.1 tionswereperformedonasystemwithinstantL-Rsymmetry
(ILR),in whichwidthsofthefourpotentialbarriersarekept
-0.2 equal. The numerical results are plotted in Fig.5, Fig.6 and
-π -π/2 φ0 π/2 π Fig.7.
0.4 12
b)
IUD
nadIL00..02 IGLIRV-0.8 d current01..50 a) T5100(E**IIFan)add c) T1100(E0*IF0n)*aIdad 01..80
e
-0.2 p
m0.0
Pu 0.6 ent
-0-0.4.0010 -0.0005 0.0000 0.0005 0.0010 -0.5 curr
00..48 c) GGULRD*10 B current01..50 b) T5500(E*0IF0a)*dInad 00..24 Pumped
B=-0.001 ed
d mp0.0 0.0
na L0.0 Pu
I
B=0.001 -0.5
-0.4 -0.2
0.010 0.015 0.020 0.025 0.030 0.01150.01200.01250.0130
GUD*10
E E
GLR F F
-0.8
-π -π/2 0 π/2 π
φ FIG. 5: Panel (a) and (b): The pumped current as well as trans-
12 missioncoefficientasafunctionofFermienergyatstaticpotential
heightV0 = 1.0andV0 = 5.0,respectively. Forvisualizationpur-
FIG.4:Panel(a):Non-adiabaticallypumpedcurrentasafunctionof pose,afactorismultipliedtothepumpedcurrentinFig.5. ForIad
phasedifferenceφ12atafixedmagneticfieldB =0.001forspatial thefactoris50inbothpanels. ForInadthisfactoris10inpanel(a)
symmetriesIUD,ILR,andGIV.Bluecurvewithup-triangleforGIV and5000inpanel(b). Otherparameters: B = 0.001, φ12 = π/2.
is multiplied by a factor of 0.1. Panel (b): Inad versus magnetic Vp = 0.5, ω = 0.002 in panel (a) and Vp = 4.5, ω = 0.002 in
fieldstrengthB forthe abovethree setupswithφ12 = π/2. Blue panel(b). Panel(c)highlightsthepumpedcurrentatsmallpumping
curve with up-triangle for GIV is offset by −0.8. Panel (c): The amplitudes at the first resonant peak. V0 = 1.0 in this panel and
pumpedcurrentforsystemsetupsGUDandGLR.CurvesforGUD Vp =0.05,ω =0.0002. ThefactorsforIadandInadare1000and
aremultipliedby0.1. Calculationparameters: EF =0.62,V0 =1, 10,respectively.
Vp =40.
Inpanel(a)ofFig.5weplotthepumpedcurrentinthepres-
enceofmagneticfieldasafunctionofFermienergyE , to-
F
gether with the transmission coefficient T(E ) at static po-
F
tential barrier V0 = 1.0. The sharp tips of transmission co-
6
efficient suggests that quantum resonance effect dominates becomes17,
the transport process. When a dc bias is applied, the tun-
neling currentis calculated from transmission profile. How- 1 τ dQα(t)
I = dt
α
ever, the pumped current is generated as zero bias by peri- τ Z0 dt
odicallyvaryingacgatevoltages. Althoughoriginatingfrom qω ∂ dN ∂ dN
α α
differentphysical mechanisms, we see that the pumped cur- = dV1dV2 − (8)
2π Z (cid:18)∂V1 dV2 ∂V2 dV1 (cid:19)
rent clearly show resonance characteristics both in adiabatic
and non-adiabatic cases near the resonant energy of static FromEq.(7),itiseasytoseethattheintegrandiszero.Hence
transmission coefficient. These resonance-assisted behavior I = 0ifS = 0. Fornon-adiabaticcase,thepumpedcur-
α αα
of pumped currentis a generic propertyof electron pump.27 rentatcompletetransmissionisingeneralnonzero.However,
Operatingatthecoherentregime,quantuminterferencenatu- ifthefrequencyisverysmall,theadiabaticcaseisrecovered.
rally results in its resonant behavior. It is worth mentioning ThisisnumericallysupportedbyFig.5(c),wherethetwocur-
that near the sharp resonance at EF = 0.0118 the adiabatic rentcurvesareverysimilar.
pumpedcurrentchangessign. Thisisunderstandable. Inthe
presenceofdcbias,thedirectionofthecurrentisdetermined

by the bias. For parametric electron pump at zero bias, the
1.0
direction of the pumped currentdependsonly on the system T(E)
F
parameters such as Fermi energy and magnetic field. Varia- 50*Iad
tionoftheseparameterscanchangethecurrentdirection.For p
Inad
the non-adiabaticpump, the pumpedcurrentchangesslowly nt p
e
neartheresonancebutthereisnosignchangesforthepumped urr
current. InFig.5(a),wealsoseeasecondresonantpointwith d c 0.5
muchbroaderpeak. Nearthisresonantlevel,we seethatthe mpe
transmissioncoefficientandthepumpedcurrentsarewellcor-
u
related. The resonant feature of the pumped current is also P
related to the width of the resonantpeak in the transmission
coefficient.Similarbehaviorsarefoundforahigherstaticpo- 0.0
tentialbarrierV0 = 5.0 in panel(b). A largerbarriermakes
the resonantpeaks much sharper, but it doesn’t qualitatively
affectthe pumpedcurrent. The noticeabledifferenceis that, 0.07 0.08 0.09 0.10 0.11 0.12
E
thepumpedcurrentpeaksareshiftedwiththatofthetransmis- F
sion coefficient. In addition, it seems that the non-adiabatic
pumpedcurrentdevelopsaplateauregionnearthefirstreso- FIG.6: ThepumpedcurrentversusFermienergyinaT-shapedsys-
nantpeak. tem. The side bar is of length 20 and width 30. Two gray blocks
indicatethepositionswherethepumpingpotentialsareappliedand
the static potential is set to be zero. A factor of 50 is multiplied
When zooming in at this resonant peak (E = 0.0118
in Fig.5(a)), we found that at small pumping amplitude to Iad. Other parameters: B = 0.001, φ12 = π/2. Vp = 0.05,
ω=0.002.
the pumped currents are zeros for both adiabatic and non-
adiabatic cases when a complete transmission occurs (trans-
Furthermore,thebehaviorofpumpedcurrentforastructure
missioncoefficientT =1). Thenumericalevidenceisshown
exhibitinganti-resonancephenomenawasstudiedandthenu-
in panel (c) of Fig.5. Note that there is only one transmis-
mericalresultsisshowninFig.6. Toestablishanti-resonance,
sion channel for the incident energy so that T = 1 corre-
we use a T-junction32, which is schematically plotted in the
spondstocompletetransmission.Weemphasizethatthenon-
insetofFig.6. Thesidebarhaslongitudinaldimension20and
adiabaticpumpedcurrentgoestozeroneartheresonanceonly
transverse dimension 30. Pumping potentials are placed on
forverysmallfrequency.Atlargerfrequencysuchasthecase
the two arms of the device and there are no static potential
inFig.5(a)or(b),itisnonzero.ForthecaseofIad,itiseasyto
barriers in the system. From the transmission curve shown
understandwhyitiszeroatT =1.Foraperfecttransmission,
in Fig.6, one clearly finds that T(E ) drops sharply to zero
thediagonaltermsS andS ofthefour-blockscattering F
LL RR aroundE =0.112,whichisthesignatureofanti-resonance.
matrix are zero. Hence we have S = S = exp(iθ). F
LR RL Atthispoint,boththeadiabaticandnonadiabaticpumpedcur-
FromEq.(2),wehave
rentarezero.Differentfromtheresonantcase,heretherange
wherethecurrentiszeroornearlyzeroismuchbroader. The
phenomenaareattributedto reasonssimilar to thosewe pre-
dNα/dVj =(i∂θ/∂Vj)/π (7) sented above, but in this case SLR and SRL are zero at the
anti-resonance point. We also see that at transmission min-
imum E = 0.10 with small transmission coefficient the
F
pumpedcurrentisnonzero.
For two pumping potentials, the current can be expressed In panel(a) and (b)of Fig.7, we examinethe influenceof
in parameter space. Using the Green’s theorem, Eq.(1) pumpingamplitudeV onthepumpedcurrentswiththestatic
p
7

0.02 0.10
a) b)
2.0
0.01 VB=0=01.0.00 1 Eφ1F2= =0 .π0/1218 0.05 TT0
adI0L.00 B=0.001 E=0.0242 -00.0.005nad IL 1.5 110010*I*nIaadd
-0.01 V0=1.0 φ1F2 = π/2 -0.10 urrent1.0
0.0 0.3 0.6Vp0.9 1.2 0 .000 0.005 0.0ω10 0.015 0.020 d c
e
c) mp0.5
2 EVF==10..012 Pu
nt T(B) φ0 = π/2
urre 100*Iad 12 0.0
c L
ped 1 10*ILnad
m -0.5
u
P 0.0060 0.0065 0.0070 0.0075 0.0080

0
B
0.000 0.002 0.004 0.006 0.008
B
FIG. 8: The pumped current as well as transmission coefficient as
afunctionofmagneticfieldB atFermienergyEF = 0.12. Other
FIG. 7: Panel (a): Adiabatic current vs pumping potential Vp at parameters: φ12 = π/2, V0 = 1, Vp = 0.5, ω = 0.002. Iad and
EF =0.0118.Panel(b):Non-adiabaticcurrentversuspumpingfre- Inadarescaledbyfactorsof100and10,respectively.
quencyωatEF =0.0242.Systemparameters:φ12 =π/2,V0 =1,
B = 0.001. Panel(c)showsthepumped currentandtransmission
coefficientversusmagneticfieldBatFermienergyEF =0.12,with system. Mathematically it is also easy to show from Eq.(1)
otherparameters: φ12 = π/2,V0 = 1,Vp = 0.5,ω = 0.002. For
illustrationafactorof100ismultipliedtoIadanditis10forInad. that for a two-probe system as long as the instantaneous re-
flectioncoefficientvanishes(inthe case of edgestate) in the
wholepumpingperiodthereisnoadiabaticpumpedcurrent.
We providea numericalevidenceforthe abovestatement,
potential barrier fixed at V0 = 1, which corresponds to the whichisshowninFig.8. Incontrasttothecalculationofpanel
caseshowninpanel(a)ofFig.5. Atthefirstresonantenergy (c)ofFig.7,thestaticpotentialbarriersextendtoawidth40,
EF = 0.0118we plot Iad versusVp, the pumpingpotential which is exactly the width of the scattering region. At the
amplitude.Thenon-adiabaticpumpedcurrentInadasafunc- same time, the pumping barriers remain the same as before
tion of the pumping frequency ω is evaluated at the second (with width 10). Now the static transmission coefficient, la-
resonant peak EF = 0.0242. In both cases, magnitudes of beledT1 in thefigure,doesnothavequantizedvaluebutex-
the pumpedcurrentchangesin a oscillatoryfashionwith the hibitsa resonantbehavior. T0 is copiedfromFig.7for com-
increasing of Vp or ω. The pumped current can change its parison. As longasthe edgestate ofan electronis scattered
sign, which also reflects the nature of the parametric pump withtransmittedandreflectedmodes,thepumpedcurrentwill
andmanifestdistinctionbetweenthepumpedcurrentandthe begeneratedwithvaryingsystemparameters.
conventionalresonanttunnelingcurrent.
Theresonancebehaviorofpumpedcurrentisalsovisiblein
panel(c)ofFig.7,inwhichwedepictI andtransmissionco- IV. CONCLUSION
p
efficientversusmagneticfieldB atFermienergyE =0.12.
F
Sweeping throughmagnetic field, there is a sharp change of In conclusion, we have studied the pumped current as a
transmissioncoefficientnearB ∼0.003andthepumpedcur- function of pumping potential, magnetic field and pumping
rentchangesaccordingly. With increasing magneticfield, T frequencyintheresonantandanti-resonanttunnelingregimes.
becomesquantized(thereisonlyonetransmissionchannelat Resonantfeaturesareclearlyobservedforadiabaticandnon-
this magnetic field) indicating the occurrence of edge states adiabatic pumpedcurrent. We foundthat when the resonant
inthequantumHallregimeandthepumpedcurrentvanishes. peakissharptheadiabaticpumpedcurrentchangessignnear
In our setup, electron pump operates by cycling modulation the resonance while non-adiabatic pumpedcurrent does not.
ofelectronpassingthroughthepumpingpotentialswhichare When the resonant peak is broad the behaviors of pumped
ontopofstaticbarriersdefiningthesystem. Withincreasing currentinadiabaticandnon-adiabaticregimesaresimilarand
ofthemagneticfield,electronwavefunctiontendstolocalize bothchangesignneartheresonance.Atanti-resonance,how-
near the edge, which decreases the modulation efficiency of ever, both adiabatic and non-adiabatic pumped current are
the pumpingpotentials. As the edge state emerges, electron zero. AsthesystementersthequantumHallregime,pumped
will circumvent the confining potentials with no reflection currents vanishes in all the setups shown in Fig.1, since the
duringtheirdeformations.Inthiscase,thevariationofpump- pumpingpotentialscannotmodulatetheelectronwavefunc-
ing potential has no effect on the moving electron. Hence tion.Furthermore,wehavenumericallyinvestigatedthesym-
there is no pumpedcurrentwhenedge state is formedin the metry of the adiabatic and non-adiabatic pumped current of
8
systems with different symmetries placed in magnetic field. V. ACKNOWLEDGMENTS
The calculated results are listed in Table.I and most of them
are in agreement with the former theoretical results derived
fromFloquetscatteringmatrixtheory.Differentfromthethe-
oreticalprediction,wefoundthatthesystemwithgeneralspa- This work is supported by RGC grant (HKU 705409P),
tial inversion symmetry (GIV) gives rise to a finite pumped UniversityGrantCouncil(ContractNo. AoE/P-04/08)ofthe
currentatadiabaticregime. Atsmallmagneticfield,boththe GovernmentofHKSAR,andLuXinEnergyGroup.Thecom-
adiabatic and non-adiabatic currents have an approximation putationalworkispartiallyperformedonHPCPOWER2sys-
relationI(B)≈I(−B). temofthecomputercenter,HKU.
∗ Electronicaddress:[email protected] 16 M.Swikes,Ph.Dthesis,StanfordUniversity,1999.
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