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doi:10.1016/j.ph

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Physica E 40 (2008) 2988–2994

www.elsevier.com/locate/physe

Time-dependent electron transport through an Aharonov–Bohm ringembedded with two quantum dots

Hui Pana,�, Li-Na Zhaob, Rong Luc

aDepartment of Physics, Beijing University of Aeronautics and Astronautics, Beijing 100083, ChinabAtomistix Asia Pacific Pte Ltd, Unit 106, Innovation Center Block 1, 16 NanYang Drive, Singapore 637722, Singapore

cCenter for Advanced Study, Tsinghua University, Beijing 100084, China

Received 15 December 2007; received in revised form 11 February 2008; accepted 4 March 2008

Available online 7 March 2008

Abstract

The time-dependent electron transport through an Aharonov–Bohm ring embedded with two quantum dots in the presence of external

microwave (MW) fields are investigated theoretically by using the nonequilibrium Green’s function method. Whether the MW field can

induce or suppress the Fano resonance depends on the part to which the field is applied. When the MW field is applied only to the two

quantum dots, the photon-assisted Fano peaks appear at the sidebands of the original Fano peak. The existence of the original Fano

peak or the photon-assisted ones can be controlled by the field strength. When the MW field is applied only to one lead, the original Fano

peak is suppressed by the MW field, and the negative current caused by the electron–photon pump effects is found.

r 2008 Elsevier B.V. All rights reserved.

PACS: 73.23.�b; 73.63.Kv; 73.23.Hk; 73.40.Gk

Keywords: Photon assisted tunneling; Fano effect; Parallel-coupled double quantum dots

1. Introduction

Recently, the time-dependent transport of electronsthrough a mesoscopic system has received more and moreattention. An essential feature is the well-known photon-assisted tunneling (PAT). The electron can tunnel throughthe system by emitting or absorbing multiple photons, andthen the new inelastic tunneling channels are opened. Theobservations of PAT have been reported in single quantumdot (QD) systems experimentally [1–3]. Time-dependenttunneling through coupled quantum dots in series has alsoreceived large attention both experimentally and theoreti-cally. Experimentally, the PAT current through seriallycoupled double quantum dots has been observed, andthe predicted extra resonance peaks under irradiation ofmicrowave are clearly discovered [4–6]. Theoretically, dif-ferent theoretical approaches have been developed. Basedon the nonequilibrium Green’s function (NGF) method [7],

e front matter r 2008 Elsevier B.V. All rights reserved.

yse.2008.03.002

ing author. Tel./fax: +86 10 82317935.

ess: hpan@buaa.edu.cn (H. Pan).

the photon–electron pumping current in serially coupleddouble quantum dots is studied [8].The electron phase coherence can be sustained during

the tunneling process through the QD systems [9]. Fanoeffect [10], i.e., the asymmetric line shapes in conductance,is a good tool to investigate the electron phase coherence inthe QD system. The Fano-type line shapes in conductancestems from quantum interference between resonant andnonresonant processes [11–15]. The discrete QD energylevel is broadened by a factor due to the couplings with theleads. The phase of the electron smoothly changes by p onthe Fano resonance within the broadened factor of the QDenergy level [16]. In the Fano effect of these system, thenonresonant channel is served by a quantum point contact,which can detect the p phase shift around the resonanttunneling channel through the QD [13–15]. More recently,an AB interferometer embedded with two quantum dotshas been realized [17–20]. The parallel-coupled doublequantum dot (DQD) system makes the quantum transportphenomena rich and varied [4,21,22]. Inspired by theserecent experiments, several groups have attempted to

ARTICLE IN PRESSH. Pan et al. / Physica E 40 (2008) 2988–2994 2989

address this parallel DQD system theoretically and pre-dicted the existence of the Fano resonance [22–27]. In theFano effect of the parallel-coupled DQD system, the twomolecular states play an important rule. The referencechannel is the molecular state with a wide band, and theother channel is the molecular state with a narrow bandaccompanied with a swift p phase shift [28].

So far, the time-dependent electron transport behaviorsfor the parallel-coupled DQD system are still less studied.How will an external MW fields affect the Fano effects?Will the MW field suppress the Fano effects? To answerthese questions, we theoretically study the MW field effectson the electron transport in a parallel coupled quantum dotsystem. We assume that the electron tunneling through theDQD is coherent and that only one electronic state in eachdot is involved, both facts consistent with the experiment.The MW field applied on the system induces an adiabaticchange for the energy of the leads and the quantum dots[7,8,29,30]. Then the energy levels of the leads and thequantum dots oscillate under the influence of the MW field.Our results show that the MW fields can affect the photon-assisted Fano resonance differently when the field isapplied on different parts of the system. The rest of thispaper is organized as follows. In Section 2 we present themodel Hamiltonian and derive the formula of the photon-assisted current by using the NGF technique. In Section 3we study the photon-assisted current with the field appliedon different parts of the system. Finally, a brief summary isgiven in Section 4.

2. Physical model and formula

The parallel-coupled DQD connected with two normal-metal leads under an external MW field is described by thefollowing Hamiltonian:

H ¼Xa¼L;R

Ha þHD þHT , (1)

with

Ha ¼X

k

�a;kðtÞay

a;kaa;k, (2)

HD ¼Xi¼1;2

eiðtÞdy

i di � ðtcdy

1d2 þH:c:Þ, (3)

HT ¼X

a;k;i¼1;2

taidy

i aa;k þH:c. (4)

Ha ða ¼ L;RÞ describes the left and right normal-metalleads. HD models the parallel-coupled double quantumdots where d

y

i ðdiÞ represents the creation (annihilation)operator of the electron with energy ei in the QD i

ði ¼ 1; 2Þ. The energy levels in the dots are measured fromthe Fermi energy of the leads. tc denotes the interdotcoupling strength. Under the adiabatic approximation, thetime-dependent microwave field can be reflected in thesingle-electron energies which can be separated into two

parts: �a;kðtÞ ¼ �a;k þ DaðtÞ for the leads, and eiðtÞ ¼ ei þ

D0ðtÞ for the central conductor. Da;0ðtÞ is a time-dependentpart from the external microwave fields, which can bewritten as Da;0ðtÞ ¼ Da;0 cosoa;0t. The tunneling matrix ofthe hopping elements t�a is the 2� 2 matrix for the presenttwo DQD system with tai the hopping strength between theith QD and the a lead.The current Ia from the a lead to the central region

can be calculated from standard NGF techniques, andcan be expressed in terms of the dot’s Green functionas [7,8].

IaðtÞ ¼2e

_Re

Zdt0 Trf½Gr

ðt; t0ÞRoa ðt0; tÞ

þGoðt; t0ÞRa

aðt0; tÞ�g. (5)

Here, the Green’s function Gr;o and the self-energy Ra;o

are all two-dimensional matrices for the DQD system. Theretarded and lesser Green functions are defined asGrðt; t0Þ ¼ �iyðt� t0ÞhfCðtÞ;Cyðt0Þgi and Go

ðt; t0Þ ¼ ihCyðt0ÞCðtÞi, respectively, with the operator C ¼ ðdy1; d

y

2Þy. The

retarded Green’s functions can be calculated by usingDyson equation

Grðt; t0Þ ¼ grðt; t0Þ þ

Zdt1

Zdt2G

rðt; t1ÞR

rðt1; t2Þgrðt2; t

0Þ,

(6)

where grðt; t0Þ is the retarded Green’s function for thequantum dots without the coupling to the leads and Rr ¼P

a Rra. Under the wide-band approximation, the retarded

self-energy due to the a lead can be derived as

Rraðt; t

0Þ ¼X

k

t�agrkaðt; t

0Þta ¼ �i

2dðt� t0ÞCa (7)

¼ �i

2dðt� t0Þ

Ga1

ffiffiffiffiffiffiffiffiffiffiffiGa1G

a2

pffiffiffiffiffiffiffiffiffiffiffiGa1G

a2

pGa2

0@

1A, (8)

where Gai is the linewidth function defined by Ga

i ¼

2prat�aitai with ra being the density of states of thecorresponding a lead. Ga

i describes the coupling be-tween the ith QD and the a lead. gr

kaðt; t0Þ is the exact

Green’s function of the a lead without the coupling to theDQD defined as gr

kaðt;t0Þ � �iyðt� t0Þhfaa;kðtÞ; a

y

a;kðt0Þgi ¼

�iyðt� t0Þ exp½�iR t

t0dt1 �a;kðt1Þ�. The advanced self-energy

can be obtained from the relation Raaðt; t

0Þ ¼ ðRraðt; t

0ÞÞy.

Due to the ac MW field, the gauge transformation isperformed as [31]

Grðt; t0Þ ¼ ~Gr

ðt� t0Þ exp �i

Z t

t0dt1D0 cosðo0t1Þ

� �, (9)

where ~Grðt� t0Þ is the Green’s function without the MW

field. After some algebra, we can obtain

Grðt; t0Þ ¼

Zd�

2pexp � i�ðt� t0Þ

�

� i

Z t

t0dt1D0 cosðo0t1Þ

�Grð�Þ, (10)

ARTICLE IN PRESSH. Pan et al. / Physica E 40 (2008) 2988–29942990

where Grð�Þ can be derived from the Dyson equation

Grð�Þ ¼ ½gr�1ð�Þ � Rrð�Þ��1 (11)

and grð�Þ is the Green’s function of the DQD systemwithout the coupling to the leads

grð�Þ ¼�� e1 tc

tc �� e2

!�1. (12)

The lesser Green functions can be calculated by usingKeldysh equation

Goðt; t0Þ ¼

Zdt1

Zdt2G

rðt; t1ÞR

oðt1; t2ÞGaðt2; t

0Þ, (13)

where Roðt1; t2Þ ¼P

a Roa ðt1; t2Þ. Under the wide-band

approximation, the lesser self-energy due to the a leadcan be derived as

Roa ðt; t

0Þ ¼X

k

t�agokaðt; t

0Þta

¼ i

Zd�

2pexp �i�ðt� t0Þ � i

Z t

t0dt1Da cosðoat1Þ

� �� f ð�a;ksÞCa, (14)

where f ð�a;kÞ ¼ 1=ðeð�a;k�maÞ=kBT þ 1Þ is the Fermi-distribu-tion function of the a lead with mL ¼ eV and mR ¼ 0.go

kaðt; t0Þ is the exact Green’s function of the a lead without

the coupling to the DQD defined as gokaðt; t

0Þ � ihaya;kðt0Þaa;kðtÞi ¼ if ð�a;kÞ exp½�i

R t

t0dt1�a;kðt1Þ�.

Substituting the retarded and lesser Greens functionsinto the current formula, the current becomes [7,30]

Ia ¼ �e

_

Zd�

2pTr 2f að�ÞIm½CaAað�; tÞ�

(

þRe Ca

Xb

f bð�ÞAbð�; tÞCbAy

bð�; tÞ

" #), (15)

where

Aað�; tÞ ¼

Zdt0 exp i�ðt� t0Þ þ i

Z t

t0dt1Da cosðoat1Þ

� ��Grðt; t0Þ. (16)

The average current is

hIi ¼2e

_

Zd�

2p

Xm;n;m0 ;n0

Tr JLmn

m0n0

� �f Lð�ÞCLG

rð�mn0 Þ

(

�CRGað�mn0 Þ � JR

mn

m0n0

� �f Rð�ÞCRG

rð�mn0 Þ

�CLGað�mn0 Þ

), (17)

where Jaðmn

m0n0Þ ¼ JmðDa=oaÞJnðDa=oaÞJm0 ðD0=o0ÞJn0 ðD0=o0Þ

and �mn0 ¼ �þmoa � n0o0. For the case in which the ex-

ternal MW field in each region has the same frequency,o0 ¼ oL ¼ oR, hIi reduces to

hIi ¼2e

_

Zd�

2p

Xm

Tr J2m

D0 � DL

o0

� �f Lð�Þ

�

�J2m

D0 � DR

o0

� �f Rð�Þ

�CLG

rð�mÞCRG

að�mÞ

, (18)

where �m ¼ ��mo0. In the absence of the MW fields withDa ¼ D0 ¼ 0, the current becomes time-independent asfollows

I ¼2e

_

Zd�

2pTrf½f Lð�Þ � f Rð�Þ�CLG

rð�ÞCRG

að�Þg. (19)

In the following, we perform the calculations at zerotemperature in units of _ ¼ e ¼ 1, and the energy ismeasured in units of the MW frequency with o0 ¼ 1. Weconsider two cases where the MW field is applied only tothe central double quantum dots or only to the left lead.The ratio of the field strength to the frequency are definedas aL;0 ¼ DL;0=o0.

3. Numerical results and discussions

The Hamiltonian HD for the coupled DQD can be

diagonalized as ~HD ¼ eþ ~dyþ~dþ þ e� ~d

y�~d� with e� ¼ 1

2½e1 þ

e2 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðe1 � e2Þ

2þ 4t2c

q� [28]. ~d� and ~dþ are the annihila-

tion operators for the bonding and antibonding states of

the QD molecule expressed as ~dþ ¼ d1 cos b� d2 sin b and~d� ¼ d1 sin bþ d2 cos b with b ¼ 1

2tan�1½2tc=ðe1 � e2Þ�.

The linewidth matrix corresponding to the two mole-

cular states coupled to the a lead are Gaþ ¼ Ga

1 cos2bþ

Ga2 sin

2b�ffiffiffiffiffiffiffiffiffiffiffiGa1G

a2

psin 2b and Ga

� ¼ Ga1 sin

2bþ Ga2 cos

2bþffiffiffiffiffiffiffiffiffiffiffiGa1G

a2

psin 2b, respectively. Then, the DQD system is

mapped onto a system of two independent molecularstates with band Ga

� connected to leads. The state

associated with a wide band can be referred to as thestrongly coupled one, while that with a narrow band isreferred to as the weakly coupled state. In the absence ofthe MW fields with Da ¼ D0 ¼ 0, the current shows aBreit–Wigner peak around the strongly coupled state(V g ¼ e�) and a Fano peak around the weakly coupled

state (V g ¼ eþ) as shown in Fig. 1. The two QD energy

levels are set as e1 ¼ e2 ¼ 0, and thus the bonding andantibonding state are ate� ¼ �tc and eþ ¼ tc, respectively.The Fano resonance results from the interference of theelectrons tunneling through the two molecular states. Sincethe band of the strongly coupled channel is much widerthan that of the weakly coupled channel, the phase shift forthe strongly coupled channel is negligibly small around theweakly coupled channel. Then, a phase shift of p across theweakly coupled level can be detected with characteristic ofthe Fano line shape. As a result, the Fano line appearsaround the weakly coupled state.

ARTICLE IN PRESSH. Pan et al. / Physica E 40 (2008) 2988–2994 2991

In the following, we investigate the MW field effects onthe average current through the system. First, we considerthe case in which the MW field is applied on the doublequantum dots. Fig. 2 displays the current as a function ofgate voltage V g at different MW field strength D0. Due tothe interference between the weakly and strongly coupledstates, the average current exhibits a Breit–Wigner and aFano line shape centered at the bonding and antibondingstates, respectively. Some new Fano peaks appear at theside bands of the antibonding state ðV g ¼ eþ � oÞ asshown in Fig. 2(a). The reason is related to the photon-assisted tunneling caused by the MW field. The bondingand antibonding state each has the side binds at e� � o andeþ � o, respectively. The interference can exist betweenthese side bands, resulting in the photon-assisted Fanopeaks. The amplitude of the nth sideband is in the form of anontrivial dependence on the Bessel function J2

nða0Þ. The

Fig. 2. hIi versus V g with a MW field applied on the double quantum dots at

same as those in Fig. 1.

0.003

0.002

0.001

0.000-2 -1 0 1 2

Vg

I

Fig. 1. I versus Vg in the absence of the MW fields. The couplings between

the two quantum dots and leads are set as GL1 ¼ 2GL2 ¼ 2GR1 ¼ GR2 ¼

tc ¼ 0:2, and the bias is V ¼ 0:005.

number of the Fano peaks can thus be controlled by thefield strength. When the MW field strength is increased, themiddle Fano peak disappears as shown in Fig. 2(b). Withfurther increasing D0, the two satellite Fano peaks at theside bands disappear, but the original middle one appearsagain as shown in Fig. 2(c). These characters are theconsequences of the photonic sidebands induced by theMW field. The position of the resonant current peaksare modulated by a displacement when electrons emit orabsorb photons in energy of no0. The amplitude of thecurrent peaks are characterized by the square of the nthBessel function Jnða0Þ for the nth side band.Next we consider the case in which the ac field is applied

only on the left lead, the average current becomes quitedifferent from the above case. Fig. 3 shows the averagecurrent at different field strength DL. At small DL, one mainresonant peak and one Fano peak exist at V g ¼ e� andVg ¼ eþ, respectively, as shown in Fig. 3(a). Furthermore,there appears a positive shoulder and a negative valleyon the left and right side of the main resonant currentpeak, respectively. The negative current is caused by theelectron–photon pump effects. When the energy level islower than the mL, it can be occupied by one electron.Under the influence of the MW field, the electron canabsorb a photon and tunnel to the left lead but not to theright lead, since the MW field is only applied on the leftlead. This results in a negative current. With increasing DL,the Fano peak still exists at the antibonding state ofVg ¼ eþ, as shown in Fig. 3(b). However, with furtherincreasing aL, the Fano peak disappears as shown inFig. 3(c). At the same time, the shoulder becomes higher,and the magnitude of the negative current larger. In fact,the MW field applied on the quantum dot causes thesidebands, and the MW field applied on the left lead inon asymmetric way induces the electron–photon pumpeffect.As mentioned above, the resonant amplitude of the

current is sensitive to the ratio of the strength and the

(a) D0 ¼ 1:2, (b) D0 ¼ 2:405, and (c) D0 ¼ 3:75. Other parameters are the

ARTICLE IN PRESS

Fig. 4. hIi versus (a) a0 and (b) aL at Vg ¼ e0 (solid line), V g ¼ e0 þ o0 (dashed line) and Vg ¼ e0 �o0 (dotted line). Other parameters are the same as

those in Fig. 1.

Fig. 3. hIi versus V g with a MW field applied on the left lead at (a) DL ¼ 0:2, (b) DL ¼ 0:4, and (c) DL ¼ 1:0. Other parameters are the same as those in

Fig. 1.

H. Pan et al. / Physica E 40 (2008) 2988–29942992

frequency of the MW field, which appears as an argumentof the Bessel function. Fig. 4 clearly shows the dependenceof hIi on the ratio a0 ðaLÞ at different V g. When the MWfield is only applied to the central quantum dots, theaverage current hIi oscillates damply with increasing a0 asshown in Fig. 4(a). The current hIi at Vg ¼ 0 shows thebehavior of the 0 order Bessel function J0ða0Þ, and thecurrent I at V g ¼ �o0 shows the behavior of the 1st orderBessel function J1ða0Þ. This clearly explain the numberchange of the Fano peaks in Fig. 2 at different D0. Ata0 ¼ 1:2, the current at V g ¼ 0 and �o0 all have the largevalues, resulting three Fano peaks. At a0 ¼ 2:405, thecurrent at Vg ¼ 0 is nearly zero, but the current at Vg ¼

�o0 still have a large value. Thus, only the current at theside bands appear and the current at the main resonancedisappears, resulting the two Fano peak. When the MWfield is only applied to the left lead, the average current hIiat Vg ¼ 0 and o0 oscillates and to a large negative value

with increasing aL as shown in Fig. 4(b). This explains thatthe negative current becomes larger with increasing aL asshown in Fig. 3.The average current versus the energy difference de with

the MW field applied to the central quantum dots or to theleft lead are plotted in Fig. 5, respectively. The molecularstates are approximately eþ ¼ jdej=2 and e� ¼ �jdej=2. Inthe presence of a field applied to the central quantum dots,if the photon has the energy o ¼ jdej=2, an electron cantunnel from the left lead to state eþ or from state e� to theright lead by absorbing one photon. Thus, two satelliteresonance peaks emerge at de ¼ �2o as shown in Fig. 5(a).When the two uncoupled states are lined up ðde ¼ 0Þ, ahigh resonance emerges due to the main resonance. In thepresence of a field applied to the left lead, the mainresonance peak ðde ¼ 0Þ still exists. The negative currentvalley due to the photon–electron pumping effect can occuras shown in Fig. 5(b).

ARTICLE IN PRESS

δ ε δ ε

Fig. 5. (a) hIi versus de with a MW field applied on the double quantum dots at D0 ¼ 0:8. (b) hIi versus de and with a MW field applied on the left lead at

DL ¼ 0:2. Other parameters are the same as those in Fig. 1.

Fig. 6. (a) hIi versus V with a MW field applied on the double quantum dots at D0 ¼ 1:2 (solid line), D0 ¼ 2:405 (dashed line), and D0 ¼ 3:75 (dotted line).

(b) hIi versus V with a MW field applied on the left lead at DL ¼ 0:2 (solid line), DL ¼ 0:4 (dashed line), and DL ¼ 1:0 (dotted line). The gate voltage is

Vg ¼ 0 and other parameters are the same as those in Fig. 1.

H. Pan et al. / Physica E 40 (2008) 2988–2994 2993

The average current hIi versus the bias V for the MWField applied to the central quantum dots or to the left leadunder different field strength are plotted in Fig. 6(a) and(b), respectively. In Fig. 6(a), the hIi–V relationship givessymmetric behaviors in positive and negative domains ofV. The different quantities of inflection points in thecurrent curves under different D0 reflect different PATsituations during the transports. The current reachessaturation with different values when bias becomes largeenough. In Fig. 6(b), the current curve is definitelyasymmetric when the MW field is only applied to the leftlead. There is a negative current even at positive bias. Withthe field strength increasing, the asymmetric behaviorappears more significant. All these result from theelectron–photon pump effects mentioned above.

4. Conclusions

In this paper, we have investigated the time-dependentelectron transport through an Aharonov–Bohm ringembedded with two quantum dots in the presence ofexternal MW fields. By using the nonequilibrium Green’sfunction method, the time-dependent current IðtÞ and theaverage current hIi are derived. We find that whether theMW field can induce or suppress the Fano resonancedepends on the part to which the field is applied. Whenthe MW field is applied only to the two quantum dots, thephoton-assisted Fano peaks are found to be at thesidebands of the original Fano peak. The existence ofthe original Fano peak and the photon-assisted ones can becontrolled by the field strength. When the MW field is

ARTICLE IN PRESSH. Pan et al. / Physica E 40 (2008) 2988–29942994

applied only to the left lead, the original Fano peak issuppressed by the MW field, and the negative currentcaused by the electron–photon pump effects is found. Bytuning the energy difference of the two quantum dots,satellite resonance peaks emerge at de ¼ �2o when thefield is only applied to the quantum dots, and the negatecurrent due to the photon–electron pumping effects appearwhen the field is only applied to the left lead.

Acknowledgments

This work was supported by the National NaturalScience Foundation of China (Grant No. 10704005), theBeijing Municipal Science and Technology Commission(Grant No. 2007B017). R.L. was supported by the MOE ofChina (Grant No. 200221), and the Ministry of Science andTechnology of China (Grant No. 2006CB605105).

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