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Electronic transport through a T-shaped four-quantum dot system
ARTICLE IN PRESS
Microelectronics Journal 38 (2007) 570–575 www.elsevier.com/locate/mejo
Electronic transport through a T-shaped four-quantum dot system Haitao Yina,b, Tianquan Lu¨a,, Hua Lic, Zelong Hea a
Center for Condensed-Matter Science and Technology, Harbin Institute of Technology, Harbin 150001, PR China b Hulan School, Harbin Normal University, Harbin 150500, PR China c Department of Physics, Harbin Normal University, Harbin 150080, PR China Received 23 December 2006; accepted 3 March 2007 Available online 30 April 2007
Abstract Making use of the equation of motion method and Keldysh Green’s function technique, we have developed a calculation method by which an analytical formula for the current under dc bias is obtained and the transmission probabilities are numerically studied for a T-shaped four-quantum dot (QD) system connected to three terminals. The results are explained in terms of the localization properties of the confined states in the system. r 2007 Elsevier Ltd. All rights reserved. Keywords: Quantum dot; Transmission probability; Electronic transport
1. Introduction Quantum dots (QDs) are highly tunable artificial mesoscopic structures, usually called artificial atoms or artificial molecules. A great advantage of this artificial atom-like system is that one can easily change its internal states by adjusting the gate voltages imposed on its external conducting leads. The recent experimental advances in the nanofabrication of quantum devices make it possible to fabricate QD array [1–3]. Also, these systems reveal very rich physics, such as Coulomb blockade [4], Coulomb staircase [5], the nonequilibrium Kondo effect [6–8], and Aharonov–Bohm oscillations in the conductance and persistent currents, etc.[9–13]. QDs, are very promising systems due to their physical properties as well as their potential applications in electronic devices, have drawn considerable attentions. In the last decade, much work has been contributed to the system of a QD array connected to two-electron reservoirs. Many authors have studied the transport properties and capacitance spectroscopy in order to probe the discrete electronic spectrum of the system [14–18]. Corresponding author. Tel.: +86 451 86412770; fax: +86 451 86412828. E-mail address: [email protected] (T. Lu¨).
0026-2692/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2007.03.003
Recently, some attention has been paid to transport of a single- or double-QD system connected to multiterminal, and here we mention only a few of them. In Ref. [19] the conductance of the N-lead system was considered showing that the Kondo resonance at equilibrium is split into N1 peaks. The electron transport and shot noise in a multiterminal coupled QD system in which each lead was disturbed by classical microwave fields were studied [20]. Multi-terminal QD systems or magnetic junctions were also intensively investigated in context of the spindependent transport [21,22]. Leturcq et al. [23] probed the Kondo density of states in a three-terminal quantum ring. In our previous works [24–26], we have studied the transport properties of a ring-shaped array of QDs connected to two terminals and found some unique properties other than that of a linear array of QDs, which indicate the geometry structure is very important to the transport properties of the coupled QD system. What’ more, motivated in part by the works cited in the preceding paragraph, in this paper we investigate transport properties of a T-shaped four-QD system connected to three terminals based on the nonequilibrium Green function proposed by Jauho et al. [27]. The formulation we derive for the current can also be easily generalized to any other multiterminal system. The rest of the paper is organized as follows. In
ARTICLE IN PRESS H. Yin et al. / Microelectronics Journal 38 (2007) 570–575
Section 2, we derive the formula of the current. Numerical results and analysis are given in Section 3. Our concluding remarks are given in Section 4. 2. Model Hamiltonian and the formula for the DC current We consider a T-shaped four-QD system connected to three terminals, as depicted schematically in Fig. 1. For simplicity, we ignore the intradot and interdot Coulomb interactions between two electrons, and assume that just one energy level is relevant at each dot. The full system is moduled by a tight-binding Hamiltonian within a noninteracting picture, which can be written as 4 X
H dotlead ¼
þ V 24 d þ 2 d 4 þ h:c:Þ, X X kL C þ kR C þ kL C kL þ kR C kR k k X þ kM C þ kM C kM , k X X V LCþ V RCþ kL d 1 þ kR d 3 k k X þ V MCþ kM d 4 þ h:c:, k
L
V12 1
V23 2
(4)
Our calculation is limited to the steady state where the Green’s functions only depend on the variable Dt ¼ t t0 . By Fourier transforming with respect to variable tt0 in Gi;i ðt t0 ; 0Þ, we can obtain Gi;i ðoÞ. Making use of the motion equation and the Keldysh’s contour integration, we obtain the following equations for the retarded Green functions: o 01 SrL G r1;1 ðoÞ V 12 Gr2;1 ðoÞ ¼ 1, (5)
ð6Þ
o 03 SrR Gr3;1 ðoÞ V 32 Gr2;1 ðoÞ ¼ 0,
(7)
o 04 SrM Gr4;1 ðoÞ V 42 G r2;1 ðoÞ ¼ 0
(8)
o o a ðo 01 SrL ÞGo 1;1 ðoÞ V 12 G 2;1 ðoÞ ¼ SL G 1;1 ðoÞ,
(9)
o o ðo 02 ÞGo 2;1 ðoÞ V 21 G 1;1 ðoÞ V 23 G 3;1 ðoÞ
ð1Þ
VR 3
V24
þ 0 0 Go i;i ðt; t Þ ¼ i d i ðt Þd i ðtÞ .
and those for the ‘‘lesser’’ Green functions
where C þ ka ðC ka Þ ða 2 L; M; RÞ are the creation (annihilation) operators of electron in the left (L) lead, the middle (M) lead and the right (R) lead; d þ i ðd i Þ is the creation (annihilation) operator of QD i; ka ða ¼ L; M; RÞ is the single-electron energy in the a lead; 0i is the energy level in the ith dot; Vij and Va are the interdot and dot–lead coupling matrix elements, respectively. For simplicity, only a single pair of spin degenerate levels is included in each dot, and here the electron spin index is suppressed. To describe the nonequilibrium states in the system we introduce the retarded, the advanced and the ‘‘lesser’’ Green’s functions as follows: D E 0 G ri;i ðt; t0 Þ ¼ iyðt t0 Þ d i ðtÞ; d þ ðt Þ , (2) i þ VL
(3)
V 24 Gr4;1 ðoÞ ¼ 0,
þ þ 0i d þ i d i þ ðV 12 d 1 d 2 þ V 23 d 2 d 3
i¼1
H lead ¼
D E 0 , ðt Þ Gai;i ðt; t0 Þ ¼ iyðt0 tÞ d i ðtÞ; d þ i þ
ðo 02 ÞGr2;1 ðoÞ V 21 G r1;1 ðoÞ V 23 G r3;1 ðoÞ
H ¼ H dot þ H lead þ H dotlead , H dot ¼
571
R L
V 24 Go 4;1 ðoÞ ¼ 0,
ð10Þ
o o a ðo 03 SrR ÞG o 3;1 ðoÞ V 32 G 2;1 ðoÞ ¼ SR G 1;1 ðoÞ,
(11)
o o a ðo 04 SrM ÞG o 4;1 ðoÞ V 42 G 2;1 ðoÞ ¼ SM G 4;1 ðoÞ.
(12)
In the above Xr;a;o X r;a;o 2 ðoÞ ¼ V gka , a a
is the self-energy associated with the tunneling between dot and a lead which is connected to the dot. The Green functions gr;a;o ka ðoÞ in Eq. (13) correspond to free-electron Green functions in lead a and have the following relations: grka ðoÞ ¼ ðo ka þ iZÞ1 ,
(14)
gaka ðoÞ ¼ ðo ka iZÞ1 ,
(15)
go ka ðoÞ ¼ 2pif a dðo ka Þ.
(16)
Eqs. (5)–(12) form a closed equation set for final results are achieved as follows:
4 VM
(13)
k
Gr1;1 ðoÞ ¼
Fig. 1. Schematic plot of a T-shaped four-quantum dot system with three terminals.
and the
,
(17)
1 o 01
Pr
L
V2
12
2 V 223 Pr 0V 24Pr 2 o0 o 3 4
o0
R
M
Gr;o 11
M
r 2 L R Go 1;1 ðoÞ ¼ iG ðoÞf L ðoÞ G 1;1 ðoÞ þ iG ðoÞf R ðoÞ r 2 2 G 3;1 ðoÞ þ iGM ðoÞf M ðoÞG r4;1 ðoÞ ,
ð18Þ
ARTICLE IN PRESS H. Yin et al. / Microelectronics Journal 38 (2007) 570–575
572
with
3. Numerical results
o
02
V2
23P
o0 3
r R
V2
24 Pr o0 M 4
Gr1;1 ðoÞ,
(19)
G r3;1 ðoÞ ¼
V 23 P Gr ðoÞ, o 03 rR 2;1
(20)
G r4;1 ðoÞ ¼
V 24 P Gr ðoÞ. o 04 rM 2;1
(21)
In the above derivations, we assume the coupling matrix elements (Vij and Va) are real since the tunnel rates just depend on the amplitude of the coupling matrix elements. We can then calculate the current flowing from the left lead to the structure is Z i eX h o V L Go JL ¼ (22) 1;kL ðoÞ V L G kL;1 ðoÞ do. h k Making use of the Dyson equation h i r o o a Go 1;kL ðoÞ ¼ V L G 1;1 ðoÞgkL ðoÞ G 1;1 ðoÞgkL ðoÞ
(23)
one can express the current with the Green function G r1;1 ðoÞ. After a simple calculation, we get Z n h io i2e r a GL ðoÞ G o JL ¼ do, 1;1 ðoÞ þ f L ðoÞ G 1;1 ðoÞ G 1;1 ðoÞ h (24) where GL ðoÞ ¼ 2p
X
jV L j2 dðo kL Þ,
(25)
k
is the coupling of dot 1 to the left lead. We have similar definitions forGRðMÞ ðoÞ for the 1right lead and the middle lead. f a ðoÞ ¼ 1 þ eðoma Þ=kB T is Fermi–Dirac distribution function of the a lead with chemical potential ma. The factor ‘‘2’’ in the Eq. (24) is due to the spin degeneracy. Substituting Eq. (18) into Eq. (24), After a simple calculation, we can then calculate the current between dot 1 and the left lead as Z 2e JL ¼ T LR ðoÞ f L ðoÞ f R ðoÞ , h þT LM ðoÞ f L ðoÞ f M ðoÞ do, ð26Þ where (27)
2 T LM ðoÞ ¼ GL ðoÞGM ðoÞGr4;1 ðoÞ .
(28)
R
a
b
c
2 T LR ðoÞ ¼ G ðoÞG ðoÞG r3;1 ðoÞ , L
In this section, we perform the detailed calculations of T LR ðoÞ and T LM ðoÞ, for various single-QD energy level arrangements and various interdot coupling amplitudes. For simplicity, a constant density of states is endowed to each lead. In the wide-band limiting approximation, the line-width functions Ga ðoÞða 2 L; M; RÞ are energy-independent constant, and the level shift is zero. Throughout this paper, we assume GL ¼ G. All energy and coupling strengths are measured by units of G. Our interest is mainly to find out how parameters determine the number and position of resonant transmission peaks. In the case of a constant density of states in the leads, T LRðMÞ ðoÞ is proportional to the transmission probability of an electron with energy o. Here we present some numerical results of T LRðMÞ ðoÞ, which not only provide the information of the resonant states of the system but also indicate how each state contributes to the electron transport. In a QD described in Eq. (1), one possible resonant state is presented with the QD energy level. When the four dots are coupled, four states are mixed and become the states spreading throughout the whole system. Fig. 2 shows the T LR ðoÞ2o curves and T LM ðoÞ2o curves for V ¼ 0:1,1:0,5:0. Here we assume V 12 ¼ V 23 ¼ V 24 ¼ V . As shown in the plot, the peaks of TLR and TLM match absolutely with the same height and the same positions. By carefully examining the level positions of all the peaks, we see that one level always lies at the single-QD quantum energy level and the other two are distributed symmetrically with respect to the energy. The distance of the levels is determined by the coupling between dots. With decreasing
TLR(M) (ω)
V 12
G r2;1 ðoÞ ¼
Here T La ða 2 R; MÞ represent the transmission probabilities from the left lead to the a lead. Thus, the electronic transport in a three-terminal system is reformulated into the electron transport in an effective two-terminal system and the transmission probabilities can then be treated, respectively.
ω/Γ Fig. 2. Transmission probability T LRðMÞ ðoÞ as a function energy o when the energy levels of the four dots are all relevant parameters are chosen as follows: GL ¼ GM ¼ GR for all i ¼ 1; 2; 3; 4, and (a) V ¼ 0:1; (b) V ¼ 1:0; and (c) V
of electronic aligned. The ¼ 1:0, 0i ¼ 0 ¼ 5:0.
ARTICLE IN PRESS H. Yin et al. / Microelectronics Journal 38 (2007) 570–575
TLT(M) (ω)
a
b
ω/Γ
Fig. 4. T LRðMÞ ðoÞ2o curves for parameters GL ¼ GM ¼ GR ¼ 1:0, V ¼ 5:0, 01 ¼ 02 ¼ 0 and (a) 03 ¼ 3:0, 04 ¼ 0 and (b) 03 ¼ 2:0, 04 ¼ 6:0.
a
b LDOS
of the coupling between dots, as shown in Fig. 2(c)–(a), the number of the peaks changes from three to one, i.e. when the tunnel coupling strength becomes small enough, we cannot differentiate these peaks. In addition, the number of peaks in the TLR(M)(o) plots does not agree with the number of dots in the system. This seems to be inconsistent with what one would expect for the system the system has four localized states and should have four peaks. To understand the origin of this discrepancy, we have calculated the electronic structure of the system and have plotted the local densities of states (LDOS) in Fig. 3. By comparison of Figs. 2(c) and 3, we can find one-to-one correspondences between the peaks in the transmission and the peaks in the LDOS. Thus, the electrons incident from the left lead can be more easily coupled into the system and transmitted into the right and/ or middle lead, if there is a state, at the energy of the incident electrons, which has a large localization in the QD 1. It is easy to see that only three-resonant states can be completely isolated and energy degeneracy partly remains at the single-QD energy level, e.g. the four apparent resonant states do not ensure four peaks in the plot. In Fig. 4 we further present the T LR ðoÞ2o and T LM ðoÞ2o curves when the energy levels in the third dot and the fourth dot split up. This can be realized experimentally by modulating the gate voltage of each dot. The transmission probabilities show complicated spectra characterized by various sharp peaks. In Fig. 4(a), there are three peaks in the plot of TLR and four peaks in TLM, respectively. A peak in the transmission probability can be seen only when there exists a state, at the energy of the incident electrons, with large localizations in both the QDs connected to the lead where the electrons are incident from and transmitted to, respectively. For
573
c
a
b LDOS
ω/Γ
Fig. 5. Local density of states (LDOS) calculated for the three-terminal T-shaped system. The relevant parameters are the same as in Fig. 4a. (a)LDOS of the quantum dot 4; (b)LDOS of the quantum dot 3; and (c)LDOS of the quantum dot 1.
c
ω/Γ
Fig. 3. Local density of states (LDOS) calculated for the three-terminal T-shaped system. The relevant parameters are the same as in Fig. 2c. (a) LDOS of the quantum dot 4; (b) LDOS of the quantum dot 3; and (c) LDOS of the quantum dot 1.
example, the transmission probability TLR does not show a peak at o ¼ 0, although there is a state at this energy, which has a large localization in the QD 1 (see Fig. 5(a). This is because the state has a low LDOS on the QD 3 (see Fig. 5(b). However, this state is strongly localized in the QD 4. As a consequence, the transmission probability TLM from the left lead to the middle lead shows a strong peak at o ¼ 0 (see Fig. 4(a)). Thus similar explanations can be made for the transport properties shown in Fig. 4(b) by the LDOS given in Fig. 6. We have found that the
ARTICLE IN PRESS H. Yin et al. / Microelectronics Journal 38 (2007) 570–575
574
a
a
b
b
c
c
Fig. 6. Local density of states(LDOS) calculated for the three-terminal T-shaped system. The relevant parameters are the same as in Fig. 4b. (a)LDOS of the quantum dot 4; (b)LDOS of the quantum dot 3; and (c)LDOS of the quantum dot 1.
a
b
c
Fig. 8. T LR ðoÞ2o curves for different GM. The relevant parameters are chosen as follows: GL ¼ GR ¼ 1:0, V ¼ 5:0, 0i ¼ 0 for all i ¼ 1; 2; 3; 4, and (a) GM ¼ 0; (b) GM ¼ 1.0; and (c) GM ¼ 5.0.
the dots 2 and 4, a much smaller transmission probability is presented in Fig. 7(b) and (c). In all cases, the spectra keep symmetric structure with respect to the single-QD energy level. Fig. 8 shows the T LR ðoÞ2o curves with V 12 ¼ V 23 ¼ V 24 ¼ 5:0 and 0i ¼ 0 for different GM. In the case of GM ¼ 0, there are four peaks in the TLR plot (see Fig. 8(a). There exists one narrow antiresonance at the single-QD energy level. It means that the electron tunneling from the left lead to the right lead is absolutely forbidden if its energy is just equal to the single-QD energy level. This is consistent with result of a quantum wire with a side QD [18]. However, with the increasing of GM, here we set GM ¼ 1.0 (see Fig. 8(b), it is easy to see that there exists a peak at single-QD energy level. When GM is large enough, the height of the peak at single-QD energy level is close to 1.0 (see Fig. 8(c), the value for an ideal channel. 4. Summary
Fig. 7. Transmission probability T LRðMÞ ðoÞ as a function of electronic energy o when the energy levels of the four dots are all aligned. The relevant parameters are chosen as follows: GL GM ¼ GR ¼ 1:0; V 12 ¼ V 23 ¼ 5:0, 0i ¼ 0 for all i ¼ 1,2,3,4 and (a) V 24 ¼ 0; (b) V 24 ¼ 1:5; and (c) V 24 ¼ 2:5.
characteristics of the transmission spectra arise from the fact that the confined states in the systems, localized in different QDs, can contribute very differently to transmission probabilities of the systems Fig. 7 gives the plots of TLR(M) with V 12 ¼ V 23 ¼ 5:0, V 24 ¼ 0; 1:5; 2:5. As shown in the plot, the height of each peak is moduled by the value of V24. When V24 ¼ 0, there are three peaks for TLR. The height of each peak is equal to 1.0, the value for an ideal channel. In the cases of V 24 ¼ 1:5; 2:5, due to the weak effective coupling between
This work deals with the electronic transport in a system of a T-shaped four-QD connected to three terminals. Both Dyson equations and equations of motion method for Green functions are used to obtain the current formula. By employing the Green’s functions approach, the electron transport in a three-terminal system can be reformulated into the electron transport in an effective two-terminal system. The calculations reveal that the transmission probabilities display oscillation structures. To understand the results of the calculations for the structure, we have calculated the local densities of states. We have found that the characteristics of the transmission spectra arise from the fact that the confined states in the systems, localized in different QDs, can contribute very differently to transmission probabilities of the systems. In addition, the tunnel
ARTICLE IN PRESS H. Yin et al. / Microelectronics Journal 38 (2007) 570–575
coupling between dots, the energy level of each dot and the coupling dot and lead play crucial roles in the transport properties of the system, especially when the energy levels of the four dots are aligned, with the increasing of GM from zero to some proper value, the antiresonance at the singleQD energy level will be transferred to resonance, which property can be used as a principle to design a switch. Acknowledgements The authors acknowledge the support by Heilongjiang Province Educational Department Item under Grant No.11511121 and Harbin Normal University Natural Science Foundation under Grant No. KM2006-30. References [1] M. Tewordt, H. Asahi, V.J. Law, R.T. Syme, M.J. Kelly, D.A. Ritchie, A. Churchill, J.E.F. Frost, R.H. Hughes, Appl. Phys. Lett. 60 (1992) 595. [2] Q. Xie, A. Madhukar, P. Chen, N.P. Kobayashi, Phys. Rev. Lett. 75 (1995) 2542. [3] A. Shailos, C. Prasad, M. Elhassan, R. Akis, D.K. Ferry, J.P. Bird, Phys. Rev. B 64 (2001) 193302. [4] J.H.F. Scott-Thomas, S.B. Field, M.A. Kastner, H.I. Smith, D.A. Antoniadis, T.K. Ng, P.A. Lee, Phys. Rev. Lett. 62 (1989) 583. [5] L.P. Kouwenhoven, N.C. van der Vaart, A.T. Johnson, W. Kool, C.J.P.M. Harmans, J.G. Williamson, A.A.M. Staring, C.T. Foxon, Z. Phys. B 85 (1991) 367. [6] T.K. Ng, P.A. Lee, Phys. Rev. Lett. 61 (1988) 1768.
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[7] L.I. Glazman, M.E. Raikh, JETP Lett. 47 (1988) 453. [8] Y. Meir, N.S. Wingreen, P.A. Lee, Phys. Rev. Lett. 70 (1993) 2601. [9] V. Chandrasekhar, R.A. Webb, M.J. Brady, M.B. Ketchen, W.J. Gallagher, A. Kleinsasser, Phys. Rev. Lett. 67 (1991) 3578. [10] S. Wu, W. Sun, S. Wang, W. Yu, Phys. Stat. Sol. (b) 241 (2004) 1299–1305. [11] D. Mailly, C. Chapelier, A. Benoit, Phys. Rev. Lett. 70 (1993) 2020. [12] S. Wu, S. Wang, W. Sun, W. Yu, Phys. Stat. Sol. (b) 239 (2003) 367–375. [13] Z.Y. Zeng, F. Claro, A. Pe´rez, Phys. Rev. B 65 (2002) 085308. [14] C.A. Stafford, N.S. Wingreen, Phys. Rev Lett. 76 (1996) 1916. [15] A. Georges, Y. Meir, Phys. Rev. Lett. 82 (1999) 3508. [16] K. Kawamura, T. Aono, Jpn. J. Appl. Phys. 36 (1999) 3951. [17] P. Stefan´ski, B.R. Bulka, Phys. Stat. Sol. (b) 236 (2003) 388–391. [18] P.A. Orellana, F. Domı´ nguez-Adame, I. Go´mez, M.L. Ladro´n de Guevara, Phys. Rev. B 67 (2003) 085321. [19] S.Y. Cho, H.Q. Zhou, R.H. McKenzie, Phys. Rev. B 68 (2003) 125327. [20] H.K. Zhao, Phys. Rev. B 63 (2001) 205327. [21] A. Cottet, W. Belzig, C. Bruder, Phys. Rev. Lett. 92 (2004) 206801. [22] Z.G. Zhu, G. Su, Q.R. Zheng, B. Jin, Phys. Rev. B 70 (2004) 174403. [23] R. Leturcq, L. Schmid, K. Ensslin, Y. Meir, D.C. Driscoll, A.C. Gossard, Phys. Rev. Lett. 95 (2005) 126603. [24] H. Li, T. Lu¨, P. Sun, Z. He, H. Yin, Microelectron. J. 37 (2006) 958–962. [25] H. Li, T. Lu¨, P. Sun, Phys. Stat. Sol. (b) 242 (2005) 1679–1686. [26] H. Li, T. Lu¨, P. Sun, Phys. Lett. A 343 (2005) 403–410. [27] A.P. Jauho, N.S. Wingreen, Y. Meir, Phys. Rev. B 50 (1994) 5528.
Microelectronics Journal 38 (2007) 570–575 www.elsevier.com/locate/mejo
Electronic transport through a T-shaped four-quantum dot system Haitao Yina,b, Tianquan Lu¨a,, Hua Lic, Zelong Hea a
Center for Condensed-Matter Science and Technology, Harbin Institute of Technology, Harbin 150001, PR China b Hulan School, Harbin Normal University, Harbin 150500, PR China c Department of Physics, Harbin Normal University, Harbin 150080, PR China Received 23 December 2006; accepted 3 March 2007 Available online 30 April 2007
Abstract Making use of the equation of motion method and Keldysh Green’s function technique, we have developed a calculation method by which an analytical formula for the current under dc bias is obtained and the transmission probabilities are numerically studied for a T-shaped four-quantum dot (QD) system connected to three terminals. The results are explained in terms of the localization properties of the confined states in the system. r 2007 Elsevier Ltd. All rights reserved. Keywords: Quantum dot; Transmission probability; Electronic transport
1. Introduction Quantum dots (QDs) are highly tunable artificial mesoscopic structures, usually called artificial atoms or artificial molecules. A great advantage of this artificial atom-like system is that one can easily change its internal states by adjusting the gate voltages imposed on its external conducting leads. The recent experimental advances in the nanofabrication of quantum devices make it possible to fabricate QD array [1–3]. Also, these systems reveal very rich physics, such as Coulomb blockade [4], Coulomb staircase [5], the nonequilibrium Kondo effect [6–8], and Aharonov–Bohm oscillations in the conductance and persistent currents, etc.[9–13]. QDs, are very promising systems due to their physical properties as well as their potential applications in electronic devices, have drawn considerable attentions. In the last decade, much work has been contributed to the system of a QD array connected to two-electron reservoirs. Many authors have studied the transport properties and capacitance spectroscopy in order to probe the discrete electronic spectrum of the system [14–18]. Corresponding author. Tel.: +86 451 86412770; fax: +86 451 86412828. E-mail address: [email protected] (T. Lu¨).
0026-2692/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2007.03.003
Recently, some attention has been paid to transport of a single- or double-QD system connected to multiterminal, and here we mention only a few of them. In Ref. [19] the conductance of the N-lead system was considered showing that the Kondo resonance at equilibrium is split into N1 peaks. The electron transport and shot noise in a multiterminal coupled QD system in which each lead was disturbed by classical microwave fields were studied [20]. Multi-terminal QD systems or magnetic junctions were also intensively investigated in context of the spindependent transport [21,22]. Leturcq et al. [23] probed the Kondo density of states in a three-terminal quantum ring. In our previous works [24–26], we have studied the transport properties of a ring-shaped array of QDs connected to two terminals and found some unique properties other than that of a linear array of QDs, which indicate the geometry structure is very important to the transport properties of the coupled QD system. What’ more, motivated in part by the works cited in the preceding paragraph, in this paper we investigate transport properties of a T-shaped four-QD system connected to three terminals based on the nonequilibrium Green function proposed by Jauho et al. [27]. The formulation we derive for the current can also be easily generalized to any other multiterminal system. The rest of the paper is organized as follows. In
ARTICLE IN PRESS H. Yin et al. / Microelectronics Journal 38 (2007) 570–575
Section 2, we derive the formula of the current. Numerical results and analysis are given in Section 3. Our concluding remarks are given in Section 4. 2. Model Hamiltonian and the formula for the DC current We consider a T-shaped four-QD system connected to three terminals, as depicted schematically in Fig. 1. For simplicity, we ignore the intradot and interdot Coulomb interactions between two electrons, and assume that just one energy level is relevant at each dot. The full system is moduled by a tight-binding Hamiltonian within a noninteracting picture, which can be written as 4 X
H dotlead ¼
þ V 24 d þ 2 d 4 þ h:c:Þ, X X kL C þ kR C þ kL C kL þ kR C kR k k X þ kM C þ kM C kM , k X X V LCþ V RCþ kL d 1 þ kR d 3 k k X þ V MCþ kM d 4 þ h:c:, k
L
V12 1
V23 2
(4)
Our calculation is limited to the steady state where the Green’s functions only depend on the variable Dt ¼ t t0 . By Fourier transforming with respect to variable tt0 in Gi;i ðt t0 ; 0Þ, we can obtain Gi;i ðoÞ. Making use of the motion equation and the Keldysh’s contour integration, we obtain the following equations for the retarded Green functions: o 01 SrL G r1;1 ðoÞ V 12 Gr2;1 ðoÞ ¼ 1, (5)
ð6Þ
o 03 SrR Gr3;1 ðoÞ V 32 Gr2;1 ðoÞ ¼ 0,
(7)
o 04 SrM Gr4;1 ðoÞ V 42 G r2;1 ðoÞ ¼ 0
(8)
o o a ðo 01 SrL ÞGo 1;1 ðoÞ V 12 G 2;1 ðoÞ ¼ SL G 1;1 ðoÞ,
(9)
o o ðo 02 ÞGo 2;1 ðoÞ V 21 G 1;1 ðoÞ V 23 G 3;1 ðoÞ
ð1Þ
VR 3
V24
þ 0 0 Go i;i ðt; t Þ ¼ i d i ðt Þd i ðtÞ .
and those for the ‘‘lesser’’ Green functions
where C þ ka ðC ka Þ ða 2 L; M; RÞ are the creation (annihilation) operators of electron in the left (L) lead, the middle (M) lead and the right (R) lead; d þ i ðd i Þ is the creation (annihilation) operator of QD i; ka ða ¼ L; M; RÞ is the single-electron energy in the a lead; 0i is the energy level in the ith dot; Vij and Va are the interdot and dot–lead coupling matrix elements, respectively. For simplicity, only a single pair of spin degenerate levels is included in each dot, and here the electron spin index is suppressed. To describe the nonequilibrium states in the system we introduce the retarded, the advanced and the ‘‘lesser’’ Green’s functions as follows: D E 0 G ri;i ðt; t0 Þ ¼ iyðt t0 Þ d i ðtÞ; d þ ðt Þ , (2) i þ VL
(3)
V 24 Gr4;1 ðoÞ ¼ 0,
þ þ 0i d þ i d i þ ðV 12 d 1 d 2 þ V 23 d 2 d 3
i¼1
H lead ¼
D E 0 , ðt Þ Gai;i ðt; t0 Þ ¼ iyðt0 tÞ d i ðtÞ; d þ i þ
ðo 02 ÞGr2;1 ðoÞ V 21 G r1;1 ðoÞ V 23 G r3;1 ðoÞ
H ¼ H dot þ H lead þ H dotlead , H dot ¼
571
R L
V 24 Go 4;1 ðoÞ ¼ 0,
ð10Þ
o o a ðo 03 SrR ÞG o 3;1 ðoÞ V 32 G 2;1 ðoÞ ¼ SR G 1;1 ðoÞ,
(11)
o o a ðo 04 SrM ÞG o 4;1 ðoÞ V 42 G 2;1 ðoÞ ¼ SM G 4;1 ðoÞ.
(12)
In the above Xr;a;o X r;a;o 2 ðoÞ ¼ V gka , a a
is the self-energy associated with the tunneling between dot and a lead which is connected to the dot. The Green functions gr;a;o ka ðoÞ in Eq. (13) correspond to free-electron Green functions in lead a and have the following relations: grka ðoÞ ¼ ðo ka þ iZÞ1 ,
(14)
gaka ðoÞ ¼ ðo ka iZÞ1 ,
(15)
go ka ðoÞ ¼ 2pif a dðo ka Þ.
(16)
Eqs. (5)–(12) form a closed equation set for final results are achieved as follows:
4 VM
(13)
k
Gr1;1 ðoÞ ¼
Fig. 1. Schematic plot of a T-shaped four-quantum dot system with three terminals.
and the
,
(17)
1 o 01
Pr
L
V2
12
2 V 223 Pr 0V 24Pr 2 o0 o 3 4
o0
R
M
Gr;o 11
M
r 2 L R Go 1;1 ðoÞ ¼ iG ðoÞf L ðoÞ G 1;1 ðoÞ þ iG ðoÞf R ðoÞ r 2 2 G 3;1 ðoÞ þ iGM ðoÞf M ðoÞG r4;1 ðoÞ ,
ð18Þ
ARTICLE IN PRESS H. Yin et al. / Microelectronics Journal 38 (2007) 570–575
572
with
3. Numerical results
o
02
V2
23P
o0 3
r R
V2
24 Pr o0 M 4
Gr1;1 ðoÞ,
(19)
G r3;1 ðoÞ ¼
V 23 P Gr ðoÞ, o 03 rR 2;1
(20)
G r4;1 ðoÞ ¼
V 24 P Gr ðoÞ. o 04 rM 2;1
(21)
In the above derivations, we assume the coupling matrix elements (Vij and Va) are real since the tunnel rates just depend on the amplitude of the coupling matrix elements. We can then calculate the current flowing from the left lead to the structure is Z i eX h o V L Go JL ¼ (22) 1;kL ðoÞ V L G kL;1 ðoÞ do. h k Making use of the Dyson equation h i r o o a Go 1;kL ðoÞ ¼ V L G 1;1 ðoÞgkL ðoÞ G 1;1 ðoÞgkL ðoÞ
(23)
one can express the current with the Green function G r1;1 ðoÞ. After a simple calculation, we get Z n h io i2e r a GL ðoÞ G o JL ¼ do, 1;1 ðoÞ þ f L ðoÞ G 1;1 ðoÞ G 1;1 ðoÞ h (24) where GL ðoÞ ¼ 2p
X
jV L j2 dðo kL Þ,
(25)
k
is the coupling of dot 1 to the left lead. We have similar definitions forGRðMÞ ðoÞ for the 1right lead and the middle lead. f a ðoÞ ¼ 1 þ eðoma Þ=kB T is Fermi–Dirac distribution function of the a lead with chemical potential ma. The factor ‘‘2’’ in the Eq. (24) is due to the spin degeneracy. Substituting Eq. (18) into Eq. (24), After a simple calculation, we can then calculate the current between dot 1 and the left lead as Z 2e JL ¼ T LR ðoÞ f L ðoÞ f R ðoÞ , h þT LM ðoÞ f L ðoÞ f M ðoÞ do, ð26Þ where (27)
2 T LM ðoÞ ¼ GL ðoÞGM ðoÞGr4;1 ðoÞ .
(28)
R
a
b
c
2 T LR ðoÞ ¼ G ðoÞG ðoÞG r3;1 ðoÞ , L
In this section, we perform the detailed calculations of T LR ðoÞ and T LM ðoÞ, for various single-QD energy level arrangements and various interdot coupling amplitudes. For simplicity, a constant density of states is endowed to each lead. In the wide-band limiting approximation, the line-width functions Ga ðoÞða 2 L; M; RÞ are energy-independent constant, and the level shift is zero. Throughout this paper, we assume GL ¼ G. All energy and coupling strengths are measured by units of G. Our interest is mainly to find out how parameters determine the number and position of resonant transmission peaks. In the case of a constant density of states in the leads, T LRðMÞ ðoÞ is proportional to the transmission probability of an electron with energy o. Here we present some numerical results of T LRðMÞ ðoÞ, which not only provide the information of the resonant states of the system but also indicate how each state contributes to the electron transport. In a QD described in Eq. (1), one possible resonant state is presented with the QD energy level. When the four dots are coupled, four states are mixed and become the states spreading throughout the whole system. Fig. 2 shows the T LR ðoÞ2o curves and T LM ðoÞ2o curves for V ¼ 0:1,1:0,5:0. Here we assume V 12 ¼ V 23 ¼ V 24 ¼ V . As shown in the plot, the peaks of TLR and TLM match absolutely with the same height and the same positions. By carefully examining the level positions of all the peaks, we see that one level always lies at the single-QD quantum energy level and the other two are distributed symmetrically with respect to the energy. The distance of the levels is determined by the coupling between dots. With decreasing
TLR(M) (ω)
V 12
G r2;1 ðoÞ ¼
Here T La ða 2 R; MÞ represent the transmission probabilities from the left lead to the a lead. Thus, the electronic transport in a three-terminal system is reformulated into the electron transport in an effective two-terminal system and the transmission probabilities can then be treated, respectively.
ω/Γ Fig. 2. Transmission probability T LRðMÞ ðoÞ as a function energy o when the energy levels of the four dots are all relevant parameters are chosen as follows: GL ¼ GM ¼ GR for all i ¼ 1; 2; 3; 4, and (a) V ¼ 0:1; (b) V ¼ 1:0; and (c) V
of electronic aligned. The ¼ 1:0, 0i ¼ 0 ¼ 5:0.
ARTICLE IN PRESS H. Yin et al. / Microelectronics Journal 38 (2007) 570–575
TLT(M) (ω)
a
b
ω/Γ
Fig. 4. T LRðMÞ ðoÞ2o curves for parameters GL ¼ GM ¼ GR ¼ 1:0, V ¼ 5:0, 01 ¼ 02 ¼ 0 and (a) 03 ¼ 3:0, 04 ¼ 0 and (b) 03 ¼ 2:0, 04 ¼ 6:0.
a
b LDOS
of the coupling between dots, as shown in Fig. 2(c)–(a), the number of the peaks changes from three to one, i.e. when the tunnel coupling strength becomes small enough, we cannot differentiate these peaks. In addition, the number of peaks in the TLR(M)(o) plots does not agree with the number of dots in the system. This seems to be inconsistent with what one would expect for the system the system has four localized states and should have four peaks. To understand the origin of this discrepancy, we have calculated the electronic structure of the system and have plotted the local densities of states (LDOS) in Fig. 3. By comparison of Figs. 2(c) and 3, we can find one-to-one correspondences between the peaks in the transmission and the peaks in the LDOS. Thus, the electrons incident from the left lead can be more easily coupled into the system and transmitted into the right and/ or middle lead, if there is a state, at the energy of the incident electrons, which has a large localization in the QD 1. It is easy to see that only three-resonant states can be completely isolated and energy degeneracy partly remains at the single-QD energy level, e.g. the four apparent resonant states do not ensure four peaks in the plot. In Fig. 4 we further present the T LR ðoÞ2o and T LM ðoÞ2o curves when the energy levels in the third dot and the fourth dot split up. This can be realized experimentally by modulating the gate voltage of each dot. The transmission probabilities show complicated spectra characterized by various sharp peaks. In Fig. 4(a), there are three peaks in the plot of TLR and four peaks in TLM, respectively. A peak in the transmission probability can be seen only when there exists a state, at the energy of the incident electrons, with large localizations in both the QDs connected to the lead where the electrons are incident from and transmitted to, respectively. For
573
c
a
b LDOS
ω/Γ
Fig. 5. Local density of states (LDOS) calculated for the three-terminal T-shaped system. The relevant parameters are the same as in Fig. 4a. (a)LDOS of the quantum dot 4; (b)LDOS of the quantum dot 3; and (c)LDOS of the quantum dot 1.
c
ω/Γ
Fig. 3. Local density of states (LDOS) calculated for the three-terminal T-shaped system. The relevant parameters are the same as in Fig. 2c. (a) LDOS of the quantum dot 4; (b) LDOS of the quantum dot 3; and (c) LDOS of the quantum dot 1.
example, the transmission probability TLR does not show a peak at o ¼ 0, although there is a state at this energy, which has a large localization in the QD 1 (see Fig. 5(a). This is because the state has a low LDOS on the QD 3 (see Fig. 5(b). However, this state is strongly localized in the QD 4. As a consequence, the transmission probability TLM from the left lead to the middle lead shows a strong peak at o ¼ 0 (see Fig. 4(a)). Thus similar explanations can be made for the transport properties shown in Fig. 4(b) by the LDOS given in Fig. 6. We have found that the
ARTICLE IN PRESS H. Yin et al. / Microelectronics Journal 38 (2007) 570–575
574
a
a
b
b
c
c
Fig. 6. Local density of states(LDOS) calculated for the three-terminal T-shaped system. The relevant parameters are the same as in Fig. 4b. (a)LDOS of the quantum dot 4; (b)LDOS of the quantum dot 3; and (c)LDOS of the quantum dot 1.
a
b
c
Fig. 8. T LR ðoÞ2o curves for different GM. The relevant parameters are chosen as follows: GL ¼ GR ¼ 1:0, V ¼ 5:0, 0i ¼ 0 for all i ¼ 1; 2; 3; 4, and (a) GM ¼ 0; (b) GM ¼ 1.0; and (c) GM ¼ 5.0.
the dots 2 and 4, a much smaller transmission probability is presented in Fig. 7(b) and (c). In all cases, the spectra keep symmetric structure with respect to the single-QD energy level. Fig. 8 shows the T LR ðoÞ2o curves with V 12 ¼ V 23 ¼ V 24 ¼ 5:0 and 0i ¼ 0 for different GM. In the case of GM ¼ 0, there are four peaks in the TLR plot (see Fig. 8(a). There exists one narrow antiresonance at the single-QD energy level. It means that the electron tunneling from the left lead to the right lead is absolutely forbidden if its energy is just equal to the single-QD energy level. This is consistent with result of a quantum wire with a side QD [18]. However, with the increasing of GM, here we set GM ¼ 1.0 (see Fig. 8(b), it is easy to see that there exists a peak at single-QD energy level. When GM is large enough, the height of the peak at single-QD energy level is close to 1.0 (see Fig. 8(c), the value for an ideal channel. 4. Summary
Fig. 7. Transmission probability T LRðMÞ ðoÞ as a function of electronic energy o when the energy levels of the four dots are all aligned. The relevant parameters are chosen as follows: GL GM ¼ GR ¼ 1:0; V 12 ¼ V 23 ¼ 5:0, 0i ¼ 0 for all i ¼ 1,2,3,4 and (a) V 24 ¼ 0; (b) V 24 ¼ 1:5; and (c) V 24 ¼ 2:5.
characteristics of the transmission spectra arise from the fact that the confined states in the systems, localized in different QDs, can contribute very differently to transmission probabilities of the systems Fig. 7 gives the plots of TLR(M) with V 12 ¼ V 23 ¼ 5:0, V 24 ¼ 0; 1:5; 2:5. As shown in the plot, the height of each peak is moduled by the value of V24. When V24 ¼ 0, there are three peaks for TLR. The height of each peak is equal to 1.0, the value for an ideal channel. In the cases of V 24 ¼ 1:5; 2:5, due to the weak effective coupling between
This work deals with the electronic transport in a system of a T-shaped four-QD connected to three terminals. Both Dyson equations and equations of motion method for Green functions are used to obtain the current formula. By employing the Green’s functions approach, the electron transport in a three-terminal system can be reformulated into the electron transport in an effective two-terminal system. The calculations reveal that the transmission probabilities display oscillation structures. To understand the results of the calculations for the structure, we have calculated the local densities of states. We have found that the characteristics of the transmission spectra arise from the fact that the confined states in the systems, localized in different QDs, can contribute very differently to transmission probabilities of the systems. In addition, the tunnel
ARTICLE IN PRESS H. Yin et al. / Microelectronics Journal 38 (2007) 570–575
coupling between dots, the energy level of each dot and the coupling dot and lead play crucial roles in the transport properties of the system, especially when the energy levels of the four dots are aligned, with the increasing of GM from zero to some proper value, the antiresonance at the singleQD energy level will be transferred to resonance, which property can be used as a principle to design a switch. Acknowledgements The authors acknowledge the support by Heilongjiang Province Educational Department Item under Grant No.11511121 and Harbin Normal University Natural Science Foundation under Grant No. KM2006-30. References [1] M. Tewordt, H. Asahi, V.J. Law, R.T. Syme, M.J. Kelly, D.A. Ritchie, A. Churchill, J.E.F. Frost, R.H. Hughes, Appl. Phys. Lett. 60 (1992) 595. [2] Q. Xie, A. Madhukar, P. Chen, N.P. Kobayashi, Phys. Rev. Lett. 75 (1995) 2542. [3] A. Shailos, C. Prasad, M. Elhassan, R. Akis, D.K. Ferry, J.P. Bird, Phys. Rev. B 64 (2001) 193302. [4] J.H.F. Scott-Thomas, S.B. Field, M.A. Kastner, H.I. Smith, D.A. Antoniadis, T.K. Ng, P.A. Lee, Phys. Rev. Lett. 62 (1989) 583. [5] L.P. Kouwenhoven, N.C. van der Vaart, A.T. Johnson, W. Kool, C.J.P.M. Harmans, J.G. Williamson, A.A.M. Staring, C.T. Foxon, Z. Phys. B 85 (1991) 367. [6] T.K. Ng, P.A. Lee, Phys. Rev. Lett. 61 (1988) 1768.
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