NONSTATIONARY EFFECTS IN THE SYSTEM OF COUPLED QUANTUM DOTS INFLUENCED BY COULOMB CORRELATIONS
V. N. Mantsevicha* N. S. Maslova", P. I. Arseevh
Lomonosov Moscow State University 119991, Moscow, Russia
Li In ,lt <• Physical Institute, Russian Academy of Sciences 119991, Moscow, Russia
Received May 6, 2013
We investigate the time evolution of filling numbers of localized electrons in the system of two coupled single-level quantum dots (QDs) connected with the continuous-spectrum states in the presence of Coulomb interaction. We considered correlation functions of all orders for electrons in the QDs by decoupling higher-order correlations between localized and band electrons in the reservoir. We analyze different initial charge configurations and consider Coulomb correlations between localized electrons both within the dots and between the different dots. We reveal the presence of a dynamical charge trapping effect in the first QD in the situation where both dots are occupied at the initial instant. We also find an analytic solution for the time-dependent filling numbers of the localized electrons for a particular configuration of the dots.
DOI: 10.7868/S0044451014010167
1. INTRODUCTION
The control and manipulation of localized charge in small-size systems is one of the most important issues in nanoelectronics [1,2]. Single semiconductor quantum dots (QDs), which are referred to as "artificial" atoms [3,4], and coupled QDs "artificial" molecules [5,6] are promising structures to serve for creation of extremely small devices. Several coupled QDs can be used in manufacturing electronic devices dealing with quantum kinetics of individual localized states [7 9]. Therefore, the behavior of coupled QDs in different configurations is currently under careful experimental [10,11] and theoretical investigation [12,13].
During the last decade, vertically aligned QDs (for example, indium arsenide QDs in gallium arsenide) have been fabricated and widely studied with great success [14 16]. Such an experimental realization allows organizing a strongly interacting system of QDs with only one of them coupled to the continuous-spectrum states. Consequently, vertically aligned QDs give an opportunity to analyze nonstationary effects in various
E-mail: vmantsev'fflspmlab.phys.msu.ru
charge and spin configurations formation in small-size structures [17].
Lateral QDs seems to be better candidates for controllable electronic coupling between two or several QDs by applying individual lateral gates. That is why they are intensively studied during the last several years both experimentally and theoretically [18,19].
Investigation of relaxation processes, nonequilib-rium charge distribution and nonstationary effects in the electron transport through a system of QDs are vital problems that have to be solved in order to integrate QDs in small quantum circuits [20 26]. Electron transport in such systems is strongly governed by the Coulomb interaction between localized electrons and, of course, by the ratio between the tunneling transfer amplitudes and the QD coupling. Correct interpretation of quantum effects in nanoscale systems gives an opportunity to create high-speed electronic and logic devices [27,28]. In some of the recent realizations, Coulomb interaction is weak [29], but for small-size QDs, the on-site Coulomb repulsion is in general strong [30], and it is therefore important to take it into account. In some cases, Coulomb correlations can determine time-dependent phenomena [31]. Hence, the problem of time evolution of charge in coupled QDs connected with continuous-spectrum states in the pres-
once of Coulomb correlations between localized electrons is indeed quite topical.
Time evolution of charge states in a semiconductor double quantum well in the presence of Coulomb interaction was experimentally studied in [32]. The authors manipulated the localized charge by the initial pulses and observed pulse-induced tunneling electron oscillations. Localized charge relaxation in the single and coupled quantum wells in the absence of Coulomb interaction was theoretically analyzed by Gurvitz [33, 34]. The author took only two time scales governing the charge time evolution into account and neglected the third time scale that is responsible for charge redistribution between different wells. Time dependence of the accumulated charge and the tunneling current through a single QD in the presence of Coulomb interaction were theoretically analyzed in [35]. The authors described relaxation processes and revealed three time rates for localized charge relaxation in the QD coupled to a thermostat. Several different time rates were also found in the system of two and three interacting QDs coupled to the reservoir [36 38]. For simplicity, on-site Coulomb repulsion was considered only in a single QD. Such a model is suitable in the case where one of the dots is narrow and the second is rather wide. In [39, 40], the authors derived rate equations to analyze the case of resonant transport in QDs linked by ballistic channels with high density of states and revealed the role of interference effects.
In this paper, we consider charge relaxation in double QDs due to the coupling to the continuous spectrum states. Tunneling from the first QD to the continuum is possible only through the second dot. We obtain a closed system of equations for the time evolution of the localized-electron filling numbers that exactly takes all-order correlation functions for localized electrons into account. We decouple the higher-order correlation functions between conduction electrons in the reservoir (band electrons) and electrons localized in the QDs. In such an approximation, the electron distribution in the reservoir is not influenced by changing the electronic states in the coupled QDs. For QDs weakly coupled to the reservoir, the proposed decoupling scheme is a good approximation. We consider different initial charge configurations and take Coulomb correlations into account both within QDs and between electrons localized in different dots. We find some peculiarities in the dynamics of electron filling numbers arising due to the Coulomb correlation effects. We demonstrate that depending on the initial charge configurations, the effect of dynamical charge trapping can be observed in the proposed system.
Ul2
T
si
U11
U22
Ef
Fig. 1. Scheme of the proposed model. The system of interacting QDs is coupled to the continuous-spectrum states by the tunneling rate 7 = ituut2
2. THE PROPOSED MODEL
We consider a system of coupled QDs with the single-particle levels e 1 and £2 coupled to an electronic reservoir (Fig. 1). We discuss three different initial charge configurations that are possible in the proposed system. The first deals with the initial charge localized in the first QD on the energy level £1 (/¿nff(0) = 1). The second corresponds to the situation where localized charge is accumulated in the second QD on the energy level e2 ('¿22<t(0) = 1)- And the last possible initial charge configuration refers to the case where the initial charge is localized on both electron levels equally (/tiiff(O) = 1122(j(0) = 1). The second QD with the energy level £2 is connected with the continuous spectrum states (ep). Relaxation of the localized charge is governed by the Hamiltonian
H — HD + H tun + Hr
The Hamiltonian Ho of interacting QDs,
(1)
HD =
n "11 n \ I > ■> ■> n I > ■> ■> n \
j=l,2<r
+ U12(nlla + nii_cr)(n22cr + n22-a) + (:lcr(:2cr + i'lcrf'Jcr
), (2)
fX
contains the spin-degenerate levels e, (indices t = 1 and t = 2 correspond to the first and second QDs), the on-site Coulomb repulsion for the double occupation of the QDs, and Coulomb interaction between electrons in different dots. The creation/annihilation of an electron with spin a = ±1 within a dot is denoted by (\aj(-'¿cr and n,j is the corresponding filling number operator.
The coupling between the dots is described by the tunneling transfer amplitude T, which is assumed to be independent of momentum and spin.
The continuous-spectrum states are modeled by the Hamiltonian
IIi t s — /* ^ • (3)
per
where c|)IT/cpcr creates/annihilates an electron with spin (T and momentum p in the lead. The coupling between the second dot and the continuous-spectrum states is described by the Hamiltonian
Hlun = f + <Vi2cr ). (4)
per
where t is the tunneling amplitude, which we assume to be independent of momentum and spin. With a constant density of states, v0 assumed in the reservoir, the tunnel rate 7 is defined as 7 = ttPot2.
Because we are interested in the specific features of the nonstationary time evolution of the initially localized charge in coupled QDs, we consider the situation where the condition (e, — ep)/7 1 is fulfilled. It means that the initial energy levels are situated well above the Fermi level and stationary occupation numbers in the second QD in the absence of coupling between the QDs is of the order of 7/(6-2 — £f) -C 1 and can be omitted. Consequently, the Ivondo effect is also negligible in the proposed model.
Our investigation deals with the low-temperature regime where the Fermi level is well defined and the temperature is much lower than all typical relaxation rates in the system. Consequently, the distribution function of electrons in the leads (band electrons) is a Fermi step.
We set h = 1, and therefore the kinetic equations for bilinear combinations of Heisenberg operators c^/c^,
Acr^icr = ni(i), f-'L^ir = n%(t), t t ^
which describe time evolution of the filling numbers for the electrons, can be written as
i-^uZ^-nn^-nlz), d
'¿ — /¿22 = T(h%-L — »12) — 2t7»22.
+ [i + (£/ii-£/2iKTKi- (6) — (U-22 — Ul2)n2lf>'22 ~
d
i~7yj^12 ~ ~T{fl22 ~ ^11) ~~
+ (U22 - Ui2K2n^ - iyhl2,
where £ = £1 — e2 is the detuning between the energy-levels in the QDs. System of equations (6) contains expressions for the pair correlators n^h^ and h^<Th1-2, which also determine relaxati
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