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Charge trapping in the system of interacting quantum dots
Solid State Communications 168 (2013) 36–41
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Charge trapping in the system of interacting quantum dots V.N. Mantsevich a,n, N.S. Maslova b, P.I. Arseyev c a b c
Moscow State University, Faculty of Physics, Chair of Semiconductors, Moscow 119991, Russia Moscow State University, Faculty of Physics, Chair of Quantum Electronics, Moscow 119991, Russia P.N. Lebedev Physical Institute of RAS, Moscow 119991, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 24 January 2013 Received in revised form 6 May 2013 Accepted 14 June 2013 by P. Hawrylak Available online 26 June 2013
We analyzed the localized charge dynamics in the system of N interacting single-level quantum dots (QDs) coupled to the continuous spectrum states in the presence of Coulomb interaction between electrons within the dots. Different dots geometry and initial charge configurations were considered. The analysis was performed by means of Heisenberg equations for localized electron pair correlators. We revealed that charge trapping takes place for a wide range of system parameters and we suggested the QDs geometry for experimental observations of this phenomenon. We demonstrated significant suppression of Coulomb correlations with the increasing QDs number. We found the appearance of several time scales with the strongly different relaxation rates for a wide range of the Coulomb interaction values. & 2013 Elsevier Ltd. All rights reserved.
Keywords: D. Quantum dots D. Charge trapping D. Non-stationary effects D. Coulomb correlations
1. Introduction Coupled quantum dots (QDs) are recently under numerous experimental [1–3] and theoretical investigations [4–10] due to their potential application in modern nanoscale devices dealing with quantum kinetics of individual localized states [9–16]. The kinetic properties of coupled QDs [1] are governed by the Coulomb interaction between the localized electrons [3,10] and depend strongly on the dots topology, which determines energy levels spacing and the coupling rates [17–19]. During the last decade experimental technique gives possibility of creating vertically aligned strongly interacting QDs with only one of them coupled to the continuous spectrum states [20,21]. This so-called sidecoupled geometry gives an opportunity to analyze non-stationary effects in formation of various charge and spin configurations in the small size structures [7,10]. Lateral QDs are extremely tunable by means of individual electrical gates [22,23]. This advantage reveals the possibility of single electron localization in the system of several coupled dots [24] and charge state manipulations in the artificial molecules. Therefore lateral QDs are considered to be ideal candidates for creation of an efficient charge traps. Previous studies demonstrated long-lived charge occupation trap states in the single QDs [25–27] and single electron spin trapping [28]. Single electron trapping in the double dot system was performed
n
Corresponding author. Tel.: +7 495 939 25 02. E-mail addresses: [email protected], [email protected] (V.N. Mantsevich), [email protected] (N.S. Maslova), [email protected] (P.I. Arseyev). 0038-1098/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2013.06.017
in [29]. The temperature of the trapped electron was measured and tunnel coupling energy was extracted by charge sensing measurements. A full configuration–interaction study on a square QD containing several electrons in the presence of an attractive impurity was performed in [30]. Authors demonstrated that the impurity changes significantly the charge densities of the twoelectron QD excited states. The effect of correlations was revealed in the enhancement of the charge density localization within the dot. QDs were investigated theoretically by various methods such as Keldysh non-equilibrium Green-function formalism [31], re-normalization group theory [32], specific approach suggested by Coleman [33], spin-density-functional theory [34] or quantum Monte-Carlo calculations [35]. In this paper we consider charge relaxation in the system of N interacting QDs with on-site Coulomb repulsion coupled to the reservoir (continuous spectrum states). The analysis was performed by means of Heisenberg equations of motion for the localized electrons pair correlators. We demonstrated the presence of strong charge trapping effects for the lateral QDs geometry. We found that on-site Coulomb repulsion within the dots results in the significant changing of the localized charge relaxation dynamics and leads to the formation of several time ranges with strongly different values of the relaxation rates.
2. Theoretical model Let us consider relaxation processes in the system of N identical (for simplicity) lateral QDs which are situated in the different
V.N. Mantsevich et al. / Solid State Communications 168 (2013) 36–41
space points and are connected only with the single-central QD by means of electron tunneling processes with the same tunneling transfer amplitudes T (to minimize the number of free parameters, all tunneling rates are put equal). We assume that the level spacing in the dots is larger than all other energy scales, so only one spin-degenerate level within the QD spectrum is accessible (ε0 in the N identical dots and ε in the central one). QD with energy level ε is also connected with the continuous spectrum states (drain). Moreover we take into account Coulomb interaction between the localized electrons within the dots (U in the central one and U0 in the surrounding dots) (see Fig. 1). Hamiltonian of the system under investigation has the form N
N
s;j ¼ 1
s;j ¼ 1
þ þ þ H^ ¼ εaþ s as þ ∑ ε0 bjs bjs þ ∑ ðTbjs as þ Tas bjs Þ þ þ þ∑εk cþ ks cks þ ∑T k ðcks as þ as cks Þ ks
k;s
þUnas na−s þ U 0 njs nj−s
ð1Þ
Tk—tunneling amplitude between the central dot and the continuous spectrum states. We assume T and Tk to be independent of momentum and spin. By considering a constant density of states in the reservoir ν0 (which is not a function of energy), the tunneling coupling strength γ is defined as γ ¼ πν0 T 2k . þ aþ s =as (bjs =bjs )—electrons creation/annihilation operators in the central dot (in the surrounding QDs). cþ =ck —electrons creation/ k annihilation operators in the continuous spectrum states (k) and nas ðnjs Þ are electron filling numbers in the dots. As we are interested in the specific features of the nonstationary time evolution of the initially localized charge in the coupled QDs, we will consider the situation when condition ðεi −εF Þ=γ⪢1 is fulfilled. Our investigations deal with the low temperature regime when Fermi level in the reservoir 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 reservoir is a Fermi step and all the states of the reservoir with energies above the Fermi energy are empty. We will at first analyze filling numbers relaxation processes in the case when on-site Coulomb repulsion is absent in the whole system (U ¼ U 0 ¼ 0). Let us assume that at the initial moment all charge densities in the system are localized only in one of the N QDs and have the value n1 ð0Þ ¼ n0 . The filling numbers time evolution can be analyzed by means of kinetic Heisenberg equations for bilinear combinations of Heisenberg operators aþ s =as and þ bjs =bjs [36,37]: þ
bjs bj′s ¼ Gjj′ ðtÞ; þ
bjs as ¼ Gja ðtÞ;
aþ s as ¼ Gaa ðtÞ aþ s bjs ¼ Gaj ðtÞ
ð2Þ
37
We set ℏ ¼ 1 and therefore localized charge time evolution can be obtained from the system of kinetic equations for the Green functions Gjj′ , Gaj, Gja and Gaa [36,37]: ∂ G ¼ T Gja −T Gaj′ ∂t jj′ ∂ i Gaa ¼ ∑ðT Gaj′ −T Gj′a Þ−i2γ Gaa ∂t j′
i
i
∂ G ¼ −Δ Gaj þ T Gaa −∑T Gj′j −iγ Gaj ∂t aj j′
i
∂ G ¼ Δ Gja −T Gaa þ ∑T Gjj′ −iγ Gja ∂t ja j′
ð3Þ
where Δ ¼ ε−ε0 is the detuning between the energy levels in the dots. The system of equations (3) describes the evolution of the pair correlation functions and we consider the situation when the states of electrons in the reservoir are not influenced by the relaxation processes. So we fulfill averaging over electronic states in the reservoir using equilibrium Fermi distribution function. As we are interested in the situation when condition ðεi −εF Þ=γ⪢1 is fulfilled, the reservoir is used as a drain and relaxation term looks as if electron tunnels only in one direction. System of equations (3) can be re-written in the compact matrix form (symbol ½ means commutation): ∂ ^ ^ ^ A−ið ^ G ¼ ½G; B^ G^ þ G^ BÞ ∂t where G^ is the pair correlators matrix: 0 1 Gaa Ga1 … GaN B G1a … … G1N C B C G^ ¼ B C ⋮ ⋮ ⋮ A @ ⋮ GNa GN1 … GNN
i
and matrix A^ has the form 0 1 Δ T1 … TN B n C B T1 0 … 0 C C A^ ¼ B B ⋮ 0 … 0 C @ A T nN 0 … 0
ð4Þ
ð5Þ
ð6Þ
The tunneling coupling matrix B^ has only one nonzero element ‖B‖11 ¼ γ. The formal solution of the system (4) can be found with the help of evolution operator ^ ^ ^ ^ ^ ½Bt−i At ^ GðtÞ ¼ e½−BtþiAt Gð0Þe
ð7Þ
Consequently, the average filling numbers value time evolution in one of the surrounding QDs (nj(t)) can be found from the following expression: ^ ^ ^ ^ 〈nj ðtÞ〉 ¼ G^ jj ðtÞ ¼ ∑½e−BtþiAt ja G^ ab ð0Þ½eBt−iAt bj
ð8Þ
a;b
U0
U0
0
0
U0
U0
0
U0 T
U
0
U0
0
0
Due to the condition that initial charge is localized only in one QD with number j, the following initial conditions are fulfilled: 〈njs ð0Þ〉 ¼ Gjj ð0Þ ¼ n0 , 〈nas ð0Þ〉 ¼ 0, 〈nj′s ð0Þ〉 ¼ 0, if j≠j′ and Gjj′ ð0Þ ¼ Gaj ð0Þ ¼ Gja ð0Þ ¼ 0. Let us analyze filling numbers time evolution in the central QD and in the dot with initial charge. Concerning initial conditions one can easily find the following expressions for filling numbers relaxation: ^
U0
0
N
Fig. 1. Schematic view of the proposed model. System of QDs coupled to the continuous spectrum states in the reservoir.
~^
njs ðtÞ ¼ ½eiH t jj n0j ½e−iH t jj ^
~^
nas ðtÞ ¼ ½eiH t aj n0j ½e−iH t ja
ð9Þ
^ ^ B^ are included. Further where operators H^ ¼ A^ þ iB^ and H~ ¼ A−i analysis deals with the calculation of matrix exponent's elements. One can easily perform this procedure with the help of recurrent
38
V.N. Mantsevich et al. / Solid State Communications 168 (2013) 36–41
ratio similar to the procedure suggested by Cummings [38]. The following ratios for operator H^ elements are fulfilled: n n−1 Þaj ðH^ Þjj ¼ H^ ja ðH^ n n−1 n−1 ðH^ Þaj ¼ ∑H^ aj′ ðH^ Þj′j þ H^ aa ðH^ Þaj
ðH^ Þjj′ ¼ T nj′ ðH^
n−1
Þaj
ð10Þ
System of equations (10) enables to get recurrent equation for the matrix elements: n n−1 n−2 Þaj þ NjTj2 ðH^ Þaj ðH^ Þaj ¼ ðΔ þ iγÞðH^
ð11Þ n
^ analogous equations can be obtained for matrix elements ðH~ Þaj . Consequently, after some calculations one can get n Tj n ðH^ Þaj ¼ pffiffiffiffi ðan þ b Þ 2 D T nj n ^n n ðH~ Þaj ¼ pffiffiffiffi ða~ þ b~ Þ ~ 2 D
~ a, a, ~ b and b~ are determined as Where coefficients D, D, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðΔ þ iγÞ2 þ NjTj2 D¼ 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ~ ¼ ðΔ−iγÞ þ NjTj2 D 4 Δ þ iγ Δ−iγ ~ þ D; a~ ¼ þD a¼ 2 2 Δ þ iγ Δ−iγ ~ b¼ −D; b~ ¼ −D 2 2
ð12Þ
ð13Þ
After substituting (14) into the expressions (9) one gets filling numbers time evolution in the central QD na(t) and in the QD with the initial charge nj(t) in the case when Coulomb correlations are absent:
~^
ð18Þ
Finally, solution will have the form NT 2 eiat −eiðΔþiγÞt eibt −eiðΔþiγÞt − ψ¼ 2D a−ðΔ þ iγÞ b−ðΔ þ iγÞ
ð19Þ
where coefficients a, b and D are determined by the expressions (13). It is clearly evident that the charge trapping effect is absent for such initial conditions. We now consider the situation when Coulomb interaction between localized electrons exists within the QDs. In this case it is necessary to take into account the following interaction part of the system Hamiltonian (1): H int ¼ U ð0Þ nαs nα−s
ð20Þ
∂ ^ ^ A^ þ C^ þ iðΓ^ G^ þ G^ Γ^ Þ G ¼ ½G; ð21Þ ∂t where matrices A^ and G^ are determined by the expressions (5) and (6) correspondingly, and matrices Γ^ and C^ can be written as i
‖Γ‖ij ¼ δi1 δj1 γ and ‖C‖ij ¼ δij U 0 Gjj . The formal solution of the system for pair correlators (21) can be again found with the help of evolution operator Rt Rt ^ ^ C^ ðt′ÞÞ dt′ ^ ½−i ðAðt′Þþ C^ ðt′ÞÞ dt′ ^ ¼ Te½i 0 ðAðt′Þþ 0 GðtÞ Gð0ÞT e ð22Þ As initial charge is localized in the QD with number j, the initial conditions are ni ð0Þ ¼ nj0 δij , na ð0Þ ¼ 0. Then one can obtain the expressions 2 Gj′j′ ðtÞ ¼ ∑ Ω−1 j′k Gkk′ ð0ÞΩk′j′ ¼ n0j jΩjj′ j k;k′
½−i
ð15Þ
Simultaneously for the large number of QDs N, electron filling pffiffiffiffi numbers in the dots reveal oscillations frequency increase as T N . If initial charge is localized in the central QD, which is coupled to the continuous spectrum states, one should solve system (3) with the initial conditions: 〈njs ð0Þ〉 ¼ 0, 〈nas ð0Þ〉 ¼ Gaa ð0Þ ¼ n0 , 〈nj′s ð0Þ〉 ¼ 0. Consequently, the charge time evolution can be described by means of expressions ^
can be obtained from the following
Gaa ðtÞ ¼ n0j jΩaj j2
It is clearly evident that with the increasing QDs number N initial charge n0j is quite fully confined in the initial QD even in the presence of dissipation in the system due to the interaction with the reservoir 1 2 lim njs ðtÞ ¼ n0j 1− ð16Þ t-∞ N
nas ðtÞ ¼ ½eiH t aa n0a ½e−iH t aa
The function ψ ¼ e equation:
ð17Þ
where index α ¼ aðjÞ and Coulomb interaction values U ð0Þ correspond to the central dot (surrounding dots). We will take into account Coulomb interaction by means of self-consistent mean field approximation [10]. It means that the initial energy level value ε has to be substituted by the value ε~ ¼ ε þ U 〈nαs ðtÞ〉 in the final expressions for the filling numbers nαs time evolution (15). So one should solve self-consistent system of equations. In the presence of Coulomb interaction system of equations for pair correlators can also be written in the compact matrix form:
^ Expanding exponents in the expression (9) in a power H^ and H~ series one can easily obtain the following expressions: jTj2 eiat eibt 1 1 ^ − þ1 − − ½eiH t jj ¼ a b 2D a b " # ~ ~ ^ jTj2 e−iat e−ibt 1 1 ½−iH~ tjj − − − þ1 ð14Þ e ¼ ~ a~ a~ b~ 2D b~
T2 ~ ~ ðeiat −eibt Þ ðe−iat nas ðtÞ ¼ −e−ibt Þ ~ 4DD 1 jTj2 eiat eibt − njs ðtÞ ¼ n0j 1− þ N 2D a b " !# ~ ~ 1 jTj2 e−iat e−ibt − 1− þ ~ a~ N 2D b~
½iH^ taa
∂ψ ^ ¼ iðΔ þ iγÞ ψ þ NT ½eiH t ja ∂t
j′
n
~^
^
njs ðtÞ ¼ ½eiH t ja n0a ½e−iH t aj
Rt
ð23Þ ^ ðAðt′Þþ C^ ðt′ÞÞ dt′
where evolution operator Ω ¼ Te 0 is considered. So, one can get equations for the matrix elements of the evolution operator Ω: _ aj ¼ iΔ Ωaj þ i∑T Ωj′j Ω j′
_ j′j ¼ iT Ωaj þ iU 0 n0j jΩj′j j2 Ωj′j Ω
ð24Þ
with the following initial conditions for the functions Ωaj ð0Þ ¼ 0, Ωjj ð0Þ ¼ 1, Ωj′j ð0Þ ¼ 0. If we are interested in the collective effects connected with the presence of N coupled QDs, the Coulomb interaction between localized electrons within the initially empty dots can be neglected, because the filling numbers amplitudes are proportional to 1=N 2 . So, taking into account only Coulomb correlations within the dot with the initial charge, one can simplify the system (24) and analytically obtain the following system: € aj ¼ −ðN−1ÞT 2 Ωaj þ iðΔ−γÞ Ω _ aj þ iT Ω _ jj Ω _ jj ¼ iT Ωaj þ iU 0 n0j jΩjj j2 Ωjj Ω
ð25Þ
System (25) can be easily solved numerically and consequently one can analyze localized charge relaxation dynamics. Results
V.N. Mantsevich et al. / Solid State Communications 168 (2013) 36–41
obtained in Section 3 correspond to the situation when on-site Coulomb repulsion is considered in all the surrounding QDs.
3. Calculation results and discussion Filling numbers time evolution within the dot with initial charge nj(t) and within the central QD na(t) in the absence of onsite Coulomb repulsion is presented in Fig. 2. The non-resonant tunneling between the dots is considered (ðε−ε0 Þ=γ ¼ −1).
Fig. 2. Filling numbers time evolution in the case when Coulomb interaction between localized electrons is absent (a) in the QD with the initial charge and (b) in the central QD for the different number of QDs N. Parameters T=γ ¼ 0:6 and ðε−ε0 Þ=γ ¼ −1 are the same for all the figures.
39
It is clearly evident that filling numbers relaxation changes significantly with the increasing QDs number N. When initial charge is localized in one of the surrounding dots, it remains confined in the initial dot for the large number of dots even in the presence of relaxation processes from the central dot to the reservoir. When one considers two surrounding dots, only 20% of the charge continue being localized in the initial QD (Fig. 2a). For ten interacting QDs more than eighty percent of the charge is confined in the initial dot (Fig. 2a). This effect can be called “charge trapping” and the proposed system of coupled QDs can be considered as a “charge trap”. Increasing QDs number N also leads to the decreasing charge amplitude in the central QD na(t) due to the effective growth of tunneling coupling (Fig. 2b). Similar radiation trapping effects can be observed in the quantum optics: the system of oscillators interacting through the cavity mode [39]. Typical calculation results, in the case when on-site Coulomb repulsion is considered only within the central QD where localized charge is absent at the initial time moment, are demonstrated in Fig. 3. “Charge trapping” effect is clearly evident with the increasing QDs number even in the presence of Coulomb interaction between localized electrons (Fig. 3). For two surrounding QDs Coulomb correlations strongly influence the filling numbers relaxation (Fig. 3a). A critical value of onsite Coulomb repulsion exists for a given set of system parameters. This value corresponds to the full compensation of the initial negative detuning [10]. For the smaller values of Coulomb interaction, correlations lead to the increasing relaxation rate in the QD with the initial charge on the first stage of relaxation process before the beginning of oscillations formation (Fig. 3a, grey line) in comparison with the case when Coulomb interaction is absent (Fig. 3a, black line), due to the decreasing initial detuning value. For the values of on-site Coulomb repulsion larger than the critical one, positive detuning occurs and filling numbers relaxation rate decreases as a result of positive detuning value increase (Fig. 3a, black-dashed line). With the increasing QDs number all the effects mentioned above are still valid, but they are less pronounced (Fig. 3c). So the role of Coulomb correlations is suppressed for the large number of QDs due to the decreasing electrons occupation in the central dot. Let us now analyze the situation when Coulomb interaction between localized electrons is taken into account within all the N QDs which interact with the central one. Calculation results are presented in Fig. 4a and demonstrate “charge trapping” effect with the increasing QDs number. In the case of two QDs interacting with the central one, Coulomb correlations reveal significantly stronger influence on the filling numbers relaxation processes in comparison with the geometry when five dots are considered (Fig. 4c). Again two typical relaxation regimes were revealed. The first one corresponds to the decreasing initial negative detuning value. In this regime Coulomb correlations lead to the increasing
Fig. 3. Filling numbers time evolution in the presence of Coulomb interaction for the different number of QDs N (a and c) in the QD with initial charge and (b) in the central QD. (a and b) N¼ 2 and (c) N ¼ 5. Coulomb interaction is considered only in the central QD: U=γ ¼ 0black line, U=γ ¼ 10grey line and U=γ ¼ 30blackdashed line for all the figures. Parameters T=γ ¼ 0:6 and ðε−ε0 Þ=γ ¼ −1 are the same for all the figures.
40
V.N. Mantsevich et al. / Solid State Communications 168 (2013) 36–41
Fig. 4. Filling numbers time evolution in the presence of Coulomb interaction for the different number of QDs N (a and c) in the QD with initial charge and (b) in the central QD. (a and b) N ¼ 2 and (c) N ¼ 5. Coulomb interaction is taken into account in all the surrounding QDs: U 0 ¼ 0black line, U 0 =γ ¼ 2grey line and U 0 =γ ¼ 7blackdashed line. Parameters T=γ ¼ 0:6 and ðε−ε0 Þ=γ ¼ −1 are the same for all the figures.
(Fig. 4a and c). We found the growth of charge amplitude in the central QD with the increasing dots number when the negative detuning value decreases and amplitude decreases when positive detuning value increases (Fig. 4b). When initial charge is localized in the central QD which is connected not only to the surrounding QDs but also to the continuous spectrum states “charge trapping” effect does not exist at all (Fig. 5). We now introduce the possible QDs geometry which allows to perform an experimental observation of charge trapping effects within the single dot. The most simple configuration is N similar lateral QDs interacting only with the single vertically aligned dot. Vertical dot is also connected to the reservoir. But this geometry reveals a problem of initial charge localization. Initial charge can be localized in the single QD in the most simple way by means of the gate voltage. So it is convenient to have a system with N−1 lateral dots and single vertical dot with the localized charge. These N dots interact only with the single central vertically aligned QD also connected to the continuous spectrum states.
4. Conclusion
Fig. 5. (a) Filling numbers time evolution in the presence of Coulomb interaction for the different number of QDs N (a and c) in the central QD with initial charge and (b and d) in one of the N surrounding QDs for (a and b). N ¼ 2 and (c and d) N ¼ 10. Coulomb interaction is taken into account in the central QD: U=γ ¼ 0black line, U=γ ¼ 4grey line and U=γ ¼ 8blackdashed line. Parameters T=γ ¼ 0:6 and ðε−ε0 Þ=γ ¼ −1 are the same for all the figures.
relaxation rate in the QD with initial charge in comparison with the case when Coulomb interaction is absent (Fig. 4a, grey and black lines correspondingly). The second one deals with the Coulomb energy values large enough to compensate negative detuning and to form the positive one. In this regime filling numbers relaxation rate decreases as a result of the positive detuning value increase caused by the Coulomb interaction (Fig. 4a, grey and black-dashed lines correspondingly). QDs number increase also results in the growth of filling numbers oscillations frequency. Oscillations frequency for the small Coulomb values decreases corresponding to the detuning decrease and increases as a result of positive detuning formation
To conclude, we have analyzed time evolution of the electron filling numbers in the system of N interacting QDs both in the absence and in the presence of Coulomb interaction between localized electrons within the dots. It was found that Coulomb interaction modifies the relaxation rates and the character of localized charge time evolution. It was shown that several time ranges with considerably different relaxation rates arise in the system of coupled QDs. We demonstrated and carefully analyzed the presence of strong charge trapping effects in the proposed systems. It was found that interacting dots can form an effective high quality charge trap. We also revealed the Coulomb correlations suppression with the increasing QDs number. The QDs geometry which allows to perform an experimental observation of charge trapping effects was suggested.
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Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Charge trapping in the system of interacting quantum dots V.N. Mantsevich a,n, N.S. Maslova b, P.I. Arseyev c a b c
Moscow State University, Faculty of Physics, Chair of Semiconductors, Moscow 119991, Russia Moscow State University, Faculty of Physics, Chair of Quantum Electronics, Moscow 119991, Russia P.N. Lebedev Physical Institute of RAS, Moscow 119991, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 24 January 2013 Received in revised form 6 May 2013 Accepted 14 June 2013 by P. Hawrylak Available online 26 June 2013
We analyzed the localized charge dynamics in the system of N interacting single-level quantum dots (QDs) coupled to the continuous spectrum states in the presence of Coulomb interaction between electrons within the dots. Different dots geometry and initial charge configurations were considered. The analysis was performed by means of Heisenberg equations for localized electron pair correlators. We revealed that charge trapping takes place for a wide range of system parameters and we suggested the QDs geometry for experimental observations of this phenomenon. We demonstrated significant suppression of Coulomb correlations with the increasing QDs number. We found the appearance of several time scales with the strongly different relaxation rates for a wide range of the Coulomb interaction values. & 2013 Elsevier Ltd. All rights reserved.
Keywords: D. Quantum dots D. Charge trapping D. Non-stationary effects D. Coulomb correlations
1. Introduction Coupled quantum dots (QDs) are recently under numerous experimental [1–3] and theoretical investigations [4–10] due to their potential application in modern nanoscale devices dealing with quantum kinetics of individual localized states [9–16]. The kinetic properties of coupled QDs [1] are governed by the Coulomb interaction between the localized electrons [3,10] and depend strongly on the dots topology, which determines energy levels spacing and the coupling rates [17–19]. During the last decade experimental technique gives possibility of creating vertically aligned strongly interacting QDs with only one of them coupled to the continuous spectrum states [20,21]. This so-called sidecoupled geometry gives an opportunity to analyze non-stationary effects in formation of various charge and spin configurations in the small size structures [7,10]. Lateral QDs are extremely tunable by means of individual electrical gates [22,23]. This advantage reveals the possibility of single electron localization in the system of several coupled dots [24] and charge state manipulations in the artificial molecules. Therefore lateral QDs are considered to be ideal candidates for creation of an efficient charge traps. Previous studies demonstrated long-lived charge occupation trap states in the single QDs [25–27] and single electron spin trapping [28]. Single electron trapping in the double dot system was performed
n
Corresponding author. Tel.: +7 495 939 25 02. E-mail addresses: [email protected], [email protected] (V.N. Mantsevich), [email protected] (N.S. Maslova), [email protected] (P.I. Arseyev). 0038-1098/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2013.06.017
in [29]. The temperature of the trapped electron was measured and tunnel coupling energy was extracted by charge sensing measurements. A full configuration–interaction study on a square QD containing several electrons in the presence of an attractive impurity was performed in [30]. Authors demonstrated that the impurity changes significantly the charge densities of the twoelectron QD excited states. The effect of correlations was revealed in the enhancement of the charge density localization within the dot. QDs were investigated theoretically by various methods such as Keldysh non-equilibrium Green-function formalism [31], re-normalization group theory [32], specific approach suggested by Coleman [33], spin-density-functional theory [34] or quantum Monte-Carlo calculations [35]. In this paper we consider charge relaxation in the system of N interacting QDs with on-site Coulomb repulsion coupled to the reservoir (continuous spectrum states). The analysis was performed by means of Heisenberg equations of motion for the localized electrons pair correlators. We demonstrated the presence of strong charge trapping effects for the lateral QDs geometry. We found that on-site Coulomb repulsion within the dots results in the significant changing of the localized charge relaxation dynamics and leads to the formation of several time ranges with strongly different values of the relaxation rates.
2. Theoretical model Let us consider relaxation processes in the system of N identical (for simplicity) lateral QDs which are situated in the different
V.N. Mantsevich et al. / Solid State Communications 168 (2013) 36–41
space points and are connected only with the single-central QD by means of electron tunneling processes with the same tunneling transfer amplitudes T (to minimize the number of free parameters, all tunneling rates are put equal). We assume that the level spacing in the dots is larger than all other energy scales, so only one spin-degenerate level within the QD spectrum is accessible (ε0 in the N identical dots and ε in the central one). QD with energy level ε is also connected with the continuous spectrum states (drain). Moreover we take into account Coulomb interaction between the localized electrons within the dots (U in the central one and U0 in the surrounding dots) (see Fig. 1). Hamiltonian of the system under investigation has the form N
N
s;j ¼ 1
s;j ¼ 1
þ þ þ H^ ¼ εaþ s as þ ∑ ε0 bjs bjs þ ∑ ðTbjs as þ Tas bjs Þ þ þ þ∑εk cþ ks cks þ ∑T k ðcks as þ as cks Þ ks
k;s
þUnas na−s þ U 0 njs nj−s
ð1Þ
Tk—tunneling amplitude between the central dot and the continuous spectrum states. We assume T and Tk to be independent of momentum and spin. By considering a constant density of states in the reservoir ν0 (which is not a function of energy), the tunneling coupling strength γ is defined as γ ¼ πν0 T 2k . þ aþ s =as (bjs =bjs )—electrons creation/annihilation operators in the central dot (in the surrounding QDs). cþ =ck —electrons creation/ k annihilation operators in the continuous spectrum states (k) and nas ðnjs Þ are electron filling numbers in the dots. As we are interested in the specific features of the nonstationary time evolution of the initially localized charge in the coupled QDs, we will consider the situation when condition ðεi −εF Þ=γ⪢1 is fulfilled. Our investigations deal with the low temperature regime when Fermi level in the reservoir 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 reservoir is a Fermi step and all the states of the reservoir with energies above the Fermi energy are empty. We will at first analyze filling numbers relaxation processes in the case when on-site Coulomb repulsion is absent in the whole system (U ¼ U 0 ¼ 0). Let us assume that at the initial moment all charge densities in the system are localized only in one of the N QDs and have the value n1 ð0Þ ¼ n0 . The filling numbers time evolution can be analyzed by means of kinetic Heisenberg equations for bilinear combinations of Heisenberg operators aþ s =as and þ bjs =bjs [36,37]: þ
bjs bj′s ¼ Gjj′ ðtÞ; þ
bjs as ¼ Gja ðtÞ;
aþ s as ¼ Gaa ðtÞ aþ s bjs ¼ Gaj ðtÞ
ð2Þ
37
We set ℏ ¼ 1 and therefore localized charge time evolution can be obtained from the system of kinetic equations for the Green functions Gjj′ , Gaj, Gja and Gaa [36,37]: ∂ G ¼ T Gja −T Gaj′ ∂t jj′ ∂ i Gaa ¼ ∑ðT Gaj′ −T Gj′a Þ−i2γ Gaa ∂t j′
i
i
∂ G ¼ −Δ Gaj þ T Gaa −∑T Gj′j −iγ Gaj ∂t aj j′
i
∂ G ¼ Δ Gja −T Gaa þ ∑T Gjj′ −iγ Gja ∂t ja j′
ð3Þ
where Δ ¼ ε−ε0 is the detuning between the energy levels in the dots. The system of equations (3) describes the evolution of the pair correlation functions and we consider the situation when the states of electrons in the reservoir are not influenced by the relaxation processes. So we fulfill averaging over electronic states in the reservoir using equilibrium Fermi distribution function. As we are interested in the situation when condition ðεi −εF Þ=γ⪢1 is fulfilled, the reservoir is used as a drain and relaxation term looks as if electron tunnels only in one direction. System of equations (3) can be re-written in the compact matrix form (symbol ½ means commutation): ∂ ^ ^ ^ A−ið ^ G ¼ ½G; B^ G^ þ G^ BÞ ∂t where G^ is the pair correlators matrix: 0 1 Gaa Ga1 … GaN B G1a … … G1N C B C G^ ¼ B C ⋮ ⋮ ⋮ A @ ⋮ GNa GN1 … GNN
i
and matrix A^ has the form 0 1 Δ T1 … TN B n C B T1 0 … 0 C C A^ ¼ B B ⋮ 0 … 0 C @ A T nN 0 … 0
ð4Þ
ð5Þ
ð6Þ
The tunneling coupling matrix B^ has only one nonzero element ‖B‖11 ¼ γ. The formal solution of the system (4) can be found with the help of evolution operator ^ ^ ^ ^ ^ ½Bt−i At ^ GðtÞ ¼ e½−BtþiAt Gð0Þe
ð7Þ
Consequently, the average filling numbers value time evolution in one of the surrounding QDs (nj(t)) can be found from the following expression: ^ ^ ^ ^ 〈nj ðtÞ〉 ¼ G^ jj ðtÞ ¼ ∑½e−BtþiAt ja G^ ab ð0Þ½eBt−iAt bj
ð8Þ
a;b
U0
U0
0
0
U0
U0
0
U0 T
U
0
U0
0
0
Due to the condition that initial charge is localized only in one QD with number j, the following initial conditions are fulfilled: 〈njs ð0Þ〉 ¼ Gjj ð0Þ ¼ n0 , 〈nas ð0Þ〉 ¼ 0, 〈nj′s ð0Þ〉 ¼ 0, if j≠j′ and Gjj′ ð0Þ ¼ Gaj ð0Þ ¼ Gja ð0Þ ¼ 0. Let us analyze filling numbers time evolution in the central QD and in the dot with initial charge. Concerning initial conditions one can easily find the following expressions for filling numbers relaxation: ^
U0
0
N
Fig. 1. Schematic view of the proposed model. System of QDs coupled to the continuous spectrum states in the reservoir.
~^
njs ðtÞ ¼ ½eiH t jj n0j ½e−iH t jj ^
~^
nas ðtÞ ¼ ½eiH t aj n0j ½e−iH t ja
ð9Þ
^ ^ B^ are included. Further where operators H^ ¼ A^ þ iB^ and H~ ¼ A−i analysis deals with the calculation of matrix exponent's elements. One can easily perform this procedure with the help of recurrent
38
V.N. Mantsevich et al. / Solid State Communications 168 (2013) 36–41
ratio similar to the procedure suggested by Cummings [38]. The following ratios for operator H^ elements are fulfilled: n n−1 Þaj ðH^ Þjj ¼ H^ ja ðH^ n n−1 n−1 ðH^ Þaj ¼ ∑H^ aj′ ðH^ Þj′j þ H^ aa ðH^ Þaj
ðH^ Þjj′ ¼ T nj′ ðH^
n−1
Þaj
ð10Þ
System of equations (10) enables to get recurrent equation for the matrix elements: n n−1 n−2 Þaj þ NjTj2 ðH^ Þaj ðH^ Þaj ¼ ðΔ þ iγÞðH^
ð11Þ n
^ analogous equations can be obtained for matrix elements ðH~ Þaj . Consequently, after some calculations one can get n Tj n ðH^ Þaj ¼ pffiffiffiffi ðan þ b Þ 2 D T nj n ^n n ðH~ Þaj ¼ pffiffiffiffi ða~ þ b~ Þ ~ 2 D
~ a, a, ~ b and b~ are determined as Where coefficients D, D, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðΔ þ iγÞ2 þ NjTj2 D¼ 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ~ ¼ ðΔ−iγÞ þ NjTj2 D 4 Δ þ iγ Δ−iγ ~ þ D; a~ ¼ þD a¼ 2 2 Δ þ iγ Δ−iγ ~ b¼ −D; b~ ¼ −D 2 2
ð12Þ
ð13Þ
After substituting (14) into the expressions (9) one gets filling numbers time evolution in the central QD na(t) and in the QD with the initial charge nj(t) in the case when Coulomb correlations are absent:
~^
ð18Þ
Finally, solution will have the form NT 2 eiat −eiðΔþiγÞt eibt −eiðΔþiγÞt − ψ¼ 2D a−ðΔ þ iγÞ b−ðΔ þ iγÞ
ð19Þ
where coefficients a, b and D are determined by the expressions (13). It is clearly evident that the charge trapping effect is absent for such initial conditions. We now consider the situation when Coulomb interaction between localized electrons exists within the QDs. In this case it is necessary to take into account the following interaction part of the system Hamiltonian (1): H int ¼ U ð0Þ nαs nα−s
ð20Þ
∂ ^ ^ A^ þ C^ þ iðΓ^ G^ þ G^ Γ^ Þ G ¼ ½G; ð21Þ ∂t where matrices A^ and G^ are determined by the expressions (5) and (6) correspondingly, and matrices Γ^ and C^ can be written as i
‖Γ‖ij ¼ δi1 δj1 γ and ‖C‖ij ¼ δij U 0 Gjj . The formal solution of the system for pair correlators (21) can be again found with the help of evolution operator Rt Rt ^ ^ C^ ðt′ÞÞ dt′ ^ ½−i ðAðt′Þþ C^ ðt′ÞÞ dt′ ^ ¼ Te½i 0 ðAðt′Þþ 0 GðtÞ Gð0ÞT e ð22Þ As initial charge is localized in the QD with number j, the initial conditions are ni ð0Þ ¼ nj0 δij , na ð0Þ ¼ 0. Then one can obtain the expressions 2 Gj′j′ ðtÞ ¼ ∑ Ω−1 j′k Gkk′ ð0ÞΩk′j′ ¼ n0j jΩjj′ j k;k′
½−i
ð15Þ
Simultaneously for the large number of QDs N, electron filling pffiffiffiffi numbers in the dots reveal oscillations frequency increase as T N . If initial charge is localized in the central QD, which is coupled to the continuous spectrum states, one should solve system (3) with the initial conditions: 〈njs ð0Þ〉 ¼ 0, 〈nas ð0Þ〉 ¼ Gaa ð0Þ ¼ n0 , 〈nj′s ð0Þ〉 ¼ 0. Consequently, the charge time evolution can be described by means of expressions ^
can be obtained from the following
Gaa ðtÞ ¼ n0j jΩaj j2
It is clearly evident that with the increasing QDs number N initial charge n0j is quite fully confined in the initial QD even in the presence of dissipation in the system due to the interaction with the reservoir 1 2 lim njs ðtÞ ¼ n0j 1− ð16Þ t-∞ N
nas ðtÞ ¼ ½eiH t aa n0a ½e−iH t aa
The function ψ ¼ e equation:
ð17Þ
where index α ¼ aðjÞ and Coulomb interaction values U ð0Þ correspond to the central dot (surrounding dots). We will take into account Coulomb interaction by means of self-consistent mean field approximation [10]. It means that the initial energy level value ε has to be substituted by the value ε~ ¼ ε þ U 〈nαs ðtÞ〉 in the final expressions for the filling numbers nαs time evolution (15). So one should solve self-consistent system of equations. In the presence of Coulomb interaction system of equations for pair correlators can also be written in the compact matrix form:
^ Expanding exponents in the expression (9) in a power H^ and H~ series one can easily obtain the following expressions: jTj2 eiat eibt 1 1 ^ − þ1 − − ½eiH t jj ¼ a b 2D a b " # ~ ~ ^ jTj2 e−iat e−ibt 1 1 ½−iH~ tjj − − − þ1 ð14Þ e ¼ ~ a~ a~ b~ 2D b~
T2 ~ ~ ðeiat −eibt Þ ðe−iat nas ðtÞ ¼ −e−ibt Þ ~ 4DD 1 jTj2 eiat eibt − njs ðtÞ ¼ n0j 1− þ N 2D a b " !# ~ ~ 1 jTj2 e−iat e−ibt − 1− þ ~ a~ N 2D b~
½iH^ taa
∂ψ ^ ¼ iðΔ þ iγÞ ψ þ NT ½eiH t ja ∂t
j′
n
~^
^
njs ðtÞ ¼ ½eiH t ja n0a ½e−iH t aj
Rt
ð23Þ ^ ðAðt′Þþ C^ ðt′ÞÞ dt′
where evolution operator Ω ¼ Te 0 is considered. So, one can get equations for the matrix elements of the evolution operator Ω: _ aj ¼ iΔ Ωaj þ i∑T Ωj′j Ω j′
_ j′j ¼ iT Ωaj þ iU 0 n0j jΩj′j j2 Ωj′j Ω
ð24Þ
with the following initial conditions for the functions Ωaj ð0Þ ¼ 0, Ωjj ð0Þ ¼ 1, Ωj′j ð0Þ ¼ 0. If we are interested in the collective effects connected with the presence of N coupled QDs, the Coulomb interaction between localized electrons within the initially empty dots can be neglected, because the filling numbers amplitudes are proportional to 1=N 2 . So, taking into account only Coulomb correlations within the dot with the initial charge, one can simplify the system (24) and analytically obtain the following system: € aj ¼ −ðN−1ÞT 2 Ωaj þ iðΔ−γÞ Ω _ aj þ iT Ω _ jj Ω _ jj ¼ iT Ωaj þ iU 0 n0j jΩjj j2 Ωjj Ω
ð25Þ
System (25) can be easily solved numerically and consequently one can analyze localized charge relaxation dynamics. Results
V.N. Mantsevich et al. / Solid State Communications 168 (2013) 36–41
obtained in Section 3 correspond to the situation when on-site Coulomb repulsion is considered in all the surrounding QDs.
3. Calculation results and discussion Filling numbers time evolution within the dot with initial charge nj(t) and within the central QD na(t) in the absence of onsite Coulomb repulsion is presented in Fig. 2. The non-resonant tunneling between the dots is considered (ðε−ε0 Þ=γ ¼ −1).
Fig. 2. Filling numbers time evolution in the case when Coulomb interaction between localized electrons is absent (a) in the QD with the initial charge and (b) in the central QD for the different number of QDs N. Parameters T=γ ¼ 0:6 and ðε−ε0 Þ=γ ¼ −1 are the same for all the figures.
39
It is clearly evident that filling numbers relaxation changes significantly with the increasing QDs number N. When initial charge is localized in one of the surrounding dots, it remains confined in the initial dot for the large number of dots even in the presence of relaxation processes from the central dot to the reservoir. When one considers two surrounding dots, only 20% of the charge continue being localized in the initial QD (Fig. 2a). For ten interacting QDs more than eighty percent of the charge is confined in the initial dot (Fig. 2a). This effect can be called “charge trapping” and the proposed system of coupled QDs can be considered as a “charge trap”. Increasing QDs number N also leads to the decreasing charge amplitude in the central QD na(t) due to the effective growth of tunneling coupling (Fig. 2b). Similar radiation trapping effects can be observed in the quantum optics: the system of oscillators interacting through the cavity mode [39]. Typical calculation results, in the case when on-site Coulomb repulsion is considered only within the central QD where localized charge is absent at the initial time moment, are demonstrated in Fig. 3. “Charge trapping” effect is clearly evident with the increasing QDs number even in the presence of Coulomb interaction between localized electrons (Fig. 3). For two surrounding QDs Coulomb correlations strongly influence the filling numbers relaxation (Fig. 3a). A critical value of onsite Coulomb repulsion exists for a given set of system parameters. This value corresponds to the full compensation of the initial negative detuning [10]. For the smaller values of Coulomb interaction, correlations lead to the increasing relaxation rate in the QD with the initial charge on the first stage of relaxation process before the beginning of oscillations formation (Fig. 3a, grey line) in comparison with the case when Coulomb interaction is absent (Fig. 3a, black line), due to the decreasing initial detuning value. For the values of on-site Coulomb repulsion larger than the critical one, positive detuning occurs and filling numbers relaxation rate decreases as a result of positive detuning value increase (Fig. 3a, black-dashed line). With the increasing QDs number all the effects mentioned above are still valid, but they are less pronounced (Fig. 3c). So the role of Coulomb correlations is suppressed for the large number of QDs due to the decreasing electrons occupation in the central dot. Let us now analyze the situation when Coulomb interaction between localized electrons is taken into account within all the N QDs which interact with the central one. Calculation results are presented in Fig. 4a and demonstrate “charge trapping” effect with the increasing QDs number. In the case of two QDs interacting with the central one, Coulomb correlations reveal significantly stronger influence on the filling numbers relaxation processes in comparison with the geometry when five dots are considered (Fig. 4c). Again two typical relaxation regimes were revealed. The first one corresponds to the decreasing initial negative detuning value. In this regime Coulomb correlations lead to the increasing
Fig. 3. Filling numbers time evolution in the presence of Coulomb interaction for the different number of QDs N (a and c) in the QD with initial charge and (b) in the central QD. (a and b) N¼ 2 and (c) N ¼ 5. Coulomb interaction is considered only in the central QD: U=γ ¼ 0black line, U=γ ¼ 10grey line and U=γ ¼ 30blackdashed line for all the figures. Parameters T=γ ¼ 0:6 and ðε−ε0 Þ=γ ¼ −1 are the same for all the figures.
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V.N. Mantsevich et al. / Solid State Communications 168 (2013) 36–41
Fig. 4. Filling numbers time evolution in the presence of Coulomb interaction for the different number of QDs N (a and c) in the QD with initial charge and (b) in the central QD. (a and b) N ¼ 2 and (c) N ¼ 5. Coulomb interaction is taken into account in all the surrounding QDs: U 0 ¼ 0black line, U 0 =γ ¼ 2grey line and U 0 =γ ¼ 7blackdashed line. Parameters T=γ ¼ 0:6 and ðε−ε0 Þ=γ ¼ −1 are the same for all the figures.
(Fig. 4a and c). We found the growth of charge amplitude in the central QD with the increasing dots number when the negative detuning value decreases and amplitude decreases when positive detuning value increases (Fig. 4b). When initial charge is localized in the central QD which is connected not only to the surrounding QDs but also to the continuous spectrum states “charge trapping” effect does not exist at all (Fig. 5). We now introduce the possible QDs geometry which allows to perform an experimental observation of charge trapping effects within the single dot. The most simple configuration is N similar lateral QDs interacting only with the single vertically aligned dot. Vertical dot is also connected to the reservoir. But this geometry reveals a problem of initial charge localization. Initial charge can be localized in the single QD in the most simple way by means of the gate voltage. So it is convenient to have a system with N−1 lateral dots and single vertical dot with the localized charge. These N dots interact only with the single central vertically aligned QD also connected to the continuous spectrum states.
4. Conclusion
Fig. 5. (a) Filling numbers time evolution in the presence of Coulomb interaction for the different number of QDs N (a and c) in the central QD with initial charge and (b and d) in one of the N surrounding QDs for (a and b). N ¼ 2 and (c and d) N ¼ 10. Coulomb interaction is taken into account in the central QD: U=γ ¼ 0black line, U=γ ¼ 4grey line and U=γ ¼ 8blackdashed line. Parameters T=γ ¼ 0:6 and ðε−ε0 Þ=γ ¼ −1 are the same for all the figures.
relaxation rate in the QD with initial charge in comparison with the case when Coulomb interaction is absent (Fig. 4a, grey and black lines correspondingly). The second one deals with the Coulomb energy values large enough to compensate negative detuning and to form the positive one. In this regime filling numbers relaxation rate decreases as a result of the positive detuning value increase caused by the Coulomb interaction (Fig. 4a, grey and black-dashed lines correspondingly). QDs number increase also results in the growth of filling numbers oscillations frequency. Oscillations frequency for the small Coulomb values decreases corresponding to the detuning decrease and increases as a result of positive detuning formation
To conclude, we have analyzed time evolution of the electron filling numbers in the system of N interacting QDs both in the absence and in the presence of Coulomb interaction between localized electrons within the dots. It was found that Coulomb interaction modifies the relaxation rates and the character of localized charge time evolution. It was shown that several time ranges with considerably different relaxation rates arise in the system of coupled QDs. We demonstrated and carefully analyzed the presence of strong charge trapping effects in the proposed systems. It was found that interacting dots can form an effective high quality charge trap. We also revealed the Coulomb correlations suppression with the increasing QDs number. The QDs geometry which allows to perform an experimental observation of charge trapping effects was suggested.
Acknowledgments This work was partly supported by RFBR grants and by the Russian Ministry of Science and Education. Russian Federation President grant for the Young scientists MK-2780.2013.2. References [1] W.G. van der Wiel, S. De Franceschi, J.M. Elzerman, et al., Rev. Mod. Phys. 75 (2002) 1. [2] R.M. Potok, I.G. Rau, H. Shtrikman, et al., Nature 446 (2007) 167. [3] T. Hayashi, T. Fujisawa, H.D. Cheong, et al., Phys. Rev. Lett. 91 (2003) 226804.
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