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Electronic properties of coupled quantum dots
ARTICLE IN PRESS
Physica E 40 (2008) 1748–1750 www.elsevier.com/locate/physe
Electronic properties of coupled quantum dots E. Machowska-Podsiad"o, M. Ma˛ czka Rzeszo´w University of Technology, ul. W. Pola 2, 35-959 Rzeszo´w, Poland Available online 4 December 2007
Abstract In the work two coupled quantum dots (CQDs) are analyzed. They are formed electrostatically in GaAs/AlGaAs heterostructure. Calculations are performed with the use of the three-dimensional (3D) numerical model of the device. The results are obtained for the structure working under the special biasing conditions. At some characteristic values of biasing voltages the transfer of the charge from one quantum well to the other is observed. Charge density distribution in the CQDs is obtained on the basis of the electron’s wave functions. They are calculated as the self-consistent solutions of Poisson and Schro¨dinger equations in 3D space. Charge density corresponds to the effective potential distribution which contains Hartree term of the potential and the two other terms which describe the effects of exchange and correlation in electron–electron interactions. These effects are modeled by the use of the local density approximation (LDA). r 2007 Elsevier B.V. All rights reserved. PACS: 85.35.Be; 73.20.r; 41.20.Cv Keywords: Coupled quantum dots; Poisson equation; Schro¨dinger equation; Local density approximation; Electron–electron interactions
1. Introduction Modern solutions in semiconductor technology allow to create few electron quantum devices with the great success [1]. Different semiconductor structures are used to form these special, zero-dimensional (0D) systems. GaAs is often applied as a basic compound for such applications [2]. In the paper two coupled quantum dots (CQDs) are analyzed. They are formed in GaAs/AlGaAs heterostructure [3] (Fig. 1). Suitable, positive value of UGS creates two-dimensional electron gas (2DEG) at GaAs/AlGaAs heterointerface. Two remaining negative voltages, i.e. UES1 and UES2 confine electrons of 2DEG in x–y plane (parallel to the heterojunction) closing them in the two, nearly parabolic quantum wells. In the work numerical results obtained for the structure working under the special biasing conditions are reported. The voltage UGS is kept at the fixed level whereas UES1 and UES2 are changed. At some characteristic values of UES1
Corresponding author. Tel.: +48 17 865 11 12.
E-mail address: [email protected] (E. Machowska-Podsiad"o). 1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.10.118
and UES2 charge transfer from one quantum well to the other is observed. 2. Method Numerical analysis of the problem begins with the solving of Poisson equation [4]: DV ðRÞ ¼
r2DEG ðRÞ ; 0 r
R ¼ ðx; y; zÞ,
(1)
where r2DEG(R) denotes charge density in 2DEG region (Fig. 1). Permittivity er takes the value of 12.8 for GaAs layer and 13.2 for AlGaAs region. Dissolving of Eq. (1) allows to find 3D potential distribution V(R) over the structure. Afterwards, two quantum wells and their environment are studied. To obtain electron states in the system 3D Schro¨dinger equation should be solved. It was verified [5] that the analysis can be performed by the use of the simplified method in which electron gas in the quantum wells is treated separately in a plane of 2DEG and separately in z direction (perpendicular to GaAs/AlGaAs hetero-interface). In such case energy levels Ez are found
ARTICLE IN PRESS E. Machowska-Podsiad!o, M. Ma˛czka / Physica E 40 (2008) 1748–1750
1749
Fig. 1. Structure of the coupled quantum dots.
for two quantum wells as the solutions of two Schro¨dinger equations in the form:
_2 DjðzÞ þ qV ðzÞjðzÞ ¼ E z jðzÞ. 2m
Fig. 2. Top view and side view of the device. In both figures parameters of the structure are marked.
(2)
Potential V(z) is taken as V(x0L,y0L,z) or V(x0R,y0R,z) (both calculated from Eq. (1)), where (x0L, y0L) and (x0R, y0R) are coordinates of the central points in the quantum wells ‘‘on the left’’ and ‘‘on the right side’’ of the device. It is assumed that electron’s wave functions j(z) may penetrate AlGaAs layer. Effective masses m* of an electron in GaAs and AlGaAs layers are taken as 0.068m0 and 0.0919m0, respectively (m0 ¼ 9.11 1031 kg). The maximum of the electron density jj(z)j2 allows to choose the right x–y plane for further analysis of the electron states in QCDs. To describe the electrons in CQDs Schro¨dinger equations in the form:
_2 Dci ðrÞ þ qV eff i ðrÞci ðrÞ ¼ E i ci ðrÞ 2mG GaAs
(3)
must be solved. The equations are considered in the previously chosen x–y plane (r ¼ (x,y)). The solutions (energy levels Ei and wave functions ci(r)) are found in a selfconsistent way and for every ith electron in the CQDs. Charge densities ri(r) determined as qjci(r)j2 correspond to the effective potential distributions: V eff i ðrÞ ¼ V ðrÞ þ V H i ðrÞ þ V ex i ðrÞ þ V corr i ðrÞ
(4)
and regarding the region of two quantum wells (depletion area and quantum dots in Fig. 1). In Eq. (4) VH i(r) denotes the Hartree term of the potential and the two other terms (Vex i(r) and Vcorr i(r)) describe the effects of exchange and correlation in electron–electron interactions [4,6]. These effects are modeled by the use of the local density approximation (LDA) [7]. The procedure described by Eqs. (1)–(4) is performed for the set of biasing voltages UGS, UES1, UES2 and UDS=0. To ensure the transfer of the charge from one quantum well to the other voltages UES1, UES2, are changed.
Fig. 3. Cross-sections of the effective potential energy qVeff (squares and circles) and corresponding charge density distributions jri(r)j (up and down triangles) for two electrons in the wells. Filled symbols (squares and up and down triangles) regard to the distributions for the case of UES1 ¼ UES2 ¼ 205 mV. Open symbols (circles and diamonds) describe the distributions for the voltages UES1 ¼ 167 mV and UES2 ¼ 243 mV. In the insets the occupied energy levels are shown. They are presented for the respective quantum wells (left and right). In the upper inset the levels for UES1 ¼ UES2 ¼ 205 mV are illustrated. And below the levels for the voltages UES1 ¼ 167 mV and UES2 ¼ 243 mV are shown.
3. Results Numerical simulations were done for the structure with parameters as marked in Fig. 2. The structure was biased with the fixed voltages: UGS ¼ 500 mV, UDS ¼ 0 V, whereas UES1, UES2 were changed. Analysis started with simulations for the case of two identical QDs, i.e. for UES1 ¼ UES2 ¼ 205 mV. It was found that in such case two electrons are confined in the wells (Fig. 3). Both electrons occupy the lowest energy levels with values lower than 0 (which is treated as the reference level in the system). The occupied levels are
ARTICLE IN PRESS 1750
E. Machowska-Podsiad!o, M. Ma˛czka / Physica E 40 (2008) 1748–1750
wells shifted. The left QD became deeper and the right one became shallower. The lowest levels Ez, i.e. in the left (EzL) and in the right (EzR) quantum well, follow these changes (Fig. 4). The quantum wells were analyzed in the plane where the maximum of 2DEG concentration jj(z)j2 was found (i.e. at z ¼ 406 nm). For the pair (UES1, UES2) ¼ (195 mV, 215 mV) both quantum wells changed their minima qVeff (x0L, y0L, 406 nm) and qVeff (x0R, y0R, 406 nm) by the values of 2.65 and 2.95 meV, respectively. The changes were observed due to the charge transfer between the dots. 4. Conclusions
Fig. 4. The bottom of the wells qV(x0,y0,400 nm) (filled squares for the left and empty squares for the right quantum well) for different pairs of voltages (UES1, UES2). The lowest energy levels Ez calculated for both wells in the system. The levels regard to the left (filled circles) and to the right (empty circles) well in the CQDs. The triangles (filled and empty) describe the bottoms of two wells qVeff(x0,y0,406 nm) (left and right, respectively) observed at the distance of 6 nm from the hetero-interface (i.e. in the plane of z ¼ 406 nm).
denoted as E0L and E0R and they regard to the well ‘‘on the left’’ and ‘‘on the right’’ side of the device, respectively. Their energies are the same and they equal to 0.65 meV. The values were obtained as the sum of the lowest energy Ez (which is the solution of Eq. (2)) and the energies Ei obtained from Eq. (3) for the electrons in the left and in the right quantum well. Charge density distributions ri(r) of the two electrons in the wells follow the symmetry of the system (Fig. 3). Afterwards calculations for subsequent pairs (UES1, UES2) were performed. The first voltage of the pair was decreased by 2 mV and the second one was increased by 2 mV in successive simulations. As it is shown in Fig. 4 the change of the voltages caused that the bottom of both the
The method allows to calculate electronic states in CQDs. The states may be determined for various biasing conditions. Calculations performed for subsequent pairs of voltages UES1 and UES2 (where the first voltage of the pair is increased and the second one is decreased) give the possibility to observe the charge transfer from one quantum well to the other. Acknowledgment The work was supported by the State Committee for Scientific Research under Grant No. 3-T11B-093-28. References [1] U. Wilhelm, J. Weis, Physica E 6 (1–4) (2000) 668. [2] I.H. Chan, P. Fallahi, R.M. Westervelt, K.D. Maranowski, A.C. Gossard, Physica E 17 (2003) 584. [3] U. Meirav, M. Heiblum, F. Stern, Appl. Phys. Lett. 52 (15) (1988) 1268. [4] E. Machowska-Podsiad"o, M. Ma˛ czka, Electron. Technol.—Int. J. 37/38 (2005/2006) 13. [5] E. Machowska-Podsiad"o, M. Ma˛ czka, M. Bugajski, Tech. Sci. 55 (No. 2) (2007) 245. [6] E. Machowska-Podsiadlo, M. Bugajski, S. Pawlowski, Acta Phys. Polon. B 32 (2002) 503. [7] M. Macucci, K. Hess, G.J. Iafrate, Phys. Rev. B 48 (1993) 17354.
Physica E 40 (2008) 1748–1750 www.elsevier.com/locate/physe
Electronic properties of coupled quantum dots E. Machowska-Podsiad"o, M. Ma˛ czka Rzeszo´w University of Technology, ul. W. Pola 2, 35-959 Rzeszo´w, Poland Available online 4 December 2007
Abstract In the work two coupled quantum dots (CQDs) are analyzed. They are formed electrostatically in GaAs/AlGaAs heterostructure. Calculations are performed with the use of the three-dimensional (3D) numerical model of the device. The results are obtained for the structure working under the special biasing conditions. At some characteristic values of biasing voltages the transfer of the charge from one quantum well to the other is observed. Charge density distribution in the CQDs is obtained on the basis of the electron’s wave functions. They are calculated as the self-consistent solutions of Poisson and Schro¨dinger equations in 3D space. Charge density corresponds to the effective potential distribution which contains Hartree term of the potential and the two other terms which describe the effects of exchange and correlation in electron–electron interactions. These effects are modeled by the use of the local density approximation (LDA). r 2007 Elsevier B.V. All rights reserved. PACS: 85.35.Be; 73.20.r; 41.20.Cv Keywords: Coupled quantum dots; Poisson equation; Schro¨dinger equation; Local density approximation; Electron–electron interactions
1. Introduction Modern solutions in semiconductor technology allow to create few electron quantum devices with the great success [1]. Different semiconductor structures are used to form these special, zero-dimensional (0D) systems. GaAs is often applied as a basic compound for such applications [2]. In the paper two coupled quantum dots (CQDs) are analyzed. They are formed in GaAs/AlGaAs heterostructure [3] (Fig. 1). Suitable, positive value of UGS creates two-dimensional electron gas (2DEG) at GaAs/AlGaAs heterointerface. Two remaining negative voltages, i.e. UES1 and UES2 confine electrons of 2DEG in x–y plane (parallel to the heterojunction) closing them in the two, nearly parabolic quantum wells. In the work numerical results obtained for the structure working under the special biasing conditions are reported. The voltage UGS is kept at the fixed level whereas UES1 and UES2 are changed. At some characteristic values of UES1
Corresponding author. Tel.: +48 17 865 11 12.
E-mail address: [email protected] (E. Machowska-Podsiad"o). 1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.10.118
and UES2 charge transfer from one quantum well to the other is observed. 2. Method Numerical analysis of the problem begins with the solving of Poisson equation [4]: DV ðRÞ ¼
r2DEG ðRÞ ; 0 r
R ¼ ðx; y; zÞ,
(1)
where r2DEG(R) denotes charge density in 2DEG region (Fig. 1). Permittivity er takes the value of 12.8 for GaAs layer and 13.2 for AlGaAs region. Dissolving of Eq. (1) allows to find 3D potential distribution V(R) over the structure. Afterwards, two quantum wells and their environment are studied. To obtain electron states in the system 3D Schro¨dinger equation should be solved. It was verified [5] that the analysis can be performed by the use of the simplified method in which electron gas in the quantum wells is treated separately in a plane of 2DEG and separately in z direction (perpendicular to GaAs/AlGaAs hetero-interface). In such case energy levels Ez are found
ARTICLE IN PRESS E. Machowska-Podsiad!o, M. Ma˛czka / Physica E 40 (2008) 1748–1750
1749
Fig. 1. Structure of the coupled quantum dots.
for two quantum wells as the solutions of two Schro¨dinger equations in the form:
_2 DjðzÞ þ qV ðzÞjðzÞ ¼ E z jðzÞ. 2m
Fig. 2. Top view and side view of the device. In both figures parameters of the structure are marked.
(2)
Potential V(z) is taken as V(x0L,y0L,z) or V(x0R,y0R,z) (both calculated from Eq. (1)), where (x0L, y0L) and (x0R, y0R) are coordinates of the central points in the quantum wells ‘‘on the left’’ and ‘‘on the right side’’ of the device. It is assumed that electron’s wave functions j(z) may penetrate AlGaAs layer. Effective masses m* of an electron in GaAs and AlGaAs layers are taken as 0.068m0 and 0.0919m0, respectively (m0 ¼ 9.11 1031 kg). The maximum of the electron density jj(z)j2 allows to choose the right x–y plane for further analysis of the electron states in QCDs. To describe the electrons in CQDs Schro¨dinger equations in the form:
_2 Dci ðrÞ þ qV eff i ðrÞci ðrÞ ¼ E i ci ðrÞ 2mG GaAs
(3)
must be solved. The equations are considered in the previously chosen x–y plane (r ¼ (x,y)). The solutions (energy levels Ei and wave functions ci(r)) are found in a selfconsistent way and for every ith electron in the CQDs. Charge densities ri(r) determined as qjci(r)j2 correspond to the effective potential distributions: V eff i ðrÞ ¼ V ðrÞ þ V H i ðrÞ þ V ex i ðrÞ þ V corr i ðrÞ
(4)
and regarding the region of two quantum wells (depletion area and quantum dots in Fig. 1). In Eq. (4) VH i(r) denotes the Hartree term of the potential and the two other terms (Vex i(r) and Vcorr i(r)) describe the effects of exchange and correlation in electron–electron interactions [4,6]. These effects are modeled by the use of the local density approximation (LDA) [7]. The procedure described by Eqs. (1)–(4) is performed for the set of biasing voltages UGS, UES1, UES2 and UDS=0. To ensure the transfer of the charge from one quantum well to the other voltages UES1, UES2, are changed.
Fig. 3. Cross-sections of the effective potential energy qVeff (squares and circles) and corresponding charge density distributions jri(r)j (up and down triangles) for two electrons in the wells. Filled symbols (squares and up and down triangles) regard to the distributions for the case of UES1 ¼ UES2 ¼ 205 mV. Open symbols (circles and diamonds) describe the distributions for the voltages UES1 ¼ 167 mV and UES2 ¼ 243 mV. In the insets the occupied energy levels are shown. They are presented for the respective quantum wells (left and right). In the upper inset the levels for UES1 ¼ UES2 ¼ 205 mV are illustrated. And below the levels for the voltages UES1 ¼ 167 mV and UES2 ¼ 243 mV are shown.
3. Results Numerical simulations were done for the structure with parameters as marked in Fig. 2. The structure was biased with the fixed voltages: UGS ¼ 500 mV, UDS ¼ 0 V, whereas UES1, UES2 were changed. Analysis started with simulations for the case of two identical QDs, i.e. for UES1 ¼ UES2 ¼ 205 mV. It was found that in such case two electrons are confined in the wells (Fig. 3). Both electrons occupy the lowest energy levels with values lower than 0 (which is treated as the reference level in the system). The occupied levels are
ARTICLE IN PRESS 1750
E. Machowska-Podsiad!o, M. Ma˛czka / Physica E 40 (2008) 1748–1750
wells shifted. The left QD became deeper and the right one became shallower. The lowest levels Ez, i.e. in the left (EzL) and in the right (EzR) quantum well, follow these changes (Fig. 4). The quantum wells were analyzed in the plane where the maximum of 2DEG concentration jj(z)j2 was found (i.e. at z ¼ 406 nm). For the pair (UES1, UES2) ¼ (195 mV, 215 mV) both quantum wells changed their minima qVeff (x0L, y0L, 406 nm) and qVeff (x0R, y0R, 406 nm) by the values of 2.65 and 2.95 meV, respectively. The changes were observed due to the charge transfer between the dots. 4. Conclusions
Fig. 4. The bottom of the wells qV(x0,y0,400 nm) (filled squares for the left and empty squares for the right quantum well) for different pairs of voltages (UES1, UES2). The lowest energy levels Ez calculated for both wells in the system. The levels regard to the left (filled circles) and to the right (empty circles) well in the CQDs. The triangles (filled and empty) describe the bottoms of two wells qVeff(x0,y0,406 nm) (left and right, respectively) observed at the distance of 6 nm from the hetero-interface (i.e. in the plane of z ¼ 406 nm).
denoted as E0L and E0R and they regard to the well ‘‘on the left’’ and ‘‘on the right’’ side of the device, respectively. Their energies are the same and they equal to 0.65 meV. The values were obtained as the sum of the lowest energy Ez (which is the solution of Eq. (2)) and the energies Ei obtained from Eq. (3) for the electrons in the left and in the right quantum well. Charge density distributions ri(r) of the two electrons in the wells follow the symmetry of the system (Fig. 3). Afterwards calculations for subsequent pairs (UES1, UES2) were performed. The first voltage of the pair was decreased by 2 mV and the second one was increased by 2 mV in successive simulations. As it is shown in Fig. 4 the change of the voltages caused that the bottom of both the
The method allows to calculate electronic states in CQDs. The states may be determined for various biasing conditions. Calculations performed for subsequent pairs of voltages UES1 and UES2 (where the first voltage of the pair is increased and the second one is decreased) give the possibility to observe the charge transfer from one quantum well to the other. Acknowledgment The work was supported by the State Committee for Scientific Research under Grant No. 3-T11B-093-28. References [1] U. Wilhelm, J. Weis, Physica E 6 (1–4) (2000) 668. [2] I.H. Chan, P. Fallahi, R.M. Westervelt, K.D. Maranowski, A.C. Gossard, Physica E 17 (2003) 584. [3] U. Meirav, M. Heiblum, F. Stern, Appl. Phys. Lett. 52 (15) (1988) 1268. [4] E. Machowska-Podsiad"o, M. Ma˛ czka, Electron. Technol.—Int. J. 37/38 (2005/2006) 13. [5] E. Machowska-Podsiad"o, M. Ma˛ czka, M. Bugajski, Tech. Sci. 55 (No. 2) (2007) 245. [6] E. Machowska-Podsiadlo, M. Bugajski, S. Pawlowski, Acta Phys. Polon. B 32 (2002) 503. [7] M. Macucci, K. Hess, G.J. Iafrate, Phys. Rev. B 48 (1993) 17354.