- Email: [email protected]
demultiplexer
Solid State Communications 134 (2005) 571–576 www.elsevier.com/locate/ssc
Novel structure of single-electron two-channel multiplexer/demultiplexer K.-M. Hung*, Y.-S. Wu Department of Electronics Engineering, National Kaohsiung University of Applied Sciences, 415, Chien-Kung Road, Kaohsiung 807, Taiwan, ROC Received 15 July 2004; received in revised form 12 January 2005; accepted 4 February 2005 by H. Takayama Available online 10 March 2005
Abstract We propose a novel structure of single-electron two-channel multiplexer and demultiplexer based on three coupled singledopant quantum dots defined by enhancement gates on AlGaAs/GaAs heterostructure. Two side-gates next to the dots are designed for applying a lateral switching field to the structure. A simple model of spherical parabolic quantum dot within effective-mass approximation demonstrates that the coupling strengths of the dots are adjustable by applying a lateral field. This gives the promise on achieving the functions of multiplexing and demultiplexing through the proposed structure. q 2005 Elsevier Ltd. All rights reserved. PACS: 71.55.Ki; 71.55.Cn; 73.20.Hb Keywords: A. Quantum dots; A. Single electron; D. Demultiplexer; D. Multiplexer; D. Variational method
Search of a scalable solid-state quantum computer (QC) system is of great interest in the past two decades. Several advanced schemes for implementing a quantum-mechanical system based on present solid-state technologies have been proposed, such as nuclear-spin-based QC [1–4] and chargebased QC [5–7]. The information in the former system is encoded onto the nuclear spin states of buried phosphorus dopants in a silicon crystal, but encoded onto the excitation levels of an impurity or a quantum dot (QD) for the latter. Together with the charge detection of a single-electron transistor that provides an interface to access the quantum information in QCs [8,9], bring both the systems closer to reality. In addition to this, data-flow management (DFM) is another important technology in a practical computer system, which makes the computing algorithm more
* Corresponding author. Tel.: C886 7381 452 65628; fax: C886 7381 1182. E-mail address: [email protected] (K.-M. Hung).
0038-1098/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2005.02.012
structural and increases the utilization efficiency of the computer. The requirement of a data-flow manager for QCs is quite different from the traditional one; it has to be provided with the capability on manipulating the singleelectron data flow. At the same time, multiplexing and demultiplexing are two major functions to DFM. There is a similar structure design, named electron Y-branch switch (YBS) proposed by Palm and Thylen in 1992 [10], to these functionalities. In the YBS, a 1-D source is split into two 1-D drains and a lateral switching field directs the electrons in the source into either of two drains [11–13]. Nevertheless, a self-gating effect due to capacitive coupling between the branches (source and drains) strongly modifies the effective switching field [14]. On the other side, increasing current scattering leads to a randomization of electron momentum [15,16], which may result in the loss of the information carried by a stream of electrons. This paper aims at giving a novel structure of single-electron two-channel multiplexer and demultiplexer (SETCMD) based on three coupled single-dopant QDs (SDQD) defined on AlGaAs/GaAs
572
K.-M. Hung, Y.-S. Wu / Solid State Communications 134 (2005) 571–576
heterostructure. The dopant in QDs together with the dot structure design has two advantages: (1) the deeper resonant state enhances the contrast of switching-on and -off currents and (2) the greater level spacing of impurity states gives promise of high operating temperature and preventing the self-gating effect. The major part of the device structure is three coupled SDQDs defined by three enhancement gates on AlGaAs/ GaAs heterostructure. A gate with a large area is placed at the center of the structure and two small gates are next to the centered dot as depicted in Fig. 1. Three supply channels or quantum-wire leads, also defined by enhancement gates, are separately coupled to these dots. Two diffusion contacts located at both sides of the main structure play the role of ~ to control electrodes for applying a lateral electric field E the structure. Under applicable biases, the energy profiles of multiplexer and demultiplexer, as depicted in Fig. 2, are obtained. The Fermi-level of the channel S is smaller (greater) than those of the channels A and B for multiplexing (demultiplexing) operation. The switching function is achievable by utilizing the
facts: (1) the electron wavefunction for an impurity in a QD is movable under applied electric fields (2) the moving distance is proportional to both strengths of the quantum confinement and electric field, a higher confinement and/or a smaller field getting to a smaller moving distance. With the design of a strong confinement to the dots A and B and a weak confinement to the dot S, the moving distance of the wavefunction in the dot S under applied electric field is much greater than those in the dots A and B. This allows us to change the coupling strengths (CS) (or the transmission amplitudes, TA) of dot-S-to-dot-A (S–A) and dot-S-to-dotB (S–B) by justifying the field direction and strength and hence to direct the electron path from dot S to dot A in rightdirection field or from dot S to dot B in opposite direction. In order to demonstrate the ground-state wavefunction behavior of an impurity in an isolated QD, let us consider a simple model of an impurity with its location deviated (along z direction) away from the center of a parabolic QD under applied electric field (directed to z-axis). The Hamiltonian of the system within one-band effective-mass approximation (in the units of the effective Rydberg
Fig. 1. Top view (a) and side view (b) of the structure of single-electron two-channel multiplexer/demultiplexer.
K.-M. Hung, Y.-S. Wu / Solid State Communications 134 (2005) 571–576
573
Fig. 2. Energy profile of single-electron two-channel demultiplexer (a) and multiplexer (b) under biases.
RyZ m e4 =32p2 32 Z2 and the effective Bohr radius a0 Z 4p3Z2 =m e2 ) can be written as 2 H Z KV2 C Vp r 2 C VF z K ~r K~z 0
(1)
where Vp and VF are the strengths of the parabolic confinement potential and the electric-field energy, respectively. They can be related to their conventional forms of Vp Z m u2 a20 =2Ry and VF Z eFa0 =Ry, with the parabola frequency u, electric field F, effective mass m*, electron charge e and the dielectric constant 3. There are two reasons for the consideration that the impurity location is shifted away from the center of the dot along the field direction: (1) the carrier has the greatest change in its binding energy due to the interplays of the field and impurity location (2) the variational integral of total energy is analytically integrable. In such considerations, the trial wavefunction of the dopant’s ground state can be written as the product of the 1s state of hydrogenic atom and the ground state of spherical harmonic oscillator,
2 K ~r K~z 0 J Z C Exp ExpðKl~r C~z F Þ a
(2)
where C is thepnormalization constant, a the variational ffiffiffiffiffi parameter, l h Vp =2 and zFhVF/2Vp. The impurity level is obtained by minimizing the expectation value of the total energy Etot hhJjHjJi. In solving the variational integral hjjHjji, we simply take the transformation ~r /~r C~z 0 such that the Eqs. (1) and (2), then, become H Z KV2 C Vp j~r C~z 0 C~z F j2 K
VF2 2 K 4Vp r
(3)
and J Z C ExpðKr=a K lj~r C~z 0 C~z F j2 Þ
(4)
It is clear as revealed in Eq. (3) that the action of the electric field has the same effect as relatively shifting the center of the confinement potential in a distance zF, while the energy level of the dot decreases in a value of -V2F/4VP. Under the transformation, the variational integral of the total energy
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can be solved analytically in the assistance of the package MATHEMATICA for saving a great amount of CPU times. After tedious integration and algebra, the total energy of the system expressed in terms of the kinetic energy Ek, Coulomb energy ECol, the lowest energy Ec of the dot and the field energy EF is given by Etot Z EK C ECol C Ec C EF
(5)
with pffiffiffiffi 1=4 n pVp 2 eKC ð2 K 3aKCVp1=4 ÞerfðKCÞ EK Z 4abN o 2 C eKK 2erfðKKÞ C 3aKKVp1=4 erfðKKÞ C
pffiffiffiffi p Aa 16a2 bVp N KCKK
(6)
pffiffiffiffi 1=4 pVp 2 2 ECol Z ½eKC erfcðKCÞ K eKK erfcðKKKÞ bN
(7)
Ab 1 K pffiffiffiffiffi 16a2 Vp7=4 N a Vp N
Ec Z
pffiffiffiffiffiffiffiffiffi h pVp KK2 2 C e KKð3 C 2KK ÞerfðKKÞ 4bN i 2 2 ÞerfðKCÞ C Vp z20 K eKC KCð3 C 2KC
EF Z
Fig. 3. Binding energy, calculated by the variational method (solidline) and the first-order perturbation approach (broken-line), of the ground state of an impurity in a parabolic QD with respect to the confinement strength VP.
pffiffiffiffi Ab ¼ Uþ p a2 b2 þ 6a2 Vp3=2 þ 12Vp pffiffiffiffiffiffiffiffiffi 2UK pVp 2 2 þ 3a b þ 6a2 Vp3=2 þ 4Vp ab Ac Z 4bVp C 4ab2
pffiffiffiffiffi Vp K a2 4KCVp9=4 K 2bVp3=2 K b3
and (8)
pffiffiffiffi n h 2 p a2 eKK 4Vp9=4 KK K bðb2 C 2Vp3=2 Þ 2 7=4 8a Vp bN i pffiffiffiffiffi 2 C eKC 4Vp9=4 KC K bðb2 C 2Vp3=2 C 4ab Vp
Ad Z 4bVp K 4ab2
pffiffiffiffiffi Vp K a2 4KKVp9=4 K 2bVp3=2 K b3
In these equations, erf and erfc are the error function and complementary error function, respectively. Numerically minimizing the total energy Etot(min), one obtains the ground-state level of the impurity. The binding energy measured pffiffiffiffiffi from the lowest level of the dot is found to be Eb Z 3 Vp K VF2 =4Vp K EtotðminÞ , where the sum of the
! 2Vp3=4 ! pffiffiffiffi K bUK p 2 2 C Ac eKC erfðKCÞ K Ad eKC erfðKKÞ K 4bUCVp C VF z0 (9) and NZ
pffiffiffiffi h i 2 p KC2 e KCerfcðKCÞ C eKK KKerfcðKKKÞ 2b
(10)
The symbols used in Eqs. (6)–(10) are defined as follows: pffiffiffiffiffi abG2 Vp 2 2 ; UG Z eKC GeKK b Z VF C 2Vp z0 ; KG Z 2aVp3=4 2
Aa ¼ KeKK KKð4Vp K 3a2 b2 K 4ab
pffiffiffiffiffi Vp Þ
pffiffiffiffiffi Ke Kþ ð4Vp K 3a b þ 4ab Vp Þ Kþ2
2 2
Fig. 4. Binding energy as a function of impurity position for various values of confinement potential, VPZ1.0, 2.5, 4.0, 5.5 and 7.0 and the field energy VFZ0.
K.-M. Hung, Y.-S. Wu / Solid State Communications 134 (2005) 571–576
Fig. 5. Binding energy as a function of electric field for various values of confinement potential, VPZ1.0, 2.5, 4.0, 5.5 and 7.0 and the impurity location z0Z0.
first two terms is the lowest energy of the QD under electric field [17]. In the following discussions, we apply the derived theory to the system of AlGaAs/GaAs heterostructure ˚ and the with the effective Bohr radius a0y102.7 A effective Rydberg Ryy5.39 meV for GaAs material. Fig. 3 plots the binding energy with respect to the
575
confinement potential in the case of VFZ0 and z0Z0 by using the variational method (solid-line) and the firstorder perturbation approach (broken-line). Notice that the binding energy estimated by the latter approach has pffiffiffiffi the simple form of 4Vp1=4 = p. As shown in the figure, the binding energy increases with increasing the confinement strength. This character also appears in the cases of VFs0 and/or z0s0 as depicted in Figs. 4 and 5. It is mainly attributed to the increases of the wavefunction localization as the rise of the confinement strength. In addition, Figs. 4 and 5 show that the electric field takes the same effect as moved the impurity location along the field direction, as we mentioned above. Based on this simple model, we demonstrate the fieldinduced switching property with the following structure parameters: VPAZVPBZ1Ry (the confinement strengths of the dots A and B), VPSZ0.25Ry (the confinement strength of the dot S) and, z0AZK5a0 and z0BZ5a0 (the centerspof ffiffiffi the dots A and B, respectively). The radius, defined as 1= l, for pffiffiffi the dots A and B is taken to be 2a0 and one for the dot S to be 2a0. In this case, the total particle density within weak coupling region may be approximately estimated by rt Z jJA j2 C jJB j2 C jJS j2 . The calculated results of rt for yZ0 are presented in Fig. 6(a)–(d) with the field energy VF varied from K1.5Ry to C1.5Ry stepped in 1Ry, which corresponds to the electric field F ranging from 7.87!103 to
Fig. 6. Total particle density in the three QDs for yZ0.
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are calculated based on the transfer Hamiltonian approach and the simple model of an impurity in an isolated parabolic QD. The contrast of switching-on and -off currents approaches 106. The theoretical results demonstrate that a lateral switching field can be used to direct the electron path from the centered dot to either of its neighboring dots.
Acknowledgements This work was supported by the Education Ministry, Taiwan, ROC.
References Fig. 7. Transmission amplitudes (in log scale with arbitrary unit) of S–A and S–B versus applied field energy VF.
K7.87!103 V/cm. It is obvious that there is an increase in the amount of wavefunction overlap for S–A with increasing right-direction fields, while one for S–B decreases. Since, the CS is proportional to the wavefunction overlap, the changes of the overlaps the changes of the CSs. As a consequence, the lateral field can direct the electron in the dot S into either of the dots A and B, which depends on the field direction and strength. In order to quantitatively study the variations of the CSs of the dots with respect to the external control fields, we apply the transfer Hamiltonian approach [18,19] on the estimation of the TAs for S–A and S–B. TA can be calculated by the expression ð Z2 jMi/j j2 ¼ j Ji vz Jj K Jj vz Ji dSz j2 (11) 2m S
where the surface integral is evaluated at the intersection (on z-axis) of the confinement potentials of the neighboring dots. Fig. 7 plots the TAs (in log scale) of S–A and S–B with respect to the field energy VF, ranging from K1.5Ry to 1.5Ry. One can see that the TA of S–A just is a mirrored image of S–B due to the symmetric structure and the contrast, that is the ratio of the turn-on to the turn-off currents or the maximum TA to the minimum TA, for both S–A and S–B approaches w106. It is an evidence to show that the proposed configuration forms a good switch for the utilization of multiplexer and demultiplexer. In conclusion, we have proposed a novel structure of SETCMD based on three coupled SDQDs defined on AlGaAs/GaAs heterostructure. The TAs between the dots
[1] B.E. Kane, Nature (London) 393 (1998) 133. [2] J.L. O’Brien, S.R. Schofield, M.Y. Simmons, R.G. Clark, A.S. Dzurak, N.J. Curson, B.E. Kane, N.S. McAlpine, M.E. Hawley, G.W. Brown, Phys. Rev. B 64 (2001) 161401. [3] S.R. Schofield, N.J. Curson, M.Y. Simmons, F.J. Ruess, T. Hallam, L. Oberbeck, R.G. Clark, Phys. Rev. Lett. 91 (2003) 136104. [4] A.S. Dzurak, L.C.L. Hollenberg, D.N. Jamieson, F.E. Stanley, C. Yang, T.M. Buehler, V. Chan, D.J. Reilly, C. Wellard, A.R. Hamilton, C.I. Pakes, A.G. Ferguson, E. Gauja, S. Prawer, G.J. Milburn, R.G. Clark, cond-mat/0306265 (unpublished). [5] A. Barenco, D. Deutsch, A. Ekert, R. Jozsa, Phys. Rev. Lett. 74 (1995) 4083. [6] L. Fedichkin, M. Yanchenko, K.A. Valiev, Nanotechnology 11 (2000) 387. [7] L.C.L. Hollenberg, A.S. Dzurak, C. Wellard, A.R. Hamilton, D.J. Reilly, G.J. Milburn, R.G. Clark, Phys. Rev. B 69 (2004) 113301. [8] R.J. Schoelkopf, P. Wahlgren, A.A. Kozhevnikov, P. Delsing, D.E. Prober, Science 280 (1998) 1238. [9] A. Aassime, G. Johansson, G. Wendin, R.J. Schoelkopf, P. Delsing, Phys. Rev. Lett. 86 (2001) 3376. [10] T. Palm, L. Thylen, Appl. Phys. Lett. 60 (1992) 237. [11] L. Worschech, B. Weidner, S. Reitzenstein, A. Forchel, Appl. Phys. Lett. 78 (2001) 3325. [12] T. Palm, J. Appl. Phys. 74 (1993) 3351. [13] T. Palm, L. Thylen, O. Nilsson, C. Svensson, J. Appl. Phys. 74 (1993) 687. [14] J.-O.J. Wesstrom, Phys. Rev. Lett. 82 (1999) 2564. [15] U. Sivan, M. Heiblum, C.P. Umbach, Phys. Rev. Lett. 63 (1989) 992. [16] R. Gupta, N. Balkan, B.K. Ridley, Phys. Rev. B 46 (1992) 7745. [17] N. Zettili, Quantum Mechanics Concepts and Applications, Wiley, New York, 2001. [18] J. Bardeen, Phys. Rev. Lett. 6 (1961) 57. [19] M.C. Payne, J. Phys. C: Solid State Phys. 19 (1986) 1145.
Novel structure of single-electron two-channel multiplexer/demultiplexer K.-M. Hung*, Y.-S. Wu Department of Electronics Engineering, National Kaohsiung University of Applied Sciences, 415, Chien-Kung Road, Kaohsiung 807, Taiwan, ROC Received 15 July 2004; received in revised form 12 January 2005; accepted 4 February 2005 by H. Takayama Available online 10 March 2005
Abstract We propose a novel structure of single-electron two-channel multiplexer and demultiplexer based on three coupled singledopant quantum dots defined by enhancement gates on AlGaAs/GaAs heterostructure. Two side-gates next to the dots are designed for applying a lateral switching field to the structure. A simple model of spherical parabolic quantum dot within effective-mass approximation demonstrates that the coupling strengths of the dots are adjustable by applying a lateral field. This gives the promise on achieving the functions of multiplexing and demultiplexing through the proposed structure. q 2005 Elsevier Ltd. All rights reserved. PACS: 71.55.Ki; 71.55.Cn; 73.20.Hb Keywords: A. Quantum dots; A. Single electron; D. Demultiplexer; D. Multiplexer; D. Variational method
Search of a scalable solid-state quantum computer (QC) system is of great interest in the past two decades. Several advanced schemes for implementing a quantum-mechanical system based on present solid-state technologies have been proposed, such as nuclear-spin-based QC [1–4] and chargebased QC [5–7]. The information in the former system is encoded onto the nuclear spin states of buried phosphorus dopants in a silicon crystal, but encoded onto the excitation levels of an impurity or a quantum dot (QD) for the latter. Together with the charge detection of a single-electron transistor that provides an interface to access the quantum information in QCs [8,9], bring both the systems closer to reality. In addition to this, data-flow management (DFM) is another important technology in a practical computer system, which makes the computing algorithm more
* Corresponding author. Tel.: C886 7381 452 65628; fax: C886 7381 1182. E-mail address: [email protected] (K.-M. Hung).
0038-1098/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2005.02.012
structural and increases the utilization efficiency of the computer. The requirement of a data-flow manager for QCs is quite different from the traditional one; it has to be provided with the capability on manipulating the singleelectron data flow. At the same time, multiplexing and demultiplexing are two major functions to DFM. There is a similar structure design, named electron Y-branch switch (YBS) proposed by Palm and Thylen in 1992 [10], to these functionalities. In the YBS, a 1-D source is split into two 1-D drains and a lateral switching field directs the electrons in the source into either of two drains [11–13]. Nevertheless, a self-gating effect due to capacitive coupling between the branches (source and drains) strongly modifies the effective switching field [14]. On the other side, increasing current scattering leads to a randomization of electron momentum [15,16], which may result in the loss of the information carried by a stream of electrons. This paper aims at giving a novel structure of single-electron two-channel multiplexer and demultiplexer (SETCMD) based on three coupled single-dopant QDs (SDQD) defined on AlGaAs/GaAs
572
K.-M. Hung, Y.-S. Wu / Solid State Communications 134 (2005) 571–576
heterostructure. The dopant in QDs together with the dot structure design has two advantages: (1) the deeper resonant state enhances the contrast of switching-on and -off currents and (2) the greater level spacing of impurity states gives promise of high operating temperature and preventing the self-gating effect. The major part of the device structure is three coupled SDQDs defined by three enhancement gates on AlGaAs/ GaAs heterostructure. A gate with a large area is placed at the center of the structure and two small gates are next to the centered dot as depicted in Fig. 1. Three supply channels or quantum-wire leads, also defined by enhancement gates, are separately coupled to these dots. Two diffusion contacts located at both sides of the main structure play the role of ~ to control electrodes for applying a lateral electric field E the structure. Under applicable biases, the energy profiles of multiplexer and demultiplexer, as depicted in Fig. 2, are obtained. The Fermi-level of the channel S is smaller (greater) than those of the channels A and B for multiplexing (demultiplexing) operation. The switching function is achievable by utilizing the
facts: (1) the electron wavefunction for an impurity in a QD is movable under applied electric fields (2) the moving distance is proportional to both strengths of the quantum confinement and electric field, a higher confinement and/or a smaller field getting to a smaller moving distance. With the design of a strong confinement to the dots A and B and a weak confinement to the dot S, the moving distance of the wavefunction in the dot S under applied electric field is much greater than those in the dots A and B. This allows us to change the coupling strengths (CS) (or the transmission amplitudes, TA) of dot-S-to-dot-A (S–A) and dot-S-to-dotB (S–B) by justifying the field direction and strength and hence to direct the electron path from dot S to dot A in rightdirection field or from dot S to dot B in opposite direction. In order to demonstrate the ground-state wavefunction behavior of an impurity in an isolated QD, let us consider a simple model of an impurity with its location deviated (along z direction) away from the center of a parabolic QD under applied electric field (directed to z-axis). The Hamiltonian of the system within one-band effective-mass approximation (in the units of the effective Rydberg
Fig. 1. Top view (a) and side view (b) of the structure of single-electron two-channel multiplexer/demultiplexer.
K.-M. Hung, Y.-S. Wu / Solid State Communications 134 (2005) 571–576
573
Fig. 2. Energy profile of single-electron two-channel demultiplexer (a) and multiplexer (b) under biases.
RyZ m e4 =32p2 32 Z2 and the effective Bohr radius a0 Z 4p3Z2 =m e2 ) can be written as 2 H Z KV2 C Vp r 2 C VF z K ~r K~z 0
(1)
where Vp and VF are the strengths of the parabolic confinement potential and the electric-field energy, respectively. They can be related to their conventional forms of Vp Z m u2 a20 =2Ry and VF Z eFa0 =Ry, with the parabola frequency u, electric field F, effective mass m*, electron charge e and the dielectric constant 3. There are two reasons for the consideration that the impurity location is shifted away from the center of the dot along the field direction: (1) the carrier has the greatest change in its binding energy due to the interplays of the field and impurity location (2) the variational integral of total energy is analytically integrable. In such considerations, the trial wavefunction of the dopant’s ground state can be written as the product of the 1s state of hydrogenic atom and the ground state of spherical harmonic oscillator,
2 K ~r K~z 0 J Z C Exp ExpðKl~r C~z F Þ a
(2)
where C is thepnormalization constant, a the variational ffiffiffiffiffi parameter, l h Vp =2 and zFhVF/2Vp. The impurity level is obtained by minimizing the expectation value of the total energy Etot hhJjHjJi. In solving the variational integral hjjHjji, we simply take the transformation ~r /~r C~z 0 such that the Eqs. (1) and (2), then, become H Z KV2 C Vp j~r C~z 0 C~z F j2 K
VF2 2 K 4Vp r
(3)
and J Z C ExpðKr=a K lj~r C~z 0 C~z F j2 Þ
(4)
It is clear as revealed in Eq. (3) that the action of the electric field has the same effect as relatively shifting the center of the confinement potential in a distance zF, while the energy level of the dot decreases in a value of -V2F/4VP. Under the transformation, the variational integral of the total energy
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K.-M. Hung, Y.-S. Wu / Solid State Communications 134 (2005) 571–576
can be solved analytically in the assistance of the package MATHEMATICA for saving a great amount of CPU times. After tedious integration and algebra, the total energy of the system expressed in terms of the kinetic energy Ek, Coulomb energy ECol, the lowest energy Ec of the dot and the field energy EF is given by Etot Z EK C ECol C Ec C EF
(5)
with pffiffiffiffi 1=4 n pVp 2 eKC ð2 K 3aKCVp1=4 ÞerfðKCÞ EK Z 4abN o 2 C eKK 2erfðKKÞ C 3aKKVp1=4 erfðKKÞ C
pffiffiffiffi p Aa 16a2 bVp N KCKK
(6)
pffiffiffiffi 1=4 pVp 2 2 ECol Z ½eKC erfcðKCÞ K eKK erfcðKKKÞ bN
(7)
Ab 1 K pffiffiffiffiffi 16a2 Vp7=4 N a Vp N
Ec Z
pffiffiffiffiffiffiffiffiffi h pVp KK2 2 C e KKð3 C 2KK ÞerfðKKÞ 4bN i 2 2 ÞerfðKCÞ C Vp z20 K eKC KCð3 C 2KC
EF Z
Fig. 3. Binding energy, calculated by the variational method (solidline) and the first-order perturbation approach (broken-line), of the ground state of an impurity in a parabolic QD with respect to the confinement strength VP.
pffiffiffiffi Ab ¼ Uþ p a2 b2 þ 6a2 Vp3=2 þ 12Vp pffiffiffiffiffiffiffiffiffi 2UK pVp 2 2 þ 3a b þ 6a2 Vp3=2 þ 4Vp ab Ac Z 4bVp C 4ab2
pffiffiffiffiffi Vp K a2 4KCVp9=4 K 2bVp3=2 K b3
and (8)
pffiffiffiffi n h 2 p a2 eKK 4Vp9=4 KK K bðb2 C 2Vp3=2 Þ 2 7=4 8a Vp bN i pffiffiffiffiffi 2 C eKC 4Vp9=4 KC K bðb2 C 2Vp3=2 C 4ab Vp
Ad Z 4bVp K 4ab2
pffiffiffiffiffi Vp K a2 4KKVp9=4 K 2bVp3=2 K b3
In these equations, erf and erfc are the error function and complementary error function, respectively. Numerically minimizing the total energy Etot(min), one obtains the ground-state level of the impurity. The binding energy measured pffiffiffiffiffi from the lowest level of the dot is found to be Eb Z 3 Vp K VF2 =4Vp K EtotðminÞ , where the sum of the
! 2Vp3=4 ! pffiffiffiffi K bUK p 2 2 C Ac eKC erfðKCÞ K Ad eKC erfðKKÞ K 4bUCVp C VF z0 (9) and NZ
pffiffiffiffi h i 2 p KC2 e KCerfcðKCÞ C eKK KKerfcðKKKÞ 2b
(10)
The symbols used in Eqs. (6)–(10) are defined as follows: pffiffiffiffiffi abG2 Vp 2 2 ; UG Z eKC GeKK b Z VF C 2Vp z0 ; KG Z 2aVp3=4 2
Aa ¼ KeKK KKð4Vp K 3a2 b2 K 4ab
pffiffiffiffiffi Vp Þ
pffiffiffiffiffi Ke Kþ ð4Vp K 3a b þ 4ab Vp Þ Kþ2
2 2
Fig. 4. Binding energy as a function of impurity position for various values of confinement potential, VPZ1.0, 2.5, 4.0, 5.5 and 7.0 and the field energy VFZ0.
K.-M. Hung, Y.-S. Wu / Solid State Communications 134 (2005) 571–576
Fig. 5. Binding energy as a function of electric field for various values of confinement potential, VPZ1.0, 2.5, 4.0, 5.5 and 7.0 and the impurity location z0Z0.
first two terms is the lowest energy of the QD under electric field [17]. In the following discussions, we apply the derived theory to the system of AlGaAs/GaAs heterostructure ˚ and the with the effective Bohr radius a0y102.7 A effective Rydberg Ryy5.39 meV for GaAs material. Fig. 3 plots the binding energy with respect to the
575
confinement potential in the case of VFZ0 and z0Z0 by using the variational method (solid-line) and the firstorder perturbation approach (broken-line). Notice that the binding energy estimated by the latter approach has pffiffiffiffi the simple form of 4Vp1=4 = p. As shown in the figure, the binding energy increases with increasing the confinement strength. This character also appears in the cases of VFs0 and/or z0s0 as depicted in Figs. 4 and 5. It is mainly attributed to the increases of the wavefunction localization as the rise of the confinement strength. In addition, Figs. 4 and 5 show that the electric field takes the same effect as moved the impurity location along the field direction, as we mentioned above. Based on this simple model, we demonstrate the fieldinduced switching property with the following structure parameters: VPAZVPBZ1Ry (the confinement strengths of the dots A and B), VPSZ0.25Ry (the confinement strength of the dot S) and, z0AZK5a0 and z0BZ5a0 (the centerspof ffiffiffi the dots A and B, respectively). The radius, defined as 1= l, for pffiffiffi the dots A and B is taken to be 2a0 and one for the dot S to be 2a0. In this case, the total particle density within weak coupling region may be approximately estimated by rt Z jJA j2 C jJB j2 C jJS j2 . The calculated results of rt for yZ0 are presented in Fig. 6(a)–(d) with the field energy VF varied from K1.5Ry to C1.5Ry stepped in 1Ry, which corresponds to the electric field F ranging from 7.87!103 to
Fig. 6. Total particle density in the three QDs for yZ0.
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are calculated based on the transfer Hamiltonian approach and the simple model of an impurity in an isolated parabolic QD. The contrast of switching-on and -off currents approaches 106. The theoretical results demonstrate that a lateral switching field can be used to direct the electron path from the centered dot to either of its neighboring dots.
Acknowledgements This work was supported by the Education Ministry, Taiwan, ROC.
References Fig. 7. Transmission amplitudes (in log scale with arbitrary unit) of S–A and S–B versus applied field energy VF.
K7.87!103 V/cm. It is obvious that there is an increase in the amount of wavefunction overlap for S–A with increasing right-direction fields, while one for S–B decreases. Since, the CS is proportional to the wavefunction overlap, the changes of the overlaps the changes of the CSs. As a consequence, the lateral field can direct the electron in the dot S into either of the dots A and B, which depends on the field direction and strength. In order to quantitatively study the variations of the CSs of the dots with respect to the external control fields, we apply the transfer Hamiltonian approach [18,19] on the estimation of the TAs for S–A and S–B. TA can be calculated by the expression ð Z2 jMi/j j2 ¼ j Ji vz Jj K Jj vz Ji dSz j2 (11) 2m S
where the surface integral is evaluated at the intersection (on z-axis) of the confinement potentials of the neighboring dots. Fig. 7 plots the TAs (in log scale) of S–A and S–B with respect to the field energy VF, ranging from K1.5Ry to 1.5Ry. One can see that the TA of S–A just is a mirrored image of S–B due to the symmetric structure and the contrast, that is the ratio of the turn-on to the turn-off currents or the maximum TA to the minimum TA, for both S–A and S–B approaches w106. It is an evidence to show that the proposed configuration forms a good switch for the utilization of multiplexer and demultiplexer. In conclusion, we have proposed a novel structure of SETCMD based on three coupled SDQDs defined on AlGaAs/GaAs heterostructure. The TAs between the dots
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