- Email: [email protected]
Variational wave function for a two-electron quantum dot
Physica B 255 (1998) 145—149
Variational wave function for a two-electron quantum dot A. Harju *, V.A. Sverdlov , B. Barbiellini , R.M. Nieminen Laboratory of Physics, Helsinki University of Technology, SF-02150 Espoo 15, Finland Freie Universita( t Berlin, Berlin, Germany
Abstract We have applied the variational quantum Monte Carlo technique to a two-electron quantum dot. A simple variational wave function is presented and we compare the results obtained using it with exact numerical diagonalizations. The comparison shows that almost exact results are obtained for lowest energy states of different relative angular momenta. 1998 Elsevier Science B.V. All rights reserved. Keywords: Fermions in reduced dimensions; Quantum dots; Correlation
1. Introduction Quantum Monte Carlo (QMC) methods have become increasingly important tools to study correlated many-body systems. QMC has been successfully applied to find the ground state properties of different systems of interacting particles. Progress in growing of artificial atoms or quantum dots (QD) in semiconductor heterostructures with a large but finite number of electrons opens a new perspective in fabrication of nanoelectronic devices. On the other hand a QD as a gigantic atom with variable parameters is an interesting model object
* Corresponding author. E-mail: [email protected]. Permanent address: Department of Theoretical Physics, University of St. Petersburg, Universitetskaya 7/9, 199034 St. Petersburg, Russia. Present address: Solid State group, Physics Department, University of California, Los Angeles, CA 90095, USA.
to study fundamental many-body properties of interacting electrons and thus attracts much theoretical attention. In the present paper we study two interacting electrons in a QD in the presence of a magnetic field. We present a simple form of variation QMC (VMC) trial wave function that gives results in good agreement with the exact ones. The problem of two interacting electrons in a QD has been the subject of intensive investigation by using different methods [1—7]. For us, the two-electron QD is attractive and useful for several reasons. Firstly, one can easily find zeroes of the wave function which appear because of relative angular momentum of the particle pair. Secondly, the function which solves the radial Schro¨dinger equation for the relative motion can be interpreted as the twobody Jastrow correlation factor. It makes it possible to study the dependence of the Jastrow factor on the strength of the Coulomb interaction,
0921-4526/98/$ — see front matter 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 4 6 1 - X
146
A. Harju et al. / Physica B 255 (1998) 145—149
external confinement and magnetic field. This information can then be used in studies of many-electron QDs. In Section 2 we define the model used for the two-electron QD and discuss some general aspects of the wave function. Then in Section 3 we present a trial wave function and compare the QMC results with the exact numerical diagonalization ones.
2. The model and the structure of the wave function Since in experimentally realized dots the extent of the wave function in one direction is much less than in the other two, we assume that electrons move in the plane r"(x, y) under the additional parabolic confinement potential º(r)"m*ur and in the external magnetic field B described by the vector potential A in the symmetrical gauge A"1/2(!By, Bx). The Hamiltonian of the system is
u m*X r H" !
# ¸ # 2m* G 2 X G 2 G G
e # , e"r !r "
(1)
Schro¨dinger equation with the Hamiltonian of Eq. (2) is well known since it is a harmonic oscillator equation. The energy values for the center-ofmass motion are E " X (2n #"m "#1)! !+ !+ !+ (m /2) u , where n "0, 1, 2,2 and m " !+ !+ !+ 0,$1,$2,2. Since we are looking for the ground state energy, we will restrict the center-of-mass motion to the lowest level n , m "0, described !+ !+ by the unnormalized wave function
R W "exp ! , !+ l
where l"( /m*X . Next, we consider the wave function for the relative motion t(r), governed by the Hamiltonian of Eq. (3). Since this Hamiltonian conserves the angular momentum of the electron pair, the wave function t(r) can be written as a product of radial and angular parts: t(r)"g(r)exp(!im ), where m"0,$1,$2,2, is the relative angular momentum quantum number. It is convenient to introduce dimensionless variables by measuring energies in units of X and distances r in units of length l. The equation for the radial wave function of relative motion in the new variables has the form
!
where m* and e are the effective electron mass and charge, respectively, e is the dielectric constant, X"u#u/4, u ""e"B/mc is the cyclotron fre quency, and ¸ is the z-component of the angular X momentum operator. The Hamiltonian of Eq. (1) can be presented as a sum of center-of-mass and relative motion term H"HR#Hr with
u HR"!
R# ¸ #m*XR, 4m* 2 X !+
(2)
u e Hr"! r # ¸ #m*Xr# , X m* 2 er
(3)
where R"(r #r )/2; r"r !r . Because of the separation of variables in the Hamiltonian its eigenvalues E can be written as a sum E" E #E of the center-of-mass energy E and !+ !+ the relative energy E . The solution of the
(4)
1 d d m mu rL l 1 # # rL ! ! g(rL )"eg(rL ), rL 2X 4 a rL rL drL drL (5)
where r rL " , l
e a " , m*e
E e" .
X We are looking for the solution of Eq. (5) which is regular at rL "0 and rL "R. Splitting off the asymptotic behavior at rL PR by the substitution
rL g(rL )"exp ! m(rL ), 4 we obtain the following equation for m(rL ):
1 1 1 m !m # rL ! m # # m rL a rL rL mu !1 m. " e# 2X
A. Harju et al. / Physica B 255 (1998) 145—149
Next, the substitution m"rL K g with g a regular function at r"0 leads to the equation for g:
3. Trial wave function and results We build the variational wave function by approximating the function g by a simple Jastrow function J with two parameters a and b:
2"m"#1 l 1 g # !rL g ! g rL a rL
mu !1!"m" g"0. # e# 2X
(6)
The spatial part of the total wave function of the Hamiltonian of Eq. (1) can now be written in the form:
147
"z "#"z " , W(r , r )"(z !z )"m"g(rL )exp ! 2l
(7)
where z "x !iy , j"1, 2, and we will consider H H H the states with m*0 since the energy of those states is less than the energy of states with m " !m, mO0. If one restricts the Hilbert space of relative motion to the lowest Landau level, the total wave function has the form (7) with function g(rL )"1. Thus, the function g takes into account the mixing of different Landau levels for relative motion. On the other hand, the function g depends on the inter-electron distance and has a similar role as a two-body Jastrow correlation factor has in the usual VMC decomposition of the total many-body wave function. From this point of view, the mixing of the Landau levels for relative motion can be seen as correlation in a QD. The total wave function which is a product of coordinate and spin parts must be anti-symmetrical with respect to the permutation of electrons. It then follows that if the electrons have the same spin projections, the spin part is symmetrical under the permutation whereas the coordinate wave function W is anti-symmetrical: W(r , r )"!W(r , r ). If the electrons have opposite spins, the coordinate wave function is symmetrical under the coordinates permutation: W(r , r )"W(r , r ). The form of the coordinate function (7) provides a simple connection between the relative angular momentum of the pair of electrons and its spin projections: the relative angular momentum is equal to m"(2l#1) for spin-polarized and m"2l for spin-unpolarized two-electron states (l"0, 1, 2,2).
J(rL )"exp
l arL . a (2"m"#1)(1#brL )
(8)
The Jastrow factor fits the cusp condition if the parameter a"1. The parameter b provides the best shape of the Jastrow function at relatively small values of rL in order to obtain the minimal value of energy. One should note that the Jastrow ansatz (8) does not have the right asymptotic behavior as compared with exact solutions at rL PR. On the other hand, it has less influence in the energy because of the additional exponential decrease in the wave function (7) at large rL , which is not present at small rL , where the Jastrow factor has the correct behavior. Successful application of VMC for finding the ground state energy of many-electron system requires, besides the good trial wave function, an efficient optimization scheme. The stochastic gradient approximation (SGA) applied to different systems shows an excellent performance in finding the minimum with respect to the parameters of the trial wave function [8]. The method utilizes the noise inherent in stochastic functions. To demonstrate the efficiency of the presented form of the wave function, we consider two strengths of the Coulomb interaction, namely 0.5 and 10.0. The comparisons of exact diagonalization results and VMC for these two strengths of Coulomb interaction and for different values of relative angular momentum are shown in Figs. 1 and 2. We do not see any deviations from the exact diagonalization results within the computational error. Thus, the SGA optimization scheme applied to the wave function with ansatz (8) shows excellent performance for calculating the lowest energy values for fixed relative angular momentum for two interacting electrons in a QD in the region, where neither the convenient perturbation theory nor the quasi-classical approximation could be applied. It is interesting to compare the energies obtained previously with their values with J(rL )"1. This
148
A. Harju et al. / Physica B 255 (1998) 145—149 Table 1 Jastrow and LLL total energies as a function of the angular momentum m for a two-electron quantum dot for the strength of Coulomb interaction equal to 0.5 and to 10.0. The statistical uncertainty is one for the last digit shown for the Jastrow energies
Fig. 1. Total energy as a function of the total angular momentum, with the Coulomb interaction strength 0.5. The VMC results are indistinguishable from the exact numerical diagonalizations.
m
LLL (0.5)
Jastrow (0.5)
LLL (10)
Jastrow (10)
0 1 2 3 4 5 6 7 8
2.6267 2.3133 2.2350 2.1958 2.1714 2.1542 2.1414 2.1313 2.1231
2.553 2.305 2.232 2.194 2.1705 2.1536 2.1409 2.1309 2.1228
14.5331 8.2666 6.6999 5.9166 5.4270 5.0843 4.8273 4.6254 4.4613
7.384 6.528 5.918 5.478 5.148 4.892 4.687 4.5187 4.3775
exactly the mixing with higher Landau levels for a two-electron QD. We have tested the performance of the simple Jastrow factor for a larger number of electrons. For example, in the three-electron quantum dot with GaAs parameters, the simple Jastrow factor added to the proper LLL wave function is able to take care of nearly all (from 96% up to 98%) of the Landau level mixing [9].
4. Conclusions
Fig. 2. Same as Fig. 1 but Coulomb interaction equal to 10.0.
corresponds to the case when the Hilbert space for relative motion is restricted to the lowest Landau level (LLL). As seen from Table 1, the restriction to the LLL is a good approximation for large relative angular momenta of the pair of particles when the Coulomb strength is about unity. On the other hand, the LLL wave function gives a rather qualitative description at larger Coulomb strengths and relatively small angular momenta. The introduction of a Jastrow factor allows to capture nearly
In summary, we have shown that excellent agreement with exact numercial results for a two-electron quantum dot can be obtained by a simple variational wave function. The Jastrow factor proposed allows to capture nearly exactly the mixing with higher Landau levels for a two-electron QD. It shows the efficiency of the VMC in obtaining the ground state properties of the many-body electron system in a QD. In addition, the good results obtained strongly indicate the possibilities of VMC in the study of larger QDs.
Acknowledgements We would like to thank Veikko Halonen and Pekka Pietila¨inen for the exact diagonalization
A. Harju et al. / Physica B 255 (1998) 145—149
results and helpful conversations. One of us (V.A.S.) thanks the Laboratory of Physics, Helsinki University of Technology and the Institute for Theoretical Physics, Freie Universita¨t Berlin for hospitality. V.A.S. is supported by the Grant for Young Scientists of the Ministry of Education of Russia and the Grant for Young Researchers of the Government of St. Petersburg.
References [1] R.B. Laughlin, Phys. Rev. B 27 (1983) 3383. [2] D. Pfannkuche, V. Gudmundsson, P.A. Maksym, Phys. Rev. B. 47 (1993) 2244.
149
[3] M.J. Taut, J. Phys. A 27 (1994) 1045. [4] Ya.M. Blanter, N.E. Kaputkina, Yu.E. Lozovik, Physica Scripta 54 (1996) 539. [5] J.-L. Zhu, J.-Z. Yu, Z.-Q. Li, Y. Kawazoe, J. Phys. 8 (1996) 7857. [6] F. Bolton, Phys. Rev. B 54 (1996) 4780. [7] M. Dineykhan, R.G. Nazmitdinov, Phys. Rev. B 55 (1997) 20. [8] A. Harju, B. Barbiellini, S. Siljama¨ki, R.M. Nieminen, G. Ortiz, Phys. Rev. Lett. 79 (1997) 1173. [9] A. Harju, V.A. Sverdlov, R.M. Nieminen, Europhys. Lett. 41 (1998) 407.
Variational wave function for a two-electron quantum dot A. Harju *, V.A. Sverdlov , B. Barbiellini , R.M. Nieminen Laboratory of Physics, Helsinki University of Technology, SF-02150 Espoo 15, Finland Freie Universita( t Berlin, Berlin, Germany
Abstract We have applied the variational quantum Monte Carlo technique to a two-electron quantum dot. A simple variational wave function is presented and we compare the results obtained using it with exact numerical diagonalizations. The comparison shows that almost exact results are obtained for lowest energy states of different relative angular momenta. 1998 Elsevier Science B.V. All rights reserved. Keywords: Fermions in reduced dimensions; Quantum dots; Correlation
1. Introduction Quantum Monte Carlo (QMC) methods have become increasingly important tools to study correlated many-body systems. QMC has been successfully applied to find the ground state properties of different systems of interacting particles. Progress in growing of artificial atoms or quantum dots (QD) in semiconductor heterostructures with a large but finite number of electrons opens a new perspective in fabrication of nanoelectronic devices. On the other hand a QD as a gigantic atom with variable parameters is an interesting model object
* Corresponding author. E-mail: [email protected]. Permanent address: Department of Theoretical Physics, University of St. Petersburg, Universitetskaya 7/9, 199034 St. Petersburg, Russia. Present address: Solid State group, Physics Department, University of California, Los Angeles, CA 90095, USA.
to study fundamental many-body properties of interacting electrons and thus attracts much theoretical attention. In the present paper we study two interacting electrons in a QD in the presence of a magnetic field. We present a simple form of variation QMC (VMC) trial wave function that gives results in good agreement with the exact ones. The problem of two interacting electrons in a QD has been the subject of intensive investigation by using different methods [1—7]. For us, the two-electron QD is attractive and useful for several reasons. Firstly, one can easily find zeroes of the wave function which appear because of relative angular momentum of the particle pair. Secondly, the function which solves the radial Schro¨dinger equation for the relative motion can be interpreted as the twobody Jastrow correlation factor. It makes it possible to study the dependence of the Jastrow factor on the strength of the Coulomb interaction,
0921-4526/98/$ — see front matter 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 4 6 1 - X
146
A. Harju et al. / Physica B 255 (1998) 145—149
external confinement and magnetic field. This information can then be used in studies of many-electron QDs. In Section 2 we define the model used for the two-electron QD and discuss some general aspects of the wave function. Then in Section 3 we present a trial wave function and compare the QMC results with the exact numerical diagonalization ones.
2. The model and the structure of the wave function Since in experimentally realized dots the extent of the wave function in one direction is much less than in the other two, we assume that electrons move in the plane r"(x, y) under the additional parabolic confinement potential º(r)"m*ur and in the external magnetic field B described by the vector potential A in the symmetrical gauge A"1/2(!By, Bx). The Hamiltonian of the system is
u m*X r H" !
# ¸ # 2m* G 2 X G 2 G G
e # , e"r !r "
(1)
Schro¨dinger equation with the Hamiltonian of Eq. (2) is well known since it is a harmonic oscillator equation. The energy values for the center-ofmass motion are E " X (2n #"m "#1)! !+ !+ !+ (m /2) u , where n "0, 1, 2,2 and m " !+ !+ !+ 0,$1,$2,2. Since we are looking for the ground state energy, we will restrict the center-of-mass motion to the lowest level n , m "0, described !+ !+ by the unnormalized wave function
R W "exp ! , !+ l
where l"( /m*X . Next, we consider the wave function for the relative motion t(r), governed by the Hamiltonian of Eq. (3). Since this Hamiltonian conserves the angular momentum of the electron pair, the wave function t(r) can be written as a product of radial and angular parts: t(r)"g(r)exp(!im ), where m"0,$1,$2,2, is the relative angular momentum quantum number. It is convenient to introduce dimensionless variables by measuring energies in units of X and distances r in units of length l. The equation for the radial wave function of relative motion in the new variables has the form
!
where m* and e are the effective electron mass and charge, respectively, e is the dielectric constant, X"u#u/4, u ""e"B/mc is the cyclotron fre quency, and ¸ is the z-component of the angular X momentum operator. The Hamiltonian of Eq. (1) can be presented as a sum of center-of-mass and relative motion term H"HR#Hr with
u HR"!
R# ¸ #m*XR, 4m* 2 X !+
(2)
u e Hr"! r # ¸ #m*Xr# , X m* 2 er
(3)
where R"(r #r )/2; r"r !r . Because of the separation of variables in the Hamiltonian its eigenvalues E can be written as a sum E" E #E of the center-of-mass energy E and !+ !+ the relative energy E . The solution of the
(4)
1 d d m mu rL l 1 # # rL ! ! g(rL )"eg(rL ), rL 2X 4 a rL rL drL drL (5)
where r rL " , l
e a " , m*e
E e" .
X We are looking for the solution of Eq. (5) which is regular at rL "0 and rL "R. Splitting off the asymptotic behavior at rL PR by the substitution
rL g(rL )"exp ! m(rL ), 4 we obtain the following equation for m(rL ):
1 1 1 m !m # rL ! m # # m rL a rL rL mu !1 m. " e# 2X
A. Harju et al. / Physica B 255 (1998) 145—149
Next, the substitution m"rL K g with g a regular function at r"0 leads to the equation for g:
3. Trial wave function and results We build the variational wave function by approximating the function g by a simple Jastrow function J with two parameters a and b:
2"m"#1 l 1 g # !rL g ! g rL a rL
mu !1!"m" g"0. # e# 2X
(6)
The spatial part of the total wave function of the Hamiltonian of Eq. (1) can now be written in the form:
147
"z "#"z " , W(r , r )"(z !z )"m"g(rL )exp ! 2l
(7)
where z "x !iy , j"1, 2, and we will consider H H H the states with m*0 since the energy of those states is less than the energy of states with m " !m, mO0. If one restricts the Hilbert space of relative motion to the lowest Landau level, the total wave function has the form (7) with function g(rL )"1. Thus, the function g takes into account the mixing of different Landau levels for relative motion. On the other hand, the function g depends on the inter-electron distance and has a similar role as a two-body Jastrow correlation factor has in the usual VMC decomposition of the total many-body wave function. From this point of view, the mixing of the Landau levels for relative motion can be seen as correlation in a QD. The total wave function which is a product of coordinate and spin parts must be anti-symmetrical with respect to the permutation of electrons. It then follows that if the electrons have the same spin projections, the spin part is symmetrical under the permutation whereas the coordinate wave function W is anti-symmetrical: W(r , r )"!W(r , r ). If the electrons have opposite spins, the coordinate wave function is symmetrical under the coordinates permutation: W(r , r )"W(r , r ). The form of the coordinate function (7) provides a simple connection between the relative angular momentum of the pair of electrons and its spin projections: the relative angular momentum is equal to m"(2l#1) for spin-polarized and m"2l for spin-unpolarized two-electron states (l"0, 1, 2,2).
J(rL )"exp
l arL . a (2"m"#1)(1#brL )
(8)
The Jastrow factor fits the cusp condition if the parameter a"1. The parameter b provides the best shape of the Jastrow function at relatively small values of rL in order to obtain the minimal value of energy. One should note that the Jastrow ansatz (8) does not have the right asymptotic behavior as compared with exact solutions at rL PR. On the other hand, it has less influence in the energy because of the additional exponential decrease in the wave function (7) at large rL , which is not present at small rL , where the Jastrow factor has the correct behavior. Successful application of VMC for finding the ground state energy of many-electron system requires, besides the good trial wave function, an efficient optimization scheme. The stochastic gradient approximation (SGA) applied to different systems shows an excellent performance in finding the minimum with respect to the parameters of the trial wave function [8]. The method utilizes the noise inherent in stochastic functions. To demonstrate the efficiency of the presented form of the wave function, we consider two strengths of the Coulomb interaction, namely 0.5 and 10.0. The comparisons of exact diagonalization results and VMC for these two strengths of Coulomb interaction and for different values of relative angular momentum are shown in Figs. 1 and 2. We do not see any deviations from the exact diagonalization results within the computational error. Thus, the SGA optimization scheme applied to the wave function with ansatz (8) shows excellent performance for calculating the lowest energy values for fixed relative angular momentum for two interacting electrons in a QD in the region, where neither the convenient perturbation theory nor the quasi-classical approximation could be applied. It is interesting to compare the energies obtained previously with their values with J(rL )"1. This
148
A. Harju et al. / Physica B 255 (1998) 145—149 Table 1 Jastrow and LLL total energies as a function of the angular momentum m for a two-electron quantum dot for the strength of Coulomb interaction equal to 0.5 and to 10.0. The statistical uncertainty is one for the last digit shown for the Jastrow energies
Fig. 1. Total energy as a function of the total angular momentum, with the Coulomb interaction strength 0.5. The VMC results are indistinguishable from the exact numerical diagonalizations.
m
LLL (0.5)
Jastrow (0.5)
LLL (10)
Jastrow (10)
0 1 2 3 4 5 6 7 8
2.6267 2.3133 2.2350 2.1958 2.1714 2.1542 2.1414 2.1313 2.1231
2.553 2.305 2.232 2.194 2.1705 2.1536 2.1409 2.1309 2.1228
14.5331 8.2666 6.6999 5.9166 5.4270 5.0843 4.8273 4.6254 4.4613
7.384 6.528 5.918 5.478 5.148 4.892 4.687 4.5187 4.3775
exactly the mixing with higher Landau levels for a two-electron QD. We have tested the performance of the simple Jastrow factor for a larger number of electrons. For example, in the three-electron quantum dot with GaAs parameters, the simple Jastrow factor added to the proper LLL wave function is able to take care of nearly all (from 96% up to 98%) of the Landau level mixing [9].
4. Conclusions
Fig. 2. Same as Fig. 1 but Coulomb interaction equal to 10.0.
corresponds to the case when the Hilbert space for relative motion is restricted to the lowest Landau level (LLL). As seen from Table 1, the restriction to the LLL is a good approximation for large relative angular momenta of the pair of particles when the Coulomb strength is about unity. On the other hand, the LLL wave function gives a rather qualitative description at larger Coulomb strengths and relatively small angular momenta. The introduction of a Jastrow factor allows to capture nearly
In summary, we have shown that excellent agreement with exact numercial results for a two-electron quantum dot can be obtained by a simple variational wave function. The Jastrow factor proposed allows to capture nearly exactly the mixing with higher Landau levels for a two-electron QD. It shows the efficiency of the VMC in obtaining the ground state properties of the many-body electron system in a QD. In addition, the good results obtained strongly indicate the possibilities of VMC in the study of larger QDs.
Acknowledgements We would like to thank Veikko Halonen and Pekka Pietila¨inen for the exact diagonalization
A. Harju et al. / Physica B 255 (1998) 145—149
results and helpful conversations. One of us (V.A.S.) thanks the Laboratory of Physics, Helsinki University of Technology and the Institute for Theoretical Physics, Freie Universita¨t Berlin for hospitality. V.A.S. is supported by the Grant for Young Scientists of the Ministry of Education of Russia and the Grant for Young Researchers of the Government of St. Petersburg.
References [1] R.B. Laughlin, Phys. Rev. B 27 (1983) 3383. [2] D. Pfannkuche, V. Gudmundsson, P.A. Maksym, Phys. Rev. B. 47 (1993) 2244.
149
[3] M.J. Taut, J. Phys. A 27 (1994) 1045. [4] Ya.M. Blanter, N.E. Kaputkina, Yu.E. Lozovik, Physica Scripta 54 (1996) 539. [5] J.-L. Zhu, J.-Z. Yu, Z.-Q. Li, Y. Kawazoe, J. Phys. 8 (1996) 7857. [6] F. Bolton, Phys. Rev. B 54 (1996) 4780. [7] M. Dineykhan, R.G. Nazmitdinov, Phys. Rev. B 55 (1997) 20. [8] A. Harju, B. Barbiellini, S. Siljama¨ki, R.M. Nieminen, G. Ortiz, Phys. Rev. Lett. 79 (1997) 1173. [9] A. Harju, V.A. Sverdlov, R.M. Nieminen, Europhys. Lett. 41 (1998) 407.