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Binding energy of hydrogenic impurities in a quantum well under the tilted magnetic field
Solid State Communications 125 (2003) 429–434 www.elsevier.com/locate/ssc
Binding energy of hydrogenic impurities in a quantum well under the tilted magnetic field E. Kasapoglua,*, H. Sarıa, I. So¨kmenb a
Department of Physics, Cumhuriyet University, 58140 Sivas, Turkey b Department of Physics, Dokuz Eylu¨l University, I˙zmir, Turkey
Received 31 October 2002; received in revised form 8 November 2002; accepted 10 November 2002 by K.-A. Chao
Abstract This paper treats theoretically the angle dependence of the ground state binding energy of a shallow donor impurity in semiconductor quantum-well systems on the tilted magnetic field. By making an appropriate coordinate transform we have calculated the ground state binding energy of a shallow donor impurity at the center of GaAs/Ga12xAlxAs quantum well in the effective-mass approximations and variationally. We show that the binding energy depends strongly not only on quantum ˚ , the change of the binding energy confinement, but also on the direction of the magnetic field. For example; for L0 ¼ 100 A between u ¼ 15 and 458 approximately is 2:5Ry (,13 meV). We expect that this change will be useful in designing the quantum-well structure in which the impurity effects play important role. q 2003 Elsevier Science Ltd. All rights reserved. PACS: 71.55.Eq; 71.55. 2 i Keywords: A. Quantum wells; C. Impurities in semiconductors
1. Introduction With the development of several experimental techniques, such as molecular beam epitaxy and metal organic chemical – vapor deposition, there has been a lot of work devoted to the understanding of hydrogenic impurity states in low-dimensional semiconductor heterostructures such as quantum wells [1– 6], quantum-well wires [7– 13], and quantum dots [14– 16]. Studies of semiconductor multilayer quasi-two dimensional system as well as single quantum wells of a GaAs/GaAlAs crystal type shows that the carriers caught by impurity centers effect essentially on the electronic properties of such system. Magnetic or electric fields are effective tools for studying these properties. A number of papers are devoted to the theoretical studies of the impurity states in the quantum wells when the external fields are applied. The use of the tilted magnetic fields is of * Corresponding author. Tel.: þ90-346-21910101937; fax: þ 90346-219 11 86. E-mail addresses: [email protected] (H. Sari), [email protected] (E. Kasapoglu).
interest theoretically as it illustrates confinement effects. If the magnetic field is tilted in respect to the interface, the variables in Schro¨dinger equation cannot be seperated and variational [17,18] or perturbation [19,20] methods have been used. So far only the eigenenergies of two-dimensional electrons subjected to a tilted magnetic field have been solved analytically using a parabolic potential well [21]. In our previous studies [22,23], however, we have completely solved the Schro¨dinger equation using a square well potential as confining potential and obtained analytical solutions without making any approximations for twodimensional semiconductor heterostructures under the tilted magnetic field. In this study, we report a calculation, with the use of a variational approximation, of the ground state binding energy of a hydrogenic donor impurity at the center of a GaAs quantum well in the presence of a magnetic field applied tilted to the growth direction. To solve the Schro¨dinger equation we apply an orthogonal transformation, and then we use a tricky substitution into the potential, that makes the Hamiltonian seperable in terms of the new coordinates. The general solution smoothly goes to the
0038-1098/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 8 - 1 0 9 8 ( 0 2 ) 0 0 7 7 4 - 3
430
E. Kasapoglu et al. / Solid State Communications 125 (2003) 429–434
results of two limits where the magnetic field is either parallel or perpendicular to the layers. Our results are given as a function of the well width, magnetic field strength and tilt angle. An interaction of carrier with the impurity centre is considered to be one of the Coulomb potential type.
We define the z-axis to be along the growth axis, and take the magnetic field to be applied in the x– z plane at angle-u to the x-axis. We choose a gauge for the magnetic field in which the vector potential A is written form A ¼ ð0; xB sin u 2 zB cos u; 0Þ using the 7·A ¼ 0 gauge, where B ¼ ðB cos u; 0; B sin uÞ and u is the angle between the direction of the magnetic field and x-axis. Within the framework of an effective-mass approximation, the Hamiltonian of a hydrogenic donor in a GaAs quantum well, in the presence of an applied magnetic field, can be written as 1 e~ 2 e2 H¼ p~ þ A 2 ð1Þ þ Vðze Þ; 2me c 10 l~re 2 ~ri l where me is the effective mass, e is the elementary charge, p~ is the momentum, 10 is the dielectric constant, and Vðze Þ is the confinement potential profile for the electron in the z-direction. The functional form of the confinement potential is given as Vðze Þ ¼ V0 ½SðzL 2 ze Þ þ Sðze 2 zR Þ;
ð2Þ
where S is the step function, and the left and right boundaries of the well are located at z ¼ zL ¼ 2L0 =2 and z ¼ zR ¼ L0 =2; respectively. By using the following transformation, ! ! ! cos u 2 sin u z z0 ¼ ð3Þ 0 x sin u cos u x the Hamiltonian can be written as below 1 1 2 e2 B2 0 2 z þ Vðx0e ; z0e Þ ðp2x0 þ p2z0 Þ þ py þ 2me 2me 2me c
ð6Þ
e 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 10 ðx0e 2 x0i Þ2 þ ðy 2 yi Þ2 þ ðz0e 2 z0i Þ2
By considering the above equations, we can separate the potential as (see Appendix A) Vðx0e ; z0e Þ ¼ Vðx0e Þ þ Vðz0e Þ
ð4Þ
where Eq. (4) does not contain the term ðeB=me cÞz0 py because, the expectation value of this term is identically zero for the chosen trial wave function in Eq. (11). After the coordinate transformation, the left and right boundaries of wells on the x0 and z0 axes are ð5Þ
z0L;R ¼ zL;R cos u 2 x sin u; respectively. The solution of the corresponding Schro¨dinger equation is not straightforward since, after the coordinate
ð7Þ
where Vðx0e Þ ¼ V0 sin2 u½Sðx0L 2 x0e Þ þ Sðx0e 2 x0R Þ;
ð8Þ
Vðz0e Þ ¼ V0 cos2 u½Sðz0L 2 z0e Þ þ Sðz0e 2 z0R Þ: Notice that applied magnetic field is parallel to the growth direction at the u ¼ 908 and the electron becomes free in the z0 direction and the eigenvalues do not depend on z0 , and that applied magnetic field is perpendicular to the growth direction at u ¼ 08 and the electron becomes free in the x0 direction and the eigenvalues do not depend on x0 . As known, for these values of u, the Schro¨dinger equation can be solve exactly and we not need such a transformation to solve the problem. By scaling all lengths in effective Bohr radius ðaB ¼ 10 "2 =me e2 Þ; and energies in effective Rydberg ðRy ¼ me e4 =2120 "2 Þ; and considering above results, we can rewrite the dimensionless Hamiltonian of the system as, 2 2 2 2 2 ~ ¼ 2 d þ Vð~ ~ x0e Þ 2 d þ Vð~ ~ z0e Þ þ e B " z~0 2 H 0 2 0 2 d~x d~z 4m2e c2 R2y
2
d2 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; d~y2 x~ 0e 2 þ y~ 2 þ z~0e 2
ð9Þ
where x~ 0i ; y~ 0i and z~0i is equal the zero since, donor impurity is located on center of the well. We propose the following variational trial wave function for the electron bound to impurity
c ¼ cð~x0 Þcð~z0 Þwðy; aÞ
2
x0L;R ¼ zL;R sin u þ x cos u;
SðzL 2 ze Þ ¼ cos2 uSðz0L 2 z0e Þ þ sin2 uSðx0L 2 x0e Þ; Sðze 2 zR Þ ¼ cos2 uSðz0e 2 z0R Þ þ sin2 uSðx0e 2 x0R Þ:
2. Theory
H¼
transformation the potential energy of the electron in the well Vðx0e ; z0e Þ couples x0 and z0 variables. In order to decompose the potential energy of the electron, we rewrite the step functions in Eq. (2) as follows:
ð10Þ
where the wave function in the y-direction wðy; aÞ is chosen to be Gaussian-type orbital function: 1 2 1=4 2y2 =a2 wðy; aÞ ¼ pffiffi e ; ð11Þ a p in which a is a variational parameter, cðx0 Þ is the wave function of the electron in the x0 direction which is exactly obtained from the Schro¨dinger equation in the x0 direction. cðz0 Þ; the wave function of the electron in the z0 direction. To solve the Schro¨dinger equation in the z0 direction, we choose as base the eigenfunction of the infinite potential well with the Lb width. We have also used this technique in our previous studies [22,23]. In calculating the wave functions cðz0 Þ; we have ensured that the eigenvalues are independent
E. Kasapoglu et al. / Solid State Communications 125 (2003) 429–434
431
of the choice infinite potential well width Lb and that the wave functions are localized in the well region. The total energy of the system is evaluated by minimizing the expectation value of the Hamiltonian in Eq. (9) with respect to a: ~ cl ¼ E: ~ min kclHl
ð12Þ
a
The binding energy of the donor impurity ground state is given by ~ E~ B ¼ E~ 0 2 E;
ð13Þ
where E~ 0 is the lowest electron total subband energy in the x0 and z0 directions, respectively. Substituting the expectation value of the Hamiltonian into the Eq.(12), we get the ground state binding energy of the donor impurity.
3. Results and discussions The values of the physical parameters used in our calculations are me ¼ 0:0665m0 (m0 is the free electron mass), 10 ¼ 12:58 (static dielectric constant is assumed to be same GaAs and GaAlAs), V0 ¼ 228 meV. These parameters are suitable in GaAs/Ga12xAlxAs heterostructures with an Al concentration of x ø 0:3: Without losing generality, and for simplicity in numerical calculations, we have chosen the boundaries of the well at x0LðRÞ ¼ ^L0 =2 sin u and z0LðRÞ ¼ ^L0 =2 cos u; and after the coordinate transformation which satisfies the following equation:
Fig. 1. The variation of the binding energy of the ground state for a donor at the center of a GaAs quantum well as a function of the well width for u ¼ 158 and three different magnetic field values.
ð14Þ
it can be observed that, for strong magnetic fields the binding energy reaches a constant value for large well width. The results in the large L0 limit and for magnetic fields in the experimental range are compared with the hydrogenic atom limit [24]. The comparison shows that in the range studied the present calculation is quite accurate. In Fig. 2, we show the variation of the impurity binding
derived from Eq. (5). In Fig. 1, we display the variation of the binding energy of the ground state for a donor at the center of a GaAs quantum well as a function of the well width for different magnetic field values and u ¼ 158: As seen in this figure impurity binding energy increases as the well size increases as the independent of all magnetic field values, since the geometric confinement predominates at small L0 values ˚ ). For B ¼ 1T; the binding energy (100 # L0 # 200 A increases as L0 increases and reaches a maximum value. Where the binding energy is maximum the system has quasi-two-dimensional character. After the certain L0 value ˚ ), impurity binding energy decreases as L0 (L0 ø 250 A increases, since the confinement of the electron in the z0 direction decreases i.e. the influence of the Coulomb field of the impurity center on the electron weakens. This behaviour reproduces several results previously reported [3,4,11]. For ˚ , at large magnetic field values, magnetic L0 . 200 A confinement becomes stronger and the impurity binding energy increases as the magnetic field increases. In this limit, the extension of the wave function in the plane which is perpendicular to the magnetic field is determined primarily by the magnetic field and the barrier potential is a small perturbation on the magnetic term. Also from Fig. 1
Fig. 2. The variation of the binding energy of the ground state for a donor at the center of a GaAs quantum well as a function of the well width for u ¼ 308 and three different magnetic field values.
z0R 2 z0L ¼ L0 cos u;
x0R 2 x0L ¼ L0 sin u;
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energy versus the well width for u ¼ 308 and different magnetic field values. Impurity binding energy decreases for all magnetic field values as L0 increases. When we compare the results are obtained in this case with that of u ¼ 158, we see that the binding energy of impurity at the center of the well increases with increasing tilt angle, since the localisation of the electron increases in both x0 and z0 direction, and the electronic probability density around the impurity is higher than in the previous case. In Fig. 3, we present the variation of the impurity binding energy versus the well width for u ¼ 458 and different magnetic field values. For u ¼ 458; binding energy becomes maximum, since the effective well widths and potential heights of electron in both x0 and z0 directions are equal, electron is under the effect of the same geometric confinement in both directions. If we compare the results of the binding energy for u ¼ 458 with that of u ¼ 158; we see that the binding energy changes from , 1:5Ry to , 4Ry ˚ and all magnetic field values. So, tilt angle is for L0 ¼ 100 A a good tunable parameter providing a change on the impurity binding energy for especially small L0 values. In Figs. 4 and 5, we present the variation of the impurity binding energy versus the well width for u ¼ 608 and u ¼ 758; respectively. For these tilt angles, the well width in which the binding energy begins to be sensitive to the magnetic field becomes smaller than the previous tilt angles. This case is evident in Fig. 5. In these tilt angles values, the well width in which the electron is confined in the z0 direction became so small that the electron is delocalised in all L0 values and an interaction of the electron with the impurity centre is completely provided with the magnetic confinement.
The variation of the impurity binding energy versus the ˚ tilt angle for different magnetic field values and L0 ¼ 200 A is given In Fig. 6. As seen in this figure, in the range of 0 # u # 458 impurity binding energy increases as tilt angle increases and reaches a maximum value at u ¼ 458; and then in the range of 45 # u # 908 the binding energy decreases up to u ¼ 758 and for further tilt angle value it converges to a constant value. The observations are that; in the range
Fig. 3. The variation of the binding energy of the ground state for a donor at the center of a GaAs quantum well as a function of the well width for u ¼ 458 and three different magnetic field values.
Fig. 5. The variation of the binding energy of the ground state for a donor at the center of a GaAs quantum well as a function of the well width for u ¼ 758 and three different magnetic field values.
Fig. 4. The variation of the binding energy of the ground state for a donor at the center of a GaAs quantum well as a function of the well width for u ¼ 608 and three different magnetic field values.
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these results will be of importance in the understanding of experimental studies related with donor impurities in GaAs quantum wells under the external tilted magnetic field.
Appendix A In order to decompose the potential energy Vðx0e ; z0e Þ which is defined in Eq. (4), let us introduce a function f ðx0 ; z0 Þ ¼ cos2 uSðz0L 2 z0 Þ þ sin2 uSðx0L 2 x0 Þ 0
ðA1Þ
0
By using the variables x , and z in terms of x and z we can write the step function Sðz0L 2 z0 Þ as follows: Sðz0L 2 z0 Þ ¼ SðzL cos u 2 x sin u 2 z cos u þ x sin uÞ ¼ SððzL 2 zÞcos uÞ
Fig. 6. The variation of the binding energy of the ground state for a donor at the center of a GaAs quantum well as a function of the tilt angle for three different magnetic field values and well width˚. L0 ¼ 200 A
0 # u # 458, as the localization of the donor electron in the z0 direction and Coulombic interaction increases, the binding energy increases, and in the range of 45 # u # 908, the well width becomes smaller in the z0 direction and interaction decreases between donor electron and ion as delocalisation for donor electron begins, and thus the binding energy also decreases. It is important to observe that very similar magnetic field dependence has the ground state exciton binding energy in quantum wells [22]. As a consequence of this analysis it can be seen that the direction of the magnetic field plays an essential role in the determination of the binding energy. In conclusion, this paper presents the solution of the Schro¨dinger equation of a square well potential problem under the influence of an externally applied tilted magnetic field by making the Hamiltonian separable via a coordinate transformation. In this study, we calculated by using a variational approximation the ground state impurity binding energy in the single square quantum well under the externally applied tilted magnetic field as a function of the tilt angle, magnetic field and the well width. It is seen that the direction of the magnetic field causes important changes ˚ , the in the binding energy. For example; for L0 ¼ 100 A change of the binding energy between u ¼ 15 and 458 approximately is 2:5Ry (, 13 meV). So, we can say that the quantum well structure is reduced to the quantum wire from the results are obtained at u ¼ 458, since the system is under the effect of the same geometric confinement in both x0 and z0 directions. To the best of our knowledge, so far studies about the shallow donor impurity binding energy under the tilted magnetic field have not been reported. We expect that
ðA2Þ
and since cos u . 0; one can rewrite the last term in Eq. (A2) as SððzL 2 zÞcos uÞ ¼ SððzL 2 zÞÞ: Similarly, by considering the above results we can write Sðx0L 2 x0 Þ ¼ SððzL 2 zÞsin uÞ ¼ SððzL 2 zÞÞ: Thus f ðx0 ; z0 Þ takes the following form: f ðx0 ; z0 Þ ¼ ðcos2 u þ sin2 uÞ SðzL 2 zÞ ¼ SðzL 2 zÞ
ðA3Þ
By using the same procedure for Sðz 2 zR Þ we can write the step function as in Eq. (6), and by considering the above equations we can decompose the potential Vðx0e ; z0e Þ as Eq. (7).
References [1] G. Bastard, Phys. Rev. B 24 (1981) 4714. [2] C. Mailhiot, Y.C. Chang, T.C. McGill, Phys. Rev. B 26 (1982) 4449. [3] S. Chaudhuri, K.K. Bajaj, Phys. Rev. B 29 (1984) 1803. [4] R.L. Greene, K.K. Bajaj, Phys. Rev. B 31 (1985) 913. [5] B.S. Monozon, P. Schmelcher, J. Phys.: Condens. Matter 13 (2001) 3727. [6] M. Pacheco, Z. Barticevic, A. Latge, Physica B 302–303 (2001) 77. [7] J.W. Brown, H.N. Spector, J. Appl. Phys. 59 (1986) 1179. [8] G.W. Bryant, Phys. Rev. B 29 (1984) 6632. [9] N. Porras-Montenegro, J. Lopez-Gondar, L.E. Oliveira, Phys. Rev. B 43 (1991) 1824. [10] N. Porras-Montenegro, A. Latge, S.T. Perez-Merchancano, Phys. Rev. B 46 (1992) 9780. [11] M. Ulas¸, H. Akbas¸, M. Tomak, Phys. Status Solidi B 200 (1997) 67. [12] G. Weber, P.A. Schultz, L.E. Oliveira, Phys. Rev. B 38 (1988) 2179. [13] A. Montes, C.A. Duque, N. Porras-Montenegro, J. Appl. Phys. 84 (1998) 1421. [14] D.S. Chuu, C.M. Hsiao, W.N. Mei, Phys. Rev. B 46 (1992) 3898. [15] F.J. Ribeiro, A. Latge, Phys. Rev. B 50 (1994) 4913. [16] N. Porras-Montenegro, S.T. Perez-Merchancano, A. Latge, J. Appl. Phys. 74 (1993) 7624.
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[17] F. Stern, Phys. Rev. B 5 (1972) 4891. [18] T. Chakraborty, P. Pietila¨ineu, Phys. Rev. B 39 (1989) 7971. [19] M.K. Bose, C. Majumder, A.B. Maity, A.N. Chakravarty, Phys. Status Solidi 54 (1982) 437. [20] M.A. Brummel, M.A. Hopkins, R.J. Nicholast, J.C. Portal, K.Y. Cheng, A.Y. Cho, J. Phys. C 19 (1986) L107. [21] J.C. Maan, in: G. Bauer, F. Kuchar, H. Heinrich (Eds.), SolidState Sciences, vol. 53, 1984.
[22] E. Kasapoglu, H. Sari, I. So¨kmen, J. Appl. Phys. 88 (2000) 2671. [23] E. Kasapoglu, H. Sari, I. So¨kmen, Superlatt. Microstruct. 29 (2001) 1. [24] G. Fonte, P. Falsaperla, G. Schiffrer, D. Stanzial, Phys. Rev. A 41 (1990) 5087.
Binding energy of hydrogenic impurities in a quantum well under the tilted magnetic field E. Kasapoglua,*, H. Sarıa, I. So¨kmenb a
Department of Physics, Cumhuriyet University, 58140 Sivas, Turkey b Department of Physics, Dokuz Eylu¨l University, I˙zmir, Turkey
Received 31 October 2002; received in revised form 8 November 2002; accepted 10 November 2002 by K.-A. Chao
Abstract This paper treats theoretically the angle dependence of the ground state binding energy of a shallow donor impurity in semiconductor quantum-well systems on the tilted magnetic field. By making an appropriate coordinate transform we have calculated the ground state binding energy of a shallow donor impurity at the center of GaAs/Ga12xAlxAs quantum well in the effective-mass approximations and variationally. We show that the binding energy depends strongly not only on quantum ˚ , the change of the binding energy confinement, but also on the direction of the magnetic field. For example; for L0 ¼ 100 A between u ¼ 15 and 458 approximately is 2:5Ry (,13 meV). We expect that this change will be useful in designing the quantum-well structure in which the impurity effects play important role. q 2003 Elsevier Science Ltd. All rights reserved. PACS: 71.55.Eq; 71.55. 2 i Keywords: A. Quantum wells; C. Impurities in semiconductors
1. Introduction With the development of several experimental techniques, such as molecular beam epitaxy and metal organic chemical – vapor deposition, there has been a lot of work devoted to the understanding of hydrogenic impurity states in low-dimensional semiconductor heterostructures such as quantum wells [1– 6], quantum-well wires [7– 13], and quantum dots [14– 16]. Studies of semiconductor multilayer quasi-two dimensional system as well as single quantum wells of a GaAs/GaAlAs crystal type shows that the carriers caught by impurity centers effect essentially on the electronic properties of such system. Magnetic or electric fields are effective tools for studying these properties. A number of papers are devoted to the theoretical studies of the impurity states in the quantum wells when the external fields are applied. The use of the tilted magnetic fields is of * Corresponding author. Tel.: þ90-346-21910101937; fax: þ 90346-219 11 86. E-mail addresses: [email protected] (H. Sari), [email protected] (E. Kasapoglu).
interest theoretically as it illustrates confinement effects. If the magnetic field is tilted in respect to the interface, the variables in Schro¨dinger equation cannot be seperated and variational [17,18] or perturbation [19,20] methods have been used. So far only the eigenenergies of two-dimensional electrons subjected to a tilted magnetic field have been solved analytically using a parabolic potential well [21]. In our previous studies [22,23], however, we have completely solved the Schro¨dinger equation using a square well potential as confining potential and obtained analytical solutions without making any approximations for twodimensional semiconductor heterostructures under the tilted magnetic field. In this study, we report a calculation, with the use of a variational approximation, of the ground state binding energy of a hydrogenic donor impurity at the center of a GaAs quantum well in the presence of a magnetic field applied tilted to the growth direction. To solve the Schro¨dinger equation we apply an orthogonal transformation, and then we use a tricky substitution into the potential, that makes the Hamiltonian seperable in terms of the new coordinates. The general solution smoothly goes to the
0038-1098/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 8 - 1 0 9 8 ( 0 2 ) 0 0 7 7 4 - 3
430
E. Kasapoglu et al. / Solid State Communications 125 (2003) 429–434
results of two limits where the magnetic field is either parallel or perpendicular to the layers. Our results are given as a function of the well width, magnetic field strength and tilt angle. An interaction of carrier with the impurity centre is considered to be one of the Coulomb potential type.
We define the z-axis to be along the growth axis, and take the magnetic field to be applied in the x– z plane at angle-u to the x-axis. We choose a gauge for the magnetic field in which the vector potential A is written form A ¼ ð0; xB sin u 2 zB cos u; 0Þ using the 7·A ¼ 0 gauge, where B ¼ ðB cos u; 0; B sin uÞ and u is the angle between the direction of the magnetic field and x-axis. Within the framework of an effective-mass approximation, the Hamiltonian of a hydrogenic donor in a GaAs quantum well, in the presence of an applied magnetic field, can be written as 1 e~ 2 e2 H¼ p~ þ A 2 ð1Þ þ Vðze Þ; 2me c 10 l~re 2 ~ri l where me is the effective mass, e is the elementary charge, p~ is the momentum, 10 is the dielectric constant, and Vðze Þ is the confinement potential profile for the electron in the z-direction. The functional form of the confinement potential is given as Vðze Þ ¼ V0 ½SðzL 2 ze Þ þ Sðze 2 zR Þ;
ð2Þ
where S is the step function, and the left and right boundaries of the well are located at z ¼ zL ¼ 2L0 =2 and z ¼ zR ¼ L0 =2; respectively. By using the following transformation, ! ! ! cos u 2 sin u z z0 ¼ ð3Þ 0 x sin u cos u x the Hamiltonian can be written as below 1 1 2 e2 B2 0 2 z þ Vðx0e ; z0e Þ ðp2x0 þ p2z0 Þ þ py þ 2me 2me 2me c
ð6Þ
e 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 10 ðx0e 2 x0i Þ2 þ ðy 2 yi Þ2 þ ðz0e 2 z0i Þ2
By considering the above equations, we can separate the potential as (see Appendix A) Vðx0e ; z0e Þ ¼ Vðx0e Þ þ Vðz0e Þ
ð4Þ
where Eq. (4) does not contain the term ðeB=me cÞz0 py because, the expectation value of this term is identically zero for the chosen trial wave function in Eq. (11). After the coordinate transformation, the left and right boundaries of wells on the x0 and z0 axes are ð5Þ
z0L;R ¼ zL;R cos u 2 x sin u; respectively. The solution of the corresponding Schro¨dinger equation is not straightforward since, after the coordinate
ð7Þ
where Vðx0e Þ ¼ V0 sin2 u½Sðx0L 2 x0e Þ þ Sðx0e 2 x0R Þ;
ð8Þ
Vðz0e Þ ¼ V0 cos2 u½Sðz0L 2 z0e Þ þ Sðz0e 2 z0R Þ: Notice that applied magnetic field is parallel to the growth direction at the u ¼ 908 and the electron becomes free in the z0 direction and the eigenvalues do not depend on z0 , and that applied magnetic field is perpendicular to the growth direction at u ¼ 08 and the electron becomes free in the x0 direction and the eigenvalues do not depend on x0 . As known, for these values of u, the Schro¨dinger equation can be solve exactly and we not need such a transformation to solve the problem. By scaling all lengths in effective Bohr radius ðaB ¼ 10 "2 =me e2 Þ; and energies in effective Rydberg ðRy ¼ me e4 =2120 "2 Þ; and considering above results, we can rewrite the dimensionless Hamiltonian of the system as, 2 2 2 2 2 ~ ¼ 2 d þ Vð~ ~ x0e Þ 2 d þ Vð~ ~ z0e Þ þ e B " z~0 2 H 0 2 0 2 d~x d~z 4m2e c2 R2y
2
d2 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; d~y2 x~ 0e 2 þ y~ 2 þ z~0e 2
ð9Þ
where x~ 0i ; y~ 0i and z~0i is equal the zero since, donor impurity is located on center of the well. We propose the following variational trial wave function for the electron bound to impurity
c ¼ cð~x0 Þcð~z0 Þwðy; aÞ
2
x0L;R ¼ zL;R sin u þ x cos u;
SðzL 2 ze Þ ¼ cos2 uSðz0L 2 z0e Þ þ sin2 uSðx0L 2 x0e Þ; Sðze 2 zR Þ ¼ cos2 uSðz0e 2 z0R Þ þ sin2 uSðx0e 2 x0R Þ:
2. Theory
H¼
transformation the potential energy of the electron in the well Vðx0e ; z0e Þ couples x0 and z0 variables. In order to decompose the potential energy of the electron, we rewrite the step functions in Eq. (2) as follows:
ð10Þ
where the wave function in the y-direction wðy; aÞ is chosen to be Gaussian-type orbital function: 1 2 1=4 2y2 =a2 wðy; aÞ ¼ pffiffi e ; ð11Þ a p in which a is a variational parameter, cðx0 Þ is the wave function of the electron in the x0 direction which is exactly obtained from the Schro¨dinger equation in the x0 direction. cðz0 Þ; the wave function of the electron in the z0 direction. To solve the Schro¨dinger equation in the z0 direction, we choose as base the eigenfunction of the infinite potential well with the Lb width. We have also used this technique in our previous studies [22,23]. In calculating the wave functions cðz0 Þ; we have ensured that the eigenvalues are independent
E. Kasapoglu et al. / Solid State Communications 125 (2003) 429–434
431
of the choice infinite potential well width Lb and that the wave functions are localized in the well region. The total energy of the system is evaluated by minimizing the expectation value of the Hamiltonian in Eq. (9) with respect to a: ~ cl ¼ E: ~ min kclHl
ð12Þ
a
The binding energy of the donor impurity ground state is given by ~ E~ B ¼ E~ 0 2 E;
ð13Þ
where E~ 0 is the lowest electron total subband energy in the x0 and z0 directions, respectively. Substituting the expectation value of the Hamiltonian into the Eq.(12), we get the ground state binding energy of the donor impurity.
3. Results and discussions The values of the physical parameters used in our calculations are me ¼ 0:0665m0 (m0 is the free electron mass), 10 ¼ 12:58 (static dielectric constant is assumed to be same GaAs and GaAlAs), V0 ¼ 228 meV. These parameters are suitable in GaAs/Ga12xAlxAs heterostructures with an Al concentration of x ø 0:3: Without losing generality, and for simplicity in numerical calculations, we have chosen the boundaries of the well at x0LðRÞ ¼ ^L0 =2 sin u and z0LðRÞ ¼ ^L0 =2 cos u; and after the coordinate transformation which satisfies the following equation:
Fig. 1. The variation of the binding energy of the ground state for a donor at the center of a GaAs quantum well as a function of the well width for u ¼ 158 and three different magnetic field values.
ð14Þ
it can be observed that, for strong magnetic fields the binding energy reaches a constant value for large well width. The results in the large L0 limit and for magnetic fields in the experimental range are compared with the hydrogenic atom limit [24]. The comparison shows that in the range studied the present calculation is quite accurate. In Fig. 2, we show the variation of the impurity binding
derived from Eq. (5). In Fig. 1, we display the variation of the binding energy of the ground state for a donor at the center of a GaAs quantum well as a function of the well width for different magnetic field values and u ¼ 158: As seen in this figure impurity binding energy increases as the well size increases as the independent of all magnetic field values, since the geometric confinement predominates at small L0 values ˚ ). For B ¼ 1T; the binding energy (100 # L0 # 200 A increases as L0 increases and reaches a maximum value. Where the binding energy is maximum the system has quasi-two-dimensional character. After the certain L0 value ˚ ), impurity binding energy decreases as L0 (L0 ø 250 A increases, since the confinement of the electron in the z0 direction decreases i.e. the influence of the Coulomb field of the impurity center on the electron weakens. This behaviour reproduces several results previously reported [3,4,11]. For ˚ , at large magnetic field values, magnetic L0 . 200 A confinement becomes stronger and the impurity binding energy increases as the magnetic field increases. In this limit, the extension of the wave function in the plane which is perpendicular to the magnetic field is determined primarily by the magnetic field and the barrier potential is a small perturbation on the magnetic term. Also from Fig. 1
Fig. 2. The variation of the binding energy of the ground state for a donor at the center of a GaAs quantum well as a function of the well width for u ¼ 308 and three different magnetic field values.
z0R 2 z0L ¼ L0 cos u;
x0R 2 x0L ¼ L0 sin u;
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energy versus the well width for u ¼ 308 and different magnetic field values. Impurity binding energy decreases for all magnetic field values as L0 increases. When we compare the results are obtained in this case with that of u ¼ 158, we see that the binding energy of impurity at the center of the well increases with increasing tilt angle, since the localisation of the electron increases in both x0 and z0 direction, and the electronic probability density around the impurity is higher than in the previous case. In Fig. 3, we present the variation of the impurity binding energy versus the well width for u ¼ 458 and different magnetic field values. For u ¼ 458; binding energy becomes maximum, since the effective well widths and potential heights of electron in both x0 and z0 directions are equal, electron is under the effect of the same geometric confinement in both directions. If we compare the results of the binding energy for u ¼ 458 with that of u ¼ 158; we see that the binding energy changes from , 1:5Ry to , 4Ry ˚ and all magnetic field values. So, tilt angle is for L0 ¼ 100 A a good tunable parameter providing a change on the impurity binding energy for especially small L0 values. In Figs. 4 and 5, we present the variation of the impurity binding energy versus the well width for u ¼ 608 and u ¼ 758; respectively. For these tilt angles, the well width in which the binding energy begins to be sensitive to the magnetic field becomes smaller than the previous tilt angles. This case is evident in Fig. 5. In these tilt angles values, the well width in which the electron is confined in the z0 direction became so small that the electron is delocalised in all L0 values and an interaction of the electron with the impurity centre is completely provided with the magnetic confinement.
The variation of the impurity binding energy versus the ˚ tilt angle for different magnetic field values and L0 ¼ 200 A is given In Fig. 6. As seen in this figure, in the range of 0 # u # 458 impurity binding energy increases as tilt angle increases and reaches a maximum value at u ¼ 458; and then in the range of 45 # u # 908 the binding energy decreases up to u ¼ 758 and for further tilt angle value it converges to a constant value. The observations are that; in the range
Fig. 3. The variation of the binding energy of the ground state for a donor at the center of a GaAs quantum well as a function of the well width for u ¼ 458 and three different magnetic field values.
Fig. 5. The variation of the binding energy of the ground state for a donor at the center of a GaAs quantum well as a function of the well width for u ¼ 758 and three different magnetic field values.
Fig. 4. The variation of the binding energy of the ground state for a donor at the center of a GaAs quantum well as a function of the well width for u ¼ 608 and three different magnetic field values.
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these results will be of importance in the understanding of experimental studies related with donor impurities in GaAs quantum wells under the external tilted magnetic field.
Appendix A In order to decompose the potential energy Vðx0e ; z0e Þ which is defined in Eq. (4), let us introduce a function f ðx0 ; z0 Þ ¼ cos2 uSðz0L 2 z0 Þ þ sin2 uSðx0L 2 x0 Þ 0
ðA1Þ
0
By using the variables x , and z in terms of x and z we can write the step function Sðz0L 2 z0 Þ as follows: Sðz0L 2 z0 Þ ¼ SðzL cos u 2 x sin u 2 z cos u þ x sin uÞ ¼ SððzL 2 zÞcos uÞ
Fig. 6. The variation of the binding energy of the ground state for a donor at the center of a GaAs quantum well as a function of the tilt angle for three different magnetic field values and well width˚. L0 ¼ 200 A
0 # u # 458, as the localization of the donor electron in the z0 direction and Coulombic interaction increases, the binding energy increases, and in the range of 45 # u # 908, the well width becomes smaller in the z0 direction and interaction decreases between donor electron and ion as delocalisation for donor electron begins, and thus the binding energy also decreases. It is important to observe that very similar magnetic field dependence has the ground state exciton binding energy in quantum wells [22]. As a consequence of this analysis it can be seen that the direction of the magnetic field plays an essential role in the determination of the binding energy. In conclusion, this paper presents the solution of the Schro¨dinger equation of a square well potential problem under the influence of an externally applied tilted magnetic field by making the Hamiltonian separable via a coordinate transformation. In this study, we calculated by using a variational approximation the ground state impurity binding energy in the single square quantum well under the externally applied tilted magnetic field as a function of the tilt angle, magnetic field and the well width. It is seen that the direction of the magnetic field causes important changes ˚ , the in the binding energy. For example; for L0 ¼ 100 A change of the binding energy between u ¼ 15 and 458 approximately is 2:5Ry (, 13 meV). So, we can say that the quantum well structure is reduced to the quantum wire from the results are obtained at u ¼ 458, since the system is under the effect of the same geometric confinement in both x0 and z0 directions. To the best of our knowledge, so far studies about the shallow donor impurity binding energy under the tilted magnetic field have not been reported. We expect that
ðA2Þ
and since cos u . 0; one can rewrite the last term in Eq. (A2) as SððzL 2 zÞcos uÞ ¼ SððzL 2 zÞÞ: Similarly, by considering the above results we can write Sðx0L 2 x0 Þ ¼ SððzL 2 zÞsin uÞ ¼ SððzL 2 zÞÞ: Thus f ðx0 ; z0 Þ takes the following form: f ðx0 ; z0 Þ ¼ ðcos2 u þ sin2 uÞ SðzL 2 zÞ ¼ SðzL 2 zÞ
ðA3Þ
By using the same procedure for Sðz 2 zR Þ we can write the step function as in Eq. (6), and by considering the above equations we can decompose the potential Vðx0e ; z0e Þ as Eq. (7).
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