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Impurity states in a narrow band gap semiconductor quantum dot with parabolic confinement potential
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Physica E 16 (2003) 174 – 178 www.elsevier.com/locate/physe
Impurity states in a narrow band gap semiconductor quantum dot with parabolic con&nement potential E.M. Kazaryan, L.S. Petrosyan, H.A. Sarkisyan∗ Department of Physics, Yerevan State University, Al. Manoukyan Street 1, 375025 Yerevan, Armenia Received 20 March 2002; accepted 22 August 2002
Abstract Impurity states in a narrow band gap semiconductor quantum dot are studied theoretically. The quantum dot con&nement potential is approximated by the three-dimensional spherically symmetric parabolic potential. The electron dispersion law is considered within the framework of two-band Kane model. It is shown that the nonparabolicity leads to the Schroedinger equation with the e6ective oscillatory potential containing terms proportional to 1=r, r, r 4 . Within the framework of the stationary theory of perturbation the energy corrections due to these terms are calculated. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 71.20.Nr; 7.61.Ey Keywords: Parabolic potential; Impurity states; Quantum dot
1. Introduction Theoretical investigation of Coulomb problems in semiconductor quantum dots (QD) is called not only for academic interest, but also because of necessity to take into account the Coulomb e6ects at experimental analysis of optical and kinetic properties of QD. The broad class of Coulomb problems consists of problems connected with the investigation of impurity and exciton states in QD. The speci&c character of similar problems consist in the fact, that there is a necessity to compare two parameters of quantizing: Bohr radius and QD radius. Depending on the ratio between them, the main contribution to energy is determined by Coulomb center (at aB r0 ) or dot wall repulsing
∗
Corresponding author. Tel.: +3742-265-068. E-mail address: [email protected] (H.A. Sarkisyan).
potential (at r0 aB ). QD wall inBuence on impurity states is conditioned by the term, describing the interaction of impurity electron with QD walls. In di6erent articles the di6erent models of QD con&nement potential were considered. So in Ref. [1] the impurity states in spherical GaAs=Ga1−x Alx As QD were investigated, the QD con&nement potential was modeled by spherical symmetrical rectangular dot of &nal altitude. The similar problem, but already with in&nitely high rectangular con&nement potential has been solved in Ref. [2]. The authors [3] investigated impurity states in spherical QD with a potential similar to one, that was considered in Ref. [1] but taking into account the di6erence of e6ective mass inside and outside of QD. Another widespread model of con&nement potential is the parabolic model. Such approximation arised in connection with the generalization of Kohn theorem in the case of quantized semiconductor systems [4–6]. Impurity and exciton states in
1386-9477/03/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 2 ) 0 0 6 6 2 - 8
E.M. Kazaryan et al. / Physica E 16 (2003) 174 – 178
spherical and cylindrical QD with parabolic con&nement potential were investigated in articles [7–10]. So, for example [7] Ferreyra et al. have reported a study of the dependence of the impurity related energy corrections on the impurity position and the magnetic &eld in parabolic quantum dot by using the strong con&nement approach. They have shown that the impurity related binding energy depends strongly on the impurity position in the dot and the magnetic &eld strength. In Ref. [8] the binding energy of shallow hydrogenic impurities in spherical QDs with parabolic con&nement potential is calculated, using a variational approach within the e6ective mass approximation. Charrour et al. [9] have reported a study of the ground state binding energy of a hydrogenic impurity in cylindrical QD subjected to an external strong magnetic &eld. Calculations are performed within the e6ective-mass approximation using the variational procedure and considering an in&nite con&nement potential on all surfaces of the system. Using the method of series expansion the authors Jia-Lin Zhu et al. [10] have obtained the exact series forms in di6erent regions of the radial equation for donor states in spherically rectangular quantum well. In all the above cited articles the dispersion law for charge carriers was considered parabolic. Whereas it was mentioned in articles [11,12], there are narrow band gap semiconductors in which charge carrier dispersion law is non-parabolic (Kane dispersion law [13] for A3 B5 -type semiconductors). In the case of two-band approximation, valid for InSb [14], this dispersion law is similar to the relativistic dispersion law E = p2 s 2 + 2 s 4 ; where is the electron e6ective mass at the conduction band bottom (for A3 B5 compounds ∼ 10−2 me ), s = ( g =2 )1=2 is the non-parabolicity parameter proportional to the band interaction matrix element (s ∼ 108 cm s−1 for A3 B5 ), g is the forbidden band width. That is why the SchrNodinger equation, corresponding to this dispersion law is de&ned by analogy with Klein–Gordon equation. Therefore, the investigation of impurity levels in QD with non-parabolic dispersion law of electron is interesting. In this article the impurity states in spherical QD with parabolic con&nement potential are investigated. Thus the electron dispersion law is described
175
within the framework of two-band Kane model [14]. 2. Theory Let us study the impurity states of electron with Kane’s dispersion law in a spherical QD. Considering con&nement potential as three-dimensional, spherically symmetric parabolic one, the SchrNodinger equation (in this case it looks similar to Klein–Gordon equation) for the electron envelope wave function takes the form: 2
!2 r 2 Ze2 ; (1) + (pˆ2 s2 + 2 s4 ) = E − 2
el r where Z is core charge, e is electron charge, el is the dielectric constant of the semiconductor (for InSb el = 16), ! is the frequency of QD con&nement potential, which can be presented as [11] !∼
˝ 1 ;
r02 4 1 + ˝2 = 2 r 2 s2 0
(2)
where r0 is the radius of QD. Introducing resignations
=
E 2 − 2 s4 ; 2 s2
Ze2 = e12
el
(3)
Eq. (1) can be presented in standard form pˆ2 + U ∗ (r) = ; 2
(4)
where the e6ective potential is U ∗ (r) =
!2 E 2 e14 1
!2 e12 r − + r 2 s2 2 s2 r 2 2 s2 −
2e12 E 1 2 !4 4 r : − 2 s2 r 8 s2
(5)
Eq. (4) with e6ective potential U ∗ (r) does not have an exact solution, because it includes the term, proportional to 1=r, r, r 4 . Therefore we shall solve the problem in two stages. (a) First we consider the terms proportional to 1=r, r, r 4 as a perturbation, to obtain the exact solution of the problem with the unperturbed Hamiltonian. (b) Second we &nd the corrections to the unperturbed levels.
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E.M. Kazaryan et al. / Physica E 16 (2003) 174 – 178
2.1. Exactly solvable problem
Let us &rst consider the equation pˆ2 e14 1
!2 E 2 = ; r − + 2
2 s2 2 s2 r 2
(6)
where E is energy levels for unperturbed problem. Introducing new variables !∗ = ! E= s2 , e14 =2 s2 = Z 2 e4 =2 el2 s2 = 2 , we get the following equation: 2 pˆ
!∗2 r 2 2 + − 2 = : (7) 2
2 r The same equation was solved in Ref. [15] and its solutions can be presented in form
2
(r; ; ’) = Cnr l r l e−(=2)r 1 3 3 2 ×1 F1 − −l − ; l + ; r 2 ˝!∗ 2 2 ×Ylm (; ’); where =
!∗ ; ˝
0 =
1 l = − + 2
R we where Es = s2 ∼ 0:081 ev, rs = ˝= s ∼ 100 A, write Eq. (10) as 2
−1 =2w
l+
1 2
2
√
n−l+1+
l+
1 2 2
−
02
:
(12)
It should be noted that Eq. (10) at 0 = 0 (electronic states without impurity) turns into the expression for the energy of electron with Kane’s dispersion law in the parabolic QD [11]. Expanding expression (10) into series by 1= s2 small parameter we obtain 2 l + 12 − 02 Enl = ˝! n − l + 1 +
(8)
−18
3 1 2 2
2 s4 : ˝! n −l + 1 + l + 2 −0 (13)
From this expression follows, that taking into account the non-parabolicity of dispersion law results in energy level decreasing of unperturbed problem, in comparison with the same problem, but with parabolic dispersion law [15].
− 02 ;
1 2 = Zfs∗ ¡ ⇒ Z 6 3 ˝ 2
Cn2r l
where n = 2nr + l = 0; 1; 2; : : : is the principal quantum number. Introducing dimensionless parameters ˝! r0 E ; "= ; (11) = ; w= Es Es rs
2l +3=2 (l + 32 )(l + 52 ) : : : (l + = (l + 32 )nr !
2nr +1 2 )
;
(9)
where nr ; l; m are radial, orbital and magnetic quantum numbers, respectively, fs∗ = e2 =˝s el is analogous to the e6ective &ne structure constant in Kane’s semiconductor (for InSb fs∗ =0:14), c is light velocity, Cnr l is the normalizing constant. Using Eqs. (3), (8) and (9) we &nd that the energy levels for the unperturbed problem are given by 2 − 2 s4 Enl Enl = ˝! 2 s2
s2 2 1 l+ × n − l + 1 + − 02 ; 2 (10)
2.2. Corrections to the energy levels Let us consider the total Hamiltonian of the problem Hˆ = Hˆ 0 + V (r);
(14)
where pˆ2
!∗2 r 2 2 Hˆ 0 = + − 2; 2
2 r V (r) = V1 (r) + V2 (r) + V3 (r); V1 (r) = − V2 (r) =
2 !4 4 r = Ar 4 ; 8 s2
!2 e12 r = Br; 2 s2
V3 (r) = −
A=− B=
2e12 Enl 1 1 =C ; 2 s2 r r
2 !4 ; 8 s2
!2 e12 ; 2 s2 C =−
2e2 Enl : 2 s2
(15)
E.M. Kazaryan et al. / Physica E 16 (2003) 174 – 178
For the energy corrections we have (1) ∗
nl = nlm V (r)nlm dv:
(16)
177
it is visible, that the perturbation results in decreasing energy levels of electron. Thus along with decreasing QD radius the contribution of perturbation arises and
After lengthy calculation it is possible to show that nr (nr + l + 32 ) ((2l + 7)=2) 1 (1)
nl = A 1+6 3 5 ((2l + 3)=2) 2 (l + 2 )(l + 2 ) 1 (l + 2) √ +B ((2l + 3)=2)
1+
n r −1 i=1
nr (nr − 1) · · · (nr − i)(− 32 − i) · · · (− 32 )(− 12 ) · · · (− 12 + i) 12 22 · · · (i + 1)2 (l + 32 ) · · · (l + 32 + i)
n −1
r nr (nr − 1) · · · (nr − i)(− 12 − i) · · · (− 12 )( 12 ) · · · ( 12 + i) (l + 1) √ 1 + +C ((2l + 3)=2) 12 22 : : : (i + 1)2 (l + 32 ) · · · (l + 32 + i) i=1
where (x) is Euler’s gamma-function. (1) we have Using (3) for Enl 2 (1)
2 s (1) nl Enl : = −Enl 1 − 1 + 2 Enl
(18)
;
(17)
is exhibited brighter for more high levels. With the increase of r0 the energy decreases because the QD wall inBuence becomes less essential. Let us also mention, that as against the problem without impurity [11]
Abs (E(1)/E
3. Discussion of results
0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01
3 2 1 2
0
4 r0/rs
6
8
Fig. 1. Absolute value of perturbation energy dependence on QD radius for (1) n = l = 0, (2) n = l = 1, (3) n = 2, l = 0 states (in Es and rs units for 0 = 0:14).
3 2.75 2.5 E/Es
All the above mentioned results are right in case of such values of parameters of the problem (QD radius, n and l quantum numbers) at which perturbation theory is applicable. As against to cases, observed in Refs. [11,12] in given case we cannot show the simple analytical condition of application. In general case for its application we need energetic corrections magnitudes, conditioned by this perturbation, to be much smaller than the distance between unperturbed levels. Fig. 1 introduces curves of absolute value of perturbation energy dependence on QD radius. At once it is visible, that this perturbation is too small and it decreases with the increase of QD radius. As was above mentioned the perturbations that correspond to higher levels are exhibited brighter. For example, for n=l=0 states at " = 1:5, = 1:6 and for perturbed energy at (1) the same " = 1:5 |E00 | = 0:005E00 , while for the state (1) n = 2, l = 0, = 2:25, |E20 | = 0:025E20 . In Fig. 2 the diagrams of impurity electron energy dependence on QD radius are shown. The curves 1, 2, 3 correspond to energies of unperturbed Hamiltonian for n = l = 0; n = l = 1; n = 2, l = 0 states in Es and rs units for 0 = 0:14 values. The curves 1 , 2 , 3 correspond to total energy of electron. From this diagram
1′
3′
1
2′
2.25 2 1.75 1.5
3 2
1.25 0.5
1
1.5
2
2.5
3
r0/rs Fig. 2. Impurity electron energy (unperturbed and total) dependence on QD radius for (1; 1 ) n = l = 0, (2; 2 ) n = l = 1, (3; 3 ) n = 2, l = 0 states (in Es and rs units for 0 = 0:14).
178
E.M. Kazaryan et al. / Physica E 16 (2003) 174 – 178
electron in narrow band gap semiconductor QD without impurity e = 0 (0 = 0) [11].
1
Eb/Es
0. 8 0. 6
Acknowledgements
0. 4 1 0. 2
2 0
0.2
0. 4
0. 6
0. 8
1
This work was supported by INTAS #99-00928, INTAS #0175WP, and the Armenian National Science and Education Fund PS24-01.
r0/rs Fig. 3. Binding energies of a hydrogen-like impurity as a function of QD radius for: (1) n = l = 0, (2) n = l = 1 states (in Es and rs units for 0 = 0:14).
the energy of electron that corresponds to unperturbed Hamiltonian is not degenerated by l orbital quantum number any more, and, consequently, these levels at given l are degenerated only by m magnetic quantum number with s = 2l + 1 degeneracy multiplicity. In Fig. 3 the dependencies of the binding energy of the impurity on the QD radius for states with n = l = 0 and n = l = 1 are shown. De&ning binding energy as a di6erence between energy of considered problem and energy of problem without impurity we can see, that it decreases with the increase of r0 . As it should be, the binding energy of ground state is more than binding energies of excited states. At last we shall mention, that expressions (10) for unperturbed energy of electron turns to the appropriate expressions for unperturbed energy of
References [1] J.-L. Zhu, J.-J. Xiong, B.-L. Gu, Phys. Rev. B 41 (1990) 6001. [2] D.S. Chuu, C.M. Hsiao, W.N. Mei, Phys. Rev. B 46 (1992) 3898. [3] V. Ranjan, Vijay A. Singh, J. Appl. Phys. 89 (2001) 6415. [4] P.A. Maksym, T. Chakraborty, Phys. Rev. Lett. 65 (1990) 108. [5] F.M. Peeters, Phys. Rev. B 42 (1990) 1486. [6] Q. Li, et al., Phys. Rev. B 43 (1991) 5151. [7] J.M. Ferreyra, P. Bosshard, C.R. Proetto, Phys. Rev. B 55 (1997) 13 682. [8] C. Bose, J. Appl. Phys. 83 (1998) 3089. [9] R. Charrour, et al., Physica B 293 (2000) 137. [10] J.-L. Zhu, Phys. Rev. B 39 (1989) 8780. [11] E.M. Kazaryan, L.S. Petrosyan, H.A. Sarkisyan, Physica E 8 (2000) 19. [12] E.M. Kazaryan, L.S. Petrosyan, H.A. Sarkisyan, Physica E 11 (2001) 362. [13] E.O. Kane, J. Phys. Chem. Solids 1 (1957) 249. [14] M. Cidil’kovski, Electrons and Holes in Semiconductors, Nauka, Moscow, 1972. [15] L.D. Landau, E.M. Lifshic, Quantum Mechanics, 4th ed. Nauka, Moscow, 1989, p. 161 (in Russian).
Physica E 16 (2003) 174 – 178 www.elsevier.com/locate/physe
Impurity states in a narrow band gap semiconductor quantum dot with parabolic con&nement potential E.M. Kazaryan, L.S. Petrosyan, H.A. Sarkisyan∗ Department of Physics, Yerevan State University, Al. Manoukyan Street 1, 375025 Yerevan, Armenia Received 20 March 2002; accepted 22 August 2002
Abstract Impurity states in a narrow band gap semiconductor quantum dot are studied theoretically. The quantum dot con&nement potential is approximated by the three-dimensional spherically symmetric parabolic potential. The electron dispersion law is considered within the framework of two-band Kane model. It is shown that the nonparabolicity leads to the Schroedinger equation with the e6ective oscillatory potential containing terms proportional to 1=r, r, r 4 . Within the framework of the stationary theory of perturbation the energy corrections due to these terms are calculated. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 71.20.Nr; 7.61.Ey Keywords: Parabolic potential; Impurity states; Quantum dot
1. Introduction Theoretical investigation of Coulomb problems in semiconductor quantum dots (QD) is called not only for academic interest, but also because of necessity to take into account the Coulomb e6ects at experimental analysis of optical and kinetic properties of QD. The broad class of Coulomb problems consists of problems connected with the investigation of impurity and exciton states in QD. The speci&c character of similar problems consist in the fact, that there is a necessity to compare two parameters of quantizing: Bohr radius and QD radius. Depending on the ratio between them, the main contribution to energy is determined by Coulomb center (at aB r0 ) or dot wall repulsing
∗
Corresponding author. Tel.: +3742-265-068. E-mail address: [email protected] (H.A. Sarkisyan).
potential (at r0 aB ). QD wall inBuence on impurity states is conditioned by the term, describing the interaction of impurity electron with QD walls. In di6erent articles the di6erent models of QD con&nement potential were considered. So in Ref. [1] the impurity states in spherical GaAs=Ga1−x Alx As QD were investigated, the QD con&nement potential was modeled by spherical symmetrical rectangular dot of &nal altitude. The similar problem, but already with in&nitely high rectangular con&nement potential has been solved in Ref. [2]. The authors [3] investigated impurity states in spherical QD with a potential similar to one, that was considered in Ref. [1] but taking into account the di6erence of e6ective mass inside and outside of QD. Another widespread model of con&nement potential is the parabolic model. Such approximation arised in connection with the generalization of Kohn theorem in the case of quantized semiconductor systems [4–6]. Impurity and exciton states in
1386-9477/03/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 2 ) 0 0 6 6 2 - 8
E.M. Kazaryan et al. / Physica E 16 (2003) 174 – 178
spherical and cylindrical QD with parabolic con&nement potential were investigated in articles [7–10]. So, for example [7] Ferreyra et al. have reported a study of the dependence of the impurity related energy corrections on the impurity position and the magnetic &eld in parabolic quantum dot by using the strong con&nement approach. They have shown that the impurity related binding energy depends strongly on the impurity position in the dot and the magnetic &eld strength. In Ref. [8] the binding energy of shallow hydrogenic impurities in spherical QDs with parabolic con&nement potential is calculated, using a variational approach within the e6ective mass approximation. Charrour et al. [9] have reported a study of the ground state binding energy of a hydrogenic impurity in cylindrical QD subjected to an external strong magnetic &eld. Calculations are performed within the e6ective-mass approximation using the variational procedure and considering an in&nite con&nement potential on all surfaces of the system. Using the method of series expansion the authors Jia-Lin Zhu et al. [10] have obtained the exact series forms in di6erent regions of the radial equation for donor states in spherically rectangular quantum well. In all the above cited articles the dispersion law for charge carriers was considered parabolic. Whereas it was mentioned in articles [11,12], there are narrow band gap semiconductors in which charge carrier dispersion law is non-parabolic (Kane dispersion law [13] for A3 B5 -type semiconductors). In the case of two-band approximation, valid for InSb [14], this dispersion law is similar to the relativistic dispersion law E = p2 s 2 + 2 s 4 ; where is the electron e6ective mass at the conduction band bottom (for A3 B5 compounds ∼ 10−2 me ), s = ( g =2 )1=2 is the non-parabolicity parameter proportional to the band interaction matrix element (s ∼ 108 cm s−1 for A3 B5 ), g is the forbidden band width. That is why the SchrNodinger equation, corresponding to this dispersion law is de&ned by analogy with Klein–Gordon equation. Therefore, the investigation of impurity levels in QD with non-parabolic dispersion law of electron is interesting. In this article the impurity states in spherical QD with parabolic con&nement potential are investigated. Thus the electron dispersion law is described
175
within the framework of two-band Kane model [14]. 2. Theory Let us study the impurity states of electron with Kane’s dispersion law in a spherical QD. Considering con&nement potential as three-dimensional, spherically symmetric parabolic one, the SchrNodinger equation (in this case it looks similar to Klein–Gordon equation) for the electron envelope wave function takes the form: 2
!2 r 2 Ze2 ; (1) + (pˆ2 s2 + 2 s4 ) = E − 2
el r where Z is core charge, e is electron charge, el is the dielectric constant of the semiconductor (for InSb el = 16), ! is the frequency of QD con&nement potential, which can be presented as [11] !∼
˝ 1 ;
r02 4 1 + ˝2 = 2 r 2 s2 0
(2)
where r0 is the radius of QD. Introducing resignations
=
E 2 − 2 s4 ; 2 s2
Ze2 = e12
el
(3)
Eq. (1) can be presented in standard form pˆ2 + U ∗ (r) = ; 2
(4)
where the e6ective potential is U ∗ (r) =
!2 E 2 e14 1
!2 e12 r − + r 2 s2 2 s2 r 2 2 s2 −
2e12 E 1 2 !4 4 r : − 2 s2 r 8 s2
(5)
Eq. (4) with e6ective potential U ∗ (r) does not have an exact solution, because it includes the term, proportional to 1=r, r, r 4 . Therefore we shall solve the problem in two stages. (a) First we consider the terms proportional to 1=r, r, r 4 as a perturbation, to obtain the exact solution of the problem with the unperturbed Hamiltonian. (b) Second we &nd the corrections to the unperturbed levels.
176
E.M. Kazaryan et al. / Physica E 16 (2003) 174 – 178
2.1. Exactly solvable problem
Let us &rst consider the equation pˆ2 e14 1
!2 E 2 = ; r − + 2
2 s2 2 s2 r 2
(6)
where E is energy levels for unperturbed problem. Introducing new variables !∗ = ! E= s2 , e14 =2 s2 = Z 2 e4 =2 el2 s2 = 2 , we get the following equation: 2 pˆ
!∗2 r 2 2 + − 2 = : (7) 2
2 r The same equation was solved in Ref. [15] and its solutions can be presented in form
2
(r; ; ’) = Cnr l r l e−(=2)r 1 3 3 2 ×1 F1 − −l − ; l + ; r 2 ˝!∗ 2 2 ×Ylm (; ’); where =
!∗ ; ˝
0 =
1 l = − + 2
R we where Es = s2 ∼ 0:081 ev, rs = ˝= s ∼ 100 A, write Eq. (10) as 2
−1 =2w
l+
1 2
2
√
n−l+1+
l+
1 2 2
−
02
:
(12)
It should be noted that Eq. (10) at 0 = 0 (electronic states without impurity) turns into the expression for the energy of electron with Kane’s dispersion law in the parabolic QD [11]. Expanding expression (10) into series by 1= s2 small parameter we obtain 2 l + 12 − 02 Enl = ˝! n − l + 1 +
(8)
−18
3 1 2 2
2 s4 : ˝! n −l + 1 + l + 2 −0 (13)
From this expression follows, that taking into account the non-parabolicity of dispersion law results in energy level decreasing of unperturbed problem, in comparison with the same problem, but with parabolic dispersion law [15].
− 02 ;
1 2 = Zfs∗ ¡ ⇒ Z 6 3 ˝ 2
Cn2r l
where n = 2nr + l = 0; 1; 2; : : : is the principal quantum number. Introducing dimensionless parameters ˝! r0 E ; "= ; (11) = ; w= Es Es rs
2l +3=2 (l + 32 )(l + 52 ) : : : (l + = (l + 32 )nr !
2nr +1 2 )
;
(9)
where nr ; l; m are radial, orbital and magnetic quantum numbers, respectively, fs∗ = e2 =˝s el is analogous to the e6ective &ne structure constant in Kane’s semiconductor (for InSb fs∗ =0:14), c is light velocity, Cnr l is the normalizing constant. Using Eqs. (3), (8) and (9) we &nd that the energy levels for the unperturbed problem are given by 2 − 2 s4 Enl Enl = ˝! 2 s2
s2 2 1 l+ × n − l + 1 + − 02 ; 2 (10)
2.2. Corrections to the energy levels Let us consider the total Hamiltonian of the problem Hˆ = Hˆ 0 + V (r);
(14)
where pˆ2
!∗2 r 2 2 Hˆ 0 = + − 2; 2
2 r V (r) = V1 (r) + V2 (r) + V3 (r); V1 (r) = − V2 (r) =
2 !4 4 r = Ar 4 ; 8 s2
!2 e12 r = Br; 2 s2
V3 (r) = −
A=− B=
2e12 Enl 1 1 =C ; 2 s2 r r
2 !4 ; 8 s2
!2 e12 ; 2 s2 C =−
2e2 Enl : 2 s2
(15)
E.M. Kazaryan et al. / Physica E 16 (2003) 174 – 178
For the energy corrections we have (1) ∗
nl = nlm V (r)nlm dv:
(16)
177
it is visible, that the perturbation results in decreasing energy levels of electron. Thus along with decreasing QD radius the contribution of perturbation arises and
After lengthy calculation it is possible to show that nr (nr + l + 32 ) ((2l + 7)=2) 1 (1)
nl = A 1+6 3 5 ((2l + 3)=2) 2 (l + 2 )(l + 2 ) 1 (l + 2) √ +B ((2l + 3)=2)
1+
n r −1 i=1
nr (nr − 1) · · · (nr − i)(− 32 − i) · · · (− 32 )(− 12 ) · · · (− 12 + i) 12 22 · · · (i + 1)2 (l + 32 ) · · · (l + 32 + i)
n −1
r nr (nr − 1) · · · (nr − i)(− 12 − i) · · · (− 12 )( 12 ) · · · ( 12 + i) (l + 1) √ 1 + +C ((2l + 3)=2) 12 22 : : : (i + 1)2 (l + 32 ) · · · (l + 32 + i) i=1
where (x) is Euler’s gamma-function. (1) we have Using (3) for Enl 2 (1)
2 s (1) nl Enl : = −Enl 1 − 1 + 2 Enl
(18)
;
(17)
is exhibited brighter for more high levels. With the increase of r0 the energy decreases because the QD wall inBuence becomes less essential. Let us also mention, that as against the problem without impurity [11]
Abs (E(1)/E
3. Discussion of results
0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01
3 2 1 2
0
4 r0/rs
6
8
Fig. 1. Absolute value of perturbation energy dependence on QD radius for (1) n = l = 0, (2) n = l = 1, (3) n = 2, l = 0 states (in Es and rs units for 0 = 0:14).
3 2.75 2.5 E/Es
All the above mentioned results are right in case of such values of parameters of the problem (QD radius, n and l quantum numbers) at which perturbation theory is applicable. As against to cases, observed in Refs. [11,12] in given case we cannot show the simple analytical condition of application. In general case for its application we need energetic corrections magnitudes, conditioned by this perturbation, to be much smaller than the distance between unperturbed levels. Fig. 1 introduces curves of absolute value of perturbation energy dependence on QD radius. At once it is visible, that this perturbation is too small and it decreases with the increase of QD radius. As was above mentioned the perturbations that correspond to higher levels are exhibited brighter. For example, for n=l=0 states at " = 1:5, = 1:6 and for perturbed energy at (1) the same " = 1:5 |E00 | = 0:005E00 , while for the state (1) n = 2, l = 0, = 2:25, |E20 | = 0:025E20 . In Fig. 2 the diagrams of impurity electron energy dependence on QD radius are shown. The curves 1, 2, 3 correspond to energies of unperturbed Hamiltonian for n = l = 0; n = l = 1; n = 2, l = 0 states in Es and rs units for 0 = 0:14 values. The curves 1 , 2 , 3 correspond to total energy of electron. From this diagram
1′
3′
1
2′
2.25 2 1.75 1.5
3 2
1.25 0.5
1
1.5
2
2.5
3
r0/rs Fig. 2. Impurity electron energy (unperturbed and total) dependence on QD radius for (1; 1 ) n = l = 0, (2; 2 ) n = l = 1, (3; 3 ) n = 2, l = 0 states (in Es and rs units for 0 = 0:14).
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E.M. Kazaryan et al. / Physica E 16 (2003) 174 – 178
electron in narrow band gap semiconductor QD without impurity e = 0 (0 = 0) [11].
1
Eb/Es
0. 8 0. 6
Acknowledgements
0. 4 1 0. 2
2 0
0.2
0. 4
0. 6
0. 8
1
This work was supported by INTAS #99-00928, INTAS #0175WP, and the Armenian National Science and Education Fund PS24-01.
r0/rs Fig. 3. Binding energies of a hydrogen-like impurity as a function of QD radius for: (1) n = l = 0, (2) n = l = 1 states (in Es and rs units for 0 = 0:14).
the energy of electron that corresponds to unperturbed Hamiltonian is not degenerated by l orbital quantum number any more, and, consequently, these levels at given l are degenerated only by m magnetic quantum number with s = 2l + 1 degeneracy multiplicity. In Fig. 3 the dependencies of the binding energy of the impurity on the QD radius for states with n = l = 0 and n = l = 1 are shown. De&ning binding energy as a di6erence between energy of considered problem and energy of problem without impurity we can see, that it decreases with the increase of r0 . As it should be, the binding energy of ground state is more than binding energies of excited states. At last we shall mention, that expressions (10) for unperturbed energy of electron turns to the appropriate expressions for unperturbed energy of
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