- Email: [email protected]
Interband light absorption in parabolic quantum dot in the presence of electrical and magnetic fields
ARTICLE IN PRESS
Physica E 31 (2006) 83–85 www.elsevier.com/locate/physe
Interband light absorption in parabolic quantum dot in the presence of electrical and magnetic fields M.S. Atoyan, E.M. Kazaryan, H.A. Sarkisyan Department of Solid State Physics, Yerevan State University, 1. Al. Manoukyan 375049 Armenia Received 12 September 2005; accepted 4 October 2005 Available online 15 December 2005
Abstract An analytical expression for the light interband absorption coefficient in parabolic GaAs quantum dot in the presence of codirected electric and magnetic fields is obtained. An analytical dependence of the absorption threshold frequency on values of electric and magnetic fields is determined. It is shown that application of electric field makes the selection rules for the dipole transitions obsolete at the field direction. r 2005 Elsevier B.V. All rights reserved. PACS: 71.20.Nr; 73.61.Ey Keywords: Absorption coefficient; Selection rule; Quantum dot
1. Introduction Physical properties of quantum dots (QDs) are investigated both theoretically and experimentally. For instance, electronic and impurity states in QDs are investigated in detail (see, e.g. Refs. [1–3]). As a result of these investigations, the strong interdependence between the character of QD energy spectrum and its geometrical parameters (size, shape) has been found. These physical–chemical properties of QD and surrounding medium form the character of potential well. On the other side, QD shape and size conditions the height and symmetry of this potential. Thus, it is natural to suppose, that optical, kinetic, etc. properties of QD depend on abovementioned properties as well. In particular, it is well known, that the spectrum of interband optical absorption in semiconductors is conditioned by wave functions and energy spectra of charge carriers, present inside them [4]. The paper after Efros’s [5] was one of the firsts in which the optical absorption in QD was theoretically investigated. They investigated the peculiarities of interband optical absorption in spherical QD with confinement potential, described in the scopes of spherical symmetrical infinitely high potential well. The authors of Corresponding author.
E-mail address: [email protected] (H.A. Sarkisyan). 1386-9477/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2005.10.008
Ref. [6] considered light absorption in spherical QDs, taking into account the anisotropy of zonal structure. It was shown that the account of anisotropy results in appearance of optical transitions, forbidden in isotropic approximation. In Ref. [7], the authors considered light absorption in cylindrical QD in the presence of magnetic field. It should be mentioned that the parabolic approximation of the confinement potential of QD introduced in works of Maksym, Chakraborty [8] and Peeters [9] enables to carry out detailed analytic analysis of the one-particle energetic spectrum and wave functions at the presence of codirected electric and magnetic fields. In its turn, that serves as a base for calculation of corresponding interband optical absorption for the given system. In this short article, interband optical absorption in GaAs spherical shape parabolic QD in the presence of electrical and magnetic fields is investigated. 2. Theory Let us present the confinement potential of QD V conf ð~ rÞ in the form V conf ðr; j; zÞ ¼
mo2 r2 mo2 ðr2 þ z2 Þ , ¼ 2 2
(1)
ARTICLE IN PRESS M.S. Atoyan et al. / Physica E 31 (2006) 83–85
84
where o is the frequency of QD confinement potential. For o, we have o
_ , mr20
(2)
where r0 is the radius of spherical QD. Suppose the fields are directed along the OZ axis, the Schroedinger equation will be written as 1 ~ ^ e~ 2 P A c eezc þ V conf ð~ rÞc ¼ Ec, (3) 2m c ~ is the vector potential of the magnetic field where A ~ ¼ AðA ~ r ¼ Az ¼ 0; Aj ¼ Hr=2Þ, ~ e is the electrostatic A intensity, and m is the effective electronic mass (for GaAs, m ¼ 0:067m0 ). The Schroedinger equation that corresponds to potential (1) in cylindrical coordinates has a form _2 1 q q q2 1 q2 r þ 2þ 2 2 c qr r qj 2m r qr qz i_oc qc mo2c r2 þ c 2 qj 8 mo2 r2 mo2 z2 eezc þ cþ c ¼ ðE r þ E z Þc, 2 2 where oc ¼ eH=mc is the cyclotron frequency. Let us present a wave function of an electron as
ð4Þ
cðr; j; zÞ ¼ f ðr; j; ÞwðzÞ
(5)
and by separating variables in Eq. (5), we shall obtain equations that determine f ðr; jÞ and wðzÞ _2 1 q qf 1 q2 f r þ 2 2 qr r qj 2m r qr i_oc qf mO2 r2 þ f ¼ Erf , 2 qj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 ffi
where O ¼
ð6Þ
2
o2c þ ð2oÞ and
_2 d2 w mo2 z2 w eezw ¼ E z w. þ 2m dz2 2
(7)
Energy levels of electron are determined by formula (7) jmj þ 1 _oc m þ E nr ;m;n ¼ _O nr þ 2 2 2 2 1 ee þ _o n þ . ð10Þ 2 2mo2 As to the wave function, it will be the product of functions (8) and (9) " #1=2 ðjmj þ nr Þ! 1 1 cðr; j; zÞ ¼ pffiffiffiffiffiffi 1þjmj jmj 2 nr !jmj! 2p a r2 2 2 eimj er =4a rjmj F nr ; jmj þ 1; 2 2a mo1=2 1 2 2 pffiffiffiffiffiffiffiffi ffi emo=_ðzðe=mo ÞÞ p_ 2n n! rffiffiffiffiffiffiffi mo e H n z 2 . ð11Þ _ mo Expressions (10) and (11), obtained above for charge carriers energy spectrum and wave functions in spherical QD under the influence of external electrical and magnetic fields, allow to calculate the direct interband light absorption coefficient Kð$Þ in such system. In case of strong size quantization, when is possible to neglect electron–hole interactions (taking into account exciton effects needs apart consideration and in the scopes of given article is not observed) according to Ref. [5] for light absorption coefficient we have expression: 2 X Z h ce Kð$Þ ¼ A c d~ r 0 0 0 nr ;m;n nr ;m ;n nr n0r mm0 nn0
d D E enr ;m;n E hn0r ;m0 ;n0 ,
ð12Þ
where D ¼ _$ E g , E g is the width of forbidden zone, $ is the frequency of incident light, ceðhÞ is the wave function of electron (hole), E eðhÞ is the corresponding energy of electron (hole), A is the quantity, proportional to the square of dipole moment matrix element modulus, taken on Bloch functions. Inserting expressions for wave functions into Eq. (11) and integrating, we obtain 2 jmjþ1 X ah a2e nr nn0 Kð$Þ ¼ A Bnr n0r I nn0 ð1Þ Gðjmj þ 1Þ 4a2h a2e nr n0
The solutions of Eqs. (6) and (7) are known and look like [10] " #1=2 ðjmj þ nr Þ! 1 1 f ðr; jÞ ¼ pffiffiffiffiffiffi 1þjmj jmj 2 nr !jmj! 2p a r2 imj r2 =4a2 jmj e e r F nr ; jmj þ 1; 2 , ð8Þ 2a pffiffiffiffiffiffiffiffiffiffiffi where a ¼ _=mO, F ða; b; xÞ is the confluent hypergeometric function, m is the magnetic quantum number, nr the radial quantum number and mo1=2 1 2 2 pffiffiffiffiffiffiffiffi ffi emo=_ðzðe=mo ÞÞ wðzÞ ¼ p_ 2n n! rffiffiffiffiffiffiffi mo e H n z 2 , ð9Þ _ mo
where Bnn nr n0r are some constants expressed through the
where H n ðxÞ is the Hermite polynomial and n is the quantum number.
normalization constants of wave functions ce and ch , 2 F 1 ða; b; g; xÞ is the hypergeometric function, GðxÞ is the
r mm0 nn0
4a2h a2e 2F 1 þ 1; ða2e a2h Þ2 d D E enr ;m;n E hn0r ;m;n0 ,
!
nr ; n0r ; jmj
ð13Þ
0
ARTICLE IN PRESS M.S. Atoyan et al. / Physica E 31 (2006) 83–85
Euler function, aeðhÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffi
_ meðhÞ OeðhÞ ,
and I nn0 is the integral of
the form Z 1 2 2 2 2 eme oe =_ðzðee=me oe ÞÞ emh oh =_ðzþðee=mh oh ÞÞ I nn0 ¼ 1 rffiffiffiffiffiffiffiffiffiffi me oe ee z H n me o2e _ ffiffiffiffiffiffiffiffiffiffi ffi r mh oh ee zþ H n0 dz. _ mh o2h
ð14Þ
85
~ It is important to mention, that cated dependence on H. the Expression (16) makes sense at such ~ e fields, that the particle in Z direction is localized inside QD. In conclusion, let us turn to the analysis of nature of electron energy level shift under electrical field. In the considered case, owing to the complete nondegeneracy of energy levels, the dipole approach is absent. It explains Stark splitting quadratic dependence on ~ e DE z ¼
e2 2 . 2me o2
(17)
3. Conclusion For magnetic quantum number m, we have the following selection rule
Acknowledgements
m ¼ m0 .
This work was supported by the Armenian State Program ‘‘Semiconductor Nanoelectronics’’.
(15)
In direction of electric field the selection rule for dipole transitions obsolete. Here, it is worth to mention that in the absence of the field, in case of parabolic approximation of the confinement potential, the transitions along the OZaxis direction take place between levels of the same parity only [11]. The argument of Dirak d-function allows to define the threshold frequency of absorption $00 $00 ¼
E g Oe þ Oh oe þ oh þ þ _ 2 2 e2 e 2 1 1 þ . 2_ me o2e mh o2h
References [1] [2] [3] [4] [5] [6] [7]
ð16Þ
e From the Expression (16), it follows that $00 rises on field~ by quadratic law, at the same time having more compli-
[8] [9] [10] [11]
J.-L. Zhu, J.-J. Xiong, B.-L. Gu, Phys. Rev. B 41 (1990) 6001. C. Bose, C. Sarkar, Physica B 253 (1998) 238. K.G. Dvoyan, E.M. Kazaryan, Phys. Status Solidi B 228 (2001) 695. D. Bimberg, M. Grundman, N. Ledentsov, Quantum Dot Heterostructures, Wiley, New York, 1999. Al.L. Efros, A.L. Efros, Semiconductors 16 (1982) 1209. A.D. Andreev, A.A. Lipovskii, Semiconductors 33 (1999) 1450. M.S. Atoyan, E.M. Kazaryan, H.A. Sarkisyan, Physica E 22 (2004) 860. P. Maksym, T. Chakraborty, Phys. Rev. Lett. 65 (1990) 108. F. Peeters, Phys. Rev. B 42 (1990) 1486. S. Flugge, Practical Quantum Mechanics 1, Springer, Berlin, 1971. H.A. Sarkisyan, Mod. Phys. Lett. B 18 (2004) 443.
Physica E 31 (2006) 83–85 www.elsevier.com/locate/physe
Interband light absorption in parabolic quantum dot in the presence of electrical and magnetic fields M.S. Atoyan, E.M. Kazaryan, H.A. Sarkisyan Department of Solid State Physics, Yerevan State University, 1. Al. Manoukyan 375049 Armenia Received 12 September 2005; accepted 4 October 2005 Available online 15 December 2005
Abstract An analytical expression for the light interband absorption coefficient in parabolic GaAs quantum dot in the presence of codirected electric and magnetic fields is obtained. An analytical dependence of the absorption threshold frequency on values of electric and magnetic fields is determined. It is shown that application of electric field makes the selection rules for the dipole transitions obsolete at the field direction. r 2005 Elsevier B.V. All rights reserved. PACS: 71.20.Nr; 73.61.Ey Keywords: Absorption coefficient; Selection rule; Quantum dot
1. Introduction Physical properties of quantum dots (QDs) are investigated both theoretically and experimentally. For instance, electronic and impurity states in QDs are investigated in detail (see, e.g. Refs. [1–3]). As a result of these investigations, the strong interdependence between the character of QD energy spectrum and its geometrical parameters (size, shape) has been found. These physical–chemical properties of QD and surrounding medium form the character of potential well. On the other side, QD shape and size conditions the height and symmetry of this potential. Thus, it is natural to suppose, that optical, kinetic, etc. properties of QD depend on abovementioned properties as well. In particular, it is well known, that the spectrum of interband optical absorption in semiconductors is conditioned by wave functions and energy spectra of charge carriers, present inside them [4]. The paper after Efros’s [5] was one of the firsts in which the optical absorption in QD was theoretically investigated. They investigated the peculiarities of interband optical absorption in spherical QD with confinement potential, described in the scopes of spherical symmetrical infinitely high potential well. The authors of Corresponding author.
E-mail address: [email protected] (H.A. Sarkisyan). 1386-9477/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2005.10.008
Ref. [6] considered light absorption in spherical QDs, taking into account the anisotropy of zonal structure. It was shown that the account of anisotropy results in appearance of optical transitions, forbidden in isotropic approximation. In Ref. [7], the authors considered light absorption in cylindrical QD in the presence of magnetic field. It should be mentioned that the parabolic approximation of the confinement potential of QD introduced in works of Maksym, Chakraborty [8] and Peeters [9] enables to carry out detailed analytic analysis of the one-particle energetic spectrum and wave functions at the presence of codirected electric and magnetic fields. In its turn, that serves as a base for calculation of corresponding interband optical absorption for the given system. In this short article, interband optical absorption in GaAs spherical shape parabolic QD in the presence of electrical and magnetic fields is investigated. 2. Theory Let us present the confinement potential of QD V conf ð~ rÞ in the form V conf ðr; j; zÞ ¼
mo2 r2 mo2 ðr2 þ z2 Þ , ¼ 2 2
(1)
ARTICLE IN PRESS M.S. Atoyan et al. / Physica E 31 (2006) 83–85
84
where o is the frequency of QD confinement potential. For o, we have o
_ , mr20
(2)
where r0 is the radius of spherical QD. Suppose the fields are directed along the OZ axis, the Schroedinger equation will be written as 1 ~ ^ e~ 2 P A c eezc þ V conf ð~ rÞc ¼ Ec, (3) 2m c ~ is the vector potential of the magnetic field where A ~ ¼ AðA ~ r ¼ Az ¼ 0; Aj ¼ Hr=2Þ, ~ e is the electrostatic A intensity, and m is the effective electronic mass (for GaAs, m ¼ 0:067m0 ). The Schroedinger equation that corresponds to potential (1) in cylindrical coordinates has a form _2 1 q q q2 1 q2 r þ 2þ 2 2 c qr r qj 2m r qr qz i_oc qc mo2c r2 þ c 2 qj 8 mo2 r2 mo2 z2 eezc þ cþ c ¼ ðE r þ E z Þc, 2 2 where oc ¼ eH=mc is the cyclotron frequency. Let us present a wave function of an electron as
ð4Þ
cðr; j; zÞ ¼ f ðr; j; ÞwðzÞ
(5)
and by separating variables in Eq. (5), we shall obtain equations that determine f ðr; jÞ and wðzÞ _2 1 q qf 1 q2 f r þ 2 2 qr r qj 2m r qr i_oc qf mO2 r2 þ f ¼ Erf , 2 qj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 ffi
where O ¼
ð6Þ
2
o2c þ ð2oÞ and
_2 d2 w mo2 z2 w eezw ¼ E z w. þ 2m dz2 2
(7)
Energy levels of electron are determined by formula (7) jmj þ 1 _oc m þ E nr ;m;n ¼ _O nr þ 2 2 2 2 1 ee þ _o n þ . ð10Þ 2 2mo2 As to the wave function, it will be the product of functions (8) and (9) " #1=2 ðjmj þ nr Þ! 1 1 cðr; j; zÞ ¼ pffiffiffiffiffiffi 1þjmj jmj 2 nr !jmj! 2p a r2 2 2 eimj er =4a rjmj F nr ; jmj þ 1; 2 2a mo1=2 1 2 2 pffiffiffiffiffiffiffiffi ffi emo=_ðzðe=mo ÞÞ p_ 2n n! rffiffiffiffiffiffiffi mo e H n z 2 . ð11Þ _ mo Expressions (10) and (11), obtained above for charge carriers energy spectrum and wave functions in spherical QD under the influence of external electrical and magnetic fields, allow to calculate the direct interband light absorption coefficient Kð$Þ in such system. In case of strong size quantization, when is possible to neglect electron–hole interactions (taking into account exciton effects needs apart consideration and in the scopes of given article is not observed) according to Ref. [5] for light absorption coefficient we have expression: 2 X Z h ce Kð$Þ ¼ A c d~ r 0 0 0 nr ;m;n nr ;m ;n nr n0r mm0 nn0
d D E enr ;m;n E hn0r ;m0 ;n0 ,
ð12Þ
where D ¼ _$ E g , E g is the width of forbidden zone, $ is the frequency of incident light, ceðhÞ is the wave function of electron (hole), E eðhÞ is the corresponding energy of electron (hole), A is the quantity, proportional to the square of dipole moment matrix element modulus, taken on Bloch functions. Inserting expressions for wave functions into Eq. (11) and integrating, we obtain 2 jmjþ1 X ah a2e nr nn0 Kð$Þ ¼ A Bnr n0r I nn0 ð1Þ Gðjmj þ 1Þ 4a2h a2e nr n0
The solutions of Eqs. (6) and (7) are known and look like [10] " #1=2 ðjmj þ nr Þ! 1 1 f ðr; jÞ ¼ pffiffiffiffiffiffi 1þjmj jmj 2 nr !jmj! 2p a r2 imj r2 =4a2 jmj e e r F nr ; jmj þ 1; 2 , ð8Þ 2a pffiffiffiffiffiffiffiffiffiffiffi where a ¼ _=mO, F ða; b; xÞ is the confluent hypergeometric function, m is the magnetic quantum number, nr the radial quantum number and mo1=2 1 2 2 pffiffiffiffiffiffiffiffi ffi emo=_ðzðe=mo ÞÞ wðzÞ ¼ p_ 2n n! rffiffiffiffiffiffiffi mo e H n z 2 , ð9Þ _ mo
where Bnn nr n0r are some constants expressed through the
where H n ðxÞ is the Hermite polynomial and n is the quantum number.
normalization constants of wave functions ce and ch , 2 F 1 ða; b; g; xÞ is the hypergeometric function, GðxÞ is the
r mm0 nn0
4a2h a2e 2F 1 þ 1; ða2e a2h Þ2 d D E enr ;m;n E hn0r ;m;n0 ,
!
nr ; n0r ; jmj
ð13Þ
0
ARTICLE IN PRESS M.S. Atoyan et al. / Physica E 31 (2006) 83–85
Euler function, aeðhÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffi
_ meðhÞ OeðhÞ ,
and I nn0 is the integral of
the form Z 1 2 2 2 2 eme oe =_ðzðee=me oe ÞÞ emh oh =_ðzþðee=mh oh ÞÞ I nn0 ¼ 1 rffiffiffiffiffiffiffiffiffiffi me oe ee z H n me o2e _ ffiffiffiffiffiffiffiffiffiffi ffi r mh oh ee zþ H n0 dz. _ mh o2h
ð14Þ
85
~ It is important to mention, that cated dependence on H. the Expression (16) makes sense at such ~ e fields, that the particle in Z direction is localized inside QD. In conclusion, let us turn to the analysis of nature of electron energy level shift under electrical field. In the considered case, owing to the complete nondegeneracy of energy levels, the dipole approach is absent. It explains Stark splitting quadratic dependence on ~ e DE z ¼
e2 2 . 2me o2
(17)
3. Conclusion For magnetic quantum number m, we have the following selection rule
Acknowledgements
m ¼ m0 .
This work was supported by the Armenian State Program ‘‘Semiconductor Nanoelectronics’’.
(15)
In direction of electric field the selection rule for dipole transitions obsolete. Here, it is worth to mention that in the absence of the field, in case of parabolic approximation of the confinement potential, the transitions along the OZaxis direction take place between levels of the same parity only [11]. The argument of Dirak d-function allows to define the threshold frequency of absorption $00 $00 ¼
E g Oe þ Oh oe þ oh þ þ _ 2 2 e2 e 2 1 1 þ . 2_ me o2e mh o2h
References [1] [2] [3] [4] [5] [6] [7]
ð16Þ
e From the Expression (16), it follows that $00 rises on field~ by quadratic law, at the same time having more compli-
[8] [9] [10] [11]
J.-L. Zhu, J.-J. Xiong, B.-L. Gu, Phys. Rev. B 41 (1990) 6001. C. Bose, C. Sarkar, Physica B 253 (1998) 238. K.G. Dvoyan, E.M. Kazaryan, Phys. Status Solidi B 228 (2001) 695. D. Bimberg, M. Grundman, N. Ledentsov, Quantum Dot Heterostructures, Wiley, New York, 1999. Al.L. Efros, A.L. Efros, Semiconductors 16 (1982) 1209. A.D. Andreev, A.A. Lipovskii, Semiconductors 33 (1999) 1450. M.S. Atoyan, E.M. Kazaryan, H.A. Sarkisyan, Physica E 22 (2004) 860. P. Maksym, T. Chakraborty, Phys. Rev. Lett. 65 (1990) 108. F. Peeters, Phys. Rev. B 42 (1990) 1486. S. Flugge, Practical Quantum Mechanics 1, Springer, Berlin, 1971. H.A. Sarkisyan, Mod. Phys. Lett. B 18 (2004) 443.