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Electronic states and interband light absorption in semi-spherical quantum dot under the influence of strong magnetic field
Solid State Communications 139 (2006) 537–540 www.elsevier.com/locate/ssc
Electronic states and interband light absorption in semi-spherical quantum dot under the influence of strong magnetic field L.A. Juharyan a , E.M. Kazaryan b , L.S. Petrosyan a,b,∗ a Department of Solid State Physics, Yerevan State University, 1 Alex Manoogian str., Yerevan 375025, Armenia b Physicotechnical Department, Russian-Armenian (Slavonic) State University, 123 Hovseph Emin str., Yerevan 375051, Armenia
Received 25 November 2005; received in revised form 13 March 2006; accepted 7 July 2006 by B.-F. Zhu Available online 21 July 2006
Abstract In the framework of effective mass approximation the electronic states in semi-spherical quantum lens under the influence of strong magnetic field are investigated. We have used the adiabatic approximation for the case of strong magnetic field. The eigenfunctions and eigenvalues of this problem are determined. In strong confinement regime interband optical absorption of light is investigated in quantum lens from InAs. The threshold frequencies of absorption are determined. The comparison with the case of film under the influence of strong magnetic field with infinitely high confinement potential is performed. c 2006 Elsevier Ltd. All rights reserved.
PACS: 71.20.Nr; 7.61 Ey Keywords: A. Quantum lens; D. Absorption coefficient; E. Adiabatic approximation
1. Introduction Several progresses in growth techniques during the last few years allow one to obtain semiconductor low-dimensional systems with a high degree of accuracy and crystalline quality. Because of their applications in nano-electronic devices, such artificial structures have been intensively studied. In particular, quantum dots (QDs) obtained by interrupted growth in stained semiconductor interfaces (as InAs/GaAs, for example) constitute, at present, a research subject for many experimental and theoretical groups [1]. As is known, optoelectronic properties of QD strongly depend on their geometrical shapes and sizes. Experiments, performed by different methods, investigating QD geometrical shapes and sizes showed that self-assembled QDs have lens-like surface (so called quantum lenses (QLs)). Different techniques used to study the shape and size of self-assembled QDs suggest the lens
∗ Corresponding author at: Physicotechnical Department, Russian-Armenian (Slavonic) State University, 123 Hovseph Emin str., Yerevan 375051, Armenia. E-mail address: [email protected] (L.S. Petrosyan).
c 2006 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2006.07.012
format as the one which resembles in the best way the actual geometry of such structures [2,3], it seems to be reasonable that, when the dot diameter is small enough, it can be approached by a semi-sphere. As is known, the application of magnetic field could provide relevant information about the behavior of electrons in solids. For this reason, in the present paper we try to develop the main features of the electronic structure of self-assembled QDs under the effects of magnetic field applied perpendicularly to the dot base. Our theoretical model is based on a simplified semi-spherical QD in the strong confinement regime. A similar problem was solved in [4]. But the authors of [4] considered weak magnetic fields and, using the secondorder approximation of perturbation theory, calculated energy states. In our paper we consider the case of strong magnetic field, which allows us to use the adiabatic approximation and obtain energy states. For example in Ref. [5] the adiabatic approximation was used to solve the problem of electronic state energies in flat QL. But there the validity of adiabatic approximation is conditioned by the geometry of QL (the height of QL is much smaller than the radius of QL). In contrast, in our case the validity is provided by strong magnetic field.
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L.A. Juharyan et al. / Solid State Communications 139 (2006) 537–540
It is also given by the dependence of interband absorption frequency on the geometrical parameters of QD and on the magnetic field. 2. Theory We consider electronic states in a semi-spherical QD, under the influence of strong magnetic field. The confining potential of QD is approximated by an indefinitely high potential well and can be written as p ρ ≤ ρ = R2 − z2 0 0, U (ρ, z) = (1) z≥0 ∞, otherwise. It is proper to mention that by modeling the confining potential as (1) we neglected the existence of strain effect, which appears on the edge of QD-surrounding area. It comes from the fact that the potential would bring about the change of the shape of indefinitely high potential well, however this would not have any effect on qualitative outcomes. Taking into account that studying the effect of strain potential was not the goal of our paper we neglected this factor. It is moreover approved, because the authors in Ref. [5] included the existence of strain effect in Hamiltonian by including that potential into the value of confining potential. In the framework of the effective mass approximation, the Schr¨odinger equation is given by ˆE 2 ˆE − e A/c) ( p + U (ρ, z) Ψ = EΨ , (2) 2µ where µ is the carrier effective mass, e is the electron charge, c is the speed of light, AE is vector-potential of magnetic field HE . Calibration of vector-potential AE in cylindrical coordinates can be written as Aϕ = Hρ/2, Aρ = A z = 0, so that magnetic field is applied in the direction of QL rotation axis. Due to the symmetry of this problem, it is convenient to solve it in cylindrical coordinates. We will seek the solution of Eq. (2) as Ψ=
f (ρ, z)eimϕ , √ 2π
(3)
where m = 0, ±1, ±2, . . ., is magnetic quantum number. For the function f (ρ, z) we have: h¯ 2 ∂ 2 1 ∂ m2 h¯ 2 ∂ 2 − + − f − f 2 2 2µ ∂ρ ρ ∂ρ 2µ ∂z 2 ρ +
µω2H ρ 2 f + U (ρ, z) f = ε f, 8
(4)
where ε = E − h¯ ω2H m and ω H = eH µc is cyclotron frequency. The Eq. (4) does not have analytical solution. When the magnetic field is sufficiently strong, the problem can be solved in adiabatic approximation [6]. The Hamiltonian of the whole system can be expressed as Hˆ (ρ, z) = Hˆ 1 (ρ, z) + Hˆ 2 (z),
(5)
where m2 h¯ 2 ∂ 2 1 ∂ − Hˆ 1 (ρ, z) = − + 2µ ∂ρ 2 ρ ∂ρ ρ2 2 2 µω H ρ + + U1 (ρ, z), 8 h¯ 2 ∂ 2 Hˆ 2 (z) = − + U2 (z) + εn 1 ,m (z) 2µ ∂z 2 are the Hamiltonians of “quick” and “slow” subsystems, correspondingly, the potential U (ρ, z) = U1 (ρ, z) + U2 (z), ( p 0, ρ ≤ ρ0 = pR 2 − z 2 U1 (ρ, z) = ∞, ρ ≥ ρ0 = R 2 − z 2 , 0, z ≥ 0 U2 (z) = ∞, z ≤ 0.
(6)
In adiabatic approximation the eigenfunctions of Hamiltonian for Eq. (4) can be written as f ∼ = f n 1 ,n 2 ,m (z) f n 1 ,m (ρ, z),
(7)
where f n 1 ,m (ρ, z) is the wave function of “quick” and f n 1 ,n 2 ,m (z) is the wave function of “slow” subsystem. At first we will solve the Schr¨odinger equation for “quick” subsystem: Hˆ 1 f n 1 ,m (ρ, z) = εn 1 ,m f n 1 ,m (ρ, z),
(8)
where the z-coordinate is a fixed parameter. Then in potential U1 (ρ, z), at fixed value of the z-coordinate, the ρ0 -coordinate is the radius of area, where U1 ≡ 0. It is easy to see, that inside QD (U1 (ρ, z) = 0) Eq. (8) coincides with the radial part of the Schr¨odinger equation for electron motion in the magnetic field. Then the wave function will be ! |m| 2 2 2 − ρ2 ρ f n 1 ,m (ρ) = Ce 4a H 2a 2H # " εn 1 ,m |m| + 1 ρ2 − (9) × 1 F1 − , |m| + 1, 2 , h¯ ω H 2 2a H where C is the normalization constant. The eigenvalues of Eq. (8) are found from the boundary condition 1 F 1 [−(σn 1 ,m
− (|m| + 1)/2), |m| + 1, ρ˜02 /2] = 0,
(10)
where σn 1 ,m = εn 1 ,m h¯ ω H , ρ˜0 = ρ0 /a H . In dimensional variables the energy is given by ρ0 εn 1 ,m = h¯ ω H σn 1 ,m . (11) aH In adiabatic approximation the energies εn 1 ,m are effective potentials for the “slow” subsystem. The z-coordinate varies from zero to R. In Fig. 1 the dependence of σn 1 ,m on z is presented at fixed radius of QD R˜ = aRH = 5 and for quantum numbers m = 0, 1, n 1 = 1, 2, 3. When z → 0 for the “quick” subsystem we have two-dimensional (2D) oscillatory problem in indefinitely high potential well with the radius ρ0 = R and
L.A. Juharyan et al. / Solid State Communications 139 (2006) 537–540
Fig. 1. Solution of expression (10) σn 1 ,m and Hulthen approximation depending on coordinate az , when the radius of QD is fixed R˜ = 5. H
fixed energy value. When R a H , these energies (Eq. (11)) are approximately the same as the energies of 2D oscillatory problem h¯ ω H (n+1) with frequency ω = ω2H (see Fig. 1). And at 2 z → R we have the same problem with the condition ρ0 → 0, for which the energies tend to infinity. According to adiabatic approximation, the Hamiltonian of the “slow” subsystem can be written as h¯ 2 ∂ 2 + U2 (z) + εn 1 ,m (z). Hˆ 2 (z) = − 2µ ∂z 2
(12)
The eigenvalue equation is Hˆ 2 f n 1 ,n 2 ,m (z) = εn 1 ,n 2 ,m f n 1 ,n 2 ,m (z),
(13)
where εn 1 ,n 2 ,m = E − h¯ ω2H m . The Eq. (13) with the potential εn 1 ,m has no exact solution. But for example in the Ref. [5] this problem is solved by approximating εn 1 ,m by the n-step piecewise potential. The numerical calculation shows that the function εn 1 ,m can be approximated to a high accuracy with the Hulthen potential [7] Un 1 ,m (z) = −V0
e−
1−e
(z−z 0 ) a
,
(14)
where V0 , z 0 , a are free parameters determined from approximating conditions; for example, in this case z 0 coincides with the radius of QD (z 0 = R). In Fig. 1 these approximations are given for some functions σn 1 ,m (z). In Eq. (13) after the replacement of εn 1 ,m (z) by the Hulthen potential, we obtain the following Schr¨odinger equation: h¯ 2 ∂ 2 f n ,n ,m (z) 2µ ∂z 2 1 2
where q χ1 = y −iα 2 F1 −iα − −α 2 + β 2 , −iα q + −α 2 + β 2 , 1 − 2iα; y , q χ2 = y iα 2 F 1 iα − −α 2 + β 2 , iα q 2 2 + −α + β , 1 + 2iα; y , s s (z 0 −z ) 2µV0 2µεn 1 ,n 2 ,m y=e a , β=a , α=a . 2 h¯ h¯ 2 The solution of Eq. (15) should satisfy the boundary conditions and the normalization requirement for the wave function f n 1 ,n 2 ,m (0) = 0, f n 1 ,n 2 ,m (R) = 0, Z R f n21 ,n 2 ,m (z) dz = 1.
(17)
From the last two equations of (17) one can find the normalization constants. And from the first transcendental equation of (17) we obtain the eigen-energies εn 1 ,n 2 ,m χ1 (0)χ2 (z 0 ) − χ1 (z 0 )χ2 (0) = 0.
(18)
Finally the energy of the system can be written as E n 1 ,n 2 ,m = εn 1 ,n 2 ,m +
h¯ ω H m . 2
(19)
The criterion of usage of the adiabatic approximation is (z 0 −z)
V0 e a + εn 1 ,n 2 ,m − − f n 1 ,n 2 ,m (z) = 0. (z 0 −z) 1−e a
(15)
The exact solution of Eq. (15) is given by the hypergeometric function of the second kind 2 F 1 [a, b, c, z] f n 1 ,n 2 ,m (z) = Aχ1 + Bχ2 ,
Fig. 2. The dependencies of electron state energies on parameter γ = aR (for H the levels m = 0, n 1 = 0, n 2 = 1 and m = 0, n 1 = 0, n 2 = 2).
0
(z−z 0 ) a
−
539
(16)
h¯ ω H m + 2Un 1 ,m (0) εn 1 ,n 2 ,m . 2
(20)
3. Discussion of results In Fig. 2 explicit energy states are presented as a function of the parameter γ = R/a H . For comparison it has shown the
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L.A. Juharyan et al. / Solid State Communications 139 (2006) 537–540
analogous dependency for the ground state energy of the film under influence of the perpendicular magnetic field. As we had expected, the increasing of parameter γ lead to the decreasing of state energies. If γ 1 the problem comes to the problem of the film energy states under the effect of the perpendicular |m|+m+1 π 2 h¯ 2 2 magnetic field E nfilm = h ω (n + n . ) + ¯ H 1 2 1 ,n 2 ,m 2µR 2 2 From Fig. 2 one can see that the curves fit together. In Eq. (20) we used the values of Un 1 ,m (0) and εn 1 ,n 2 ,m obtained by numerical calculations. In view of assessment for quantum numbers the explicit criterion for use of the adiabatic approximation can be expressed for a film: n2
R p 2n 1 + |m| + m + 1. πa H
(21)
It could be satisfied, if the quantum numbers of electron radial motion get large values and at the same time the quantum numbers describing the z-coordinate take small values. For fixed quantum numbers we can obtain the value of magnetic field, so the multiplier R/a H satisfies the condition (21). Thus a strong magnetic field allowed us to apply the adiabatic approximation and give energy states and wave functions of QL. The expressions (3), (7), (9) and (16), obtained above for charge carrier’s energy spectrum and wave functions in semi-spherical QL under the influence of external magnetic field, allow the calculation of the interband light absorption coefficient K (ω) in such systems. In case of strong size quantization R a eB , when it is possible to neglect electronhole interactions (for example in InAs, where the effective Bohr ˚ effective electron mass is µe = 0.023 m e radius is a eB = 350 A, and effective mass hole µe = 0.33 m e ) according to [8,9] we have: 2 X Z e h K (ω) = A r Ψn 1 ,n 2 ,m Ψn 0 ,n 0 ,m 0 dE n 1 n 01 mm 0 n 2 n 02
1
2
× δ ∆ − E ne1 ,n 2 ,m − E nh0 ,n 0 ,m 0 , 1
2
(22)
Fig. 3. The dependence of W on aR for fixed value of magnetic field a H = 1 e 7 aB .
B
m = m0.
(23)
The argument of Dirac δ-function allows the definition of the threshold frequency of absorption ω00 1 e h {E g + E 0,1,0 + E 0,1,0 }. (24) h¯ For the case of strong size quantization this dependence is 2 (h ω −εg ) units, where E eR = h¯ e2 presented in Fig. 3 (in W = ¯ 00 Ee ω00 =
R
2µa B
the Rydberg energy for InAs). Curve 1 corresponds to the case of QL and curve 2 corresponds to the interband absorption in film. As it follows from Fig. 3, the curve of the W versus R/a B dependence for QL case is higher then the curve for film case. Along with increasing R, when the role of size quantization decreases, curves 1 and 2 decrease and come close to each other. Vice versa, at small R the role of size quantization sharply increases and therefore the effective width of forbidden band increases. Acknowledgements
2
where ∆ = h¯ ω − E g , E g is the width of forbidden zone, ω is the frequency of incident light, Ψ e(h) is the wave function of electron (hole), E e(h) 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. It is significant, that for quantum structures like InAs/In1−x Alx As valence band formed bands of light and heavy holes and also the bands of spin-orbital shift. In connection with this the value of spin-orbital shift is bigger than the energy of size quantization, and so we do not consider the transition from this band to conduction band [10]. In the other case, the bands of heavy and light holes quantized separately, so we consider transmissions between the band of heavy hole (for that band the square rule of dispersion law is more realistic) and the conduction band. After integrating we found, that the selection rule is only for magnetic quantum number
This work was supported by Armenian State Program “Semiconductor Nanoelectronics”. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
P.M. Petroff, G. Medeiros-Ribero, Mater. Res. Bull. 21 (1996) 50. D.J. Eaglesham, M. Cerullo, Phys. Rev. Lett. 64 (1990) 1942. J.H. Zhu, K. Brunner, G. Abstreiter, Appl. Phys. Lett. 72 (1998) 424. J. Lo’pez Gondar, B. Costa, C. Trallero-Giner, G. Marques, Phys. Status Solidi (b) 230 (2002) 437. A. Wojs, P. Hawerylac, S. Fafard, L. Jacak, Phys. Rev. B 54 (8) (1996) 5604. V.M. Galitsky, B.M. Karnakov, V.I. Kogan, Problems of Quantum Mechanics, M. Nauka, 1981. S. Fl¨ugge, Practical Quantum Mechanics, Berlin, 1971. Al.L. Efros, A.L. Efros, Semiconductors 16 (1982) 1209. M.S. Atoyan, E.M. Kazaryan, H.A. Sarkisyan, Physica E 31 (2006) 83. V. Demikhovskii, G. Vugalter, Physics of quantum low dimensional structures, M. Logos, 2000.
Electronic states and interband light absorption in semi-spherical quantum dot under the influence of strong magnetic field L.A. Juharyan a , E.M. Kazaryan b , L.S. Petrosyan a,b,∗ a Department of Solid State Physics, Yerevan State University, 1 Alex Manoogian str., Yerevan 375025, Armenia b Physicotechnical Department, Russian-Armenian (Slavonic) State University, 123 Hovseph Emin str., Yerevan 375051, Armenia
Received 25 November 2005; received in revised form 13 March 2006; accepted 7 July 2006 by B.-F. Zhu Available online 21 July 2006
Abstract In the framework of effective mass approximation the electronic states in semi-spherical quantum lens under the influence of strong magnetic field are investigated. We have used the adiabatic approximation for the case of strong magnetic field. The eigenfunctions and eigenvalues of this problem are determined. In strong confinement regime interband optical absorption of light is investigated in quantum lens from InAs. The threshold frequencies of absorption are determined. The comparison with the case of film under the influence of strong magnetic field with infinitely high confinement potential is performed. c 2006 Elsevier Ltd. All rights reserved.
PACS: 71.20.Nr; 7.61 Ey Keywords: A. Quantum lens; D. Absorption coefficient; E. Adiabatic approximation
1. Introduction Several progresses in growth techniques during the last few years allow one to obtain semiconductor low-dimensional systems with a high degree of accuracy and crystalline quality. Because of their applications in nano-electronic devices, such artificial structures have been intensively studied. In particular, quantum dots (QDs) obtained by interrupted growth in stained semiconductor interfaces (as InAs/GaAs, for example) constitute, at present, a research subject for many experimental and theoretical groups [1]. As is known, optoelectronic properties of QD strongly depend on their geometrical shapes and sizes. Experiments, performed by different methods, investigating QD geometrical shapes and sizes showed that self-assembled QDs have lens-like surface (so called quantum lenses (QLs)). Different techniques used to study the shape and size of self-assembled QDs suggest the lens
∗ Corresponding author at: Physicotechnical Department, Russian-Armenian (Slavonic) State University, 123 Hovseph Emin str., Yerevan 375051, Armenia. E-mail address: [email protected] (L.S. Petrosyan).
c 2006 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2006.07.012
format as the one which resembles in the best way the actual geometry of such structures [2,3], it seems to be reasonable that, when the dot diameter is small enough, it can be approached by a semi-sphere. As is known, the application of magnetic field could provide relevant information about the behavior of electrons in solids. For this reason, in the present paper we try to develop the main features of the electronic structure of self-assembled QDs under the effects of magnetic field applied perpendicularly to the dot base. Our theoretical model is based on a simplified semi-spherical QD in the strong confinement regime. A similar problem was solved in [4]. But the authors of [4] considered weak magnetic fields and, using the secondorder approximation of perturbation theory, calculated energy states. In our paper we consider the case of strong magnetic field, which allows us to use the adiabatic approximation and obtain energy states. For example in Ref. [5] the adiabatic approximation was used to solve the problem of electronic state energies in flat QL. But there the validity of adiabatic approximation is conditioned by the geometry of QL (the height of QL is much smaller than the radius of QL). In contrast, in our case the validity is provided by strong magnetic field.
538
L.A. Juharyan et al. / Solid State Communications 139 (2006) 537–540
It is also given by the dependence of interband absorption frequency on the geometrical parameters of QD and on the magnetic field. 2. Theory We consider electronic states in a semi-spherical QD, under the influence of strong magnetic field. The confining potential of QD is approximated by an indefinitely high potential well and can be written as p ρ ≤ ρ = R2 − z2 0 0, U (ρ, z) = (1) z≥0 ∞, otherwise. It is proper to mention that by modeling the confining potential as (1) we neglected the existence of strain effect, which appears on the edge of QD-surrounding area. It comes from the fact that the potential would bring about the change of the shape of indefinitely high potential well, however this would not have any effect on qualitative outcomes. Taking into account that studying the effect of strain potential was not the goal of our paper we neglected this factor. It is moreover approved, because the authors in Ref. [5] included the existence of strain effect in Hamiltonian by including that potential into the value of confining potential. In the framework of the effective mass approximation, the Schr¨odinger equation is given by ˆE 2 ˆE − e A/c) ( p + U (ρ, z) Ψ = EΨ , (2) 2µ where µ is the carrier effective mass, e is the electron charge, c is the speed of light, AE is vector-potential of magnetic field HE . Calibration of vector-potential AE in cylindrical coordinates can be written as Aϕ = Hρ/2, Aρ = A z = 0, so that magnetic field is applied in the direction of QL rotation axis. Due to the symmetry of this problem, it is convenient to solve it in cylindrical coordinates. We will seek the solution of Eq. (2) as Ψ=
f (ρ, z)eimϕ , √ 2π
(3)
where m = 0, ±1, ±2, . . ., is magnetic quantum number. For the function f (ρ, z) we have: h¯ 2 ∂ 2 1 ∂ m2 h¯ 2 ∂ 2 − + − f − f 2 2 2µ ∂ρ ρ ∂ρ 2µ ∂z 2 ρ +
µω2H ρ 2 f + U (ρ, z) f = ε f, 8
(4)
where ε = E − h¯ ω2H m and ω H = eH µc is cyclotron frequency. The Eq. (4) does not have analytical solution. When the magnetic field is sufficiently strong, the problem can be solved in adiabatic approximation [6]. The Hamiltonian of the whole system can be expressed as Hˆ (ρ, z) = Hˆ 1 (ρ, z) + Hˆ 2 (z),
(5)
where m2 h¯ 2 ∂ 2 1 ∂ − Hˆ 1 (ρ, z) = − + 2µ ∂ρ 2 ρ ∂ρ ρ2 2 2 µω H ρ + + U1 (ρ, z), 8 h¯ 2 ∂ 2 Hˆ 2 (z) = − + U2 (z) + εn 1 ,m (z) 2µ ∂z 2 are the Hamiltonians of “quick” and “slow” subsystems, correspondingly, the potential U (ρ, z) = U1 (ρ, z) + U2 (z), ( p 0, ρ ≤ ρ0 = pR 2 − z 2 U1 (ρ, z) = ∞, ρ ≥ ρ0 = R 2 − z 2 , 0, z ≥ 0 U2 (z) = ∞, z ≤ 0.
(6)
In adiabatic approximation the eigenfunctions of Hamiltonian for Eq. (4) can be written as f ∼ = f n 1 ,n 2 ,m (z) f n 1 ,m (ρ, z),
(7)
where f n 1 ,m (ρ, z) is the wave function of “quick” and f n 1 ,n 2 ,m (z) is the wave function of “slow” subsystem. At first we will solve the Schr¨odinger equation for “quick” subsystem: Hˆ 1 f n 1 ,m (ρ, z) = εn 1 ,m f n 1 ,m (ρ, z),
(8)
where the z-coordinate is a fixed parameter. Then in potential U1 (ρ, z), at fixed value of the z-coordinate, the ρ0 -coordinate is the radius of area, where U1 ≡ 0. It is easy to see, that inside QD (U1 (ρ, z) = 0) Eq. (8) coincides with the radial part of the Schr¨odinger equation for electron motion in the magnetic field. Then the wave function will be ! |m| 2 2 2 − ρ2 ρ f n 1 ,m (ρ) = Ce 4a H 2a 2H # " εn 1 ,m |m| + 1 ρ2 − (9) × 1 F1 − , |m| + 1, 2 , h¯ ω H 2 2a H where C is the normalization constant. The eigenvalues of Eq. (8) are found from the boundary condition 1 F 1 [−(σn 1 ,m
− (|m| + 1)/2), |m| + 1, ρ˜02 /2] = 0,
(10)
where σn 1 ,m = εn 1 ,m h¯ ω H , ρ˜0 = ρ0 /a H . In dimensional variables the energy is given by ρ0 εn 1 ,m = h¯ ω H σn 1 ,m . (11) aH In adiabatic approximation the energies εn 1 ,m are effective potentials for the “slow” subsystem. The z-coordinate varies from zero to R. In Fig. 1 the dependence of σn 1 ,m on z is presented at fixed radius of QD R˜ = aRH = 5 and for quantum numbers m = 0, 1, n 1 = 1, 2, 3. When z → 0 for the “quick” subsystem we have two-dimensional (2D) oscillatory problem in indefinitely high potential well with the radius ρ0 = R and
L.A. Juharyan et al. / Solid State Communications 139 (2006) 537–540
Fig. 1. Solution of expression (10) σn 1 ,m and Hulthen approximation depending on coordinate az , when the radius of QD is fixed R˜ = 5. H
fixed energy value. When R a H , these energies (Eq. (11)) are approximately the same as the energies of 2D oscillatory problem h¯ ω H (n+1) with frequency ω = ω2H (see Fig. 1). And at 2 z → R we have the same problem with the condition ρ0 → 0, for which the energies tend to infinity. According to adiabatic approximation, the Hamiltonian of the “slow” subsystem can be written as h¯ 2 ∂ 2 + U2 (z) + εn 1 ,m (z). Hˆ 2 (z) = − 2µ ∂z 2
(12)
The eigenvalue equation is Hˆ 2 f n 1 ,n 2 ,m (z) = εn 1 ,n 2 ,m f n 1 ,n 2 ,m (z),
(13)
where εn 1 ,n 2 ,m = E − h¯ ω2H m . The Eq. (13) with the potential εn 1 ,m has no exact solution. But for example in the Ref. [5] this problem is solved by approximating εn 1 ,m by the n-step piecewise potential. The numerical calculation shows that the function εn 1 ,m can be approximated to a high accuracy with the Hulthen potential [7] Un 1 ,m (z) = −V0
e−
1−e
(z−z 0 ) a
,
(14)
where V0 , z 0 , a are free parameters determined from approximating conditions; for example, in this case z 0 coincides with the radius of QD (z 0 = R). In Fig. 1 these approximations are given for some functions σn 1 ,m (z). In Eq. (13) after the replacement of εn 1 ,m (z) by the Hulthen potential, we obtain the following Schr¨odinger equation: h¯ 2 ∂ 2 f n ,n ,m (z) 2µ ∂z 2 1 2
where q χ1 = y −iα 2 F1 −iα − −α 2 + β 2 , −iα q + −α 2 + β 2 , 1 − 2iα; y , q χ2 = y iα 2 F 1 iα − −α 2 + β 2 , iα q 2 2 + −α + β , 1 + 2iα; y , s s (z 0 −z ) 2µV0 2µεn 1 ,n 2 ,m y=e a , β=a , α=a . 2 h¯ h¯ 2 The solution of Eq. (15) should satisfy the boundary conditions and the normalization requirement for the wave function f n 1 ,n 2 ,m (0) = 0, f n 1 ,n 2 ,m (R) = 0, Z R f n21 ,n 2 ,m (z) dz = 1.
(17)
From the last two equations of (17) one can find the normalization constants. And from the first transcendental equation of (17) we obtain the eigen-energies εn 1 ,n 2 ,m χ1 (0)χ2 (z 0 ) − χ1 (z 0 )χ2 (0) = 0.
(18)
Finally the energy of the system can be written as E n 1 ,n 2 ,m = εn 1 ,n 2 ,m +
h¯ ω H m . 2
(19)
The criterion of usage of the adiabatic approximation is (z 0 −z)
V0 e a + εn 1 ,n 2 ,m − − f n 1 ,n 2 ,m (z) = 0. (z 0 −z) 1−e a
(15)
The exact solution of Eq. (15) is given by the hypergeometric function of the second kind 2 F 1 [a, b, c, z] f n 1 ,n 2 ,m (z) = Aχ1 + Bχ2 ,
Fig. 2. The dependencies of electron state energies on parameter γ = aR (for H the levels m = 0, n 1 = 0, n 2 = 1 and m = 0, n 1 = 0, n 2 = 2).
0
(z−z 0 ) a
−
539
(16)
h¯ ω H m + 2Un 1 ,m (0) εn 1 ,n 2 ,m . 2
(20)
3. Discussion of results In Fig. 2 explicit energy states are presented as a function of the parameter γ = R/a H . For comparison it has shown the
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analogous dependency for the ground state energy of the film under influence of the perpendicular magnetic field. As we had expected, the increasing of parameter γ lead to the decreasing of state energies. If γ 1 the problem comes to the problem of the film energy states under the effect of the perpendicular |m|+m+1 π 2 h¯ 2 2 magnetic field E nfilm = h ω (n + n . ) + ¯ H 1 2 1 ,n 2 ,m 2µR 2 2 From Fig. 2 one can see that the curves fit together. In Eq. (20) we used the values of Un 1 ,m (0) and εn 1 ,n 2 ,m obtained by numerical calculations. In view of assessment for quantum numbers the explicit criterion for use of the adiabatic approximation can be expressed for a film: n2
R p 2n 1 + |m| + m + 1. πa H
(21)
It could be satisfied, if the quantum numbers of electron radial motion get large values and at the same time the quantum numbers describing the z-coordinate take small values. For fixed quantum numbers we can obtain the value of magnetic field, so the multiplier R/a H satisfies the condition (21). Thus a strong magnetic field allowed us to apply the adiabatic approximation and give energy states and wave functions of QL. The expressions (3), (7), (9) and (16), obtained above for charge carrier’s energy spectrum and wave functions in semi-spherical QL under the influence of external magnetic field, allow the calculation of the interband light absorption coefficient K (ω) in such systems. In case of strong size quantization R a eB , when it is possible to neglect electronhole interactions (for example in InAs, where the effective Bohr ˚ effective electron mass is µe = 0.023 m e radius is a eB = 350 A, and effective mass hole µe = 0.33 m e ) according to [8,9] we have: 2 X Z e h K (ω) = A r Ψn 1 ,n 2 ,m Ψn 0 ,n 0 ,m 0 dE n 1 n 01 mm 0 n 2 n 02
1
2
× δ ∆ − E ne1 ,n 2 ,m − E nh0 ,n 0 ,m 0 , 1
2
(22)
Fig. 3. The dependence of W on aR for fixed value of magnetic field a H = 1 e 7 aB .
B
m = m0.
(23)
The argument of Dirac δ-function allows the definition of the threshold frequency of absorption ω00 1 e h {E g + E 0,1,0 + E 0,1,0 }. (24) h¯ For the case of strong size quantization this dependence is 2 (h ω −εg ) units, where E eR = h¯ e2 presented in Fig. 3 (in W = ¯ 00 Ee ω00 =
R
2µa B
the Rydberg energy for InAs). Curve 1 corresponds to the case of QL and curve 2 corresponds to the interband absorption in film. As it follows from Fig. 3, the curve of the W versus R/a B dependence for QL case is higher then the curve for film case. Along with increasing R, when the role of size quantization decreases, curves 1 and 2 decrease and come close to each other. Vice versa, at small R the role of size quantization sharply increases and therefore the effective width of forbidden band increases. Acknowledgements
2
where ∆ = h¯ ω − E g , E g is the width of forbidden zone, ω is the frequency of incident light, Ψ e(h) is the wave function of electron (hole), E e(h) 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. It is significant, that for quantum structures like InAs/In1−x Alx As valence band formed bands of light and heavy holes and also the bands of spin-orbital shift. In connection with this the value of spin-orbital shift is bigger than the energy of size quantization, and so we do not consider the transition from this band to conduction band [10]. In the other case, the bands of heavy and light holes quantized separately, so we consider transmissions between the band of heavy hole (for that band the square rule of dispersion law is more realistic) and the conduction band. After integrating we found, that the selection rule is only for magnetic quantum number
This work was supported by Armenian State Program “Semiconductor Nanoelectronics”. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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