- Email: [email protected]
Direct interband light absorption in the cylindrical quantum dot with modified Pöschl–Teller potential
Physica E 46 (2012) 274–278
Contents lists available at SciVerse ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Direct interband light absorption in the cylindrical quantum dot ¨ with modified Poschl–Teller potential D.B. Hayrapetyan a,b,n, E.M. Kazaryan a, H.Kh. Tevosyan a a b
Russian-Armenian (Slavic) University, 123 Hovsep Emin Str., Yerevan 0051, Armenia State Engineering University of Armenia, 105 Terian Str., Yerevan 0009, Armenia
H I G H L I G H T S c c c c
¨ Light absorption in cylindrical quantum dot with modified Poschl–Teller potential is studied. Analytical expressions for the particle energy spectrum are obtained. We obtained absorption coefficient and threshold frequencies as a function of geometrical sizes. We have found selection rules corresponding to different transitions between quantum levels.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 August 2012 Received in revised form 19 September 2012 Accepted 2 October 2012 Available online 12 October 2012
¨ In this paper the direct interband light absorption in cylindrical quantum dot with modified Poschl–Teller potential made of GaAs is studied. For the regime of strong size quantization analytical expressions for the particle energy spectrum, absorption coefficient and dependencies of effective threshold frequencies of absorption on the geometrical sizes of quantum dot are obtained. The selection rules corresponding to different transitions between quantum levels are found. To facilitate the comparison of obtained results with the probable experimental data, size dispersion distribution of growing quantum dots by the geometrical sizes using two experimentally realizing distribution functions has been taken into account. Distribution functions of Lifshits–Slezov and Gaussian have been considered. & 2012 Elsevier B.V. All rights reserved.
1. Introduction Progress in semiconductor nanoelectronics has made it possible to fabricate zero-dimensional structures, the so-called quantum dots (QDs) or ‘‘artificial atoms’’ [1–3]. These structures are interesting because of the fact that charge carrier (CC) motions are restricted in all three directions, which gives the possibility for physical characteristic effective control of those structures. A strong dependence of the energy spectrum of CC on the geometrical shape and sizes of QD allows to manipulate the energy spectrum and consequently, the physical characteristics of QD [4–7]. The controlling QD physical properties are attractive not only from the fundamental science point of view, but also because of its potential application in the development of semiconductor optoelectronics devices. A correct approximation of quantum dots confinement potential plays the relevant role at the investigation of physical characteristics
n Corresponding author at: Russian-Armenian (Slavic) University, 123 Hovsep Emin Str., Yerevan 0051, Armenia. Tel.: þ374 93 289 451. E-mail address: [email protected] (D.B. Hayrapetyan).
1386-9477/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2012.10.005
of such systems. It is important to note that the modeling of this potential in many aspects depends on the method of size-quantized semiconductor growth. Various approximations of confinement potential were proposed in the literature [8–10]. As a result of diffusion, the confining potential, formed during the growth process, in most cases can be approximated with a high accuracy by a parabolic potential. It should be mentioned that the parabolic approximation of the confinement potential of QD introduced in works of Maksym and Chakraborty [8] and Peeters [9] enables to carry out the detailed analytic analysis of the one-particle energetic spectrum and wave functions in the presence of codirected electric and magnetic fields. However, in reality, the parabolic potential is realized only for the lowest energy levels. It is obvious that with the increase of quantum number the course of confining potential diverges from the parabolic potential. For a more successful and realistic approximation of the formed confining potential in z direction ¨ the use of the modified potential Poschl–Teller (MPPT) has been proposed [11–23]. In the radial direction the parabolic confining potential has been used. Investigations of the optical absorption spectrum of various semiconductor structures are a powerful tool for determination of many characteristics of these systems: forbidden band gaps,
D.B. Hayrapetyan et al. / Physica E 46 (2012) 274–278
effective masses of electrons and holes, their mobilities, dielectric permittivities, etc. There are many works devoted to the theoretical and experimental studies of the optical absorption in the size-quantized systems [24–26]. The presence of size quantization essentially influences the absorption mechanism. In fact, the formation of new energy levels of the size quantization makes possible new interlevel transitions. In this paper the electronic states and direct interband absorption of light in cylindrical QD with MPPT are discussed. Absorption edge and absorption coefficient are obtained in a strong size quantization regime. To facilitate the comparison of obtained results with the probable experimental data, size dispersion distributions of growing QDs by the geometrical parameters of QD have been taken into account for two experimentally realizing distribution functions. Distribution function of Gauss has been considered in the first model and distribution function of Lifshits– Slezov has been considered in the second case.
2. Energy spectra and wave functions Consider the motion of particle (electron and hole) in a cylindrical QD, which is limited in z axis of MPPT, and in a radial direction by the parabolic potential. In the regime of strong size quantization the energy of Coulomb interaction between an electron and a hole is much smaller than the energy caused by the walls of the QD. In this approximation, the Coulomb interaction between particles can be neglected. Then the problem reduces to finding the energy states of electrons and holes separately. In this case, the potential energy of the particle in cylindrical coordinates can be written as mn o2 r2 U~ 0 þ p U r,Z ¼ U~ 0 , 2 2 ch Z=b~
275
Then the problem is reduced to finding the electronic states of the subsystems separately. After simple transformations of the wave function, which is given by the degenerate hypergeometric function, and by the energy of the radial subsystem, we obtain
pffiffiffijmj=2 pffiffiffi er jmjþ 1 2 pffiffi jmjþ 1, , r2 g , Rðr Þ ¼ er g=2 r 2 g pffiffiffi 1F1 2 4 g ð5Þ pffiffiffi er ¼ 8 gðN þ 1Þ,
ð6Þ
where we use the following notations: N ¼2nr þ9m9, N ¼ 0, 1, 2,... . Here nr and N are respectively the radial and oscillatory quantum numbers. The wave function of the second subsystem is given by Gauss hypergeometric functions and has the following form:
ez =2 1th z=b z , wðzÞ ¼ 1th2 F e s, e þs þ 1, e þ 1, z z z 2 1 b 2 ð7Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where s ¼ 1=2 ½1 þ 1 þ 4b U 0 . For the energy of the same subsystem, we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 ez ¼ U 0 2 ð1 þ2nz Þ þ 1þ 4b2 U 0 , ð8Þ 4b where nz is the quantum number in z direction, which takes values from nz ¼0, 1, 2,... . Note that for lower energy levels MPPT is coincided with parabolic potential, because if we expand MPPT into series we will obtain parabolic case. The total energy of the system is the sum of the energies of two subsystems
e ¼ er þ ez :
ð9Þ
ð1Þ 3. Direct interband light absorption
where U~ 0 and b~ are respectively the depth and half-width of the MPPT, mnp is the effective mass of the particle. The relationship between height of the cylindrical QD and half-width of potential well is given as h0 ¼ dh b~ , and frequencyo is connected with radius of cylinder R0 via equation o ¼ do ð_Þ= mnp R20 , where parameters dh and do are determined from the experiment. The Hamiltonian of system in cylindrical coordinates and dimensionless variables can be presented as the sum of Hamiltonians of the subsystems ^ ¼H ^ 1 þH ^ 2, H
ð2Þ
Consider the direct interband light absorption in cylindrical quantum dot with MPPT in the strong size quantization regime, when the Coulomb interaction between electron and hole can be neglected. Furthermore, consider the case of a heavy hole with mne 5mnh , where mne and mnh are effective masses of the electron and hole, respectively. Then the absorption coefficient is given by X
Z e h !
2
F F 0 d r d _OEg Ee Eh0 , ð10Þ K ¼A n n n n
n, n0
0
where 2 2 ^ 1 ¼ @ 1 @ þ 1 @ þ gr 2 H 2 2 r @r @r r @j2
and
2 ^ 2 ¼ @ þ U 0 U0 : H 2 @z2 ch z
b
ð3Þ Here the following notations are made: r¼
r aB
,
z¼
Z , aB
U0 ¼
2
n 2
U~ 0 , ER
b¼
b~ aB
and
g¼
mnp oa2B
!2
_
where n and n are sets of quantum numbers corresponding to the electron and the heavy hole, respectively, Eg is the band gap of massive semiconductor, O is the frequency of the incident light, A is a quantity proportional to the square of the matrix element taken by Bloch functions [27]. In the regime of strong size quantization for the absorption edge we finally get 0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi !2 u 2 pffiffiffiffiffi d d d Bm u b~ U~ 0 mne t W 000 ¼ 1 þ8 g0 þ þ 1þ4 2 @mn ae ah d Eg m e 2b~ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 !2 u b~ U~ 0 mnh C 1A, þ n 1þ 4 mh d Eg m
m u t
where ER ¼ _ =2mp aB is the Rydberg effective energy, aB ¼ ðk_2 Þ=ðmnp e2 Þ is the Bohr effective radius of the particle, k is the dielectric constant, and e is the charge of the particle. We shall search the total wave function of the system in the following form: F r, j,z ¼ Ceimj RðrÞwðzÞ, ð4Þ
where
where m is the magnetic quantum number and C is the normalization coefficient.
ah ¼
W 000 ¼
_O000 , Eg
k_2 mnh e2
and
d¼
sffiffiffiffiffiffiffiffiffiffiffi _2 , 2mEg
m¼
g0 ¼
mne mne mne þ mne
ð11Þ
mod2 _
!2 ,
ae ¼
k_2 mne e2
,
276
D.B. Hayrapetyan et al. / Physica E 46 (2012) 274–278
the reduced electron–hole pair effective mass. Consider selection rules for transitions between levels with different quantum numbers. For the magnetic quantum number transitions between 0 the levels with m¼ m are allowed, and for the quantum number nz any transitions are allowed. For the oscillator quantum number 0 transitions for the levels with N ¼N are allowed. Note that in the case of cylindrical quantum dot with parabolic confining potentials selection rule for the magnetic quantum number is the same, but for quantum number nz transitions are allowed for the levels with 9nz n0z 9 ¼ 2t, where t is an integer [26]. So using of MPPT leads to the change of quantum transitions for nz. Until now all the above mentioned calculations have been performed for the semiconductor QD’s of one size. To compare the results with the experimental results the dispersion of the geometric sizes of cylindrical QD (height and radius of the cylinder) should be taken into account. In other words, we must take into account variance of QD halfwidth b~ and frequency o. It should be noted that taking into account the geometric dispersion, the study of light absorption will lead to a series of blurring lines instead of clear lines of maxima of the absorption frequencies. During the growing process of QD’s ensemble, depending on the technological parameters of the growth, symmetrical or asymmetrical distribution of the geometric parameters of QD around certain average values may occur. To describe the symmetric distribution of the geometrical parameters around the average value, the Gaussian distribution function is used, and for the asymmetrical the Lifshits–Slezov function (see Fig. 1). Forms of these functions are given below ( ) 1 ðu1Þ2 P G ðuÞ ¼ pffiffiffiffiffiffi exp 2 , s 2p 2 s=Z 8 4 2 ÞÞ < 3 eu expð1=ð12u=3 u o3=2, 11=3 , 5=3 7=3 ð12Þ P LS ðuÞ ¼ 2 ðu þ 3Þ ð3=2uÞ : 0, u 43=2, where the variable u is the parameters to their ratio of dispersive and uR0 ¼ o= average values, uh0 ¼ b~ =/b~ S ¼ h0 =/h0 S /oSÞ ¼ R0 =/R0 S , and all other parameters are determined from the experiment. Taking into account the dispersion of b~ and o parameters will replace Eq. (10) by this X ZZ Ph0 uh0 P R0 uR0 d _OEg Een Ehn duh0 duR0 : ð13Þ K ¼A m,N nz ,n0z
Fig. 1. Gauss and Lifshits–Slezov distribution functions.
Taking into account the selection rules for the absorption coefficient we get K¼
A X Eg m,N
1
/R0 S2
l2 l3
l1
l3 /h0 S2
1
P R0
!
l3 l1 l2 =/R0 S2
!
l2
P h0
1
l2
nz ,n0z pffiffiffiffiffiffiffiffiffiffi l1
l1 l3 =/R0 S2
,
ð14Þ
where l1 ¼ l2 ¼
_OEg , Eg 8do
m d3
/R0 S2
mne ae
l3 ¼ U 0
m d2 mne a2e
þ
þ
m d3 mnh ah
m d2 mnh a2h
0
!
! ðN þ 1Þ
and 0
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 ! u
u m @ b~ U~ 0 mne A ð2nz þ1Þ þ t1 þ 4 d Eg m 4/h0 S2 mne
d2h d2
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 !
u n ~ ~ u m @ 0 t1 þ 4 b U 0 mh A : 2n þ 1 þ z 2 mn d E m 4/h0 S g h
d2h d2
4. Discussion Let us proceed to the discussion of the results. Note that the numerical calculations are made for the cylindrical QD from GaAs with the following parameters: me ¼ 0.067me, k ¼13.8, ER ¼ ˚ 5.275 meV, and aB ¼104 A. The important difference between the direction of limiting potential axis of the cylinder (MPPT) and the radial direction (quadratic potential) is that the first potential is finite, while the second one is infinite. This difference leads to the fact that on the z direction implementation of quantum electron emission is possible. The criterion of the existence of energy levels in a 2 cylindrical QD with MPPT is U 0 Z 1=b . As we see from Eq. (6), the energy of the radial part is equidistant, in other words it means that for fixed value of the quantum number nz the equidistant families of energy levels are obtained. The energy levels of the system decrease when the geometric dimensions of the cylindrical QD increase: height and radius of the base (the same as setting o and semiaxis b). With increasing semiaxis b interlevel distances are also reduced. In the regime of strong quantum transition the frequency between the equidistant levels (for the value N ¼0) for the first group (family), with fixed values U0 ¼40, b ¼0.5 and g ¼1 we get o00 ¼5.61 1013 C 1, which corresponds to the infrared range of the spectrum. With the same value of quantum numbers, but already with U0 ¼70, b ¼1 and g ¼1 we get o00 ¼6.27 1013 C 1. Fig. 2(a) and (b) shows the dependence of absorption edge on the radius and height of the cylindrical QD for the fixed values of height and radius, respectively, in the regime of strong size quantization. As can be seen from the figures, with decreasing the radius of the base and height of QD, the absorption edge increases. It is the consequence of the following: with the decrease of R0 and h0 parameters the effective width of the band gap increases by reducing the influence of the QD’s walls. For the same reason, the curves corresponding to the large fixed values of these parameters are lower than those for the smaller values. Note that allowed transitions are between levels with quantum 0 0 numbers m ¼ m and N ¼N ; any transitions are allowed for nz 0 and nz .
D.B. Hayrapetyan et al. / Physica E 46 (2012) 274–278
277
Fig. 4. Dependence of the absorption coefficient on the frequency of incident light for the case of Gauss–Lifshits–Slezov distribution functions.
Fig. 2. Dependence of absorption edge (a) on the radius of cylindrical QD for the fixed value of height and (b) on the height of cylindrical QD for the fixed value of radius.
numbers for different values of the depth of the potential well U0. In other words, each line represents one individual transition between certain levels, the quantum numbers of which are fixed in the figures. Fig. 3 corresponds to the case when the distribution of the geometric parameters of the cylindrical QD in all directions is symmetrical. It means that for calculations the Gaussian distribution function is used both in radial and also in z directions. As can be seen from the figure, with an increase in the oscillator quantum number the absorption coefficient decreases, but the absorption maximum does not change the position. In the right corner of the figure the dependence of the total absorption coefficient on the frequency of incident light at different values of the parameter U0 is shown. Another interesting case is when during the growth of an ensemble of cylindrical QDs the size distribution of the grown objects in different directions is described by different distribution functions. Such a mixed case of distribution is shown in Fig. 4. In contrast to the case of symmetric distribution, for the asymmetric distribution the absorption coefficient changes the position of the maximum with the change of the quantum numbers; meanwhile the intensity of the absorption, as in the previous case, is also reduced. In the right corner of the figure the total absorption coefficient of the frequency of the incident light is shown.
5. Conclusion
Fig. 3. Dependence of the absorption coefficient on the frequency of incident light for the case of Gauss–Gauss distribution functions.
Summarizing, the direct interband light absorption in cylind¨ rical quantum dot with modified Poschl–Teller potential is studied. In the regime of strong size quantization analytical expressions for the particle energy spectrum, absorption coefficient and dependencies of effective threshold frequencies of absorption on the geometrical sizes of quantum dot are obtained. The selection rules corresponding to different transitions between quantum levels are found. Size dispersion distribution of growing quantum dots by the geometrical sizes using two experimentally realizing symmetric and asymmetric distribution functions has been taken into account.
References In Fig. 3 the dependence of the absorption coefficient on the frequency of the incident light for the cylindrical QDs is given for the regime of strong size quantization for two sets of quantum
[1] P. Harrison, Quantum Wells, Wires and Dots: Theoretical and Computational Physics, John Wiley & Sons Ltd., New York, 2005.
278
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[2] D. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, Les Editions de Physique, Paris, 1989. [3] E.M. Kazaryan, S.G. Petrosyan, Physical Principles of Semiconductor Nanoelectronics, RAU, Yerevan, 2005. [4] H.A. Sarkisyan, Modern Physics Letters B 16 (2001) 835. [5] D.B. Hayrapetyan, Journal of Contemporary Physics 42 (2007) 292. [6] K.G. Dvoyan, D.B. Hayrapetyan, E.M. Kazaryan, A.A. Tshantshapanyan, Nanoscale Research Letters 4 (2009) 130. [7] T. Chakraborty, P. Pietilainen, Physical Review B 52 (1995) 1932. [8] P. Maksym, T. Chakraborty, Physical Review Letters 65 (1990) 108. [9] F. Peeters, Physical Review B 42 (1990) 1486. [10] E.M. Kazaryan, L.S. Petrosyan, H.A. Sarkisyan, Physica E 16 (2003) 174. [11] Y. Wu, Journal of Mathematical Physics 31 (1990) 2586. [12] J.-P. Antoine, J.-P. Gazeau, P. Monceau, J.R. Klauder, K.A. Penson, Journal of Mathematical Physics 42 (2001) 2349. [13] A. Rodriguez, J.M. Ceveno, Physical Review B 74 (2006) 104201. [14] G. Wang, Q. Guo, L. Wu, X. Yang, Physical Review B 75 (2007) 205337.
[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
G. Wei, Sh. Dong, Physics Letters A 373 (2009) 2428. M. Rey, F. Michelot, Physics Letters A 374 (2010) 4761. T. Chen, Y. Diao, Ch. Jia, Physica Scripta 79 (2009) 065014. H.Kh. Tevosyan, D.B. Hayrapetyan, K.G. Dvoyan, E.M. Kazaryan, International Journal of Modern Physics: Conference Series 15 (2012) 204. A. Hakimyfard, Journal of Contemporary Physics 45 (2010) 42. D. Agboola, Chinese Physics Letters 27 (2010) 040301. M.G. Barseghyan, A. Hakimyfard, A.A. Kirakosyan, M.E. Mora-Ramos, C.A. Duque, Superlattices and Microstructures 51 (2012) 119. O. Aytekin, S. Turgut, M. Tomak, Physica E 44 (2012) 1612. M.G. Barseghyan, A. Hakimyfard, M. Zuhair, C.A. Duque, A.A. Kirakosyan, Physica E 44 (2011) 419. Al.L. Efros, A.L. Efros, Semiconductors 16 (1982) 772. K.G. Dvoyan, D.B. Hayrapetyan, E.M. Kazaryan, Nanoscale Research Letters 4 (2009) 106. H.A. Sarkisyan, Modern Physics Letters B 18 (2004) 443. A.I. Anselm, Introduction of Semiconductors Theory, Nauka, Moscow, 1978.
Contents lists available at SciVerse ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Direct interband light absorption in the cylindrical quantum dot ¨ with modified Poschl–Teller potential D.B. Hayrapetyan a,b,n, E.M. Kazaryan a, H.Kh. Tevosyan a a b
Russian-Armenian (Slavic) University, 123 Hovsep Emin Str., Yerevan 0051, Armenia State Engineering University of Armenia, 105 Terian Str., Yerevan 0009, Armenia
H I G H L I G H T S c c c c
¨ Light absorption in cylindrical quantum dot with modified Poschl–Teller potential is studied. Analytical expressions for the particle energy spectrum are obtained. We obtained absorption coefficient and threshold frequencies as a function of geometrical sizes. We have found selection rules corresponding to different transitions between quantum levels.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 August 2012 Received in revised form 19 September 2012 Accepted 2 October 2012 Available online 12 October 2012
¨ In this paper the direct interband light absorption in cylindrical quantum dot with modified Poschl–Teller potential made of GaAs is studied. For the regime of strong size quantization analytical expressions for the particle energy spectrum, absorption coefficient and dependencies of effective threshold frequencies of absorption on the geometrical sizes of quantum dot are obtained. The selection rules corresponding to different transitions between quantum levels are found. To facilitate the comparison of obtained results with the probable experimental data, size dispersion distribution of growing quantum dots by the geometrical sizes using two experimentally realizing distribution functions has been taken into account. Distribution functions of Lifshits–Slezov and Gaussian have been considered. & 2012 Elsevier B.V. All rights reserved.
1. Introduction Progress in semiconductor nanoelectronics has made it possible to fabricate zero-dimensional structures, the so-called quantum dots (QDs) or ‘‘artificial atoms’’ [1–3]. These structures are interesting because of the fact that charge carrier (CC) motions are restricted in all three directions, which gives the possibility for physical characteristic effective control of those structures. A strong dependence of the energy spectrum of CC on the geometrical shape and sizes of QD allows to manipulate the energy spectrum and consequently, the physical characteristics of QD [4–7]. The controlling QD physical properties are attractive not only from the fundamental science point of view, but also because of its potential application in the development of semiconductor optoelectronics devices. A correct approximation of quantum dots confinement potential plays the relevant role at the investigation of physical characteristics
n Corresponding author at: Russian-Armenian (Slavic) University, 123 Hovsep Emin Str., Yerevan 0051, Armenia. Tel.: þ374 93 289 451. E-mail address: [email protected] (D.B. Hayrapetyan).
1386-9477/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2012.10.005
of such systems. It is important to note that the modeling of this potential in many aspects depends on the method of size-quantized semiconductor growth. Various approximations of confinement potential were proposed in the literature [8–10]. As a result of diffusion, the confining potential, formed during the growth process, in most cases can be approximated with a high accuracy by a parabolic potential. It should be mentioned that the parabolic approximation of the confinement potential of QD introduced in works of Maksym and Chakraborty [8] and Peeters [9] enables to carry out the detailed analytic analysis of the one-particle energetic spectrum and wave functions in the presence of codirected electric and magnetic fields. However, in reality, the parabolic potential is realized only for the lowest energy levels. It is obvious that with the increase of quantum number the course of confining potential diverges from the parabolic potential. For a more successful and realistic approximation of the formed confining potential in z direction ¨ the use of the modified potential Poschl–Teller (MPPT) has been proposed [11–23]. In the radial direction the parabolic confining potential has been used. Investigations of the optical absorption spectrum of various semiconductor structures are a powerful tool for determination of many characteristics of these systems: forbidden band gaps,
D.B. Hayrapetyan et al. / Physica E 46 (2012) 274–278
effective masses of electrons and holes, their mobilities, dielectric permittivities, etc. There are many works devoted to the theoretical and experimental studies of the optical absorption in the size-quantized systems [24–26]. The presence of size quantization essentially influences the absorption mechanism. In fact, the formation of new energy levels of the size quantization makes possible new interlevel transitions. In this paper the electronic states and direct interband absorption of light in cylindrical QD with MPPT are discussed. Absorption edge and absorption coefficient are obtained in a strong size quantization regime. To facilitate the comparison of obtained results with the probable experimental data, size dispersion distributions of growing QDs by the geometrical parameters of QD have been taken into account for two experimentally realizing distribution functions. Distribution function of Gauss has been considered in the first model and distribution function of Lifshits– Slezov has been considered in the second case.
2. Energy spectra and wave functions Consider the motion of particle (electron and hole) in a cylindrical QD, which is limited in z axis of MPPT, and in a radial direction by the parabolic potential. In the regime of strong size quantization the energy of Coulomb interaction between an electron and a hole is much smaller than the energy caused by the walls of the QD. In this approximation, the Coulomb interaction between particles can be neglected. Then the problem reduces to finding the energy states of electrons and holes separately. In this case, the potential energy of the particle in cylindrical coordinates can be written as mn o2 r2 U~ 0 þ p U r,Z ¼ U~ 0 , 2 2 ch Z=b~
275
Then the problem is reduced to finding the electronic states of the subsystems separately. After simple transformations of the wave function, which is given by the degenerate hypergeometric function, and by the energy of the radial subsystem, we obtain
pffiffiffijmj=2 pffiffiffi er jmjþ 1 2 pffiffi jmjþ 1, , r2 g , Rðr Þ ¼ er g=2 r 2 g pffiffiffi 1F1 2 4 g ð5Þ pffiffiffi er ¼ 8 gðN þ 1Þ,
ð6Þ
where we use the following notations: N ¼2nr þ9m9, N ¼ 0, 1, 2,... . Here nr and N are respectively the radial and oscillatory quantum numbers. The wave function of the second subsystem is given by Gauss hypergeometric functions and has the following form:
ez =2 1th z=b z , wðzÞ ¼ 1th2 F e s, e þs þ 1, e þ 1, z z z 2 1 b 2 ð7Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where s ¼ 1=2 ½1 þ 1 þ 4b U 0 . For the energy of the same subsystem, we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 ez ¼ U 0 2 ð1 þ2nz Þ þ 1þ 4b2 U 0 , ð8Þ 4b where nz is the quantum number in z direction, which takes values from nz ¼0, 1, 2,... . Note that for lower energy levels MPPT is coincided with parabolic potential, because if we expand MPPT into series we will obtain parabolic case. The total energy of the system is the sum of the energies of two subsystems
e ¼ er þ ez :
ð9Þ
ð1Þ 3. Direct interband light absorption
where U~ 0 and b~ are respectively the depth and half-width of the MPPT, mnp is the effective mass of the particle. The relationship between height of the cylindrical QD and half-width of potential well is given as h0 ¼ dh b~ , and frequencyo is connected with radius of cylinder R0 via equation o ¼ do ð_Þ= mnp R20 , where parameters dh and do are determined from the experiment. The Hamiltonian of system in cylindrical coordinates and dimensionless variables can be presented as the sum of Hamiltonians of the subsystems ^ ¼H ^ 1 þH ^ 2, H
ð2Þ
Consider the direct interband light absorption in cylindrical quantum dot with MPPT in the strong size quantization regime, when the Coulomb interaction between electron and hole can be neglected. Furthermore, consider the case of a heavy hole with mne 5mnh , where mne and mnh are effective masses of the electron and hole, respectively. Then the absorption coefficient is given by X
Z e h !
2
F F 0 d r d _OEg Ee Eh0 , ð10Þ K ¼A n n n n
n, n0
0
where 2 2 ^ 1 ¼ @ 1 @ þ 1 @ þ gr 2 H 2 2 r @r @r r @j2
and
2 ^ 2 ¼ @ þ U 0 U0 : H 2 @z2 ch z
b
ð3Þ Here the following notations are made: r¼
r aB
,
z¼
Z , aB
U0 ¼
2
n 2
U~ 0 , ER
b¼
b~ aB
and
g¼
mnp oa2B
!2
_
where n and n are sets of quantum numbers corresponding to the electron and the heavy hole, respectively, Eg is the band gap of massive semiconductor, O is the frequency of the incident light, A is a quantity proportional to the square of the matrix element taken by Bloch functions [27]. In the regime of strong size quantization for the absorption edge we finally get 0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi !2 u 2 pffiffiffiffiffi d d d Bm u b~ U~ 0 mne t W 000 ¼ 1 þ8 g0 þ þ 1þ4 2 @mn ae ah d Eg m e 2b~ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 !2 u b~ U~ 0 mnh C 1A, þ n 1þ 4 mh d Eg m
m u t
where ER ¼ _ =2mp aB is the Rydberg effective energy, aB ¼ ðk_2 Þ=ðmnp e2 Þ is the Bohr effective radius of the particle, k is the dielectric constant, and e is the charge of the particle. We shall search the total wave function of the system in the following form: F r, j,z ¼ Ceimj RðrÞwðzÞ, ð4Þ
where
where m is the magnetic quantum number and C is the normalization coefficient.
ah ¼
W 000 ¼
_O000 , Eg
k_2 mnh e2
and
d¼
sffiffiffiffiffiffiffiffiffiffiffi _2 , 2mEg
m¼
g0 ¼
mne mne mne þ mne
ð11Þ
mod2 _
!2 ,
ae ¼
k_2 mne e2
,
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the reduced electron–hole pair effective mass. Consider selection rules for transitions between levels with different quantum numbers. For the magnetic quantum number transitions between 0 the levels with m¼ m are allowed, and for the quantum number nz any transitions are allowed. For the oscillator quantum number 0 transitions for the levels with N ¼N are allowed. Note that in the case of cylindrical quantum dot with parabolic confining potentials selection rule for the magnetic quantum number is the same, but for quantum number nz transitions are allowed for the levels with 9nz n0z 9 ¼ 2t, where t is an integer [26]. So using of MPPT leads to the change of quantum transitions for nz. Until now all the above mentioned calculations have been performed for the semiconductor QD’s of one size. To compare the results with the experimental results the dispersion of the geometric sizes of cylindrical QD (height and radius of the cylinder) should be taken into account. In other words, we must take into account variance of QD halfwidth b~ and frequency o. It should be noted that taking into account the geometric dispersion, the study of light absorption will lead to a series of blurring lines instead of clear lines of maxima of the absorption frequencies. During the growing process of QD’s ensemble, depending on the technological parameters of the growth, symmetrical or asymmetrical distribution of the geometric parameters of QD around certain average values may occur. To describe the symmetric distribution of the geometrical parameters around the average value, the Gaussian distribution function is used, and for the asymmetrical the Lifshits–Slezov function (see Fig. 1). Forms of these functions are given below ( ) 1 ðu1Þ2 P G ðuÞ ¼ pffiffiffiffiffiffi exp 2 , s 2p 2 s=Z 8 4 2 ÞÞ < 3 eu expð1=ð12u=3 u o3=2, 11=3 , 5=3 7=3 ð12Þ P LS ðuÞ ¼ 2 ðu þ 3Þ ð3=2uÞ : 0, u 43=2, where the variable u is the parameters to their ratio of dispersive and uR0 ¼ o= average values, uh0 ¼ b~ =/b~ S ¼ h0 =/h0 S /oSÞ ¼ R0 =/R0 S , and all other parameters are determined from the experiment. Taking into account the dispersion of b~ and o parameters will replace Eq. (10) by this X ZZ Ph0 uh0 P R0 uR0 d _OEg Een Ehn duh0 duR0 : ð13Þ K ¼A m,N nz ,n0z
Fig. 1. Gauss and Lifshits–Slezov distribution functions.
Taking into account the selection rules for the absorption coefficient we get K¼
A X Eg m,N
1
/R0 S2
l2 l3
l1
l3 /h0 S2
1
P R0
!
l3 l1 l2 =/R0 S2
!
l2
P h0
1
l2
nz ,n0z pffiffiffiffiffiffiffiffiffiffi l1
l1 l3 =/R0 S2
,
ð14Þ
where l1 ¼ l2 ¼
_OEg , Eg 8do
m d3
/R0 S2
mne ae
l3 ¼ U 0
m d2 mne a2e
þ
þ
m d3 mnh ah
m d2 mnh a2h
0
!
! ðN þ 1Þ
and 0
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 ! u
u m @ b~ U~ 0 mne A ð2nz þ1Þ þ t1 þ 4 d Eg m 4/h0 S2 mne
d2h d2
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 !
u n ~ ~ u m @ 0 t1 þ 4 b U 0 mh A : 2n þ 1 þ z 2 mn d E m 4/h0 S g h
d2h d2
4. Discussion Let us proceed to the discussion of the results. Note that the numerical calculations are made for the cylindrical QD from GaAs with the following parameters: me ¼ 0.067me, k ¼13.8, ER ¼ ˚ 5.275 meV, and aB ¼104 A. The important difference between the direction of limiting potential axis of the cylinder (MPPT) and the radial direction (quadratic potential) is that the first potential is finite, while the second one is infinite. This difference leads to the fact that on the z direction implementation of quantum electron emission is possible. The criterion of the existence of energy levels in a 2 cylindrical QD with MPPT is U 0 Z 1=b . As we see from Eq. (6), the energy of the radial part is equidistant, in other words it means that for fixed value of the quantum number nz the equidistant families of energy levels are obtained. The energy levels of the system decrease when the geometric dimensions of the cylindrical QD increase: height and radius of the base (the same as setting o and semiaxis b). With increasing semiaxis b interlevel distances are also reduced. In the regime of strong quantum transition the frequency between the equidistant levels (for the value N ¼0) for the first group (family), with fixed values U0 ¼40, b ¼0.5 and g ¼1 we get o00 ¼5.61 1013 C 1, which corresponds to the infrared range of the spectrum. With the same value of quantum numbers, but already with U0 ¼70, b ¼1 and g ¼1 we get o00 ¼6.27 1013 C 1. Fig. 2(a) and (b) shows the dependence of absorption edge on the radius and height of the cylindrical QD for the fixed values of height and radius, respectively, in the regime of strong size quantization. As can be seen from the figures, with decreasing the radius of the base and height of QD, the absorption edge increases. It is the consequence of the following: with the decrease of R0 and h0 parameters the effective width of the band gap increases by reducing the influence of the QD’s walls. For the same reason, the curves corresponding to the large fixed values of these parameters are lower than those for the smaller values. Note that allowed transitions are between levels with quantum 0 0 numbers m ¼ m and N ¼N ; any transitions are allowed for nz 0 and nz .
D.B. Hayrapetyan et al. / Physica E 46 (2012) 274–278
277
Fig. 4. Dependence of the absorption coefficient on the frequency of incident light for the case of Gauss–Lifshits–Slezov distribution functions.
Fig. 2. Dependence of absorption edge (a) on the radius of cylindrical QD for the fixed value of height and (b) on the height of cylindrical QD for the fixed value of radius.
numbers for different values of the depth of the potential well U0. In other words, each line represents one individual transition between certain levels, the quantum numbers of which are fixed in the figures. Fig. 3 corresponds to the case when the distribution of the geometric parameters of the cylindrical QD in all directions is symmetrical. It means that for calculations the Gaussian distribution function is used both in radial and also in z directions. As can be seen from the figure, with an increase in the oscillator quantum number the absorption coefficient decreases, but the absorption maximum does not change the position. In the right corner of the figure the dependence of the total absorption coefficient on the frequency of incident light at different values of the parameter U0 is shown. Another interesting case is when during the growth of an ensemble of cylindrical QDs the size distribution of the grown objects in different directions is described by different distribution functions. Such a mixed case of distribution is shown in Fig. 4. In contrast to the case of symmetric distribution, for the asymmetric distribution the absorption coefficient changes the position of the maximum with the change of the quantum numbers; meanwhile the intensity of the absorption, as in the previous case, is also reduced. In the right corner of the figure the total absorption coefficient of the frequency of the incident light is shown.
5. Conclusion
Fig. 3. Dependence of the absorption coefficient on the frequency of incident light for the case of Gauss–Gauss distribution functions.
Summarizing, the direct interband light absorption in cylind¨ rical quantum dot with modified Poschl–Teller potential is studied. In the regime of strong size quantization analytical expressions for the particle energy spectrum, absorption coefficient and dependencies of effective threshold frequencies of absorption on the geometrical sizes of quantum dot are obtained. The selection rules corresponding to different transitions between quantum levels are found. Size dispersion distribution of growing quantum dots by the geometrical sizes using two experimentally realizing symmetric and asymmetric distribution functions has been taken into account.
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