- Email: [email protected]
ZnO quantum dot
Physica B 406 (2011) 3666–3670
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Physica B journal homepage: www.elsevier.com/locate/physb
Interband optical absorption in a strained CdxZn1 xO/ZnO quantum dot A. John Peter n, Chang Woo Lee nn Department of Chemical Engineering and Green Energy Center, College of Engineering, Kyung Hee University, 1Seochun, Gihung, Yongin, Gyeonggi 446-701, South Korea
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 May 2011 Accepted 28 June 2011 Available online 6 July 2011
Numerical calculations of the excitonic absorption spectra in a strained CdxZn1 xO/ZnO quantum dot are investigated for various Cd contents. We calculate the quantized energies of the exciton as a function of dot radius for various confinement potentials and thereby the interband emission energy is computed considering the internal electric field induced by the spontaneous and piezoelectric polarizations. The optical absorption as a function of photon energy for different dot radii is discussed. Decrease of exciton binding energy and the corresponding optical band gap with the Cd concentration imply that the confinement of carriers decreases with composition x. The main results show that the confined energies and the transition energies between the excited levels are significant for smaller dots. Non-linearity band gap with the increase in Cd content is observed for smaller dots in the strong confinement region and the magnitude of the absorption spectra increases for the transitions between the higher excited levels. & 2011 Elsevier B.V. All rights reserved.
Keywords: Quantum dot Donor bound excitons Oscillator strength
1. Introduction Present decade is witnessing some novel wide band gap materials such as GaN and ZnO play a vitol role for making multilayered heterostrucutres in the green region of spectrum. The wide band gap semiconductors have attracted much attention due to their potential applications and device fabrications for electro- and opto-electronic devices in blue and ultraviolet regions. The ZnO material is preferred because it can deliver low cost with high brightness devices, further the Gallium nitride mateial is toxic. ZnO based compounds have a larger exciton binding energy roughly twice than that of GaN, a high radiation resistance and high chemical stability and they are compatible with both wet chemical and dry etching [1]. And also, all the wurtzite materials will exhibit both pyroelectric and piezoelectric behavior. CdZnO materials have internal electric field and will also contribute both spoantaneous and piezoelectric polarizations. This property will change the behavior of the ZnO based heterostructures [2]. Hence, it is imperative to understand the behavior of its excitonic recombination. CdO is an n-type material, a conducting and a transperarent element with the direct band gap energy approximately 2.5 eV while the Eg for ZnO is around 3.4 eV. The band gap of ZnO can be red-shifted to blue and green spectral range with the suitable incorporation of Cd so one can tune the band gap for some potenital applications and it has been used for device fabrications such as
n Corresponding author. Permanent address: Govt. Arts College, Melur 625106, Madurai, Tamil Nadu, India. Tel.: þ91 9786141966; fax: þ91 4522415467. nn Corresponding author. E-mail address: [email protected] (A. John Peter).
0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.06.067
phototransistors, solar cells, gas sensors IR detectors and liquid crystal displays based on ZnO/ZnCdO heterojunctions and superlattice structures [3–5]. Thus ZnCdO is considered to be an ideal material for ZnO-based devices, which are given due attention to study their excitonic electro-optical properties especially for making laser diodes wherein the band gap engineering is necessary. In the present work, we have calculated the eigen energy values of the confined exciton as a function of dot radius in a strained CdZnO/ZnO quantum dot with various values of Cd content. The exciton binding energy with different confinment is computed and thereby interband emission energy is calculated. The interband absorption spectra of the exciton in a CdxZn1 xO/ ZnO quantum dot have been investigated considering the internal electric field induced by the spontaneous and piezoelectric polarizations. We have used the variational technique within the single band effective mass approximation to calculate the binding energy and thereby their optical band gap as afuntion of dot radius. We investigate the exciton oscillator strength and the radiative lifetime of confined exciton as a function of dot radius of Cd0.2Zn0.8O/ZnO quantum dot with and without strain effect. The optical absorption as a function of photon energy for different dot radii is discussed. This paper is organized as follows: the method followed is presented in Sections 2 and 2.1 while the results and discussion are presented in Section 3. Finally, we summarize the main conclusions obtained in this paper in Section 4.
2. Theoretical Within the framework of single band effective mass approximation, the Hamiltonian of the exciton considering a strained
A. John Peter, C. Woo Lee / Physica B 406 (2011) 3666–3670
CdxZn1 xO/ZnO quantum dot with the radius R, can be written as " # p2rj X P2 p2 e2 þ fðrÞ þ Vj ðrj Þ þ Z þ z ð1Þ Hexc ¼ 2m 2M 2 m e r rh j e j where j¼ e and h refer to the electron and hole, respectively, e is the dielectric constant for the material inside the quantum dot and Vj(rj) is the strain induced confinement potential of the particle j. The band gap of the material is given by [6] Eg ðCdx Zn1x OÞ ¼ 1:760:11x þ 1:75x2 ðeVÞ
ð2Þ
The above expression of band gap has been obtained by varying Cd concentration in ZnO quantum dot. The distribution of the band gap discontinuities is taken as V(z) ¼0.8DEg. PZ and pz are the exciton center-of mass momentum and relative motion momentum position along the z-direction, respectively. M ¼me þmh and m ¼memh/(me þmh) are the total and reduced effective masses of the exciton, respectively. f(r) is the electrostatic potential induced by the internal electric field. The internal electric field in CdxZn1 xO is characterized with the linear variation of Cd content up to x¼ 0.35 (Eq. (2)) beyond this level there occurs a phase change. As the CdxZn1 xO material changes a phase transition when x40.32 in the wurtzite phase [10,11]. The strength of the built-in electric field caused by the spontaneous and piezoelectric polarizations in the ZnxCd1 xO/ ZnO strained quantum dot expressed as [12] 8 CdO CdO Znx Cd1x O < PSP þ PPE PSP roR e0 eCdO e fðrÞ ¼ ð3Þ : 0 rZR CdO is the electronic dielectric constant of CdO and PPE , here, eCdO e Zns Cd1x O CdO PSP and PSP are the piezoelectric polarizations, spontaneous polarizations of CdO and the spontaneous polarization of ZnxCd1 xO, respectively. The above values can be generally calculated by the polarity of the crystal and the strains of the quantum nanostructure. Since the wurtzite crystal lattice of CdO and ZnO lack inversion symmetry, the heterostructure will have spontaneous polarization (PSP) and the piezoelectric polarization (PPZ) due to strain caused by the lattice mismatch between CdO and ZnO material. The piezoelectric polarization along the c axis is given by
P PZ ¼ e31 ðexx þ eyy Þ þ e33 ezz
ð4Þ
with exx ¼ eyy ¼a(CdO) a(ZnCdO)/a(CdO) and ezz ¼ 2(C13)/ (C33)exx, the values of parameters are given in Table 1. The piezoelectric polarization is given by C13 P PZ ¼ 2exx e31 e33 ð5Þ C33 Table 1 Material parameters for ZnO, CdO and CdxZn1 xO (linearly interpolated from the data of ZnO and CdO) used in the calculations. Parameters
ZnO
CdO
Cd0.1Zn0.9O
Cd0.2Zn0.8O
Cd0.3Zn0.7O
Eg (eV) me (m0) mh (m0)
3.29 0.24 0.78 6.8 3.20 0.62 0.96 105.1 0.2 3.1
2.2 0.21 0.78 18.1 4.70 0.48 0.96 1.67 61 0 0.4
2.89 0.237 0.78 7.93 4.54 0.49 1.59 65.22 0.023 0.66
2.73 0.234 0.78 9.06 4.40 0.50 1.53 69.64 0.042 0.93
2.49 0.231 0.78 10.9 4.24 0.51 1.46 74.55 0.064 1.16
e a (nm) e31 e33 C13 av ac
Parameters are taken from Refs. [17,18].
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Thus the total polarization is given by ! !PZ !SP P ¼ P þP
ð6Þ
2.1. Energy bands under strain Hydrostatic and uniaxial strains will be induced due to the lattice constant mismatch between different materials. Former will influence the lattice volume, which results in energy level change of materials whereas the later will remove the degeneracy. The strain-induced potential for the conduction band can be expressed as [7] VCstrian ¼ ac ðexx þ eyy þ ezz Þ
ð7Þ
where ac is the deformation potential constant of conduction band, exx ¼ eyy ¼ a0 a=a where a0 is the equilibrium lattice constant for the strained layer and a is the unstrained lattice constant and ezz ¼ 2ððC12 Þ=ðC11 ÞÞexx , C11 and C12 are the components of the two order tensor of elastic moduli. Semiconductor alloys have been proposed to obey Vegard’s law, revealing the linear relationship between the lattice constants and composition as given by [8] a0Cdx Zn1x O ðxÞ ¼ xa0CdO þ ð1xÞa0ZnO
ð8Þ
The strain-induced potential for the valence band can then be written as [9] b VVstrain ¼ av ðexx þ eyy þ ezz Þ ðexx þ eyy 2ezz Þ 2
ð9Þ
where av and b are the deformation potential constants of valence band. Some alloy parameters of CdZnO are interpolated from the values of ZnO and CdO and the values of parameters are given in Table 1. ¨ The time independent Schrodinger equation for this system is given by Hexc c ¼ Eexc c
ð10Þ
where E is the energy and c is the wave function for the eigen energy of the system. Thus the ground state of the electron–hole pair can be written as " L2e,h P2 _2 1 @2 2 re,h þ þ Ve,h ðre,h Þ Z n 2 2 2mne,h re,h @re,h 2M 2me,h re,h # p2z e2 E ce,h ðre,h Þ ¼ 0 ð11Þ þ 2m e9re rh 9 where _ is Planck’s constant, mn is the effective mass of electron (hole) defined as mne ¼ mne? ¼ mneJ and 1=mnh ¼ 1=3ððe? =eJ Þ ð1=mnhJ Þ þ ð2=mnh? ÞÞ where the e?,e99 are the dielectric constants in the different directions in the c axis of the quantum dot. L the azimuthal momentum operator, V is the confinement potential as defined earlier, E is the confinement energy and c(r) is the eigen wave function. The electron (hole) effective mass mnj is given by ( n mIj r o R n mj ¼ ð12Þ mnIIj r Z R mnIj and mnIIj denote the reduced effective mass of the inside and outside the quantum dot, respectively. The material parameters are given in Table 1. We choose a trial wave function as follows:
Cðre ,rh Þ ¼ 9cðre ÞS9cðrh ÞScðzÞS
ð13Þ
The ground state wave function of the electron (hole) confined in the strained CdxZn1 xO/ZnO quantum dot can be written as
cðrj Þ ¼ r1j jðrj ÞYlm ðy, fÞ
ð14Þ
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A. John Peter, C. Woo Lee / Physica B 406 (2011) 3666–3670
where Ylm ðy, fÞ are the complex spherical harmonics and j(r) is the radial wave function and the corresponding confinement energy equation of the electron (hole) can be obtained using the m-order Bessel function Jm and the modified Bessel function Km as ( roR N1 e 7 ilf Jl ðrnl rÞ cðr, y, jÞ ¼ ð15Þ N1 KJl ðrðbnl RÞRÞ e 7 ilf Kl ðbnl rÞ r Z R l
nl
where N1 is the normalization constant sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mnj Enlk rnl ¼ _2
ð16Þ
0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mnj ðVEnlk Þ _2
where rnl is the nth root satisfying the equation dJ ðr rÞ J ðr rÞ dKl ðbnl rÞ rnl l nl ¼ bnl l nl dðrnl rÞ r ¼ R Kl ðbnl rÞ dðbnl rÞ r ¼ R
ð17Þ
ð18Þ
And Enlk is the lowest binding energy calculated by solving the transcendental equation pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffi ð19Þ EJ1 ð ERÞK0 ð ðVEÞRÞ ¼ ð ðVEÞK1 ð ðVEÞRÞJ0 ð ERÞ This fixes the values of rnl and bnl for the lowest values of Enlk after matching the wave functions and their derivatives at boundaries of the quantum dot along with the normalization. The variation trial wave function 9c(z)S describing the internal motion between the electron and the hole in the system, taken to be the lowest state of the exciton energy is given by 9cðzÞS ¼ expða2 r2 b2 z2 Þ
ð20Þ
where a2 and b2 are the variational parameters. The anisotropic two-parameters relative to the trial wave function describe the internal motion between the electron and the hole in the complex systems. We calculate the ground state energy Eexc by finding out the expectation value of the energy of the Hamiltonian, Eq. (11), as /Eexc S ¼ min a, b
/cre 9Hexc 9cre S /cre 9cre S
ðjfr 4 R Þr cos yjir 4 R dt
For any electronic system transitions, these calculations are imperative to compute the different optical properties. However, the dipole transition transitions are allowed using the selection rules Dl ¼ 71 where l is the angular momentum quantum number. In addition to that the oscillator strength which is related to the dipole transition, expressed as 2mn _2
2
DEfi 9Mfi 9
ð25Þ
where DEfi ¼Ef Ei refers the difference of the energy between the ! lower and upper states. Mfi ¼ 2/f9 R 9iS is the electric dipole moment of the transition from i state to f state in the quantum dot. The observation of oscillator strength is imperative especially in the study of optical properties and they are related to the electronic dipole allowed absorptions. Moreover, the outcome of the results will viewed on the fine structure of the optical absorption. The optical absorption coefficient is given by [13] 8 9 2 rffiffiffiffiffiffiffiffiffi< = Mfi r_t m aðoÞ ¼ o ð26Þ 2 2 er e0 : ðEfi _oÞ þ ð_tÞ ; where m is the permeability of the system defined as m ¼1/e0c2, r is electron density of the quantum dot, o the angular frequency of the incident photon energy, c is the speed of light in the free space and e0 is the electrical permittivity of the vacuum. Ei and Ef denote the confinement energy levels for the ground and the first excited state, respectively, The radiative lifetime can be calculated as [14] 2pe m c3 h2 t ¼ pffiffiffi 02 02 ee Eexc Pfi
ð27Þ
where Pfi is the oscillator strength, Eexc is the exciton binding energy and all the other parameters are universal physical constants.
ð21Þ 3. Results and discussion
The interband emission associated with the exciton is calculated using the following equation: Eph ¼ Ee þEh þ Eg Eexc
R
1
ð24Þ
Pfi ¼
and bnl ¼
final state is given by Z R Z /Mz Sfi ¼ e ðjrf o R Þr cos yjir o R dt þ
ð22Þ
where Ee and Eh are the electron and hole quantum dot confined states, respectively, Eexc is ground state energy of the exciton and Eg is the band gap energy of CdZnO material. For the excited states, Eq. (11) may be written as " P2 l ðl þ 1Þ2 _2 _2 1 @2 2 re,h þ e,h e,hn 2 þVe,h ðre,h Þ Z 2 2mne,h re,h @re,h 2M 2me,h re,h # p2 e2 E ce,h ðre,h Þ ¼ 0 ð23Þ þ z 2m e re rh where l denotes the quantum numbers associated with the operator L. The wave function j(r) is considered to be zero at the center of the nanosphere. For the excited states, l a0, Eq. (19) may be solved using the Bessel functions as an orthonormal basis set as done earlier. We consider the polarization of the electromagnetic radiation in the electric dipole approximation, chosen in the z-direction. The dipole transition matrix element between the initial state and
Numerical calculations have been carried out to compute the donor exciton binding energy in the cylindrical CdxZn1 xO/ZnO quantum dot with the heavy hole mass as the heavy excitons are more common in experimental results. The binding energy values and thereby the interband emission energy are computed as a function of dot radius for various values of Cd content of the host material. The effect of z-confinement has been calculated through a finite quantum dot model with confinement potential determined by the band offsets and strain effects. The variation of the composition x of the quantum dot material modifies the energy gap. We observe that the band gap decreases as the Cd content increases and it can be estimated by the expression Eg(CdxZn1 xO) ¼1.76 0.11xþ1.753x2 (eV). The band gap of Cd is observed to decrease from 3.4 to 1.76 eV. It is noticed that the barrier height increases as the Cd content increases. Variation of exciton binding energy of the CdxZn1 xO/ZnO quantum dot as a function of dot radius of three different concentration with and without strain is shown in Fig. 1 and the inset figure shows the corresponding shift in transition energy for Cd concentration for two different dot radii. In all the cases, the exciton binding energy increases with a decrease of dot radius, reaching a maximum value and then decreases when the
A. John Peter, C. Woo Lee / Physica B 406 (2011) 3666–3670
65
Exciton binding energy (meV)
60 55 50 45 40
with strain
35
(1) --- Cd Zn O 30
(2) --- Cd Zn O (3) --- Cd Zn O
25
dashed lines---without strain
20 0
20
40
60
80
100
120
140
160
180
200
Dot Radius (A) Fig. 1. Variation of exciton binding energy of the CdZnO/ZnO quantum dot as a function of dot radius of three different concentrations with and without strain; the inset figure shows the shift in transition energy for Cd concentration for two different dot radii.
3669
different content of Cd. It is noted that the interband emission energy decreases monotonically as the dot radius is increased. This is due to the confinement of electron–hole with respect to z-plane when the dot radius is increased. Also it is observed that as concentration of Cd increases the optical band gap decreases due to the reduction of barrier height. Moreover it is clearly shown that the effect of donor bound exciton has influence on the interband emission energy. This representation clearly brings out the quantum size effect. Non-linearity band gap with the increase in Cd content is observed for smaller dots in the strong confinement region. Our results are closely resulted with the other results [16]. Also it is observed that the exciton transition energy lowers for all the quantum dot sizes if the strain effect due to the built-in electrostatic field is included, it is because decrease in the conduction band energy with the inclusion of biaxial strain. So it is important to include the effect of strain induced by spontaneous and piezoelectric polarization. We display the exciton oscillator strength and the radiative lifetime of confined exciton as a function of dot radius for Cd0.2Zn0.8O/ZnO with and without strain effect in Fig. 3. It is observed that the oscillator strength increases with the dot radius. The radiative lifetime can be calculated using Eq. (27). It is found that the radiative lifetime of exciton decreases with the dot radius.
b,c with strain 8
(d)
a,d without strain 9000
(c)
3.0
2.8
2.6
6
8000
7000
4 6000
2
(b)
5000
2.4
all dashed lines- without strain effect
Radiactive Life Time (10-9s)
10000
3.2
Oscillator Strength
Interband emission energy (eV)
3.4
(a) 4000
20
40
60
80
100
120
140
160
180
0
200
0
50
Dot Radius (A0)
dot radius still decreases. The Coulomb interaction between the electron and hole is increased, which ultimately causes the decrease in binding energy when the dot radius decreases. As the dot radius approaches zero the confinement becomes negligibly small, and in the finite barrier problem the tunneling becomes huge. Further, we find that the exciton binding energy is higher when the dot radius becomes smaller it is due to the confinement. From the inset figure, we find the shift in transition energy increases with the Cd content with the inclusion of strain effect induced by the spontaneous and piezoelectric polarization [15]. Further we observe that the band gap decreases as the Cd content increases and it can be estimated by Eq. (2). Red shift in emission energy is observed while including the strain effect induced by the spontaneous and piezoelectric effect. Moreover, the strain effect lowers larger dot size due to the quantum confinement. Further, we observe that the linear variation is found for lower concentration and slightly non-linearity for higher concentration [16]. In Fig. 2, we present the variation of optical band gap as a function of dot radius of CdxZn1 xO/ZnO quantum dot with
150
200
Dot Radius (Α0) Fig. 3. Exciton oscillator strength and the radiative lifetime of confined exciton as a function of dot radius of CdZnO/ZnO quantum dot with and without the strain effect.
8000
Absorption coefficient (10 /m)
Fig. 2. Variation of interband emission energy as a function of dot radius of CdxZn1 xO/ZnO quantum dot with different contents of Cd with and without strain.
100
60 A0 6000
1p-1d 1s-1p
4000
100 A0 60 A0
2000
100 A0 0 0
100
200
300
400
500
600
Photon energy (eV) Fig. 4. Variation of absorption coefficients for 1s–1p and 1p–1d states for two ˚ in a CdZnO/ZnO quantum dot as function of different dot radii (60 and 100 A) photon energy with and without the strain effect; the dashed curve represents the effect without the strain inclusion.
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A. John Peter, C. Woo Lee / Physica B 406 (2011) 3666–3670
The values of radiative lifetime are almost constant beyond the dot ˚ This is the manifestation of shortening of the lifetime radius 100 A. due to the strong overlap between electron and hole wave function. The strain induced by the polarization effect is more influenced for the exciton and it is clearly observed from the figure. Fig. 4 displays the variation of linear optical absorption coefficients for 1s–1p and 1p–1d states for two different dot radii ˚ as function of incident photon energy with and (60 and 100 A) without the strain effect. It is inferred from the figure that the maximum absorption coefficient occurs at the corresponding value of the threshold photon energy ðEfi ¼ _oÞ in all the two cases. We notice that the linear optical absorption coefficient shows larger values for smaller dots than for larger dots due to the confinement. Further, it shows the stronger behavior for smaller quantum dots since the absorption spectrum depends on the quantum dot volume. Moreover, we notice that the electronic dipolar transition matrix elements are found to be high for the transitions between the higher levels of the intensity of the total absorption coefficient which increases for the transitions between higher excited levels.
4. Conclusion In conclusion, we have calculated the eigen energy values of the confined exciton of a strained CdZnO/ZnO quantum dot as a function of dot radius with various values of Cd content using single band effective mass approximation. We have also calculated the oscillator strengths for the transitions and the linear optical absorption coefficients as a function of incident photon energy for 1s–1p and 1p–1d transitions. We have found that the confined energies and the transition energies between the excited levels are significant for smaller dots and the magnitude of the absorption spectra increases for the transitions between higher excited levels. Further, we believe that our results will lend support for further calculations in extensive optical device applications especially in the optical switching and optical communication.
Acknowledgment One of the authors (AJP) thanks the University Grants Commission, India, for the Grant, MRP (F. no. 38–78/2009((SR)) for the financial support of this work. References [1] J.W. Bae, C.H. Jeong, H.K. Kim, K.K. Kim, N.G. Cho, T.Y. Seong, S.J. Park, I. Adesida, G.Y. Yeom, Jpn. J. Appl. Phys. 42 (Part 2) (2003) L535. ¨ [2] S. Kalusniak, S. Sadofev, J. Puls, H.J. Wunsche, F. Henneberger, Phys. Rev. B 77 (2008) 113312. [3] R.R. Salunkhe, C.D. Lohande, Sensors and Actuators B 129 (2008) 345. [4] G. Yogeeswaran, C.R. Chenthamarakshan, A. Seshadri, N.R. De Tacconi, K. Rajeshwar, Thin Solid Films 515 (2006) 2464. [5] R. Vinodkumara, K.J. Lethya, P.R. Arunkumara, Renju R. Krishnana, N. Venugopalan Pillai, V.P.Mahadevan Pillai, Reji Philip, Mater. Chem. Phys. 121 (2010) 406. [6] S. Adachi, Properties of Semiconductor Alloys Group-IV, III–V and II–VI Semiconductors, John Wiley and Sons Ltd., 2009. [7] G.L. Bir, E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors, Wiley, New York, 1974. [8] J. Singh, Optoelectronics: An Introduction to Materials and Devices, Tata McGraw Hill, New Delhi, 1996. [9] S.L. Chuang, Physics of Optoelectronic Devices, John Wiley and Sons, New York, 1995. ¨ [10] S. Sadofev, S. Blumstengel, J. Cui, J. Puls, S. Rogaschewski, P. Schafer, F. Henneberger, Appl. Phys. Lett. 89 (2006) 201907. [11] B.D. Cullity, S.R. Stock, Elements of X-Ray Diffraction, third ed., Englewood Cliffs, NJ: Prentice Hall, 2001. [12] J.J. Shi, Z.Z. Gan, J. Appl. Phys. 94 (2003) 407. [13] E.M. Goldys, J.J. Shi, Phys. Status Solidi B 210 (1998) 237. [14] A. Vladimir, Fonoberov, Alexander A. Balandin, Appl. Phys. Lett. 85 (2004) 5971. [15] F. Benharrats, K. Zitouni, A. Kadri, B. Gil, Superlattices and Microstructures 47 (2010) 592. ¨ [16] S. Kalusniak, S. Sadofev, J. Puls, H.J. Wunsche, F. Henneberger, Phys. Rev. B 77 (2008) 113312. [17] A. Janotti, C.G. Van de Walle, Rep. Prof. Phys. 72 (2009) 126501. [18] Takafumi Yao, Soon-Ku Hong, Oxide and Nitride Semiconductors: Processing, Properties, and Applications Advances in Materials Research, Springer, Berlin Heidelberg, 2009 vol. 12; T. Makino, Y. Segawa, M. Kawasaki, A. Ohtomo, R. Shiroki, K. Tamura, T. Yasuda, H. Koinuma, Appl. Phys. Lett. 78 (2001) 1237.
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Physica B journal homepage: www.elsevier.com/locate/physb
Interband optical absorption in a strained CdxZn1 xO/ZnO quantum dot A. John Peter n, Chang Woo Lee nn Department of Chemical Engineering and Green Energy Center, College of Engineering, Kyung Hee University, 1Seochun, Gihung, Yongin, Gyeonggi 446-701, South Korea
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 May 2011 Accepted 28 June 2011 Available online 6 July 2011
Numerical calculations of the excitonic absorption spectra in a strained CdxZn1 xO/ZnO quantum dot are investigated for various Cd contents. We calculate the quantized energies of the exciton as a function of dot radius for various confinement potentials and thereby the interband emission energy is computed considering the internal electric field induced by the spontaneous and piezoelectric polarizations. The optical absorption as a function of photon energy for different dot radii is discussed. Decrease of exciton binding energy and the corresponding optical band gap with the Cd concentration imply that the confinement of carriers decreases with composition x. The main results show that the confined energies and the transition energies between the excited levels are significant for smaller dots. Non-linearity band gap with the increase in Cd content is observed for smaller dots in the strong confinement region and the magnitude of the absorption spectra increases for the transitions between the higher excited levels. & 2011 Elsevier B.V. All rights reserved.
Keywords: Quantum dot Donor bound excitons Oscillator strength
1. Introduction Present decade is witnessing some novel wide band gap materials such as GaN and ZnO play a vitol role for making multilayered heterostrucutres in the green region of spectrum. The wide band gap semiconductors have attracted much attention due to their potential applications and device fabrications for electro- and opto-electronic devices in blue and ultraviolet regions. The ZnO material is preferred because it can deliver low cost with high brightness devices, further the Gallium nitride mateial is toxic. ZnO based compounds have a larger exciton binding energy roughly twice than that of GaN, a high radiation resistance and high chemical stability and they are compatible with both wet chemical and dry etching [1]. And also, all the wurtzite materials will exhibit both pyroelectric and piezoelectric behavior. CdZnO materials have internal electric field and will also contribute both spoantaneous and piezoelectric polarizations. This property will change the behavior of the ZnO based heterostructures [2]. Hence, it is imperative to understand the behavior of its excitonic recombination. CdO is an n-type material, a conducting and a transperarent element with the direct band gap energy approximately 2.5 eV while the Eg for ZnO is around 3.4 eV. The band gap of ZnO can be red-shifted to blue and green spectral range with the suitable incorporation of Cd so one can tune the band gap for some potenital applications and it has been used for device fabrications such as
n Corresponding author. Permanent address: Govt. Arts College, Melur 625106, Madurai, Tamil Nadu, India. Tel.: þ91 9786141966; fax: þ91 4522415467. nn Corresponding author. E-mail address: [email protected] (A. John Peter).
0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.06.067
phototransistors, solar cells, gas sensors IR detectors and liquid crystal displays based on ZnO/ZnCdO heterojunctions and superlattice structures [3–5]. Thus ZnCdO is considered to be an ideal material for ZnO-based devices, which are given due attention to study their excitonic electro-optical properties especially for making laser diodes wherein the band gap engineering is necessary. In the present work, we have calculated the eigen energy values of the confined exciton as a function of dot radius in a strained CdZnO/ZnO quantum dot with various values of Cd content. The exciton binding energy with different confinment is computed and thereby interband emission energy is calculated. The interband absorption spectra of the exciton in a CdxZn1 xO/ ZnO quantum dot have been investigated considering the internal electric field induced by the spontaneous and piezoelectric polarizations. We have used the variational technique within the single band effective mass approximation to calculate the binding energy and thereby their optical band gap as afuntion of dot radius. We investigate the exciton oscillator strength and the radiative lifetime of confined exciton as a function of dot radius of Cd0.2Zn0.8O/ZnO quantum dot with and without strain effect. The optical absorption as a function of photon energy for different dot radii is discussed. This paper is organized as follows: the method followed is presented in Sections 2 and 2.1 while the results and discussion are presented in Section 3. Finally, we summarize the main conclusions obtained in this paper in Section 4.
2. Theoretical Within the framework of single band effective mass approximation, the Hamiltonian of the exciton considering a strained
A. John Peter, C. Woo Lee / Physica B 406 (2011) 3666–3670
CdxZn1 xO/ZnO quantum dot with the radius R, can be written as " # p2rj X P2 p2 e2 þ fðrÞ þ Vj ðrj Þ þ Z þ z ð1Þ Hexc ¼ 2m 2M 2 m e r rh j e j where j¼ e and h refer to the electron and hole, respectively, e is the dielectric constant for the material inside the quantum dot and Vj(rj) is the strain induced confinement potential of the particle j. The band gap of the material is given by [6] Eg ðCdx Zn1x OÞ ¼ 1:760:11x þ 1:75x2 ðeVÞ
ð2Þ
The above expression of band gap has been obtained by varying Cd concentration in ZnO quantum dot. The distribution of the band gap discontinuities is taken as V(z) ¼0.8DEg. PZ and pz are the exciton center-of mass momentum and relative motion momentum position along the z-direction, respectively. M ¼me þmh and m ¼memh/(me þmh) are the total and reduced effective masses of the exciton, respectively. f(r) is the electrostatic potential induced by the internal electric field. The internal electric field in CdxZn1 xO is characterized with the linear variation of Cd content up to x¼ 0.35 (Eq. (2)) beyond this level there occurs a phase change. As the CdxZn1 xO material changes a phase transition when x40.32 in the wurtzite phase [10,11]. The strength of the built-in electric field caused by the spontaneous and piezoelectric polarizations in the ZnxCd1 xO/ ZnO strained quantum dot expressed as [12] 8 CdO CdO Znx Cd1x O < PSP þ PPE PSP roR e0 eCdO e fðrÞ ¼ ð3Þ : 0 rZR CdO is the electronic dielectric constant of CdO and PPE , here, eCdO e Zns Cd1x O CdO PSP and PSP are the piezoelectric polarizations, spontaneous polarizations of CdO and the spontaneous polarization of ZnxCd1 xO, respectively. The above values can be generally calculated by the polarity of the crystal and the strains of the quantum nanostructure. Since the wurtzite crystal lattice of CdO and ZnO lack inversion symmetry, the heterostructure will have spontaneous polarization (PSP) and the piezoelectric polarization (PPZ) due to strain caused by the lattice mismatch between CdO and ZnO material. The piezoelectric polarization along the c axis is given by
P PZ ¼ e31 ðexx þ eyy Þ þ e33 ezz
ð4Þ
with exx ¼ eyy ¼a(CdO) a(ZnCdO)/a(CdO) and ezz ¼ 2(C13)/ (C33)exx, the values of parameters are given in Table 1. The piezoelectric polarization is given by C13 P PZ ¼ 2exx e31 e33 ð5Þ C33 Table 1 Material parameters for ZnO, CdO and CdxZn1 xO (linearly interpolated from the data of ZnO and CdO) used in the calculations. Parameters
ZnO
CdO
Cd0.1Zn0.9O
Cd0.2Zn0.8O
Cd0.3Zn0.7O
Eg (eV) me (m0) mh (m0)
3.29 0.24 0.78 6.8 3.20 0.62 0.96 105.1 0.2 3.1
2.2 0.21 0.78 18.1 4.70 0.48 0.96 1.67 61 0 0.4
2.89 0.237 0.78 7.93 4.54 0.49 1.59 65.22 0.023 0.66
2.73 0.234 0.78 9.06 4.40 0.50 1.53 69.64 0.042 0.93
2.49 0.231 0.78 10.9 4.24 0.51 1.46 74.55 0.064 1.16
e a (nm) e31 e33 C13 av ac
Parameters are taken from Refs. [17,18].
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Thus the total polarization is given by ! !PZ !SP P ¼ P þP
ð6Þ
2.1. Energy bands under strain Hydrostatic and uniaxial strains will be induced due to the lattice constant mismatch between different materials. Former will influence the lattice volume, which results in energy level change of materials whereas the later will remove the degeneracy. The strain-induced potential for the conduction band can be expressed as [7] VCstrian ¼ ac ðexx þ eyy þ ezz Þ
ð7Þ
where ac is the deformation potential constant of conduction band, exx ¼ eyy ¼ a0 a=a where a0 is the equilibrium lattice constant for the strained layer and a is the unstrained lattice constant and ezz ¼ 2ððC12 Þ=ðC11 ÞÞexx , C11 and C12 are the components of the two order tensor of elastic moduli. Semiconductor alloys have been proposed to obey Vegard’s law, revealing the linear relationship between the lattice constants and composition as given by [8] a0Cdx Zn1x O ðxÞ ¼ xa0CdO þ ð1xÞa0ZnO
ð8Þ
The strain-induced potential for the valence band can then be written as [9] b VVstrain ¼ av ðexx þ eyy þ ezz Þ ðexx þ eyy 2ezz Þ 2
ð9Þ
where av and b are the deformation potential constants of valence band. Some alloy parameters of CdZnO are interpolated from the values of ZnO and CdO and the values of parameters are given in Table 1. ¨ The time independent Schrodinger equation for this system is given by Hexc c ¼ Eexc c
ð10Þ
where E is the energy and c is the wave function for the eigen energy of the system. Thus the ground state of the electron–hole pair can be written as " L2e,h P2 _2 1 @2 2 re,h þ þ Ve,h ðre,h Þ Z n 2 2 2mne,h re,h @re,h 2M 2me,h re,h # p2z e2 E ce,h ðre,h Þ ¼ 0 ð11Þ þ 2m e9re rh 9 where _ is Planck’s constant, mn is the effective mass of electron (hole) defined as mne ¼ mne? ¼ mneJ and 1=mnh ¼ 1=3ððe? =eJ Þ ð1=mnhJ Þ þ ð2=mnh? ÞÞ where the e?,e99 are the dielectric constants in the different directions in the c axis of the quantum dot. L the azimuthal momentum operator, V is the confinement potential as defined earlier, E is the confinement energy and c(r) is the eigen wave function. The electron (hole) effective mass mnj is given by ( n mIj r o R n mj ¼ ð12Þ mnIIj r Z R mnIj and mnIIj denote the reduced effective mass of the inside and outside the quantum dot, respectively. The material parameters are given in Table 1. We choose a trial wave function as follows:
Cðre ,rh Þ ¼ 9cðre ÞS9cðrh ÞScðzÞS
ð13Þ
The ground state wave function of the electron (hole) confined in the strained CdxZn1 xO/ZnO quantum dot can be written as
cðrj Þ ¼ r1j jðrj ÞYlm ðy, fÞ
ð14Þ
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where Ylm ðy, fÞ are the complex spherical harmonics and j(r) is the radial wave function and the corresponding confinement energy equation of the electron (hole) can be obtained using the m-order Bessel function Jm and the modified Bessel function Km as ( roR N1 e 7 ilf Jl ðrnl rÞ cðr, y, jÞ ¼ ð15Þ N1 KJl ðrðbnl RÞRÞ e 7 ilf Kl ðbnl rÞ r Z R l
nl
where N1 is the normalization constant sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mnj Enlk rnl ¼ _2
ð16Þ
0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mnj ðVEnlk Þ _2
where rnl is the nth root satisfying the equation dJ ðr rÞ J ðr rÞ dKl ðbnl rÞ rnl l nl ¼ bnl l nl dðrnl rÞ r ¼ R Kl ðbnl rÞ dðbnl rÞ r ¼ R
ð17Þ
ð18Þ
And Enlk is the lowest binding energy calculated by solving the transcendental equation pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffi ð19Þ EJ1 ð ERÞK0 ð ðVEÞRÞ ¼ ð ðVEÞK1 ð ðVEÞRÞJ0 ð ERÞ This fixes the values of rnl and bnl for the lowest values of Enlk after matching the wave functions and their derivatives at boundaries of the quantum dot along with the normalization. The variation trial wave function 9c(z)S describing the internal motion between the electron and the hole in the system, taken to be the lowest state of the exciton energy is given by 9cðzÞS ¼ expða2 r2 b2 z2 Þ
ð20Þ
where a2 and b2 are the variational parameters. The anisotropic two-parameters relative to the trial wave function describe the internal motion between the electron and the hole in the complex systems. We calculate the ground state energy Eexc by finding out the expectation value of the energy of the Hamiltonian, Eq. (11), as /Eexc S ¼ min a, b
/cre 9Hexc 9cre S /cre 9cre S
ðjfr 4 R Þr cos yjir 4 R dt
For any electronic system transitions, these calculations are imperative to compute the different optical properties. However, the dipole transition transitions are allowed using the selection rules Dl ¼ 71 where l is the angular momentum quantum number. In addition to that the oscillator strength which is related to the dipole transition, expressed as 2mn _2
2
DEfi 9Mfi 9
ð25Þ
where DEfi ¼Ef Ei refers the difference of the energy between the ! lower and upper states. Mfi ¼ 2/f9 R 9iS is the electric dipole moment of the transition from i state to f state in the quantum dot. The observation of oscillator strength is imperative especially in the study of optical properties and they are related to the electronic dipole allowed absorptions. Moreover, the outcome of the results will viewed on the fine structure of the optical absorption. The optical absorption coefficient is given by [13] 8 9 2 rffiffiffiffiffiffiffiffiffi< = Mfi r_t m aðoÞ ¼ o ð26Þ 2 2 er e0 : ðEfi _oÞ þ ð_tÞ ; where m is the permeability of the system defined as m ¼1/e0c2, r is electron density of the quantum dot, o the angular frequency of the incident photon energy, c is the speed of light in the free space and e0 is the electrical permittivity of the vacuum. Ei and Ef denote the confinement energy levels for the ground and the first excited state, respectively, The radiative lifetime can be calculated as [14] 2pe m c3 h2 t ¼ pffiffiffi 02 02 ee Eexc Pfi
ð27Þ
where Pfi is the oscillator strength, Eexc is the exciton binding energy and all the other parameters are universal physical constants.
ð21Þ 3. Results and discussion
The interband emission associated with the exciton is calculated using the following equation: Eph ¼ Ee þEh þ Eg Eexc
R
1
ð24Þ
Pfi ¼
and bnl ¼
final state is given by Z R Z /Mz Sfi ¼ e ðjrf o R Þr cos yjir o R dt þ
ð22Þ
where Ee and Eh are the electron and hole quantum dot confined states, respectively, Eexc is ground state energy of the exciton and Eg is the band gap energy of CdZnO material. For the excited states, Eq. (11) may be written as " P2 l ðl þ 1Þ2 _2 _2 1 @2 2 re,h þ e,h e,hn 2 þVe,h ðre,h Þ Z 2 2mne,h re,h @re,h 2M 2me,h re,h # p2 e2 E ce,h ðre,h Þ ¼ 0 ð23Þ þ z 2m e re rh where l denotes the quantum numbers associated with the operator L. The wave function j(r) is considered to be zero at the center of the nanosphere. For the excited states, l a0, Eq. (19) may be solved using the Bessel functions as an orthonormal basis set as done earlier. We consider the polarization of the electromagnetic radiation in the electric dipole approximation, chosen in the z-direction. The dipole transition matrix element between the initial state and
Numerical calculations have been carried out to compute the donor exciton binding energy in the cylindrical CdxZn1 xO/ZnO quantum dot with the heavy hole mass as the heavy excitons are more common in experimental results. The binding energy values and thereby the interband emission energy are computed as a function of dot radius for various values of Cd content of the host material. The effect of z-confinement has been calculated through a finite quantum dot model with confinement potential determined by the band offsets and strain effects. The variation of the composition x of the quantum dot material modifies the energy gap. We observe that the band gap decreases as the Cd content increases and it can be estimated by the expression Eg(CdxZn1 xO) ¼1.76 0.11xþ1.753x2 (eV). The band gap of Cd is observed to decrease from 3.4 to 1.76 eV. It is noticed that the barrier height increases as the Cd content increases. Variation of exciton binding energy of the CdxZn1 xO/ZnO quantum dot as a function of dot radius of three different concentration with and without strain is shown in Fig. 1 and the inset figure shows the corresponding shift in transition energy for Cd concentration for two different dot radii. In all the cases, the exciton binding energy increases with a decrease of dot radius, reaching a maximum value and then decreases when the
A. John Peter, C. Woo Lee / Physica B 406 (2011) 3666–3670
65
Exciton binding energy (meV)
60 55 50 45 40
with strain
35
(1) --- Cd Zn O 30
(2) --- Cd Zn O (3) --- Cd Zn O
25
dashed lines---without strain
20 0
20
40
60
80
100
120
140
160
180
200
Dot Radius (A) Fig. 1. Variation of exciton binding energy of the CdZnO/ZnO quantum dot as a function of dot radius of three different concentrations with and without strain; the inset figure shows the shift in transition energy for Cd concentration for two different dot radii.
3669
different content of Cd. It is noted that the interband emission energy decreases monotonically as the dot radius is increased. This is due to the confinement of electron–hole with respect to z-plane when the dot radius is increased. Also it is observed that as concentration of Cd increases the optical band gap decreases due to the reduction of barrier height. Moreover it is clearly shown that the effect of donor bound exciton has influence on the interband emission energy. This representation clearly brings out the quantum size effect. Non-linearity band gap with the increase in Cd content is observed for smaller dots in the strong confinement region. Our results are closely resulted with the other results [16]. Also it is observed that the exciton transition energy lowers for all the quantum dot sizes if the strain effect due to the built-in electrostatic field is included, it is because decrease in the conduction band energy with the inclusion of biaxial strain. So it is important to include the effect of strain induced by spontaneous and piezoelectric polarization. We display the exciton oscillator strength and the radiative lifetime of confined exciton as a function of dot radius for Cd0.2Zn0.8O/ZnO with and without strain effect in Fig. 3. It is observed that the oscillator strength increases with the dot radius. The radiative lifetime can be calculated using Eq. (27). It is found that the radiative lifetime of exciton decreases with the dot radius.
b,c with strain 8
(d)
a,d without strain 9000
(c)
3.0
2.8
2.6
6
8000
7000
4 6000
2
(b)
5000
2.4
all dashed lines- without strain effect
Radiactive Life Time (10-9s)
10000
3.2
Oscillator Strength
Interband emission energy (eV)
3.4
(a) 4000
20
40
60
80
100
120
140
160
180
0
200
0
50
Dot Radius (A0)
dot radius still decreases. The Coulomb interaction between the electron and hole is increased, which ultimately causes the decrease in binding energy when the dot radius decreases. As the dot radius approaches zero the confinement becomes negligibly small, and in the finite barrier problem the tunneling becomes huge. Further, we find that the exciton binding energy is higher when the dot radius becomes smaller it is due to the confinement. From the inset figure, we find the shift in transition energy increases with the Cd content with the inclusion of strain effect induced by the spontaneous and piezoelectric polarization [15]. Further we observe that the band gap decreases as the Cd content increases and it can be estimated by Eq. (2). Red shift in emission energy is observed while including the strain effect induced by the spontaneous and piezoelectric effect. Moreover, the strain effect lowers larger dot size due to the quantum confinement. Further, we observe that the linear variation is found for lower concentration and slightly non-linearity for higher concentration [16]. In Fig. 2, we present the variation of optical band gap as a function of dot radius of CdxZn1 xO/ZnO quantum dot with
150
200
Dot Radius (Α0) Fig. 3. Exciton oscillator strength and the radiative lifetime of confined exciton as a function of dot radius of CdZnO/ZnO quantum dot with and without the strain effect.
8000
Absorption coefficient (10 /m)
Fig. 2. Variation of interband emission energy as a function of dot radius of CdxZn1 xO/ZnO quantum dot with different contents of Cd with and without strain.
100
60 A0 6000
1p-1d 1s-1p
4000
100 A0 60 A0
2000
100 A0 0 0
100
200
300
400
500
600
Photon energy (eV) Fig. 4. Variation of absorption coefficients for 1s–1p and 1p–1d states for two ˚ in a CdZnO/ZnO quantum dot as function of different dot radii (60 and 100 A) photon energy with and without the strain effect; the dashed curve represents the effect without the strain inclusion.
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The values of radiative lifetime are almost constant beyond the dot ˚ This is the manifestation of shortening of the lifetime radius 100 A. due to the strong overlap between electron and hole wave function. The strain induced by the polarization effect is more influenced for the exciton and it is clearly observed from the figure. Fig. 4 displays the variation of linear optical absorption coefficients for 1s–1p and 1p–1d states for two different dot radii ˚ as function of incident photon energy with and (60 and 100 A) without the strain effect. It is inferred from the figure that the maximum absorption coefficient occurs at the corresponding value of the threshold photon energy ðEfi ¼ _oÞ in all the two cases. We notice that the linear optical absorption coefficient shows larger values for smaller dots than for larger dots due to the confinement. Further, it shows the stronger behavior for smaller quantum dots since the absorption spectrum depends on the quantum dot volume. Moreover, we notice that the electronic dipolar transition matrix elements are found to be high for the transitions between the higher levels of the intensity of the total absorption coefficient which increases for the transitions between higher excited levels.
4. Conclusion In conclusion, we have calculated the eigen energy values of the confined exciton of a strained CdZnO/ZnO quantum dot as a function of dot radius with various values of Cd content using single band effective mass approximation. We have also calculated the oscillator strengths for the transitions and the linear optical absorption coefficients as a function of incident photon energy for 1s–1p and 1p–1d transitions. We have found that the confined energies and the transition energies between the excited levels are significant for smaller dots and the magnitude of the absorption spectra increases for the transitions between higher excited levels. Further, we believe that our results will lend support for further calculations in extensive optical device applications especially in the optical switching and optical communication.
Acknowledgment One of the authors (AJP) thanks the University Grants Commission, India, for the Grant, MRP (F. no. 38–78/2009((SR)) for the financial support of this work. References [1] J.W. Bae, C.H. Jeong, H.K. Kim, K.K. Kim, N.G. Cho, T.Y. Seong, S.J. Park, I. Adesida, G.Y. Yeom, Jpn. J. Appl. Phys. 42 (Part 2) (2003) L535. ¨ [2] S. Kalusniak, S. Sadofev, J. Puls, H.J. Wunsche, F. Henneberger, Phys. Rev. B 77 (2008) 113312. [3] R.R. Salunkhe, C.D. Lohande, Sensors and Actuators B 129 (2008) 345. [4] G. Yogeeswaran, C.R. Chenthamarakshan, A. Seshadri, N.R. De Tacconi, K. Rajeshwar, Thin Solid Films 515 (2006) 2464. [5] R. Vinodkumara, K.J. Lethya, P.R. Arunkumara, Renju R. Krishnana, N. Venugopalan Pillai, V.P.Mahadevan Pillai, Reji Philip, Mater. Chem. Phys. 121 (2010) 406. [6] S. Adachi, Properties of Semiconductor Alloys Group-IV, III–V and II–VI Semiconductors, John Wiley and Sons Ltd., 2009. [7] G.L. Bir, E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors, Wiley, New York, 1974. [8] J. Singh, Optoelectronics: An Introduction to Materials and Devices, Tata McGraw Hill, New Delhi, 1996. [9] S.L. Chuang, Physics of Optoelectronic Devices, John Wiley and Sons, New York, 1995. ¨ [10] S. Sadofev, S. Blumstengel, J. Cui, J. Puls, S. Rogaschewski, P. Schafer, F. Henneberger, Appl. Phys. Lett. 89 (2006) 201907. [11] B.D. Cullity, S.R. Stock, Elements of X-Ray Diffraction, third ed., Englewood Cliffs, NJ: Prentice Hall, 2001. [12] J.J. Shi, Z.Z. Gan, J. Appl. Phys. 94 (2003) 407. [13] E.M. Goldys, J.J. Shi, Phys. Status Solidi B 210 (1998) 237. [14] A. Vladimir, Fonoberov, Alexander A. Balandin, Appl. Phys. Lett. 85 (2004) 5971. [15] F. Benharrats, K. Zitouni, A. Kadri, B. Gil, Superlattices and Microstructures 47 (2010) 592. ¨ [16] S. Kalusniak, S. Sadofev, J. Puls, H.J. Wunsche, F. Henneberger, Phys. Rev. B 77 (2008) 113312. [17] A. Janotti, C.G. Van de Walle, Rep. Prof. Phys. 72 (2009) 126501. [18] Takafumi Yao, Soon-Ku Hong, Oxide and Nitride Semiconductors: Processing, Properties, and Applications Advances in Materials Research, Springer, Berlin Heidelberg, 2009 vol. 12; T. Makino, Y. Segawa, M. Kawasaki, A. Ohtomo, R. Shiroki, K. Tamura, T. Yasuda, H. Koinuma, Appl. Phys. Lett. 78 (2001) 1237.