- Email: [email protected]
MgxZn1−xO quantum wells with an external electric field
Superlattices and Microstructures 56 (2013) 92–98
Contents lists available at SciVerse ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
A simulation of intersubband absorption in ZnO/MgxZn1xO quantum wells with an external electric field J. Zhu a, S.L. Ban a,⇑, S.H. Ha b a
School of Physical Science and Technology, Inner Mongolia University, Key Laboratory of Semiconductor Photovoltaic Technology at Universities of Inner Mongolia Autonomous Region, Hohhot 010021, PR China b Department of Physics, College of Sciences, Inner Mongolia University of Technology, Hohhot 010051, PR China
a r t i c l e
i n f o
Article history: Received 23 May 2012 Received in revised form 9 October 2012 Accepted 9 November 2012 Available online 21 November 2012 Keywords: Intersubband transition ZnO/MgxZn1xO quantum well Electric field
a b s t r a c t A detailed simulation on the intersubband absorption for 1–2, 2–3, and 1–3 optical transitions in ZnO/MgxZn1xO quantum wells is presented. The quantum-confined Stark effect induced by the internal polarization field on the absorption process is effectively controlled through an external electric field. It is easy to obtain the structural optimization of light absorption in different terahertz ranges in our numerical analysis. The absorption wavelengths corresponding to almost all of the transitions increase as the applied field varies from 500 kV/cm to 500 kV/cm. The small absorption coefficient corresponding to the 1–3 optical transition is increased by enhancing the structural asymmetry of quantum-well potential. The present work can be extended to the structural design and simulation of new optoelectronic devices based on other low-dimensional structures such as quantum wires and quantum dots. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Recently, much attention has been paid on intersubband transitions (ISBTs) in quantum wells (QWs) composed of wurtzite semiconductors due to their device applications such as blue/green/ ultraviolet light emitting diodes, laser diodes, photodetectors and all-optical switches [1–4]. Existing experiments [5–8] showed that the shape of absorption line exhibits either a pure Lorentzian shape or multiple peaks attributed to 1–2, 1–3 transitions and so on. The absorption coefficients associated with ISBTs in these low-dimensional structures are very large due to the fast relaxation process. Some ⇑ Corresponding author. Tel.: +86 471 4992237; fax: +86 471 4951761. E-mail address: [email protected] (S.L. Ban). 0749-6036/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.spmi.2012.11.011
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93
authors studied theoretically the linear or nonlinear intersubband absorption in III-nitride QWs [9,10]. The quasi-bound electron states were obtained by solving the Schrödinger equation using the analytical Airy functions without considering the many-electron effect. Within the framework of the Hartree–Fock approximation, an iterative procedure was developed to investigate the ISBTs in doped GaN/AlxGa1xN QWs [11] and multiple QWs (MQWs) [12]. The calculated absorption peaks show a quantum-confined Stark effect due to the built-in electric field (BEF) produced by the spontaneous polarization and the strain-induced piezoelectric polarization. However, less quantized electron states can exist in these single QWs with a very strong BEF (in order of MV/cm) which was pointed out in our previous works [13,14]. This is unfavorable for the device applications. Shen et al. [15,16] reported that a four-energy-level system was achieved in nitride double QWs with an applied electric field. However, the BEF was underestimated without considering the piezoelectric polarization in the well regions, and the applied field was still too large to be realized in actual conditions. Ohno et al. [17,18] experimentally studied the ISBTs in ZnO/MgxZn1xO MQWs and found that there are certain unique properties for intersubband absorption in ZnO-based QWs. As known, ZnO is a semiconductor with a wide band-gap of 3.37 eV, making it an ideal candidate for optoelectronic device applications in the blue to ultraviolet regions [19–22]. The transition wavelength in ZnO-based QWs can be tailored in a wide spectral range for potential applications in optical communication by structural optimization due to the large conduction band offset at heterointerfaces. Moreover, the intersubband relaxation time is expected to be very fast because of large effective masses of electrons and the ionicity of ZnO [23]. A comparatively large spontaneous polarization exists in ZnO material due to the ionic nature of Zn–O bond and low symmetry crystal structure [20–22]. This, together with the straininduced piezoelectric polarization, produces a BEF in ZnO/MgxZn1xO QWs, whereas the field strength is estimated to be smaller than that of nitride QWs [20–22] and thus can be restrained by applying an external electric field. As far as we known, there were few of detailed investigations concerning about the ISBTs in ZnO/MgZnO QWs. In this work, we adopt a numerical method proposed by Li et al. [11] to solve self-consistently the discrete electron states in ZnO/MgxZn1xO QWs under an external electric field and then provide a theoretical analysis on the intersubband absorption in this system. 2. Model Let us consider a ZnO/MgxZn1xO QW with barrier and well width denoted as db and dw. Within the framework of the effective mass approximation, the Schrödinger equation in the z direction for a confined electron with charge e is given by
(
) 2 @ h 1 @ þ VðzÞ þ e½F ex þ F in ðzÞz þ V H ðzÞ þ V xc ðzÞ wi ðzÞ ¼ Ei wi ðzÞ: 2 @z mðzÞ @z
ð1Þ
In above equation, m(z) and V(z) are the electronic effective mass and the height of potential barriers, respectively. Fex is the applied external electric field and Fin(z) is the BEF given by
8 > < F b ; dw =2 < jzj < dw =2 þ db ; FðzÞ ¼ F w ; jzj 6 dw =2; > :
ð2Þ
where
Fw ¼
2db P ; 2db ew þ dw eb
ð3Þ
and
Fb ¼
dw Fw: 2db
ð4Þ
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J. Zhu et al. / Superlattices and Microstructures 56 (2013) 92–98
In above equations, ew and eb are the relative dielectric constant. P is the polarization including spontaneous and piezoelectric polarization. In ZnO/MgxZn1xO QWs, the spontaneous and piezoelectric polarizations are in opposite directions, PPE = 0.034x cm2 and PSP = 0.066x cm2, thus P = PPE + PSP [22]. The Hartree electrostatic potential VH(z) in Eq. (1) can be determined by the Poisson equation
d dz
e0 ðzÞ
d V H ðzÞ ¼ e2 ½NþD ðzÞ NðzÞ; dz
ð5Þ
with the density of ionized donors written as
NþD ðzÞ ¼
ND 1 þ 2 exp
h
Ef ED kB T
i;
ð6Þ
and the density of free electrons
NðzÞ ¼
X mðzÞkB T
ph2
i
Ef Ei ; jwi ðzÞj2 ln 1 þ exp kB T
ð7Þ
where ND is the donor density and ED the donor ionization energy. kB is the Boltzmann constant and T is the temperature. Ef is the Fermi energy which can be derived from the condition of charge neutrality
Z
þ1
1
½N þD ðzÞ NðzÞdz ¼ 0:
ð8Þ
Using the local density approximation, the exchange–correlation potential Vxc(z) in Eq. (1) can be formulated in an analytic form [24]
1=3 e2 9 11:4 p 1 þ 0:0545r ðzÞ ln 1 þ ; s r s ðzÞ 4p2 e0 ðzÞaB ðzÞrs ðzÞ 4
V xc ðzÞ ¼
ð9Þ
where the effective Bohr radius is defined as aB(z) = e0(z) h2/e2m(z) and the radius of a sphere contain1 1=3 3 ing one electron as r s ðzÞ ¼ fð3=4pÞ½aB ðzÞNðzÞ g . The eigen energies and wave functions of the electron can be obtained by self-consistently solving the Schrödinger Eq. (1) and the Poisson Eq. (5) via an iterative procedure. The optical ISBT absorption coefficients are calculated using the following formula:
x
aðhxÞ ¼
rffiffiffiffiffiffiffiffiffiffiX
dw
l0 ðNn N m Þ h= s jM j2 ; e0 ew m>n mn ðEm En hxÞ2 þ ðh=sÞ2
ð10Þ
where l0 and e0 are the permeability and dielectric constant of vacuum, Mmn is the dipole matrix element, Ni(z) (i = m, n) is the electron density in the ith subband level, and s is the relaxation time. For an intersubband optical transition with a polarization along the z direction, the dipole matrix element Mmn in Eq. (10) is given by
M mn ¼
Z
þ1
1
wm ðzÞezwn ðzÞdz:
ð11Þ
3. Numerical results and discussion The material parameters used in the calculation are cited from Ref. [20]. The donor density and the temperature are fixed at 1 1019 cm3 and 300 K, respectively. The dephasing time s is taken as 20 fs [23]. It is worthwhile to mention that s is a complex parameter affected by many scattering mechanisms including the scattering from different kinds of phonons, alloy disorder, donors and so on [25]. The numerical results are shown in Figs. 1–5. Fig. 1 gives the conduction band structures, energy levels and electron wavefunctions for a typical ZnO/MgxZn1xO QW in the cases of different external electric fields. The BEFs in the well and barriers are calculated as 0.82 MV/cm and 0.41 MV/cm with the positive direction assumed to be along the
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95
Fig. 1. Calculated conduction band structures, energy levels and electron wavefunctions for a 3aB wide ZnO/Mg0.3Zn0.7O QW with different external electric fields. Here, the effective Bohr radius aB is taken as 1.65 nm.
Fig. 2. Dipole matrix elements as functions of external electric field F for 1–2, 1–3 and 2–3 ISBTs in ZnO/Mg0.3Zn0.7O QWs with well width dw = 3aB.
growth direction. As shown in Fig. 1a, the potential of conduction band is tilted by the BEF and bent by the Hartree–Fock potential, and thus the electron wavefunctions slightly move close to the left interface. We also show that the influence from the exchange–correlation potential is more obvious than that from the Hartree term in the Poisson equation if more electron states are involved in the computation. It can be seen from Fig. 1b that the BEFs in the barriers are suppressed but the BEF in the well increases with applying a negative electric field. Consequently, the energy levels collectively increase while the wavefunctions get more close to the left interface and easily penetrate into the left barrier. Oppositely, Fig. 1c indicates that the electron states can be controlled like the case of a square QW with a positive external electric field. The dipole matrix elements Mmn as functions of external electric field F for 1–2, 1–3 and 2–3 ISBTs in ZnO/MgxZn1xO QWs are plotted in Fig. 2. As F increases from 500 kV/cm to 500 kV/cm, M12 and M23 always increase but M13 decreases. That is to say, the strengths of 1–2 and 2–3 optical transitions increase. The 1–3 transition becomes more difficult as the BEF in the well is gradually restrained by increasing F although the BEF allows this so-called forbidden optical transition. The absorption wavelength of each transition as a function of F is shown in Fig. 3. It is found that the absorption wavelengths corresponding to almost all of the ISBTs increase with F. However, the amplitude variation discriminates from each transition. The absorption wavelength of 1–2 transition rapidly increases
96
J. Zhu et al. / Superlattices and Microstructures 56 (2013) 92–98
Fig. 3. Absorption wavelengths as functions of external electric field F for 1–2, 1–3 and 2–3 ISBTs in ZnO/Mg0.3Zn0.7O QWs with well width dw = 3aB.
Fig. 4. Absorption wavelengths as functions of well width dw and Mg composition x for 1–2 ISBTs in ZnO/MgxZn1xO QWs with F = 500 kV/cm. Each kind of color, i.e., the grad scale, represents a special wavelength range, e.g., red indicates 0–1 lm and blue indicates 15–16 lm.
with F but the absorption wavelength of 2–3 transition is insensitive to F since the lower excited states are more likely influenced by the quantum-confined Stark effect. In order to obtain the structural optimization of light absorption in different terahertz ranges as required as one can, Fig. 4 plots the absorption wavelength versus well width dw and Mg composition x for 1–2 ISBTs with F = 500 kV/cm. As seen, the absorption wavelength decreases with increasing x; but it increases if dw becomes larger due to the enhancement of the quantum confinement on electrons from higher barriers and wider well. The favorable wavelength region can be achieved by choosing suitable x and dw. For example, the near-infrared light absorption occurs if dw < 2 nm and x > 0.7, and the light absorption in mid-infrared range can be achieved when x > 0.15. If taking advantage of the absorption wavelength originated from 1–3 transition, it only needs to apply a negative electric
J. Zhu et al. / Superlattices and Microstructures 56 (2013) 92–98
97
Fig. 5. Absorption coefficients as functions of external electric field F and incident photon energy ⁄x for 1–2, 1–3 and 2–3 ISBTs in ZnO/Mg0.3Zn0.7O QWs with well width dw = 3aB.
field to strengthen the BEF in the well or alter the Mg composition in one of the barriers to enhance the structural asymmetry. Fig. 5 shows the total absorption coefficient as functions of F and incident photon energy for 1–2, 1–3 and 2–3 ISBTs. It can be seen that the peak value contributed from 1–2 transition is largest, whereas the peak value of 1–3 transition is very imperceptible since its dipole matrix element is still very small although the symmetry of the wave functions is broken by the BEF. In fact, the optical absorption coefficient mainly depends on the optical dipole matrix element and the position of energy levels relative to the Fermi energy. The former reflects the selection rule of different transitions, whereas the latter determines the amount of carriers locating at each energy level. With increasing F, the peak positions are shifted to lower energies; the peak value contributed from 1–2 transition monotonically increases while the peak value originated from 1–3 transition decreases. In summary, we have investigated the optical intersubband absorption for 1–2, 2–3, and 1–3 ISBTs in ZnO/MgxZn1xO QWs. The quantum-confined Stark effect on the absorption process can be effectively controlled by applying an external electric field. The structural optimization of light absorption in different terahertz ranges can be easily obtained. As the field strength increases from 500 kV/cm to 500 kV/cm, the absorption wavelengths increase. The absorption coefficient of the 1–3 transition is comparatively small and can be increased by enhancing the structural asymmetry of QW potential. The present work can be extended to the structural design and simulation of new optoelectronic devices based on other low-dimensional structures such as quantum wires and quantum dots. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 60966001). One of the authors (J. Zhu) wishes to thank Dr. L.B. Cen for helpful discussions. References [1] S. Nakamura, Science 281 (1998) 956. [2] B. Monemar, G. Pozina, Prog. Quantum Electron. 24 (2000) 239. [3] Ü. Özgür, Ya.I. Alivov, C. Liu, A. Teke, M.A. Reshchikov, S. Dog˘an, V. Avrutin, S.-J. Cho, H. Morkoç, J. Appl. Phys. 98 (2005) 041301. [4] J.Q. Wu, J. Appl. Phys. 106 (2009) 011101. [5] N. Iizuka, K. Kaneko, N. Suzuki, Appl. Phys. Lett. 81 (2002) 1803. [6] M. Tchernycheva, L. Nevou, L. Doyennette, F.H. Julien, E. Warde, F. Guillot, E. Monroy, E. Bellet-Amalric, T. Remmele, M. Albrecht, Phys. Rev. B 73 (2006) 125347. [7] H.L. Zhou, W. Liu, S.J. Chua, Jpn. J. Appl. Phys. 46 (2007) 5128.
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[8] K. Berland, M. Stattin, R. Farivar, D.M.S. Sultan, P. Hyldgaard, A. Larsson, S.M. Wang, T.G. Andersson, Appl. Phys. Lett. 97 (2010) 043507. [9] V. Jovanovic´, D. Indjin, Z. Ikonic´, V. Milanovic´, J. Radovanovic, Solid State Commun. 121 (2002) 619. [10] Y.M. Chi, J.J. Shi, J. Lumin. 128 (2002) 1836. [11] J.M. Li, Y.W. Lü, D.B. Li, X.X. Han, Q.S. Zhu, X.L. Liu, Z.G. Wang, J. Vac. Sci. Technol. B 22 (2004) 2568. [12] N. Suzuki, N. Iizuka, K. Kaneko, Jpn. J. Appl. Phys. 42 (2003) 132. [13] S.H. Ha, S.L. Ban, J. Zhu, Physica B 406 (2011) 3640. [14] J. Zhu, S.L. Ban, S.H. Ha, Superlatt. Microstruct. 51 (2012) 471. [15] S.Y. Lei, Z.G. Dong, B. Shen, G.Y. Zhang, Phys. Lett. A 373 (2008) 136. [16] L.B. Cen, B. Shen, Z.X. Qin, G.Y. Zhang, J. Appl. Phys. 104 (2008) 063114. [17] M. Belmoubarik, K. Ohtani, H. Ohno, Appl. Phys. Lett. 92 (2008) 191906. [18] K. Ohtani, M. Belmoubarik, H. Ohno, J. Cryst. Growth 311 (2009) 2176. [19] C. Morhain, X. Tang, M. Teisseire-Doninelli, B. Lo, M. Laügt, J.-M. Chauveau, B. Vinter, O. Tottereau, P. Vennéguès, C. Deparis, G. Neu, Superlatt. Microstruct. 38 (2005) 455. [20] M. Yano, K. Hashimoto, K. Fujimoto, K. Koike, S. Sasa, M. Inoue, Y. Uetsuji, T. Ohnishi, K. Inaba, J. Cryst. Growth 301–302 (2007) 353. [21] M.W. Allen, P. Miller, R.J. Reeves, S.M. Durbin, Appl. Phys. Lett. 90 (2007) 062104. [22] J.A. Davis, L.V. Dao, X. Wen, C. Ticknor, P. Hannaford, V.A. Coleman, H.H. Tan, C. Jagadish, K. Koike, S. Sasa, M. Inoue, M. Yano, Nanotechnology 19 (2008) 055205. [23] D. Ahn, S.H. Park, J. Semicond. Technol. Sci. 6 (2006) 125. [24] X.G. Guo, Z.Y. Tan, J.C. Cao, H.C. Liu, Appl. Phys. Lett. 94 (2009) 201101. [25] D.N. Quang, V.N. Tuoc, N.H. Tung, N.V. Minh, P.N. Phong, Phys. Rev. B 72 (2005) 245303.
Contents lists available at SciVerse ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
A simulation of intersubband absorption in ZnO/MgxZn1xO quantum wells with an external electric field J. Zhu a, S.L. Ban a,⇑, S.H. Ha b a
School of Physical Science and Technology, Inner Mongolia University, Key Laboratory of Semiconductor Photovoltaic Technology at Universities of Inner Mongolia Autonomous Region, Hohhot 010021, PR China b Department of Physics, College of Sciences, Inner Mongolia University of Technology, Hohhot 010051, PR China
a r t i c l e
i n f o
Article history: Received 23 May 2012 Received in revised form 9 October 2012 Accepted 9 November 2012 Available online 21 November 2012 Keywords: Intersubband transition ZnO/MgxZn1xO quantum well Electric field
a b s t r a c t A detailed simulation on the intersubband absorption for 1–2, 2–3, and 1–3 optical transitions in ZnO/MgxZn1xO quantum wells is presented. The quantum-confined Stark effect induced by the internal polarization field on the absorption process is effectively controlled through an external electric field. It is easy to obtain the structural optimization of light absorption in different terahertz ranges in our numerical analysis. The absorption wavelengths corresponding to almost all of the transitions increase as the applied field varies from 500 kV/cm to 500 kV/cm. The small absorption coefficient corresponding to the 1–3 optical transition is increased by enhancing the structural asymmetry of quantum-well potential. The present work can be extended to the structural design and simulation of new optoelectronic devices based on other low-dimensional structures such as quantum wires and quantum dots. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Recently, much attention has been paid on intersubband transitions (ISBTs) in quantum wells (QWs) composed of wurtzite semiconductors due to their device applications such as blue/green/ ultraviolet light emitting diodes, laser diodes, photodetectors and all-optical switches [1–4]. Existing experiments [5–8] showed that the shape of absorption line exhibits either a pure Lorentzian shape or multiple peaks attributed to 1–2, 1–3 transitions and so on. The absorption coefficients associated with ISBTs in these low-dimensional structures are very large due to the fast relaxation process. Some ⇑ Corresponding author. Tel.: +86 471 4992237; fax: +86 471 4951761. E-mail address: [email protected] (S.L. Ban). 0749-6036/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.spmi.2012.11.011
J. Zhu et al. / Superlattices and Microstructures 56 (2013) 92–98
93
authors studied theoretically the linear or nonlinear intersubband absorption in III-nitride QWs [9,10]. The quasi-bound electron states were obtained by solving the Schrödinger equation using the analytical Airy functions without considering the many-electron effect. Within the framework of the Hartree–Fock approximation, an iterative procedure was developed to investigate the ISBTs in doped GaN/AlxGa1xN QWs [11] and multiple QWs (MQWs) [12]. The calculated absorption peaks show a quantum-confined Stark effect due to the built-in electric field (BEF) produced by the spontaneous polarization and the strain-induced piezoelectric polarization. However, less quantized electron states can exist in these single QWs with a very strong BEF (in order of MV/cm) which was pointed out in our previous works [13,14]. This is unfavorable for the device applications. Shen et al. [15,16] reported that a four-energy-level system was achieved in nitride double QWs with an applied electric field. However, the BEF was underestimated without considering the piezoelectric polarization in the well regions, and the applied field was still too large to be realized in actual conditions. Ohno et al. [17,18] experimentally studied the ISBTs in ZnO/MgxZn1xO MQWs and found that there are certain unique properties for intersubband absorption in ZnO-based QWs. As known, ZnO is a semiconductor with a wide band-gap of 3.37 eV, making it an ideal candidate for optoelectronic device applications in the blue to ultraviolet regions [19–22]. The transition wavelength in ZnO-based QWs can be tailored in a wide spectral range for potential applications in optical communication by structural optimization due to the large conduction band offset at heterointerfaces. Moreover, the intersubband relaxation time is expected to be very fast because of large effective masses of electrons and the ionicity of ZnO [23]. A comparatively large spontaneous polarization exists in ZnO material due to the ionic nature of Zn–O bond and low symmetry crystal structure [20–22]. This, together with the straininduced piezoelectric polarization, produces a BEF in ZnO/MgxZn1xO QWs, whereas the field strength is estimated to be smaller than that of nitride QWs [20–22] and thus can be restrained by applying an external electric field. As far as we known, there were few of detailed investigations concerning about the ISBTs in ZnO/MgZnO QWs. In this work, we adopt a numerical method proposed by Li et al. [11] to solve self-consistently the discrete electron states in ZnO/MgxZn1xO QWs under an external electric field and then provide a theoretical analysis on the intersubband absorption in this system. 2. Model Let us consider a ZnO/MgxZn1xO QW with barrier and well width denoted as db and dw. Within the framework of the effective mass approximation, the Schrödinger equation in the z direction for a confined electron with charge e is given by
(
) 2 @ h 1 @ þ VðzÞ þ e½F ex þ F in ðzÞz þ V H ðzÞ þ V xc ðzÞ wi ðzÞ ¼ Ei wi ðzÞ: 2 @z mðzÞ @z
ð1Þ
In above equation, m(z) and V(z) are the electronic effective mass and the height of potential barriers, respectively. Fex is the applied external electric field and Fin(z) is the BEF given by
8 > < F b ; dw =2 < jzj < dw =2 þ db ; FðzÞ ¼ F w ; jzj 6 dw =2; > :
ð2Þ
where
Fw ¼
2db P ; 2db ew þ dw eb
ð3Þ
and
Fb ¼
dw Fw: 2db
ð4Þ
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J. Zhu et al. / Superlattices and Microstructures 56 (2013) 92–98
In above equations, ew and eb are the relative dielectric constant. P is the polarization including spontaneous and piezoelectric polarization. In ZnO/MgxZn1xO QWs, the spontaneous and piezoelectric polarizations are in opposite directions, PPE = 0.034x cm2 and PSP = 0.066x cm2, thus P = PPE + PSP [22]. The Hartree electrostatic potential VH(z) in Eq. (1) can be determined by the Poisson equation
d dz
e0 ðzÞ
d V H ðzÞ ¼ e2 ½NþD ðzÞ NðzÞ; dz
ð5Þ
with the density of ionized donors written as
NþD ðzÞ ¼
ND 1 þ 2 exp
h
Ef ED kB T
i;
ð6Þ
and the density of free electrons
NðzÞ ¼
X mðzÞkB T
ph2
i
Ef Ei ; jwi ðzÞj2 ln 1 þ exp kB T
ð7Þ
where ND is the donor density and ED the donor ionization energy. kB is the Boltzmann constant and T is the temperature. Ef is the Fermi energy which can be derived from the condition of charge neutrality
Z
þ1
1
½N þD ðzÞ NðzÞdz ¼ 0:
ð8Þ
Using the local density approximation, the exchange–correlation potential Vxc(z) in Eq. (1) can be formulated in an analytic form [24]
1=3 e2 9 11:4 p 1 þ 0:0545r ðzÞ ln 1 þ ; s r s ðzÞ 4p2 e0 ðzÞaB ðzÞrs ðzÞ 4
V xc ðzÞ ¼
ð9Þ
where the effective Bohr radius is defined as aB(z) = e0(z) h2/e2m(z) and the radius of a sphere contain1 1=3 3 ing one electron as r s ðzÞ ¼ fð3=4pÞ½aB ðzÞNðzÞ g . The eigen energies and wave functions of the electron can be obtained by self-consistently solving the Schrödinger Eq. (1) and the Poisson Eq. (5) via an iterative procedure. The optical ISBT absorption coefficients are calculated using the following formula:
x
aðhxÞ ¼
rffiffiffiffiffiffiffiffiffiffiX
dw
l0 ðNn N m Þ h= s jM j2 ; e0 ew m>n mn ðEm En hxÞ2 þ ðh=sÞ2
ð10Þ
where l0 and e0 are the permeability and dielectric constant of vacuum, Mmn is the dipole matrix element, Ni(z) (i = m, n) is the electron density in the ith subband level, and s is the relaxation time. For an intersubband optical transition with a polarization along the z direction, the dipole matrix element Mmn in Eq. (10) is given by
M mn ¼
Z
þ1
1
wm ðzÞezwn ðzÞdz:
ð11Þ
3. Numerical results and discussion The material parameters used in the calculation are cited from Ref. [20]. The donor density and the temperature are fixed at 1 1019 cm3 and 300 K, respectively. The dephasing time s is taken as 20 fs [23]. It is worthwhile to mention that s is a complex parameter affected by many scattering mechanisms including the scattering from different kinds of phonons, alloy disorder, donors and so on [25]. The numerical results are shown in Figs. 1–5. Fig. 1 gives the conduction band structures, energy levels and electron wavefunctions for a typical ZnO/MgxZn1xO QW in the cases of different external electric fields. The BEFs in the well and barriers are calculated as 0.82 MV/cm and 0.41 MV/cm with the positive direction assumed to be along the
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95
Fig. 1. Calculated conduction band structures, energy levels and electron wavefunctions for a 3aB wide ZnO/Mg0.3Zn0.7O QW with different external electric fields. Here, the effective Bohr radius aB is taken as 1.65 nm.
Fig. 2. Dipole matrix elements as functions of external electric field F for 1–2, 1–3 and 2–3 ISBTs in ZnO/Mg0.3Zn0.7O QWs with well width dw = 3aB.
growth direction. As shown in Fig. 1a, the potential of conduction band is tilted by the BEF and bent by the Hartree–Fock potential, and thus the electron wavefunctions slightly move close to the left interface. We also show that the influence from the exchange–correlation potential is more obvious than that from the Hartree term in the Poisson equation if more electron states are involved in the computation. It can be seen from Fig. 1b that the BEFs in the barriers are suppressed but the BEF in the well increases with applying a negative electric field. Consequently, the energy levels collectively increase while the wavefunctions get more close to the left interface and easily penetrate into the left barrier. Oppositely, Fig. 1c indicates that the electron states can be controlled like the case of a square QW with a positive external electric field. The dipole matrix elements Mmn as functions of external electric field F for 1–2, 1–3 and 2–3 ISBTs in ZnO/MgxZn1xO QWs are plotted in Fig. 2. As F increases from 500 kV/cm to 500 kV/cm, M12 and M23 always increase but M13 decreases. That is to say, the strengths of 1–2 and 2–3 optical transitions increase. The 1–3 transition becomes more difficult as the BEF in the well is gradually restrained by increasing F although the BEF allows this so-called forbidden optical transition. The absorption wavelength of each transition as a function of F is shown in Fig. 3. It is found that the absorption wavelengths corresponding to almost all of the ISBTs increase with F. However, the amplitude variation discriminates from each transition. The absorption wavelength of 1–2 transition rapidly increases
96
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Fig. 3. Absorption wavelengths as functions of external electric field F for 1–2, 1–3 and 2–3 ISBTs in ZnO/Mg0.3Zn0.7O QWs with well width dw = 3aB.
Fig. 4. Absorption wavelengths as functions of well width dw and Mg composition x for 1–2 ISBTs in ZnO/MgxZn1xO QWs with F = 500 kV/cm. Each kind of color, i.e., the grad scale, represents a special wavelength range, e.g., red indicates 0–1 lm and blue indicates 15–16 lm.
with F but the absorption wavelength of 2–3 transition is insensitive to F since the lower excited states are more likely influenced by the quantum-confined Stark effect. In order to obtain the structural optimization of light absorption in different terahertz ranges as required as one can, Fig. 4 plots the absorption wavelength versus well width dw and Mg composition x for 1–2 ISBTs with F = 500 kV/cm. As seen, the absorption wavelength decreases with increasing x; but it increases if dw becomes larger due to the enhancement of the quantum confinement on electrons from higher barriers and wider well. The favorable wavelength region can be achieved by choosing suitable x and dw. For example, the near-infrared light absorption occurs if dw < 2 nm and x > 0.7, and the light absorption in mid-infrared range can be achieved when x > 0.15. If taking advantage of the absorption wavelength originated from 1–3 transition, it only needs to apply a negative electric
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Fig. 5. Absorption coefficients as functions of external electric field F and incident photon energy ⁄x for 1–2, 1–3 and 2–3 ISBTs in ZnO/Mg0.3Zn0.7O QWs with well width dw = 3aB.
field to strengthen the BEF in the well or alter the Mg composition in one of the barriers to enhance the structural asymmetry. Fig. 5 shows the total absorption coefficient as functions of F and incident photon energy for 1–2, 1–3 and 2–3 ISBTs. It can be seen that the peak value contributed from 1–2 transition is largest, whereas the peak value of 1–3 transition is very imperceptible since its dipole matrix element is still very small although the symmetry of the wave functions is broken by the BEF. In fact, the optical absorption coefficient mainly depends on the optical dipole matrix element and the position of energy levels relative to the Fermi energy. The former reflects the selection rule of different transitions, whereas the latter determines the amount of carriers locating at each energy level. With increasing F, the peak positions are shifted to lower energies; the peak value contributed from 1–2 transition monotonically increases while the peak value originated from 1–3 transition decreases. In summary, we have investigated the optical intersubband absorption for 1–2, 2–3, and 1–3 ISBTs in ZnO/MgxZn1xO QWs. The quantum-confined Stark effect on the absorption process can be effectively controlled by applying an external electric field. The structural optimization of light absorption in different terahertz ranges can be easily obtained. As the field strength increases from 500 kV/cm to 500 kV/cm, the absorption wavelengths increase. The absorption coefficient of the 1–3 transition is comparatively small and can be increased by enhancing the structural asymmetry of QW potential. The present work can be extended to the structural design and simulation of new optoelectronic devices based on other low-dimensional structures such as quantum wires and quantum dots. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 60966001). One of the authors (J. Zhu) wishes to thank Dr. L.B. Cen for helpful discussions. References [1] S. Nakamura, Science 281 (1998) 956. [2] B. Monemar, G. Pozina, Prog. Quantum Electron. 24 (2000) 239. [3] Ü. Özgür, Ya.I. Alivov, C. Liu, A. Teke, M.A. Reshchikov, S. Dog˘an, V. Avrutin, S.-J. Cho, H. Morkoç, J. Appl. Phys. 98 (2005) 041301. [4] J.Q. Wu, J. Appl. Phys. 106 (2009) 011101. [5] N. Iizuka, K. Kaneko, N. Suzuki, Appl. Phys. Lett. 81 (2002) 1803. [6] M. Tchernycheva, L. Nevou, L. Doyennette, F.H. Julien, E. Warde, F. Guillot, E. Monroy, E. Bellet-Amalric, T. Remmele, M. Albrecht, Phys. Rev. B 73 (2006) 125347. [7] H.L. Zhou, W. Liu, S.J. Chua, Jpn. J. Appl. Phys. 46 (2007) 5128.
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