- Email: [email protected]
AlGaN asymmetric coupled quantum wells
ARTICLE IN PRESS Physica B 405 (2010) 3272–3275
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Effect of hydrostatic pressure on the donor impurity states in GaN/AlGaN asymmetric coupled quantum wells Congxin Xia, Zaiping Zeng n, Shuyi Wei Department of Physics, Henan Normal University, Xinxiang 453007, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 22 March 2010 Received in revised form 14 April 2010 Accepted 26 April 2010
Based on the effective-mass approximation, the ground-state donor binding energy in zinc-blende(ZB) GaN/AlGaN asymmetric coupled quantum wells(QWs) is investigated variationally, considering the hydrostatic pressure effect. Numerical results show that the donor binding energy is highly dependent on the impurity positions, the asymmetric coupled QWs structure parameters and the hydrostatic pressure. It is found that the donor binding energy increases with increase in the hydrostatic pressure for any impurity position. The hydrostatic pressure has an obvious influence on the donor binding energy of impurity localized inside the wide well of the asymmetric coupled QWs. For any hydrostatic pressure, our results show that the donor binding energy is distributed asymmetrically with respect to the center of the asymmetric coupled QWs. In particular, for the impurity located inside the wide well, the donor binding energy is insensitive to the middle barrier width in ZB GaN/AlGaN asymmetric coupled QWs if the middle barrier width is large. & 2010 Elsevier B.V. All rights reserved.
Keywords: Quantum wells Hydrogenic impurity
1. Introduction Wide-band-gap GaN and related nanostructures have attracted much attention due to their unique electronic and optical properties as well as potential applications in electronics and optoelectronic devices [1–3]. In the past years, the electronic states and optical properties in wurtzite (WZ) GaN-based semiconducting heterostructures have been studied extensively [4–6]. These studies show that the electronic states and optical properties in WZ GaN-based semiconducting heterostructures are highly affected by the built-in electric fields due to spontaneous and piezoelectric polarizations. However, the spontaneous polarization does not exist in the zinc-blende(ZB) GaN due to higher crystal symmetry, and piezoelectric polarization in ZB GaN-based semiconducting heterostructures can be negligible due to the (0 0 0 1) growth direction of the epitaxial layers [7–8]. Therefore, the strong built-in electric field is absent in ZB GaN-based semiconducting heterostructures. In addition, it is also found that ZB GaN has the ability to produce cleaved laser cavities and to be doped easily [9–10]. Thus, interest in ZB GaN-based quantum structure has been grown recently [11–13]. It is well known that impurity states play an important role in the semiconducting optoelectronic devices. Without impurities, there would be no diode, no transistor or no semiconducting science and technology. For this reason, in the past many years,
n
Corresponding author. E-mail addresses: [email protected] (C. Xia), [email protected] (Z. Zeng). 0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.04.058
much theoretical work is involved in investigating the impurity states in different QWs by using external perturbations, such as hydrostatic pressure [14–20]. These results show that the impurity states in the QWs are highly dependent on the materials and impurity positions. Besides, the hydrostatic pressure also has a remarkable influence on the semiconductor band structure and leads to change in the properties of the elementary excitations of the QWs. However, to the best of our knowledge, there are few papers involved in investigating the impurity states in ZB GaNbased QWs. Fan et al. [21] have investigated the electronic structures of ZB GaN/AlGaN compressively strained superlattices and QWs. Optical gain in ZB GaN/GaAlN strained QW laser has been studied theoretically [22]. The electronic structure of cubic GaN/AlGaN QWs has also been investigated using a tight binding approximation [23]. However, as we known, coupled QWs, especially asymmetric coupled QWs, rather than single quantum well(QW) are adopted in the commonly used GaN-based optoelectronic devices. Thus, it is necessary to study the impurity states in ZB GaN coupled QWs. In this paper, we will investigate the donor binding energy of a hydrogenic impurity in ZB GaN/ AlGaN asymmetric coupled QWs by means of a variational procedure, considering the hydrostatic pressure effect.
2. Theoretical framework In the present paper, let us now consider an asymmetric AlxGa1 xN/GaN/AlxGa1 xN/GaN/AlxGa1 xN coupled QWs with the corresponding layer thickness N/Llw/Lmb/Lrw/N (see Fig. 1).
ARTICLE IN PRESS C. Xia et al. / Physica B 405 (2010) 3272–3275
3273
Fig. 1. Schematic representation of ZB GaN/AlxGa1 xN asymmetric coupled QWs with the left well width Llw, the middle barrier width Lmb and the right well width Lrw.
Within the framework of effective-mass approximation, the Hamiltonian for a hydrogenic impurity in ZB GaN/AlGaN asymmetric coupled QWs, under the influence of hydrostatic pressure, may be written as ^ ¼H ^ 0 H
e2 4pkeðPÞr
ð1Þ
with p2 þVðzÞ ð2Þ 2m ðPÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ x2 þy2 þ ðzzi Þ2 is the distance between the electron and the impurity site, x(0), y(0) and z(zi) are the coordinates of the electron (impurity) in ZB GaN/AlGaN asymmetric coupled QWs. It needs to point out that the reference of the asymmetric coupled QWs system is taken at the center of the middle barrier layer (see Fig. 1). e is the absolute value of the electron charge. V(z) is the z-direction confinement potential due to the conduction band offset in ZB GaN/AlGaN asymmetric coupled QWs. k is the permittivity of free space. eðPÞ is the pressure-dependent effective mean relative dielectric constant of ZB GaN material. In order to consider the hydrostatic pressure effect on the effective mean relative dielectric constant, the band gap, the electron effective mass and the geometric dimensions of ZB GaN/AlGaN asymmetric coupled QWs, the theoretical and the computational methods used in the present paper are the same as in Ref. [24]. In order to calculate the ground-state donor binding energy of the hydrogenic impurity in ZB GaN/AlGaN QWs, the trial wave function may be written as [14–17,19]
^0¼ H
F ¼ hðzÞexpðlrÞ
ð13Þ
where h(z) is the eigenfunction of the Hamiltonian described in Eq. (2), which can be solved using the transfer matrix method [25]. l is the variational parameter. We would like to point out that the trial wave function used in the present paper is a reasonable choice for studying impurity states in ZB GaN/AlGaN QWs [14–17,19]. The ground-state energy of a hydrogenic impurity in ZB GaN/ AlGaN QW may be obtained by minimizing E ¼ min l
^ FS /F9H9 /F9FS
ð14Þ
The ground-state donor binding energy Eb can be represented as follows: Eb ¼ E0 E
ð15Þ
where E0 is the ground-state energy for the Hamiltonian of Eq. (2).
3. Numerical results and discussion We have calculated the ground-state donor binding energy Eb as functions of the impurity positions zi, the right well width Lrw,
Fig. 2. The ground-state donor binding energy Eb as a function of the impurity positions zi for different hydrostatic pressures in ZB GaN/Al0.15Ga0.85N asymmetric coupled QWs with the middle barrier width Lmb ¼ 1 nm. (a) left well width Llw ¼ 4 nm, right well width Lrw ¼ 2 nm; and (b) left well width Llw ¼ 2 nm, right well width Lrw ¼ 4 nm.
the middle barrier width Lmb and the hydrostatic pressure P in ZB GaN/AlGaN asymmetric coupled QWs. All material parameters used in the present paper are the same as in Ref. [24]. In Fig. 2, the ground-state donor binding energy is investigated as a function of the impurity positions in ZB GaN/Al0.15Ga0.85N asymmetric coupled QWs for different hydrostatic pressures. Numerical results show that the donor binding energy is distributed asymmetrically with respect to the center of the asymmetric coupled QWs in all cases. This result is different from that in symmetric coupled QWs, in which the donor binding energy is distributed symmetrically around the center of the coupled quantum QWs [26]. Moreover, the position of the maximum value is located inside the wide well. This is because the electron wave function is mainly localized inside the wide well in the asymmetric coupled QWs. It is also found from Fig. 2 that the donor binding energy is increased when the hydrostatic pressure is considered for any impurity position. Moreover, the pressure effect has an obvious influence on the donor binding energy of the impurity located inside the wide well. The main reasons can be given as follows: as the hydrostatic pressure is considered, the dielectric constant and well size are decreased; in addition, the electron effective mass is increased, then the electron wave function is more localized inside the asymmetric coupled QWs. Thus, the electron–impurity distance is reduced and the donor binding energy is increased when the hydrostatic pressure effect is considered. In addition, Fig. 2 (a) and (b) show that when either well is wide in the asymmetric coupled QWs, the behaviors of the donor binding energy are equivalent. Thus, in the following part, we only discuss the effect of the variation of
ARTICLE IN PRESS 3274
C. Xia et al. / Physica B 405 (2010) 3272–3275
Fig. 3. The ground-state donor binding energy Eb as a function of the right well width Lrw for different impurity positions zi and hydrostatic pressures P in the ZB GaN/Al0.15Ga0.85N asymmetric coupled QWs with well width Llw ¼ 2 nm and middle barrier width Lmb ¼ 1 nm. Curves A, B and C are for the impurity located at zi ¼ (Lmb + Llw/2), 0 and Lmb + Lrw/2, respectively.
Fig. 4. The ground-state donor binding energy Eb as a function of the middle barrier width Lmb for different impurity positions zi and hydrostatic pressures P in ZB GaN/Al0.15Ga0.85N asymmetric coupled QWs with the left well width Llw ¼ 2 nm and the right well width Lrw ¼ 4 nm. The curves A, B and C are for the same meaning as in Fig. 3.
the right well width on the donor binding energy in ZB GaN/ Al0.15Ga0.85N asymmetric coupled QWs. In Fig. 3, the ground-state donor binding energy is investigated as a function of the right well width for different impurity positions in ZB GaN/Al0.15Ga0.85N asymmetric coupled QWs. Numerical results show that the donor binding energy is increased for any impurity position when the hydrostatic pressure effect is considered, as expected. In addition, it can also be seen from Fig. 2 that if the right well width Lrw r2 nm, the hydrostatic pressure is more pronounced for the impurity located at the center of the left well (see curve A). However, when the right well width Lrw Z2 nm, the hydrostatic pressure is more pronounced for the impurity located at the center of the right well (see curve C). These behaviors indicate that the hydrostatic pressure increased the donor binding energy for any well width and impurity position. In particular, one can conclude that the hydrostatic pressure has an obvious influence on the binding energy in the wide well of the asymmetric coupled QWs. The reasons are that the electron wave function is mainly localized inside the wide well. These results are in agreement with that of Fig. 2. In the following, we can further understand the characteristic behaviors of curves A, B and C of Fig. 3. It can be found from curve A of Fig. 3 that for impurity located at the center of the left well, the donor binding energy is decreased with increase in the right well width for any hydrostatic pressures. This is because the electron wave function located inside the left well is reduced with increase in the right well width in ZB GaN/Al0.15Ga0.85N asymmetric coupled QWs. However, for curve B (impurity located at the center of the middle barrier layer), the donor binding energy has a maximum value with increase in the right well width in the case of any pressure. This behavior can be explained as follows: when the right well width is increased from zero, the electron wave function begins to penetrate into the middle barrier layer and the donor binding energy increases. And if the right well width continuously increases, the electron wave function gradually penetrates into the right well. Thus, the donor binding energy along curve B is decreased by continuously increasing the right well width. Moreover, we can also see from curve C of Fig. 3 that for the impurity located at the center of the right well, the donor binding energy also has a maximum value in the case of any hydrostatic pressure. This can be understood as follows: for small right well width, the electron wave function localized inside the right well is increased with increase in the right well width of the asymmetric
coupled QWs. Therefore, the electron–impurity distance is decreased for the impurity located at the center of the right well. Whereas, for large right well width, the electron wave function is mainly localized inside the right well. The electron–impurity distance is increased with the increased right well width. Thus, both the Coulomb interaction between the electron and the impurity and the donor binding energy are reduced with an increment in the right well width. In addition, Fig. 3 shows that the donor binding energy along curves A and C reaches the same value when the right well width Lrw ¼ 2 nm (see dotted line). The physical reasons can be given as follows: when the right well width Lrw ¼Llw ¼2 nm, symmetric coupled QWs with well width Lw ¼2 nm is produced. Thus, the donor binding energy of impurities located at the symmetric positions with respect to the center of the symmetric coupled QWs has the same value, which is in agreement with the results in the symmetric coupled QWs [26]. In Fig. 4, the middle barrier width effect on the ground-state donor binding energy in ZB GaN/Al0.15Ga0.85N asymmetric coupled QWs is investigated for different impurity positions and hydrostatic pressures. Numerical results show that the donor binding energy is the largest when the impurity is located at the center of the right well in the case of any pressure. This is because the electron wave function is localized around the center of the right well. It can also be seen from Fig. 4 that the donor binding energy is increased when the hydrostatic pressure is considered for any impurity position, as expected. In particular, Fig. 4 also shows that the hydrostatic pressure has an obvious influence on the donor binding energy of impurity located inside the wide well, which agrees with the results of Fig. 2. In the following, we can further understand the characteristic behaviors of curves A, B and C of Fig. 4. For curve C, it can be found that for large middle barrier width (Lmb Z2 nm), the donor binding energy is insensitive to the middle barrier width in all cases. The physical reasons can be illustrated as follows: when the middle barrier width Lmb Z2 nm, the coupled QWs are decoupled. Thus, the electron–impurity distance for the impurity located at the center of the right well is insensitive to the variation of the middle barrier width. However, for small middle barrier width (Lmb o2 nm), the donor binding energy increases with increase in the middle barrier width for any pressure. The reason can be understood as follows: when the middle barrier width appears, the asymmetric coupled QWs system is produced. The electron wave function is more strongly localized inside the wide well when the middle barrier width is
ARTICLE IN PRESS C. Xia et al. / Physica B 405 (2010) 3272–3275
Fig. 5. The ground-state donor binding energy Eb as a function of the hydrostatic pressure P for different impurity positions in ZB GaN/Al0.15Ga0.85N asymmetric coupled QWs with the left well width Llw ¼ 2 nm, middle barrier width Lmb ¼ 1 nm and the right well width Lrw ¼ 4 nm. Curves A, B and C are for the same meaning as in Fig. 3.
increased form zero in the asymmetric coupled QWs. For curves A and B, the donor binding energy is decreased with increase in the middle barrier width for any hydrostatic pressure. The physical reasons can be explained as follows: the electron wave function is mainly localized inside the wide well of the asymmetric coupled QWs. Thus, for the impurity located inside the narrow well and the middle barrier layer, the electron–impurity distance is increased and the donor binding energy is decreased when the middle barrier width is increased. Specially, it needs to point out that when the left well width Llw ¼2 nm and the right well width Lrw ¼4 nm, and single QW with the well width Lw ¼6 nm are produced in the case of the middle barrier width Lmb ¼0. The impurity positions of curves B and C are symmetric with respect to the center of the single QW. Thus, the impurity binding energy along curves B and C has the same value for any hydrostatic pressure when Lmb ¼0. And it is also found from Fig. 4 that the difference in the donor binding energy along curves B and C is increased with increase in the middle barrier width. The reason is that the electron wave function is mainly localized inside the wide well of the asymmetric coupled QWs. The donor binding energy of impurity localized inside the wide (narrow) well is increased (decreased) when the middle barrier wide is increased. In Fig. 5, the ground-state donor binding energy is investigated as a function of the hydrostatic pressure for different impurity positions in ZB GaN/Al0.15Ga0.85N asymmetric coupled QWs. Numerical result shows that the hydrostatic pressure increases the donor binding energy for any impurity position, which is in agreement with that in a single QW and symmetric coupled QWs. This is because the electron wave function is more confined inside the asymmetric coupled QWs when the hydrostatic pressure is increased. In particular, it is found from curve C of Fig. 5 that the donor binding energy is more sensitive to the hydrostatic pressure for impurity located at the center of the right well. This is because the electron wave function is mainly localized around the center of the right well.
4. Conclusions In conclusion, we have investigated variationally the groundstate donor binding energy as functions of the impurity positions, the right well width and the middle barrier width in ZB GaN/ AlGaN asymmetric coupled QWs, considering the hydrostatic pressure effect. Numerical results show that the donor binding
3275
energy is highly dependent on the impurity positions, asymmetric coupled QWs structure parameters and the hydrostatic pressure. It is found that the hydrostatic pressure increases the donor binding energy for any impurity position. The hydrostatic pressure has an obvious influence on the impurity localized inside the wide well of the asymmetric coupled QWs. In the case of any hydrostatic pressure, our results show that the donor binding energy is distributed asymmetrically with respect to the center of the asymmetric coupled QWs. When the right well width is increased, the donor binding energy of impurity localized inside the narrow well is decreased; the donor binding energy of impurity localized inside the middle barrier layer has a maximum. In particular, for the impurity localized inside the wide well, the donor binding energy is insensitive to the middle barrier width in ZB GaN/AlGaN asymmetric coupled QWs if the middle barrier width is large. Moreover, to our knowledge, there are few papers on the impurity states in other such semiconducting structures. Thus, it is also interesting to point out that our calculation can be applied to other asymmetric coupled QWs structure just with different numbers of material parameters. Experimental studies concerned the effect of hydrostatic pressure on hydrogenic impurity states in ZB GaN/AlGaN asymmetric coupled QWs, which are still lacking at present. We hope that our calculation can stimulate further investigations of the related physics, as well as device applications of group-III nitrides.
Acknowledgment This work was supported by the National Natural Science Foundation of China under Grant no. 60906044.
References [1] B. Gil, in: Group III Nitride Semiconductor Compounds, Clarendon Press, Oxford, 1998. [2] S. Nakamura, S.F. Chichibu, in: Introduction to Nitride Semiconductor Blue Lasers and Light Emitting Diodes, Taylor and Francis, London, 2000. [3] S.H. Han, D.Y. Lee, S.J. Lee, C.Y. Cho, M.K. Kwon, S.P. Lee, D.Y. Noh, D.J. Kim, Y.C. Kim, S.J. Park, Appl. Phys. Lett. 94 (2009) 231123. [4] C.X. Xia, S.Y. Wei, Phys. Lett. A 359 (2006) 161. [5] J.J. Shi, Z.Z. Gan, J. Appl. Phys. 94 (2003) 407. [6] J.J. Shi, C.X. Xia, S.Y. Wei, Z.X. Liu, J. Appl. Phys. 97 (2005) 083705. [7] O. Husberg, A. Khartchenko, D.J. As, H. Vogelsang, T. Frey, D. Schikora, K. Lischka, O.C. Noriega, A. Tabata, J.R. Leite, Appl. Phys. Lett. 79 (2001) 1243. [8] S.F. Chichibu, T. Onuma, T. Sota, J. Appl. Phys. 93 (2003) 2051. [9] S. Strite, H. Morkoc, J. Vac. Sci. Technol. B 10 (1992) 1237. [10] E. Martinez-Guerrero, C. Adelmann, F. Chabuel, J. Simon, N.T. Pelekanos, G. Mula, B. Daudin, G. Feuillet, H. Mariette, Appl. Phys. Lett. 77 (2000) 809. [11] B.M. Shi, M.H. Xie, H.S. Wu, N. Wang, S.Y. Tong, Appl. Phys. Lett. 89 (2006) 151921. [12] Y. Fu, Y.Q. Wei, X.D. Wang, M. Sadeghi, S.M. Wang, A. Larsson, J. Appl. Phys. 100 (2006) 073105. ¨ [13] C. Brimont, M. Gallart, O. Cre´gut, B. Honerlage, P. Gilliot, D. Lagarde, A. Balocchi, T. Amand, X. Marie, S. Founta, H. Mariette, J. Appl. Phys. 106 (2009) 053514. [14] G. Bastard, Phys. Rev. B 24 (1981) 4714. [15] B.D. Byungsu Yoo, McCombe, W. Schaff, Phys. Rev. B 44 (1991) 13152. [16] H. Chen, X.D. Li, S.X. Zhou, Phys. Rev. B 44 (1991) 6220. [17] B. Sunder, Phys. Rev. B 45 (1992) 8562. [18] A.L. Morales, A. Montes, S.Y. Lo´pez, C.A. Duque, J. Phys.: Condens. Matter 14 (2002) 987. [19] N. Raigoza, A.L. Morales, A. Montes, N. Porras-Montenegro, C.A. Duque, Phys. Rev. B 69 (2004) 045323. [20] E. Kasapoglu, Phys. Lett. A 373 (2008) 140. [21] W.J. Fan, M.F. Li, T.C. Chong, J.B. Xia, Superlattices Microstruct. 19 (1996) 251. [22] W.J. Fan, M.F. Li, T.C. Chong, J.B. Xia, Solid State Commun. 98 (1996) 737. [23] J. Arriaga, H. Hernandez-Cocoletzi, D.A. Contreras-Solorioa, Physica E 17 (2003) 238. [24] C.X. Xia, Z.P. Zeng, S.Y. Wei, J. Appl. Phys. 107 (2010) 014305. [25] S. Vatannia, G. Gildenblat, IEEE J. Quantum Electron. 32 (1996) 1093. [26] E. Kasapoglu, Phys. Lett. A 373 (2008) 140.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Effect of hydrostatic pressure on the donor impurity states in GaN/AlGaN asymmetric coupled quantum wells Congxin Xia, Zaiping Zeng n, Shuyi Wei Department of Physics, Henan Normal University, Xinxiang 453007, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 22 March 2010 Received in revised form 14 April 2010 Accepted 26 April 2010
Based on the effective-mass approximation, the ground-state donor binding energy in zinc-blende(ZB) GaN/AlGaN asymmetric coupled quantum wells(QWs) is investigated variationally, considering the hydrostatic pressure effect. Numerical results show that the donor binding energy is highly dependent on the impurity positions, the asymmetric coupled QWs structure parameters and the hydrostatic pressure. It is found that the donor binding energy increases with increase in the hydrostatic pressure for any impurity position. The hydrostatic pressure has an obvious influence on the donor binding energy of impurity localized inside the wide well of the asymmetric coupled QWs. For any hydrostatic pressure, our results show that the donor binding energy is distributed asymmetrically with respect to the center of the asymmetric coupled QWs. In particular, for the impurity located inside the wide well, the donor binding energy is insensitive to the middle barrier width in ZB GaN/AlGaN asymmetric coupled QWs if the middle barrier width is large. & 2010 Elsevier B.V. All rights reserved.
Keywords: Quantum wells Hydrogenic impurity
1. Introduction Wide-band-gap GaN and related nanostructures have attracted much attention due to their unique electronic and optical properties as well as potential applications in electronics and optoelectronic devices [1–3]. In the past years, the electronic states and optical properties in wurtzite (WZ) GaN-based semiconducting heterostructures have been studied extensively [4–6]. These studies show that the electronic states and optical properties in WZ GaN-based semiconducting heterostructures are highly affected by the built-in electric fields due to spontaneous and piezoelectric polarizations. However, the spontaneous polarization does not exist in the zinc-blende(ZB) GaN due to higher crystal symmetry, and piezoelectric polarization in ZB GaN-based semiconducting heterostructures can be negligible due to the (0 0 0 1) growth direction of the epitaxial layers [7–8]. Therefore, the strong built-in electric field is absent in ZB GaN-based semiconducting heterostructures. In addition, it is also found that ZB GaN has the ability to produce cleaved laser cavities and to be doped easily [9–10]. Thus, interest in ZB GaN-based quantum structure has been grown recently [11–13]. It is well known that impurity states play an important role in the semiconducting optoelectronic devices. Without impurities, there would be no diode, no transistor or no semiconducting science and technology. For this reason, in the past many years,
n
Corresponding author. E-mail addresses: [email protected] (C. Xia), [email protected] (Z. Zeng). 0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.04.058
much theoretical work is involved in investigating the impurity states in different QWs by using external perturbations, such as hydrostatic pressure [14–20]. These results show that the impurity states in the QWs are highly dependent on the materials and impurity positions. Besides, the hydrostatic pressure also has a remarkable influence on the semiconductor band structure and leads to change in the properties of the elementary excitations of the QWs. However, to the best of our knowledge, there are few papers involved in investigating the impurity states in ZB GaNbased QWs. Fan et al. [21] have investigated the electronic structures of ZB GaN/AlGaN compressively strained superlattices and QWs. Optical gain in ZB GaN/GaAlN strained QW laser has been studied theoretically [22]. The electronic structure of cubic GaN/AlGaN QWs has also been investigated using a tight binding approximation [23]. However, as we known, coupled QWs, especially asymmetric coupled QWs, rather than single quantum well(QW) are adopted in the commonly used GaN-based optoelectronic devices. Thus, it is necessary to study the impurity states in ZB GaN coupled QWs. In this paper, we will investigate the donor binding energy of a hydrogenic impurity in ZB GaN/ AlGaN asymmetric coupled QWs by means of a variational procedure, considering the hydrostatic pressure effect.
2. Theoretical framework In the present paper, let us now consider an asymmetric AlxGa1 xN/GaN/AlxGa1 xN/GaN/AlxGa1 xN coupled QWs with the corresponding layer thickness N/Llw/Lmb/Lrw/N (see Fig. 1).
ARTICLE IN PRESS C. Xia et al. / Physica B 405 (2010) 3272–3275
3273
Fig. 1. Schematic representation of ZB GaN/AlxGa1 xN asymmetric coupled QWs with the left well width Llw, the middle barrier width Lmb and the right well width Lrw.
Within the framework of effective-mass approximation, the Hamiltonian for a hydrogenic impurity in ZB GaN/AlGaN asymmetric coupled QWs, under the influence of hydrostatic pressure, may be written as ^ ¼H ^ 0 H
e2 4pkeðPÞr
ð1Þ
with p2 þVðzÞ ð2Þ 2m ðPÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ x2 þy2 þ ðzzi Þ2 is the distance between the electron and the impurity site, x(0), y(0) and z(zi) are the coordinates of the electron (impurity) in ZB GaN/AlGaN asymmetric coupled QWs. It needs to point out that the reference of the asymmetric coupled QWs system is taken at the center of the middle barrier layer (see Fig. 1). e is the absolute value of the electron charge. V(z) is the z-direction confinement potential due to the conduction band offset in ZB GaN/AlGaN asymmetric coupled QWs. k is the permittivity of free space. eðPÞ is the pressure-dependent effective mean relative dielectric constant of ZB GaN material. In order to consider the hydrostatic pressure effect on the effective mean relative dielectric constant, the band gap, the electron effective mass and the geometric dimensions of ZB GaN/AlGaN asymmetric coupled QWs, the theoretical and the computational methods used in the present paper are the same as in Ref. [24]. In order to calculate the ground-state donor binding energy of the hydrogenic impurity in ZB GaN/AlGaN QWs, the trial wave function may be written as [14–17,19]
^0¼ H
F ¼ hðzÞexpðlrÞ
ð13Þ
where h(z) is the eigenfunction of the Hamiltonian described in Eq. (2), which can be solved using the transfer matrix method [25]. l is the variational parameter. We would like to point out that the trial wave function used in the present paper is a reasonable choice for studying impurity states in ZB GaN/AlGaN QWs [14–17,19]. The ground-state energy of a hydrogenic impurity in ZB GaN/ AlGaN QW may be obtained by minimizing E ¼ min l
^ FS /F9H9 /F9FS
ð14Þ
The ground-state donor binding energy Eb can be represented as follows: Eb ¼ E0 E
ð15Þ
where E0 is the ground-state energy for the Hamiltonian of Eq. (2).
3. Numerical results and discussion We have calculated the ground-state donor binding energy Eb as functions of the impurity positions zi, the right well width Lrw,
Fig. 2. The ground-state donor binding energy Eb as a function of the impurity positions zi for different hydrostatic pressures in ZB GaN/Al0.15Ga0.85N asymmetric coupled QWs with the middle barrier width Lmb ¼ 1 nm. (a) left well width Llw ¼ 4 nm, right well width Lrw ¼ 2 nm; and (b) left well width Llw ¼ 2 nm, right well width Lrw ¼ 4 nm.
the middle barrier width Lmb and the hydrostatic pressure P in ZB GaN/AlGaN asymmetric coupled QWs. All material parameters used in the present paper are the same as in Ref. [24]. In Fig. 2, the ground-state donor binding energy is investigated as a function of the impurity positions in ZB GaN/Al0.15Ga0.85N asymmetric coupled QWs for different hydrostatic pressures. Numerical results show that the donor binding energy is distributed asymmetrically with respect to the center of the asymmetric coupled QWs in all cases. This result is different from that in symmetric coupled QWs, in which the donor binding energy is distributed symmetrically around the center of the coupled quantum QWs [26]. Moreover, the position of the maximum value is located inside the wide well. This is because the electron wave function is mainly localized inside the wide well in the asymmetric coupled QWs. It is also found from Fig. 2 that the donor binding energy is increased when the hydrostatic pressure is considered for any impurity position. Moreover, the pressure effect has an obvious influence on the donor binding energy of the impurity located inside the wide well. The main reasons can be given as follows: as the hydrostatic pressure is considered, the dielectric constant and well size are decreased; in addition, the electron effective mass is increased, then the electron wave function is more localized inside the asymmetric coupled QWs. Thus, the electron–impurity distance is reduced and the donor binding energy is increased when the hydrostatic pressure effect is considered. In addition, Fig. 2 (a) and (b) show that when either well is wide in the asymmetric coupled QWs, the behaviors of the donor binding energy are equivalent. Thus, in the following part, we only discuss the effect of the variation of
ARTICLE IN PRESS 3274
C. Xia et al. / Physica B 405 (2010) 3272–3275
Fig. 3. The ground-state donor binding energy Eb as a function of the right well width Lrw for different impurity positions zi and hydrostatic pressures P in the ZB GaN/Al0.15Ga0.85N asymmetric coupled QWs with well width Llw ¼ 2 nm and middle barrier width Lmb ¼ 1 nm. Curves A, B and C are for the impurity located at zi ¼ (Lmb + Llw/2), 0 and Lmb + Lrw/2, respectively.
Fig. 4. The ground-state donor binding energy Eb as a function of the middle barrier width Lmb for different impurity positions zi and hydrostatic pressures P in ZB GaN/Al0.15Ga0.85N asymmetric coupled QWs with the left well width Llw ¼ 2 nm and the right well width Lrw ¼ 4 nm. The curves A, B and C are for the same meaning as in Fig. 3.
the right well width on the donor binding energy in ZB GaN/ Al0.15Ga0.85N asymmetric coupled QWs. In Fig. 3, the ground-state donor binding energy is investigated as a function of the right well width for different impurity positions in ZB GaN/Al0.15Ga0.85N asymmetric coupled QWs. Numerical results show that the donor binding energy is increased for any impurity position when the hydrostatic pressure effect is considered, as expected. In addition, it can also be seen from Fig. 2 that if the right well width Lrw r2 nm, the hydrostatic pressure is more pronounced for the impurity located at the center of the left well (see curve A). However, when the right well width Lrw Z2 nm, the hydrostatic pressure is more pronounced for the impurity located at the center of the right well (see curve C). These behaviors indicate that the hydrostatic pressure increased the donor binding energy for any well width and impurity position. In particular, one can conclude that the hydrostatic pressure has an obvious influence on the binding energy in the wide well of the asymmetric coupled QWs. The reasons are that the electron wave function is mainly localized inside the wide well. These results are in agreement with that of Fig. 2. In the following, we can further understand the characteristic behaviors of curves A, B and C of Fig. 3. It can be found from curve A of Fig. 3 that for impurity located at the center of the left well, the donor binding energy is decreased with increase in the right well width for any hydrostatic pressures. This is because the electron wave function located inside the left well is reduced with increase in the right well width in ZB GaN/Al0.15Ga0.85N asymmetric coupled QWs. However, for curve B (impurity located at the center of the middle barrier layer), the donor binding energy has a maximum value with increase in the right well width in the case of any pressure. This behavior can be explained as follows: when the right well width is increased from zero, the electron wave function begins to penetrate into the middle barrier layer and the donor binding energy increases. And if the right well width continuously increases, the electron wave function gradually penetrates into the right well. Thus, the donor binding energy along curve B is decreased by continuously increasing the right well width. Moreover, we can also see from curve C of Fig. 3 that for the impurity located at the center of the right well, the donor binding energy also has a maximum value in the case of any hydrostatic pressure. This can be understood as follows: for small right well width, the electron wave function localized inside the right well is increased with increase in the right well width of the asymmetric
coupled QWs. Therefore, the electron–impurity distance is decreased for the impurity located at the center of the right well. Whereas, for large right well width, the electron wave function is mainly localized inside the right well. The electron–impurity distance is increased with the increased right well width. Thus, both the Coulomb interaction between the electron and the impurity and the donor binding energy are reduced with an increment in the right well width. In addition, Fig. 3 shows that the donor binding energy along curves A and C reaches the same value when the right well width Lrw ¼ 2 nm (see dotted line). The physical reasons can be given as follows: when the right well width Lrw ¼Llw ¼2 nm, symmetric coupled QWs with well width Lw ¼2 nm is produced. Thus, the donor binding energy of impurities located at the symmetric positions with respect to the center of the symmetric coupled QWs has the same value, which is in agreement with the results in the symmetric coupled QWs [26]. In Fig. 4, the middle barrier width effect on the ground-state donor binding energy in ZB GaN/Al0.15Ga0.85N asymmetric coupled QWs is investigated for different impurity positions and hydrostatic pressures. Numerical results show that the donor binding energy is the largest when the impurity is located at the center of the right well in the case of any pressure. This is because the electron wave function is localized around the center of the right well. It can also be seen from Fig. 4 that the donor binding energy is increased when the hydrostatic pressure is considered for any impurity position, as expected. In particular, Fig. 4 also shows that the hydrostatic pressure has an obvious influence on the donor binding energy of impurity located inside the wide well, which agrees with the results of Fig. 2. In the following, we can further understand the characteristic behaviors of curves A, B and C of Fig. 4. For curve C, it can be found that for large middle barrier width (Lmb Z2 nm), the donor binding energy is insensitive to the middle barrier width in all cases. The physical reasons can be illustrated as follows: when the middle barrier width Lmb Z2 nm, the coupled QWs are decoupled. Thus, the electron–impurity distance for the impurity located at the center of the right well is insensitive to the variation of the middle barrier width. However, for small middle barrier width (Lmb o2 nm), the donor binding energy increases with increase in the middle barrier width for any pressure. The reason can be understood as follows: when the middle barrier width appears, the asymmetric coupled QWs system is produced. The electron wave function is more strongly localized inside the wide well when the middle barrier width is
ARTICLE IN PRESS C. Xia et al. / Physica B 405 (2010) 3272–3275
Fig. 5. The ground-state donor binding energy Eb as a function of the hydrostatic pressure P for different impurity positions in ZB GaN/Al0.15Ga0.85N asymmetric coupled QWs with the left well width Llw ¼ 2 nm, middle barrier width Lmb ¼ 1 nm and the right well width Lrw ¼ 4 nm. Curves A, B and C are for the same meaning as in Fig. 3.
increased form zero in the asymmetric coupled QWs. For curves A and B, the donor binding energy is decreased with increase in the middle barrier width for any hydrostatic pressure. The physical reasons can be explained as follows: the electron wave function is mainly localized inside the wide well of the asymmetric coupled QWs. Thus, for the impurity located inside the narrow well and the middle barrier layer, the electron–impurity distance is increased and the donor binding energy is decreased when the middle barrier width is increased. Specially, it needs to point out that when the left well width Llw ¼2 nm and the right well width Lrw ¼4 nm, and single QW with the well width Lw ¼6 nm are produced in the case of the middle barrier width Lmb ¼0. The impurity positions of curves B and C are symmetric with respect to the center of the single QW. Thus, the impurity binding energy along curves B and C has the same value for any hydrostatic pressure when Lmb ¼0. And it is also found from Fig. 4 that the difference in the donor binding energy along curves B and C is increased with increase in the middle barrier width. The reason is that the electron wave function is mainly localized inside the wide well of the asymmetric coupled QWs. The donor binding energy of impurity localized inside the wide (narrow) well is increased (decreased) when the middle barrier wide is increased. In Fig. 5, the ground-state donor binding energy is investigated as a function of the hydrostatic pressure for different impurity positions in ZB GaN/Al0.15Ga0.85N asymmetric coupled QWs. Numerical result shows that the hydrostatic pressure increases the donor binding energy for any impurity position, which is in agreement with that in a single QW and symmetric coupled QWs. This is because the electron wave function is more confined inside the asymmetric coupled QWs when the hydrostatic pressure is increased. In particular, it is found from curve C of Fig. 5 that the donor binding energy is more sensitive to the hydrostatic pressure for impurity located at the center of the right well. This is because the electron wave function is mainly localized around the center of the right well.
4. Conclusions In conclusion, we have investigated variationally the groundstate donor binding energy as functions of the impurity positions, the right well width and the middle barrier width in ZB GaN/ AlGaN asymmetric coupled QWs, considering the hydrostatic pressure effect. Numerical results show that the donor binding
3275
energy is highly dependent on the impurity positions, asymmetric coupled QWs structure parameters and the hydrostatic pressure. It is found that the hydrostatic pressure increases the donor binding energy for any impurity position. The hydrostatic pressure has an obvious influence on the impurity localized inside the wide well of the asymmetric coupled QWs. In the case of any hydrostatic pressure, our results show that the donor binding energy is distributed asymmetrically with respect to the center of the asymmetric coupled QWs. When the right well width is increased, the donor binding energy of impurity localized inside the narrow well is decreased; the donor binding energy of impurity localized inside the middle barrier layer has a maximum. In particular, for the impurity localized inside the wide well, the donor binding energy is insensitive to the middle barrier width in ZB GaN/AlGaN asymmetric coupled QWs if the middle barrier width is large. Moreover, to our knowledge, there are few papers on the impurity states in other such semiconducting structures. Thus, it is also interesting to point out that our calculation can be applied to other asymmetric coupled QWs structure just with different numbers of material parameters. Experimental studies concerned the effect of hydrostatic pressure on hydrogenic impurity states in ZB GaN/AlGaN asymmetric coupled QWs, which are still lacking at present. We hope that our calculation can stimulate further investigations of the related physics, as well as device applications of group-III nitrides.
Acknowledgment This work was supported by the National Natural Science Foundation of China under Grant no. 60906044.
References [1] B. Gil, in: Group III Nitride Semiconductor Compounds, Clarendon Press, Oxford, 1998. [2] S. Nakamura, S.F. Chichibu, in: Introduction to Nitride Semiconductor Blue Lasers and Light Emitting Diodes, Taylor and Francis, London, 2000. [3] S.H. Han, D.Y. Lee, S.J. Lee, C.Y. Cho, M.K. Kwon, S.P. Lee, D.Y. Noh, D.J. Kim, Y.C. Kim, S.J. Park, Appl. Phys. Lett. 94 (2009) 231123. [4] C.X. Xia, S.Y. Wei, Phys. Lett. A 359 (2006) 161. [5] J.J. Shi, Z.Z. Gan, J. Appl. Phys. 94 (2003) 407. [6] J.J. Shi, C.X. Xia, S.Y. Wei, Z.X. Liu, J. Appl. Phys. 97 (2005) 083705. [7] O. Husberg, A. Khartchenko, D.J. As, H. Vogelsang, T. Frey, D. Schikora, K. Lischka, O.C. Noriega, A. Tabata, J.R. Leite, Appl. Phys. Lett. 79 (2001) 1243. [8] S.F. Chichibu, T. Onuma, T. Sota, J. Appl. Phys. 93 (2003) 2051. [9] S. Strite, H. Morkoc, J. Vac. Sci. Technol. B 10 (1992) 1237. [10] E. Martinez-Guerrero, C. Adelmann, F. Chabuel, J. Simon, N.T. Pelekanos, G. Mula, B. Daudin, G. Feuillet, H. Mariette, Appl. Phys. Lett. 77 (2000) 809. [11] B.M. Shi, M.H. Xie, H.S. Wu, N. Wang, S.Y. Tong, Appl. Phys. Lett. 89 (2006) 151921. [12] Y. Fu, Y.Q. Wei, X.D. Wang, M. Sadeghi, S.M. Wang, A. Larsson, J. Appl. Phys. 100 (2006) 073105. ¨ [13] C. Brimont, M. Gallart, O. Cre´gut, B. Honerlage, P. Gilliot, D. Lagarde, A. Balocchi, T. Amand, X. Marie, S. Founta, H. Mariette, J. Appl. Phys. 106 (2009) 053514. [14] G. Bastard, Phys. Rev. B 24 (1981) 4714. [15] B.D. Byungsu Yoo, McCombe, W. Schaff, Phys. Rev. B 44 (1991) 13152. [16] H. Chen, X.D. Li, S.X. Zhou, Phys. Rev. B 44 (1991) 6220. [17] B. Sunder, Phys. Rev. B 45 (1992) 8562. [18] A.L. Morales, A. Montes, S.Y. Lo´pez, C.A. Duque, J. Phys.: Condens. Matter 14 (2002) 987. [19] N. Raigoza, A.L. Morales, A. Montes, N. Porras-Montenegro, C.A. Duque, Phys. Rev. B 69 (2004) 045323. [20] E. Kasapoglu, Phys. Lett. A 373 (2008) 140. [21] W.J. Fan, M.F. Li, T.C. Chong, J.B. Xia, Superlattices Microstruct. 19 (1996) 251. [22] W.J. Fan, M.F. Li, T.C. Chong, J.B. Xia, Solid State Commun. 98 (1996) 737. [23] J. Arriaga, H. Hernandez-Cocoletzi, D.A. Contreras-Solorioa, Physica E 17 (2003) 238. [24] C.X. Xia, Z.P. Zeng, S.Y. Wei, J. Appl. Phys. 107 (2010) 014305. [25] S. Vatannia, G. Gildenblat, IEEE J. Quantum Electron. 32 (1996) 1093. [26] E. Kasapoglu, Phys. Lett. A 373 (2008) 140.