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The variational method has been widely used to study the electronic structures of heterostructure materials in spite of this method being less accurate than the numerical method, because analytical formulas for some electrical parameters can be derived using this method. However, effects of surface states on the two-dimensional electron gas (2DEG) have not been taken into account in the variational studies of GaN-based heterostructures. In the present study, analytical formulas for the electron wave function and ground state energy level of the 2DEG in InAlN/GaN heterostructures are derived using the variational method, and the influences of structural parameters of InAlN/GaN heterostructures on the electrical properties are discussed. In the theoretical model, evenly distributed surface states below the conduction band are assumed to be the origin of the 2DEG, and the polarization charges at the InAlN surface and the InAlN/GaN interface due to spontaneous and piezoelectric polarization effects in InAlN/GaN heterostructures are taken into account. A trial envelope wave function with two variational parameters is used to derive the expectation value of the total energy per electron. The variational parameters are determined by minimizing the expectation value. The model predicts a linear conduction band profile in InAlN barrier layer and an approximately triangular-shaped potential well on the GaN side of the InAlN/GaN interface. Electrons released from the surface states are confined in the potential well, forming the 2DEG. The 2DEG sheet density for the lattice-matched InAlN/GaN heterostructure with a 15 nm InAlN layer is 1.961013 cm-2, and the average distance from the InAlN/GaN interface of electrons is 2.23 nm. The 2DEG sheet density increases rapidly with InAlN thickness increasing when the InAlN layer exceeds the critical thickness, and starts to be saturated above 15 nm. The dependence of the calculated 2DEG sheet density on the InAlN thickness quantitatively agrees with recently reported experimental data. The increasing 2DEG sheet density results in increasing the ground state energy level and Fermi energy, and the energy spacing between the two also increases for containing more electrons. The polarization discontinuity at the InAlN/GaN interface decreases with increasing In mole fraction, causing the 2DEG sheet density to decrease, and thus the ground state energy level and the Fermi energy to decrease. This model is conducive to understanding the electrical behaviors of InAlN/GaN heterostructures and providing readily applicable formulas for studying the electron transport and optical transitions.
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Keywords:
- InAlN/GaN heterostructure /
- two-dimensional electron gas /
- variational method /
- wave function
[1] Li Q, Zhang J W, Meng L, Hou X 2014 Phys. Status Solidi B 251 755
[2] Zhang Y, Gu S L, Ye J D, Huang S M, Gu R, Chen B, Zhu S M, Zheng Y D 2013 Acta Phys. Sin. 62 150202 (in Chinese)[张阳, 顾书林, 叶建东, 黄时敏, 顾然, 陈斌, 朱顺明, 郑有炓 2013 物理学报 62 150202]
[3] Tien N T, Thao D N, Thao P T B, Quang D N 2015 Physica B 479 62
[4] Manouchehri F, Valizadeh P, Kabir M Z 2014 J. Vac. Sci. Technol. A 32 021104
[5] Stern F 1972 Phys. Rev. B 5 4891
[6] Fouillant C, Alibert C 1994 Am. J. Phys. 62 564
[7] Hao Y, Zhang J F, Zhang J C, Ma X H, Zheng X F 2015 Chin. Sci. Bull. 60 874 (in Chinese)[郝跃, 张金风, 张进成, 马晓华, 郑雪峰 2015 科学通报 60 874]
[8] Fang Y L, Feng Z H, Yin J Y, Zhang Z R, Lv Y J, Dun S B, Liu B, Li C M, Cai S J 2015 Phys. Status Solidi B 252 1006
[9] Arulkumaran S, Ng G I, Ranjan K, Kumar C M M, Foo S C, Ang K S, Vicknesh S, Dolmanan S B, Bhat T, Tripathy S 2015 Jpn. J. Appl. Phys. 54 04DF12
[10] Goyal N, Fjeldly T A 2016 IEEE Trans. Electron Dev. 63 881
[11] Jiao W, Kong W, Li J, Collar K, Kim T H, Losurdo M, Brown A S 2016 Appl. Phys. Lett. 109 082103
[12] Gordon L, Miao M S, Chowdhury S, Higashiwaki M, Mishra U K, van de Walle C G 2010 J. Phys. D: Appl. Phys. 43 505501
[13] Quang D N, Tung N H, Tuoc V N, Minh N V, Huy H A, Hien D T 2006 Phys. Rev. B 74 205312
[14] Ando T 1982 J. Phys. Soc. Jpn. 51 3900
[15] Yang P, L Y W, Wang X B 2015 Acta Phys. Sin. 64 197303 (in Chinese)[杨鹏, 吕燕伍, 王鑫波 2015 物理学报 64 197303]
[16] Cao Y, Jena D 2007 Appl. Phys. Lett. 90 182112
[17] Kaun S W, Ahmadi E, Mazumder B, Wu F, Kyle E C H, Burke P G, Mishra U K, Speck J S 2014 Semicond. Sci. Technol. 29 045011
[18] Dong X, Li Z H, Li Z Y, Zhou J J, Li L, Li Y, Zhang L, Xu X J, Xu X, Han C L 2010 Chin. Phys. Lett. 27 037102
[19] Zhang J F, Wang P Y, Xue J S, Zhou Y B, Zhang J C, Hao Y 2011 Acta Phys. Sin. 60 117305 (in Chinese)[张金风, 王平亚, 薛军帅, 周勇波, 张进成, 郝跃 2011 物理学报 60 117305]
[20] Xue J S, Zhang J C, Zhang W, Li L, Meng F N, Lu M, Jing N, Hao Y 2012 J. Cryst. Growth 343 110
[21] Čičo K, Greguov D, Gaži, oltys J, Kuzmk J, Carlin J F, Grandjean N, Pogany D, Frhlich K 2010 Phys. Status Solidi C 7 108
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[1] Li Q, Zhang J W, Meng L, Hou X 2014 Phys. Status Solidi B 251 755
[2] Zhang Y, Gu S L, Ye J D, Huang S M, Gu R, Chen B, Zhu S M, Zheng Y D 2013 Acta Phys. Sin. 62 150202 (in Chinese)[张阳, 顾书林, 叶建东, 黄时敏, 顾然, 陈斌, 朱顺明, 郑有炓 2013 物理学报 62 150202]
[3] Tien N T, Thao D N, Thao P T B, Quang D N 2015 Physica B 479 62
[4] Manouchehri F, Valizadeh P, Kabir M Z 2014 J. Vac. Sci. Technol. A 32 021104
[5] Stern F 1972 Phys. Rev. B 5 4891
[6] Fouillant C, Alibert C 1994 Am. J. Phys. 62 564
[7] Hao Y, Zhang J F, Zhang J C, Ma X H, Zheng X F 2015 Chin. Sci. Bull. 60 874 (in Chinese)[郝跃, 张金风, 张进成, 马晓华, 郑雪峰 2015 科学通报 60 874]
[8] Fang Y L, Feng Z H, Yin J Y, Zhang Z R, Lv Y J, Dun S B, Liu B, Li C M, Cai S J 2015 Phys. Status Solidi B 252 1006
[9] Arulkumaran S, Ng G I, Ranjan K, Kumar C M M, Foo S C, Ang K S, Vicknesh S, Dolmanan S B, Bhat T, Tripathy S 2015 Jpn. J. Appl. Phys. 54 04DF12
[10] Goyal N, Fjeldly T A 2016 IEEE Trans. Electron Dev. 63 881
[11] Jiao W, Kong W, Li J, Collar K, Kim T H, Losurdo M, Brown A S 2016 Appl. Phys. Lett. 109 082103
[12] Gordon L, Miao M S, Chowdhury S, Higashiwaki M, Mishra U K, van de Walle C G 2010 J. Phys. D: Appl. Phys. 43 505501
[13] Quang D N, Tung N H, Tuoc V N, Minh N V, Huy H A, Hien D T 2006 Phys. Rev. B 74 205312
[14] Ando T 1982 J. Phys. Soc. Jpn. 51 3900
[15] Yang P, L Y W, Wang X B 2015 Acta Phys. Sin. 64 197303 (in Chinese)[杨鹏, 吕燕伍, 王鑫波 2015 物理学报 64 197303]
[16] Cao Y, Jena D 2007 Appl. Phys. Lett. 90 182112
[17] Kaun S W, Ahmadi E, Mazumder B, Wu F, Kyle E C H, Burke P G, Mishra U K, Speck J S 2014 Semicond. Sci. Technol. 29 045011
[18] Dong X, Li Z H, Li Z Y, Zhou J J, Li L, Li Y, Zhang L, Xu X J, Xu X, Han C L 2010 Chin. Phys. Lett. 27 037102
[19] Zhang J F, Wang P Y, Xue J S, Zhou Y B, Zhang J C, Hao Y 2011 Acta Phys. Sin. 60 117305 (in Chinese)[张金风, 王平亚, 薛军帅, 周勇波, 张进成, 郝跃 2011 物理学报 60 117305]
[20] Xue J S, Zhang J C, Zhang W, Li L, Meng F N, Lu M, Jing N, Hao Y 2012 J. Cryst. Growth 343 110
[21] Čičo K, Greguov D, Gaži, oltys J, Kuzmk J, Carlin J F, Grandjean N, Pogany D, Frhlich K 2010 Phys. Status Solidi C 7 108
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