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采用第一性原理方法系统地研究了沿（001）、（101）和（111）面施加晶面内各方向应变不相等的双轴张应变，即非对称双轴张应变对锗能带结构的影响.结果表明：对于沿（001）面施加非对称双轴张应变，至少某一个方向应变大于2.95%，间接-直接带隙转变才能发生；对于沿（101）面施加非对称双轴张应变，至少某一个方向应变大于3.44%，间接-直接带隙转变才能发生；然而，沿（111）面施加非对称双轴张应变，不发生间接-直接带隙转变.另外，研究还发现无论是施加对称双轴应变还是非对称双轴应变，间接-直接带隙转变得到的应变Ge带隙值都与应变前后拉伸面面积变化大小成反比.The strain engineering is an effective method to modulate the optical properties of germanium. The biaxial tensile strain has been extensively studied, most of the investigations focusing on biaxial tensile strain with equal in-plane strain at different crystal orientations, namely symmetric biaxial tensile strain. However, the effect of biaxial tensile strain with unequal in-plane strain at different crystal orientations, namely asymmetric biaxial tensile strain, has not been reported. In this paper, we systematically investigate the effect of asymmetric biaxial tensile strain on the band structure of Ge by using first-principle calculation.#br#We firstly calculate and analyze the dependence of band gap on strain for Ge with asymmetric biaxial tensile strain along three low Miller index planes, i.e., (001), (101) and (111). Then, we present the values of band gap and strain for some typical indirect-to-direct bandgap-transition-points under asymmetric biaxial tensile strain. Finally, we analyze the influence of biaxial tensile strain on the valance band structure. For the asymmetric biaxial tensile strain along the (001) plane, the indirect-to-direct band gap transition only occurs when the strain of one orientation is larger than 2.95%. For asymmetric biaxial tensile strain along the (101) plane, the indirect-to-direct band gap transition only occurs when the strain of one orientation is larger than 3.44%. Asymmetric biaxial tensile strain along the (111) plane cannot transform Ge into direct band gap material.#br#For asymmetric biaxial tensile strains along the (001) and (101) plane, the indirect-to-direct band gap transition points can be adjusted by changing the combination of in-plane strain at different crystal orientations. The value of bandgap of direct-band-gap Ge under biaxial tensile strain is inversely proportional to the area variation induced by application of strain. The asymmetric biaxial tensile strain along the (001) plane is the most effective to transform Ge into direct band gap material among the three types of biaxial strains, which are similar to the symmetric biaxial tensile strains.#br#In addition, the symmetric biaxial tensile strain will remove the three-fold degenerate states of valance band maximum, leading to a removal of the degeneracy between one heavy hole band and the light hole band. For biaxial tensile strain along the (001) and (101) plane, the asymmetric biaxial tensile strain could further remove the degeneracy between another heavy hole band and the light hole band.
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Keywords:
- strained Ge /
- biaxial tensile strain /
- band structure /
- first-principle methods
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[18] Inaoka T, Furukawa T, Toma R, Yanagisawa S 2015 J. Appl. Phys. 118 105704
[19] Dai X Y, Yang C, Song J J, Zhang H M, Hao Y, Zheng R C 2012 Acta Phys. Sin. 61 237102 (in Chinese)[戴显英, 杨程, 宋建军, 张鹤鸣, 郝跃, 郑若川2012物理学报61 237102]
[20] Kresse G, Furthmller J 1996 Phys. Rev. B 54 11169
[21] Kresse G, Joubert D 1999 Phys. Rev. B 59 1758
[22] Perdew J P, Burke K, Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
[23] Blöchl P E 1994 Phys. Rev. B 50 17953
[24] Tahini H, Chroneos A, Grimes R W, Schwingenschlögl U, Bracht H 2011 Appl. Phys. Lett. 99 072112
[25] Tahini H, Chroneos A, Grimes R W, Schwingenschlögl U 2011 Appl. Phys. Lett. 99 162103
[26] Hummer K, Harl J, Kresse G 2009 Phys. Rev. B 80 115205
[27] Kittel C (translated by Xiang J Z, Wu X H) 2012 Introduction to Solid State Physics (8th Ed.) (Beijing:Chemical Industry Press) p133(in Chinese)[基泰尔C著(项金钟, 吴兴惠译) 2012固体物理导论第八版(北京:化学工业出版社)第133页]
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[1] Soref R 2006 IEEE J. Sel. Top. Quant. Electron. 12 1678
[2] Michel J, Liu J, Kimerling L C 2010 Nature Photon. 4 527
[3] Kasper E 2010 Front. Optoelectron. China 3 143
[4] Reed G T, Mashanovich G, Gardes F Y, Thomson D J 2010 Nature Photon. 4 518
[5] Sun X, Liu J, Kimerling L C, Michel J 2009 Opt. Lett. 34 1198
[6] Liu J, Sun X, Kimerling L C, Michel J 2009 Opt. Lett. 34 1738
[7] Jain J R, Hryciw A, Baer T M, Miller D A B, Brongersma M L, Howe R T 2012 Nature Photon. 6 398
[8] Huang W Q, Liu S R 2005 Acta Phys. Sin. 54 972 (in Chinese)[黄伟其, 刘世荣2005物理学报54 972]
[9] Ma S Y, Qin G G, You L P, Wang Y Y 2001 Acta Phys. Sin. 50 1580 (in Chinese)[马书懿, 秦国刚, 尤力平, 王印月2001物理学报50 1580]
[10] Boucaud P, Kurdi M E, Ghrib A, Prost M, Kersauson M, Sauvage S, Aniel F, Checoury X, Beaudoin G, Largeau L, Sagnes I, Ndong G, Chaigneau M, Ossikovski R 2013 Photon. Res. 1 102
[11] Chen M J, Tsai C S, Wu M K 2006 Jpn. J. Appl. Phys. 45 6576
[12] Sánchez-Péreza J R, Boztug C, Chen F, Sudradjat F F, Paskiewicz D M, Jacobson R B, Lagally M G, Paiella R 2011 Proc. Natl. Acad. Sci. USA 108 18893
[13] Huo Y, Lin H, Chen R, Makarova M, Rong Y, Li M, Kamins T I, Vuckovic J, Harris J S 2011 Appl. Phys. Lett. 98 011111
[14] Hoshina Y, Iwasaki K, Yamada A, Konagai M 2009 Jpn. J. Appl. Phys. 48 04C125
[15] Tahini H, Chroneos A, Grimes R W, Schwingenschlogl U, Dimoulas A 2012 J. Phys.:Condens. Matter 24 195802
[16] Yang C H, Yu Z Y, Liu Y M, Lu P F, Gao T, Li M, Manzoor S 2013 Phys. B:Condens. Matter 427 62
[17] Liu L, Zhang M, Hu L, Di Z, Zhao S J 2014 J. Appl. Phys. 116 113105
[18] Inaoka T, Furukawa T, Toma R, Yanagisawa S 2015 J. Appl. Phys. 118 105704
[19] Dai X Y, Yang C, Song J J, Zhang H M, Hao Y, Zheng R C 2012 Acta Phys. Sin. 61 237102 (in Chinese)[戴显英, 杨程, 宋建军, 张鹤鸣, 郝跃, 郑若川2012物理学报61 237102]
[20] Kresse G, Furthmller J 1996 Phys. Rev. B 54 11169
[21] Kresse G, Joubert D 1999 Phys. Rev. B 59 1758
[22] Perdew J P, Burke K, Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
[23] Blöchl P E 1994 Phys. Rev. B 50 17953
[24] Tahini H, Chroneos A, Grimes R W, Schwingenschlögl U, Bracht H 2011 Appl. Phys. Lett. 99 072112
[25] Tahini H, Chroneos A, Grimes R W, Schwingenschlögl U 2011 Appl. Phys. Lett. 99 162103
[26] Hummer K, Harl J, Kresse G 2009 Phys. Rev. B 80 115205
[27] Kittel C (translated by Xiang J Z, Wu X H) 2012 Introduction to Solid State Physics (8th Ed.) (Beijing:Chemical Industry Press) p133(in Chinese)[基泰尔C著(项金钟, 吴兴惠译) 2012固体物理导论第八版(北京:化学工业出版社)第133页]
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