三价镨离子掺杂对铽镓石榴石晶体磁光性能影响的量子计算
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摘要
在铽镓石榴石(TGG)晶体中掺杂Pr~(3+)离子能够有效提升材料的磁光性能,但目前缺乏系统的理论计算阐明此问题.本文根据量子理论,分析了掺杂Pr~(3+)离子的影响机理并进行了定量计算.根据微扰理论解算久期方程,得到自旋-轨道耦合、晶场、有效场及离子之间的超交换作用下, Tb~(3+), Pr~(3+)离子的能级位移及波函数;进一步解算出Tb~(3+), Pr~(3+)离子自基态4f至5d的跃迁电偶极矩、各能级上的分布概率及平均磁矩;获得了Pr:TGG晶体的维尔德常数和磁化率,以及维尔德常数与Pr3+离子掺杂量之间的关系.研究结果表明:由于Pr~(3+)离子引起的法拉第旋转角较Tb~(3+)离子大,同时Tb~(3+)离子和Pr~(3+)离子之间强烈的超交换作用引起了能级的进一步分裂,导致Pr:TGG晶体维尔德常数明显提升;掺杂Pr~(3+)离子后,晶体内部有效磁矩增高,磁化率增大,且温度依赖性降低;维尔德常数数与Pr~(3+)离子掺杂量成分段线性关系,当晶体内部的Tb~(3+)离子和Pr~(3+)离子含量相等时,达到最大值.本文的计算结果与已有的实验数据符合较好.
Compared with those materials with superior magneto-optical properties, such as YIG, Ce:YIG and Ba_3Tb(PO_4)_3, pure terbium gallium garnet(TGG) crystal has comparative low Verdet constant and cannot meet the requirements of some high-power devices. Doping Pr~(3+) ions in TGG crystal can remarkably enhance its magneto-optical properties and expand its application scope, but there are still lack of systematic theoretical calculations to clarify this phenomenon. Based on the quantum theory, this paper presents the influence of doping Pr~(3+) ions on the magneto-optical performance and the corresponding quantitative calculation results.Firstly, taking various effects on Tb~(3+) ions and Pr~(3+) ions in the crystal into consideration, the Hamiltonian is modeled and discussed in detail. The secular equations are solved by applying the perturbation method, and then the energy level shifts and wave functions of the Tb~(3+) ions and Pr~(3+) ions are worked out, where the spinorbit coupling, crystal field, effective field and super-exchange interaction between the two types of ions are considered. Furthermore, the transition dipole moments of Tb~(3+) ions and Pr~(3+) ions from the 4f ground state to higher level 5d, together with the distribution probability at each energy level and the average magnetic moment, are resolved. Finally, the Verdet constants and magnetic susceptibilities of pure TGG crystal and Pr:TGG crystal are calculated and compared with each other. Moreover, the relationship between the Verdet constant of Pr:TGG crystal and the Pr~(3+)-doping amount is derived. The results show that the Faraday rotation angle caused by Pr~(3+) ions is larger than that of Tb~(3+) ions, meanwhile, the strong super-exchange between Tb~(3+)ions and Pr~(3+) ions causes further splitting of the 4 f energy level, resulting in a significant increasement of the Verdet constant of the Pr:TGG crystal, which reaches 313.4 rad/m·T, 191.2 rad/m·T and 60.4 rad/m·T at the wavelengths of 532 nm, 632.8 nm and 1064 nm, respectively. In addition, doping Pr~(3+) ions inside the crystal improves the internal effective magnetic moment, which can reach 9.92 μB at 10 K. At the same time, the magnetic susceptibility increases, while the temperature interdependency decreases. The linear relationship between the reciprocal of magnetic susceptibility and temperature reduces from 4.41/K to 3.92/K.The Verdet constant of the Pr:TGG crystal is linear with the amount of Pr~(3+) ions doping. When the contents of Tb~(3+) ions and Pr~(3+) ions inside the crystal are equal, the maximum value is reached, which is about2913.4 rad/m·T. The calculation results in this paper are in good agreement with the existing experimental data.
Compared with those materials with superior magneto-optical properties, such as YIG, Ce:YIG and Ba_3Tb(PO_4)_3, pure terbium gallium garnet(TGG) crystal has comparative low Verdet constant and cannot meet the requirements of some high-power devices. Doping Pr~(3+) ions in TGG crystal can remarkably enhance its magneto-optical properties and expand its application scope, but there are still lack of systematic theoretical calculations to clarify this phenomenon. Based on the quantum theory, this paper presents the influence of doping Pr~(3+) ions on the magneto-optical performance and the corresponding quantitative calculation results.Firstly, taking various effects on Tb~(3+) ions and Pr~(3+) ions in the crystal into consideration, the Hamiltonian is modeled and discussed in detail. The secular equations are solved by applying the perturbation method, and then the energy level shifts and wave functions of the Tb~(3+) ions and Pr~(3+) ions are worked out, where the spinorbit coupling, crystal field, effective field and super-exchange interaction between the two types of ions are considered. Furthermore, the transition dipole moments of Tb~(3+) ions and Pr~(3+) ions from the 4f ground state to higher level 5d, together with the distribution probability at each energy level and the average magnetic moment, are resolved. Finally, the Verdet constants and magnetic susceptibilities of pure TGG crystal and Pr:TGG crystal are calculated and compared with each other. Moreover, the relationship between the Verdet constant of Pr:TGG crystal and the Pr~(3+)-doping amount is derived. The results show that the Faraday rotation angle caused by Pr~(3+) ions is larger than that of Tb~(3+) ions, meanwhile, the strong super-exchange between Tb~(3+)ions and Pr~(3+) ions causes further splitting of the 4 f energy level, resulting in a significant increasement of the Verdet constant of the Pr:TGG crystal, which reaches 313.4 rad/m·T, 191.2 rad/m·T and 60.4 rad/m·T at the wavelengths of 532 nm, 632.8 nm and 1064 nm, respectively. In addition, doping Pr~(3+) ions inside the crystal improves the internal effective magnetic moment, which can reach 9.92 μB at 10 K. At the same time, the magnetic susceptibility increases, while the temperature interdependency decreases. The linear relationship between the reciprocal of magnetic susceptibility and temperature reduces from 4.41/K to 3.92/K.The Verdet constant of the Pr:TGG crystal is linear with the amount of Pr~(3+) ions doping. When the contents of Tb~(3+) ions and Pr~(3+) ions inside the crystal are equal, the maximum value is reached, which is about2913.4 rad/m·T. The calculation results in this paper are in good agreement with the existing experimental data.
引文
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[3]Zhang F, Tian Y, Yi Z, Gu S H 2016 Chin. Phys. B 25094206
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[6]Yasuhara R, Furuse H 2013 Opt. Lett. 38 1751
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[16]Chen Z, Yang L, Wang X Y, Hang Y 2016 Opt. Mater. 62475
[17]Zhu N F, Li Y X, Yu X F 2008 Mater. Lett. 62 2355
[18]Sugar J 1965 JOSA 55 1058
[19]Yang C H 2004 M. S. Thesis(Yangzhou:Yangzhou University)(in Chinese)[杨翠红2004硕士学位论文(扬州:扬州大学)]
[20]Suits J 1972 IEEE Trans. Magn. 8 95
[21]Shen Y R 1964 Phys. Rev. B 133 A511
[22]Cai W, Xing J H, Yang Z Y 2017 Acta Phys. Sin. 66 187801(in Chinese)[蔡伟,邢俊辉,杨志勇2017物理学报66 187801]
[23]Villaverde A B, Donatti D A, Bozinis D G 1978 J. Phys. C:Solid State Phys. 11 L495
[24]Kiyoshi S 2010 Crystal Growth&Design 10 3466
[25]L?w U, Zvyagin S, Ozerov M, Schaufuss U, Kataev V, Wolf B, Lüthi B 2013 Eur. Phys. J. B 86 87
[2]Kaminskii A A, Eichler H J, Reiche P, Uecker R 2005 Laser Phys. Lett. 2 489
[3]Zhang F, Tian Y, Yi Z, Gu S H 2016 Chin. Phys. B 25094206
[4]Li C S 2015 Acta Phys. Sin. 64 047801(in Chinese)[李长胜2015物理学报64 047801]
[5]Yasuhara R, Tokita S, Kawanaka J, Kawashima T 2007 Opt.Express 15 11264
[6]Yasuhara R, Furuse H 2013 Opt. Lett. 38 1751
[7]Yasuhara R, Tokita S, Kawanaka J 2007 Rev. Laser Eng. 35806
[8]Liu L, Yu Y D 1985 J. Synth. Cryst. 1 27(in Chinese)[刘琳,俞育德1985人工晶体学报1 27]
[9]Chani V I, Takeda H, Fukuda T 1999 J. Alloy. Compd. 60212
[10]Chen J B, Lin Y, Li G H, Chen J S, Teng S, Yao Y G 2014J. Synth. Cryst. 43 8(in Chinese)[陈建斌,林羽,李国辉,陈建珊,滕硕,姚元根2014人工晶体学报43 8]
[11]Xu J L, Dong W L, Peng H Y, Liu W, Jin W Z, Lin H, Li C2015 J. Changchun Univ. Technol. 3 20(in Chinese)[徐嘉林,董玮利,彭海益,刘旺,金维召,林海,李春2015长春理工大学学报3 20]
[12]Long Y, Xu Y, Shi Z B, Ding Y T, Wang J, Fu C L 2015Piezoelectric and Sound and Light 37 277(in Chinese)[龙勇,徐扬,石自彬,丁雨憧,王佳,付昌禄2015压电与声光37 277]
[13]Pei G Q, Zhang Y, Liu Z P 2015 J. Synth. Cryst. 44 885(in Chinese)[裴广庆,张艳,柳祝平2015人工晶体学报44 885]
[14]Long Y, Shi Z B, Ding Y D 2016 Piezoelectr. Acoustoopt. 38433(in Chinese)[龙勇,石自彬,丁雨憧2016压电与声光38433]
[15]Chen Z, Hang Y, Yang L, Wang J, Wang X Y, Zhang P X,Hong J Q, Shi C J, Wang Y Q 2015 Mater. Lett. 145 171
[16]Chen Z, Yang L, Wang X Y, Hang Y 2016 Opt. Mater. 62475
[17]Zhu N F, Li Y X, Yu X F 2008 Mater. Lett. 62 2355
[18]Sugar J 1965 JOSA 55 1058
[19]Yang C H 2004 M. S. Thesis(Yangzhou:Yangzhou University)(in Chinese)[杨翠红2004硕士学位论文(扬州:扬州大学)]
[20]Suits J 1972 IEEE Trans. Magn. 8 95
[21]Shen Y R 1964 Phys. Rev. B 133 A511
[22]Cai W, Xing J H, Yang Z Y 2017 Acta Phys. Sin. 66 187801(in Chinese)[蔡伟,邢俊辉,杨志勇2017物理学报66 187801]
[23]Villaverde A B, Donatti D A, Bozinis D G 1978 J. Phys. C:Solid State Phys. 11 L495
[24]Kiyoshi S 2010 Crystal Growth&Design 10 3466
[25]L?w U, Zvyagin S, Ozerov M, Schaufuss U, Kataev V, Wolf B, Lüthi B 2013 Eur. Phys. J. B 86 87