Quantum calculation of the influence of trivalent praseodymium ions doping on the magneto-optical properties of terbium gallium garnet crystal

Cai Wei; Xu You-An; Yang Zhi-Yong

doi:10.7498/aps.68.20190576

Abstract

Compared with those materials with superior magneto-optical properties, such as YIG, Ce:YIG and Ba₃Tb(PO₄)₃, pure terbium gallium garnet (TGG) crystal has comparative low Verdet constant and cannot meet the requirements of some high-power devices. Doping Pr³⁺ 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 Pr³⁺ ions on the magneto-optical performance and the corresponding quantitative calculation results. Firstly, taking various effects on Tb³⁺ ions and Pr³⁺ 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 Tb³⁺ ions and Pr³⁺ ions are worked out, where the spin-orbit coupling, crystal field, effective field and super-exchange interaction between the two types of ions are considered. Furthermore, the transition dipole moments of Tb³⁺ ions and Pr³⁺ 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 Pr³⁺-doping amount is derived. The results show that the Faraday rotation angle caused by Pr³⁺ ions is larger than that of Tb³⁺ ions, meanwhile, the strong super-exchange between Tb³⁺ ions and Pr³⁺ ions causes further splitting of the 4f 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 Pr³⁺ ions inside the crystal improves the internal effective magnetic moment, which can reach 9.92 μ_B 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 Pr³⁺ ions doping. When the contents of Tb³⁺ ions and Pr³⁺ ions inside the crystal are equal, the maximum value is reached, which is about 2913.4 rad/m·T. The calculation results in this paper are in good agreement with the existing experimental data.

Keywords:

Authors and contacts

1.
Armament Launch Theory and Technology Key Discipline Laboratory of China, Rocket Force University of Engineering, Xi′an 710025, China
2.
Science and Technology on Electro-optic Control Laboratory, Luoyang 471000, China

Corresponding author: Xu You-An, 408091240@qq.com

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61505254).

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References

[1]	Tian Y, Tan B Z, Yang J, Zhang Y, Gu S H 2015 Chin. Phys. B 24 063302 Google Scholar
[2]	Kaminskii A A, Eichler H J, Reiche P, Uecker R 2005 Laser Phys. Lett. 2 489 Google Scholar
[3]	Zhang F, Tian Y, Yi Z, Gu S H 2016 Chin. Phys. B 25 094206 Google Scholar
[4]	李长胜 2015 64 047801 Google Scholar Li C S 2015 Acta Phys. Sin. 64 047801 Google Scholar
[5]	Yasuhara R, Tokita S, Kawanaka J, Kawashima T 2007 Opt. Express 15 11264
[6]	Yasuhara R, Furuse H 2013 Opt. Lett. 38 1751 Google Scholar
[7]	Yasuhara R, Tokita S, Kawanaka J 2007 Rev. Laser Eng. 35 806 Google Scholar
[8]	刘琳, 俞育德 1985 人工晶体学报 1 27 Liu L, Yu Y D 1985 J. Synth. Cryst. 1 27
[9]	Chani V I, Takeda H, Fukuda T 1999 J. Alloy. Compd. 60 212
[10]	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 Chen J B, Lin Y, Li G H, Chen J S, Teng S, Yao Y G 2014 J. Synth. Cryst. 43 8
[11]	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 Google Scholar Xu J L, Dong W L, Peng H Y, Liu W, Jin W Z, Lin H, Li C 2015 J. Changchun Univ. Technol. 3 20 Google Scholar
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[13]	裴广庆, 张艳, 柳祝平 2015 人工晶体学报 44 885 Google Scholar Pei G Q, Zhang Y, Liu Z P 2015 J. Synth. Cryst. 44 885 Google Scholar
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[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 Google Scholar
[16]	Chen Z, Yang L, Wang X Y, Hang Y 2016 Opt. Mater. 62 475 Google Scholar
[17]	Zhu N F, Li Y X, Yu X F 2008 Mater. Lett. 62 2355 Google Scholar
[18]	Sugar J 1965 JOSA 55 1058 Google Scholar
[19]	杨翠红 2004 硕士学位论文(扬州: 扬州大学) Yang C H 2004 M. S. Thesis (Yangzhou: Yangzhou University) (in Chinese)
[20]	Suits J 1972 IEEE Trans. Magn. 8 95 Google Scholar
[21]	Shen Y R 1964 Phys. Rev. B 133 A511 Google Scholar
[22]	蔡伟, 邢俊辉, 杨志勇 2017 66 187801 Google Scholar Cai W, Xing J H, Yang Z Y 2017 Acta Phys. Sin. 66 187801 Google Scholar
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[24]	Kiyoshi S 2010 Crystal Growth & Design 10 3466 Google Scholar
[25]	Löw U, Zvyagin S, Ozerov M, Schaufuss U, Kataev V, Wolf B, Lüthi B 2013 Eur. Phys. J. B 86 87 Google Scholar

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图 1 维尔德常数的波长特性

Figure 1. Wavelength characteristics of the Verdet constant.

DownLoad: Full-Size Img PowerPoint

图 2 磁化率的温度特性

Figure 2. Temperature characteristics of the magnetic susceptibility.

DownLoad: Full-Size Img PowerPoint

图 3 维尔德常数随Pr³⁺离子含量(y)的变化情况

Figure 3. The variation of Verdet constant with Pr³⁺ ions content (y).

DownLoad: Full-Size Img PowerPoint

表 1 作用于Tb³⁺, Pr³⁺离子的晶场参数(cm^–1)

Table 1. Crystal field parameters acting on Tb³⁺ and Pr³⁺ ions (cm^–1).

	能级	${B_{2,0}}$	${B_{2,2}}$	${B_{4,0}}$	${B_{4,2}}$	${B_{4,4}}$	${B_{6,0}}$	${B_{6,2}}$	${B_{6,4}}$	${B_{6,6}}$
Tb³⁺	4f	–129.9	271.2	–2558.8	296.2	1121.6	682.3	–157.2	1048	–4.7
Tb³⁺	5d	–3063	1180	–139845	72999	5412
Pr³⁺	4f	–334	144	–2630	252	1126	932	–207	1622	–199
Pr³⁺	5d	–4162	2150	–128877	79630	8425

DownLoad: CSV

表 2 晶场及自旋轨道作用下的能级位移(cm^–1)

Table 2. Energy level shift under the action of crystal field and spin orbit (cm^–1).

		1	2	3	4	5	6	7	8
Tb³⁺	E_a₁	41.6	49.7	84.9	89.2	267.5	272	303.2	310.5
Tb³⁺	E_b₁	–863.2	–336.4	–56.3	784.6	1446.7	1996.2
Pr³⁺	E_a₁	–7.4	–8.9	55	452.8	512.4	549.9	567.1	722.4
Pr³⁺	E_b₁	–1767.6	–542.9	1115.9	1198.2	2349.5

DownLoad: CSV

表 3 有效场作用下的能级分裂(cm^–1)

Table 3. Energy level splitting under the action of effective field (cm^–1).

	1	2	3	4
Tb³⁺	(± 2.342$\mp\; 0.9516 \nu \chi $)	(± 0.463$\mp\; 0.1422 \nu \chi $)	(± 0.897$\mp\; 0.3641 \nu \chi $)	(± 1.499$\mp \;0.6561 \nu \chi $)
Pr³⁺	(± 1.641$\mp \;0.8244 \nu \chi $)	(± 0.423$\mp\; 0.0893 \nu \chi $)	(± 3.302$\mp\; 0.1176 \nu \chi $)

DownLoad: CSV

表 4 超交换作用下的能级位移(cm^–1)

Table 4. Energy level shift under the action of super-exchange interaction (cm^–1).

		1	2	3	4	5	6	7	8
Tb³⁺	E_a₃	–201.3	–152.3	–96.4	–3.2	82.4	141	168.3	210
Tb³⁺	E_b₂	–499.1	–112.7	78.8	236.7	774.1	1135.8
Pr³⁺	E_a₃	–262.1	–194.3	–32.4	56.9	61.5	176.5	211.7	387.9
Pr³⁺	E_b₂	–844.1	–10.3	743.5	882.4	1178.3

DownLoad: CSV

表 5 不同波长下的维尔德常数V (${\rm{rad/m}} \cdot {\rm{T}}$)

Table 5. Verdet constant at different wavelengths (${\rm{rad/m}} \cdot {\rm{T}}$).

	波长λ/nm	457.9	532	632.8	830	1064	1300
TGG	V_c	290.1	179.4	122.2	51.9	31.8	18.5
TGG	V_e	305.7	190	134.4	61	40.2	20
5%Pr:TGG	V_c	421.8	312.5	190.2	108.8	59.7	45.9
5%Pr:TGG	V_e	437	324.5	200.1	121.4	68.7	49.2
注: V_c为本文维尔德常数的计算值, V_e为实验值^[16,23,24].

DownLoad: CSV

表 6 不同Pr³⁺离子含量(y)下的维尔德常数V(${\rm{rad/m}} \cdot {\rm{T}}$)

Table 6. Verdet constant under different Pr³⁺ ions content (${\rm{rad/m}} \cdot {\rm{T}}$).

λ/nm y	0	0.073	1	1.5	2	2.927	3
532	179.4	312.5	2002.1	2913.4	2021.4	367.9	237.7
632.8	122.2	190.2	1099.8	1588.6	1112.9	231	161.5
1064	31.8	59.7	432.1	632.3	438	77.8	49.4

DownLoad: CSV

表 7 不同温度下磁化率的倒数1/χ

Table 7. Inverse magnetic susceptibility at different temperatures.

	温度T /K	10	100	150	200	250	300
TGG	1/χ_c	80.3	458.4	688.3	942.8	1167.8	1399.5
TGG	1/χ_e	72.2	469.3	659.2	908.9	1128.7	1349.6
5%Pr:TGG	1/χ_c	64.3	441.2	637.7	852.4	1029.3	1210.5
5%Pr:TGG	1/χ_e	56.4	419	599.2	803.7	987.1	1163.2
注: 1/χ_c为本文计算值, 1/χ_e为实验值^[16,25].

DownLoad: CSV

map

[1]	Tian Y, Tan B Z, Yang J, Zhang Y, Gu S H 2015 Chin. Phys. B 24 063302 Google Scholar
[2]	Kaminskii A A, Eichler H J, Reiche P, Uecker R 2005 Laser Phys. Lett. 2 489 Google Scholar
[3]	Zhang F, Tian Y, Yi Z, Gu S H 2016 Chin. Phys. B 25 094206 Google Scholar
[4]	李长胜 2015 64 047801 Google Scholar Li C S 2015 Acta Phys. Sin. 64 047801 Google Scholar
[5]	Yasuhara R, Tokita S, Kawanaka J, Kawashima T 2007 Opt. Express 15 11264
[6]	Yasuhara R, Furuse H 2013 Opt. Lett. 38 1751 Google Scholar
[7]	Yasuhara R, Tokita S, Kawanaka J 2007 Rev. Laser Eng. 35 806 Google Scholar
[8]	刘琳, 俞育德 1985 人工晶体学报 1 27 Liu L, Yu Y D 1985 J. Synth. Cryst. 1 27
[9]	Chani V I, Takeda H, Fukuda T 1999 J. Alloy. Compd. 60 212
[10]	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 Chen J B, Lin Y, Li G H, Chen J S, Teng S, Yao Y G 2014 J. Synth. Cryst. 43 8
[11]	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 Google Scholar Xu J L, Dong W L, Peng H Y, Liu W, Jin W Z, Lin H, Li C 2015 J. Changchun Univ. Technol. 3 20 Google Scholar
[12]	龙勇, 徐扬, 石自彬, 丁雨憧, 王佳, 付昌禄 2015 压电与声光 37 277 Google Scholar Long Y, Xu Y, Shi Z B, Ding Y T, Wang J, Fu C L 2015 Piezoelectric and Sound and Light 37 277 Google Scholar
[13]	裴广庆, 张艳, 柳祝平 2015 人工晶体学报 44 885 Google Scholar Pei G Q, Zhang Y, Liu Z P 2015 J. Synth. Cryst. 44 885 Google Scholar
[14]	龙勇, 石自彬, 丁雨憧 2016 压电与声光 38 433 Long Y, Shi Z B, Ding Y D 2016 Piezoelectr. Acoustoopt. 38 433
[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 Google Scholar
[16]	Chen Z, Yang L, Wang X Y, Hang Y 2016 Opt. Mater. 62 475 Google Scholar
[17]	Zhu N F, Li Y X, Yu X F 2008 Mater. Lett. 62 2355 Google Scholar
[18]	Sugar J 1965 JOSA 55 1058 Google Scholar
[19]	杨翠红 2004 硕士学位论文(扬州: 扬州大学) Yang C H 2004 M. S. Thesis (Yangzhou: Yangzhou University) (in Chinese)
[20]	Suits J 1972 IEEE Trans. Magn. 8 95 Google Scholar
[21]	Shen Y R 1964 Phys. Rev. B 133 A511 Google Scholar
[22]	蔡伟, 邢俊辉, 杨志勇 2017 66 187801 Google Scholar Cai W, Xing J H, Yang Z Y 2017 Acta Phys. Sin. 66 187801 Google Scholar
[23]	Villaverde A B, Donatti D A, Bozinis D G 1978 J. Phys. C: Solid State Phys. 11 L495 Google Scholar
[24]	Kiyoshi S 2010 Crystal Growth & Design 10 3466 Google Scholar
[25]	Löw U, Zvyagin S, Ozerov M, Schaufuss U, Kataev V, Wolf B, Lüthi B 2013 Eur. Phys. J. B 86 87 Google Scholar

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