Effect of excited-state population density on nonradiative multiphonon relaxation rates of rare-earth ions

F. Pellé; N. Gardant; F. Auzel

doi:10.1364/JOSAB.15.000667

Journal of the Optical Society of America B
Vol. 15,
Issue 2,
pp. 667-679
(1998)
•https://doi.org/10.1364/JOSAB.15.000667

Effect of excited-state population density on nonradiative multiphonon relaxation rates of rare-earth ions

F. Pellé, N. Gardant, and F. Auzel

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F. Pellé, N. Gardant, and F. Auzel, "Effect of excited-state population density on nonradiative multiphonon relaxation rates of rare-earth ions," J. Opt. Soc. Am. B 15, 667-679 (1998)

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Abstract

Multiphonon relaxation rate saturation is observed for rare-earth-doped glasses for high excited-state densities. This behavior is analyzed in a statistical approach; a theoretical model for the microscopic process is proposed. A phonon bottleneck effect on radiationless relaxation related to an accepting-modes saturation is suggested. At high excited-state density, ions in a phonon diffusion volume simultaneously fill the accepting modes. The critical distance below which excited ions share a common phonon bath is related to the phonon diffusion length in the host and deduced from our model. The results are in good agreement with the phonon mean free path independently deduced from sound velocity, thermal conductivity, and heat capacity.

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Glass	Composition (mol. %)
Phosphate	$50 (P_{2} O_{5}) - 16.66 (MgO) - 33.34 ({Li}_{2} O)$
Germanate	$66 ({GeO}_{2}) - 17 (BaO) - 17 (K_{2} O)$
Tellurite	$80 ({TeO}_{2}) - 20 ({Li}_{2} O)$
ZBLAN	$57.35 ({ZrF}_{4}) - 29.63 ({BaF}_{2}) - 3.54 ({LaF}_{3})$ $- 1.78 (NaF) - 5.16 ({InF}_{3}) - 2.54 ({AlF}_{3})$

$S^{'} L^{'} J^{'}$	Glass
	Phosphate		Germanate		Tellurite		ZBLAN
	$f_{exp}$ $(\times 10^{6})$	$f_{calc}$ $(\times 10^{6})$	$f_{exp}$ $(\times 10^{6})$	$f_{calc}$ $(\times 10^{6})$	$f_{exp}$ $(\times 10^{6})$	$f_{calc}$ $(\times 10^{6})$	$f_{exp}$ $(\times 10^{6})$	$f_{calc}$ $(\times 10^{6})$
$^{4} I_{13 / 2}$	${0.78}^{*}$	${0.70}^{*}$	${0.29}^{*}$	${0.39}^{*}$	${1.08}^{*}$	${0.80}^{*}$	${1.14}^{*}$	${0.90}^{*}$
$^{4} I_{11 / 2}$	0.38	0.35	0.18	0.27	0.37	0.36	0.39	0.42
$^{4} I_{9 / 2}$	0.14	0.40	0.10	0.17	0.22	0.32	0.19	0.31
$^{4} F_{9 / 2}$	1.25	1.55	0.58	0.07	1.38	1.34	1.44	1.45
$^{4} S_{3 / 2}$	0.30	0.23	0.68	0.12	0.32	0.24	0.37	0.33
$^{2} H_{11 / 2}$	5.47	5.96	5.70	6.45	5.33	5.33	3.91	4.11
$^{4} F_{7 / 2}$	1.13	1.19	0.45	0.58	1.09	1.13	1.24	1.42
$^{4} F_{5 / 2}$	0.44	0.46	0.01	0.01	0.31	0.30	0.39	0.42
$^{4} F_{3 / 2}$	–	–	0.01	0.01	0.11	0.18	0.19	0.25
$^{2} H_{9 / 2}$	0.39	0.39	0.22	0.02	0.41	0.40	0.44	0.53
$^{4} G_{11 / 2}$	10.77	10.50	11.76	11.34	–	–	7.33	7.22
$^{4} G_{9 / 2}$	1.89	1.37	–	–	–	–	–	–
$^{4} G_{9 / 2} +^{4} K_{15 / 2} +^{2} G_{7 / 2}$	–	–	1.21	0.97	–	–	1.79	1.68

$n$	1.50		1.66		2.15		1.51
$d$ $(g / {cm}^{3})$	2.53		4.04		5.1		4.24
$Ω_{2}$ $(\times 10^{20} {cm}^{- 2})$	5.01		6.33		4.61		3.31
$Ω_{4}$ $(\times 10^{20} {cm}^{- 2})$	2.20		0.93		1.74		1.63
$Ω_{6}$ $(\times 10^{20} {cm}^{- 2})$	0.86		0.46		0.91		1.25
rms $(\times 10^{- 7})$	3.10		3.61		6.73		1.41

$S^{'} L^{'} J^{'}$	Glass
	Phosphate		Germanate		Tellurite		ZBLAN
	$f_{exp}$ $(\times 10^{6})$	$f_{calc}$ $(\times 10^{6})$	$f_{exp}$ $(\times 10^{6})$	$f_{calc}$ $(\times 10^{6})$	$f_{exp}$ $(\times 10^{6})$	$f_{calc}$ $(\times 10^{6})$	$f_{exp}$ $(\times 10^{6})$	$f_{calc}$ $(\times 10^{6})$
$^{3} F_{4}$	1.63	1.75	1.17	1.34	1.61	1.72	1.34	1.43
$^{3} H_{5}$	${0.10}^{*}$	${0.89}^{*}$	${0.73}^{*}$	${0.64}^{*}$	${0.83}^{*}$	${1.00}^{*}$	${0.93}^{*}$	${0.96}^{*}$
$^{3} H_{4}$	1.88	1.55	1.44	1.27	1.72	1.62	1.52	1.44
$^{3} F_{3}$	1.59	1.43	0.68	0.81	1.82	1.74	1.89	1.84
$^{3} F_{2}$	0.14	0.25	0.14	0.18	0.17	0.34	0.17	0.39
$^{1} G_{4}$	0.66	0.53	0.40	0.40	0.75	0.52	0.51	0.42
$^{1} D_{2}$	1.44	1.37	0.72	0.61	–	–	1.45	1.45

$n$	1.50		1.66		2.15		1.51
$d$ $(g / {cm}^{3})$	2.53		4.06		5.1		4.24
$Ω_{2}$ $(\times 10^{20} {cm}^{- 2})$	3.82		3.79		3.49		2.40
$Ω_{4}$ $(\times 10^{20} {cm}^{- 2})$	1.64		0.68		1.71		1.62
$Ω_{6}$ $(\times 10^{20} {cm}^{- 2})$	0.78		0.54		1.06		1.19
rms $(\times 10^{- 7})$	4.48		1.57		2.17		1.33

$S^{'} L^{'} J^{'}$	Phosphate		Germanate		Tellurite		ZBLAN
$S^{'} L^{'} J^{'}$	$f_{exp}$ $(\times 10^{6})$	$f_{calc}$ $(\times 10^{6})$	$f_{exp}$ $(\times 10^{6})$	$f_{calc}$ $(\times 10^{6})$	$f_{exp}$ $(\times 10^{6})$	$f_{calc}$ $(\times 10^{6})$	$f_{exp}$ $(\times 10^{6})$	$f_{calc}$ $(\times 10^{6})$
$^{4} I_{15 / 2}$	0.20	0.18	0.09	0.09	0.14	0.12	0.18	0.14
$^{4} F_{3 / 2}$	1.56	1.43	1.05	1.39	1.42	1.52	1.28	1.24
$^{4} F_{5 / 2} +^{2} {H_{9 / 2}}^{(2)}$	5.67	5.40	3.35	3.43	4.37	4.28	4.70	4.54
$^{4} F_{7 / 2} +^{4} S_{3 / 2}$	5.54	5.94	3.00	3.01	4.03	4.12	4.56	4.96
$^{4} F_{9 / 2}$	0.39	0.44	0.24	0.25	0.34	0.33	0.34	0.37
$^{2} {H_{11 / 2}}^{(2)}$	0.10	0.12	0.03	0.07			0.08	0.10
$^{4} G_{5 / 2} +^{4} G_{7 / 2}$	12.83	12.88	16.22	16.24	12.24	12.24	8.87	8.95
$^{4} G_{7 / 2} +^{4} G_{9 / 2} +^{2} K_{13 / 2}$	4.57	3.96	3.98	6.36	3.93	3.74	4.27	3.21
$^{2} {G_{9 / 2}}^{(1)} +^{2} D_{3 / 2} +^{4} K_{15 / 2} +^{4} G_{11 / 2}$	1.42	0.88	0.77	0.696	–	–	1.06	0.75
$^{2} P_{1 / 2}$	0.35	0.36	0.34	0.42	0.26	0.44	0.25	0.31
$^{2} D_{5 / 2}$	0.06	0.03	0.007	0.014	0.14	0.02	0.05	0.02
$^{4} D_{3 / 2} +^{4} D_{5 / 2} +^{4} D_{1 / 2} +^{2} I_{11 / 2} +^{2} L_{15 / 2}$	6.91	7.15	–	–	–	–	5.96	6.25
$^{2} I_{13 / 2} +^{4} D_{7 / 2} +^{2} L_{17 / 2}$	–	–	–	–	–	–	0.99	0.24

Title
$n$	1.50		1.66		$2.15 / 1.99$ (1 μm)		1.51
$d$ $(g / {cm}^{3})$	2.50		4.02		5.14		4.22
$Ω_{2}$ $(\times 10^{20} {cm}^{- 2})$	3.99		6.32		3.92		2.69
$Ω_{4}$ $(\times 10^{20} {cm}^{- 2})$	3.61		4.27		4.41		3.16
$Ω_{6}$ $(\times 10^{20} {cm}^{- 2})$	5.94		2.82		4.06		4.91
rms $(\times 10^{- 7})$	3.79		1.75		0.12		4.56

	$A_{rad}$ $(s^{- 1})$
	Phosphate			Germanate			Tellurite			ZBLAN
Level	$A_{ED}$	$A_{MD}$	$A_{tot}$	$A_{ED}$	$A_{MD}$	$A_{tot}$	$A_{ED}$	$A_{MD}$	$A_{tot}$	$A_{ED}$	$A_{MD}$	$A_{tot}$
${Er}^{3 +}^{4} S_{3 / 2}$	–	–	985	–	–	755	–	–	3576	–	–	1463
$^{4} I_{11 / 2}$	107	6	113	104	8	112	377	18	395	113	45	158
$^{4} I_{13 / 2}$	68	44	112	55	47	102	239	130	369	90	6	96
${Tm}^{3 +}^{3} H_{4}$	840	16	856	902	16	918	2878	34	2912	752	11	763
${Nd}^{3 +}^{4} F_{3 / 2}$	–	–	2568	–	–	2518	–	–	5700	–	–	2218

Glass	Calculated thermal Conductivity κ [cal (cm deg s)] $(\times 10^{- 3})$	Specific Heat $c_{ν}$ [cal $/ ({cm}^{3} \deg)]$	Measured Sound Velocity $ν_{s}$ (cm/s) $(\times 10^{5})$	Mean Free Path $l_{c}$ (nm) ^a
Tellurite	1.38	0.852	2.40	2.0
Germanate	1.64	0.445	3.81	2.9
Phosphate	2.38	0.549	4.01	3.2
ZBLAN	1.673	0.152	2.80	1.2

Abstract

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Tables (6)

Equations (19)

Journal of the Optical Society of America B