The Temperature-Dependence of Multiphonon Relaxation of Rare-Earth Ions in Solid-State Hosts

Abstract

Rare-earth ions are used in a wide range of emissive devices – ranging from lasers to displays – where high optical efficiency and narrow-linewidth are important. While their radiative properties are important, nonradiative properties are also critical since they can reduce optical efficiency and generate heat. In this paper, theories for multiphonon relaxation rate are reviewed for rare-earth excited states in solid-state dielectric hosts. A range of various approaches are used to simplify the mathematical form of the rate equations. The ²H_9/2 excited state of Er³⁺, responsible for a technologically significant green emission, is modeled to show how the various theories manifest an order-of-magnitude variation in the thermal dependence of the multiphonon relaxation rate, as well as anomalous local minima in phonon scattering for temperatures above 0 K. This work proposes a corrective term of two quanta (Δν= + 2) of the mediating phonon energy to energy gap, so the calculated and the experimentally determined relaxation rates are equal. Radiative quantum efficiencies of both the ¹G₄→³H₅ ~1.3 μm and ³F₃→³F₂ ~7 μm of Pr³⁺ are calculated to show the importance of both proper measurement of phonon energy and application of the multiphonon relaxation rate theory.

Graphical abstract

I. INTRODUCTION

The increasing importance of advanced solid-state laser systems requires more and accurate information about the rate of nonradiative relaxation (NR) processes. Such NR processes, generally including concentration quenching, cross-relaxation and multiphonon emission, are necessary to complete the overall radiative emissions (ranging from ultraviolet to infrared) of rare-earth (RE) ‘activator’ centers.¹ In practice, the effects of these NR processes limit the achievable quantum efficiency (QE) of the luminescent system, which determines the necessary pump power and other design considerations.² For a host activated by a dilute RE dopant concentration, the total NR rate is only determined by the multiphonon emission rate, W_NR.³ The radiative QE, η, can be computed for an optical transition between an initial excited-state manifold, i, and a final manifold, j, as follows

$$\mathrm{\eta }=\frac{{\tau }_m}{{\tau }_i}=\frac{{\displaystyle \sum _j}{\mathrm{A}}_{i,j}}{{\displaystyle \sum _j}{\mathrm{A}}_{i,j}+{\mathrm{W}}_{\text{NR}}},$$ (1)

where τ_m is the experimentally measured lifetime of i excited state, and τ_i is theoretical radiative lifetime. The efficiency is the probability of the total radiative transition rate from the initial i energy level to the lower-lying j levels, ${\displaystyle \sum _j}{\mathrm{A}}_{i,j}$, normalized to the overall depopulation rate of the excited state, ${\displaystyle \sum _j}{\mathrm{A}}_{i,j}+{\mathrm{W}}_{\text{NR}}$. Maximizing η is typically achieved by minimizing W_NR. Hence, significant implications can be technologically derived from Eq. (1). Defining β as $\frac{{\mathrm{W}}_{\text{NR}}}{{\displaystyle \sum _j}{\mathrm{A}}_{i,j}}$, the η and its first derivative with respected to β are $\mathrm{\eta }(\mathrm{\beta })=\frac{1}{1+\mathrm{\beta }}$ and $\left|\frac{\mathrm{d}\mathrm{\eta }(\mathrm{\beta })}{\mathrm{d}\mathrm{\beta }}\right|=\frac{1}{{(1+\mathrm{\beta })}^2}$, respectively. Figure 1 shows that as the β value approaches zero, the uncertainty in W_NR increases. This uncertainty ultimately leads to large errors in the computation of η. Thus, to ensure proper quantitative evaluation of QE, an accurate determination of W_NR is imperative.

Figure 1.

Quantum efficiency (η, black solid line) and its derivative ( $\left|\frac{\mathrm{d}}{\mathrm{d}\mathrm{\beta }}\mathrm{\eta }(\mathrm{\beta })\right|$, red dash line) as a function of $\mathrm{\beta }(\frac{{\mathrm{W}}_{\text{NR}}}{{\displaystyle \sum _j}{\mathbf{A}}_{i,j}})$.

Theories and mathematical descriptions for nonradiative decay of RE excited states were first proposed in the 1960s on the basis of an approximated dynamic crystal field potential around RE ‘activator’ (perturbed by lattice vibration).⁴ Thermal dependence of the relaxation rate correlates to an integral number of monoenergetic phonons with the maximum vibrational energy of the host. In the early 1970s, Fong et al. presented a more complete relaxation rate theory by associating both thermal and electronic band gap dependence of the multiphonon emission with five phonon-scattering (Fong-model).^5,6 Yet the approximations made decades ago are still routinely used to quantify nonradiative relaxation because it is far simpler to do than the complex mathematical modeling required for the Fong-model.^7–11 Although there is a good correlation between theoretical and experimental energy level dependencies for nonradiative decay rates, these approximations have been known to create meaningless nonphysical trends, which will be discussed later.

In this paper, we outline the multiphonon relaxation theories and their approximations, reveal the nonphysical behavior induced from the incorrect application of simplified equation, and provide a corrective term that achieves better agreement between the relaxation rates calculated by theoretical and experimental treatments. Promoted by interest for visible to infrared light sources, the multiphonon relaxation rates of Er^{3+: 2}H_9/2, Pr^{3+: 1}G₄ and Pr^{3+: 3}F₂ excited states in lanthanum halides (LaF₃ and LaCl₃) are calculated as a function of temperature within the framework of more comprehensive and approximate approaches, respectively. LaF₃ and LaCl₃ are specified as representative hosts due to their low phonon energies (ħω_LaF3 ~350 cm⁻¹, ħω_LaCl3 ~260 cm⁻¹, as listed in Table 1)⁶ and their exceptional transparency at shorter wavelengths.

TABLE 1.

Numerical values of effective parameters for multiphonon relaxation rates of RE³⁺ (RE= Nd, Dy and Er) doped halides (LaF₃ and LaCl₃)

Transition	Lmgm2	Lm|Cαα′m|2 (ergs2)†	ħωmax (cm−1)‡	Phonon energy cutoff (cm−1)	Sum of squares
LaCl3:Nd3+	0.1509	2.1 × 10−33	259	260	3.37
LaCl3:Dy3+	0.1501	4.2 × 10−32	231	260	2.16
LaF3:Er3+	0.1400	5.3 × 10−29	353	360	0.226

^†N.B.: L_m|C_αα’^m|² is not a constant for a given host. Therefore, we will need to calculate a specific value for each electronic transition with differing ΔE.

^‡In multiphonon transitions, the maximum energy phonon mode contributes most significantly to the relaxation processes. Contributions from the lower energy modes do tend to yield effective phonon energy somewhat lower that the cut-off energy typically invoked in mediating the decay.

II. REVIEW OF MULTIPHONON RELAXATION MODELS

A. Approximations based on a single mediating phonon

The 4f electrons of RE species are relatively well-shielded by the 5s and 5p electrons such that RE ions only are weakly coupled to the crystal field.¹² Despite this lessened degree of interaction, perturbations in the ligand field caused by lattice vibrations promote the 4f electronic transitions from an excited state to one lowered by lattice mode energy.¹³ When energy differences between the adjacent levels are greater than the maximum phonon energy (ħω_max, where ħ is Planck constant, and ω_max is the maximum phonon frequency), higher order processes (i.e., multiphonon emission) are required by conservation laws. In the weak coupling limit between the RE and its host,^7,13 the interaction Hamiltonian, H, is given by expanding the static crystal field potential, V_o, in a Taylor series around the ionic equilibrium position

$$\mathrm{H}={\mathrm{V}}_0+\sum _{\mathrm{i}}{\mathrm{V}}_{\mathrm{i}}{\mathrm{Q}}_{\mathrm{i}}+\sum _{\text{ij}}{\mathrm{V}}_{\text{ij}}{\mathrm{Q}}_{\mathrm{i}}{\mathrm{Q}}_{\mathrm{j}}+\dots ,$$ (2)

where Q_i are the normal mode coordinates of the host lattice, and V_ij are derivatives of the crystal field with respect to their coordinates, (i, j).

Single phonon transition rates involve matrix elements of the modes that couple to RE through the second term of Eq. (2). Taking this single-phonon term in the expansion of a p-order perturbation calculation (p is the quantum-mechanical momentum operator) yields a very complicated function. This function sums over the matrix elements for all 2^p-1 sequences of intermediate electronic states normalized to the number of single phonons with energy $\hslash \omega =\frac{\mathrm{\Delta }\mathrm{E}}{\mathrm{p}}$. This energy is what is required to bridge the energy difference between the two given electronic transition levels. Assuming that each matrix element can be replaced by an average value, 〈a|V′|b〉, the multiphonon relaxation rate relative to the energy gap to be bridged, W_NR, can be written as the comparison of the decay rate for a p-phonon process with that for a p-1 phonon decay,⁷

$$W_{NR}=\frac{2\pi }{\hslash }{\left(\frac{\hslash }{2M\omega }\right)}^p{(n+1)}^p{2}^{2(p-1)}{m}^{2p}\frac{|\langle a|V^{\prime }{|b\rangle |}^{2p}}{{(\hslash \omega )}^{2(p-1)}},$$ (3)

where m is the number of oscillators coupled to the ion, n is the phonon occupation number (discussed below), M is the reduced mass of the vibrating ionic system of m coupled oscillators, a and b refer to the different intermediate electronic states, and |V′| is the average crystal field. The temperature dependence of multiphonon emission rate arises from the dependence of phonon mode on Bose-Einstein occupation number (n(T)),^7,12

$$\mathrm{n}(\mathrm{T})={\left(e^{\frac{\hslash \omega }{\text{kT}}}-1\right)}^{-1},$$ (4)

where kT is Boltzmann energy with k = 0.695 cm⁻¹/K. Therefore, for a p-order process, the (n +1)^p temperature dependence identifies the order and the energy of mediating phonon. The rate dependence on energy gap results from comparing the rate for p- with that for (p – 1)-phonon process. As the host material remains constant, that is ħω, m, and 〈a|V′|b〉 are constants, the Eqn. 3 simplifies directly to Eqn. 5 when one divides W_NR(p) by W_NR(p – 1):

$$\frac{\mathrm{W}(\mathrm{p})}{\mathrm{W}(\text{p-}1)}=4{\mathrm{m}}^2\left(\frac{\hslash }{2\mathrm{M}\omega }\right)(\mathrm{n}+1)\frac{{\left|\langle \mathrm{a}\left|{\mathrm{V}}^{\prime }\right|\mathrm{b}\rangle \right|}^2}{{(\hslash \omega )}^2},$$ (5)

Owing to the weak coupling between the RE and its crystal field, the multiphonon emission rate is $\frac{\mathrm{W}(\mathrm{p})}{\mathrm{W}(\mathrm{p}-1)}=\epsilon$, where ε is the electron-phonon coupling constant, a value much less than 1.⁴ Resultantly the rate for a p^th-order process is approximated by⁴

$${\mathrm{W}}^{(\mathrm{p})}\approx \mathrm{A}{\mathrm{\epsilon }}^{\mathrm{p}}=Aexp\left[\frac{ln(\mathrm{\epsilon })}{\hslash \omega }\right]\mathrm{\Delta }E,$$ (6)

where A is a constant. On the basis of the initial assumption that the lowest-order process is responsible for the relaxation, this order is approximately determined by the minimum number of highest energy host phonons required to bridge the energy gap, $\mathrm{p}\approx \frac{\mathrm{\Delta }\mathrm{E}}{\hslash {\omega }_{max}}$. Making use of the above equations and defining ε in terms of $\mathrm{\alpha }=-\frac{ln(\mathrm{\epsilon })}{\hslash \omega }$, the multiphonon emission rate becomes temperature-independent⁴

$$W=C{e}^{\alpha \mathrm{\Delta }\mathrm{E}},$$ (7)

where C and α (α< 0) are constant characteristic of the RE host lattice. These results were discussed in far greater detail by Riseberg and Weber¹⁴ and Reisfeld.¹⁵ With increasing temperature, the phonon emission will be stimulated by thermal phonons and the relaxation rate, W, increases according to⁴

$$\mathrm{W}(\mathrm{T})=\mathrm{W}(0){\left[1+\overline{\mathrm{n}}(\mathrm{T})\right]}^{\mathrm{p}},$$ (8)

where n̄(T) is the average occupation number of maximum phonon energy ħω_max, which follows a Bose distribution

$$\overline{n}(\hslash {\omega }_{\mathit{max}})=\frac{1}{e^{\frac{(\hslash {\omega }_{\mathit{max}})}{kT}}-1},$$ (9)

By combining the temperature independent Eq. (7) and the temperature dependence approximated by Eqs. (8) and (9), the simplified multiphonon relaxation rate is given by¹⁶

$${\mathrm{W}}_{\text{NR}}={\text{Be}}^{\mathrm{\alpha }\mathrm{\Delta }\mathrm{E}}{\left\{1-{\mathrm{e}}^{-\frac{\hslash {\omega }_{max}}{\text{kT}}}\right\}}^{-\frac{\mathrm{\Delta }\mathrm{E}}{\hslash {\omega }_{max}}},$$ (10)

The above derivation of Eq. (10) is called as the Reisfeld-model. The relative NR parameters like B and α are given for a variety of crystals are summarized in Table 2.

TABLE 2.

Comparison of nonradiative relaxation parameters for various crystals

Hosts	B (sec−1)	|α(cm)|	ħω max (cm−1)	ΔE (cm−1)	WNR (sec−1)†
SiO2	1.4 × 1012	3.8 × 10−3	1100	~2900	2.32 × 107 ‡
YAlO3	6.425 × 109	4.69 × 10−3	600 cm−1	2231	53.8 × 103
Y2O3	1.204 × 108	3.53 × 10−3	600 cm−1	~ 2230	56.9 × 103
Y3Al5O12	2.235 × 108	3.50 × 10−3	700 cm−1	2227	34.4 × 103
LaF3	3.966 × 109	6.45 × 10−3	350 cm−1	2545	1.32 × 103
LaCl3	3.008 × 1010	1.37 × 10−2	260 cm−1	2929	5.11 × 10−6
LaBr3	1.03 × 1010	1.88 × 10−2	175 cm−1	2960	1.00 × 10−10

^†NR rates calculated at room temperature for the ¹G₄ → ³H₅ transition of Pr³⁺ in a variety of hosts. ΔE corresponds to the energy difference between the lower ¹G₄ and upper ³F₄ Stark levels for each host and was calculated from data tabulated by Kaminskii.³⁵ The multiphonon B and α coefficients are from Reisfeld and Jørgensen¹⁶ with the exception of LaBr₃ being from Wright, et al.¹⁷

^‡Multiphonon relaxation dominates over the radiative de-excitation such that no 1.3 μm (¹G₄ → ³H₅) luminescence is observed in SiO₂.

B. Modified approximation for LaF₃:Er³⁺ by Riseberg et al

In practice, Eq. (10) can be taken a further step to include contributions from the Stark levels (i.e., the inner structure or multiplicities).⁴ In these cases, Eq. (10) is modified to include a term that accounts for the number and relative energy difference between sublevels with respect to the thermal distribution. For the multiphonon transition from the ²H_9/2 to ⁴F_3/2 of Er³⁺ in LaF₃, Riseberg et al. modeled the temperature dependence of multiphonon relaxation rate (sec⁻¹) of Er^{3+: 2}H_9/2 → ⁴F_3/2. In this case, thermal depopulation of higher-lying Stark levels accounted for the assumption of three nondecaying energy levels 200 cm⁻¹ above the lowest energy level.⁴ This is expressed as follows:

$${\mathrm{W}}_{\text{NR}}=1.1\times {10}^4{\left\{1-e^{-\frac{310}{\text{kT}}}\right\}}^{-6}{\left\{1+3e^{-\frac{200}{\text{kT}}}\right\}}^{-1},$$ (11)

where the 1.1 × 10⁴ is the temperature-independent prefactor that includes contributions from the appropriate C and α constants in Eq. (7) for LaF₃ host material and the energy difference (~1860 cm⁻¹) between the ²H_9/2 and ⁴F_3/2 states of Er³⁺.

The great utility of Eq. (10) and the modified version of Eq. (11) are that they both permit the determination of multiphonon relaxation rates through quantities that are fairly easy to measure. First, the energy gap, ΔE, can be measured by either absorption spectroscopy or found in the literature for the more common host materials (i.e., see Ref. 18). Second, the phonon energy, ħω, can be measured by either Raman or phonon sideband spectroscopy.¹⁹ Finally, the material constants, C and α can be quantified by curve-fitting the nonradiative decay rate from the high-lying energy states with differing ΔE to adjacent states of RE ions. The relative ease of these measurements renders Eq. (10) and Eq. (11) as the preferred approaches to determine W_NR.

C. Adiabatic approximation derived by Fong et al

In the early 1970s, Fong et al.^5,6 derived a more complete theory of nonradiative relaxation as a function of temperature and energy level difference without many approximations (the Fong-model). The investigation into the multiphonon relaxation rate of RE-doped solids made use of the Born-Oppenheimer (BO) adiabatic approximation, dealing with the separation of a system’s wavefunction into ‘faster’ electronic processes and ‘slower’ nuclear ones.^20–23 Accordingly, the kinetic energy operator for the lattice can be regarded as a small perturbation on the electronic wavefunction. More specifically, electronic energies, ε_α, of the zeroth-order BO eigenstates, |α〉, are written parametrically in terms of the normal mode coordinates of lattice phonons. In terms of the phonon creation and annihilation operators, the electron-lattice Hamiltonian can be expressed as⁵

$$\mathrm{H}=-\frac{{\hslash }^2}{2\mathrm{m}}\sum _i\frac{{\partial }^2}{\partial {{\mathrm{r}}_i}^2}+\mathrm{V}(\mathbf{r},\mathbf{R})-\frac{{\hslash }^2}{2}\sum _j\frac{{\partial }^2}{{\mathrm{M}}_j\partial {\mathrm{R}}_j^2},$$ (12)

where M_j is the jth normal mode of effective mass, and r and R are the coordinates of electrons and lattice phonons, respectively, and V(r, R) includes all electrostatic interactions (e.g., electron-ion and ion-ion). The Schrödinger equation then takes the form,

$$\mathrm{H}|\mathrm{\alpha }(\mathbf{r},\mathbf{R})\rangle ={\mathrm{\epsilon }}_{\mathrm{\alpha }}(\mathbf{R})|\mathrm{\alpha }(\mathbf{r},\mathbf{R})\rangle ,$$ (13)

At the zeroth order, the problem would simplify to solving the Schrödinger equation, where eigenvalues are expanded in power series of the ionic displacements from equilibrium. This is due to the dynamic nature of the lattice, a similar treatment to that given in Eq. (2) with respect to static crystal field potentials. Using the weak coupling limit, a saddle-point approximation enabled Fong et al. to show that time integration of the correlation function leads to a relaxation rate in which the energy gap is effectively modified by the addition or subtraction of Δν quanta of the mediating phonon mode via five contributing scattering processes.^13,24,25 The radiationless decay rate of electronically excited ions in a crystalline lattice is given by⁶

$$\begin{array}{c}{\mathrm{W}}_{\mathrm{\alpha }{\mathrm{\alpha }}^{\prime }}=\frac{{(2\mathrm{\pi })}^{1/2}}{{\hslash }^2}{\mathrm{L}}_{\mathrm{m}}{\left|{{\mathrm{C}}_{\mathrm{\alpha }{\mathrm{\alpha }}^{\prime }}}^{\mathrm{m}}\right|}^2{{\mathrm{\omega }}_{\mathrm{m}}}^{-1}\sum _{\mathrm{\Delta }\nu =-2}^{\mathrm{\Delta }\nu =+2}{\mathrm{\lambda }}_{\mathrm{\Delta }\nu }{(\mathrm{p}-\mathrm{\Delta }\nu )}^{-1/2}\times exp\left[-\frac{1}{4}{\mathrm{L}}_{\mathrm{m}}{{\mathrm{g}}_{\mathrm{m}}}^2(2{\mathrm{n}}_{\mathrm{m}}+1)\right]\\ \times exp\left\{-(\mathrm{p}-\mathrm{\Delta }\nu )\left[ln\left(\frac{4(\mathrm{p}-\mathrm{\Delta }\nu )}{{\mathrm{L}}_{\mathrm{m}}{{\mathrm{g}}_{\mathrm{m}}}^2({\mathrm{n}}_{\mathrm{m}}+1)}\right)-1\right]\right\}\end{array},$$ (14)

where $\mathrm{p}=\frac{\mathrm{\Delta }\mathrm{E}}{\hslash \omega }=\frac{({\mathrm{\epsilon }}_{\mathrm{\alpha }}-{\mathrm{\epsilon }}_{{\mathrm{\alpha }}^{\prime }})}{\hslash \omega }$, L_m is an effective degeneracy factor, n_m is the phonon occupation number for the effective L_m-fold degenerate mode m mediating the relaxation, g_m² is a (dimensionless) displacement factor related to phonon creation and annihilation operators, |C_αα′|² is the electron-phonon coupling constant and the λ_{Δν=−2,…,+2} coefficients are defined as:

$$\begin{array}{l}{\mathrm{\lambda }}_{-2}=\frac{1}{4}{\mathrm{L}}_{\mathrm{m}}{{\mathrm{g}}_{\mathrm{m}}}^2{{\mathrm{n}}_{\mathrm{m}}}^2\hfill \\ {\mathrm{\lambda }}_{-1}=-\frac{1}{2}{\mathrm{L}}_{\mathrm{m}}{{\mathrm{g}}_{\mathrm{m}}}^2(2{\mathrm{n}}_{\mathrm{m}}+1){\mathrm{n}}_{\mathrm{m}}+{\mathrm{n}}_{\mathrm{m}}\hfill \\ {\mathrm{\lambda }}_0=\frac{1}{4}{\mathrm{L}}_{\mathrm{m}}{{\mathrm{g}}_{\mathrm{m}}}^2[6{\mathrm{n}}_{\mathrm{m}}({\mathrm{n}}_{\mathrm{m}}+1)+1]\hfill \\ {\mathrm{\lambda }}_{+1}=-\frac{1}{2}{\mathrm{L}}_{\mathrm{m}}{{\mathrm{g}}_{\mathrm{m}}}^2(2{\mathrm{n}}_{\mathrm{m}}+1)({\mathrm{n}}_{\mathrm{m}}+1)+{\mathrm{n}}_{\mathrm{m}}+1\hfill \\ {\mathrm{\lambda }}_{+2}=\frac{1}{4}{\mathrm{L}}_{\mathrm{m}}{{\mathrm{g}}_{\mathrm{m}}}^2{({\mathrm{n}}_{\mathrm{m}}+1)}^2\hfill \end{array},$$ (15)

All of the above terms are determined through least-squares fitting of experimental rates as function of temperature and energy gap difference. Furthermore, by assuming only one mode dominates the RE ⇔ lattice coupling (i.e., weak-coupling limit),^24–26 the relaxation rate comprises five contributions that correspond to the scattering-related addition or subtraction of Δν quanta of the mediating phonon energy to the transitional energy gap, ε_α −ε_α′.

$${\mathrm{\epsilon }}_{\mathrm{\alpha }}-{\mathrm{\epsilon }}_{{\mathrm{\alpha }}^{\prime }}-\mathrm{\Delta }\nu \hslash {\omega }_m>{\mathrm{L}}_{\mathrm{m}}{{\mathrm{g}}_{\mathrm{m}}}^2\hslash {\omega }_{\mathrm{m}},$$ (16)

As a consequence, determination of the relaxation rate would follow from time integration²⁵ in the weak-coupling limit by the saddle-point approximation.²⁴ Table 1 lists the corresponding parameters for LaF₃ and LaCl₃ host materials.^5,6

III. QUANTITATIVE COMPARISON BETWEEN MODELS, UNIFICATION AND RESULTS

A. Inaccuracy induced by rough employment of ${\mathrm{W}}_{\text{NR}}^{\text{Fong}}(\mathrm{T})$ and ${\mathrm{W}}_{\text{NR}}^{\text{Reisfeld}}(\mathrm{\Delta }\mathrm{E})$

In practice, it is obvious that inaccuracy would be induced by simply employing the different models of W_NR(T) and W_NR(ΔE), respectively. In the low-temperature limit (kT≪ħω_max), Eq. (14) can be reduced to

$${\mathrm{W}}_{\text{NR}}^{\text{Fong}}(\mathrm{T})={\mathrm{W}}_{\text{NR}}(\mathrm{T}=0)\left[1+{\text{pe}}^{\frac{-h{\omega }_{max}}{\text{kT}}}\right],$$ (17)

which is functionally similar to that of Eq. (10), ${\mathrm{W}}_{\text{NR}}^{\text{Reisfeld}}(\mathrm{\Delta }\mathrm{E})$. However, at higher temperatures when kT~ħω_max, the computed result of ${\mathrm{W}}_{\text{NR}}^{\text{Fong}}(\mathrm{T})$ and that of ${\mathrm{W}}_{\text{NR}}^{\text{Reisfeld}}(\mathrm{\Delta }\mathrm{E})$ would differ significantly from each other. For instance, for LaF₃ host lattice with ħω_max =350 cm⁻¹, the ratio of Eq. (10) to Eq. (16) at T=100 K is essentially unity for equal W_NR(T=0), while, at T=300 K, the ratio becomes > 2, invalidating the approximation. Thus, great calculation must be taken when using these approximations.

B. Inconsistent multiphonon relaxation rates and non-physical trends

In order to clearly illustrate the possible inaccuracies, the temperature dependence of relaxation rate, W(T), for a given ²H_9/2 → ⁴F_3/2 transition of Er³⁺ is comparatively calculated through the three models of Eqs. 10, 11 and 14. In practice, Fong’s theory for a multiphonon rate, $\mathrm{W}{(\mathrm{T})}_{\text{NR}}^{\text{Fong}}$, can be calculated from

$$\mathrm{W}{(\mathrm{T})}_{\text{NR}}^{\text{Fong}}={\mathrm{\Gamma }}_{-2}+{\mathrm{\Gamma }}_{-1}+{\mathrm{\Gamma }}_0+{\mathrm{\Gamma }}_{+1}+{\mathrm{\Gamma }}_{+2},$$ (18)

where Γ_n(n= −2, −1, 0, +1, +2) is the individual contribution of each W_αα′ (λ_Δν) effective phonon. All the parameters for nonradiative relaxation are provided in Tables 1 and 2. Figure 2 shows the multiphonon relaxation rate from the ²H_9/2 exited state to the following ⁴F_3/2 state of Er³⁺ in LaF₃. At room temperature, there is over an order-of-magnitude difference between curves because the W(T)_NR values are estimated by curve-fitting experimental data using the more approximate model, ${\mathrm{W}}_{\text{NR}}^{\text{Riseberg}}$, and the most rigorous model, ${\mathrm{W}}_{\text{NR}}^{\text{Fong}}$. Accordingly, this discrepancy greatly influences the computation of η, the radiative quantum efficiency. This is especially pronounced in the case of multiphonon relaxation with a relatively large energy gap, such as for the 2545 cm⁻¹ emission from the ¹G₄ level to the ³F₄ level for Pr³⁺ in LaF₃. At room temperature, a difference of 400% in the radiative QE of the ¹G₄ → ³H₅ transition at about 1.3 μm can be obtained by using ${\mathrm{W}}_{\text{NR}}^{\text{Riseberg}}$ and ${\mathrm{W}}_{\text{NR}}^{\text{Fong}}$ calculation modes,²⁷ respectively.

Figure 2.

Multiphonon emission rate (W_NR, sec⁻¹) as a function of temperature (K) for Er^{3+: 2}H_9/2 → ⁴F_3/2 transition in LaF₃, which is calculated from the models of Reisfeld [Eq. (10)], Riseberg [Eq. (11)], and Fong [Eq. (14)], respectively.

Further inspection of the Riseberg-model [Eq. (11)] reveals another serious problem, namely the anomalous appearance of local minima. The problem is the anomalous appearance of local minima in the multiphonon emission rate at T > 0 K (Figure 3). The local minima have the physical basis to exist. The anomalous appearance solely results from the functional form of the mathematical simulation through the selection of fitting parameters, and lead to large errors. The problem is that it causes large errors in subsequently calculating the radiative QE. For instance, LaCl₃:Pr³⁺ crystal is of great commercial interest for room temperature solid-state infrared sources and sensors.^28–30 If the radiative properties of Pr^{3+: 3}F₃ → ³F₂ at 7 μm are predicted by simply applying the Riseberg-model, the radiative QE of Pr³⁺ ~7 μm calculated at 77 K (liquid nitrogen cooling) increases by a factor of 5 relative to that evaluated at temperature cooled from 300 to 150 K or below. In principle, extrapolation indicates that, for Pr^{3+: 3}F₃ → ³F₂ in LaCl₃, the point at 0 K features the minimum multiphonon relaxation rate of 1.1 × 10⁴ sec⁻¹, while the fitting of the Riseberg-model yields a minima of 1.046 × 10⁴ sec⁻¹ in the 77–150 K range.

Figure 3.

Temperature (K) dependence of multiphonon emission rate (W_NR, sec⁻¹) for Er^{3+: 2}H_9/2 → ⁴F_3/2 transition in LaF₃ host calculated by the Riseberg-model.

C. Unification of various approaches

Given the inaccuracy and nonphysical behavior of the three models, it is worth investigating if a unification of the various approaches can resolve these problems. Equation (8) represents a relaxation rate comprising penta-phononic scattering, as opposed to the mono-phononic case of Eq. (6). The polyphonic scattering yields the greatest contribution to the overall NR process through the terms related to the addition of vibrational energy to the electronically excited ion. Since the λ_Δν = +2 term provides the largest perturbation to the mono-phononic mechanism in the Fong’s approach,⁶ it is reasonable to suggest adding the Γ₊₂ to Eq. (10) as following

$$\begin{array}{l}{W}_{NR}=B{e}^{\alpha \mathrm{\Delta }E}{\left\{1-e^{-\frac{\hslash {\omega }_{max}}{kT}}\right\}}^{-p}+\frac{{(2\pi )}^{1/2}}{-{\hslash }^2}{L}_m{\mid C_m\mid }^2\frac{{\lambda }_{+2}}{{\omega }_{\mathrm{m}}{(p-2)}^{1/2}}\times exp\left[-\frac{1}{4}{L}_m{g_m}^2(2n_m+1)\right]\hfill \\ \times exp\left\{-(p-2)\left[ln\left(\frac{4(p-2)}{L_m{g_{\mathrm{m}}}^2(n_m+1)}\right)-1\right]\right\}\hfill \end{array},$$ (19)

All the terms in Eq. (19) are defined in the abovementioned sections, as listed in Table 1 and Table 2. As shown in Figure. 4(a), the large deviation between the results of the Reisfeld-model and that of the Fong-model can be eliminated by adding the Fong-model Γ₊₂ term to the Reisfeld-model (Figure 4(b)). Noted that the values listed in Table 1 for ${\mathrm{L}}_{\mathrm{m}}{\mathrm{g}}_{\mathrm{m}}^2$ and L_m|C_m|² are specifically for the ²H_9/2 state of Er³⁺ in LaF₃ because the multiphonon transition between RE electronic energy levels are just coupled to the different phonon modes.^5,6 Extension to other RE transitions requires the specific parameters either previously tabulated or calculated using the outlined methodology. For clarity, the proposed addition to the Reisfeld-model of multiphonon relaxation rates for LaF₃:Er³⁺ and LaCl₃:Er³⁺ are summarized below for the Reisfeld-model to Fong-model transition,⁵

Figure 4.

Dependence of multiphonon emission rate (W_NR, sec⁻¹) on temperature (K) in terms of (a): the Reisfeld-model of energy gap law (red dash line) and the Fong-model of penta-phonon theory (black solid line), and (b): the Fong-model (black solid line) and the unified model of energy gap law with the addition of Δν = +2 quanta (green dot line).

$$\begin{array}{c}F{(T)}_{{\mathrm{\Gamma }}_{+2}}=\left[\frac{1}{4}{L}_g{(n(T)+1)}^2\right]{(p(E)-2)}^{\frac{1}{2}}·exp\left[-\frac{1}{4}{L}_g{(2n(T)+1)}^2\right].\\ exp\left[-(p(E)-2).\left[ln\left(\frac{4.(p(E)-1)}{L_g.(n(T)+1)}\right)-1\right]\right]\end{array},$$ (20)

For example, LaF₃ host material: L_c = 7.356 × 10³; $\frac{\sqrt{\mathrm{2..}\pi }}{\omega .h^2}{L}_c=1.12\times {10}^{13}$; L_g = 0.1444; p(E) = ΔE/310, and LaCl₃ host material: L_c = 0.057; $\frac{\sqrt{\mathrm{2..}\pi }}{\omega .h^2}{L}_c=1.006\times {10}^9$; L_g = 0.1537: p(E) = ΔE /238.

D. Influence of selected transition energy

Although the specific focus of paper is on the discrepancies among the values of W_NR(T) obtained from several models, there still exists potential ambiguity in the definition of energy difference, ΔE, between the adjacent excited states of RE ions, ultimately effecting the QE calculation. Further work should be done before the applying extension of Fong’s parameters to different ΔE regimes. The Reisfeld-model [Eq. (10)] shows that the different values of ΔE would lead to a wide range of W_NR(T). For Pr³⁺, the ¹G₄ → ³H₅ at about 1.3 μm because of its potential use in the next-generation optical amplifier.^28–32 The multiphonon relaxation rate of Pr^{3+: 1}G₄ state is calculated to be 112 sec⁻¹ by Eq. (10), where the phonon energy and multiphonon relaxation parameters of LaF₃ are listed in Table 2. ΔE of ¹G₄-³F₄ of Pr³⁺ is conventionally determined by absorption spectroscopy and found to be 2966 cm⁻¹ in terms of the manifold centroid value (e.g., the energetic average of Stark sublevels).³³ Correspondingly, the radiative QE is estimated as

$${\eta }_{{\text{LaF}}_3:{Pr}^{3+}}({{}^1\mathrm{G}}_4\to {{}^3\mathrm{H}}_5)\cong \frac{321}{321+112}\cong 74\%,$$ (21)

where 321 sec⁻¹ is the spontaneous emission probability for the ¹G₄ level of Pr³⁺ in LaF₃ as given by Weber.³⁴ However, a QE of 74% is approximately 500% larger than the value determined experimentally,^29–31 or that expected based on the relatively small ΔE ratio. The correct definition of ΔE should take the specific intermanifold structure into account because the minimum energy difference would originate from the lowest Stark component of the excited state and the upper Stark component of the adjacent lowest lying energy level. Recalculation of the ¹G₄-³F₄ ΔE in this manner for LaF₃ yields a value of ~2545 cm^−1.27,35 Accordingly, the multiphonon relaxation rate of Pr^{3+: 1}G₄ state is calculated to be 1323 sec⁻¹ by Eq. (10), and then the QE is about 20%, which is much closer to those measured on LaF₃: Pr³⁺ materials prepared by various methods.^31,27,35

For LaCl₃, the multiphonon relaxation rate of Pr^{3+: 1}G₄ state is calculated to be 5.11 × 10⁻⁶ sec⁻¹ by Eq. (10) and the theoretical QE for Pr^{3+: 1}G₄ → ³H₅ ~1.3 μm is about 100%, where the spontaneous emission probability for the ¹G₄ level of Pr³⁺ in LaCl₃ is given to be 682.0 sec^−1.36 Table 3 summarizes the calculated values for the Pr³⁺ ~ 1.3 μm emission for the lanthanum halides.^34,36 The negligibly small contributions to the ${\displaystyle \sum _j}{\mathrm{A}}_{i,j}$term in the denominator of Eq. (1) for η allows its reduction to $\eta =\frac{{\displaystyle \sum _j}{\mathrm{A}}_{i,j}}{{\displaystyle \sum _j}{\mathrm{A}}_{i,j}}=1$. Obviously, this is the case for any transition where ${\mathrm{W}}_{\text{NR}}\ll {\displaystyle \sum _j}{\mathrm{A}}_{i,j}$. It can be speculated that, with very low phonon energy, the great utility of these halides (chlorides) is the lack of phonon influence on the electronic transitions of RE³⁺ ions that do not possess large radiative emission probabilities in other hosts, such as oxides.^3,28–31 In addition, halides are transparent over a much broader range than other infrared materials like the chalcogenides, which makes them more appropriate materials for active devices (e.g., UV, blue and green upconversion laser sources).

TABLE 3.

Spontaneous emission probabilities, multiphonon relaxation rates and theoretical QE of Pr^{3+: 1}G₄ state in the lanthanum halides

Host	Spontaneous emission rate (sec−1)	Multiphonon relaxation rate (sec−1)	Theoretical QE (%)
LaF3	321.15	1.323 × 103	20
LaCl3	682.03	5.11 × 10−6	100
LaBr3	---	1.00 × 10−10	100

IV. CONCLUSIONS

A review of the multiphonon relaxation processes of RE excited states has been presented along with several approximation models, which are customarily applied to simplify its mathematical description so as to permit the evaluation of characteristic parameters with relative experimental ease. Temperature dependence of multiphonon relaxation rates for the ²H_9/2 state of Er³⁺-doped LaF₃ were employed to show the significant difference by means of the typical Reisfeld-, Riseberg-, and Fong-model, respectively. Comparison of the three models gives insight to how the approximation leads to inaccuracies and the nonphysical existence of local minima in the magnitude of phonon scattering for temperature above absolute zero. A unified model with a corrective term was empirically derived to significantly improve the agreement between relaxation rates predicted using the various approximations by the measured and tabulated spectral absorption data. In addition, QE calculations can be corrected by defining the minimum energy difference of RE ions in the Stark manifold.