Competing radiative and nonradiative decay of embedded ions states in dielectric crystals: theory, and application to Co2+:AgCl0.5Br0.5

A. Shirakov; Z. Burshtein; A. Katzir; E. Frumker; A. A. Ishaaya

doi:10.1364/OE.26.011694

1. Introduction

A very large part of laser science and technology involves the study of solid-state lasers. Generally, solid-state laser gain materials are composed of a host material doped with active metallic ions. Laser action stems from stimulated transitions between electronic states of the active ions. The host material holds the ions in their respective positions. However, the host itself, via interaction with the active ions, has an important impact on the lasing process. This includes gain shifting, splitting, broadening, and enabling of heat dissipation; in some cases also assisting in draining the lower laser state to the ground state. A considerable drawback is the host effect in shortening the effective excited state lifetime via non-radiative multi-phonon transitions. One expects these processes to turn more significant at longer transition wavelengths.

Recently, there was a significant resurgence of interest in transition metal ions for ultrafast lasers in the near and mid-IR [1, 2]. Thus, understanding the effect of host matrix on the active lasing ions in general, and its effective lifetime in particular, is of fundamental significance and of utmost importance for laser physics.

In this respect, a large number of studies, both theoretical and experimental, have been devoted to the non-radiative, multi-phonon transition subject. It has been rightfully assumed that low energy transitions are dominantly non-radiative. Many theoretical studies formulated the problem in “first principles” terms, yet the expressions obtained are cumbersome and agreement with experiment is rather poor (see [3,4] and the references cited therein). Researchers often resorted to gross simplifications and assumptions, and use of simplistic empiric expressions, to enable comparison with actual physical systems. In most treatments [5–9], empiric fit parameters assigned to the host material effects exhibit no specific relation with the matrix fundamental physical properties. In the limit of high energy transitions (e.g. in the short wavelength of the visible spectrum) it is reasonable to assume that the host vibrational effects are limited only to state broadening. However, hitherto the adequate theoretical foundation in the intermediate energy transition range (the mid- and far-infrared) is yet lacking. This important energy range still awaits a comprehensive treatment.

Generally, radiative and non-radiative transitions are simultaneously operative. In other words, any atom or ion in an excited state may “select” to decay by either of the above mechanisms according to its relative probability. In Burshtein’s paper [10], it has been pointed out that radiative and non-radiative transitions are both proportional to the transitions oscillator strength f. Therefore, for the same ion and matrix, states exhibiting a faster radiative transition, simultaneously exhibit a faster non-radiative one. Furthermore, a specific expression for the non-radiative multi-phonon transition rate has been developed [10]. It relates directly to the fundamental host material properties, specifically: mass density, unit-cell dimensions, vibrational cutoff frequency, and configuration potential dissociation energy.

In the present paper, we start by modeling a simple two level transition system. Next we generalize the treatment to more realistic scenarios, where the lower and/or upper levels consist of sub-level multiplets. We then apply the modeling to recently published experimental results on Co²⁺ ions in a mixed AgCl_0.5Br_0.5 crystal [11], and obtain an excellent fit over the entire temperature range. Consequently, we obtain a prediction for the radiative lifetime at zero absolute temperature, as well as the lattice cutoff vibrational frequency. We validate the latter experimentally by optical reflection measurements in the far-IR.

2. Theoretical background

2.1. A two-state system

In Fig. 1 we show a schematic view of a two-state system with an occupied excited-state |2〉, undergoing simultaneously competing radiative and a multi-phonon non-radiative decay to the lower |1〉 state. For a homogeneously broadened emission, the zero-temperature T = 0 limit of the radiative decay rate is given by [10, 12]:

τ_{r a d}^{- 1} (0) = \frac{8 π n^{2} ν_{0}^{2}}{c^{2}} \int_{0}^{\infty} σ_{e m} (ν) d ν,

where n is the ion host matrix refractive index, ν₀ is the central radiative transition frequency, c is the vacuum speed of light constant, and σ_em is the frequency-dependent radiative transition cross section of the emission band (the fluorescence band). In the following we denote E_g = hν₀ = E₂ − E₁, to be termed “the energy gap”. In principle, the radiative decay rate is temperature dependent, stimulated by thermal photons. We shall address this issue specifically later on, to see that under a broad range of physical conditions of interest, this contribution may be safely ignored.

Fig. 1 Energy scheme describing simultanzand spontaneous emission of several phonons (nonradiative multi-phonon), indicated by wiggled arrows.

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The transition oscillator strength is given by [10]:

f = \frac{3 m c}{π n e^{2}} \int_{0}^{\infty} σ_{e m} (ν) d ν,

where m is the electronic mass and e is the electron charge. One then gets

f = \frac{3 m c}{8 π^{2} e^{2} n^{3} ν_{0}^{2}} τ_{r e d}^{- 1} (0) .

Following [10], the zero-temperature multi-phonon non-radiative transition rate is given by

τ_{n r}^{- 1} (0) = fB (υ) \frac{N_{c} - 1}{N_{c}^{2}} \cdot \frac{4 π^{2} ρ_{M} a^{3} D}{3 ℏ m} {(\frac{ℏ ω_{c o}}{D})}^{E_{g} / ℏ ω_{c o}},

where ω_co is the cut-off angular frequency of the matrix vibrations (phonons), D is the characteristic dissociation energy of a Morse-type configuration potential [13] (related to the primitive unit cell vibrating at ω_co), a³ is the primitive unit cell volume, ρ_M is the host material mass density, N_c is the number of atoms occupying the unit cell (correspondingly forming the ligand for each embedded ion). We now replace the expression

ρ_{M} a^{3} (N_{c} - 1) / N_{c}^{2}

by

{\bar{M}}_{a t} (N_{c} - 1) / N_{c}

, where

{\bar{M}}_{a t}

is the average unit cell atomic mass. For a unit cell containing a large number of atoms, this expression simply reduces to

{\bar{M}}_{a t}

. The ratio υ ≡ E_g/ħω_co represents the number of ħω_co phonons involved in the non-radiative transition. The B(υ) function specified in Eq. (5) below is a υ-dependent numerical factor. Its presence in Eq. (4) stems from expansion to a Taylor series of the assumed lattice configuration potential, as well as from other model assumptions for the υ-order multi-phonon induced transitions [10]. It is shown graphically in Fig. 2 between υ = 1 and υ = 16:

B (υ) = \frac{{[2^{υ} - 1]}^{2} υ^{υ - 2}}{{[Γ (υ + 1)]}^{2}} .

Fig. 2 Variation of the $B (υ) = \frac{{[2^{υ} - 1]}^{2} υ^{υ - 2}}{{[Γ (υ + 1)]}^{2}}$ function of Eq. (5) between υ = 1 and υ = 16.

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Note that υ is not necessarily an integer. It is a manifestation of the physical fact that lattice vibrations exhibit broad bands, practically covering the entire phonon energy scale up to ħω_co. The ratio D/ħω_co may be considered a measure of the deviation from harmonicity of the lattice vibrations, which allows phonon involvement in transitions between electronic states. In a perfectly harmonic potential, the D/ħω_co ratio is infinite; mathematically, in that limit the non-radiative transition rate diminishes (Eq. (4)).

Setting Eq.’s (3) expression for f in (4), one gets

τ_{n r}^{- 1} (0) = τ_{r a d}^{- 1} (0) \frac{ℏ c}{e^{2}} \cdot \frac{N_{c} - 1}{N_{c}} \cdot \frac{2 π^{2}}{n^{3}} \frac{[{\bar{M}}_{a t} c^{2}] D}{E_{g}^{2}} B (υ) {(\frac{ℏ ω_{c o}}{D})}^{υ} .

Hence, the combined upper state decay rate at the zero temperature limit is given by:

\begin{array}{l} τ_{e f f}^{- 1} (0) = τ_{r a d}^{- 1} (0) + τ_{n r}^{- 1} (0) \\ = τ_{r a d}^{- 1} (0) [1 + \frac{ℏ c}{e^{2}} \cdot \frac{N_{c} - 1}{N_{c}} \cdot \frac{2 π^{2}}{n^{3}} \frac{[{\bar{M}}_{a t} c^{2}] D}{E_{g}^{2}} B (υ) {(\frac{ℏ ω_{c o}}{D})}^{υ}] \\ \equiv τ_{r a d}^{- 1} (0) [1 + C_{n r} (E_{g}, ℏ ω_{c o}, D)], \end{array}

where

C_{n r} (E_{g}, ℏ ω_{c o}, D) = τ_{n r}^{- 1} (0) = \frac{ℏ c}{e^{2}} \cdot \frac{N_{c} - 1}{N_{c}} \cdot \frac{2 π^{2}}{n^{3}} \frac{[{\bar{M}}_{a t} c^{2}] D}{E_{g}^{2}} B (υ) {(\frac{ℏ ω_{c o}}{D})}^{υ},

is the non-radiative contribution constant to the decay rate between the states. Note, that in the T → 0 limit, τ_rad(0)/τ_eff (0) = 1 + C_nr(E_g, ħω_co, D).

Notably, the non-radiative constant C_nr employs parameters related to the specific host material properties, and the actual transition in the ion. Particularly, it is inversely proportional to the squared energy gap $E_{g}^{2}$ . Naturally ħω_co ≪ D; then C_nr reduces steeply as function of the number of phonons required for non-radiative decay (last factor in Eq. (8)).

For a finite absolute temperature T, the effective fluorescence decay rate becomes

τ_{e f f}^{- 1} (T) = τ_{r a d}^{- 1} (0) [1 + \frac{1}{\exp (E_{g} / k_{B} T) - 1} + C_{n r} {(1 + \frac{1}{\exp (ℏ ω_{c o} / k_{B} T) - 1})}^{υ}] .

The [exp (E_g/k_BT) − 1]⁻¹ addend inside the square brackets accounts for the thermal enhanced population effect of E_g-energy photons. The factor ${[1 + \frac{1}{\exp (ℏ ω_{c o} / k_{B} T) - 1}]}^{υ}$ (recall that υ = E_g/ħω_co), accounts for the thermal enhanced population effect of ħω_co-energy phonons by the υ-phonon transition mechanism; see [14] equation (7) and [10] equation (70), and related text. For not too far-infrared energies, say E_g ≳ 2, 000 cm⁻¹, and not too high temperatures, say T ≲ 900 K, the thermal photon population effect is negligibly small (in that particular limit it amounts to only ∼ 0.035 ≪ 1). For simplicity and brevity, we hereby ignore it. Equation (9) then reduces to

τ_{e f f}^{- 1} (T) = τ_{r a d}^{- 1} (0) [1 + C_{n r} {(1 + \frac{1}{\exp (ℏ ω_{c o} / k_{B} T) - 1})}^{υ}] .

Apparently, for the two-state system of Fig. 1 under normal conditions, the major contribution capable of shortening the effective fluorescence lifetime τ_eff upon increased temperature T is non-radiative.

2.2. A single state to a multiplet decay system

In Fig. 3 we show a schematic view of a multi-state system with an occupied single excited-state, undergoing simultaneously competing radiative, and multi-phonon non-radiative decays to a lower states multiplet. Numbering of the latter states starts from ℓ = 0 for the lowest energy one. The analysis is based on the measured low-temperature limit of the fluorescence shape. Let g_em(ν) be the fluorescence emission shape of a unity integral, which resolves into single-peak components $g_{e m}^{(ℓ)} (ν)$ (Gaussian, Lorentzian, or others) representing the different ℓ-state components, where $g_{e m} (ν) = \sum_{ℓ} g_{e m}^{(ℓ)} (ν)$ . We then denote $g^{(ℓ)} \equiv \int_{0}^{\infty} g_{e m}^{(ℓ)} (ν) d ν$ ; obviously, $\sum_{ℓ} g^{(ℓ)} = 1$ .

Fig. 3 Energy scheme describing competing radiative and multi-phonon non-radiative decay probabilities of an occupied higher electronic state to a multiplet of lower states. The partially wiggled arrows symbolize the competing nature within each inter-state decay transition. Enhanced brightness of the wiggled part symbolizes the reduced non-radiative probability for larger energy gaps. For more details, see text.

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The effective fluorescence lifetime is determined by the weighted summation on all the transition lifetimes between the upper states and all the states of the lower multiplet. Hence, the expression for the effective fluorescence lifetime becomes

τ_{e f f}^{- 1} (T) = τ_{r a d}^{- 1} (0) [1 + \sum_{ℓ} g^{(ℓ)} C_{n r}^{ℓ} {(1 + \frac{1}{\exp (ℏ ω_{c o} / k_{B} T) - 1})}^{υ (ℓ)}],

where υ⁽^ℓ⁾ ≡ E_gℓ/ħω_co, and

C_{n r}^{(ℓ)} \equiv \frac{ℏ c}{e^{2}} \cdot \frac{N_{c} - 1}{N_{c}} \cdot \frac{2 π^{2}}{n^{3}} \frac{[{\bar{M}}_{a t} c^{2}] D}{E_{g ℓ}^{2}} B (υ^{(ℓ)}) {(\frac{ℏ ω_{c o}}{D})}^{υ (ℓ)},

Note, that in the T → 0 limit, $τ_{r a d} (0) / τ_{e f f} (0) = 1 + \sum_{ℓ} g^{(ℓ)} C_{n r}^{(ℓ)} (E_{g ℓ} ℏ ω_{c o}, D)$ . The fluorescence quantum efficiency would correspondingly be

η = τ_{e f f} (0) / τ_{r a d} (0) = {[1 + \sum_{ℓ} g^{(ℓ)} C_{n r}^{(ℓ)} (E_{g ℓ} ℏ ω_{c o}, D)]}^{- 1} .

The dominant non-radiative contribution to the effective decay rate is usually the largest ℓ transition (smallest energy gap). If gaps among the multiplet states are large, the effective system reduces to a two-state case, per section 2.1.

2.3. A higher to a lower multiplet decay system

In Fig. 4 we show a schematic view of a multi-state system with an occupied excited upper states multiplet, undergoing simultaneously competing radiative and multi-phonon non-radiative decays to a lower states multiplet. Numbering of the lower multiplet states starts upwards in energy from ℓ = 0 of the lowest energy one. Numbering of the upper multiplet states starts upward in energy from u = 0^∗ for the lowest energy one. The occupation statistics of the upper multiplet is assumed to follow a quasi-thermal equilibrium distribution (effected by very fast, mostly non-radiative transitions among those states). d_u indicates the degeneracy of an upper multiplet u-state, and E_u its energy relative to E₀∗. The figure describes decay transitions from both lowest and first thermally occupied state of the upper multiplet.

Fig. 4 Energy scheme describing competing radiative and multi-phonon non-radiative decay probabilities of a higher occupied electronic multiplet to a multiplet of lower states at a finite temperature. The partially wiggled arrows symbolize the competing nature within each inter-state decay transition. Enhanced grayness of a wiggled part symbolizes the reduced non-radiative probability for larger energy gaps. Enhanced brightness of a higher multiplet state symbolizes its actual lower population compared to the lower states at the quasi-thermal equilibrium. For more details, see text.

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The analysis is based on the measured low-temperature limit of the absorption cross section and fluorescence shapes spectra, as follows:

Let σ_abs(ν) be the absorption cross-section spectrum from the ground (ℓ = 0) lower multiplet state to the entire upper multiplet. The spectral shape resolves into single-peak components $σ_{a b s}^{(u)} (ν)$ (Gaussian, Lorentzian, or others), representing the different u-state components, such that $σ_{a b s} (ν) = \sum_{u} σ_{a b s}^{(u)} (ν)$ . We denote $S_{0, u} \equiv \int_{0}^{\infty} σ_{a b s}^{(u)} (ν) d ν$ ; obviously, $\sum_{u} S_{0, u} = S \equiv \int_{0}^{\infty} σ_{a b s} (ν) d ν$ . An underlying assumption in the following analysis is that the radiative decay rate from any u-state to the lower multiplet ones is proportional to the u-state “absorption strength” S_0,_u.
The lower multiplet states characteristics are identical with those of Fig. 3. The actual radiative decay rate from any u-state to the entire lower multiplet is now also temperature dependent, related to the relative thermal occupation of the u-state. Under quasi thermal equilibrium conditions, it would be given by

τ_{r a d, u}^{- 1} (T) = τ_{r a d}^{- 1} (0) \frac{S_{0, u}}{S_{0, 0, *}} \frac{d_{u}}{Z (T)} e^{- E_{u} / k_{B} T},

where Z(T)Σ_ud_u exp (−E_u/k_BT). Accordingly, a corresponding summation on all possible transition lifetimes would be given by

τ_{e f f}^{- 1} (T) = \sum_{u} τ_{r a d, u}^{- 1} (T) [1 + \sum_{ℓ} g^{(ℓ)} C_{n r, u}^{ℓ} {(1 + \frac{1}{\exp (ℏ ω_{c o} / k_{B} T) - 1})}^{υ_{u}^{(ℓ)}}],

where

C_{n r, u}^{(ℓ)} \equiv \frac{ℏ c}{e^{2}} \cdot \frac{N_{c} - 1}{N_{c}} \cdot \frac{2 π^{2}}{n^{3}} \frac{[{\bar{M}}_{a t} c^{2}] D}{{(E_{g ℓ} + E_{u})}^{2}} B (υ_{u}^{(ℓ)}) {(\frac{ℏ ω_{c o}}{D})}^{υ_{u}^{(ℓ)}},

for

υ_{u}^{(ℓ)} \equiv (E_{g ℓ} + E_{u}) / ℏ ω_{c o}

.

Notably, increased temperature also affects the fluorescence spectral shape by added emission of higher energy photons [10]. Analysis of this important spectral effect, however, is beyond the scope of our present paper.

The dominant, both radiative and non-radiative contributions to the effective decay rate is usually the smallest u transition (smallest set of energy gaps). If energy gaps among the higher multiplet states are large, the effective system reduces to a single state to a multiplet decay case, per section 2.2.

3. Experimental methods

Single crystals of Co²⁺:AgCl_0.5Br_0.5 were grown from the melt by the Bridgman-Stockbarger method [15]. Parallel plates approximately 3 mm thick and 10 mm in diameter were cut and polished to near optical quality [16]. A room temperature reflection spectrum of a 3.03 mm thick sample was measured in the frame of our present work between 50 and 600 cm⁻¹, using a Nicolet 6700, with DTGS/PE detector at a resolution of 0.5 cm⁻¹ (Fig. 9 below). The beam incidence angle was 10°. Since sample thickness was only 3.03 mm, correction for rear surface reflectance was needed, assumed as about 35% of the front surface one. The equation used for the said correction is specified in Appendix A.

Analysis of the above reflectance spectral shape results was worked out by fitting to the classical independent oscillator model, see [17] chapter IV. For completeness of presentation, the expressions used are summarized in Appendix B.

4. Fluorescence transitions in Co²⁺:AgCl_0.5Br_0.5 crystals

4.1. Background

In this section we provide a revised analysis of the photo-excited fluorescence lifetime of Co²⁺ ions embedded in the mixed silver halide crystal AgCl_0.5Br_0.5 [11]. The 4.2 µm centered fluorescence was excited by 1.92 µm wavelength, 9 ns long laser pulses. The decay lifetime was measured between 20 and 280 K. The authors attempted to match the lifetime vs. temperature dependence to a single configuration-coordinate model, assuming a strong coupling between the ion states and the ligand vibrations of the form (Eq. 1 in [11])

τ_{e f f}^{- 1} (0) = τ_{r a d}^{- 1} + W_{a}^{- 1} \exp (- E_{a c t} / k_{B} T),

where

W_{a}^{- 1}

is the high-temperature limit non-radiative decay rate, and E_act is the energy required for non-radiative activation back to the ground state. A fit was obtained only for the low temperature range between 40 and 150 K, yet failed to fit a further shortening of the lifetime at higher temperatures by over an order of magnitude till 280 K.

4.2. Revised analysis

Our revised analysis involves the single state to a multiplet decay system model of section 2.2 (Fig. 3 above) particularly, Eqs. (11), (12). The absorption and fluorescence spectra were re-plotted and resolved each into Gaussian single-peak components, with their relevant parameters inset in the figures’ frames (Figs. 5 and 6, respectively).

Fig. 5 Experimental absorption cross section spectrum of cobalt Co²⁺ in a silver chloride-bromide AgCl_0.5Br_0.5 crystal at near absolute zero temperature per Fig. 1 in [11], resolved into Gaussian single-peak components. Parameters relevant for further analysis are inset: E_u in cm⁻¹ units; standard deviation width w_0,_u in cm⁻¹ units; and S_0,_u in cm²cm⁻¹ units.

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Fig. 6 Experimental fluorescence emission-shape spectrum of cobalt Co²⁺ in a silver chloride-bromide AgCl_0.5Br_0.5 crystal at near absolute zero temperature per Fig. 1 in [11], resolved into Gaussian single-peak components. Parameters relevant for further analysis are inset: g⁽^ℓ⁾; E_ℓ in cm⁻¹ units; and standard deviation width w_0∗,_ℓ in cm⁻¹ units. The emission cross-section scale was added to comply with our revised estimate of τ_rad(0), per Fig. 8.

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The resultant energy scheme is plotted in Fig. 7. The Co²⁺ ions are described to reside in two different sites, with some overlap. Excitation into the O-symmetry T₂(⁴F) multiplet results in a fast non-radiative intersystem crossing into the T_d-symmetry T₂(⁴F) multiplet. The measured radiative fluorescence relates to the T₂(⁴F) → T₁(⁴F) allowed transition of the Co²⁺ ions residing in the T_d-symmetry sites.

Fig. 7 Energy-level scheme of absorption and fluorescence emission transitions in doubly ionized cobalt Co²⁺ in a silver chloride-bromide AgCl_0.5Br_0.5 crystal, based on experimental results provided in [11] and [16] (Figs. 5 and 6 above).

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A fit of the effective fluorescence lifetime τ_eff as function of the absolute temperature T to Eq.(11) was obtained, and shown in Fig. 8 using the following known material parameters: ${\bar{M}}_{a t} = 82.7755 amu$ ; N_c = 8; n = 2.1. Other process parameters deducted via Figs. 4–7 are listed in Table 1, along with adjustable parameters obtained by the fit to Eq. (11). The latter are emphasized by bold-type lettering.

Fig. 8 Effective fluorescence lifetime of an excited cobalt Co²⁺ in a silver chloride-bromide AgCl_0.5Br_0.5 crystal as function of absolute temperature. Full circles are experimental results from [11], Fig. 2. Dashed line is a fit to our Eq. (11). For detailed fit parameters see text and Table 1. Dotted curve replicates the original fitting attempt by Tsur et al using Eq. (1) in [11], presented here as Eq. (17).

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Table 1. Fluorescence parameters and adjustable material and process parameters obtained by the fit to Eq. (11). The adjustable parameters are emphasized by a bold-faced lettering text.

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5. Results and discussion

Correction of the experimental reflectance results for rear surface reflection were performed in such a manner that the fit reflectance at the zero frequency limit agrees with the theoretically calculated static dielectric constant ε(0) = 13.4 [18] (equivalent to 32.6% single surface reflection); see Appendices A and B. The ε(∞) parameter was chosen as 4.41, to be consistent with the ≈2.1 optical refractive index in the near-IR [19].

A notable result of the fit summarized in Table 1 is the value of the vibrational cutoff frequency ħω_co = 180 cm⁻¹. To verify consistency of this parameter with other methods of estimating the cutoff frequency of this particular material, we have carried out a reflection spectral measurement of the crystal surface in the far-IR region (50 − 600 cm⁻¹). The results are shown in Fig. 9. The reflectance is moderate in the high frequency region with very little structure, but exhibits a steep rise at approximately 180 cm⁻¹, up to about 45 − 50% towards the low frequencies. At about 75 cm⁻¹ it appears to reduce towards about 35%, although with a considerable experimental noise level. A fit of the steep rise was obtained according to the Eq. (28) using the parameters appearing in Fig. 9. The vibrational ħω₀₁ = 97 cm⁻¹ and ħω₀₂ = 75 cm⁻¹ are quite close to the transverse optical (TO) frequencies measured at the Brillouin zone Γ point for pure AgCl (117.5 cm⁻¹) and pure AgBr (89.1 cm⁻¹) crystals quoted by Fujii et al ([20], Table. 2). Their corresponding longitudinal optical (LO) frequencies are 194.5 cm⁻¹ and 139.3 cm⁻¹. Particularly, the LO frequencies of the mixed crystals are somewhat lower than the quoted LO frequency of AgCl.

Fig. 9 Front surface reflectance of an AgCl_0.5Br_0.5 single crystal silver chloride-bromide, corrected for second surface reflection, as function of frequency. Solid line - experimental result; dashed line - fit to Eq. (28). Fit parameters are inset in the figure frame.

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Table 2. Fit parameters for reflection measurement of Fig. 9. The numerical j value represents the subscript index of ħω₀ frequencies.

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In Table 2 we summarize the parameters used in the theoretical to experimental fit of Fig. 9. The related LO frequency of each TO one is given by

ℏ ω_{L O}^{(j)} = \sqrt{ℏ^{2} ω_{0 j}^{2} + Q_{0 j} / ε (\infty)},

where Q₀_j is j’s band strength parameter (see Appendix B).

The effective oscillator strength f_eff that simulates the behavior of the molecular dipole as an electronic one, is given by

f_{e f f}^{(j)} = \frac{3 m}{2 π^{2} e^{2}} \frac{V_{c}}{2} \frac{Q_{0 j}}{ε (\infty)},

where V_c is the unit cell volume given by V_c = a³ for the crystalline period a = 5.66 Å [21].

The highest obtained LO frequency, 162.5 cm⁻¹ is only slightly lower than the ħω_co = 180 cm⁻¹ obtained for the τ_eff temperature dependence fit of Fig. 8. This, quite impressive correlation between the two entirely different methods for obtaining physical quantities, lends support to the theory of non-radiative transitions, and its application to analysis of fluorescence decay lifetime presented in section 2 above.

The enhanced reflectance between 350 and 600 cm⁻¹ has been overlooked in the fit (Fig. 9). The origin of this effect is not yet clear. We suspect that it emanates from bulk imperfections due to the polishing process that contributes a broad spectrum of higher vibrational frequencies. This particular effect certainly requires further studies; however, it has no bearing on the ħω_co estimate which is relevant to the non-radiative contribution to the decay process.

The radiative lifetime quantity is important for laser gain estimation: it appears in the denominator of the well known gain expression; see [22] Eq. (5.6-1). For the free-space wavelength λ as the variable, the gain expression reads

γ (λ) = \frac{Δ N λ^{4}}{8 π n^{2} c τ_{r a d}} g (λ),

where γ is the crossing light intensity gain, ∆N is the population inversion, n is the material refractive index, and g(λ) is the fluorescence decay intensity spectral shape.

The effective upper state lifetime at the absolute zero temperature limit has been often considered as a fair estimate of τ_rad [11, 23]. Our analysis suggests that the actual τ_rad might be substantially longer than τ_eff, and hence the gain significantly smaller. Thus, a proper estimate of τ_rad requires the measurements of τ_eff over a broad range of temperatures (starting from near zero); that, in conjunction with spectral measurements of both absorption and fluorescence.

A side conclusion of the energy-level scheme of Fig. 7, is estimation of the spin-orbit coupling that forms the energy separations among the different multiplet components. The spin-orbit coupling energy splitting Γ_SLJ is given by

Γ_{S L J} = \frac{1}{2} ξ_{L S} [J (J + 1) - L (L + 1) - S (S - 1)],

where ξ_LS is the spin-orbit coupling coefficient, and L, S and J are the orbital, spin, and total angular momentum quantum numbers, respectively; see [24] equation (4.25). Accepting that the first four lower peak positions indicate the four A₁(⁴F_9/2) → T₂(⁴F) absorption transitions of the Co²⁺ in the T_d symmetry, then ξ_LS = −90 cm⁻¹. That parameter has been estimated as −150 cm⁻¹ for Co²⁺ of a T_d symmetry residing in ceramic Co²⁺:MgAl₂O₄ (Co²⁺:spinel) [25]. For the T₂(⁴F) states in the O-symmetry (higher energy four transitions), one obtains ξ_LS = −100 cm⁻¹. For the T₁(⁴F) states in the T_d symmetry (four transitions), one obtains ξ_LS = −80 cm⁻¹.

It should be emphasized that the transitions between the excited Co²⁺ manifold states belonging to the octahedral and tetrahedral sites, as indicated in Fig. 7, exhibit no bearing on the present excited state lifetime analysis. The analysis involves only the lower excited state component in the T_d spectrum.

6. Conclusions and summary

In the present paper we have introduced the competing non-radiative multi-phonon decay transitions into the modeling of effective fluorescence lifetime of the rare-earth ions embedded in host dielectric matrices. Our theory, when applied to the Co²⁺:AgCl_0.5Br_0.5 crystal, is consistent with recently published experimental results [11] over the full measured temperature range. The original authors attempt to match the lifetime vs. temperature dependence to a single configuration-coordinate model, assuming a strong coupling between the ion states and the ligand vibrations, provided a fit only for the low temperature range. It however failed to account for a steep shortening of the lifetime at higher temperatures by over an order of magnitude.

In addition, our analysis predicts a zero absolute temperature radiative lifetime τ_rad(0) = 5.5 ms, and a lattice cutoff vibrational frequency ħω_co = 180 cm⁻¹. The latter estimate shows good agreement with experimentally measured value of ħω_co = 162.5 cm⁻¹ [20]. The configuration potential dissociation energy was estimated to be D ≅ 2500 cm⁻¹. This parameter describes the dissociation energy in the attractive configuration potential that bonds the atoms to their equilibrium positions. It should thus correspond to the host material melting characteristics, see [26] section 121. This particular aspect requires further experimental and theoretical considerations.

An interesting application of the role of multi-phohon non-radiative transition would be the design and use of fast saturable absorbers where the excited state lifetime is very short compared to the photon back-and-forth transit time between the end resonator cavity mirrors. A host of pico-second and sub-picosecond mode-locked pulsed lasers have been designed on the basis of fast saturable absorption in the laser cavity; for example semiconductor saturable absorber mirrors (SESAM), see [27] section 9.2.4.

Notably, the bulk medium parameters responsible for the non-radiative transitions are independent of its periodic nature. Therefore, the expression obtained maybe be adopted to metal ions embedded in glasses and even to electronic-vibronic internal conversion in free large molecules. The latter two issues deserve further analysis and experimental study.

Appendix A: Front surface reflectance correction for a thin slab

Relating to incoherent two-surface reflections, the expression used for the corrected front surface reflection is

R_{e x} = 1 - \frac{(1 - R_{1}) (1 - R_{2})}{(1 - R_{1} R_{2})},

where R_ex is the experimentally measured intensity reflectance, R₁ is the front surface reflectance and R₂ is the rear surface “effective” reflectance. The rear surface effective reflectance is reduced compared to the front surface one due to geometric stopping as well as some surface wedging. These effects are lumped by setting R₂ = βR₁, where β < 1 is an empiric correction factor. Solving for R₁ yields the desired front surface reflectance (only physical solution in presented):

R_{1} = \frac{(1 + β) - \sqrt{{(1 + β)}^{2} - 4 β R_{e x} (2 - R_{e x})}}{2 β (2 - R_{e x})} .

Appendix B: Classical independent oscillators expressions

The reflection of a flat dielectric surface involving several bulk oscillators; see [17] chapter IV, is related to the bulk dielectric function by [28]

ε^{'} (ω) = ε (\infty) + \sum_{j} \frac{Q_{0 j} (ω_{0 j}^{2} - ω^{2})}{(ω_{0 j}^{2} - ω^{2}) + γ_{j}^{2} ω^{2}},

ε^{″} (ω) = \sum_{j} \frac{Q_{0 j} γ_{j} ω}{(ω_{0 j}^{2} - ω^{2}) + γ_{j}^{2} ω^{2}},

where ε′(ω) and ε″(ω) are the real and imaginary parts of the dielectric function, respectively, at a frequency (energy) ω, ω₀_j is the transverse optical (TO) frequency, Q₀_j is a strength parameter, and γ_j is a damping parameter, all of the j-th oscillator. ε(∞) is the dielectric function at frequencies considerably greater than the longitudinal optical (LO) cutoff frequency of the lattice vibrations, but considerably smaller than the lowest electronic frequency. The related real n and imaginary κ parts of the refractive index are given by

n^{2} = \frac{1}{2} \sqrt{{ε^{'}}^{2} + {ε^{″}}^{2}} + \frac{1}{2} ε^{'},

κ^{2} = \frac{1}{2} \sqrt{{ε^{'}}^{2} + {ε^{″}}^{2}} - \frac{1}{2} ε^{'},

The surface reflection spectrum is then given by

R (ω) = \frac{{| n - 1 + i κ |}^{2}}{{| n + 1 + i κ |}^{2}} .

Funding

Israel Ministry of Science and Technology (grant No. MOST 1-14459).

Acknowledgments

We are indebted to Dr. S. Kolusheva of the Ilse Katz Institute for Nano-Science and Technology (IKI), Ben Gurion University of the Negev, Israel, for her skillful technical assistance in performing the optical reflectance measurements.

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ℓ	E_gℓ (cm⁻¹)	g⁽^ℓ⁾	ħω_co(cm⁻¹)	υ⁽^ℓ⁾	B(υ⁽^ℓ⁾	$g^{(ℓ)} C_{n r}^{(ℓ)}$	$C_{n r}^{(ℓ)}$	D (cm⁻¹)	τ_rad(0) (ms)
3	1997	0.0176	180	11.1	6.026	2.187	124.27	2500	5.50
2	2261	0.4641	180	12.56	3.638	0.576	1.24	2500
1	2511	0.4714	180	13.95	2.002	0.006	0.014	2500
0	2818	0.0468	180	15.65	0.83	2.5×10⁻⁶	5×10⁻⁶	2500

j		ħω₀ (cm⁻¹)	Q₀_j (cm⁻²)	γ (cm⁻¹)	ħω_LO (cm⁻¹)	f_eff (10⁻⁵)
1	AgCl	97	7.5 × 10⁴	70	162.5	1.33
2	AgBr	75	0.3 × 10⁴	20	79.4	0.053

ℓ	E_gℓ (cm⁻¹)	g⁽^ℓ⁾	ħω_co(cm⁻¹)	υ⁽^ℓ⁾	B(υ⁽^ℓ⁾	$g^{(ℓ)} C_{n r}^{(ℓ)}$	$C_{n r}^{(ℓ)}$	D (cm⁻¹)	τ_rad(0) (ms)
3	1997	0.0176	180	11.1	6.026	2.187	124.27	2500	5.50
2	2261	0.4641	180	12.56	3.638	0.576	1.24	2500
1	2511	0.4714	180	13.95	2.002	0.006	0.014	2500
0	2818	0.0468	180	15.65	0.83	2.5×10⁻⁶	5×10⁻⁶	2500

j		ħω₀ (cm⁻¹)	Q₀_j (cm⁻²)	γ (cm⁻¹)	ħω_LO (cm⁻¹)	f_eff (10⁻⁵)
1	AgCl	97	7.5 × 10⁴	70	162.5	1.33
2	AgBr	75	0.3 × 10⁴	20	79.4	0.053

Competing radiative and nonradiative decay of embedded ions states in dielectric crystals: theory, and application to Co²⁺:AgCl_0.5Br_0.5

Abstract

1. Introduction