Radiationless decay processes of Nd 3 + ions in solidsRichard C. Powell, Dean P. Neikirk, and Dhiraj Sardar

Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74074(Received 8 August 1979; revised 31 December 1979)

Laser photoacoustic spectroscopy measurements were made on Nd3 + ions in garnet, vanadate, andpentaphosphate host crystals. The variations of signal intensities with chopping frequency of theincident light are not in agreement with the predictions of standard photoacoustic signal generationtheory. The results are distinctly different for concentrated and dilute Nd-doped crystals, indicatingthat the mechanisms for generating heat have different characteristics in these two types of samples.The determination of radiative quantum efficiencies of these materials by photoacoustic spectroscopytechniques is also described. These results are compared with those obtained by other methods.

INTRODUCTION

Photoacoustic spectroscopy (PAS) techniques have recentlyreceived a great deal of interest as a method for characterizingradiationless relaxation processes of ions in solids.1- 5 Wedescribe here the results of PAS investigations of Nd3+ ionsin several different types of host crystals. The data is inter-preted in terms of radiationless decay, concentrationquenching, and energy migration processes and a method fordetermining the quantum efficiencies of the samples is de-scribed. Problems with understanding the signal generationprocess with laser excitation, and problems with the accuracyof quantitative measurements are discussed.

Our experimental setup for PAS measurements has beendescribed previously. 3' 4 The only difference for the workdescribed here is that the individual lines of an argon ion laserwere used as the excitation source. The laser power wascontinuously monitored and stabilized at a level of 0.15 W.The exciting light was chopped at frequencies varying from110 to 2700 Hz and focused onto the samples which wereplaced on the quartz exit window of the PA cell. The micro-phone was mounted at 900 to the exciting light and behind abaffle to prevent scattered laser light from reaching it. Thesignal was sent through a preamplifier to a lock-in amplifierand read out on a digital voltmeter after adjusting the lock-inphase for signal maximum. The signal to noise ratio wasbetter than 100 to 1 in all cases and the measured backgroundcell signal was at least 200 times smaller than the sample PAsignal.

The samples investigated were all single crystals between1 and 2 mm thick and between 5 and 10 mm on a side withpolished faces. Three types of samples were studied: thegarnets Y3AI 50 12 (0.85% Nd), Y3Ga 5O12 (0.25% Nd),Y3(Alo.5Gao.5)5O12 (0.85% Nd), and Nd3 Ga5Ol2 ; the vanadateYVO4 (2.0% and 3.0% Nd); and the pentaphosphatesNdY 1-xP 5 O14 with x ranging from 0.1 to 1.0.

Two types of experiments were performed: the measure-ment of PA signal intensity versus chopping frequency andthe determination of quantum efficiencies.

1. PA SIGNAL VERSUS CHOPPING FREQUENCY

The PA signals at maximum phase were recorded at 12 ormore chopping frequencies P, after excitation with the 5145-Aline of the argon laser. Examples of the results are shown inFigs. 1-3. For both samples of YVO4:Nd3 +, the PA signalvaries as P` throughout the entire range of frequencies. For

all three of the lightly doped garnet samples, a ZP` dependencewas also observed as shown for Y:3AlO52:Nd3+ in Fig. 2.However, for the concentrated neodymium gallium garnetsample the PA signal was observed to vary as v-312 throughoutthe entire frequency range. For the pentaphosphate crystalsthe 10% Nd sample exhibits a PA signal intensity which variesas P'Cl over the entire frequency range while the PA signal forthe 100% Nd sample varies as v1 up to about 450 Hz and thenvaries as V'j312 at higher frequencies. Samples with interme-diate Nd concentrations have PA signals which vary as `cwhere n is intermediate between 1.0 and 1.5.

Understanding the results described above poses a difficultproblem in the light of current theories for photoacousticsignal generation. Three parameters are necessary for the-oretical analysis of PA measurements: the sample thickness1, the optical penetration depth lop, and the thermal diffusionlength lth. For the samples investigated 1 is between 1 and2 mm. The optical penetration depth is characterized as 1/awhere a is the absorption coefficient at the wavelength of theexciting light. For the laser line used for excitation, lop is ofthe order of 80 mm for the lightly doped samples and ap-proximately 0.7 mm for the concentrated Nd samples. Theformer (lop > 1) is the "optically thin" case while the latter (lop< 1) is the "optically thick" case.

The thermal diffusion length is given by lth =\/1/7rv,where ,B is the thermal diffusivity. For garnet crystals6 ,3 isabout 5.0 X 10-2 cm2 s-1, whereas in NdP5O14, 3 is anisotropicwith the largest value being about7 1.0 X 10-2 cm2 s-1. Thethermal diffusivity of YVO4 should be of the same magnitudeas that for the garnets and pentaphosphates. Thus for therange of chopping frequencies investigated, lth varies fromabout 55 down to 10 ,Am. For all samples lth << 1, which is the"thermally thick" case. Also, Ith < lop for all samples inves-tigated.

For a thermally thick case with Ith < lop as investigated herethe Rosencwaig-Gersho (RG)8 theory predicts that the PAsignal intensity will vary as v73

/2. Only if Ith > lop should a P-

dependence be observed. Obviously there is a striking dif-ference between these theoretical predictions and the datashown in Figs. 1-3 for the lightly doped samples. A possibleexplanation for this discrepancy is the difference in dimen-sionality between theory and experiment. The RG theory isbased on a one-dimensional model where the thermal gradientat the boundary between the excited surface and the gascauses uniform heat flow back into the cell. In the experi-ments described here only a small area of the total sample

486 J. Opt. Soc. Am., Vol. 70, No. 5, May 1980 C 1980 Optical Society of America 4860030-3941/80/050486-05$00.50

500HVc (Hz)

L.6

CL(F1

FIG. 1. Photoacoustic signal intensity as a function of chopping frequencyfor Nd-doped YV0 4 crystals.

surface is excited by the laser and the heat is generated in acylindrical volume within the crystal. Near the front surfacethe dominant temperature gradient is still that between thesample and the gas, and the heat generated in this regiondiffuses to the surface and contributes to the PA signal aspredicted in the one-dimensional model. However, in theinterior region of the sample the dominant temperature gra-dient is radially outward from the heated cylinder. A three-dimensional photoacoustic theory which accurately modelsthe details of our experiment has recently been developed.9

The predictions of this theory have been shown to be in ex-cellent agreement with our data with no adjustable parame-ters.

IL. DETERMINATION OF QUANTUM EFFICIENCY

The accurate determination of absolute quantum ef-ficiencies of ions in solids has been an experimental problemof interest for many years. It has been hoped that PAStechniques will provide a method for doing this and we de-scribe there the results of measuring the quantum efficienciesof the neodymium-doped crystals. The maximum PA signalintensity and the phase at signal maximum were recorded forexcitation wavelengths of 4765 A and 5145 A at choppingfrequencies of 312 and 1000 Hz.

The PA signal at phase angle 0 can be described by

Ia(O) = C(Pa/Ea) EZ f'jEii cos[k + tan-'(27rvTi) - 0], (1)

where Pa and Ea are the power absorbed and the energy of thelevel where absorption occurs, C is a factor accounting for theproperties of the cell and detection system, and Iqr is theprobability for a nonradiative transition between levels i andj separated by energy Eij where the initial state has a lifetime-Ti. The summation is over all relaxation processes that occurafter absorption. 4.' is the phase shift due to the detectionprocess.

The summation in Eq. (1) can be evaluated by consideringthe energy-level scheme and transitions for Nd3 + ions shown

100 200 500 1000 2000 5000Vc(Hz)

FIG. 2. Photoacoustic signal intensity as a function of chopping frequencyfor Nd3+ ions in garnet crystals.

in Fig. 4. For the lines of the argon laser used for excitation,absorption occurs in either the 4G1 1/2 or 2G9/2 levels at energiesE7 and E6 . This is followed by a cascade of nonradiativedecay processes to the 4F3/2 metastable state. From this levelradiative decay can occur with a rate WR to the various 4 Ijmultiplets with branching ratios b5i. Nonradiative decayprocesses within the 4I term return the ion to the ground state.The metastable state can also relax through direct nonradi-ative processes with a rate WNR or through cross relaxationtransitions with neighboring Nd3 + ions. This process isgenerally referred to as concentration quenching and occursat a rate Wx. The quantum efficiency, probability of non-

so I ,100\80- NdxY P5014

100 400 1-0 0 1460-

40 _Vc

20

lo 10 1

-3/26 (z

k 0

4-1

2-

100 200 400 1000 2000 4000

FIG. 3. Photoacoustic signal intensity as a function of chopping frequencyfor NdxY 1-,Ps5 1 4 crystals.

487 J. Opt. Soc. Am., Vol. 70, No. 5, May 1980 Powell et al. 487

where

QE + PNR + Px = 1.

U 1 E 5 - 4FF3 /2

0 10

7 51 b NR6 Ib52 b53 15/

6 -E4

4 ~45/

0 El 4d~ 19/2

FIG. 4. Energy levels and transitions for Nd3+ ions.

radiative decay, and probability of concentration quenchingof the metastable state are given by

QE = WR/(WR + WNR + WX), (2)

PNR = WNR/(WR + WNR + WX), (3)

PX = WX/(WR + WNR + WX), (4)

_ AE 7 5 cos(4 - 07) + AE 5Mcos[i + tan-' (2wrvT 5)

QEkA cos[4 + tan-l(27rvT5)

where

A= (U6 /I7 )(E6 1/E 71)(P 7 /P6 ). (9)

The phase shift due to equipment response 4 poses anotherproblem. This was handled by deriving a second equationfound from maximizing Eq. (6) with respect to 0 and againsolving for quantum efficiency:

Ea5sin(4' - 0.) +F 1 . (0

QE = sin[ + tan'(22rvTr5)- a] )5

Similar equations exist for the two different excitationwavelengths. These two equations along with Eq. (8) can besolved simultaneously in an iterative way to obtain a uniquevalue for QE. To check the uniqueness of the result, theprocedure was repeated for data obtained at both choppingfrequencies.

The quantum efficiencies obtained by the above procedureare listed in Table I. Y3 AI5012:Nd was the only one of thegarnet samples analyzed because some of the important pa-rameters, such as branching ratios, are not yet established forthe gallium garnet and mixed garnet host crystals. A valueof 0.60 was found for the quantum efficiency of Nd 3+ in thishost. This is much smaller than the value of 0.91 that ispredicted by Judd-Ofelt theoretical calculations."1 However,it is quite close to the value of 0.56 obtained by Singh et al. 1 2

by direct measurement techniques and the values of 0.48 and0.63 obtained by using indirect laser-pumping techniques.'3 '1 "The variations in measured values may be due to sampleshaving different Nd3+ concentrations and thus different levels

(5)

All of the sample properties needed for Eqs. (1)-(5), suchas energy levels, branching ratios,10 and lifetimes are knownfor Nd3+ ions in Y3AI 5012, YVO4, and Nd.Y,-,P5O04 crystals.Since only the 4F 3/2 level has a long enough lifetime to causea phase shift in the PA signal, the evaluation of Eq. (1) withthe model of Fig. 4 gives

I.(0) = C(P0 /E.i)JE.5 cos(' - 0)

+ (E51 - kQE)cos [ + tan'I(2rv Ts) - 0], (6)

where

k = E51 - b54E 41 - b53E31 - b52E21, (7)

and Ts is the lifetime of the 4F3/2 level.

One of the problems in making absolute photoacoustic in-tensity measurements is the determination of the factor Crepresenting cell and system response characteristics. Toeliminate this factor we took the ratio of PA signals at twodifferent excitation wavelengths. If we assume that the re-sponse characteristics of the system are approximately con-stant at least over a limited spectral region, the unknownfactor cancels out and upon solving for the quantum efficiencygives

-07] - E 65cos(4 - 06) - Escos[4 + tan-1(2wvrsT) - 06 (8)

- 07] - cos[4 + tan1(2rrvrTs) - 06I}

Iof concentration quenching. Polycrystalline powders ofY3 AI5012:Nd have been found to have a quantum efficiencyclose to 1.0 but this is greatly reduced when single crystals areformed.' 5 This has been attributed to crystal defects actingas quenching centers' 5 and thus explains the apparent dis-crepancy between measured values and theoretical predic-tions.

The results obtained for the 2.0% and 3.0% Nd-doped YVO4crystals give quantum efficiencies of 9.59 and 0.54, respec-tively. Calculations by Judd-Ofelt theory' 6 predict a QE

TABLE I. Quantum efficiencies.

QESample PAS method Other methods Reference

Y3AI 5012 0.91 (10)0.56 (11)

(0.85% Nd) 0.60 0.48 (12)0.63 (13)

YV0 4 0.72 (15)(2.0% Nd) 0.59 0.53 (a)(3.0% Nd) 0.54 0.51 (a)NdxYl1 .P5O1 4 0.40 (16)(100% Nd) 0.45 0.50 (b)(50% Nd) 0.60 0.66 (b)(10% Nd) 0.90 1.00 (b)

(a) From lifetime measurements and the radiative lifetime of (15).(b) From lifetime quenching measurements of (4).

488 J. Opt. Soc. Am., Vol. 70, No. 5, May 1980 Powell et al. 488

value of 0.72. For the relatively high concentrations of Ndin our samples, concentration quenching will lower the mea-sured QE from this theoretical value. Use of the calculatedradiative lifetime and our measured fluorescence lifetimespredicts values of QE of 0.53 and 0.51 for our samples. Nodirect measurements of QE for Nd in YVO4 are available forcomparison.

For NdP5 0 4 crystals a value of QE of 0.45 was obtained,which is close to the value obtained by Auzel et al. 17 throughrelative PAS measurements on powdered samples. Thecombination of this result with the measured concentrationquenching of the fluorescence lifetime4 predicts a QE for the4F312 level in the absence of concentration quenching in thissystem of 0.90. This is consistent with the values of QE foundfor lightly doped NdXYl-,P 5 0 4 as listed in Table I. No di-rect measurements of quantum efficiency in this system isavailable for comparison.

111. DISCUSSION AND CONCLUSIONS

The results presented here on the frequency dependenceof the PA signal intensity on a variety of lightly doped andheavily doped samples show that there is still much work tobe done in developing an accurate theoretical description ofPA signal generation under typical experimental conditions.An attempt was made to better simulate the one-dimensionalmodel upon which theoretical calculations have been basedby using a lens to disperse the incident laser beam over mostof the sample surface. However, when this was done a verydifferent frequency dependence of the signal was obtainedwhich was less than linear over most of frequency range,reached a maximum at about 400 Hz, and decreased at lowerfrequencies. This erratic dependence appears to be due toscattered-light and cell-surface effects. The experiments wereagain repeated with no lens used, resulting in an excited areaof about 1 mm in radius. The results obtained under theseconditions are the same as those shown in Figs. 1-3. Finally,to ensure that scattered light was not affecting the resultsobtained with small-area excitation, the laser was tuned to4880 A. This wavelength of light is not absorbed efficientlyby Nd3+ ions and, therefore, the sample PAS signal should beessentially zero while the scattered light signal should be ap-proximately the same as for the 5145-A excitation. It wasfound that the scattered light signal was approximately one-and-one-half order of magnitude smaller than the smallestsample PAS signals. It should be emphasized that the spec-ular reflection from the polished sample surface came backout through the front window of the cell and was measured tobe approximately the same for both excitation wavelengths.These experiments indicate that experimental artifacts suchas scattered light are not affecting the data. Also for samplesof the type investigated here the small-diameter laser beamappears to be favorable to the dispersed excitation source andit is important to further develop theoretical models to ac-count for these types of experimental conditions.

Recent work has been presented which accounts for someof the three-dimensional aspects of the heat flow problem.' 8"19

The results of Quimby and Yen' 8 indicate that no matterwhere the heat enters the cell from the sample, the one- andthree-dimensional models are equivalent as long as cell-walleffects are not important. However, for the experiments

described in this paper the important point is that in truethree-dimensional heat flow inside the sample, some of theheat that is predicted to reach the surface in a one-dimen-sional flow model will actually never reach the surface withina duty cycle of the experiment. McDonald' 9 has recentlydeveloped a three-dimensional photoacoustic theory basedon a point source lying on the axis of a cylindrical sample.The new theory developed by Chow9 gives a rigorous resultfor arbitrary beam profile. This theory is the most appro-priate one for comparing with data obtained from our exper-imental geometry and the good fit that is obtained indicatesthat these data are affected by the three-dimensional aspectsof the experiment. The data of Quimby and Yen' 8 can alsobe fit well by this theory.

Another possibility for interpreting the results could be thecJominance of the thermal expansion contribution to the PAsignal.2 0 This term will dominate only if / 3

T >> yg/(lTO),where 3

T is the thermal expansion coefficient, pug is the ther-mal diffusion length in the gas, Is is the sample thickness, andTo is the ambient temperature. For an air-filled cell at afrequency of 100 Hz, Aug 2.5 X 10-2 cm. For To = 300 K anda typical sample thickness of 0.15 cm, the right-hand side ofthe inequality was found to be about 5 X 10-4 K-'. Since thethermal expansion coefficients of the type of materials in-vestigated here2 ' are of the order of 7 X 10-6 K-', the in-equality does not hold and thus thermal expansion makes anegligible contribution to the PA signal.

The method of measuring quantum efficiencies by pho-toacoustic techniques appears to give results generally con-sistent with other predictions and measurements. The pro-cedure of eliminating unknown system-response propertiesby obtaining the ratio of PA signals at two different excitationwavelengths was first suggested by Rockley and Waugh2 2 fordetermining the QE of organic dye molecules. Two otherattempts have been made to obtain the QE of Nd3 + ions insolids. Auzel et al. 17 studied powdered samples of concen-trated stoichiometric neodymium materials and eliminatedequipment response factors by forming a calibration curvefrom samples of known quantum efficiencies. Quimby andYen5 investigated Nd3 + in a glass host and eliminated theproblem of equipment response factors by measuring lifetimeand PA signal changes on a series of samples with differentconcentrations. Their results extrapolated to very low con-centrations imply that the QE for this system is between 0.65and 0.75 depending on the Nd sites selectively excited by thelaser. These values are less than the value of 0.9 measuredby other techniques and theoretically predicted.23' 24 Thechopping frequency used in this previous work was only 22 Hzand this eliminated the necessity for treating the phase-shiftdifferences of the PA signal generated by phonons given offin different relaxation processes. For our experimental setupwe found that the signal to noise ratio decreased below about100 Hz and thus in our case more accurate data could be ob-tained at the higher chopping frequencies. Experimentallythis problem of phase-shift differences could be eliminatedby pumping directly into the 4F3/2 level.

The most significant problem in the method of obtainingQE's described here is the accurate determination of the pa-rameter A given in Eq. (9). A change in this parameter by anamount of only 0.01 can change the value of QE by about 0.10.For the type of samples studied the ratio of PA signal inten-

489 J. Opt. Soc. Am., Vol. 70, No. 5, May 1980 Powell et al. 489

sities can be measured accurately and the ratio of the energiesof the excitation lines is known exactly. The problem is inaccurately determining the ratio of the power absorbed at thetwo excitation wavelengths. The major difficulties arise inmeasuring the amount of light scattered at the cell windowand sample surface and in determining the exact absorptioncoefficients for sharp absorption lines at the positions of verynarrow band laser lines. Thus the values of QE quoted inTable I should be considered to have an error range of as muchas ±0.10.

In conclusion, PAS techniques can be used to obtain in-formation on radiationless relaxation processes of ions insolids. However, it is important to develop better theoreticaldescriptions of the signal generation process when narrow-beam-diameter laser excitation sources are used. Quantumefficiencies can be determined even in complicated cases in-volving numerous radiationless processes if enough infor-mation is known concerning energy levels and branching ra-tios. However, absolute measurements are difficult and it isimportant to develop methods of eliminating cell- and sys-tem-response properties.

ACKNOWLEDGMENT

This work was supported by the U.S. Army Research Of-fice.

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