SPECTRAL-SPATIAL DIFFUSION OF PHONONS

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SPECTRAL-SPATIAL DIFFUSION OF PHONONS

U. Happek, T. Holstein, K. Renk

To cite this version:

JOURNAL DE PHYSIQUE

Colloque C7, supplément au n°10, Tome 46, octobre 1985 page C7-229

SPECTRAL-SPATIAL DIFFUSION OF PHONONS

U. Happek, T. Holstein+ and K.F. Renk

Institut fur Angewandte Physik, D-8400 Regeneburg, F.R.G.

Résumé - Une étude sur la propagation des phonons "a une fréquence près de 1 THz dans un cristal â'Al203 est présentée. Le cristal

contenait des ions (V1*+ et Cr2+) , qui donnaient lieu à des

pro-cessus de diffusion résonante-élastique et inêlastique. Nous avons effectué nos expériences à l'aide d'un laser infrarouge. Les phonons ont été générés à une fréquence et optiquement

détectés â une autre fréquence. Nos résultats montrent

l'exi-stence de la diffusion spatiale et spectrale des phonons. On présente aussi une analyse théorique, qui permet de déterminer les coefficients de la diffusion spatiale et spectrale.

Abstract - We have studied propagation of phonons with frequen-cies near 1 THz in an AI2O3 crystal that contained resonance-elastic and inresonance-elastic scattering centers (V1*"1" and C r2 + impurity

ions). By use of a pulsed far infrared laser, monochromatic pho-nons were generated at one frequency and detected optically at another frequency. Our results indicate that spectral and spatial phonon diffusion occurred. A theoretical analysis is presented which allows to determine spectral and spatial diffusion co-efficients.

By use of far infrared radiation, we have generated monochromatic pho-nons in an A l203 crystal that contained resonance-elastic and inelastic

scattering centers and studied the propagation of these phonons. Multi-ple elastic scattering leads to spatial diffusion while multiMulti-ple in-elastic scattering causes phonon diffusion in the frequency space

{spectral diffusion). We report on an experiment which demonstrates

the trapping as well as the spectral diffusion and we present a theory of spectral-spatial phonon diffusion. First results have been published

recently /I/.

The experiment was performed with an A 1203 crystal doped with V203

(0.3 mol%) and C r203 (0.05 mol%) that we have X-ray irradiated. By the

irradiation, a part of the V3 + ions was transformed into V't+ and V 2+

ions and a part of the C r3 + ions into C r2 + and Cr't+ ions. The V1*"1" ions

were used for phonon generation and acted as elastic scattering cen-ters, C r2 + acted most likely as inelastic scattering centers, and C r3 +

ions were used for phonon detection.

The V1*"1" ion in AI2O3 has a resonance line (Fig. 1) that is due to an

electronic transition from t h e E3/2 ground state to the E]y2 lowest

excited state. We have generated phonons either at a generation fre-quency v[ (=839 GHz) near the line center at frequency vo (=843 GHz) or in the wing of the line at a generation frequency v2 (-779 GHz) .

Deceased

C7-230 JOURNAL DE PHYSIQUE

By far infrared excitation and one-phonon relaxation of

v4+

ions,mono- chromatic phonons are obtained. We detected phonons at a detection frequency v d e t ('874 GHz) wich corresponds to the separation between the E and 2x levels of excited cr3+. We observed R2 and R 1 fluore- scence, that is due to transitions from 2x and to the cr3+ ground state, and obtained from the intensity ratio the phonon occupation number pdFt at the detection frequency. The corresponding E + 2 X re- sonance llne is narrow (halfwidth 0.5 GHz) and therefore phonons are detected only in a narrow frequency band.

FREQUENCY (GHzl

Fig. 1

-

Resonance line of

v4+

in A1203. Phonons are excited at frequencies vi and v; and detected at vdet.

TheA1203 crystal (size 5.5.5 mm3), immersed in liquid helium of 2 K, was continuously irradiated with radiation of a cw krypton ion laser to produce excited cr3+ ions in the metastable state. For phonon excitation, the crystal was homogenously illuminated with pulsed far infrared radiation of a C02 laser pumped D20 laser (pulse energy I O - ~ J , duration I O - ~ S , repetition rate 10 cps). Since the D20 laser emitted radiation of both frequencies vi and vh, we selected radiation at one frequency by use of a Fabry ~ 6 r o t filter.

For excitation of phonons at line center, the phonon occupation number pdet increases fast during the far infrared excitation (Fig. 2a). This shows that fast redistribution of the monochromatically generated pho- nons occurred. From the decay time of pdet, a phonon trapping time

(--2.1.10-~s) is found that is larger than the ballistic time of flight (0.5.10-~s) through the crystal. For excitation at v;, the occupation number pdet increases much slower (Fig. 2b). The signal maximum is delayed with respect to the far infrared pulse, with a delay time td=0.7-10-~s. This result indicates that spectral diffusion from the frequency v 2 to vdet takes a much longer time, according to the larger frequency distance, than spectral diffusion from v i ty vdet. The pho- non trapping time was the same as for excitation at vl.

u I

: FAR INFRARED PULSE

I I

0 5 10

TIME (10-~s)

Fig. 2

-

Phonon occupation number pdet for phonon excitation at different excitation frequencies.

reduced /2/. After reduction by a factor of 5 spectral redistribution occurred much slower, as indicated by a signal delay also for phonon excitation at frequency v; (Fig. 3)

.

The decay time of pdet was the same as before uv irradiating.

For a theoretical analysis we describe the

v4+

resonance line by a Lorentzian lineshape function g(v) = [ I

+

( ~ - v ~ ) ~ / r 2 ] - ~ where v O is the resonance frequency and 2r the halfwidth of the resonance line. Then, the diffusion constant due to resonance scattering of the phonons at

v4+

ions is given by D(v) = Do g-l(v), where Do = cs(3NoO)-' is the diffusion constant and 00 the resonance scattering cross section at line center, cs an average velocity of sound and N the concentration of

v4+

ions. Furthermore, we characterize the multiple inelastic scattering by a spectral-diffusion constant Dv = v ~ /where T ~ ~vR is an average Raman shift for an inelastic scattering process and

T ; :

an inelastic scattering rate of the phonons. For simplicity we assume that D v is independent of frequency and that processes with energy gain and loss have equal scattering rates. The spectral-spatial diffusion can then be described by the equation

JOURNAL DE PHYSIQUE AFTER UV- 2 .

_..

.

C

. .

n

5 TIME ( I O - ~ S )

Fig. 3

-

Phonon occupation number p d e t before and after uv irradiating of the crystal.

where xn(x,y,z) describes the spatial shape and where kn are appro- priate wave-vectors for the diffusive modes. We make the assumption that the phonons decay at the crystal surfaces and find the solutions Xn = A cos kxx cos kyy cos kzz where A is an amplitude, given by the strength of the initial phonon excitation, where k,=nla/d, ky=nar/d, kz=n3s/d are the wave-vectors, d the crystal dimensions and nl,n2,n3 integers; the center of the crystal is chosen as the origin x = y = z = O . F7e restrict the discussion to the main mode n = 1 ( n l = n 2 = n 3 = I ) and write f - g l exp(-Do k2t) where k = 1.7 a/d. We introduce dimensionless time anh -frequency variables T = t/T and 5 = (v

-

vo) / y where

and

and obtain the equation

I

--

(6) G

-

(2n s i n h ~ ) exp

2)

where G(S,St,r) is the probability to find, at time T , a phonon at fre-

quency 5 that was originally generated at frequency 5'. For small times ( ~ < 1 ) , eq.(6) has the approximate solution

which shows that the delay time between phonon excitation and the signal maximum at the detection frequency is given by

This relation describes well the observation and allows to determine D,. We obtain D, = 6.10~' s - ~ before the uv irradiating and a value which is by a factor of 4 smaller after irradiating. For large times

(r>l), the distribution is given by the expression

1 1

( '

) f =

-

₂ exp [ - 2 ( v l -v0)2/y2~exp [

-

3 1 ( V

-

V C I ) ~ / ~ ~ I exp(-t/~) which shows that a quasi-equilibrium distribution is reached at large times and that y is a characteristic width of a GauBian frequency distribution. Using eq.(4) we find, with T=2.1

.

10-6s from the ex- periment (Fig. 2 ) , 2y = 2 0 0 GHz. The halfwidth 2y of the distribution is large compared with the halfwidth of the resonance line (Fig. 1). Usin eq. (3), we can determine

r .

From the cross section oo = (4a/3) q-2=

l 0 - l g cm2 where qo = 2nvg/cS is the wave-vector of the phonons at resonance and cs ( e 7.10 cm/s) an average sound velocity we estimate D o = 3 cm2/s. With the value k - 1 1 cm-I we find

r

= 3 GHz. This value is by a factor of 5 smaller than the experimental halfwidth indicating that inhomogenous broadening determines the shape of the resonance line near the center.

The trapping time after uv irradiating (Fig. 3) was the same as before. We suggest therefore, by taking account of eq.(3), that the diffusion constant D o had increased by eliminations of

v4+

centers by the same factor as D v had decreased by elimination of cr2+ centers.

By use of eq.(6), the solid lines in Fig. 2 and Fig. 3 are obtained. The analysis gives a reasonable description of the experimental results. We note that phonon propagation experiments, with local phonon excitation and detection at another place /I/, support the occurrence of spectral-spatial diffusion.

The work was supported by the Deutsche Forschungsgemeinschaft.

REFERENCES

/I/ Happek, U., Holstein, T., and Renk, K.F., Phys.Rev.Lett.

54

(1985) 2091.