Solid StateCommunications, Vol. 7, pp. 917—919, 1969. PergamonPress. Printed in GreatBritain

ANISTROPY EFFECTSIN THE ACOUSTOELECTRICINTERACTION IN GaAs

R. Klein*

RCA Laboratories,Princeton,N.J. 08540

(Received21 April 1969 by J.L. Olsen)

Within the small-signalapproximationthe Hutson—Whitetheory ofthe acoustoelectriceffect hasbeenextendedto accountfor piezo-electric and elastic anisotropy. The resultsareapplied to the caseof GaAs with the static electric field alonga [1111 direction, wherethe coupling to quasitransverse‘off-axis’ wavesturns out to be ofprimary importance.

THE METHOD of light scatteringfrom acoustic propagatingalong [1111, but only to the longi-domains(especiallyBrillouin scatteringand tudinal wave. As in the caseof CdS alongtheoptical strain-birefringence)hasmade it possible c-axis, however,the electronsystem loses itsto study the growth of acoustoelectricdomains energyto off-axis shearwavesbeforeamplifi-and its propagationcharacteristicsin great cation of longitudinal wavessetsin becauseof

detail. By this techniquethe importanceof off- the higher value of the longitudinal sound velocity.axis shearwavesfor the acoustoelectriceffect, Therefore,we will disregardthe waveof (quasi)which wasfirst postulatedsomeyearsago, could longitudinal characterin what follows. The

directly be investigatedfor CdS andZnO with degeneracyof the two transversewaves is liftedthe applied electric field alongthe c-axis.23 In in a generaldirection. Considering for the sakeconnectionwith theseobservationswe haveex- of simplicity waves propagatingin the (110) plane,

tendedthe linear theory of the acoustoelectric one modeis truly transversewhile the other iseffect by White4 to accountfor the elastic and only predominantlytransverse(the correspondingpiezoelectricanisotropy.~ As an applicationthe displacementfield deviatesfrom beingat rightgain was calculatedfor all waves propagatingin angle to the wave vector). The latter modeis the

a planecontainingthe c-axisof CdS, and in acoustoelectricactive oneand correspondstoparticular it was shownthat the direction of the the T

2 modein CdS. Thereis one important

wave with maximum gain dependson the value of differencehowever, coming from the fact that thethe applied electric field. [111~ direction in a cubic lattice is a three-fold

axis whereasthe T2 mode in CdS hascylindricalIt is the purposeof this note to point out symmetryaroundthe c-axis.

that a similar situation occursfor GaAs if thefield is applied alongthe [111] direction andto Within the frameworkof the extensionof thepresentresults for the gain of off-axis waves in small-signal theory to the three-dimensional

this material.6 As is well known, thereis no anisotropiccasethe result for the dampingof a

couplingto the two degeneratetransversewaves wave propagatingin the (110) planeunderan

— angle ~ off the [00 ii direction is*Permanentaddress:LaboratoriesRCA 8005Zurich, Switzerland. a(~)= [d2v~(111)/8fDe)F(q~)K(qS).(1)

917

918 ACOUSTOELECTRICINTERACTION IN GaAs Vol. 7, No. 13

The externalelectric field is alongthe [1111direction. Here, d is the piezoelectricconstant,v8 (111) the sound velocity of the transversewaves °~- -

along [111LI is the trapping factor, D the dif- o4 - -

fusion constantand c the dielectric permittivity GaAsat constantstrain. The latter two quantities are 03 - -

hereapproximatedby sphericaltensors. This is -

well satisfied in GaAs. The factor F(çb) accounts

for the difference in directions betweenthe wave 5I~0 - os 1.05 -

underconsiderationandthe fixed electric field. .. (oo,j “ 4 50(,ti~60 Il (no]

It is important to use the true (angle dependent), C0 0 20 3 10 60 90

value of the soundvelocities in F(~)as cal- ~ - .3

culated from the known elasticconstants.7 K(~)

is proportional to the electromechanicalcoupling .5 5

coefficient and is representedin Fig. 1. The 03 -

maximumcoupling of the quasi-transversewaves I?

is along [lioL which is the reasonwhy the -0.4

majority of experimentsarecarriedout with cry-stals cut in this direction. In the caseof CdS .06 20

the transversecoupling is highest in a direction .0.8

30°off the c-axis and nearly asstrong in thebasalplane. a(~) is calculated in (1) for the .0 -

frequencyof maximum gainwhich in the present -12 -

caseis againan angle dependentquantity. For 20

this reasonthe results in Fig. 2 showthe damping(or gain) for eachvalue of ~ for a slightly dif-ferent frequency. This situation correspondsto FIG. 2. Attenuation(or gain) of the quasitransverse

wavespropagatingin a (110) planewith the staticelectric field in the [liii direction for severalvaluesof Vd/V~.[111J.

I I I I

.2

E 1.0

0

.as—a

0.6

04 KM

0.2 -

0.1 I I I I0 10 20 30 40 50 60 70 80 90

[ooi] (in] ~. —. [uoJ

FIG. 1. KL and KM areproportional to the electromechanicalcoupling expressingthe strengthsof thecouplingbetweenthepredominantly longitudinal andthe predominantlytransversewavespropagatingin a(110) planeandthe electron system respectively.

Vol. 7, No. 13 ACOUSTOELECTRICINTERACTION IN GaAs 919

many experimentsin which the system selects and [110]. Besidesthis ‘jump’ there is againaits own direction and frequencyfor the build up smoothdependenceof theangle of maximumgainof the domain. This is in contrastto a transducer on the value of the applied field like for CdS with

experimentor an experimentwith an external E° parallel the c-axis. Both effectsshould besoundfield. important for the interpretationof experimentson

the acoustoelectriceffect in GaAswith the static

It is interestingto note that for the ratio of field along [liii.drift to soundvelocity vd/vs<l.S (both quantitiestakenalong [111]) the maximum gain lies between Acknowledgements— I wish to thank L. Friedman,

[0011 and [lii] andthat for highervalues of A. Moore, andR.W. Smith for discussions.vd/vS the maximum of the gain is between[iii]

REFERENCES

1. McFEE J.H., J. Appi. Phys.34, 1548 (1963); MEYER N.I. andJORGENSENM.H., Phys.Lett. 20,450 (1966); ZUCKER J. andZEMON S., Appl. Phys.Lett. 9, 398 (1966); McFEE J.H., TIEN P.K.,and HODGESH.L., J. appl. Phys.38, 1721 (1967).

2. WETTLING W. and BRUUN M., Phys.Lett. 27A, 123 (1968).

3. MOORE A.R., Appl. Phys.Leti. 13, 1261 (1968).

4. WHITE D.L., J. appi. Phys.33, 2547 (1962).

5. KLEIN R., Phys.Leti. 28A, 428 (1968).

6. From the largenumberof experimentalstudies of the acoustoelectriceffect in GaAs we mentiononlytwo recentpapersin which more referencescan be found: SPEARSD.L. andBRAY R., J. appi. Phys.39, 5039 (1968); LEROUX HUGON P., Phys.StatusSolidi 31, 331, 339 (1969).

7. BATEMAN T.B., McSKIMIN H.J., and WHELAN J.M., J. appl. Phys.30. 544 (1959).

Im Rahmender linearenNäherungwird die Theorievon Hutson uridWhite für den akustoelektrischenEffekt so erweitert, dasspiezo-elektrischeund elastischeAnisotropienberücksichtigtwerden. DieErgebnissefinden Anwendungauf GaAs mit dem statischenelek-trischenFeld längseiner [iii] -Richtung, wobei es sich zeigt, dassdie Kopplung an quasitransversaleSchrägwellenvon grosserBedeutungist.