Solid Sta te Communica t ions , Vol 11, pp 1691-1694, 1972 Pergamon P r e s s . P r in ted in Great Bri ta in

L A T T I C E DYNA~IICS OF I I - V I , I I I - V COMPOUNDS

D N Ta lwar and Bal K A g r a ~ a l

Department ot P h > s l c s , Unlvers l t? of A l l ahabad , A l i a h a b a d - 2 . India

( R e c e i v e d 11 Septer,'ber 1972 by 4 R ~ er,na)

The l a t t i c e v lb ra tmns of I I - V I compound ZnSe and I I I - V compound InSb have been c a l c u l a t e d in the frame work of B a n e r j e e - V a r s h m ' s second nelghbour tomc [SNI] model utlhzlr~g c r i t i c a l -po in t phonons as an input to de termine the required seven pa ramete r s and the expe r imen ta l ly de termined three e l a s t i c c o n s t a n t s C,, , C,2 , Q.~ as r e s t r a in t s on the va lues of the pa ramete rs . A r e m v e s t t g a t m n of the exper imenta l e t a s t m cons t an t s pa r t i cu la r ly C,, and C44 for ZnSe has been s u g g e s t e d R e s u l t s are p resen ted for the d l s p e r s l o n cur~es along hlgh symmetry d i rec t ions -~ r e a sona b l e agreement with the recen t ly measured m e l a s t m nutrop s c a t t e r i n g da ta :s obse rved

L A T T I C E vibra t ion spec t r a of I I - V I , I I I - V compounds, that c r y s t a l l i z e in the z inc-b ie r , de s t ruc ture have been e x t e n s i v e l y s tudmd in recent yea r s . Z m c - b l e n d e c r } s t a l s have drawn the a t ten t ion of both t heo re t i ca l and exper imenta l p h y s t m s t s b e c a u s e of their r e l a t i ve ly s imple c r ' , s t a l s t ruc ture , which makes an in te rp re ta t ion fa i r ly s t ra ight forward and f a c l h t a t e s compar i sons of exper imenta l data ~vlth the a v m l a b l e t heo re t i c a l models . Phonon d i spe r s ion curves along high symmetry d i r ec t ions have been accumula ted e x t e n s i v e l y in var ious I I - V I , 7 I I I - V a '3 semi-

conduc tors u smg i n e l a s t i c neutron s c a t t e r i n g exper imen t s . Recen t ly neutron s c a t t e r i n g da ta for phonon d i spe r s ion curves along [001], [110] and [111] d l r ec tmns for ZnSe 1 and InSb ~ have been ob ta ined The theory of l a t t i c e v ib ra t ions for t he se c r y s t a l s are b a s e d on the phenomeno- l og i ca l she l I and ng ld - ion models . In InSb P r i c e et al. 3 have cons ide red ~t n e c e s s a r y to use a

14-parameter s econd nelghbour she l l model for a r e a s o n a b l e desc r ip t ion of the d I s p e r s m n curves In th is commun~catlon we have p re sen ted the phonon d i spe r s i on curves for ZnSe and InSb*, u t l h z m g a ph!~sicall.~ meaningful and most econ- omlca i (as for the number of pa ramete rs ) 7-parameter rigid mn model e s t h e r in t roduced b~ Bane r j ee and Varshm 4

The SNI model incorpora tes the short range forces up to and m c l u d m g second =elghbours in t roduced b~ Helen J Smiths for diamond l a t t i c e s and the long range coulomb forces due to an e f fec t ive charge a s s o c i a t e d with each ton orig-

ina l ly in t roduced by Kel lerman 6 for ionic c r y s t a l s h a v i n g r o c k - s a l t s t ruc ture The de[ar ts of the model are given in re ference 4

The f r equencms of v ibra t ion fo: the zinc-blend., type of c ry s t a l may be ob ta ined b} so lv ing the c h a r a c t e n s t t c equa t ion

I D=,, , ~K ' - a ) C~<K,3~ = 0, (1)

for var ious ~ a v e - v e c t o r s k in rec ip roca l s p a c e The symbols K, K' = 1,2 r ep resen t the type of

B a n e r j e e and Varshni 4 used their model to fit the d i s p e r s i o n curves of InSb along [001] and [111] d i r ec t i ons measured by Jouffroy [ JOUFFROY J . , C.r. hebd. S#anc. Acad. Scz. Pares 265, 67 ( 1967)] by di f fuse s c a t t e r i ng of X-rays . Nearly a l l the zone bo '~,dary f requencms measured b!~ neutron s c a t t e r i n g exper iment have turned out to be very d i f ferent from those measured by X-ray s ca t t e r i ng exper iment . That is why we cons ide r it nece s sa r ? to r e inves t i - ga te .

1691

1692 LATTICE DYNAWCS OF II-VI, III-V COMPOUNDS Vol 11, No 12

atom and &,, ’ ( ) & are the elements of the

Fourier-trans ormed dqnamlcal matrlv These

matrix elements are related to the secen para-

meters 9 fl, ,z, , p, , A, , .\r and y and their

expressions have been given by Banerlee and

Varshni.

In order to calculate the parameters for the

SNI model the crltlcal-point phonons are used as

an Input whrle the three elastrc constants are employed as restraints on their values. The two

parameters aand x are calculated using the TO

(r ) and LO(r) modes for long wavelength vi-

bratlons by the following eupresslons.

phonon frequencies found by neut:ons with the

calculated ones uslsg the experimental values

of the elastic constants suggested that the value

of tie elastic constant C,, should be somewhat

greater Since there 1s no a priori reason to sus-

pect that ZnSe would be different in crystal

structure from its family crystals, the present

results suggest that an experimental relnvestl-

gatlon of the elastic constants for ZnSe should

be made

Table i. -I frst oi treyuencres ohsrrx,ed h neutror; data ror ZnSe by Hennlon et al , rererence 1, are presented In column 3 In cm-’ The zone boundary irequencles determined bv Price et al ,’

jar InSb rn THz are glcen VI column 4

and Crltical Point 1lode ZnSe’ InSb’

obserl ed observed

. where

m, VI2 m = ----, m, : m,

m, - m2

r- LO 253 5 90

TO 213 5 51

and m, , m, are the masses of the tuo atoms

The parameters ,ti, and are calculated +“z

using LO(U) and LA(i) zone boundary phonons

In the [OOl] direction by using the relations

Ml & iL< = 1- i

4 4 - z - 0 54125 x , (4)

! LO 213 1 _- I3

LA 194 4 30

TO 219 5 38

TA 70 1 12

L LO 1 82

L.-i 166 3 82

TO 234 5 31

! ml 4.~ TA 57 0 98

and p, = d - 4

- c - 0.54125 y (5)

The values of the parameters C, y, ii, and

+* are then used to evaluate approvlmate values

of fi and (h, + h,) from their relatlonshlp with

the three elastic constants The values of /3, ,i,

and A, are varied slightly to obtain a good fit of

the calculated TO(X) and TA(X) zone boundary

frequencies In the [OOl] dIrection with the neutron

data. During this fitting procedure attention has

been pald to malntaln the calculated and exper-

imentally determlned elastic constants wlthln the

llmlt of 10 per cent error However, this model

leads to marked dlscrepancles between the cal-

culated and observed values of elastic constants

particularly C,, and C,, for ZnSe. MartIn’ while

dIscussIng the elastic propertles of ZnSe also

pointed out that the experlnentally determlned

elastic constant C,, may be In error Hennlon

et al.’ when comparlng the long wavelength

The macroscopic data used to evaluate the

parameters for the model are presented In Table 1

and the derived values of the parameters along

with the experimental and calculated elastic con-

stants are given in Table 2- The calculated dls-

perslon tunes are plotted in Figs 1 and 2 for

ZnSe and InSb respectively along with the neutron

data The continuous curves represent SNI-model

fits discussed earlier Full dots represent the

unconfirmed points whereas the hollow circles

show the neutron scattering points

Our calculated phonon dlsperslon curves are

in good qualltatlve agreement with those of the

neutron data They explain well the crossing of

longltudlnal optlcal and transverse optical curves

along [OOl] dlrectlon but devlatlons from the ob-

served frequencres are seen at the zone boundarles

Vol. 11, No. 12 LATTICE DYNA\IICS OF I I -VI , I I I -V CO'vlPOUNDS 1693

T a b l e 2 E~perzmen ta l and theore t zca l v a l u e s ot some m a c r o s c o p z c quan t z t t e s . E ! a s t z c c o n s : a n t s are Ln the unz ts or 10 u d } n / c m a -Ill the la t t zce a~,namzcai mode l parame ter s are zn the unzts o f l O : d y n / c m

Experimental Quantlt} Exper lmental Calcula ted Ca lcu la ted ZnSe InSb

C~, 8 10 ~ 9 57 6 . 6 6 9 ' 6 85 C2 -t.88" 5 1 3 645 T 3 35 C=, 4 41 ~ 1 95 3.02" 2 8 a 27 216 31 0589

30 5 28.5

~, 3.2897 2.50616 ~, 3 308 1 11646 'h -0 61 1 1043 A, -3 .394 -6 1474 \ 6.2582 2.5377

* BERLINCOURT D., J A F F E H. andSHIZAWA L R , P h y s . R e v 129, 1009 (1963).

; SLUTSKY L.J and GARLAND C.W , Ph>s. R e v . 113, 167 (1959).

200

C m -~

f

10o

T 0

o'~ 0'~ d, o'2 -)0~

i I

0 Q2 ~ 1 3 5

FIG. 1. Phonon d lspers lon curves for ZnSe in the [001], [110] and [111] d l rec t lons . The phonon fre- quenc ies are in the unit of cm-' The le t ters i-, \ , K and L refer to symmetr1~ points m the B n I l o u m - z o n e

e spec ia l ly for t r ansverse acous t i c modes whtch

are m the same order of agreement obtained by Baner jee and Varshm 4 wlth the neutron sca t te r ing data for GaAs

M X X ~, r ~

~ t T ~

L ~

'3 0 2 O a 0 6 0 8 0~, 3 6 Od. 3Z $ 0 0 1 0a. 05

)or_ ~,o] D',]

FIG 2 Phonon d ispers ion curve for InSb m the [001], [110] and [111] d l r ecnons The phonon f requenctes are in the umts of THz

- t c k n o ~ l e d g e m e n t s - The authors express their s ince re thanks to Dr Y P Varshm, Ottawa Umvers l ty , Canada for pro~tdmg us the necessa ry de ta i l s of their SNI modeI and the Counczl of Scient l f lc and Indust r la l Research, New Delhi for i ts f inanc ia l a s s l s t a n c e All ca l cu !a t lons were performed at IIT Kanpur on IBM 704-t-1401 Computers

1694 L A T T I C E DYNA'dlCS OF T ,T . I - \ . . I I I - V CO;IPOUNDS X, ol 11, No 12

R E F E R E N C E S

1 HENNION B , MOUSSA F , P E P Y G and KUNC K , Ph~s Le f t 36A. 376 (1971)

2 ~AUGH J L T and DOLLING G , Phvs ~.ev 132. 2410 11963), Proc lnt Corzt La tr lce D,.~zae'tc~, Copenhagen (1963)

3 PRICE D L , RO~E J .~l and NICKLO~ R ,',I , Phvs Rev B3, 1268 (1971)

4 B A N E R J E E R and V4RSHNI Y P , Cat, ] P~z:s -17. 451 (1969)

5 SMITH H ,',1 J . Pha! Trans R Soc (Lo'Tdon; A2,11, 105 (1948)

6 KELLERMAN E W . Phzl. Yr~.r's R Soc rLondon;A238, 513 (1940)

7 MARTIN R ~I , Phvs Re,' BI, 4005 (1970)

On a ca lcu le ies v ibra t ions du reseau du compose I I - V I ZnSe et du compose I I I -V InSb dans le cadre du mode[e (SNI) du deux16me

m s m iomque, propose par B a n e r ] e e - V a r s h m , en ernployan: des phonons de poin t -c r i t ique comme entr6e , afro de determiner les sept param~tres ~oulus et les trols cons tan te s ~ l a s t l ques C. , C~ , C~a - qul avalent ~te t rouv~es par vine e x p d n m e n t a l e - comme facteurs hml ta t l f s des ~aleurs des paramStres. On a recommand~ une nouvel le dtude des valeurs e x p d n ~ e n t a l e s des cons t an t e s ~ las t lques C,~ et C . pour ZnSe. On a p resen te [es resu l ta t s se rapportant aux courbes d e ' d i s p e r s i o n s dans des &rec t lons d e h a u t e symmetrm On a t r o u ~ un accord s a t t s f a l s a n t avec les mesures de la diffusion du neutron non-~las t lque