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