A CALCULATION OF THE PHONON DISPERSION CURVES IN TRIGONAL AND MONOCLINIC SELENIUM WITH THE HELP OF A SINGLE VIBRATIONAL POTENTIAL

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A CALCULATION OF THE PHONON DISPERSION

CURVES IN TRIGONAL AND MONOCLINIC

SELENIUM WITH THE HELP OF A SINGLE

VIBRATIONAL POTENTIAL

M. Merian, J. Etchepare

To cite this version:

JOURNAL DE PHYSIQUE

CoZZoque C6, suppZe'ment au n012, Tome 42, de'cembre 1981 page C6-608

A CALCULATION OF T H E PHONON DISPERSION CURVES IN TRIGONAL A N D

MONOCLINIC SELENIUM WITH T H E HELP O F A SINGLE VIBRATIONAL POTENTIAL

M. Merian and J. Etchepare

Laboratoire drOptique AppZique'e, EcoZe PoZytechnique, EcoZe NationaZe Supe'rieure de Techniques Avance'es, Batterie de Z'Yvette, 91120 PaZaiseau, France

Abstract.- We present phonon d i s p e r s i o n curves i n t r i g o n a l Selenium (chains) and zone c e n t e r phonons i n a monoclinic Selenium ( r i n g s ) c a l c u l a t e d w i t h t h e same i n t e r n a l f o r c e Constants. Most o f t h e i r d i f f e r e n c es i n phonon energies c a n be explained by i n t e r m o l e c u l a r i n t e r a c t i o n s which i n both case promote a t e t r a h e d r a l n e i hbouring l e a v i n g some d a n g l i n bonds.in monoclinic Se. Long range f o r c e s of tile Van der Waals type were neg?ected i n f i r s t approximation.

H e l i c o i d a l i n f i n i t e chains i n t r i g o n a l Se (t-Se) and e i g h t membered r i n g s Se8 i n b o t h a and 6 monoclinic Se ( a and 6-Se) e x h i b i t v e r y c l o s e values o f t h e i r f i r s t neighbours bond l e n g t h s r klk+I and bond angles ek = _($k,k-I¶ -+ r _k,k+, ) and very

CTQXQ

absolute values of t h e i r d i h e d r a l angles rk. An harmonic p o t e n t i a l

v!:!,

which i s i n t e r n a l t o the molecular u n i t and w r i t t e n i n t h e Wilson coordinates

_~

_e

_~

_,

A r k should then c o n t a i n s s i m i l a r f o r c e constants f :

v ( 2 ) - 2

i n t -

:

fr

1'

+ f e ( ~ 0 ~ ) + f T ( ~ i+ ~cross terms ) ~ (1 To describe t h e i n t e r m o l e c u l a r b i n d i n g energy, we take :

A, B and C a r e constants. The RkK a r e t h e e x t e r n a l i n t e r a t o m i c distances between k

0

i n one molecule and K i n t h e others, up t o RkK = 8 A. V(akK) i s a f o u r atoms term 0

due t o angles akK between t h e RkK

<

4.366 A (second e x t e r n a l neighbour l e n g t h i n t-Se) and t h e i r e q u i l i b r i u m p o s i t i o n . I t w i l l be explained elsewhere why t h e f i r s t terms i n ( 2 ) p l a y l i t t l e r o l e on phonons frequencies i n t-Se, a and 6 -Se ; i t w i l l t h e r e f o r e be ignored here.

The f i r s t order v a r i a t i o n s sn o f akK may be expressed as l i n e a r combinationsof Wilson coordinates and thus a r e l i n e a r combinationsof f i r s t order c a r t e s i a n atomic displacements

?ik

:

.

k - I

-+ _+-

Sn = AakK = I,Bnkl

.

U k t where k ' =Ik;'

k

K

Then t h e f o r c e a c t i n g on atom k i s :

We almost o b t a i n zero f o r c e s on each atom k i n a-Se (and f3-Se) by choosing V a (l-cos2akK) and by t a k i n g i n t o account o n l y those angles akK which a r e w i t h i n

-f +

10" from t h e two e q u i l i b r i u m d i r e c t i o n s Dk and Dlk. These d i r e c t i o n s a r e i n t h e b i s s e c t o r plane o f t h e angle ek, a t an angle y - t h e same f o r a l l k

-

o f 12" from

-f

+

t h e normal Nk ( o r -Nk) t o t h e i n t r a molecular plane ( k - I ,

k,

k + l ) ( f i g u r e 1 ).

The

+

_{+ +}

Dk

+ =

*

cos ek,k-l A ek,k+l

s i n

ek

+

e

-

s i n cp ek,k-l k,k+l ek 2 cos - 2 + ₊ Ek = (;k,k+l

-

ek,k-l)

/

( 2 s i n F i g u r e 1 +

ekk, i n (5) and ( 6 ) a r e u n i t v e c t o r s along t h e bonds. The appearance o

+

f Ek w i l l be explained below.

I n t-Se b o t h o f these d i r e c t i o n s a r e occupied by a bond w i t h i n 3' ( f i g . 2). I n monoclinic Se a t l e a s t one o f them i s occupied by a bond w i t h i n 10". This i s t r u e f o r each atom ( f i g . 3 f o r a-Se). No i n - p l a n e e q u i l i b r i u m d i r e c t i o n r o u g h l y along f i r s t i n t e r n a l neighbours bonds i s found f o r a-Se as opposed t o t h e case o f t-Se.

F i u r e 2 : p r o j e c t i o n on t h e

(z.5)

F i g u r e 3 : p r o j e c t i o n on t h e x y plane h o r m a l t o t h e screw a x i s c ) o f o f t h e 8 atoms k = 1 t o 8 o f one r i n g o f

t h e 3 atoms k = 1,2,3 of one u n i t c e l l the a-Se u n i t c e l l and o f t h e e x t e r n a l of t-Se and of t h e i r two f i r s $ e x t e r - bonds R which a r e ~ o u g t ~ l y along Nk o r n a l neighbours roughly along Nk and

-fik

( f u r y l i n e s ) o r NK1-NK ( d o t t e d lines).

C6-6 10 JOURNAL DE PHYSIQUE

Using f i r s t o r d e r f o r A a k K o n l y g i v e s those v a r i a t i o n s o f akK which a r e i n

-f -f

L

+

t h e plane (Dk, Rk(;) o r (D;, RkK). For very small akK t h e v a r i a t i o n s o f t h e angle o f

+-

RkK w i t h t h e normal t o these plane must a l s o be used w i t h about t h e same weight as ActkK. We then decided t o use twp angular coordinates i n s t e a d o f t h e AakK : t h e ₊

-f -f

Wilson out-of-plane v a r i a t i o n o f t h e angle bkK between !kK an$ Nk o r -Nkandthe

+

v a r i a t i o n aykK o f t h e angle between RkK and t h e u n i t v e c t o r E o r -Ek normal t o t h e b i s s e c t o r plane ( f i g . 1 and Eq. 6). The e x t e r n a l v i b r a t i o n n a l p o t e n t i a l then becomes:

The l a s t term w i t h t h e f o r c e constant fra i n Eq. 7 i s t h e b e s t one we found t o des- c r i b e t h e q u i t e s t r o n g d i s p e r s i o n o f the h i g h e s t frequency branche r-K i n t-Se ( f i g . 4). The signs

+

o r

-

has t o be taken such t h a t t h e f o r c e s push

K

i n t h e bissec-

-f -f

t o r plane toward Dk o r D l k ( f i g . 2). The f i r s t term i n Eq. 2 does n o t g i v e t h i s d i s - p e r s i o n w i t h t h e c o e f f i c i e n t s which were c a l c u l a t e d f o r e q u i l i b r i u m i n a-Se and

0

"

₆

t-Se : p 2 .54 A and C

-

3.00 A

.

( 2 ) (2) S o l v i n g t h e eigen values equation o f t h e t o t a l dynamical m a t r j x Vint + Vext,we obtained t h e f o l l o w i n g r e s u l t s o w i t h t h e same f o r c e constant: f o r t-Se and a-Se :

fr = 1.256 and frr, = .047 md/A, f o = .949 and

fa=

. I 5 0 md.~/rd', fra=.092 md/rd.

Observed frequencies('

)

254 R (E2)

Fig. 4 : phonons i n t-Se. The dashes w i t h symmetry t a b l e 1 : The f i r s t four co- assignements a r e e x p h i m e n t a l data on t h e

mive

the Jane center 5 zone edges ( 2 )

.

frequencies o f a-Se i n i t s Cph

space group.

I n t h e f i f t h column o f t a b l e 1 a r e t h e experimental r e s u l t s and assignements f o r i s o l a t e d r i n g s w i t h Dad symmetry.

The o t h e r f o r c e constants w i l l be used t o o b t a i n a b e t t e r fit between theore- t i c a l and experimental phonon energies f o r b o t h t-Se and a-Se.

References

1 , G . LUCOVSKY et a l . S o l i d S t a t e Corn. 5, 113, 1967.