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Submitted on 1 Jan 1981
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SIMULATION OF LONG-WAVELENGTH OPTICAL
PHONONS BY GENERALIZED INTERNAL
STRAINS
E. Anastassakis, M. Cardona
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C6, suppl6ment au n o 12, Tome 42, d6cembre 1981 page C6-537
SIMULATION O F LONG-WAVELENGTH OPTICAL PHONONS BY GENERALIZED SNTERNAL STRAINS
E.M. ~nastassakis+and M. ~ardona*
NationaZ Technical University, Athens, Greece.
*
Mm-PZanck-Institute for Solid State Research Stuttgart, F. R. G.Abstract.- A generalized external force applied to a crystal can introduce new atomic position parameters known as internal strains. Each internal strain may be looked as a "frozen" q = 0 optical phonon of appropriate symmetry. Explicit forms can be worked out for the internal strains produced by any type of force and crystal class. Related crystal parameters can be determined in any case with X-ray diffraction techniques.
The concept of internal strain will be generalized to include all possibilities of microscopic relative displacements of the various Bravais sublattices which are produced by any type of external force
F. When F is applied to the crystal a macroscopic strain tl may be pro- duced. The mechanism which produces 11 depends on the nature of F [e.g. piezoelectricity (electrostriction) when F is an electric field E in first (second) order, piezomagnetism (magnetostriction) when F is a magnetic field h in first (second) order, etc]. Because of F the k th atom in the dth cell moves from its initial position r,,, to a new one
=: (l + tl r,,,+ U, ( 1 )
The first term is the macroscopic contribution to r;+ while the second term (uk) represents a microscopic strain due to F . It is the same for all atoms of type k in the crystal, i.e., it represents a microscopic displacement of the entire kth Bravais sublattice and as such it con- stitutes an internal strain1. Both terms in ( 1 ) depend on the symmetry of the crystal independently of each other.
There are 3(N-1) different sublattice displacements, where N the no. of atoms per unit cell. 3(N-l) also gives the no. of q
-
0 optical phonons of the crystal. The normal coordinate U of the phonon j andj
the internal strainu, (or u ~ , ~ , X=x,y,z) are qualitatively similar in the sense that both represent sublattice displacements and transform alike under the crystal symmetry operations. When "frozen" each phonon will look exactly like one of the uis. The symmetry of U will deter-
j
mine the type of F which is necessary to produce its internal strain
U , . This symmetry is described by one of the point group irreducible
representations
r
(j ).
Thusr
( j) Er
(uk).
The variousr
(j) 'S are obtai- 'partial support by the National Research Counsil and by the ResearchCommittee of the Technical University of Athens.
C6-538 JOURNAL DE PHYSIQUE
ned from the reduction of the reducible reD of the most general displacement of the q 0 optical phonon of the crystal.
The connection between F and u k i s derived thermodynamically. Let
f be the intensive parameter which corresponds to F. ( F is treated 2
as an extensive parameter ) . To first order of u,we can write F= G
.
uk or f = g . u k ( 2 )This results in a contribution to the internal energy of the crystal 6 ~ ' ~ ) = F . f = F . g - U , = guy), Uk,A (3) where ~.r may be a composite index. Since 6u(k) is a scalar, the only products F U entering (3) are those which transform like scalars.
P k,A
This allows one to determine the type F necessary to produce a speci- fic U, and vice versa. The criterion is that
r
( j ),
orr
(uk),
be inclu-ded in the reduction of r(F) at least once, in general n times. The j
number n indicates the number of independent components of the ten- j
sor coefficient g
.
The form of g is identical to that of known mo- 11,Ade coef f icients3. Typical examples for various forces are given next.
1 . Electric field E . The reduction r(E) includes only IR-active pho- nons.Eq. (3) yields g = (a2~(k)/a~Pauk?A), i.e., gu,i transforms like
11,X
the phonon effective charge tensor3 e!!l*=
(a
'~(j)/a&
au j,~).2. Elastic stress or strainq. The reduction
r(n)
includes only Raman- (k)active phonons in symmetric scatterind4.since gp,3A=(a2~
/anp,auk,,) it transforms like the Raman tensor a(j) = (a3u(j) /a& abaau. )
.
P ~ , A P ?,A 3. Electric field gradient V E = (aE,/a
.
The reduction r(VE) inclu-P 0 XP)
des only Raman-active phonons in non-symmetric scattering.9 trans-
3 ap,A
forms like the non-symmetric Raman tensor
.
4. Strain gradient V r)
.
The reduction of r(Vq) is the same as that D b Tof the piezo-electric tensor. transforms like the electric 3 guPp,A
field-induced Raman tensor
.
5. Hydrostatic pressure P. The reduction r(P) coincides with A l l the totally symmetric rep. Hence there are as many internal strainsuk pro- duced by P as the number of phonons Al. This also is the same as the number of atomic position parameters which cannot be uniquely determi- ned by symmetry arguments4. The tensor g becomes a scalar.
1 1 5 ,
6. Magnetic field H . U will transform like an axial vector.
g ~ is . ~
k,A
like a second-rank polar tensor which reverses sign upcn time rever-
5
sal (c-type tensor ) . I can only exist for the 90 magnetic crystal classes. The matrices of such tensors are given in tables 7 and 4 of Ref .5.
As an example we consider a-quartz (N=9 point group D3). We have 3
rapt
= 4A1+
4A2+
8E r(BE) = 2 A 1+
A2 +3E r(P) = A l r(E) = A2+
Er
(H) = A~+
EAccording t o t h e s e r e d u c t i o n s t h e r e a r e 16 i n d e p e n d e n t i n t e r n a l s t r a i n s grouped i n t o 3 t y p e s , 4 of t y p e AI ( n o n - d e g e n e r a t e ) , 4 o f t y - p e A2 (non-degenerate) and 8 o f t y p e E (doubly d e g e n e r a t e ) . Type AI can be produced by ~ , V E # P a n d v q w i t h 2 , 2 , 1 , 2 i n d e p e n d e n t components e a c h , f o r e a c h of t h e c o r r e s p o n d i n g t e n s o r s g
.
Type A2 c a n b e produ- ced by E , V E and V0 w i t h 1 , l ,4 components e a c h f o r t h e t e n s o r s 9 , e t c . For a magnetic c r y s t a l w i t h N=9 and magnetic s t r u c t u r e D3, a f i e l d H w i l l produce 4 i n t e r n a l s t r a i n s o f t y p e A2 and 8 o f t y p e E w i t h o n l y one component e a c h f o r t h e t e n s o r g . Once t h e forms of g a r e w r i t t e n , s i m i l a r o n e s h o l d f o r G of ( 2 ) and i t s i n v e r s e y ,where U,= y.
F.T h i s i s because g , G and y have t h e same symmetry4. P1e w i l l c o n s i d e r t h e c a s e of F = Vq f o r t h e A2 t y p e o f i n t e r n a l s t r a i n s of a - q u a r t z . Y f o l - lows from t h e e l e c t r i c f i e l d - i n d u c e d Raman t e n s o r , a c c o r d i n g t o t h e scheme 3
a E =
( a * u ( j ) / a ~ ~ a \ a ~ , a ~ ~ , ~ ) j
( a 2 u ( " ) / a v
n
a u k , ,o - c ~ , ~ P ' n ~ p , ~ ( 4 )
The r e s u l t s f o r y and U , = y.vq a r e
There a r e 4 sets o f v a l u e s f o r yl...y4 s i n c e we e x p e c t 4 i n d e p e n d e n t i n t e r n a l s t r a i n s of t y p e A2 due t o Vq. From ( 6 ) one can e a s i l y d e r i v e t h e s e l e c t i o n r u l e s f o r i n d u c i n g a d e s i r e d t e r m o f u A
.
2
,=
I n t e r n a l s t r a i n s due t o e x t e r n a l s t r e s s e s have been s t u d i e d f o r some m a t e r i a l s t h r o u g h t h e o b s e r v a t i o n o f f o r b i d d e n X-ray d i f f r a - c t i o n s 4 . I n p r i n c i p l e s i m i l a r t e c h n i q u e s can be employed t o m a n i f e s t i n t e r n a l s t r a i n s due t o f o r c e s o t h e r t h a n s t r e s s e s .
R e f e r e n c e s
1 . M. Born and K. Huang, Dynamical Theory o f C r y s t a l l a t t i c e s , Claren- don P r e s s Oxford 1962.
2. H . B . C a l l e n , Thermodynamics, J. Wiley, N.Y. 1960.
3. E. A n s t a s s a k i s , i n Dynamical P r o p e r t i e s o f S o l i d s , v o l . 4 e d . G . K . Horton and A.A. Maradudin, N. Holland 1980 and r e f e r e n c e s t h e r e i n . 4 . E. A n a s t a s s a k i s and M. ~ a r d o n a , phys. s t a t . s o l . ( b )
104,
589( 1 9 8 1 ) .
