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Weak field, low frequency third harmonic generation by nonlocalized dipoles in solids
Volume 26A, number 7
WEAK
P H Y SIC S L E T T E R S
FIELD,
LOW FREQUENCY BY N O N L O C A L I Z E D
26 February 1968
THIRD HARMONIC GENERATION DIPOLES IN SOLIDS
J. AMRAN SUSSMANN Department of Solid State Physics, Israel Atomic Energy Commission, Soreq Nuclear Research Center, Yavne, Israel Received 23 January 1968
It is shown that the polarizability of paraelectric centers in alkali halides and other nonlocalized dipoles deviates from linearity already in weak electric fields. Third harmonic generation and spurious dielectric losses result therefrom.
The p u r p o s e of this l e t t e r is to point out that the p o l a r i z a t i o n of delocalized dipoles d e p a r t s f r o m l i n e a r i t y a l r e a d y in:weak fields. The lowest h a r m o n i c g e n e r a t e d i s the third, and should be easily detected e x p e r i m e n t a l l y . S e v e r a l s y s t e m s have r e c e n t l y been shown to p r e s e n t delocalized e l e c t r i c dipoles, as for i n s t a n c e p a r a e l e c t r i c defects in alkali h a l i d e s [1-4]. Colour c e n t e r s in smoky q u a r t z [5,6], may be another example. The ground state of the delocalized dipoles i s split by the c r y s t a l field s y m m e t r y into states which - in the a b s e n c e of an e l e c t r i c field - have no dipole. However, off diagonal dipole m a t r i x - e l e m e n t s do connect them and thus an e l e c t r i c field can mix them, leading to new s t a t e s p r e s e n t ing dipoles. Weak fields a r e quite effective, b e c a u s e the splitting between l e v e l s is s m a l l (of the o r d e r of l ° K ) . Again just b e c a u s e of this s m a l l n e s s of the splitting, none of the s t a t e s i s empty. All this leads at low t e m p e r a t u r e s to field induced changes in the population d i s t r i b u t i o n . We shall take as a model a charge in a s y m m e t r i c a l double potential well. The splitting between the lowest s y m m e t r i c a l and a n t i - s y m m e t r i c a l wave functions being 5 and the off diagonal e l e c t r i c dipole m a t r i x e l e m e n t being ~ the e n e r g i e s of t h e s e two levels in the p r e s e n c e of a weak e l e c t r i c field (P ~ << 5) b e c o m e s E l , 2 = E o + ½5(1 + 2 ( # ~ / 5 ) 2) ,
(1)
~1,2 = ~ ( 2 p 2 ~ / 5 ) (1 - 2(U~/5) 2) ,
(2)
and the e l e c t r i c dipole
where ~ is the local field acting on the p o l a r i z a b l e c e n t e r . The a v e r a g e p o l a r i z a t i o n i s given by N1/I 1 + N2~ 2. The occupation f r a c t i o n s N1, N 2 a r e obtained f r o m N 1 + N 2 = 1 and N1 = - NI(W12+ W21) + W21 •
(3)
The t r a n s i t i o n p r o b a b i l i t i e s Wij a r e affected by the energy changes induced by the e l e c t r i c field. We a s s u m e 5 << kT. When the t r a n s i t i o n p r o c e s s can be d e s c r i b e d by the A r r h e n i u s law, we get W21(~) = W21(0) (1 +U2~2/6kT) , (4) W12(~) + W21(~) = (W12(0) + W21(0)) (1 +/~2~ 2/2k2T2 ) , w h e r e a s in the c a s e of single phonon p r o c e s s e s [7] which p r e d o m i n a t e at liquid h e l i u m t e m p e r a t u r e s
298
Volume 26A, number 7
PHYSICS LETTERS
W21(~) = W21(0)
1+2---~2--
26 February 1968
~
12k2T2
' (5)
W12(C) + W21(C) = (W12(0) + W21(0)) rL1 + 2 w h e r e n i s e i t h e r 1 o r 3 [7]~ depending on the coupling between the phonons and the dipole. With ~ 2 = ~ o2 c o s 2 o ) t = ~ o 2 ( 1 + c o s 2oJt) and using eqs. (4) and (5) eq. (3) t a k e s the f o r m ,,2~2
, 2~2
N1 = - N1~'-1(1+ "2¢02 +--/~2d °2COS 2wt) + ,-1(1+ exp (-5 kT))-I (1 + -~"- ~ o + --~--~ ~ o cos 2 w t \] ~2
and k being obtained f r o m eq • (4) o r (5), and T-1 = W12 + W21" F o r s m a l l ~ o / 7 l e a d s to the a p p r o x i m a t e solution for the population d i f f e r e n c e 5
1 _ 1
~z2~2 n-1 rt=1
and # ~ o / k ,
eq. (6)
n =
with tan 51 = 2WT . The a v e r a g e p o l a r i z a t i o n i s M = ~I(N1 -N2) and f r o m eqs. (2) and (7) we get
-v"2°2 I
(6
.2o2 (8) + / 1 ~2~2+4WZTz ( 1 1 1-. ~
(cos
+
+ c o s (3
t+
t
Thus we can s e e that the t h i r d h a r m o n i c of the a p p l i e d e l e c t r i c f i e l d f r e q u e n c y (and h i g h e r ones, which have been n e g l e c t e d in (8)) i s g e n e r a t e d w h e n e v e r the population can follow the changes of the e n e r g y , that i s f o r 2wr << 1. In the s y s t e m we a r e c o n s i d e r i n g , a new s o u r c e of d i e l e c t r i c l o s s a p p e a r s , b e s i d e s the u s u a l ones. T h e t h i r d (and higher) h a r m o n i c t e r m s in the p o l a r i z a t i o n p r o d u c e a field, and consequently a c u r r e n t in the e x p e r i m e n t a l setup. We s u g g e s t this m e c h a n i s m to be r e s p o n s i b l e f o r the a n o m a l o u s l o s s e s o b s e r v e d by K~nzig [8] in the KBr:KOH s y s t e m , at audio f r e q u e n c i e s . In the beginning of t h i s l e t t e r , c o l o u r c e n t e r s in s m o k y q u a r t z w e r e mentioned a s constituting a n o t h e r p o s s i b l e s y s t e m with d e l o c a l i z e d d i p o l e s . T h i s a s s u m p t i o n i s j u s t i f i e d by the t e m p e r a t u r e d e p e n dence of the p o l a r i z a b i l i t y a s obtained by De V o s and V o l g e r [5] which h a s the s a m e f o r m a s the f i r s t t e r m in e x p r e s s i o n (8) of t h i s l e t t e r . The fitting of the c u r v e s l e a d s to a splittIng 5 = 5°K. The e l e c t r i c r e s o n a n c e e x p e r i m e n t [6] i s a p p a r e n t l y in c o n t r a d i c t i o n with t h i s conclusion l e a d i n g to a much s m a l l e r splitting. H o w e v e r , d i f f e r e n t s e t s of s t a t e s may be involved in each of the two e x p e r i m e n t s . The a u t h o r thanks P r o f e s s o r W. K ~ z i g for his s t i m u l a t i n g question and to Dr. R. Englman v o r v e r y helpful d i s c u s s i o n s .
1. 2. 3. 4. 5. 6. 7. 8.
W.K~nzig, H.R.Hart Jr. and S.Roberts, Phys. Rev. Letters 13 (1964)543. U. Boshard. R.W. Dreyfus and W.Kffnzig, Phys. Kondens. Materie 4 (1965) 254. H.S. Sack and M. C. Moriarty, Solid State Comm. 3 (1965) 93. M.E. Baur and W.R. Salzman, Phys. Rev. Letters 18 (1967) 590. W.J.De Vos and J. Volger, Phys. Letters 24A (1967) 539. J.Kersen and J.Volger, Phys. Letters 24A (1967) 647. J.Amran Sussmann, J. Phys. Chem. Solids 28 (1967) 1643. W.K~nzig, private communication. * * * * *
299
WEAK
P H Y SIC S L E T T E R S
FIELD,
LOW FREQUENCY BY N O N L O C A L I Z E D
26 February 1968
THIRD HARMONIC GENERATION DIPOLES IN SOLIDS
J. AMRAN SUSSMANN Department of Solid State Physics, Israel Atomic Energy Commission, Soreq Nuclear Research Center, Yavne, Israel Received 23 January 1968
It is shown that the polarizability of paraelectric centers in alkali halides and other nonlocalized dipoles deviates from linearity already in weak electric fields. Third harmonic generation and spurious dielectric losses result therefrom.
The p u r p o s e of this l e t t e r is to point out that the p o l a r i z a t i o n of delocalized dipoles d e p a r t s f r o m l i n e a r i t y a l r e a d y in:weak fields. The lowest h a r m o n i c g e n e r a t e d i s the third, and should be easily detected e x p e r i m e n t a l l y . S e v e r a l s y s t e m s have r e c e n t l y been shown to p r e s e n t delocalized e l e c t r i c dipoles, as for i n s t a n c e p a r a e l e c t r i c defects in alkali h a l i d e s [1-4]. Colour c e n t e r s in smoky q u a r t z [5,6], may be another example. The ground state of the delocalized dipoles i s split by the c r y s t a l field s y m m e t r y into states which - in the a b s e n c e of an e l e c t r i c field - have no dipole. However, off diagonal dipole m a t r i x - e l e m e n t s do connect them and thus an e l e c t r i c field can mix them, leading to new s t a t e s p r e s e n t ing dipoles. Weak fields a r e quite effective, b e c a u s e the splitting between l e v e l s is s m a l l (of the o r d e r of l ° K ) . Again just b e c a u s e of this s m a l l n e s s of the splitting, none of the s t a t e s i s empty. All this leads at low t e m p e r a t u r e s to field induced changes in the population d i s t r i b u t i o n . We shall take as a model a charge in a s y m m e t r i c a l double potential well. The splitting between the lowest s y m m e t r i c a l and a n t i - s y m m e t r i c a l wave functions being 5 and the off diagonal e l e c t r i c dipole m a t r i x e l e m e n t being ~ the e n e r g i e s of t h e s e two levels in the p r e s e n c e of a weak e l e c t r i c field (P ~ << 5) b e c o m e s E l , 2 = E o + ½5(1 + 2 ( # ~ / 5 ) 2) ,
(1)
~1,2 = ~ ( 2 p 2 ~ / 5 ) (1 - 2(U~/5) 2) ,
(2)
and the e l e c t r i c dipole
where ~ is the local field acting on the p o l a r i z a b l e c e n t e r . The a v e r a g e p o l a r i z a t i o n i s given by N1/I 1 + N2~ 2. The occupation f r a c t i o n s N1, N 2 a r e obtained f r o m N 1 + N 2 = 1 and N1 = - NI(W12+ W21) + W21 •
(3)
The t r a n s i t i o n p r o b a b i l i t i e s Wij a r e affected by the energy changes induced by the e l e c t r i c field. We a s s u m e 5 << kT. When the t r a n s i t i o n p r o c e s s can be d e s c r i b e d by the A r r h e n i u s law, we get W21(~) = W21(0) (1 +U2~2/6kT) , (4) W12(~) + W21(~) = (W12(0) + W21(0)) (1 +/~2~ 2/2k2T2 ) , w h e r e a s in the c a s e of single phonon p r o c e s s e s [7] which p r e d o m i n a t e at liquid h e l i u m t e m p e r a t u r e s
298
Volume 26A, number 7
PHYSICS LETTERS
W21(~) = W21(0)
1+2---~2--
26 February 1968
~
12k2T2
' (5)
W12(C) + W21(C) = (W12(0) + W21(0)) rL1 + 2 w h e r e n i s e i t h e r 1 o r 3 [7]~ depending on the coupling between the phonons and the dipole. With ~ 2 = ~ o2 c o s 2 o ) t = ~ o 2 ( 1 + c o s 2oJt) and using eqs. (4) and (5) eq. (3) t a k e s the f o r m ,,2~2
, 2~2
N1 = - N1~'-1(1+ "2¢02 +--/~2d °2COS 2wt) + ,-1(1+ exp (-5 kT))-I (1 + -~"- ~ o + --~--~ ~ o cos 2 w t \] ~2
and k being obtained f r o m eq • (4) o r (5), and T-1 = W12 + W21" F o r s m a l l ~ o / 7 l e a d s to the a p p r o x i m a t e solution for the population d i f f e r e n c e 5
1 _ 1
~z2~2 n-1 rt=1
and # ~ o / k ,
eq. (6)
n =
with tan 51 = 2WT . The a v e r a g e p o l a r i z a t i o n i s M = ~I(N1 -N2) and f r o m eqs. (2) and (7) we get
-v"2°2 I
(6
.2o2 (8) + / 1 ~2~2+4WZTz ( 1 1 1-. ~
(cos
+
+ c o s (3
t+
t
Thus we can s e e that the t h i r d h a r m o n i c of the a p p l i e d e l e c t r i c f i e l d f r e q u e n c y (and h i g h e r ones, which have been n e g l e c t e d in (8)) i s g e n e r a t e d w h e n e v e r the population can follow the changes of the e n e r g y , that i s f o r 2wr << 1. In the s y s t e m we a r e c o n s i d e r i n g , a new s o u r c e of d i e l e c t r i c l o s s a p p e a r s , b e s i d e s the u s u a l ones. T h e t h i r d (and higher) h a r m o n i c t e r m s in the p o l a r i z a t i o n p r o d u c e a field, and consequently a c u r r e n t in the e x p e r i m e n t a l setup. We s u g g e s t this m e c h a n i s m to be r e s p o n s i b l e f o r the a n o m a l o u s l o s s e s o b s e r v e d by K~nzig [8] in the KBr:KOH s y s t e m , at audio f r e q u e n c i e s . In the beginning of t h i s l e t t e r , c o l o u r c e n t e r s in s m o k y q u a r t z w e r e mentioned a s constituting a n o t h e r p o s s i b l e s y s t e m with d e l o c a l i z e d d i p o l e s . T h i s a s s u m p t i o n i s j u s t i f i e d by the t e m p e r a t u r e d e p e n dence of the p o l a r i z a b i l i t y a s obtained by De V o s and V o l g e r [5] which h a s the s a m e f o r m a s the f i r s t t e r m in e x p r e s s i o n (8) of t h i s l e t t e r . The fitting of the c u r v e s l e a d s to a splittIng 5 = 5°K. The e l e c t r i c r e s o n a n c e e x p e r i m e n t [6] i s a p p a r e n t l y in c o n t r a d i c t i o n with t h i s conclusion l e a d i n g to a much s m a l l e r splitting. H o w e v e r , d i f f e r e n t s e t s of s t a t e s may be involved in each of the two e x p e r i m e n t s . The a u t h o r thanks P r o f e s s o r W. K ~ z i g for his s t i m u l a t i n g question and to Dr. R. Englman v o r v e r y helpful d i s c u s s i o n s .
1. 2. 3. 4. 5. 6. 7. 8.
W.K~nzig, H.R.Hart Jr. and S.Roberts, Phys. Rev. Letters 13 (1964)543. U. Boshard. R.W. Dreyfus and W.Kffnzig, Phys. Kondens. Materie 4 (1965) 254. H.S. Sack and M. C. Moriarty, Solid State Comm. 3 (1965) 93. M.E. Baur and W.R. Salzman, Phys. Rev. Letters 18 (1967) 590. W.J.De Vos and J. Volger, Phys. Letters 24A (1967) 539. J.Kersen and J.Volger, Phys. Letters 24A (1967) 647. J.Amran Sussmann, J. Phys. Chem. Solids 28 (1967) 1643. W.K~nzig, private communication. * * * * *
299