TEMPERATURE DEPENDENCE OF PHONON-PHONON INTERACTION IN SILICON

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TEMPERATURE DEPENDENCE OF

PHONON-PHONON INTERACTION IN SILICON

M. Breazeale, J. Philip

To cite this version:

M. Breazeale, J. Philip. TEMPERATURE DEPENDENCE OF PHONON-PHONON INTER- ACTION IN SILICON. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-134-C6-136.

�10.1051/jphyscol:1981640�. �jpa-00221576�

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JOURNAL DE PHYSIQUE

CoZZoque C6, suppZe'ment au n012, Tome 42, de'cembre 1981 page 126- 134

TEMPERATURE DEPENDENCE OF PHONON-PHONON INTERACTION IN SILICON

M.A. Breazeale and J. Philip

Department of Physics, The Unioersity o f Tennessee, fiomiZZe, TN 37926, U. S. A.

Abstract.- The generation o f a second harmonic of an i n i t i a l l y s i n u s o i d a l u l t r a s o n i c wave can be considered as c o l l i n e a r i n t e r a c t i o n between two i d e n t i - c a l phonons t o produce t h e sum frequency phonon. This phenomenon has been s t u d i e d i n considerable d e t a i l , and t h e s t u d i e s have l e d t o a technique f o r measuring harmonic amplitudes and c a l c u l a t i n g anharmonic c o u o l i n g c o e f f i c i e n t s i n terms o f second- and t h i r d - o r d e r e l a s t i c constants. Yeasurements havebeen made on s i l i c o n , and have r e s u l t e d f o r t h e f i r s t time i n a complete s e t o f t h i r d - o r d e r e l a s t i c constants a t temperatures between 3°K and 300°K. Results a r e presented f o r s i l i c o n between 3°K and 300°K and a comparison i s made between TOE constants c a l c u l a t e d by using o u r data i n t h e Keating theory and TOE constants evaluated by combining o u r data and those o f B e a t t i e and Schirber.

I n t r o d u c t i o n . - Nonlinear i n t e r a c t i o n o f two phonons o f frequency wl and w2 produces both sum and d i f f e r e n c e frequency phonons. This i n t e r a c t i o n can be described i n t e r n s o f an i n t e r a c t i o n Hami 1 t o n i a n which contains c o u ~ l i n g parameters t h a t can be expressed i n terms o f higher-order e l a s t i c constants. S p e c i a l i z a t i o n t o c o l l i n e a r propagation o f phonons of t h e same frequency leads t o a sum frequency which i s t h e second harmonic. Coing t o t h e continuum l i m i t leads t o a s e t o f equations which con- t a i n n o n l i n e a r c o e f f i c i e n t s which can be measured a t u l t r a s o n i c frequencies. Ke have measured t h e n o n l i n e a r i t y oarameters o f s i l i c o n and have i n t e r p r e t e d our r e s u l t s i n terms o f the Keating l a t t i c e dynamical model t o o b t a i n a l l s i x t h i r d - o r d e r e l a s t i c constants as a f u n c t i o n o f temperature between 3°K and 300°K. I n a d d i t i o n , we have combined our data w i t h t h a t o f B e a t t i e and Schirber t o o b t a i n t h e TOE constants a t 300°K, 77"K, and 4°K.

Method.- The experimental setup and t h e procedure t o measure t h e absolute amplitudes o f t h e fundamental and generated second harmonic o f a n i n i t i a l l y s i n u s o i d a l u l t r a s o n i c wave propagating through a n o n l i n e a r medium has been described i n e a r l i e r pub1 ic a - t i o n s . 1 y 2 y 3 I n t h e s p e c i a l d i r e c t i o n s along which pure mode l o n g i t u d i n a l wave propa- g a t i o n i s possible, t h e n o n l i n e a r wave equation f o r a c u b i c c r y s t a l reduces t o t h e form 4

where K2 and K3 a r e l i n e a r combinations 0.f second- and t h i r d - o r d e r e l a s t i c constants r e s p e c t i v e l y given by

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981640

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K2[111] = 7 1 (CI1 + 2C12 + 4C44) ,

and

Considering an i n i t i a l l y s i n u s o i d a l disturbance a t a=O, t h e s o l u t i o n t o Ea. (1) i s o f t h e form

2 2

A1 k a 3K2 + K3

u = Alsin(ka-at) - - ₈ ₍

K2 )cos 2(ka-wt) + .. .

i n which a i s t h e propagation d i s t a n c e and k i s t h e propagation constant. Measuring t h e amplitude o f t h e fundamental wave and t h e amplitude o f t h e second harmonicallows one t o c a l c u l a t e t h e Kg parameters along t h e pure mode d i r e c t i o n s . Eleasurements have been made f o r s i l i c o n as a f u n c t i o n o f temperature between 3 and 300°K by measuring t h e absolute A1 and A2 values using a v a r i a b l e gap c a p a c i t i v e d e t e c t o r . The K2 parameters have been determined by measuring t h e l o n g i t u d i n a l wave v e l o c i t i e s along t h e symmetry d i r e c t i o n s .

~ e a t i n ~ ' s ~ t h e o r y o f t h e TOE constants o f diamond-1 i k e s o l i d s i s b a s i c a l l y e q u i v a l e n t t o t h e Born-Huang approach o f imposing t h e i n v a r i a n c e requirements on t h e anharmonic s t r a i n energy o f t h e c r y s t a l . Keating has d e r i v e d t h e expressions f o r t h e s i x TOE constants i n t e r n s o f t h r e e anharmonic f i r s t and second neighbor f o r c e con- s t a n t s y, 6 and E and two harmonic f o r c e constants a and 6 :

Combining t h e s e t o f equations ( 3 ) and ( 5 ) , we can w r i t e t h e t h r e e anhamonic f o r c e constants E, 6 and y i n terms o f t h e K3 narameters. E v a l u a t i o n o f ^E, 6 and y as a f u n c t i o n o f temperature from t h e temperature dependent K3 parameters leads t o t h e determination o f a l l t h e TOE constants as a f u n c t i o n o f temperature.

clll = Y - 6 + 9E Cl12 = y - ⁶ + ^E Clz3 = Y + 36 - 3 E

2 2

C l 4 = Y t 6 ( l + O + E ( l + ( ) ( 3 < - 1 ) t c12f2

Results and Discussion.- The TOE constants o f s i l i c o n p l o t t e d as a f u n c t i o n o f temperature a r e reproduced i n Figs. 1 ( a ) and 2(a). We have p l o t t e d both t h e calcu- l a t e d data p o i n t s as w e l l as t h e best f i t curve through them t o show t h e e f f e c t o f e r r o r ~ r o p a g a t i o n . B e a t t i e and Schirber have measured t h e pressure d e r i v a t i v e s o f 6 t h e SOE constants o f s i l i c o n a t 4, 77 and 300°K. Combining t h e i r r e s u l t s w i t h o u r K3 values, we have i s o l a t e d a l l t h e s i x TOE constants a t these s p e c i f i c temperatures.

where

a-B - 2C12

< = - -

a+6 Cll+C12 ' ( 5 )

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C6- 136 JOURNAL DE PHYSIQUE

- 4 '

S TEMPERATURE I'KI

5 ^,^{40 ,}⁸⁰^,^l?O^,¹⁶⁰^,²⁰⁰^,²⁴⁰^,^280,

,

Q

7 w (b) Our data p l u s B e a t t i e and Schirber

( a ) Our data p l u s Keating theory data

Fig. 1. The TOE constants C1ll, ClI2, and d166 of s i l i c o n p l o t t e d as a f u n c t i o n o f temperature.

TEMPERATURE ⁽^{O K}1 .-. 0 40 8 0 1 2 0 IGO 2 0 0 2 4 0 2 8 0

"= ^1.2

0

(b) Our data p l u s B e a t t i e and S c h i r b e r ( a ) Our data p l u s Keating theory data

F i g . 2. The TOE constants ClZ3, C144, and C456 o f s i l i c o n p l o t t e d as a f u n c t i o n o f temperature.

The c a l c u l a t e d p o i n t s and t h e curves j o i n i n g them a r e given i n Figs. 1 ( b ) and 2(b) A comparison between l ( a ) and l ( b ) and between 2(a) and 2(b) shows t h e e x t e n t t o which t h e Keating model and experimental r e s u l t s agree f o r s i l i c o n . The f a c t t h a t we can measure t h r e e q u a n t i t i e s and t h a t t h e Keating model i n v o l v e s o n l y t h r e e anharmonic f o r c e constants has made t h i s approach p o s s i b l e . The Grcneisen parameters o f s i l i c o n evaluated as a f u n c t i o n o f temperature using our TOE constant data a r e i n b e t t e r agreement w i t h experimental curves than those r e p o r t e d by e a r l i e r workers who t r e a t e d TOE constants as temperature independent.

Acknowledgment.- Research supported by t h e O f f i c e o f Naval Research.

References .- ^1. ^El.B. Gauster and M. A. Breazeale, Phys. Rev. 168, 655 (1968).

2. J. A. Bains, Jr., and M. A. Breazeale, Phys. R% G, 3623 (1976).

3. Jacob P h i l i p and M. A. Breazeale, J. Appl. Phys. 52, 3383 (1981 ) . 4. M. A. Breazeale and J. Ford, J. Appl . ^Phys.36, 3486 (1965).

5. P. N. Keating, Phys. Rev. 149, 674 (1966).

6. A. G. B e a t t i e and J. E. S c m b e r , Phys. Rev. 87, 1548 (1970).