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Calculation of steady state distribution functions by exploiting stability
Volume 26A, number 9
PHYSICS LETTERS
CALCULATION
OF
25 March 1968
STEADY STATE DISTRIBUTION EXPLOITING STABILITY
FUNCTIONS
BY
H. D. R E E S
Royal Radar Establishment, Malvern, Worcestershire, UK Received 8 February 1968
The stability of the steady state may be exploited to calculate the distribution function for free carriers in an electric field without introducing approximations in the form of the distribution function.
T h e B o l t z m a n n equation is u s u a l l y u s e d to c a l c u l a t e the d i s t r i b u t i o n function f o r f r e e c a r r i e r s in a s e m i c o n d u c t o r in an e l e c t r i c field, but app r o x i m a t i o n s in the f o r m of the d i s t r i b u t i o n function a r e often n e c e s s a r y to obtain a solution. T h i s l e t t e r o u t l i n e s an a l t e r n a t i v e a p p r o a c h , f r e e of such a p p r o x i m a t i o n s , b a s e d on exploiting the s t a b i l i t y of the s t e a d y state, i.e. a f t e r long enough the d i s t r i b u t i o n function will tend to the steady s t a t e f o r m i r r e s p e c t i v e of i ts initial value. T h i s fact m ay be d e s c r i b e d by the equation oO
f(k)=
lira n'-~
°°
jdk'g(k')J dtPn(k',k,t) (1) 0
w h e r e Pn(k', R , t ) i s the p r o b a b i l i t y that an e l e c t r o n i n i t i a l l y at k ' is at k at t i m e t a f t e r b e i n g s c a t t e r e d n t i m e s , f ( R ) i s the steady state d i s t r i bution function and g ( k ' ) is a r b i t r a r y . The j u s t i f i c a t i o n f o r eq. (1) i s that f o r l a r g e e n o u g h n , Pn ( R ' , R , t ) i s n e g l i g i b l e except f o r v a l u e s of t l a r g e c o m p a r e d with the t i m e for the d i s t r i b u t i o n function to s e t t l e to the steady s t a t e v a l u e ; f o r t h e s e v a l u e s of t, Pn (k', R, t) will be independent of k ' and w i l l v a r y with k in the s a m e way as the s t e a d y s t at e d i s t r i b u t i o n function. Now denoting the t i n t e g r a l of Pn (R', k, t) by Pn ( k ' , k ) it i s s t r a i g h t f o r w a r d to e s t a b l i s h the recurrence relation
Pn+l(k',k) =Pn(k',k")S(k",k")Pg(k'" k) w h e r e S(R', k) is the s c a t t e r i n g r a t e f r o m k '
(2)
to k and i n t e g r a t i o n o v e r r e p e a t e d k v a r i a b l e s on the r i g h t i s i m p l i e d . Po(k', k) is the p r o b a b i l i t y that an e l e c t r o n i s not s c a t t e r e d d u r i n g i ts d r i f t in the f i e l d f r o m k ' to k. Using the r e c u r r e n c e r e l a t i o n (2) in conjunction with the l i m i t (1) shows f(k) to be the r e s u l t of the c o n v e r g e n t i t e r a t i v e 416
p r o c e s s , s t a r t i n g f r o m an a r b i t r a r y initial function, each i t e r a t i o n c o n s i s t i n g of i n t e g r a t i n g the p r o d u ct of the function f i r s t with S(k, k ' ) and then
with Po(k', k). Now i n t e g r a t i n g the product of the function with is s t r a i g h t f o r w a r d since the s c a t t e r i n g r a t e contains 6 functions in the initial and final e n e r g i e s . H o w e v e r P o ( k ' , k) contains the f a c t o r
S(k', k)
T
which is the p r o b a b i l i t y that an e l e c t r o n is not s c a t t e r e d during i t s flight f r o m k ' to R in the f i el d F . T is the t i m e of flight f r o m k ' to R and )t(k) is the s c a t t e r i n g r a t e S(k, R') i n t e g r a t e d o v e r all final s t a t e s k ' . Although ~.(R) is not difficult to e v a l u a t e f o r m o s t p h y s i c a l p r o c e s s e s , i n t e g r a t i n g it is u s u a l l y difficult and t h e r e f o r e the second p a r t of each i t e r a t i o n can be difficult. Howe v e r , adding a t e r m of the f o r m s ( k ) 5 ( k - k ' ) to S(k', k) will be of no p h y s i c a l c o n s e q u e n c e . T h i s t e r m can be r e g a r d e d as a " s e l f " s c a t t e r i n g r at e. I n c o r p o r a t i n g this e x t r a t e r m in the f i r s t p a r t of each i t e r a t i o n is e l e m e n t a r y , but, since k(k) m u s t be r e p l a c e d by S ( k ) + k(k), suitable definition of S(k) can much s i m p l i f y the evaluation of Po(k',k) and make the whole i t e r a t i o n n u m e r i c a l l y s t r a i g t h forward * A s i m i l a r i t e r a t i v e t e c h n i q u e was u s e d by Budd [1] to c a l c u l a t e d i s t r i b u t i o n functions in the s p e c i a l c a s e of s c a t t e r i n g by n o n - p o l a r phonons, but the p r e s e n t a n a l y s i s d i f f e r s f i r s t l y by a p p e a l ing to the s t a b i l i t y of the steady state r a t h e r than u si n g the B o l t z m a n n equation and secondly by in* It can be shown that S(k) must not be negative for iterative convergence to be assured.
Volume 26A, number 9
PHYSICS LETTERS
t r o d u c i n g the s e l f s c a t t e r i n g r a t e S(k) to s i m p l i f y the n u m e r i c a l a n a l y s i s . N u m e r i c a l c a l c u l a t i o n s have been c a r r i e d out with the definition S(k) = F - k(k), w h e r e r is a p o s i t i v e constant, r d e t e r m i n e s the r a t e of c o n v e r g e n c e of the i t e r a t i o n , but not the final r e s u l t . As an i l l u s t r a t i o n , c a l c u l a t i o n s f o r e l e c t r o n s in g a l l i u m a r s e n i d e at 300°K, t r e a t i n g the k = 0 c o n duction band m i n i m u m as i s o t r o p i c and p a r a b o l i c , but allowing f o r both p o l a r o p t ic a l and a c o u s t i c d e f o r m a t i o n p o t en t i a l phonon s c a t t e r i n g , gave a low fi el d mobility of 10 000 c m 2 / V s e c and a m o bility at 1 k V / c m of 9100 c m 2 / V s e c . T h e low fie ld mobility i s c l o s e to the 9500 c m 2 / V s e c Hall mobility r e p o r t e d by Kang et al. [2] f o r t h e i r h i g h e s t p u r i t y GaAs s p e c i m e n . Th e r e a s o n a b l e a g r e e m e n t with e x p e r i m e n t in t h i s i l l u s t r a t e e x a m p l e s u g g e s t s that the t h e o r e -
25 March 1968
t i c a l method m ay be u s e f u l f o r c a l c u l a t i n g d i s t r i bution functions in p r o b l e m s w h e r e analytic s o l u tions a r e not a v a i l a b l e . Thanks a r e . d u e to J. B. A r t h u r who w r o t e c e r tain s p e c i a l s e c t i o n s of the c o m p u t e r p r o g r a m m e and to W. F a w c e t t f o r u s e f u l d i s c u s s i o n s . C o n t r i b u t e d by p e r m i s s i o n of the D i r e c t o r of R. R. E. Co p y r i g h t C o n t r o l l e r H. M. S. O.
References 1. H. Budd, Phys. Rev. 158 (1969) 7~8. 2. C.S. Kang, P.E. Greene and M.~CLAtalta, Devices Research Conference, Santa Barba~ra, 1967.
* * * * *
PHOTON STATISTICS FOR THRESHOLD LASER LIGHT WITH FINITE COUNTING TIME * R. F. CHANG, V. KORENMAN
Department of Physics and Astronomy, University of Maryland, College Park, Maryland, USA and R. W. D E T E N B E C K
Department of Physics, University of Vermont, Burlington, Vermont, USA Received 20 January 1968
The 2nd, 3rd and 4th cumulants of the distribution of light intensity from a He-Ne laser operating just below threshold have been measured as counting time is increased. The results agree with predictions compute! under the assumption that the dependence of the correlation functions on the time variables is the same as for Gaussian light.
E a r l y m e a s u r e m e n t s of c o r r e l a t i o n s of i n t e n sity f l u c t u a t i o n s in well s t a b i l i z e d H e - N e l a s e r s o p e r a t i n g n e a r t h r e s h o l d r e p o r t e d only the c o r r e l a t i o n s of fluctuations all at the s a m e t i m e [1,2]. T h e s e d e t e r m i n e d the p r o b a b i l i t y d e n s it y of inst a nt an eo u s s a m p l e s of the intensity (I. P. D.). R e c e n t s t u d i es [3,4] have p r o v i d e d the t i m e d e p e n d e n c e of the t w o - t i m e c o r r e l a t i o n function in the t h r e s h o l d region. All t h e s e r e s u l t s have been c o n s i s t e n t with c u r r e n t l a s e r t h e o r i e s b a s e d on the F o k k e r - P l a n c k equation with Markoffian n o i s e s o u r c e s [5-10].
We add h e r e m e a s u r e m e n t s of the t i m e d e p e n d en ce of the f i r s t f o u r n o r m a l i z e d c u m u l a n t s f o r counting t i m e s f r o m 3 to 1000 m i c r o s e c o n d s , f o r a l a s e r o p e r a t i n g s l i g h t l y below t h r e s h o l d w h e r e the fluctuation t i m e is about 100 m i c r o s e c o n d s . What is m e a s u r e d is an a v e r a g e o v e r e a c h of the * Work supported by NASA (grant NGR 21-002-022), ARPA (Contract SD-101) and the U. S. Army Research Office (contract DAHC 04 67 C 0023, under Project Defender) and the University of Maryland Computer Science Center under NASA grant NsG-398. 417
PHYSICS LETTERS
CALCULATION
OF
25 March 1968
STEADY STATE DISTRIBUTION EXPLOITING STABILITY
FUNCTIONS
BY
H. D. R E E S
Royal Radar Establishment, Malvern, Worcestershire, UK Received 8 February 1968
The stability of the steady state may be exploited to calculate the distribution function for free carriers in an electric field without introducing approximations in the form of the distribution function.
T h e B o l t z m a n n equation is u s u a l l y u s e d to c a l c u l a t e the d i s t r i b u t i o n function f o r f r e e c a r r i e r s in a s e m i c o n d u c t o r in an e l e c t r i c field, but app r o x i m a t i o n s in the f o r m of the d i s t r i b u t i o n function a r e often n e c e s s a r y to obtain a solution. T h i s l e t t e r o u t l i n e s an a l t e r n a t i v e a p p r o a c h , f r e e of such a p p r o x i m a t i o n s , b a s e d on exploiting the s t a b i l i t y of the s t e a d y state, i.e. a f t e r long enough the d i s t r i b u t i o n function will tend to the steady s t a t e f o r m i r r e s p e c t i v e of i ts initial value. T h i s fact m ay be d e s c r i b e d by the equation oO
f(k)=
lira n'-~
°°
jdk'g(k')J dtPn(k',k,t) (1) 0
w h e r e Pn(k', R , t ) i s the p r o b a b i l i t y that an e l e c t r o n i n i t i a l l y at k ' is at k at t i m e t a f t e r b e i n g s c a t t e r e d n t i m e s , f ( R ) i s the steady state d i s t r i bution function and g ( k ' ) is a r b i t r a r y . The j u s t i f i c a t i o n f o r eq. (1) i s that f o r l a r g e e n o u g h n , Pn ( R ' , R , t ) i s n e g l i g i b l e except f o r v a l u e s of t l a r g e c o m p a r e d with the t i m e for the d i s t r i b u t i o n function to s e t t l e to the steady s t a t e v a l u e ; f o r t h e s e v a l u e s of t, Pn (k', R, t) will be independent of k ' and w i l l v a r y with k in the s a m e way as the s t e a d y s t at e d i s t r i b u t i o n function. Now denoting the t i n t e g r a l of Pn (R', k, t) by Pn ( k ' , k ) it i s s t r a i g h t f o r w a r d to e s t a b l i s h the recurrence relation
Pn+l(k',k) =Pn(k',k")S(k",k")Pg(k'" k) w h e r e S(R', k) is the s c a t t e r i n g r a t e f r o m k '
(2)
to k and i n t e g r a t i o n o v e r r e p e a t e d k v a r i a b l e s on the r i g h t i s i m p l i e d . Po(k', k) is the p r o b a b i l i t y that an e l e c t r o n i s not s c a t t e r e d d u r i n g i ts d r i f t in the f i e l d f r o m k ' to k. Using the r e c u r r e n c e r e l a t i o n (2) in conjunction with the l i m i t (1) shows f(k) to be the r e s u l t of the c o n v e r g e n t i t e r a t i v e 416
p r o c e s s , s t a r t i n g f r o m an a r b i t r a r y initial function, each i t e r a t i o n c o n s i s t i n g of i n t e g r a t i n g the p r o d u ct of the function f i r s t with S(k, k ' ) and then
with Po(k', k). Now i n t e g r a t i n g the product of the function with is s t r a i g h t f o r w a r d since the s c a t t e r i n g r a t e contains 6 functions in the initial and final e n e r g i e s . H o w e v e r P o ( k ' , k) contains the f a c t o r
S(k', k)
T
which is the p r o b a b i l i t y that an e l e c t r o n is not s c a t t e r e d during i t s flight f r o m k ' to R in the f i el d F . T is the t i m e of flight f r o m k ' to R and )t(k) is the s c a t t e r i n g r a t e S(k, R') i n t e g r a t e d o v e r all final s t a t e s k ' . Although ~.(R) is not difficult to e v a l u a t e f o r m o s t p h y s i c a l p r o c e s s e s , i n t e g r a t i n g it is u s u a l l y difficult and t h e r e f o r e the second p a r t of each i t e r a t i o n can be difficult. Howe v e r , adding a t e r m of the f o r m s ( k ) 5 ( k - k ' ) to S(k', k) will be of no p h y s i c a l c o n s e q u e n c e . T h i s t e r m can be r e g a r d e d as a " s e l f " s c a t t e r i n g r at e. I n c o r p o r a t i n g this e x t r a t e r m in the f i r s t p a r t of each i t e r a t i o n is e l e m e n t a r y , but, since k(k) m u s t be r e p l a c e d by S ( k ) + k(k), suitable definition of S(k) can much s i m p l i f y the evaluation of Po(k',k) and make the whole i t e r a t i o n n u m e r i c a l l y s t r a i g t h forward * A s i m i l a r i t e r a t i v e t e c h n i q u e was u s e d by Budd [1] to c a l c u l a t e d i s t r i b u t i o n functions in the s p e c i a l c a s e of s c a t t e r i n g by n o n - p o l a r phonons, but the p r e s e n t a n a l y s i s d i f f e r s f i r s t l y by a p p e a l ing to the s t a b i l i t y of the steady state r a t h e r than u si n g the B o l t z m a n n equation and secondly by in* It can be shown that S(k) must not be negative for iterative convergence to be assured.
Volume 26A, number 9
PHYSICS LETTERS
t r o d u c i n g the s e l f s c a t t e r i n g r a t e S(k) to s i m p l i f y the n u m e r i c a l a n a l y s i s . N u m e r i c a l c a l c u l a t i o n s have been c a r r i e d out with the definition S(k) = F - k(k), w h e r e r is a p o s i t i v e constant, r d e t e r m i n e s the r a t e of c o n v e r g e n c e of the i t e r a t i o n , but not the final r e s u l t . As an i l l u s t r a t i o n , c a l c u l a t i o n s f o r e l e c t r o n s in g a l l i u m a r s e n i d e at 300°K, t r e a t i n g the k = 0 c o n duction band m i n i m u m as i s o t r o p i c and p a r a b o l i c , but allowing f o r both p o l a r o p t ic a l and a c o u s t i c d e f o r m a t i o n p o t en t i a l phonon s c a t t e r i n g , gave a low fi el d mobility of 10 000 c m 2 / V s e c and a m o bility at 1 k V / c m of 9100 c m 2 / V s e c . T h e low fie ld mobility i s c l o s e to the 9500 c m 2 / V s e c Hall mobility r e p o r t e d by Kang et al. [2] f o r t h e i r h i g h e s t p u r i t y GaAs s p e c i m e n . Th e r e a s o n a b l e a g r e e m e n t with e x p e r i m e n t in t h i s i l l u s t r a t e e x a m p l e s u g g e s t s that the t h e o r e -
25 March 1968
t i c a l method m ay be u s e f u l f o r c a l c u l a t i n g d i s t r i bution functions in p r o b l e m s w h e r e analytic s o l u tions a r e not a v a i l a b l e . Thanks a r e . d u e to J. B. A r t h u r who w r o t e c e r tain s p e c i a l s e c t i o n s of the c o m p u t e r p r o g r a m m e and to W. F a w c e t t f o r u s e f u l d i s c u s s i o n s . C o n t r i b u t e d by p e r m i s s i o n of the D i r e c t o r of R. R. E. Co p y r i g h t C o n t r o l l e r H. M. S. O.
References 1. H. Budd, Phys. Rev. 158 (1969) 7~8. 2. C.S. Kang, P.E. Greene and M.~CLAtalta, Devices Research Conference, Santa Barba~ra, 1967.
* * * * *
PHOTON STATISTICS FOR THRESHOLD LASER LIGHT WITH FINITE COUNTING TIME * R. F. CHANG, V. KORENMAN
Department of Physics and Astronomy, University of Maryland, College Park, Maryland, USA and R. W. D E T E N B E C K
Department of Physics, University of Vermont, Burlington, Vermont, USA Received 20 January 1968
The 2nd, 3rd and 4th cumulants of the distribution of light intensity from a He-Ne laser operating just below threshold have been measured as counting time is increased. The results agree with predictions compute! under the assumption that the dependence of the correlation functions on the time variables is the same as for Gaussian light.
E a r l y m e a s u r e m e n t s of c o r r e l a t i o n s of i n t e n sity f l u c t u a t i o n s in well s t a b i l i z e d H e - N e l a s e r s o p e r a t i n g n e a r t h r e s h o l d r e p o r t e d only the c o r r e l a t i o n s of fluctuations all at the s a m e t i m e [1,2]. T h e s e d e t e r m i n e d the p r o b a b i l i t y d e n s it y of inst a nt an eo u s s a m p l e s of the intensity (I. P. D.). R e c e n t s t u d i es [3,4] have p r o v i d e d the t i m e d e p e n d e n c e of the t w o - t i m e c o r r e l a t i o n function in the t h r e s h o l d region. All t h e s e r e s u l t s have been c o n s i s t e n t with c u r r e n t l a s e r t h e o r i e s b a s e d on the F o k k e r - P l a n c k equation with Markoffian n o i s e s o u r c e s [5-10].
We add h e r e m e a s u r e m e n t s of the t i m e d e p e n d en ce of the f i r s t f o u r n o r m a l i z e d c u m u l a n t s f o r counting t i m e s f r o m 3 to 1000 m i c r o s e c o n d s , f o r a l a s e r o p e r a t i n g s l i g h t l y below t h r e s h o l d w h e r e the fluctuation t i m e is about 100 m i c r o s e c o n d s . What is m e a s u r e d is an a v e r a g e o v e r e a c h of the * Work supported by NASA (grant NGR 21-002-022), ARPA (Contract SD-101) and the U. S. Army Research Office (contract DAHC 04 67 C 0023, under Project Defender) and the University of Maryland Computer Science Center under NASA grant NsG-398. 417