- Email: [email protected]
Thermo-electric power in semi-conductors
996
HEAT CONDUCTIVITY OF SEMI-CONDUCTORS
equal ,to N 0. H a v i n g thus obtained f we can then calculate N and from N the lattice h e a t current and thus the lattice heat conductivity. For T >~ @ electron-phonon scattering seems to be the d o m i n a n t process and we get ~ = ½ y u 0 ( ~ k O ) - I exp (-- ~/kT) +
+' ½~ay(T/O)a (DO~h) {T(O/aT) (F/T) (kT)-½ v ~ + (l/T)(kT)t ½~/~} (5) where ? is the specific heat *) and # = [2C2~oK2/9nMu~(dE / dK) 2] s). A t low t em p er atures b o u n d a r y and i m p u r i t y scattering becomes i m p o r t a n t and we get a more complicated expression for ng. Inserting numerical values we get n~/n, ~ 10-4 at T>~ O. Received 30-6-54. REFERENCES 1) W ~ 1 s'o n, A. H., Theory of Metals, 2nd edition, Cambridge 1953; especially Ch. IX. 2) S o m m e r f e l d , A. and B e t h e , H.,Handbuchder Physik, 24 ~,pp. 546-48, 1933.
Physica X X No. 11 A m s t e r d a m Conference Semiconductors
MacDonald, D. K. C. 1954
T H E R M O - E L E C T R I C P O W E R IN SEMI-CONDUCTORS
by D. K. C. MACDONALD Division of Physics, National Research Council, Ottawa, Canada
The expressions for thermo-electric power of metals and semi-conductors derived on the modern electron t h e o r y of solids x) 2) assume essentially t h a t the lattice is negligibly perturbed b y the th e r m a l gradient. If this were so, the thermo-electric power would arise only from the variation of electron density and mobility with energy. Consequently in metals, where the conduction electrons are highly degenerate the predicted thermo-electric power would be a "second-order" phenomenon, the electrons in first ap p r o x im a ti o n being insensitive to the direct influence of temperature.
G u r e v i c h 8) pointed out, however, t h a t if the heat current carried b y the lattice ("phonons") is limited in p a r t b y interaction with the conduction electrons ("phonon-electron scattering") then the t e n d e n c y of the electrons to be swept along b y the lattice waves would give rise to an additional co m p o n en t of thermo-electric field. Since this would be a first "first-order" effect -- not depending on a variation of electron properties -- it m ig h t well be significant in magnitude, particularly in metals. More recently F r e d e r i k s e 4 ) , Blatt and K l e m e n s S ) have discussed this source of thermoelectricity, basing their analysis on the earlier work of M a k i n s o n " ) . These t r e a t m e n t s depend on a fairly detailed statistical model of the electrons and " p h o n o n s " (Cf. e.g. ref. 1) and we have therefore tried to analyse the phenomenon independently from direct m o m e n t u m considerations. In accordance with the usual concepts of th e r m al c o n d u c t i v i t y in a solid lattice we m a y regard an " e l e m e n t a r y " region of thickness t (the mean free path) as being
THERMO-ELECTRIC POWER IN SEMI-CONDUCTORS
997
essentially in t h e r m a l e q u i l i b r i u m a t t e m p e r a t u r e T, w~th a t h e r m a l e n e r g y - d e n s i t y of U(T). This region will t h e n be emitting s o u n d r a d i a t i o n isotropically a n d absorbing r a d i a t i o n f r o m t h e a d j a c e n t e l e m e n t a r y e l e m e n t s a t t e m p e r a t u r e s T + 2 (dT/dx); T -- ~ (dT/dx). C o n s e q u e n t l y if t h e cross-section be A t h e n e t r a d i a t i o n force (i.e. a b s o r p t i o n of d i r e c t e d m o m e n t u m ) will be g i v e n a p p r o x i m a t e l y b y *) :
lVx ~ A { U ( T -- 2 (dT/dx)) -- U ( T -[- 2 (dT/dx))} -- A 2 (dU/dT)p d T / d x
(1)
i.e. t h e a b s o r p t i o n of m o m e n t u m per u n i t v o l u m e ~ -- Cp (dT/dx) where Cp is t h e specific h e a t a t c o n s t a n t p r e s s u r e / u n i t v o l u m e . N o w let 1/T 1 be t h e r a t e of t r a n s f e r of m o m e n t u m to t h e c o n d u c t i o n (i.e. free) charges, a n d let I/~:2 be r a t e of t r a n s f e r for all o t h e r m e c h a n i s m s ( " p h o n o n - p h o n o n " , " p h o n o n - i m p u r i t y " scattering), t h e n t h e overall r a t e of t r a n s f e r is: i/T = I/~ 1 + I/T2 t h u s T = ~IT2/(TI + ~2)" T h e f r a c t i o n of m o m e n t u m t r a n s f e r r e d to t h e charges =
1 / ~ l / 1 / T ---- ~ # I
= ~2/(TI + ~2)
H e n c e t h e m o m e n t u m / s e c o n d t r a n s f e r r e d to t h e charges -- Cp ['~2/(~'1 + "t'2)] d T / d x
(2)
If n o w these free charges, q, of d e n s i t y N / u n i t v o l u m e , axe p r e v e n t e d f r o m m o v e m e n t b y an electric field E . (the r e s u l t i n g t h e r m o e l e c t r i c field) this m o m e n t u m t r a n s f e r r e d is s u s t a i n e d b y a force N q E x a n d t h u s
E,c ~ (Cp/Nq) [~2/(T1 + ~2)] d T / d x
(3)
I n m e t a l s N will be c o n s t a n t ; a t h i g h t e m p e r a t u r e s we m a y e x p e c t 1/. 2 >~ I/z 1 (cf. Makinson, loc. cir.), a n d following P e i e r l s we m a y set 1]T2 = A T , say. H e n c e since CI~ ~ 3Nok, we h a v e
lgx ~ (k/nq.l) (A/T) d T / d x
(4)
where n is t h e n u m b e r of free charges per a t o m . This is in q u a l i t a t i v e a g r e e m e n t w i t h G u r e v i c h (loc. cir.) noticing t h a t he chooses to derive t h e t h e r m o e l e c t r i c p o w e r of a couple, r a t h e r t h a n s i m p l y t h e a b s o l u t e p o w e r of a single metal. The difficulty is, of course, t h a t A is still n o t k n o w n w i t h a n y c e r t a i n t y , t h e r e being as y e t no detailed q u a n t i t a t i v e t h e o r y of t h e r m a l c o n d u c t i v i t y in a lattice. A t low t e m p e r atures, C# ~ 200 R(T/O) 3, a n d if p h o n o n - e l e c t r o n s c a t t e r i n g becomes d o m i n a n t as has been suggested we should s i m p l y h a v e :
E x ~ (200klnq)(TIO) a d T / d x
(5)
again in q u a l i t a t i v e a g r e e m e n t w i t h G u r e v i c h. Since t h e " c o n v e n t i o n a l " t h e r m o e l e c t r i c field is of order: E x --~ (k/q) (T/To) where T O is t h e e l e c t r o n - d e g e n e r a c y t e m p e r a t u r e , (6) w o u l d be e n t i r e l y d o m i n a n t so long as (T/O) ~ ~ (O/To)t s e t t i n g n ~ 1. F o r a t y p i c a l m e t a l O ,-~ 100°K and T O ~ 10,000°K; hence this w o u l d occur for T >,~ 1°KI A l t h o u g h there is m u c h t h a t is a n o m a l o u s and r e m a r k a b l e in t h e t h e r m o e l e c t r i c b e h a v i o u r of m e t a l s at low t e m p e r a t u r e s (e.g. cf. M a c D o n a I d a n d P e a r s o n ~)), it appears v i r t u a l l y certain t h a t no t h e r m o - e l e c t r i e power of a f o r m and m a g n i t u d e p r e d i c t e d b y (5) plays a r61e a t least in p u r e metals. T u r n i n g to semi-conductors, it is e v i d e n t t h a t N will in general be m u c h smaller t h a n in m e t a l s and hence f r o m (3) this p h e n o m e n o n m a y well p l a y a d o m i n a n t part. Considering F r e d e r i k s e's results (loc. eit., figure 2), a q u a l i t a t i v e a g r e e m e n t is *) B r.i 11 o u i n gives F V (where F is the Grfineisen parameter) for the pressure due to diffuse sound radiation in the solid.
998
THERMO-ELECTRIC POWER IN SEMI-CONDUCTORS
e v i d e n t . A t h i g h t e m p e r a t u r e s we m i g h t e x p e c t (4) t o b e v a l i d a g a i n . I f we a s s u m e t h a t 1 / . 1 ~ n , t h e n Ex will b e c r u d e l y i n d e p e n d e n t of e l e c t r o n - c o n c e n t r a t i o n - - w h i c h is s e e n t o b e t h e case. A t l o w e r t e m p e r a t u r e s , if 1/x2 is n o l o n g e r very much greater t h a n 1/T 1 ( p h o n o n - p h o n o n s c a t t e r i n g n o t e n t i r e l y d o m i n a n t ) we s h o u l d e x p e c t E x t o i n c r e a s e w i t h d i m i n i s h i n g e l e c t r o n - c o n c e n t r a t i o n , as is i n d e e d o b s e r v e d . T h e d e t a i l e d t e m p e r a t u r e - v a r i a t i o n will d e p e n d o n t h e r e l a t i v e r a t e of d e c a y of Cp a n d N. T h i s m a k e s i t a p p e a r well w o r t h w h i l e t o c a r r y o u t e x p e r i m e n t s o n s e m i c o n d u c t o r s in the liquid helium region and these have been started in our laboratory. It would e v i d e n t l y also b e m o s t d e s i r a b l e t o h a v e a quantitative t h e o r y of l a t t i c e t h e r m a l conduction and to gain further information on "phonon-electron" scattering. I am most grateful to Doctors'J. v a l u a b l e d i s c u s s i o n s , etc.
S.
Dugdale
and
W . B.
Pearson
for
Received 17-6-54. REFERENCES 1) W i 1 s o n, A. H., Theory of Metals, Cambridge University Press, 1st and 2nd editions, (1936), (1953). 2) M o t t, N. F. and J o n e s, H., Theory of Properties of Metals and Alloys, Clarendon Press, Oxford (1936). 3) G u r e v i e h, L., J. Phys. U.S.S.R. 9 (1945) 477; J. Phys. U.S.S.R. 10(1946) 67. 4) F r e d e r i k s e , H . P . R . , P h y s . Rev. 02(1953) 248. 5) Private communication. 6) M a k i n s o n , R . E . B . , P r o c . Camb. phil. Soe. 34(1938) 474. 7) M a c D o n a l d , D. K.C. and P e a r s o n , W. B., Proe. roy. Soe. A ' ~ i 9 (1953) 373; Proe. roy. Soe. A 221 (1954) 534.
Friedel, J. 1954
Physica XX No. 11 Amsterdam Conference Semiconductors
E M P L O I D ' U N E MASSE E F F E C T I V E DANS LES SEMI-CONDUCTEURS par J. FRIEDEL Centre de Reeherehes M~tallurgiques de l'Ecole des Mines de Paris, France
I. Introduction. D a n s les s e m i - c o n d u c t e u r s p e r t u r b 6 s p a r l ' a g i t a t i o n t h e r m i q u e ou des i m p u r e t 6 s , u n 61ectron d ' 6 n e r g i e E v o i s i n e de celle E 0 de la l i m i t e de l a b a n d e de c o n d u c t i b i l i t 6 e s t c o u r a m m e n t t r a i t ~ c o m m e se d 6 p l a ~ a n t d a n s le p o t e n t i e l p e r t u r b a t e u r seul, a v e c u n e 6nergie E - - E 0. D a n s c e t t e a p p r o x i m a t i o n , d u e ~ T i b b s x) z) 3), c ' e s t la m a s s e 61ectronique n o r m a l e q u i d o i t ~tre utilis6e, e t n o n p a s l a m a s s e e f f e c t i v e d u e la s t r u c t u r e cristalline. L ' e m p l o i d ' u n e m a s s e e f f e c t i v e est li6e ~ u n e a p p r o x i m a t i o n diff6rente que nous analysons ensuite. 2. A p p r o x i m a t i o n de T i b b s. Si V R e s t le p o t e n t i e l d u c r i s t a l p a r f a i t e t V p l a pert u r b a t i o n , la f o n c t i o n d ' o n d e ~v de l'61ectron d ' 6 n e r g i e E s a t i s f a i t ~ l ' 6 q u a t i o n de
HEAT CONDUCTIVITY OF SEMI-CONDUCTORS
equal ,to N 0. H a v i n g thus obtained f we can then calculate N and from N the lattice h e a t current and thus the lattice heat conductivity. For T >~ @ electron-phonon scattering seems to be the d o m i n a n t process and we get ~ = ½ y u 0 ( ~ k O ) - I exp (-- ~/kT) +
+' ½~ay(T/O)a (DO~h) {T(O/aT) (F/T) (kT)-½ v ~ + (l/T)(kT)t ½~/~} (5) where ? is the specific heat *) and # = [2C2~oK2/9nMu~(dE / dK) 2] s). A t low t em p er atures b o u n d a r y and i m p u r i t y scattering becomes i m p o r t a n t and we get a more complicated expression for ng. Inserting numerical values we get n~/n, ~ 10-4 at T>~ O. Received 30-6-54. REFERENCES 1) W ~ 1 s'o n, A. H., Theory of Metals, 2nd edition, Cambridge 1953; especially Ch. IX. 2) S o m m e r f e l d , A. and B e t h e , H.,Handbuchder Physik, 24 ~,pp. 546-48, 1933.
Physica X X No. 11 A m s t e r d a m Conference Semiconductors
MacDonald, D. K. C. 1954
T H E R M O - E L E C T R I C P O W E R IN SEMI-CONDUCTORS
by D. K. C. MACDONALD Division of Physics, National Research Council, Ottawa, Canada
The expressions for thermo-electric power of metals and semi-conductors derived on the modern electron t h e o r y of solids x) 2) assume essentially t h a t the lattice is negligibly perturbed b y the th e r m a l gradient. If this were so, the thermo-electric power would arise only from the variation of electron density and mobility with energy. Consequently in metals, where the conduction electrons are highly degenerate the predicted thermo-electric power would be a "second-order" phenomenon, the electrons in first ap p r o x im a ti o n being insensitive to the direct influence of temperature.
G u r e v i c h 8) pointed out, however, t h a t if the heat current carried b y the lattice ("phonons") is limited in p a r t b y interaction with the conduction electrons ("phonon-electron scattering") then the t e n d e n c y of the electrons to be swept along b y the lattice waves would give rise to an additional co m p o n en t of thermo-electric field. Since this would be a first "first-order" effect -- not depending on a variation of electron properties -- it m ig h t well be significant in magnitude, particularly in metals. More recently F r e d e r i k s e 4 ) , Blatt and K l e m e n s S ) have discussed this source of thermoelectricity, basing their analysis on the earlier work of M a k i n s o n " ) . These t r e a t m e n t s depend on a fairly detailed statistical model of the electrons and " p h o n o n s " (Cf. e.g. ref. 1) and we have therefore tried to analyse the phenomenon independently from direct m o m e n t u m considerations. In accordance with the usual concepts of th e r m al c o n d u c t i v i t y in a solid lattice we m a y regard an " e l e m e n t a r y " region of thickness t (the mean free path) as being
THERMO-ELECTRIC POWER IN SEMI-CONDUCTORS
997
essentially in t h e r m a l e q u i l i b r i u m a t t e m p e r a t u r e T, w~th a t h e r m a l e n e r g y - d e n s i t y of U(T). This region will t h e n be emitting s o u n d r a d i a t i o n isotropically a n d absorbing r a d i a t i o n f r o m t h e a d j a c e n t e l e m e n t a r y e l e m e n t s a t t e m p e r a t u r e s T + 2 (dT/dx); T -- ~ (dT/dx). C o n s e q u e n t l y if t h e cross-section be A t h e n e t r a d i a t i o n force (i.e. a b s o r p t i o n of d i r e c t e d m o m e n t u m ) will be g i v e n a p p r o x i m a t e l y b y *) :
lVx ~ A { U ( T -- 2 (dT/dx)) -- U ( T -[- 2 (dT/dx))} -- A 2 (dU/dT)p d T / d x
(1)
i.e. t h e a b s o r p t i o n of m o m e n t u m per u n i t v o l u m e ~ -- Cp (dT/dx) where Cp is t h e specific h e a t a t c o n s t a n t p r e s s u r e / u n i t v o l u m e . N o w let 1/T 1 be t h e r a t e of t r a n s f e r of m o m e n t u m to t h e c o n d u c t i o n (i.e. free) charges, a n d let I/~:2 be r a t e of t r a n s f e r for all o t h e r m e c h a n i s m s ( " p h o n o n - p h o n o n " , " p h o n o n - i m p u r i t y " scattering), t h e n t h e overall r a t e of t r a n s f e r is: i/T = I/~ 1 + I/T2 t h u s T = ~IT2/(TI + ~2)" T h e f r a c t i o n of m o m e n t u m t r a n s f e r r e d to t h e charges =
1 / ~ l / 1 / T ---- ~ # I
= ~2/(TI + ~2)
H e n c e t h e m o m e n t u m / s e c o n d t r a n s f e r r e d to t h e charges -- Cp ['~2/(~'1 + "t'2)] d T / d x
(2)
If n o w these free charges, q, of d e n s i t y N / u n i t v o l u m e , axe p r e v e n t e d f r o m m o v e m e n t b y an electric field E . (the r e s u l t i n g t h e r m o e l e c t r i c field) this m o m e n t u m t r a n s f e r r e d is s u s t a i n e d b y a force N q E x a n d t h u s
E,c ~ (Cp/Nq) [~2/(T1 + ~2)] d T / d x
(3)
I n m e t a l s N will be c o n s t a n t ; a t h i g h t e m p e r a t u r e s we m a y e x p e c t 1/. 2 >~ I/z 1 (cf. Makinson, loc. cir.), a n d following P e i e r l s we m a y set 1]T2 = A T , say. H e n c e since CI~ ~ 3Nok, we h a v e
lgx ~ (k/nq.l) (A/T) d T / d x
(4)
where n is t h e n u m b e r of free charges per a t o m . This is in q u a l i t a t i v e a g r e e m e n t w i t h G u r e v i c h (loc. cir.) noticing t h a t he chooses to derive t h e t h e r m o e l e c t r i c p o w e r of a couple, r a t h e r t h a n s i m p l y t h e a b s o l u t e p o w e r of a single metal. The difficulty is, of course, t h a t A is still n o t k n o w n w i t h a n y c e r t a i n t y , t h e r e being as y e t no detailed q u a n t i t a t i v e t h e o r y of t h e r m a l c o n d u c t i v i t y in a lattice. A t low t e m p e r atures, C# ~ 200 R(T/O) 3, a n d if p h o n o n - e l e c t r o n s c a t t e r i n g becomes d o m i n a n t as has been suggested we should s i m p l y h a v e :
E x ~ (200klnq)(TIO) a d T / d x
(5)
again in q u a l i t a t i v e a g r e e m e n t w i t h G u r e v i c h. Since t h e " c o n v e n t i o n a l " t h e r m o e l e c t r i c field is of order: E x --~ (k/q) (T/To) where T O is t h e e l e c t r o n - d e g e n e r a c y t e m p e r a t u r e , (6) w o u l d be e n t i r e l y d o m i n a n t so long as (T/O) ~ ~ (O/To)t s e t t i n g n ~ 1. F o r a t y p i c a l m e t a l O ,-~ 100°K and T O ~ 10,000°K; hence this w o u l d occur for T >,~ 1°KI A l t h o u g h there is m u c h t h a t is a n o m a l o u s and r e m a r k a b l e in t h e t h e r m o e l e c t r i c b e h a v i o u r of m e t a l s at low t e m p e r a t u r e s (e.g. cf. M a c D o n a I d a n d P e a r s o n ~)), it appears v i r t u a l l y certain t h a t no t h e r m o - e l e c t r i e power of a f o r m and m a g n i t u d e p r e d i c t e d b y (5) plays a r61e a t least in p u r e metals. T u r n i n g to semi-conductors, it is e v i d e n t t h a t N will in general be m u c h smaller t h a n in m e t a l s and hence f r o m (3) this p h e n o m e n o n m a y well p l a y a d o m i n a n t part. Considering F r e d e r i k s e's results (loc. eit., figure 2), a q u a l i t a t i v e a g r e e m e n t is *) B r.i 11 o u i n gives F V (where F is the Grfineisen parameter) for the pressure due to diffuse sound radiation in the solid.
998
THERMO-ELECTRIC POWER IN SEMI-CONDUCTORS
e v i d e n t . A t h i g h t e m p e r a t u r e s we m i g h t e x p e c t (4) t o b e v a l i d a g a i n . I f we a s s u m e t h a t 1 / . 1 ~ n , t h e n Ex will b e c r u d e l y i n d e p e n d e n t of e l e c t r o n - c o n c e n t r a t i o n - - w h i c h is s e e n t o b e t h e case. A t l o w e r t e m p e r a t u r e s , if 1/x2 is n o l o n g e r very much greater t h a n 1/T 1 ( p h o n o n - p h o n o n s c a t t e r i n g n o t e n t i r e l y d o m i n a n t ) we s h o u l d e x p e c t E x t o i n c r e a s e w i t h d i m i n i s h i n g e l e c t r o n - c o n c e n t r a t i o n , as is i n d e e d o b s e r v e d . T h e d e t a i l e d t e m p e r a t u r e - v a r i a t i o n will d e p e n d o n t h e r e l a t i v e r a t e of d e c a y of Cp a n d N. T h i s m a k e s i t a p p e a r well w o r t h w h i l e t o c a r r y o u t e x p e r i m e n t s o n s e m i c o n d u c t o r s in the liquid helium region and these have been started in our laboratory. It would e v i d e n t l y also b e m o s t d e s i r a b l e t o h a v e a quantitative t h e o r y of l a t t i c e t h e r m a l conduction and to gain further information on "phonon-electron" scattering. I am most grateful to Doctors'J. v a l u a b l e d i s c u s s i o n s , etc.
S.
Dugdale
and
W . B.
Pearson
for
Received 17-6-54. REFERENCES 1) W i 1 s o n, A. H., Theory of Metals, Cambridge University Press, 1st and 2nd editions, (1936), (1953). 2) M o t t, N. F. and J o n e s, H., Theory of Properties of Metals and Alloys, Clarendon Press, Oxford (1936). 3) G u r e v i e h, L., J. Phys. U.S.S.R. 9 (1945) 477; J. Phys. U.S.S.R. 10(1946) 67. 4) F r e d e r i k s e , H . P . R . , P h y s . Rev. 02(1953) 248. 5) Private communication. 6) M a k i n s o n , R . E . B . , P r o c . Camb. phil. Soe. 34(1938) 474. 7) M a c D o n a l d , D. K.C. and P e a r s o n , W. B., Proe. roy. Soe. A ' ~ i 9 (1953) 373; Proe. roy. Soe. A 221 (1954) 534.
Friedel, J. 1954
Physica XX No. 11 Amsterdam Conference Semiconductors
E M P L O I D ' U N E MASSE E F F E C T I V E DANS LES SEMI-CONDUCTEURS par J. FRIEDEL Centre de Reeherehes M~tallurgiques de l'Ecole des Mines de Paris, France
I. Introduction. D a n s les s e m i - c o n d u c t e u r s p e r t u r b 6 s p a r l ' a g i t a t i o n t h e r m i q u e ou des i m p u r e t 6 s , u n 61ectron d ' 6 n e r g i e E v o i s i n e de celle E 0 de la l i m i t e de l a b a n d e de c o n d u c t i b i l i t 6 e s t c o u r a m m e n t t r a i t ~ c o m m e se d 6 p l a ~ a n t d a n s le p o t e n t i e l p e r t u r b a t e u r seul, a v e c u n e 6nergie E - - E 0. D a n s c e t t e a p p r o x i m a t i o n , d u e ~ T i b b s x) z) 3), c ' e s t la m a s s e 61ectronique n o r m a l e q u i d o i t ~tre utilis6e, e t n o n p a s l a m a s s e e f f e c t i v e d u e la s t r u c t u r e cristalline. L ' e m p l o i d ' u n e m a s s e e f f e c t i v e est li6e ~ u n e a p p r o x i m a t i o n diff6rente que nous analysons ensuite. 2. A p p r o x i m a t i o n de T i b b s. Si V R e s t le p o t e n t i e l d u c r i s t a l p a r f a i t e t V p l a pert u r b a t i o n , la f o n c t i o n d ' o n d e ~v de l'61ectron d ' 6 n e r g i e E s a t i s f a i t ~ l ' 6 q u a t i o n de