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Spin reversal transitions in impurity hop conduction
Volume 24A, number 12
SPIN
PHYSICS
REVERSAL
TRANSITIONS
LETTERS
IN IMPURITY
5 June 1967
HOP
CONDUCTION
*
J. A. C H R O B O C Z E K * * , E . W . P R O H O F S K Y and R. J . S L A D E K
Department of Physics, Purdue University. Lafayette. Indiana Received 6 May 1967
The magnetoresistance P/P O of phonon-assisted hop conduction in n-type Ge is found to have an appreciable temperature dependence at high magnetic fields. Reversal of spins of electrons while they make interdonor transitions is shown to be responsible.
M e a s u r e m e n t s on n - t y p e Ge s a m p l e s (9.0 × 1015 and 2.1× 1016 P / c m 3 a n d 9.6× 1 0 1 5 S b / c m 3) at l i q u i d h e l i u m t e m p e r a t u r e s , r e p o r t e d h e r e i n , r e v e a l that the m a g n e t o r e s i s t a n c e , P/Po h a s an a p p r e c i a b l e d e p e n d e n c e on t e m p e r a t u r e at high m a g n e t i c f i e l d s (45 - 146 kG). T h e b e h a v i o u r of Po at low t e m p e r a t u r e s , the i m p u r i t y c o n c e n trations deduced from Hall effect measurements at r o o m t e m p e r a t u r e , and p r e v i o u s m a g n e t o r e s i s t a n c e m e a s u r e m e n t s by us at l o w e r f i e l d s [1], a l l i n d i c a t e that p h o n o n - a s s i s t e d t u n n e l i n g of e l e c t r o n s b e t w e e n d o n o r s i t e s is the p r e d o m i n a n t c o n d u c t i o n m e c h a n i s m in o u r s a m p l e s . T h e a c c e p t e d t h e o r y of t h i s p r o c e s s [2,3] d o e s not p r e d i c t any a p p r e c i a b l e t e m p e r a t u r e d e p e n d e n c e of the m a g n e t o r e s i s t a n c e . We p r o p o s e an a d d i t i o n a l t r a n s p o r t m e c h a n i s m due to i n t e r - d o n o r t r a n s i t i o n s in w h i c h e l e c t r o n s r e v e r s e t h e i r s p i n s : C a l c u l a t i o n s a r e m a d e w h i c h s h o w that the o b s e r v e d t e m p e r a t u r e d e p e n d e n c e of P/Po d o e s a r i s e from this additional mechanism. E x p e r i m e n t a l r e s u l t s f o r the t e m p e r a t u r e d e p e n d e n c e of the m a g n e t o r e s i s t a n c e m e a s u r e d with the m a g n e t i c f i e l d , H / / [ 0 0 1 ] a r e shown in fig. 1 f o r one of the P - d o p e d s a m p l e s . N o t e that t h e m a g n e t o r e s i s t a n c e i s an i n c r e a s i n g f u n c t i o n of I / T , and i t s s l o p e i n c r e a s e s with H. T h e o t h e r s a m p l e s e x h i b i t e d s i m i l a r r e s u l t s (although the 9 × 1015 P / c m 3 s a m p l e could not b e m e a s -
u r e d to a s high v a l u e s of the m a g n e t i c f i e l d b e c a u s e of i t s e x t r e m e l y high r e s i s t a n c e . ) In o r d e r to s h o w that t r a n s i t i o n s in which I000
100
I0
65
n-GelP) NO
Z ff [llO], H H [OOJ] I .2
* Work supported by the Advanced Research Projects Agency. The high field data were taken at the National Magnet Laboratory which is supported at M. I. T . , Cambridge, Massachusetts, by Air Force Office of Scientific Research. ** On leave of absence from the Institute of Physics, Polish Academy of Sciences, Warsaw, Poland.
= 2.1 xlOI6cm -3
I
I
.3
.4 I/T
I
I .5
(°K-I)
Fig. 1. Temperature dependence of the magnetoresistance at various magnetic fields. Solid lines connect the experimental points. Dashed lines give the magnetoresistanee of the spin-conserving transitions deduced from the experimental data and eq. (2) of the text. 657
Volume 24A. number 12
PHYSICS
electrons reverse their spins while tunneling bet w e e n d o n o r s can a c c o u n t for the t e m p e r a t u r e d e p e n d e n c e o f / 9 / 0 o , we f i r s t d e r i v e d the f o l l o w ing f o r m u l a f o r the t r a n s i t i o n p r o b a b i l i t y f o r spin flip (SF) t r a n s i t i o n s by e x t e n d i n g the spin l a t t i c e r e l a x a t i o n t h e o r y of Roth [4] and H a s e gawa [51 to the c a s e of two d o n o r s w h o s e e n e r g y l e v e l s d i f f e r in e n e r g y by A,
5 ,June 1967
d i s t r i b u t i o n [61 if A ' s . Upon doing so we find f o r the m a g n e t o r e s t s t a n c e r a t i o of the SC p a r t of the total i m p e d a n c e p (H, T) I
[
67>-L sc
=
p ( O , T ) p ( H , T ) _ p ( H , T O) sinh (M/kT) p ( 0 , T o ) sinh ( M / k T o)
p ( 0 , T o ) p ( 0 , T)
a(H. T)
p ( 0 , T)
P( O, T o)
~ F ( W t ) -M~2 '1 (M+A) 3 n q + P ( t ) A2/~3 (M-A) 3 (nq- + I ) P ( , ) ]
(1)
w h e r e M - A~ # H a n d F(Wt ) =(IWt!2}a v
4
~,
[g" E u X 2
3~)
f
m which s5
P ( W t i 2 ) a v = s q u a r e of the r e s o n a n c e e n e r g y of the t r i p l e t m e m b e r of the d o n o r g r o u n d s t a t e a v e r a g e d o v e r d i r e c t i o n s [2] p(t) and p(*) a r e p r o b a b i l i t i e s of an e l e c t r o n b e i n g in the s t a t e with s p i n p a r a l l e l and a n t i p a r a l l e l to H, and the o t h e r s y m b o l s a r e d e f i n e d in r e f . 5. T h e v a l u e of f t s 1 f o r H / / [001] and ½ for H / / [ l i 0 I. T h i s f a c t o r can be r e s p o n s i b l e for the a n i s o t r o p y o b s e r v e d in the t e m p e r a t u r e d e p e n d e n c e of the magnetor esistance. C o m p a r i s o n of the v a l u e s of u(H, T) c a l c u l a t e d f r o m eq. (1) with t h o s e c a l c u l a t e d for s p i n c o n s e r v i n g (SC) j u m p s u s i n g a f o r m u l a b a s e d on the t h e o r y of r e f . 2 and m a g n e t o r e s i s t a n c e s t u d i e s [1,31 r e v e a l that for A < A o ( w h e r e A o / k l i e s within the r a n g e 0.5 - 2OK d e p e n d i n g on d o n o r c o n c e n t r a t i o n , T a n d H ) SF j u m p s w e r e m o r e p r o b a b l e than SC j u m p s w h i l e f o r /x > A o the o p p o s i t e was t r u e . T h i s i m p l i e s that for an e l e c t r o n to t r a v e r s e the c r y s t a l the i m p e d a n c e is g i v e n b y Z = n F Z F + n C Z C w h e r e n F is the n u m b e r of SF p r o c e s s e s e a c h h a v i n g i m p e d a n c e Z F and n C and Z C a r e c o r r e s p o n d i n g q u a n t i t i e s f o r SC p r o c e s s e s . T h e t e m p e r a t u r e d e p e n d e n c e of Z F can be e v a l u a t e d f r o m eq. (1) when A<< M e v e n though the t o t a l i m p e d a n c e cannot be c a l c u l a t e d due to a l a c k of d e t a i l e d k n o w l e d g e a b o u t the p r o b a b i l i t y ,
658
LETTERS
smh (M/kr) s i n h (M/kTo)
(2)
w h e r e all p o o r t y known p a r a m e t e r s , i n c l u d i n g W and A, h a v e c a n c e l l e d out. T h e a b o v e e q u a t i o n has b e e n s i m p l i f i e d by s e t t i n g n F ( T ) / n F ( T o ) nc(T)/nC(To) 1. T h e v a l u e s ofp(O,T)l/p(O, To) , p(H, T ) / p ( 0, T) and p(H, To)/p( O, To) w e r e taken f r o m the e x p e r i m e n t a l data. T h e d a s h e d c u r v e s in fig. 1 s h o w the m a g n e t o r e s t s t a n c e r a t i o c a l c u lated f r o m eq. (2). T h e t e m p e r a t u r e d e p e n d e n c e of the l a t t e r is m u c h w e a k e r than the t e m p e r a t u r e d e p e n d e n c e of the e x p e r i m e n t a l l y m e a s u r e d m a g n e t o r e s i s t a n c e f o r the s a m p l e h a v i n g 2.1 x 1016 P / c m 3 (shown by the d a t a p o i n t s and s o l i d c u r v e s in fig. 1). T h i s i n d i c a t e s that the m a i n p a r t of the o b s e r v e d t e m p e r a t u r e d e p e n d e n c e of the m a g n e t o r e s i s t a n c e i s due to the s p i n flip transitions. T h a n k s a r e due to L o u i s e Roth for g r o w i n g the c r y s t a l s and P r o f . H. J. Y e a r i a n for o r i e n t ing t h e m . Help and e x p e r i m e n t a l a s s i s t a n c e of L. G. R u b i n and R. A r d n t of N a t i o n a l M a g n e t L a b oratory are gratefully acknowledged.
[~CfC~'CHCCS 1. O.A.Chroboezekand R . J . S l a d e k . Phys. Rev. 151 (1966) 595. 2. A . M i l l e r and E.Abrahams, Phys. Rev. 120 (1960) 745. 3. N.Mikoshiba, Phys. Rev. 127 (1962)1962. 4. L.M.Roth, Phys. Rev. 118 (1960) 1534. 5. H.Hasegawa, Phys. Rev. 118 (1960)1523. 6. S.Golin. Phys. Rev. 132 (1(.)63) 178. We are indebted to Dr. Golin for sending us a more complete account of his calculations of the probability function.
SPIN
PHYSICS
REVERSAL
TRANSITIONS
LETTERS
IN IMPURITY
5 June 1967
HOP
CONDUCTION
*
J. A. C H R O B O C Z E K * * , E . W . P R O H O F S K Y and R. J . S L A D E K
Department of Physics, Purdue University. Lafayette. Indiana Received 6 May 1967
The magnetoresistance P/P O of phonon-assisted hop conduction in n-type Ge is found to have an appreciable temperature dependence at high magnetic fields. Reversal of spins of electrons while they make interdonor transitions is shown to be responsible.
M e a s u r e m e n t s on n - t y p e Ge s a m p l e s (9.0 × 1015 and 2.1× 1016 P / c m 3 a n d 9.6× 1 0 1 5 S b / c m 3) at l i q u i d h e l i u m t e m p e r a t u r e s , r e p o r t e d h e r e i n , r e v e a l that the m a g n e t o r e s i s t a n c e , P/Po h a s an a p p r e c i a b l e d e p e n d e n c e on t e m p e r a t u r e at high m a g n e t i c f i e l d s (45 - 146 kG). T h e b e h a v i o u r of Po at low t e m p e r a t u r e s , the i m p u r i t y c o n c e n trations deduced from Hall effect measurements at r o o m t e m p e r a t u r e , and p r e v i o u s m a g n e t o r e s i s t a n c e m e a s u r e m e n t s by us at l o w e r f i e l d s [1], a l l i n d i c a t e that p h o n o n - a s s i s t e d t u n n e l i n g of e l e c t r o n s b e t w e e n d o n o r s i t e s is the p r e d o m i n a n t c o n d u c t i o n m e c h a n i s m in o u r s a m p l e s . T h e a c c e p t e d t h e o r y of t h i s p r o c e s s [2,3] d o e s not p r e d i c t any a p p r e c i a b l e t e m p e r a t u r e d e p e n d e n c e of the m a g n e t o r e s i s t a n c e . We p r o p o s e an a d d i t i o n a l t r a n s p o r t m e c h a n i s m due to i n t e r - d o n o r t r a n s i t i o n s in w h i c h e l e c t r o n s r e v e r s e t h e i r s p i n s : C a l c u l a t i o n s a r e m a d e w h i c h s h o w that the o b s e r v e d t e m p e r a t u r e d e p e n d e n c e of P/Po d o e s a r i s e from this additional mechanism. E x p e r i m e n t a l r e s u l t s f o r the t e m p e r a t u r e d e p e n d e n c e of the m a g n e t o r e s i s t a n c e m e a s u r e d with the m a g n e t i c f i e l d , H / / [ 0 0 1 ] a r e shown in fig. 1 f o r one of the P - d o p e d s a m p l e s . N o t e that t h e m a g n e t o r e s i s t a n c e i s an i n c r e a s i n g f u n c t i o n of I / T , and i t s s l o p e i n c r e a s e s with H. T h e o t h e r s a m p l e s e x h i b i t e d s i m i l a r r e s u l t s (although the 9 × 1015 P / c m 3 s a m p l e could not b e m e a s -
u r e d to a s high v a l u e s of the m a g n e t i c f i e l d b e c a u s e of i t s e x t r e m e l y high r e s i s t a n c e . ) In o r d e r to s h o w that t r a n s i t i o n s in which I000
100
I0
65
n-GelP) NO
Z ff [llO], H H [OOJ] I .2
* Work supported by the Advanced Research Projects Agency. The high field data were taken at the National Magnet Laboratory which is supported at M. I. T . , Cambridge, Massachusetts, by Air Force Office of Scientific Research. ** On leave of absence from the Institute of Physics, Polish Academy of Sciences, Warsaw, Poland.
= 2.1 xlOI6cm -3
I
I
.3
.4 I/T
I
I .5
(°K-I)
Fig. 1. Temperature dependence of the magnetoresistance at various magnetic fields. Solid lines connect the experimental points. Dashed lines give the magnetoresistanee of the spin-conserving transitions deduced from the experimental data and eq. (2) of the text. 657
Volume 24A. number 12
PHYSICS
electrons reverse their spins while tunneling bet w e e n d o n o r s can a c c o u n t for the t e m p e r a t u r e d e p e n d e n c e o f / 9 / 0 o , we f i r s t d e r i v e d the f o l l o w ing f o r m u l a f o r the t r a n s i t i o n p r o b a b i l i t y f o r spin flip (SF) t r a n s i t i o n s by e x t e n d i n g the spin l a t t i c e r e l a x a t i o n t h e o r y of Roth [4] and H a s e gawa [51 to the c a s e of two d o n o r s w h o s e e n e r g y l e v e l s d i f f e r in e n e r g y by A,
5 ,June 1967
d i s t r i b u t i o n [61 if A ' s . Upon doing so we find f o r the m a g n e t o r e s t s t a n c e r a t i o of the SC p a r t of the total i m p e d a n c e p (H, T) I
[
67>-L sc
=
p ( O , T ) p ( H , T ) _ p ( H , T O) sinh (M/kT) p ( 0 , T o ) sinh ( M / k T o)
p ( 0 , T o ) p ( 0 , T)
a(H. T)
p ( 0 , T)
P( O, T o)
~ F ( W t ) -M~2 '1 (M+A) 3 n q + P ( t ) A2/~3 (M-A) 3 (nq- + I ) P ( , ) ]
(1)
w h e r e M - A~ # H a n d F(Wt ) =(IWt!2}a v
4
~,
[g" E u X 2
3~)
f
m which s5
P ( W t i 2 ) a v = s q u a r e of the r e s o n a n c e e n e r g y of the t r i p l e t m e m b e r of the d o n o r g r o u n d s t a t e a v e r a g e d o v e r d i r e c t i o n s [2] p(t) and p(*) a r e p r o b a b i l i t i e s of an e l e c t r o n b e i n g in the s t a t e with s p i n p a r a l l e l and a n t i p a r a l l e l to H, and the o t h e r s y m b o l s a r e d e f i n e d in r e f . 5. T h e v a l u e of f t s 1 f o r H / / [001] and ½ for H / / [ l i 0 I. T h i s f a c t o r can be r e s p o n s i b l e for the a n i s o t r o p y o b s e r v e d in the t e m p e r a t u r e d e p e n d e n c e of the magnetor esistance. C o m p a r i s o n of the v a l u e s of u(H, T) c a l c u l a t e d f r o m eq. (1) with t h o s e c a l c u l a t e d for s p i n c o n s e r v i n g (SC) j u m p s u s i n g a f o r m u l a b a s e d on the t h e o r y of r e f . 2 and m a g n e t o r e s i s t a n c e s t u d i e s [1,31 r e v e a l that for A < A o ( w h e r e A o / k l i e s within the r a n g e 0.5 - 2OK d e p e n d i n g on d o n o r c o n c e n t r a t i o n , T a n d H ) SF j u m p s w e r e m o r e p r o b a b l e than SC j u m p s w h i l e f o r /x > A o the o p p o s i t e was t r u e . T h i s i m p l i e s that for an e l e c t r o n to t r a v e r s e the c r y s t a l the i m p e d a n c e is g i v e n b y Z = n F Z F + n C Z C w h e r e n F is the n u m b e r of SF p r o c e s s e s e a c h h a v i n g i m p e d a n c e Z F and n C and Z C a r e c o r r e s p o n d i n g q u a n t i t i e s f o r SC p r o c e s s e s . T h e t e m p e r a t u r e d e p e n d e n c e of Z F can be e v a l u a t e d f r o m eq. (1) when A<< M e v e n though the t o t a l i m p e d a n c e cannot be c a l c u l a t e d due to a l a c k of d e t a i l e d k n o w l e d g e a b o u t the p r o b a b i l i t y ,
658
LETTERS
smh (M/kr) s i n h (M/kTo)
(2)
w h e r e all p o o r t y known p a r a m e t e r s , i n c l u d i n g W and A, h a v e c a n c e l l e d out. T h e a b o v e e q u a t i o n has b e e n s i m p l i f i e d by s e t t i n g n F ( T ) / n F ( T o ) nc(T)/nC(To) 1. T h e v a l u e s ofp(O,T)l/p(O, To) , p(H, T ) / p ( 0, T) and p(H, To)/p( O, To) w e r e taken f r o m the e x p e r i m e n t a l data. T h e d a s h e d c u r v e s in fig. 1 s h o w the m a g n e t o r e s t s t a n c e r a t i o c a l c u lated f r o m eq. (2). T h e t e m p e r a t u r e d e p e n d e n c e of the l a t t e r is m u c h w e a k e r than the t e m p e r a t u r e d e p e n d e n c e of the e x p e r i m e n t a l l y m e a s u r e d m a g n e t o r e s i s t a n c e f o r the s a m p l e h a v i n g 2.1 x 1016 P / c m 3 (shown by the d a t a p o i n t s and s o l i d c u r v e s in fig. 1). T h i s i n d i c a t e s that the m a i n p a r t of the o b s e r v e d t e m p e r a t u r e d e p e n d e n c e of the m a g n e t o r e s i s t a n c e i s due to the s p i n flip transitions. T h a n k s a r e due to L o u i s e Roth for g r o w i n g the c r y s t a l s and P r o f . H. J. Y e a r i a n for o r i e n t ing t h e m . Help and e x p e r i m e n t a l a s s i s t a n c e of L. G. R u b i n and R. A r d n t of N a t i o n a l M a g n e t L a b oratory are gratefully acknowledged.
[~CfC~'CHCCS 1. O.A.Chroboezekand R . J . S l a d e k . Phys. Rev. 151 (1966) 595. 2. A . M i l l e r and E.Abrahams, Phys. Rev. 120 (1960) 745. 3. N.Mikoshiba, Phys. Rev. 127 (1962)1962. 4. L.M.Roth, Phys. Rev. 118 (1960) 1534. 5. H.Hasegawa, Phys. Rev. 118 (1960)1523. 6. S.Golin. Phys. Rev. 132 (1(.)63) 178. We are indebted to Dr. Golin for sending us a more complete account of his calculations of the probability function.