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The dingle factor for quantum oscillations in the thermal magnetoresistance of tin
Volume36A, number 5
PHYSICS
DINGLE
THE
FACTOR
THERMAL
FOR
LETTERS
QUANTUM
27 September 1971
OSCILLATIONS
MAGNETORESISTANCE
OF
IN THE
TIN
R. C. YOUNG
Department of Physics, University of Birmingham, Birmingham, UK Received 14 August 1971
The field dependence of amplitude of the oscillatory thermal magnetoresistance of tin exhibits a subsidiary zero at a field proportional to the temperature, in contrast to electrical magnetoresistance. A theory developed to explain this gives m* = 0.120 mo for orbit 36.
A s t r i k i n g f e a t u r e of the m a g n e t o r e s i s t a n c e of tin is the l a r g e amplitude s i n u s o i d a l quantum o s c i l l a t i o n s which o c c u r w h e n / t is a p p r o x i m a t e l y 23 ° f r o m the c - a x i s , due to the c l o s i n g of open o r b i t s by m a g n e t i c breakdown. We have a l r e a d y shown [1] that the field and t e m p e r a t u r e dependence of the a m p l i t u d e of these o s c i l l a t i o n s is d e s c r i b e d by the usual Dingle [2] factor. In this p a p e r we r e p o r t the field and t e m p e r a t u r e d e pendence of the equivalent o s c i l l a t i o n s in the t h e r m a l m a g n e t o r e s i s t a n c e which we find to e x hibit quite a different behaviour to the o s c i l l a tions in e l e c t r i c a l r e s i s t a n c e . We d e r i v e a "Dingle f a c t o r " for o s c i l l a t o r y t h e r m a l r e s i s t a n c e , which we c o m p a r e with the e x p e r i m e n t a l results. Fig. 1 shows graphs of lOgl0(A/-/) v e r s u s I / H ,
2[
T=1.32°K
w h e r e A is the amplitude of the quantum o s c i l lations, for both the e l e c t r i c a l and t h e r m a l r e s i s t a n c e s . The graph for e l e c t r i c a l r e s i s t a n c e is a p p r o x i m a t e l y a s t r a i g h t line, as we have p r e v i o u s l y r e p o r t e d [1]. By c o n t r a s t , the t h e r m a l r e s i s t a n c e exhibits a deep m i n i m u m , at which the amplitude is u n m e a s u r a b l y s m a l l . This m i n i m u m o c c u r s at a field which v a r i e s l i n e a r l y with t e m p e r a t u r e as shown in fig. 2; for e x p e r i m e n t a l r e a s o n s we have a c t u a l l y plotted 1/T v e r s u s the 1/H value at the m i n i m u m . In the i m p u r i t y s c a t t e r i n g r e g i m e , the t h e r m a l conductivity t e n s o r is given by [3]
Kij =--fe2 T a/j(e)(E- eF )2
de,
where aij(e) is the e l e c t r i c a l conductivity of e l e c t r o n s of e n e r g y e. We make the a s s u m p t i o n a l r e a d y s u c c e s s f u l l y used for e l e c t r i c a l r e s i s t ance in ref.[1], that ayy is the only component to exhibit quantum o s c i l l a t i o n s , via a t e r m p r o p o r t i o n a l to -(b cos ~ ) / H 2, where b = exp (-2~2kT/hwc) and ~ = hA (e)/eH. A(e) is the
0.8 [ OK-I
T
4
°
.
.
$
6
7
/I
9 XIO-SG-I
Fig. 1. Experimental values of logl0(AH), plotted v e r sus l / H , for oscillations in the e l e c t r i c a l magnetoresist~nee (circles) and t h e r m a l magnetoresismnce (crosses). The solid lines are the predictions, based on setting the 1 / H v a l u e of the m i n i m u m to experiment and adjusting T ' t o give the best fit to both sets of data.
5
(I/.)MiN
6
7X10-5G'-I
Fig. 2. The inverse temperature plotted versus the
1/H value at which the amplitude of the oscillatory
thermal magnetoresistance passes through zero. 355
Volume 36A, number 5 a r e a in k - s p a c e of the o r b i t p a n d i n g A(e) about e F g i v e s f o / H + 2n(e - eF)/h¢o c. T h u s p e r a t u r e d e p e n d e n c e of Kyy b c o s (fo/H) /o
PHYSICS of e n e r g y ~, E x ~ = the f i e l d and t e m is g i v e n by
cos[F2~(e- 6F)?J
%leo
de
w h i c h is r e a d i l y i n t e g r a t e d to g i v e
LETTERS
27 September 1971
t e m p e r a t u r e , in a g r e e m e n t with fig. 2. T h e s o l i d l i n e s of fig. 1 a r e c a l c u l a t e d l o g l 0 ( A H ) v e r s u s 1 / H p l o t s , the p o s i t i o n of the m i n i m u m in the t h e r m a l a m p l i t u d e b e i n g s e t to the e x p e r i m e n t a l p o s i t i o n and the v a l u e of T* b e i n g c h o s e n to g i v e l i n e s w h i c h b e s t s i m u l a t e both the e l e c t r i c a l and t h e r m a l r e s u l t s . F r o m the s l o p e of the g r a p h in fig. 2 it is p o s s i b l e to d e r i v e the c y c l o t r o n m a s s , and the v a l u e found ( 0 . 1 2 0 + 0 . 0 0 5 ) rn o is in a g r e e m e n t with c y c l o t r o n r e s o n a n c e [4].
- (bk 2 T/H2) c o s (fo/H) d2D/da 2 , w h e r e D (=av/sinh av) is j u s t the u s u a l D i n g l e f a c t o r f o r r e s i s t a n c e o s c i l l a t i o n s and a = 2v2kT/hw c. F o l l o w i n g r e f . [1], and a s s u m i n g the W e i d e r m a n n - F r a n z law f o r n o n - o s c i l l a t o r y t e r m s , the f i e l d and t e m p e r a t u r e d e p e n d e n c e of Pxx i s g i v e n by -(b/T) c o s (fo/H)(d2D/da2). B e s i d e s f a l l i n g to z e r o when a b e c o m e s i n f i n i t e , d2D/da 2 a l s o p a s s e s t h r o u g h z e r o at ~a = 1.62, thus p r e d i c t i n g that the a m p l i t u d e f a l l s to z e r o at a f i e l d w h i c h is p r o p o r t i o n a l to the
356
T h e a s s i s t a n c e of M r . J. K. H u l b e r t in s p e c i m e n p r e p a r a t i o n and an e q u i p m e n t g r a n t f r o m t h e S. R. C. a r e g r a t e f u l l y a c k n o w l e d g e d .
References [1] R.C. Young, J. Phys. C4 (1971) 474. [2] R.B. Ding[e, Proc. Roy. Soe. A211 (1952) 517. [3] J. M. Ziman, Principles of the theory of solids (Cambridge University P r e s s , 1964) p. 196. [4] M. S°Khaikin, Soviet Phys. J E T P 15 (1963) 18.
PHYSICS
DINGLE
THE
FACTOR
THERMAL
FOR
LETTERS
QUANTUM
27 September 1971
OSCILLATIONS
MAGNETORESISTANCE
OF
IN THE
TIN
R. C. YOUNG
Department of Physics, University of Birmingham, Birmingham, UK Received 14 August 1971
The field dependence of amplitude of the oscillatory thermal magnetoresistance of tin exhibits a subsidiary zero at a field proportional to the temperature, in contrast to electrical magnetoresistance. A theory developed to explain this gives m* = 0.120 mo for orbit 36.
A s t r i k i n g f e a t u r e of the m a g n e t o r e s i s t a n c e of tin is the l a r g e amplitude s i n u s o i d a l quantum o s c i l l a t i o n s which o c c u r w h e n / t is a p p r o x i m a t e l y 23 ° f r o m the c - a x i s , due to the c l o s i n g of open o r b i t s by m a g n e t i c breakdown. We have a l r e a d y shown [1] that the field and t e m p e r a t u r e dependence of the a m p l i t u d e of these o s c i l l a t i o n s is d e s c r i b e d by the usual Dingle [2] factor. In this p a p e r we r e p o r t the field and t e m p e r a t u r e d e pendence of the equivalent o s c i l l a t i o n s in the t h e r m a l m a g n e t o r e s i s t a n c e which we find to e x hibit quite a different behaviour to the o s c i l l a tions in e l e c t r i c a l r e s i s t a n c e . We d e r i v e a "Dingle f a c t o r " for o s c i l l a t o r y t h e r m a l r e s i s t a n c e , which we c o m p a r e with the e x p e r i m e n t a l results. Fig. 1 shows graphs of lOgl0(A/-/) v e r s u s I / H ,
2[
T=1.32°K
w h e r e A is the amplitude of the quantum o s c i l lations, for both the e l e c t r i c a l and t h e r m a l r e s i s t a n c e s . The graph for e l e c t r i c a l r e s i s t a n c e is a p p r o x i m a t e l y a s t r a i g h t line, as we have p r e v i o u s l y r e p o r t e d [1]. By c o n t r a s t , the t h e r m a l r e s i s t a n c e exhibits a deep m i n i m u m , at which the amplitude is u n m e a s u r a b l y s m a l l . This m i n i m u m o c c u r s at a field which v a r i e s l i n e a r l y with t e m p e r a t u r e as shown in fig. 2; for e x p e r i m e n t a l r e a s o n s we have a c t u a l l y plotted 1/T v e r s u s the 1/H value at the m i n i m u m . In the i m p u r i t y s c a t t e r i n g r e g i m e , the t h e r m a l conductivity t e n s o r is given by [3]
Kij =--fe2 T a/j(e)(E- eF )2
de,
where aij(e) is the e l e c t r i c a l conductivity of e l e c t r o n s of e n e r g y e. We make the a s s u m p t i o n a l r e a d y s u c c e s s f u l l y used for e l e c t r i c a l r e s i s t ance in ref.[1], that ayy is the only component to exhibit quantum o s c i l l a t i o n s , via a t e r m p r o p o r t i o n a l to -(b cos ~ ) / H 2, where b = exp (-2~2kT/hwc) and ~ = hA (e)/eH. A(e) is the
0.8 [ OK-I
T
4
°
.
.
$
6
7
/I
9 XIO-SG-I
Fig. 1. Experimental values of logl0(AH), plotted v e r sus l / H , for oscillations in the e l e c t r i c a l magnetoresist~nee (circles) and t h e r m a l magnetoresismnce (crosses). The solid lines are the predictions, based on setting the 1 / H v a l u e of the m i n i m u m to experiment and adjusting T ' t o give the best fit to both sets of data.
5
(I/.)MiN
6
7X10-5G'-I
Fig. 2. The inverse temperature plotted versus the
1/H value at which the amplitude of the oscillatory
thermal magnetoresistance passes through zero. 355
Volume 36A, number 5 a r e a in k - s p a c e of the o r b i t p a n d i n g A(e) about e F g i v e s f o / H + 2n(e - eF)/h¢o c. T h u s p e r a t u r e d e p e n d e n c e of Kyy b c o s (fo/H) /o
PHYSICS of e n e r g y ~, E x ~ = the f i e l d and t e m is g i v e n by
cos[F2~(e- 6F)?J
%leo
de
w h i c h is r e a d i l y i n t e g r a t e d to g i v e
LETTERS
27 September 1971
t e m p e r a t u r e , in a g r e e m e n t with fig. 2. T h e s o l i d l i n e s of fig. 1 a r e c a l c u l a t e d l o g l 0 ( A H ) v e r s u s 1 / H p l o t s , the p o s i t i o n of the m i n i m u m in the t h e r m a l a m p l i t u d e b e i n g s e t to the e x p e r i m e n t a l p o s i t i o n and the v a l u e of T* b e i n g c h o s e n to g i v e l i n e s w h i c h b e s t s i m u l a t e both the e l e c t r i c a l and t h e r m a l r e s u l t s . F r o m the s l o p e of the g r a p h in fig. 2 it is p o s s i b l e to d e r i v e the c y c l o t r o n m a s s , and the v a l u e found ( 0 . 1 2 0 + 0 . 0 0 5 ) rn o is in a g r e e m e n t with c y c l o t r o n r e s o n a n c e [4].
- (bk 2 T/H2) c o s (fo/H) d2D/da 2 , w h e r e D (=av/sinh av) is j u s t the u s u a l D i n g l e f a c t o r f o r r e s i s t a n c e o s c i l l a t i o n s and a = 2v2kT/hw c. F o l l o w i n g r e f . [1], and a s s u m i n g the W e i d e r m a n n - F r a n z law f o r n o n - o s c i l l a t o r y t e r m s , the f i e l d and t e m p e r a t u r e d e p e n d e n c e of Pxx i s g i v e n by -(b/T) c o s (fo/H)(d2D/da2). B e s i d e s f a l l i n g to z e r o when a b e c o m e s i n f i n i t e , d2D/da 2 a l s o p a s s e s t h r o u g h z e r o at ~a = 1.62, thus p r e d i c t i n g that the a m p l i t u d e f a l l s to z e r o at a f i e l d w h i c h is p r o p o r t i o n a l to the
356
T h e a s s i s t a n c e of M r . J. K. H u l b e r t in s p e c i m e n p r e p a r a t i o n and an e q u i p m e n t g r a n t f r o m t h e S. R. C. a r e g r a t e f u l l y a c k n o w l e d g e d .
References [1] R.C. Young, J. Phys. C4 (1971) 474. [2] R.B. Ding[e, Proc. Roy. Soe. A211 (1952) 517. [3] J. M. Ziman, Principles of the theory of solids (Cambridge University P r e s s , 1964) p. 196. [4] M. S°Khaikin, Soviet Phys. J E T P 15 (1963) 18.