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Magnetothermal oscillations in pressure-annealed pyrolytic graphite
Volume 30A. number 3
PHYSICS LETTERS
of 10.9 /1 and we o b s e r v e d an i m p u ls i o n f r o m the d e t e c t o r with an o s c i l l o s c o p e (fig. 2). We e s t i m a t e the p o w e r of the i n f r a r e d e m i s s i o n , outside the c r y s t a l , to be of about ten m i l l i w a t t s . B e s i d e s its t h e o r e t i c a l i n t e r e s t , this e x p e r i m e n t l ead s to c o h e r e n t and tunable g e n e r a t i o n of i . r . radiation. The t w o - b e a m s method initially p r o p o s e d by M a u r i c e P a p o u l a r [7] and one of us [8], is p a r t i c u l a r l y convenient b e c a u s e we can v a r y the angle between the two incident light b e a m s , a c c o r d i n g l y to t h e i r f r e q u e n c y d i f f e r e n c e . The i n t e r a c t i o n is always s t i m u l a t e d , and we a r e not l i m i t e d to s t r i c t l y f o r w a r d s c a t t e r i n g , as it would be the c a s e when using only one pumping light b eam .
have been p o s s i b l e without the financial a s s i s t a n c e of the D~l~gation G~n~rale ~t la R e c h e r c h e Scientifique et Technique.
References 1. M. Born, and K. Huang, Dynamic theory of crystal lattices (Clarendon Press, Oxford, 1966). 2. C.H. Henry, and J. J. Hopfield, Phys. Rev. Letters 15 (1965) 964. 3. S. P. S. Porto, B. Tell, and T. C. Damen, Phys. Rev. Letters 16 (1966) 450. 4. J. F. Scott, L.E. Cheesman, and S. P, S. Porto, Phys. Rev. 162 (1967) 834. 5. S.K. Kurtz, and J. A. Giordmaine, Phys. Rev. Letters 22 (1969) 192. 6. G. Chartier, and S. Biraud, Phys. Rev. Letters 21 (1968) 1641. 7. M. Papoular, Solid State Comm. 4 (1966) 129. 8. G. Chartier and M. Papoular, Proc. Fourth Int. Quantum electronics eonf., Phoenix (1966).
It is a p l e a s u r e f o r us to acknowledge Mr. B e s son of the F r e n c h S. A. T. who gave us the H g - T e , C d - T e i n f r a r e d d e t e c t o r . This work would not
MAGNETOTHERMAL
6 October 1969
OSCILLATIONS IN PRESSURE-ANNEALED PYROLYTIC GRAPHITE D. J . F L O O D
Lewis Research Center, Cleveland, Ohio Received 27 August 1969
Magnetothermal oscillations have been observed in pressure-annealed pyrolytic graphite. The period of the oscillations was measured as a function of angle between magnetic field H and the c axis of the crystal. The sample had a resistance ratio of ½ from room temperature to 4.2°K. Data were obtained for the majority hole c a r r i e r s into the extreme quantum limit.
M a g n e t o t h e r m a l o s c i l l a t i o n s have been o b s e r v e d as a function of angle in a s a m p l e of p r e s s u r e - a n ne a l ed , p y r o l y t i c g r a p h i t e [1] in m a g n e t i c f i e l d s ranging f r o m 3 to 100 k i l o g a u s s . A field modulation t e c h n i q u e was u s e d in which a low amplitude, 4 h e r t z a.c m a g n e t i c f i el d was s u p e r i m p o s e d on, and p a r a l l e l to, the m o n o t o n i c a l l y swept d.c. field. M e a s u r e m e n t s w e r e m a d e in the t e m p e r a t u r e r a n g e p r o v i d e d by pumping on 4He, and w e r e t y p i c a l l y at 1.3°K. T e m p e r a t u r e changes w e r e m e a s u r e d using a l o c k - i n a m p l i f i e r to d e t e c t the v o l t a g e a c r o s s a c a r b o n t h e r m o m e t e r a t t a c h e d to the s a m p l e . The quantity a c t u a l l y o b s e r v e d by this technique is (OT/~H)s,O' given by [2]
-T~2~gf ~1-½[, ~ V OM (aT/aH)s,O : CH,0 (-~f )H,O- CH, O~H-~ l ~ where
coth
lT]l~exp(-lX/Ttt) sin(27rlf/HTHA sinh/
2~'/T:F¼rO[ I Y
J
(1) 2
1
Pl = - 2 Vk B (e/ hc) ~ ~(Ez F!I-'~ L "A ~/e jk °
178
lT
cos
(lrrgm*/2rno),
Volume 30A, number 3
P HY S I CS L E T T E R S
6 October 1969
16
14-- / 12--
1.4
l'& i -~ I.C
PERIOD IN
h
PEAK NO.
GAUSS-IxI05 .8
,6 .4
0~
.2
I I I [ J O 20 40 60 80 ANGLEBETWEEN H ANO C AXIS, DEG
Fig. 1.
-21 0
~INTERCEPT~0. 37
I
I
I
I
2
3xlO-4
I/H, GAUSS-I Fig. 2.
s the entropy; 0 the field o r i e n t a t i o n ; C H 0 the total heat capacity; TIt = 8" H/2 n2k s; ~* = ePI/m* c; X - Dingle t e m p e r a t u r e , f - ChAo~ F )'/2ue, the de H a a s - v a n Alphen frequency, A o(Ef ) is the e x t r e m ai a r e a of the F e r m i s u r f a c e located at k z = ko; rn* is the effective m a s s ; g is the spin-'splitting f a c t o r ; m o i s the f r e e m a s s ; and the other s y m b o l s have t h e i r u s u a l meaning. T is a phase c o n s t a n t whose value lies between 0 and ½. Fig. 1 is a plot of the p e r i o d a g a i n s t angle for the o s c i l l a t i o n s observed. The value of the p e r i o d for H p a r a l l e l to the c axis i s 1.51+0.06 × 10 -5 gauss - I _ T h i s a - g r e e s with the values obtained for the m a j o r ity hole c a r r i e r s by Soule et al. [3] i n single c r y s t a l graphite, and by W i l l i a m s o n et al. [4], in p y r o l y t i c graphite. O s c i l l a t i o n s a r i s i n g f r o m the m a j o r i t y e l e c t r o n pockets w e r e not s e e n at the t e m p e r a t u r e of these m e a s u r e m e n t s . B e c a u s e of the s m a l l effective m a s s of the m a j o r i t y hole c a r r i e r s , the e x t r e m e quantum l i m i t i n graphite is r e a c h e d by about 60 k i l o g a u s s for H p a r a l l e l to the c axis. Hence a s s i g n m e n t of the actual quantum n u m b e r c o r r e s p o n d i n g to each t e m p e r a t u r e peak b e c o m e s p o s s i b l e , with the r e s u l t that the phase factor ~ can be d e t e r m i n e d . Fig. 2 is a plot of voltage m a x i m a a g a i n s t 1/H. T h e s e peaks a r e shifted f r o m the t e m p e r a t u r e peaks by 180 d e g r e e s so that the condition for a m a x i m u m b e c o m e s 27rf/H - 2~7 ¼7r = 27rn - ±ff z • The i n t e r c e p t at (I/H) = 0 in fig. 2 i s at n = - 0 . 3 7 ± 0 . 0 3 . This gives a value of + 0.49 + 0.03 for 7, which is v e r y close to the value of ½ for a q u a d r a t i c d i s p e r s i o n law for the e n e r g y levels. The p r i n c i p l e u n c e r t a i n t y in the n u m b e r obtained for r is a r e s u l t of the a p p r o x i m a t e l y 4% u n c e r t a i n t y in the p e r i o d of the o s c i l l a t i o n s . A long p e r i o d m o d u l a t i o n of the hole o s c i l l a t i o n s was also o b s e r v e d which is tentatively a s c r i b e d to the pocket of m i n o r i t y e l e c t r o n s which a r e located at the c o r n e r of the B r i l l o u i n zone. The p e r i o d of this o s c i l l a t i o n i s 0 . 8 5 ± 0 . 2 0 × 10 -4 gauss -1, and i s c o n s i d e r a b l y lower than the m i n o r i t y c a r r i e r p e r i ods obtained by Soule [3] and W i l l i a m s o n [4]. B e c a u s e of the e x i s t e n c e of a r e m n a n t field i n the s u p e r conducting solenoid u s e d for the e x p e r i m e n t , m e a s u r e m e n t s could not be made below about 3 k i l o g a u s s to i n v e s t i g a t e the n a t u r e of this c o n t r i b u t i o n m o r e extensively. --
n
"
n
•
__
#
.
•
_
The author w i s h e s to thank Union C a r b i d e C o r p o r a t i o n , P a r m a , Ohio for g e n e r o u s l y supplying the s a m p l e u s e d in this e x p e r i m e n t .
References 1. A.W. Moore, Ned. Tijdschr. Natuurk. 32 (1966) 221. Pyrolytic graphite is characterized by a high degree of order along along the c axis, and complete disQrder in the basal plane. 2. B. McCombe and G. Seidel, Phys. Rev. 155 (1967) 633. 3. D.E Soule, J.W. McClure and L.B. Smith, Phys. Rev. 134 (1964)A453. 4. S.J. Williamson, S. Foner and M. Dresselhaus, Phys. Rev. 140 (1965) A1429.
179
PHYSICS LETTERS
of 10.9 /1 and we o b s e r v e d an i m p u ls i o n f r o m the d e t e c t o r with an o s c i l l o s c o p e (fig. 2). We e s t i m a t e the p o w e r of the i n f r a r e d e m i s s i o n , outside the c r y s t a l , to be of about ten m i l l i w a t t s . B e s i d e s its t h e o r e t i c a l i n t e r e s t , this e x p e r i m e n t l ead s to c o h e r e n t and tunable g e n e r a t i o n of i . r . radiation. The t w o - b e a m s method initially p r o p o s e d by M a u r i c e P a p o u l a r [7] and one of us [8], is p a r t i c u l a r l y convenient b e c a u s e we can v a r y the angle between the two incident light b e a m s , a c c o r d i n g l y to t h e i r f r e q u e n c y d i f f e r e n c e . The i n t e r a c t i o n is always s t i m u l a t e d , and we a r e not l i m i t e d to s t r i c t l y f o r w a r d s c a t t e r i n g , as it would be the c a s e when using only one pumping light b eam .
have been p o s s i b l e without the financial a s s i s t a n c e of the D~l~gation G~n~rale ~t la R e c h e r c h e Scientifique et Technique.
References 1. M. Born, and K. Huang, Dynamic theory of crystal lattices (Clarendon Press, Oxford, 1966). 2. C.H. Henry, and J. J. Hopfield, Phys. Rev. Letters 15 (1965) 964. 3. S. P. S. Porto, B. Tell, and T. C. Damen, Phys. Rev. Letters 16 (1966) 450. 4. J. F. Scott, L.E. Cheesman, and S. P, S. Porto, Phys. Rev. 162 (1967) 834. 5. S.K. Kurtz, and J. A. Giordmaine, Phys. Rev. Letters 22 (1969) 192. 6. G. Chartier, and S. Biraud, Phys. Rev. Letters 21 (1968) 1641. 7. M. Papoular, Solid State Comm. 4 (1966) 129. 8. G. Chartier and M. Papoular, Proc. Fourth Int. Quantum electronics eonf., Phoenix (1966).
It is a p l e a s u r e f o r us to acknowledge Mr. B e s son of the F r e n c h S. A. T. who gave us the H g - T e , C d - T e i n f r a r e d d e t e c t o r . This work would not
MAGNETOTHERMAL
6 October 1969
OSCILLATIONS IN PRESSURE-ANNEALED PYROLYTIC GRAPHITE D. J . F L O O D
Lewis Research Center, Cleveland, Ohio Received 27 August 1969
Magnetothermal oscillations have been observed in pressure-annealed pyrolytic graphite. The period of the oscillations was measured as a function of angle between magnetic field H and the c axis of the crystal. The sample had a resistance ratio of ½ from room temperature to 4.2°K. Data were obtained for the majority hole c a r r i e r s into the extreme quantum limit.
M a g n e t o t h e r m a l o s c i l l a t i o n s have been o b s e r v e d as a function of angle in a s a m p l e of p r e s s u r e - a n ne a l ed , p y r o l y t i c g r a p h i t e [1] in m a g n e t i c f i e l d s ranging f r o m 3 to 100 k i l o g a u s s . A field modulation t e c h n i q u e was u s e d in which a low amplitude, 4 h e r t z a.c m a g n e t i c f i el d was s u p e r i m p o s e d on, and p a r a l l e l to, the m o n o t o n i c a l l y swept d.c. field. M e a s u r e m e n t s w e r e m a d e in the t e m p e r a t u r e r a n g e p r o v i d e d by pumping on 4He, and w e r e t y p i c a l l y at 1.3°K. T e m p e r a t u r e changes w e r e m e a s u r e d using a l o c k - i n a m p l i f i e r to d e t e c t the v o l t a g e a c r o s s a c a r b o n t h e r m o m e t e r a t t a c h e d to the s a m p l e . The quantity a c t u a l l y o b s e r v e d by this technique is (OT/~H)s,O' given by [2]
-T~2~gf ~1-½[, ~ V OM (aT/aH)s,O : CH,0 (-~f )H,O- CH, O~H-~ l ~ where
coth
lT]l~exp(-lX/Ttt) sin(27rlf/HTHA sinh/
2~'/T:F¼rO[ I Y
J
(1) 2
1
Pl = - 2 Vk B (e/ hc) ~ ~(Ez F!I-'~ L "A ~/e jk °
178
lT
cos
(lrrgm*/2rno),
Volume 30A, number 3
P HY S I CS L E T T E R S
6 October 1969
16
14-- / 12--
1.4
l'& i -~ I.C
PERIOD IN
h
PEAK NO.
GAUSS-IxI05 .8
,6 .4
0~
.2
I I I [ J O 20 40 60 80 ANGLEBETWEEN H ANO C AXIS, DEG
Fig. 1.
-21 0
~INTERCEPT~0. 37
I
I
I
I
2
3xlO-4
I/H, GAUSS-I Fig. 2.
s the entropy; 0 the field o r i e n t a t i o n ; C H 0 the total heat capacity; TIt = 8" H/2 n2k s; ~* = ePI/m* c; X - Dingle t e m p e r a t u r e , f - ChAo~ F )'/2ue, the de H a a s - v a n Alphen frequency, A o(Ef ) is the e x t r e m ai a r e a of the F e r m i s u r f a c e located at k z = ko; rn* is the effective m a s s ; g is the spin-'splitting f a c t o r ; m o i s the f r e e m a s s ; and the other s y m b o l s have t h e i r u s u a l meaning. T is a phase c o n s t a n t whose value lies between 0 and ½. Fig. 1 is a plot of the p e r i o d a g a i n s t angle for the o s c i l l a t i o n s observed. The value of the p e r i o d for H p a r a l l e l to the c axis i s 1.51+0.06 × 10 -5 gauss - I _ T h i s a - g r e e s with the values obtained for the m a j o r ity hole c a r r i e r s by Soule et al. [3] i n single c r y s t a l graphite, and by W i l l i a m s o n et al. [4], in p y r o l y t i c graphite. O s c i l l a t i o n s a r i s i n g f r o m the m a j o r i t y e l e c t r o n pockets w e r e not s e e n at the t e m p e r a t u r e of these m e a s u r e m e n t s . B e c a u s e of the s m a l l effective m a s s of the m a j o r i t y hole c a r r i e r s , the e x t r e m e quantum l i m i t i n graphite is r e a c h e d by about 60 k i l o g a u s s for H p a r a l l e l to the c axis. Hence a s s i g n m e n t of the actual quantum n u m b e r c o r r e s p o n d i n g to each t e m p e r a t u r e peak b e c o m e s p o s s i b l e , with the r e s u l t that the phase factor ~ can be d e t e r m i n e d . Fig. 2 is a plot of voltage m a x i m a a g a i n s t 1/H. T h e s e peaks a r e shifted f r o m the t e m p e r a t u r e peaks by 180 d e g r e e s so that the condition for a m a x i m u m b e c o m e s 27rf/H - 2~7 ¼7r = 27rn - ±ff z • The i n t e r c e p t at (I/H) = 0 in fig. 2 i s at n = - 0 . 3 7 ± 0 . 0 3 . This gives a value of + 0.49 + 0.03 for 7, which is v e r y close to the value of ½ for a q u a d r a t i c d i s p e r s i o n law for the e n e r g y levels. The p r i n c i p l e u n c e r t a i n t y in the n u m b e r obtained for r is a r e s u l t of the a p p r o x i m a t e l y 4% u n c e r t a i n t y in the p e r i o d of the o s c i l l a t i o n s . A long p e r i o d m o d u l a t i o n of the hole o s c i l l a t i o n s was also o b s e r v e d which is tentatively a s c r i b e d to the pocket of m i n o r i t y e l e c t r o n s which a r e located at the c o r n e r of the B r i l l o u i n zone. The p e r i o d of this o s c i l l a t i o n i s 0 . 8 5 ± 0 . 2 0 × 10 -4 gauss -1, and i s c o n s i d e r a b l y lower than the m i n o r i t y c a r r i e r p e r i ods obtained by Soule [3] and W i l l i a m s o n [4]. B e c a u s e of the e x i s t e n c e of a r e m n a n t field i n the s u p e r conducting solenoid u s e d for the e x p e r i m e n t , m e a s u r e m e n t s could not be made below about 3 k i l o g a u s s to i n v e s t i g a t e the n a t u r e of this c o n t r i b u t i o n m o r e extensively. --
n
"
n
•
__
#
.
•
_
The author w i s h e s to thank Union C a r b i d e C o r p o r a t i o n , P a r m a , Ohio for g e n e r o u s l y supplying the s a m p l e u s e d in this e x p e r i m e n t .
References 1. A.W. Moore, Ned. Tijdschr. Natuurk. 32 (1966) 221. Pyrolytic graphite is characterized by a high degree of order along along the c axis, and complete disQrder in the basal plane. 2. B. McCombe and G. Seidel, Phys. Rev. 155 (1967) 633. 3. D.E Soule, J.W. McClure and L.B. Smith, Phys. Rev. 134 (1964)A453. 4. S.J. Williamson, S. Foner and M. Dresselhaus, Phys. Rev. 140 (1965) A1429.
179