- Email: [email protected]
On the piezoelectric polaron motion at low temperature
Volume 30A, number 7
PHYSICS
the s a m e single c r y s t a l of tin u s i n g a d o u b l e sided excitation by winding a coil around the tin sheet. C o n s i d e r i n g the e x p e r i m e n t a l difficulties of the different techniques for studying the radio f r e q u e n c y size effect we found the e x p e r i m e n t s on the box somewhat s i m p l e r to do than the exp e r i m e n t s with the d o u b l e - s i d e d excitation. We also b e l i e v e that they a r e much s i m p l e r than the t r a n s m i s s i o n e x p e r i m e n t s done by Walsh and C r i m e s [4]. F r o m fig. 1 it is seen that with d o u b l e - s i d e d excitation a continuous change of the o v e r a l l s i g nal with H o o c c u r s . The peaks a r e l e s s pronounced than for the s i n g l e - s i d e d excitation; this is e s p e c i a l l y seen for the peak at about 200 Oe. It also s e e m s that the peaks a r e somewhat shifted. On the s i n g l e sided excitation c u r v e it is i n t e r e s t i n g to corn-
ON T H E
PIEZOELECTRIC
POLARON
LETTERS
1 December 1969
p a r e in detail the l a r g e peak at about 200 Oe with the peak at about 400 Oe as t h e s e two peaks a r e connected with the s a m e o r b i t [1,6].
References 1. V.F. Gantmakher, Progress in low temperature physics, ed. C.J. Gorter, Vol. 5 (North-Holland, Amsterdam, 1967). 2. E.A. Kaner, V. L. Fal'ko, Soviet Phys. JETP 24 (1967) 392. 3. E.A. Kaner, V. F. Gantmakher, Soviet Phys. Uspekhi 11 (1968) 81. 4. W.M.Walsh Jr. and C.C. Crimes, Phys. Rev. Letters 13 (1964) 523. 5. C.A.A.J. Greebe, W. F. Druyvesteyn and A.J. Smets, Phys. Letters 22 (1966) 246; Philips Res. Repts. 23 (1968) 332. 6. M. M. M. P. Matthey, Thesis University of Nijmegen, to be published.
MOTION
AT
LOW
TEMPERATURE
M. PORSCH *
Sektion Physik der Humboldt-UniversiNit zu Berlin, Bereich Theoretische Halbleiterpkysik Received 23 October 1969
Temperature-dependent expressions for self-energy, effective mass, and energy-momentum relation of the piezoelectric polaron are given in the intermediate coupling region.
In t h e i r r e c e n t investigation of s e v e r a l a p p r o x i m a t i o n methods used in polaron theory, Whitfield and c o - w o r k e r s [1] have shown that the well-known L e e - L o w - P i n e s v a r i a t i o n a l Ansatz [2] p r o v i d e s a r e a sonable e n e r g y - m o m e n t u m r e l a t i o n for the p i e z o e l e c t r i c p o l a r o n at T = 0. By m e a n s of a G r e e n function approach being a p p r o p r i a t e for calculating p o l a r o n effects in the e n t i r e r a n g e of coupling s t r e n g t h s [3], we have obtained the following i n t e r m e d i a t e - c o u p l i n g e x p r e s s i o n f o r the energy of the p i e z o - p o l a r o n e (k) in the s i m p l e s t approximation: ~(k) = 2-raft2(1 - ~ ) k 2 --~-4na~ ti2 s 2
q
q
1
,
(1)
tisq + (t~2q2/2m)(2Nq+ 1) - ( h ' 2 / m ) ( 1 - y k ) k .q
w h e r e m i s the e l e c t r o n band m a s s , h-k the p o l a r o n m o m e n t u m , s an a v e r a g e speed of sound, Nq the m e a n n u m b e r of phonons with the wave n u m b e r q in the t h e r m a l e q u i l i b r i u m , a the d i m e n s i o n l e s s e l e c t r o n - p i e z o p h o n o n coupling constant [1], and ~ the c r y s t a l volume. ~?(k) is defined by the t r a n s c e n d e n t a l equation 4 ~ ~ tfl s 2
71kk
~~
q
%-
q
[lfsq + (ti2 q2/2m)(2Nq + 1) - ( t / 2 / m ) ( 1 - 71k)k "q ]2
(2)
The right hand sides of eqs. (1) and (2) can be a n a l y t i c a l l y evaluated u s i n g the h i g h - t e m p e r a t u r e a p proximation * New address: Institut fllr Elektronische Bauelemente, 117 Berlin, Bendigstrasse 11. 416
PHYSICS LETTERS
Volume 3OA, number 7
1 December 1969
(3)
Nq ~ k B T/~sq which a l r e a d y holds at h e l i u m t e m p e r a t u r e f o r a c o u s t i c a l phonons. The r e s u l t s a r e
¢(k) =
ll+kBT/ms2
+rpqm + 2(1 - Tlk)rp k) + kBT rp qm) In ( ~ r p k ( 1 - 7 / k ) I (1 + ms---2+ +kBT/ms2+rpqm -2(1 -~k)rpk ares2
kBT.
(1+%T/ms2+ 2(1
+
- ( l + m - - ~ ) In l +kBT/ms 2 -2(1 -~k)rp
(4)
K
+2(1_Tik)rpkin~(l +kBT/ms2+rpqm)~ -~4(1kg-4(1 ~-~k;2-r~ -nk)2r~k2~]l+ 2-m 1_w2)k2,
47rr~ ~
~--~s + r P q m ) i n ( l + k B T / m s 2 + r p q m - 2 ( 1 -
) +
(5)
_ (l +kB T/ ms2) ln ( l + kB T/ms2 + 2(1- ~k) rp kk) l , ~1 + kB T / m s 2 - 2(1 - ~k) rp w h e r e qm i s the cut-off phonon wave n u m b e r , and rp = Pi/2rns. In o u r a p p r o x i m a t i o n , t h e r m a l e f f e c t s will not q u a l i t a t i v e l y a l t e r the e n e r g y - m o m e n t u m r e l a t i o n given in [1], the l a t t e r being obtained by putting T = 0 in the above f o r m u l a e . In the e x t r e m e l o w - t e m p e r a t u r e l i m i t , h o w e v e r , o u r e x p r e s s i o n s do not apply s i n c e (3) no l o n g e r holds. At s m a l l v a l u e s of k, we have
~(le)=
4~ ms 2 In (1 + rp qm ~ ~2 k2 - lr 1+ kBT/ms 2] + 2m * '
m* = m(1 + (4a/3~r) y(T)),
(6) (7)
w h e r e y(T) = (1 + kBT/ms2 )-2 _ (1 + kBT/ms2 + rp qm)-2. Our r e s u l t s i n d i c a t e that t e m p e r a t u r e effects will d e c r e a s e the p i e z o - p o l a r o n s e l f - e n e r g y and e f f e c t i v e m a s s , the l a t t e r a p p r o a c h i n g m a s T -2. Both f e a t u r e s can b e i n t e r p r e t e d a s due to a t h e r m a l l o o s e n i n g of the c o h e r e n t v i r t u a l phonon cloud s u r r o u n d i n g the e l e c t r o n . Damping e f f e c t s a r i s i n g at h i g h e r - o r d e r decoupling will be d i s c u s s e d in f u t u r e p a p e r s .
References 1. G.Whitfieid, J. Gerstner and K. Tharmalingam, Phys. Rev. 165 (1968) 993. 2. T.D. Lee, F. Low and D. Pines, Phys. Rev. 90 (1953) 297. 3. M. Porsch, to be published. * * * * ~
417
PHYSICS
the s a m e single c r y s t a l of tin u s i n g a d o u b l e sided excitation by winding a coil around the tin sheet. C o n s i d e r i n g the e x p e r i m e n t a l difficulties of the different techniques for studying the radio f r e q u e n c y size effect we found the e x p e r i m e n t s on the box somewhat s i m p l e r to do than the exp e r i m e n t s with the d o u b l e - s i d e d excitation. We also b e l i e v e that they a r e much s i m p l e r than the t r a n s m i s s i o n e x p e r i m e n t s done by Walsh and C r i m e s [4]. F r o m fig. 1 it is seen that with d o u b l e - s i d e d excitation a continuous change of the o v e r a l l s i g nal with H o o c c u r s . The peaks a r e l e s s pronounced than for the s i n g l e - s i d e d excitation; this is e s p e c i a l l y seen for the peak at about 200 Oe. It also s e e m s that the peaks a r e somewhat shifted. On the s i n g l e sided excitation c u r v e it is i n t e r e s t i n g to corn-
ON T H E
PIEZOELECTRIC
POLARON
LETTERS
1 December 1969
p a r e in detail the l a r g e peak at about 200 Oe with the peak at about 400 Oe as t h e s e two peaks a r e connected with the s a m e o r b i t [1,6].
References 1. V.F. Gantmakher, Progress in low temperature physics, ed. C.J. Gorter, Vol. 5 (North-Holland, Amsterdam, 1967). 2. E.A. Kaner, V. L. Fal'ko, Soviet Phys. JETP 24 (1967) 392. 3. E.A. Kaner, V. F. Gantmakher, Soviet Phys. Uspekhi 11 (1968) 81. 4. W.M.Walsh Jr. and C.C. Crimes, Phys. Rev. Letters 13 (1964) 523. 5. C.A.A.J. Greebe, W. F. Druyvesteyn and A.J. Smets, Phys. Letters 22 (1966) 246; Philips Res. Repts. 23 (1968) 332. 6. M. M. M. P. Matthey, Thesis University of Nijmegen, to be published.
MOTION
AT
LOW
TEMPERATURE
M. PORSCH *
Sektion Physik der Humboldt-UniversiNit zu Berlin, Bereich Theoretische Halbleiterpkysik Received 23 October 1969
Temperature-dependent expressions for self-energy, effective mass, and energy-momentum relation of the piezoelectric polaron are given in the intermediate coupling region.
In t h e i r r e c e n t investigation of s e v e r a l a p p r o x i m a t i o n methods used in polaron theory, Whitfield and c o - w o r k e r s [1] have shown that the well-known L e e - L o w - P i n e s v a r i a t i o n a l Ansatz [2] p r o v i d e s a r e a sonable e n e r g y - m o m e n t u m r e l a t i o n for the p i e z o e l e c t r i c p o l a r o n at T = 0. By m e a n s of a G r e e n function approach being a p p r o p r i a t e for calculating p o l a r o n effects in the e n t i r e r a n g e of coupling s t r e n g t h s [3], we have obtained the following i n t e r m e d i a t e - c o u p l i n g e x p r e s s i o n f o r the energy of the p i e z o - p o l a r o n e (k) in the s i m p l e s t approximation: ~(k) = 2-raft2(1 - ~ ) k 2 --~-4na~ ti2 s 2
q
q
1
,
(1)
tisq + (t~2q2/2m)(2Nq+ 1) - ( h ' 2 / m ) ( 1 - y k ) k .q
w h e r e m i s the e l e c t r o n band m a s s , h-k the p o l a r o n m o m e n t u m , s an a v e r a g e speed of sound, Nq the m e a n n u m b e r of phonons with the wave n u m b e r q in the t h e r m a l e q u i l i b r i u m , a the d i m e n s i o n l e s s e l e c t r o n - p i e z o p h o n o n coupling constant [1], and ~ the c r y s t a l volume. ~?(k) is defined by the t r a n s c e n d e n t a l equation 4 ~ ~ tfl s 2
71kk
~~
q
%-
q
[lfsq + (ti2 q2/2m)(2Nq + 1) - ( t / 2 / m ) ( 1 - 71k)k "q ]2
(2)
The right hand sides of eqs. (1) and (2) can be a n a l y t i c a l l y evaluated u s i n g the h i g h - t e m p e r a t u r e a p proximation * New address: Institut fllr Elektronische Bauelemente, 117 Berlin, Bendigstrasse 11. 416
PHYSICS LETTERS
Volume 3OA, number 7
1 December 1969
(3)
Nq ~ k B T/~sq which a l r e a d y holds at h e l i u m t e m p e r a t u r e f o r a c o u s t i c a l phonons. The r e s u l t s a r e
¢(k) =
ll+kBT/ms2
+rpqm + 2(1 - Tlk)rp k) + kBT rp qm) In ( ~ r p k ( 1 - 7 / k ) I (1 + ms---2+ +kBT/ms2+rpqm -2(1 -~k)rpk ares2
kBT.
(1+%T/ms2+ 2(1
+
- ( l + m - - ~ ) In l +kBT/ms 2 -2(1 -~k)rp
(4)
K
+2(1_Tik)rpkin~(l +kBT/ms2+rpqm)~ -~4(1kg-4(1 ~-~k;2-r~ -nk)2r~k2~]l+ 2-m 1_w2)k2,
47rr~ ~
~--~s + r P q m ) i n ( l + k B T / m s 2 + r p q m - 2 ( 1 -
) +
(5)
_ (l +kB T/ ms2) ln ( l + kB T/ms2 + 2(1- ~k) rp kk) l , ~1 + kB T / m s 2 - 2(1 - ~k) rp w h e r e qm i s the cut-off phonon wave n u m b e r , and rp = Pi/2rns. In o u r a p p r o x i m a t i o n , t h e r m a l e f f e c t s will not q u a l i t a t i v e l y a l t e r the e n e r g y - m o m e n t u m r e l a t i o n given in [1], the l a t t e r being obtained by putting T = 0 in the above f o r m u l a e . In the e x t r e m e l o w - t e m p e r a t u r e l i m i t , h o w e v e r , o u r e x p r e s s i o n s do not apply s i n c e (3) no l o n g e r holds. At s m a l l v a l u e s of k, we have
~(le)=
4~ ms 2 In (1 + rp qm ~ ~2 k2 - lr 1+ kBT/ms 2] + 2m * '
m* = m(1 + (4a/3~r) y(T)),
(6) (7)
w h e r e y(T) = (1 + kBT/ms2 )-2 _ (1 + kBT/ms2 + rp qm)-2. Our r e s u l t s i n d i c a t e that t e m p e r a t u r e effects will d e c r e a s e the p i e z o - p o l a r o n s e l f - e n e r g y and e f f e c t i v e m a s s , the l a t t e r a p p r o a c h i n g m a s T -2. Both f e a t u r e s can b e i n t e r p r e t e d a s due to a t h e r m a l l o o s e n i n g of the c o h e r e n t v i r t u a l phonon cloud s u r r o u n d i n g the e l e c t r o n . Damping e f f e c t s a r i s i n g at h i g h e r - o r d e r decoupling will be d i s c u s s e d in f u t u r e p a p e r s .
References 1. G.Whitfieid, J. Gerstner and K. Tharmalingam, Phys. Rev. 165 (1968) 993. 2. T.D. Lee, F. Low and D. Pines, Phys. Rev. 90 (1953) 297. 3. M. Porsch, to be published. * * * * ~
417