- Email: [email protected]
Influence of crystal fields on the magnetic properties of the rare-earth nitrides
Volume 14, number 4
PHYSICS LETTERS
studied the influence of s t r o n g cubic f i e l d s in o r d e r to u n d e r s t a n d s a t u r a t i o n m o m e n t s m e a s u r e d in the LnNi 2 compounds by A b r a h a m s et al. [6]. F o r a c o m p a r i s o n (the two s t r u c t u r e s a r e not identical) we give t h e i r v a l u e s in t a b l e 1 together with the m a g n e t i z a t i o n s d e t e r m i n e d for a field of 130 kOe. Child et al. [2] found i n some a n t i m o n i d e s o r d e r e d m o m e n t s well below the gJ v a l u e s for the f r e e ions and T r a m m e l l [7] gave a t h e o r e t i c a l a n a l y s i s of this fact. C h i l d ' s [2] v a l u e s a g r e e in the case of HoSb and ErSb r e m a r k ably well (within a few percent} with m a g n e t o n n u m b e r s we obtain by e x t r a p o l a t i n g m e a s u r e d m a g n e t i z a t i o n s to infinite m a g n e t i c f i e l d s . By m e a n s of the m o l e c u l a r field a p p r o x i m a tion it is p o s s i b l e to c a l c u l a t e the W e i s s - f i e l d . The r e d u c e d o r d e r e d m o m e n t s a r e t a k e n f r o m the e x p e r i m e n t a l v a l u e s of the m a g n e t i z a t i o n s . We find that the calculated W e i s s - f i e l d a g r e e s within 25% with the c r i t i c a l field s t r e n g t h Hc. We have s u m m a r i z e d our r e s u l t s and c a l c u l a t i o n s in table 1, the n o t a t i o n s being explained in the text.
INFLUENCE
OF
CRYSTAL OF THE
15 February 1965
The a u t h o r s a r e indebted to the S c h w e i z e r i s cher National fonds z u r FOrderung d e r w i s s e n schaftlichen F o r s c h u n g and to the V e r e i n zur F ~ r d e r u n g der F e s t k ~ r p e r p h y s i k an der ETH for f i n a n c i a l support. We wish to thank P r o f . Grtin e n f e l d e r f r o m the M i n e r a l o g i c a l Institute of the ETH for p e r f o r m i n g the m i c r o p r o b e a n a l y s i s and Dr. Hulliger f r o m Cyanamid European R e s e a r c h Institute for supplying the GdSb c r y s t a l s .
References 1. G.Busch, P.Junod, O.Vogt and F.Hulliger, Physics Letters 6 (1963) 79. 2. H.R.Child, M.K.Wilkinson, J.W. Cable, W.C.Koehler and E.O.Wollan, Phys.Rev. 131 (1963) 922. 3. G.Buseh, P.Junod, P.Schwob, O.Vogt and F.Hulliger, Physics Letters 9 (1964) 7. 4. E.g.S. Smart in "Magnetism HI" by G. T. Rado and H.Suhl (Academic Press, New York, 1963). 5. B.Bleany, Proc.Roy.Soc. 276 (1963) 28. 6. S. C. Abrahams, J.L. Bernstein, R.C.Sherwood, J. H. Wernick and H. J. Williams, J. Phys. Chem. Solids 25 (1964) 1069. 7. G.T.Trammell, Phys.Rev. 131 (1963) 932.
FIELDS ON RARE-EARTH
THE MAGNETIC NITRIDES
PROPERTIES
G. BUSCH, P . JUNOD, F. LEVY, A. MENTH and O. VOGT
Laboratorium flir FestkSrperphysik, ETH, Ziirich, Switzerland Received 16 January 1965
We i n v e s t i g a t e d the whole s e r i e s of the r a r e e a r t h n i t r i d e s , f r o m CeN to YbN which a l l c r y s t a l l i z e in the s i m p l e NaC1 s t r u c t u r e . Our m e a s u r e m e n t s of the m a g n e t i z a t i o n up to 130 kOe and of the m a g n e t i c s u s c e p t i b i l i t i e s between 1.5°K and 600°K allow a unique i n t e r p r e t a t i o n and can be c o m p a r e d with other works. We would like to m e n t i o n e s p e c i a l l y the n e u t r o n diffraction exp e r i m e n t s on the heavy r a r e - e a r t h n i t r i d e s by Child et al. [1] and a t h e o r e t i c a l d i s c u s s i o n by T r a m m e l l [2]. The r e s u l t s of the l a t t e r can be s u m m a r i z e d in the following way: In the case where the exchange f o r c e s a r e s m a l l , c o m p a r e d to the c r y s t a l l i n e field s p l i t t i n g s , the p r e s e n c e or a b s e n c e of an o r d e r e d m a g n e t i c g r o u n d - s t a t e will depend only on the n u m b e r of 4I e l e c t r o n s , a s long as the c r y s t a l line field splittings a r e l a r g e r than kT. Under 264
this a s s u m p t i o n r a r e - e a r t h n i t r i d e s with an odd n u m b e r of 4/ e l e c t r o n s will be m a g n e t i c a l l y o r dered. If the 4£ shell c o n t a i n s an even n u m b e r of e l e c t r o n s , the s t a t i o n a r y state in the cubic c r y s t a l field m a y or m a y not be o r d e r e d by the exchange f o r c e s ; this r e s u l t i s a d i r e c t c o n s e quence of K r a m e r ' s t h e o r e m . We have p r e p a r e d s t o i c h i o m e t r i c s p e c i m e n s of eleven n i t r i d e s by d i r e c t r e a c t i o n of the comp o n e n t s [3]. The e x p e r i m e n t a l r e s u l t s for the whole s e r i e s of compounds f r o m P r N to TmN a r e s u m m a r i z e d in table 1. The two n i t r i d e s CeN and YbN will be d i s c u s s e d l a t e r . In the t h i r d row of the table we indicate by " y e s " or "no" the p r e s e n c e or a b s e n c e of a l o n g - r a n g e m a g n e t i c o r d e r . It i s r e m a r k a b l e that a l l the compounds which do not o r d e r at low t e m p e r a t u r e s have an even n u m b e r of 4f e l e c t r o n s and that all the n i -
Volume 14, number 4
PHYSICS LETTERS
,/x.
•
T m N (4£12). At low t e m p e r a t u r e s the C u r i e W e i s s law does not hold any longer and the s u s c e p t i b i l i t i e s tend to a t e m p e r a t u r e independent value. In the high t e m p e r a t u r e r e g i o n , where the c r y s t a l l i n e field s p l i t t i n g s b e c o m e s m a l l e r than kT, the n i t r i d e s of P r , Gd, Tb, Dy, Ho, Er and T m obey the C u r i e - W e i s s law. The e x p e r i m e n tal m a g n e t o n n u m b e r s a g r e e with the t h e o r e t i c a l v a l u e s for the t r i v a l e n t ions. The other c o m pounds NdN, StuN and EuN do not follow the C u r i e - W e i s s law a s a c o n s e q u e n c e of the t e m p e r a t u r e dependence of the m a g n e t i c m o m e n t s , as d i s c u s s e d by Van Vleck [4] for the ions Sin3+ and Eu 3+. The f e r r i m a g n e t i c p r o p e r t i e s of the compounds f r o m TbN to ErN a r e also s t r o n g l y influenced by the c r y s t a l l i n e field torques. The spontaneous m o m e n t s a r e always l o w e r than the expected gJ v a l u e s (see table 1) and a complete s a t u r a t i o n i s only r e a c h e d in m a g n e t i c f i e l d s higher than 100 kOe. The two compounds CeN and YbN, not d i s c u s s e d t i l l now, a r e m o r e difficult to include in the scheme. The lattice constant for CeN (a = 5.023 A) i s much lower than the expected value for the Ce 3+ ion and c o r r e s p o n d s a p p r o x i m a t e l y with a c o n c e n t r a t i o n of 70% Ce 4+, in agreem e n t with the v a l u e found by Didchenko [6]. The s u s c e p t i b i l i t y of CeN i n c r e a s e s continuously as the t e m p e r a t u r e a p p r o a c h e s zero. No l o n g - r a n g e m a g n e t i c o r d e r could be detected, although we have some i n d i c a t i o n s for a N4el-point at 2.5°K. At high t e m p e r a t u r e s the s u s c e p t i b i l i t y of YbN (4£13) obeys a C u r i e - W e i s s law with a p a -
I PrN
LSO
H'm'~"3.7p'O
I00
:>0_0..-0"~// 50 /.L~f • T,¢ peOp • -18"K
K~
200
~
400
15 February 1965
TpK]
Reciprocal Molar Susceptibility of PrN ond TmN F i g . 1.
t r i d e s with an odd n u m b e r of 4f e l e c t r o n s exhibit a l o n g - r a n g e m a g n e t i c o r d e r at sufficiently low t e m p e r a t u r e s . More s p e c i f i c a l l y , the behaviour of the light n i t r i d e s (from P r N to EuN) c o n f i r m s exactly what we d e s c r i b e d at the beginning. In the case of the heavy n i t r i d e s the situation is more complicated, as T r a m m e l l already dem o n s t r a t e d . The f e r r i m a g n e t i c o r d e r i n g of TbN (4£8) and HoN (4£10) at low t e m p e r a t u r e s i s the r e s u l t of a l a r g e exchange e n e r g y exceeding the c r y s t a l l i n e field splitting. Child et al. proved by m e a n s of n e u t r o n d i f f r a c t i o n that TmN does not o r d e r down to 1.5°K, a c c o r d i n g to T r a m m e l l ' s p r e d i c t i o n . T h i s is in a g r e e m e n t with our m e a s u r e m e n t s of the m a g n e t i c s u s c e p t i b i l i t y . In fig. 1 we show two t y p i c a l c a s e s of compounds with an even n u m b e r of 4£ e l e c t r o n s , i.e. P r N (4f2) and
Table 1 Physical properties of rare earth nitrides. PrN
NdN
Lattice constant [,~]
5.155
Number of 4f electrons (Ln3+)
2
Magnetic order
%ara [OK]
no
0 3.70
~eff (free ion Ln34) Spontaneous moment
as [.B]
3.62
SmN
EuN
GdN
TbN
DyN
HoN
ErN
TmN
5.131
5.049
5.006
4.986
4.936
4.894
4.877
4.836
4.810
3
5
6
7
8
9
yes
yes
no
yes
24 3.654.0
PmN
T~4~e1
69 0-5
F(~r)
F(~
F(7)
3.68
0.84
0
yes
yes
10 yes
U yes
12 no -18
34
20
12
4
8.15
10.0
10.5
10.7
9.4
7.6
7.94
9.7
10.6
10.6
9.6
7.57
5.4
5.2
6.5
5.5
9.9
7.9
2.0
ttl30kOe [~B]
3.6
7.6
9.3
gJ[~Bl
3.27
7
9
9.0 10
10
9 265
Volume 14, number 4
PHYSICS LETTERS
r a m a g n e t i c C u r i e - p o i n t of ,130OK and an effective m o m e n t of 5.1 ~B, but we could not find a N~el-point down to 1.5OK.
References 1. H.R.Child, M.K.Wilkinson, J.W. Cable, W.C. Koehler and E.O.WoUan, Phys.Rev. 131 (1963) 922. 2. G.T.Trammell, Phys.Rev. 131 (1963) 932. 3. G.Busoh, P.Junod, F.Levy and O.Vogt, Proc. Conf. on Magnetism, Nottingham (1964). 4. J.H.Van Vleck, The theory of electric and magnetic susceptibilities (Oxford University Press, New York, 1932}. 5. R.Didohenko and F.P.Gortsema, J. Phys. Chem. Solide 24 (1963) 863.
The a u t h o r s a r e indebted to the S c h w e i z e r i s cher Natlonalfonds zur F 0 r d e r u n g d e r w i s s e n schaftlichen F o r s c h u n g and the V e r e i n z u r FOrd e r u n g d e r F e s t k 0 r p e r p h y s i k an der E i d g e n 6 s s i s chert T e c l m i s c h e n Hochschule for f i n a n c i a l supp o r t of this work. We also thank Mr. U. L a m m l i for p e r f o r m i n g some of the m e a s u r e m e n t s .
DERIVATION
OF
THE
15 February 1965
HOPPINGTHE SMALL
AND BAND-CONDUCTIVITY POLARON
OF
J. SCHNAKENBERG Institut f l i t Theoretische Physik der Universititt zu K~ln
Received 16 January 1965
Lang and F i r s o v [1] and F r i e d m a n [2] have calculated the high t e m p e r a t u r e conductivity of the s m a l l p o l a r o n in the Holstein model [3] by applying the d i a g r a m technique of Konstantinov and P e r e l [4] and the d e n s i t y m a t r i x f o r m u l a t i o n r e s p . It t u r n e d out that the s m a l l p o l a r o n has two additional conductivit i e s aH + crB where ai-I c o r r e s p o n d s to the hopping motion and aB r e s u l t s f r o m the "diagonal 't t r a n s i t i o n s of the p o l a r o n leading to a p o l a r o n band with a t e m p e r a t u r e dependent band width. The p u r p o s e of t h i s note i s to show another way of c a l c u l a t i n g oi-I and crB which avoids the r a t h e r t r o u b l e s o m e c a l c u l a t i o n s p r e s e n t e d t i l l now and r e d u c e s the p r o b l e m to the well known t e c h n i q u e s of Green's functions. The c a n o n i c a l l y t r a n s f o r m e d H a m i l t o n i a n H = H 0 + H 1 and the c u r r e n t o p e r a t o r j of the s m a l l p o l a r o n in the Holstein m o d e l a r e given by ~a+a n nn
H0
H i =-
~
nnT
+ ~ co b + b q , q qq
K n , ~ a+a , B n T_l ~ n n
(1)
j = i e ~ K n,_n Rn ' - n a + a n ' B n ~ n nn ~
where B n ~n = exp
~Xq -Xq)
,
Xq = aq
.
R n d e n o t e s the l a t t i c e points and 7/is the sum of a t e m p e r a t u r e independent s e l f - e n e r g y and the c e n t r e of the e l e c t r o n i c e n e r g y band E(k) the F o u r i e r coefficients of which a r e denoted by K n. The K n a r e v e r y s m a l l in the t i g h t - b i n d i n g case of the s m a l l p o l a r o n and will thus be taken as expansion p a r a m e t e r s in the p e r t u r b a t i o n t h e o r e t i c a l t r e a t m e n t i n s t e a d of the d i m e n s i o n l e s s e l e c t r o n phonon coupling c o n s t a n t aq We shall r e s t r i c t o u r s e l v e s to the diagonal e l e m e n t s of the conductivity t e n s o r which a r e given by the Kubo f o r m u l a
266
PHYSICS LETTERS
studied the influence of s t r o n g cubic f i e l d s in o r d e r to u n d e r s t a n d s a t u r a t i o n m o m e n t s m e a s u r e d in the LnNi 2 compounds by A b r a h a m s et al. [6]. F o r a c o m p a r i s o n (the two s t r u c t u r e s a r e not identical) we give t h e i r v a l u e s in t a b l e 1 together with the m a g n e t i z a t i o n s d e t e r m i n e d for a field of 130 kOe. Child et al. [2] found i n some a n t i m o n i d e s o r d e r e d m o m e n t s well below the gJ v a l u e s for the f r e e ions and T r a m m e l l [7] gave a t h e o r e t i c a l a n a l y s i s of this fact. C h i l d ' s [2] v a l u e s a g r e e in the case of HoSb and ErSb r e m a r k ably well (within a few percent} with m a g n e t o n n u m b e r s we obtain by e x t r a p o l a t i n g m e a s u r e d m a g n e t i z a t i o n s to infinite m a g n e t i c f i e l d s . By m e a n s of the m o l e c u l a r field a p p r o x i m a tion it is p o s s i b l e to c a l c u l a t e the W e i s s - f i e l d . The r e d u c e d o r d e r e d m o m e n t s a r e t a k e n f r o m the e x p e r i m e n t a l v a l u e s of the m a g n e t i z a t i o n s . We find that the calculated W e i s s - f i e l d a g r e e s within 25% with the c r i t i c a l field s t r e n g t h Hc. We have s u m m a r i z e d our r e s u l t s and c a l c u l a t i o n s in table 1, the n o t a t i o n s being explained in the text.
INFLUENCE
OF
CRYSTAL OF THE
15 February 1965
The a u t h o r s a r e indebted to the S c h w e i z e r i s cher National fonds z u r FOrderung d e r w i s s e n schaftlichen F o r s c h u n g and to the V e r e i n zur F ~ r d e r u n g der F e s t k ~ r p e r p h y s i k an der ETH for f i n a n c i a l support. We wish to thank P r o f . Grtin e n f e l d e r f r o m the M i n e r a l o g i c a l Institute of the ETH for p e r f o r m i n g the m i c r o p r o b e a n a l y s i s and Dr. Hulliger f r o m Cyanamid European R e s e a r c h Institute for supplying the GdSb c r y s t a l s .
References 1. G.Busch, P.Junod, O.Vogt and F.Hulliger, Physics Letters 6 (1963) 79. 2. H.R.Child, M.K.Wilkinson, J.W. Cable, W.C.Koehler and E.O.Wollan, Phys.Rev. 131 (1963) 922. 3. G.Buseh, P.Junod, P.Schwob, O.Vogt and F.Hulliger, Physics Letters 9 (1964) 7. 4. E.g.S. Smart in "Magnetism HI" by G. T. Rado and H.Suhl (Academic Press, New York, 1963). 5. B.Bleany, Proc.Roy.Soc. 276 (1963) 28. 6. S. C. Abrahams, J.L. Bernstein, R.C.Sherwood, J. H. Wernick and H. J. Williams, J. Phys. Chem. Solids 25 (1964) 1069. 7. G.T.Trammell, Phys.Rev. 131 (1963) 932.
FIELDS ON RARE-EARTH
THE MAGNETIC NITRIDES
PROPERTIES
G. BUSCH, P . JUNOD, F. LEVY, A. MENTH and O. VOGT
Laboratorium flir FestkSrperphysik, ETH, Ziirich, Switzerland Received 16 January 1965
We i n v e s t i g a t e d the whole s e r i e s of the r a r e e a r t h n i t r i d e s , f r o m CeN to YbN which a l l c r y s t a l l i z e in the s i m p l e NaC1 s t r u c t u r e . Our m e a s u r e m e n t s of the m a g n e t i z a t i o n up to 130 kOe and of the m a g n e t i c s u s c e p t i b i l i t i e s between 1.5°K and 600°K allow a unique i n t e r p r e t a t i o n and can be c o m p a r e d with other works. We would like to m e n t i o n e s p e c i a l l y the n e u t r o n diffraction exp e r i m e n t s on the heavy r a r e - e a r t h n i t r i d e s by Child et al. [1] and a t h e o r e t i c a l d i s c u s s i o n by T r a m m e l l [2]. The r e s u l t s of the l a t t e r can be s u m m a r i z e d in the following way: In the case where the exchange f o r c e s a r e s m a l l , c o m p a r e d to the c r y s t a l l i n e field s p l i t t i n g s , the p r e s e n c e or a b s e n c e of an o r d e r e d m a g n e t i c g r o u n d - s t a t e will depend only on the n u m b e r of 4I e l e c t r o n s , a s long as the c r y s t a l line field splittings a r e l a r g e r than kT. Under 264
this a s s u m p t i o n r a r e - e a r t h n i t r i d e s with an odd n u m b e r of 4/ e l e c t r o n s will be m a g n e t i c a l l y o r dered. If the 4£ shell c o n t a i n s an even n u m b e r of e l e c t r o n s , the s t a t i o n a r y state in the cubic c r y s t a l field m a y or m a y not be o r d e r e d by the exchange f o r c e s ; this r e s u l t i s a d i r e c t c o n s e quence of K r a m e r ' s t h e o r e m . We have p r e p a r e d s t o i c h i o m e t r i c s p e c i m e n s of eleven n i t r i d e s by d i r e c t r e a c t i o n of the comp o n e n t s [3]. The e x p e r i m e n t a l r e s u l t s for the whole s e r i e s of compounds f r o m P r N to TmN a r e s u m m a r i z e d in table 1. The two n i t r i d e s CeN and YbN will be d i s c u s s e d l a t e r . In the t h i r d row of the table we indicate by " y e s " or "no" the p r e s e n c e or a b s e n c e of a l o n g - r a n g e m a g n e t i c o r d e r . It i s r e m a r k a b l e that a l l the compounds which do not o r d e r at low t e m p e r a t u r e s have an even n u m b e r of 4f e l e c t r o n s and that all the n i -
Volume 14, number 4
PHYSICS LETTERS
,/x.
•
T m N (4£12). At low t e m p e r a t u r e s the C u r i e W e i s s law does not hold any longer and the s u s c e p t i b i l i t i e s tend to a t e m p e r a t u r e independent value. In the high t e m p e r a t u r e r e g i o n , where the c r y s t a l l i n e field s p l i t t i n g s b e c o m e s m a l l e r than kT, the n i t r i d e s of P r , Gd, Tb, Dy, Ho, Er and T m obey the C u r i e - W e i s s law. The e x p e r i m e n tal m a g n e t o n n u m b e r s a g r e e with the t h e o r e t i c a l v a l u e s for the t r i v a l e n t ions. The other c o m pounds NdN, StuN and EuN do not follow the C u r i e - W e i s s law a s a c o n s e q u e n c e of the t e m p e r a t u r e dependence of the m a g n e t i c m o m e n t s , as d i s c u s s e d by Van Vleck [4] for the ions Sin3+ and Eu 3+. The f e r r i m a g n e t i c p r o p e r t i e s of the compounds f r o m TbN to ErN a r e also s t r o n g l y influenced by the c r y s t a l l i n e field torques. The spontaneous m o m e n t s a r e always l o w e r than the expected gJ v a l u e s (see table 1) and a complete s a t u r a t i o n i s only r e a c h e d in m a g n e t i c f i e l d s higher than 100 kOe. The two compounds CeN and YbN, not d i s c u s s e d t i l l now, a r e m o r e difficult to include in the scheme. The lattice constant for CeN (a = 5.023 A) i s much lower than the expected value for the Ce 3+ ion and c o r r e s p o n d s a p p r o x i m a t e l y with a c o n c e n t r a t i o n of 70% Ce 4+, in agreem e n t with the v a l u e found by Didchenko [6]. The s u s c e p t i b i l i t y of CeN i n c r e a s e s continuously as the t e m p e r a t u r e a p p r o a c h e s zero. No l o n g - r a n g e m a g n e t i c o r d e r could be detected, although we have some i n d i c a t i o n s for a N4el-point at 2.5°K. At high t e m p e r a t u r e s the s u s c e p t i b i l i t y of YbN (4£13) obeys a C u r i e - W e i s s law with a p a -
I PrN
LSO
H'm'~"3.7p'O
I00
:>0_0..-0"~// 50 /.L~f • T,¢ peOp • -18"K
K~
200
~
400
15 February 1965
TpK]
Reciprocal Molar Susceptibility of PrN ond TmN F i g . 1.
t r i d e s with an odd n u m b e r of 4f e l e c t r o n s exhibit a l o n g - r a n g e m a g n e t i c o r d e r at sufficiently low t e m p e r a t u r e s . More s p e c i f i c a l l y , the behaviour of the light n i t r i d e s (from P r N to EuN) c o n f i r m s exactly what we d e s c r i b e d at the beginning. In the case of the heavy n i t r i d e s the situation is more complicated, as T r a m m e l l already dem o n s t r a t e d . The f e r r i m a g n e t i c o r d e r i n g of TbN (4£8) and HoN (4£10) at low t e m p e r a t u r e s i s the r e s u l t of a l a r g e exchange e n e r g y exceeding the c r y s t a l l i n e field splitting. Child et al. proved by m e a n s of n e u t r o n d i f f r a c t i o n that TmN does not o r d e r down to 1.5°K, a c c o r d i n g to T r a m m e l l ' s p r e d i c t i o n . T h i s is in a g r e e m e n t with our m e a s u r e m e n t s of the m a g n e t i c s u s c e p t i b i l i t y . In fig. 1 we show two t y p i c a l c a s e s of compounds with an even n u m b e r of 4£ e l e c t r o n s , i.e. P r N (4f2) and
Table 1 Physical properties of rare earth nitrides. PrN
NdN
Lattice constant [,~]
5.155
Number of 4f electrons (Ln3+)
2
Magnetic order
%ara [OK]
no
0 3.70
~eff (free ion Ln34) Spontaneous moment
as [.B]
3.62
SmN
EuN
GdN
TbN
DyN
HoN
ErN
TmN
5.131
5.049
5.006
4.986
4.936
4.894
4.877
4.836
4.810
3
5
6
7
8
9
yes
yes
no
yes
24 3.654.0
PmN
T~4~e1
69 0-5
F(~r)
F(~
F(7)
3.68
0.84
0
yes
yes
10 yes
U yes
12 no -18
34
20
12
4
8.15
10.0
10.5
10.7
9.4
7.6
7.94
9.7
10.6
10.6
9.6
7.57
5.4
5.2
6.5
5.5
9.9
7.9
2.0
ttl30kOe [~B]
3.6
7.6
9.3
gJ[~Bl
3.27
7
9
9.0 10
10
9 265
Volume 14, number 4
PHYSICS LETTERS
r a m a g n e t i c C u r i e - p o i n t of ,130OK and an effective m o m e n t of 5.1 ~B, but we could not find a N~el-point down to 1.5OK.
References 1. H.R.Child, M.K.Wilkinson, J.W. Cable, W.C. Koehler and E.O.WoUan, Phys.Rev. 131 (1963) 922. 2. G.T.Trammell, Phys.Rev. 131 (1963) 932. 3. G.Busoh, P.Junod, F.Levy and O.Vogt, Proc. Conf. on Magnetism, Nottingham (1964). 4. J.H.Van Vleck, The theory of electric and magnetic susceptibilities (Oxford University Press, New York, 1932}. 5. R.Didohenko and F.P.Gortsema, J. Phys. Chem. Solide 24 (1963) 863.
The a u t h o r s a r e indebted to the S c h w e i z e r i s cher Natlonalfonds zur F 0 r d e r u n g d e r w i s s e n schaftlichen F o r s c h u n g and the V e r e i n z u r FOrd e r u n g d e r F e s t k 0 r p e r p h y s i k an der E i d g e n 6 s s i s chert T e c l m i s c h e n Hochschule for f i n a n c i a l supp o r t of this work. We also thank Mr. U. L a m m l i for p e r f o r m i n g some of the m e a s u r e m e n t s .
DERIVATION
OF
THE
15 February 1965
HOPPINGTHE SMALL
AND BAND-CONDUCTIVITY POLARON
OF
J. SCHNAKENBERG Institut f l i t Theoretische Physik der Universititt zu K~ln
Received 16 January 1965
Lang and F i r s o v [1] and F r i e d m a n [2] have calculated the high t e m p e r a t u r e conductivity of the s m a l l p o l a r o n in the Holstein model [3] by applying the d i a g r a m technique of Konstantinov and P e r e l [4] and the d e n s i t y m a t r i x f o r m u l a t i o n r e s p . It t u r n e d out that the s m a l l p o l a r o n has two additional conductivit i e s aH + crB where ai-I c o r r e s p o n d s to the hopping motion and aB r e s u l t s f r o m the "diagonal 't t r a n s i t i o n s of the p o l a r o n leading to a p o l a r o n band with a t e m p e r a t u r e dependent band width. The p u r p o s e of t h i s note i s to show another way of c a l c u l a t i n g oi-I and crB which avoids the r a t h e r t r o u b l e s o m e c a l c u l a t i o n s p r e s e n t e d t i l l now and r e d u c e s the p r o b l e m to the well known t e c h n i q u e s of Green's functions. The c a n o n i c a l l y t r a n s f o r m e d H a m i l t o n i a n H = H 0 + H 1 and the c u r r e n t o p e r a t o r j of the s m a l l p o l a r o n in the Holstein m o d e l a r e given by ~a+a n nn
H0
H i =-
~
nnT
+ ~ co b + b q , q qq
K n , ~ a+a , B n T_l ~ n n
(1)
j = i e ~ K n,_n Rn ' - n a + a n ' B n ~ n nn ~
where B n ~n = exp
~Xq -Xq)
,
Xq = aq
.
R n d e n o t e s the l a t t i c e points and 7/is the sum of a t e m p e r a t u r e independent s e l f - e n e r g y and the c e n t r e of the e l e c t r o n i c e n e r g y band E(k) the F o u r i e r coefficients of which a r e denoted by K n. The K n a r e v e r y s m a l l in the t i g h t - b i n d i n g case of the s m a l l p o l a r o n and will thus be taken as expansion p a r a m e t e r s in the p e r t u r b a t i o n t h e o r e t i c a l t r e a t m e n t i n s t e a d of the d i m e n s i o n l e s s e l e c t r o n phonon coupling c o n s t a n t aq We shall r e s t r i c t o u r s e l v e s to the diagonal e l e m e n t s of the conductivity t e n s o r which a r e given by the Kubo f o r m u l a
266