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Critical behaviour of the resistivity of a dilute Pd-Mn alloy
Volume33A, number 5
CRITICAL
PHYSICS LETTERS
BEHAVIOUR
OF
THE
RESISTIVITY
16 November 1970
OF
A DILUTE
ALLOY*
Pd-Mn
G. J. NIEUWENHUYS and B. M. B O E R S T O E L Karnerlingh Onnes Laboratory, Leiden, The Netherlands Received 12 October 1970
Measurements of the resistivity, p, of Pd-Mn 1.0 at. % are presented. The resistivity is shown to fit a phenomenological model using a Gaussian distribution of transition temperatures.
F i s h e r and L a n g e r [1] have p r e d i c t e d the c r i t i c a l b e h a v i o u r of d p / d T in m a g n e t i c a l l y o r d e n i n g s y s t e m s to be, to a c e r t a i n extent, the s a m e as for the s p e c i f i c heat. C o n s i d e r i n g the b e h a v i o u r of the s p e c i f i c heat of dilute P d - C o , P d - F e and, p a r t i c u l a r l y , P d - M n a l l o y s , r e c e n t ly r e p o r t e d by B o e r s t o e l et al. [2-4], it s e e m e d worthwhile to i n v e s t i g a t e the r e s i s t i v i t y of t h e s e a l l o y s , e s p e c i a l l y at t e m p e r a t u r e s c l o s e to the c r i t i c a l t e m p e r a t u r e , T c. P r e v i o u s l y , the r e s i s t i v i t y of t h e s e a l l o y s has been r e p o r t e d by many w o r k e r s [5-10], but no detailed work has been c a r r i e d out in the v i c i n i t y of Tc, except on a n u m b e r of P d - F e a l l o y s [9,10]. B e c a u s e of the r a t h e r p e c u l i a r b e h a v i o u r of the s p e c i f i c heat of dilute P d - M n a l l o y s [3,4] it was decided to f i r s t i n v e s t i g a t e a P d - M n s p e c i m e n containing 1.0 at.% Mn, of which Tc is a p p r o x i m a t e l y 3.4 K. The s a m p l e was p r e p a r e d by induction heating f r o m J o h n s o n - M a t t h e y 99.999% p u r e Pd and J o h n s o n - M a t t h e y 99.99% pure Mn. The s a m p l e has b e e n h o m o g e n i z e d f o r ten days at 1000oc; a f t e r drawing, the w i r e has been annealed f o r s ix h o u r s at 500oc. Subsequent to the heat t r e a t m e n t the s a m p l e and the w i r e w e r e quenched in w a t e r . The Mn c o n c e n t r a t i o n , obtained f r o m a c h e m i c a l a n a l y s i s , was found to be 0.95 at.%. The r e s i s t i v i t y m e a s u r e m e n t s w e r e c a r r i e d out by m e a n s of a conventional f o u r - p r o b e technique. The t e m p e r a t u r e was r e c o r d e d and c o n t r o l l e d using a c a r b o n t h e r m o m e t e r . In the t e m p e r a t u r e r a n g e f r o m 1.3 K to 4.2 K the r e s i s t i v i t y was m e a s u r e d at 0.1 K i n t e r v a l s ; in the v i c i n i t y of Tc (i. e. between 3.26 K and 3.56 K) data w e r e obtained at 1.2 mK intervals. During the l a t t e r m e a s u r e m e n t s the t e m p e r a t u r e was s t a b i l i z e d to within 0.02 inK. * This work is part of the research program of F.O.M., supported by Z.W.O. and T.N.O.
The i n c r e m e n t a l r e s i s t i v i t y , A p ( T ) = = Palloy(T) - PPd(T), is shown in fig. 1; data o b tained at t e m p e r a t u r e s c l o s e to T c a r e shown in detail by the c u r v e in the l o w e r i n s e r t ( cl o sed c i r c l e s in this i n s e r t a r e r e s i s t i v i t y v a l u e s obtained f r o m a c a l c u l a t i o n p r e s e n t e d below). F o r p u r e Pd u se has been m a d e of data obtained by Star [11], which can be r e p r e s e n t e d by ppd(T) = =0.0686 + 2 . 6 8 × 10 -5 T 2 ~ 2 c m . At T < 1.6 K Ap s a t i s f i e s the r e l a t i o n Ap = a + b T 3/2, with a = 1.647 #~2cm and b=0.01506 #~2 c m K -3/2, in a c c o r d a n c e with r e s u l t s of W i l l i a m s and L o r a m [6]. At h i g h e r t e m p e r a t u r e s , i.e. above as well as below Tc, A p v a r i e s p r o p o r t i o n a l l y to T ex cep t ,
..~=
,.7,
'
K
..
,.73 ,.7~
. . . . . . . .
o~2
i1 . i
,
li
3.3
3-
3,4
Pd-Mn
:
[
!
i
1o.o, /
,
i
,
3.5
K
o
10t.9~
,7, ~3
,.7o
,
T "
3.4 I
3.5 '
I
"
-
t74 ~cm
u59
t.738
,~7 i
,
23,~
+
;
;
i
Fig. 1. The incremental resistivity of Pd-Mn 1.0 at.% versus T. The meaning of the inserts is explained in the text. 281
Volume 33A, n u m b e r 5
PHYSICS
h o w e v e r , a t t e m p e r a t u r e s v e r y c l o s e to Tc, where some rounding has been observed. In o u r o p i n i o n t h i s r o u n d i n g a r i s e s p r i m a r i l y f r o m l o c a l c o n c e n t r a t i o n v a r i a t i o n s in t h e s p e c i m e n . B e c a u s e of t h i s , we a s s u m e t h a t Ap of a ' p e r f e c t l y h o m o g e n e o u s ' a l l o y v a r i e s a s Ap(T, Tc) = Ap(Tc) + 5p(T, Tc), w h e r e 5p( T, T c) = = h l ( T - T c) a t T--< T c a n d S p ( T , T c ) =k2(T-Tc) a t T >i Tc, h 1 = 0 . 0 3 4 3 g ~ c m K -1 a n d h 2 = = 0.0035 g~cmK -1, being the temperature coefficients actually obtained from the measurem e n t s at t e m p e r a t u r e s n o t t o o c l o s e to T c. O n l y a t T < T c s u c h a b e h a v i o u r of A p ( T , T c ) i s p r e d i c t e d b y t h e o r y [12,13]. In o r d e r to a c c o u n t f o r t h e o b s e r v e d r o u n d i n g of t h e e x p e r i m e n t a l c u r v e we a s s u m e t h a t t h e v a r i a t i o n s of t h e c o n c e n t r a t i o n c a n b e r e p r e s e n t e d b y a d i s t r i b u t i o n of t r a n s i t i o n temperatures, t c , w h i c h i s t a k e n to b e G a u s s i a n . H e n c e , 5 p ( T , , T c) c a n b e r e p r e s e n t e d , at temperat u r e s b e t w e e n 2.5 K a n d 4.2 K, b y 5p(
T, Tc,F ) :
16 November 1970
t h e p o i n t of i n f l e c t i o n of t h e A p v e r s u s T c u r v e , w h i c h i s e q u a l to t h a t of t h e m a x i m u m of dAp/dT, i s l o w e r t h a n Tc. T h i s b e h a v i o u r of t h e m a x i m u m of d A p / d T i s a n a l o g o u s t o t h a t of t h e s p e c i f i c h e a t m a x i m u m [4]. S e v e r a l w o r k e r s [9,10] h a v e a s s u m e d Tc to b e e q u a l to t h e t e m p e r a t u r e of t h e m a x i m u m of t h e d A p / d T v e r s u s T c u r v e . In v i e w of t h e r e s u l t s of t h e p r e s e n t a n a l y s i s s u c h a n a s s u m p t i o n a p p e a r s not to b e j u s t i f i e d . P r e s u m a b l y , this fact accounts for the rather peculiar beh a v i o u r of d 2 p / d T 2 o b t a i n e d b y K a w a t r a et al. [9] f o r P d - F e a l l o y s a n d b y C r a i g et a l . [14] f o r Ni a n d t h e b e h a v i o u r of d p / d T o b t a i n e d b y L o n g w o r t h a n d T s u e i [10] f o r P d - F e a l l o y s . T h e a u t h o r s w i s h to t h a n k M i s s M. F. P i k a r t a n d M r . H. D. D o k t e r f o r t h e i r a s s i s t a n c e d u r i n g the experiments.
References
(1)
oO
C -1 f 5 p ( T , t c) e x p - { ( / c - T c ) 2 / 2 ( F T c ) 2} d / c , 0 where C is the normalization constant. A b e s t fit to t h e e x p e r i m e n t a l c u r v e w a s o b tained using F = 0.022. Resistivity values based upon this fit are shown by closed circles in the l o w e r i n s e r t of fig. 1. T h e m e a n t r a n s i t i o n t e m p e r a t u r e d e r i v e d f r o m t h e fit i s 3.414 K a n d A p ( T c) - Ap(0) = 0 . 0 9 1 9 ~ c m . Another indication, which supports our model has been obtained from d~o/dT. It i s e a s y t o d e r i v e f r o m eq. (1) t h a t (dAp/dT)T=T~ = ( h 1 + h 2 ) / 2 . V a l u e s of ~AP/dT d e r i v e d f r o m t h e e x p e r i m e n t s a r e s h o w n i n t h e u p p e r i n s e r t of fig. 1. A s c a n b e s e e n t h e e x p e r i m e n t a l v a l u e of d~p/dT a t Tc i s i n f a i r a g r e e m e n t w i t h t h e c a l c u l a t e d o n e , which is indicated by dash-dotted lines. A s i s e v i d e n t f r o m fig. 1 t h e t e m p e r a t u r e of
282
LETTERS
[1] M. E. F i s h e r and J. S. Langer, Phys. Rev. L e t t e r s 20 (1968) 665. [2] B. M. Boerstoel, G.J. Nieuwenhuys and G. J. Van den Berg, Phys. L e t t e r s 29A (1969) 526. [3] B. M. Boerstoel, J . J . Zwart and C. van Baarle, Phys. L e t t e r s 31A (1970) 378. [4] B. M. Boerstoel and R. F. Wielinga, Phys. L e t t e r s 31A (1970) 359. [5] M. P. Sarachik and D. Shaltiel, J. Appl. Phys. 38 (1967) 1155. [6] G. Williams and J. W. Loram, Solid State Comm. 7 (1969) 1261. [7] G. Williams and J. W. Loram, J. Phys. Chem. Solids 30 (1969) 1827. [8] G. Williams, J. Phys. Chem. Solids 31 (1970) 529. [9] M. P. Kawatra, S. Skalski, J . A . Mydosh and J. I. Budnick, J. Appl. Phys. 40 (1969) 1202. [10] G. Longworth and C. C. Tsue~, Phys. L e t t e r s 27A (1968) 258. [11] W. M. Star, private communication. [12] P. D. Long and R. E. Turner, p r e p r i n t (1970). [13] A. Theumann, Phys. Rev. B1 (1970) 4400. [14] P. P. Craig, W.I. Goldburg, T.A. Kitchens and J. I. Budnick, Phys. Rev. L e t t e r s 19 (1967) 1334.
CRITICAL
PHYSICS LETTERS
BEHAVIOUR
OF
THE
RESISTIVITY
16 November 1970
OF
A DILUTE
ALLOY*
Pd-Mn
G. J. NIEUWENHUYS and B. M. B O E R S T O E L Karnerlingh Onnes Laboratory, Leiden, The Netherlands Received 12 October 1970
Measurements of the resistivity, p, of Pd-Mn 1.0 at. % are presented. The resistivity is shown to fit a phenomenological model using a Gaussian distribution of transition temperatures.
F i s h e r and L a n g e r [1] have p r e d i c t e d the c r i t i c a l b e h a v i o u r of d p / d T in m a g n e t i c a l l y o r d e n i n g s y s t e m s to be, to a c e r t a i n extent, the s a m e as for the s p e c i f i c heat. C o n s i d e r i n g the b e h a v i o u r of the s p e c i f i c heat of dilute P d - C o , P d - F e and, p a r t i c u l a r l y , P d - M n a l l o y s , r e c e n t ly r e p o r t e d by B o e r s t o e l et al. [2-4], it s e e m e d worthwhile to i n v e s t i g a t e the r e s i s t i v i t y of t h e s e a l l o y s , e s p e c i a l l y at t e m p e r a t u r e s c l o s e to the c r i t i c a l t e m p e r a t u r e , T c. P r e v i o u s l y , the r e s i s t i v i t y of t h e s e a l l o y s has been r e p o r t e d by many w o r k e r s [5-10], but no detailed work has been c a r r i e d out in the v i c i n i t y of Tc, except on a n u m b e r of P d - F e a l l o y s [9,10]. B e c a u s e of the r a t h e r p e c u l i a r b e h a v i o u r of the s p e c i f i c heat of dilute P d - M n a l l o y s [3,4] it was decided to f i r s t i n v e s t i g a t e a P d - M n s p e c i m e n containing 1.0 at.% Mn, of which Tc is a p p r o x i m a t e l y 3.4 K. The s a m p l e was p r e p a r e d by induction heating f r o m J o h n s o n - M a t t h e y 99.999% p u r e Pd and J o h n s o n - M a t t h e y 99.99% pure Mn. The s a m p l e has b e e n h o m o g e n i z e d f o r ten days at 1000oc; a f t e r drawing, the w i r e has been annealed f o r s ix h o u r s at 500oc. Subsequent to the heat t r e a t m e n t the s a m p l e and the w i r e w e r e quenched in w a t e r . The Mn c o n c e n t r a t i o n , obtained f r o m a c h e m i c a l a n a l y s i s , was found to be 0.95 at.%. The r e s i s t i v i t y m e a s u r e m e n t s w e r e c a r r i e d out by m e a n s of a conventional f o u r - p r o b e technique. The t e m p e r a t u r e was r e c o r d e d and c o n t r o l l e d using a c a r b o n t h e r m o m e t e r . In the t e m p e r a t u r e r a n g e f r o m 1.3 K to 4.2 K the r e s i s t i v i t y was m e a s u r e d at 0.1 K i n t e r v a l s ; in the v i c i n i t y of Tc (i. e. between 3.26 K and 3.56 K) data w e r e obtained at 1.2 mK intervals. During the l a t t e r m e a s u r e m e n t s the t e m p e r a t u r e was s t a b i l i z e d to within 0.02 inK. * This work is part of the research program of F.O.M., supported by Z.W.O. and T.N.O.
The i n c r e m e n t a l r e s i s t i v i t y , A p ( T ) = = Palloy(T) - PPd(T), is shown in fig. 1; data o b tained at t e m p e r a t u r e s c l o s e to T c a r e shown in detail by the c u r v e in the l o w e r i n s e r t ( cl o sed c i r c l e s in this i n s e r t a r e r e s i s t i v i t y v a l u e s obtained f r o m a c a l c u l a t i o n p r e s e n t e d below). F o r p u r e Pd u se has been m a d e of data obtained by Star [11], which can be r e p r e s e n t e d by ppd(T) = =0.0686 + 2 . 6 8 × 10 -5 T 2 ~ 2 c m . At T < 1.6 K Ap s a t i s f i e s the r e l a t i o n Ap = a + b T 3/2, with a = 1.647 #~2cm and b=0.01506 #~2 c m K -3/2, in a c c o r d a n c e with r e s u l t s of W i l l i a m s and L o r a m [6]. At h i g h e r t e m p e r a t u r e s , i.e. above as well as below Tc, A p v a r i e s p r o p o r t i o n a l l y to T ex cep t ,
..~=
,.7,
'
K
..
,.73 ,.7~
. . . . . . . .
o~2
i1 . i
,
li
3.3
3-
3,4
Pd-Mn
:
[
!
i
1o.o, /
,
i
,
3.5
K
o
10t.9~
,7, ~3
,.7o
,
T "
3.4 I
3.5 '
I
"
-
t74 ~cm
u59
t.738
,~7 i
,
23,~
+
;
;
i
Fig. 1. The incremental resistivity of Pd-Mn 1.0 at.% versus T. The meaning of the inserts is explained in the text. 281
Volume 33A, n u m b e r 5
PHYSICS
h o w e v e r , a t t e m p e r a t u r e s v e r y c l o s e to Tc, where some rounding has been observed. In o u r o p i n i o n t h i s r o u n d i n g a r i s e s p r i m a r i l y f r o m l o c a l c o n c e n t r a t i o n v a r i a t i o n s in t h e s p e c i m e n . B e c a u s e of t h i s , we a s s u m e t h a t Ap of a ' p e r f e c t l y h o m o g e n e o u s ' a l l o y v a r i e s a s Ap(T, Tc) = Ap(Tc) + 5p(T, Tc), w h e r e 5p( T, T c) = = h l ( T - T c) a t T--< T c a n d S p ( T , T c ) =k2(T-Tc) a t T >i Tc, h 1 = 0 . 0 3 4 3 g ~ c m K -1 a n d h 2 = = 0.0035 g~cmK -1, being the temperature coefficients actually obtained from the measurem e n t s at t e m p e r a t u r e s n o t t o o c l o s e to T c. O n l y a t T < T c s u c h a b e h a v i o u r of A p ( T , T c ) i s p r e d i c t e d b y t h e o r y [12,13]. In o r d e r to a c c o u n t f o r t h e o b s e r v e d r o u n d i n g of t h e e x p e r i m e n t a l c u r v e we a s s u m e t h a t t h e v a r i a t i o n s of t h e c o n c e n t r a t i o n c a n b e r e p r e s e n t e d b y a d i s t r i b u t i o n of t r a n s i t i o n temperatures, t c , w h i c h i s t a k e n to b e G a u s s i a n . H e n c e , 5 p ( T , , T c) c a n b e r e p r e s e n t e d , at temperat u r e s b e t w e e n 2.5 K a n d 4.2 K, b y 5p(
T, Tc,F ) :
16 November 1970
t h e p o i n t of i n f l e c t i o n of t h e A p v e r s u s T c u r v e , w h i c h i s e q u a l to t h a t of t h e m a x i m u m of dAp/dT, i s l o w e r t h a n Tc. T h i s b e h a v i o u r of t h e m a x i m u m of d A p / d T i s a n a l o g o u s t o t h a t of t h e s p e c i f i c h e a t m a x i m u m [4]. S e v e r a l w o r k e r s [9,10] h a v e a s s u m e d Tc to b e e q u a l to t h e t e m p e r a t u r e of t h e m a x i m u m of t h e d A p / d T v e r s u s T c u r v e . In v i e w of t h e r e s u l t s of t h e p r e s e n t a n a l y s i s s u c h a n a s s u m p t i o n a p p e a r s not to b e j u s t i f i e d . P r e s u m a b l y , this fact accounts for the rather peculiar beh a v i o u r of d 2 p / d T 2 o b t a i n e d b y K a w a t r a et al. [9] f o r P d - F e a l l o y s a n d b y C r a i g et a l . [14] f o r Ni a n d t h e b e h a v i o u r of d p / d T o b t a i n e d b y L o n g w o r t h a n d T s u e i [10] f o r P d - F e a l l o y s . T h e a u t h o r s w i s h to t h a n k M i s s M. F. P i k a r t a n d M r . H. D. D o k t e r f o r t h e i r a s s i s t a n c e d u r i n g the experiments.
References
(1)
oO
C -1 f 5 p ( T , t c) e x p - { ( / c - T c ) 2 / 2 ( F T c ) 2} d / c , 0 where C is the normalization constant. A b e s t fit to t h e e x p e r i m e n t a l c u r v e w a s o b tained using F = 0.022. Resistivity values based upon this fit are shown by closed circles in the l o w e r i n s e r t of fig. 1. T h e m e a n t r a n s i t i o n t e m p e r a t u r e d e r i v e d f r o m t h e fit i s 3.414 K a n d A p ( T c) - Ap(0) = 0 . 0 9 1 9 ~ c m . Another indication, which supports our model has been obtained from d~o/dT. It i s e a s y t o d e r i v e f r o m eq. (1) t h a t (dAp/dT)T=T~ = ( h 1 + h 2 ) / 2 . V a l u e s of ~AP/dT d e r i v e d f r o m t h e e x p e r i m e n t s a r e s h o w n i n t h e u p p e r i n s e r t of fig. 1. A s c a n b e s e e n t h e e x p e r i m e n t a l v a l u e of d~p/dT a t Tc i s i n f a i r a g r e e m e n t w i t h t h e c a l c u l a t e d o n e , which is indicated by dash-dotted lines. A s i s e v i d e n t f r o m fig. 1 t h e t e m p e r a t u r e of
282
LETTERS
[1] M. E. F i s h e r and J. S. Langer, Phys. Rev. L e t t e r s 20 (1968) 665. [2] B. M. Boerstoel, G.J. Nieuwenhuys and G. J. Van den Berg, Phys. L e t t e r s 29A (1969) 526. [3] B. M. Boerstoel, J . J . Zwart and C. van Baarle, Phys. L e t t e r s 31A (1970) 378. [4] B. M. Boerstoel and R. F. Wielinga, Phys. L e t t e r s 31A (1970) 359. [5] M. P. Sarachik and D. Shaltiel, J. Appl. Phys. 38 (1967) 1155. [6] G. Williams and J. W. Loram, Solid State Comm. 7 (1969) 1261. [7] G. Williams and J. W. Loram, J. Phys. Chem. Solids 30 (1969) 1827. [8] G. Williams, J. Phys. Chem. Solids 31 (1970) 529. [9] M. P. Kawatra, S. Skalski, J . A . Mydosh and J. I. Budnick, J. Appl. Phys. 40 (1969) 1202. [10] G. Longworth and C. C. Tsue~, Phys. L e t t e r s 27A (1968) 258. [11] W. M. Star, private communication. [12] P. D. Long and R. E. Turner, p r e p r i n t (1970). [13] A. Theumann, Phys. Rev. B1 (1970) 4400. [14] P. P. Craig, W.I. Goldburg, T.A. Kitchens and J. I. Budnick, Phys. Rev. L e t t e r s 19 (1967) 1334.