ELECTRICAL HIGH
RESISTIVITY
OF
Co-
Ga ALLOYS AT
TEMPERATURES B . S. V o r o n t s o v , a n d P . V. G e l ' d
S.
P.
Dovgopol,
UDC 669.25'871:537.311.3
The " r o t a t i n g magnetic field" method was used to investigate the t e m p e r a t u r e dependences of the e l e c t r i c a l r e s i s t i v i t y of C o - Ga alloys in the t e m p e r a t u r e range 800-1750~ The e x p e r i m e n t a l data obtained f o r the liquid phase a r e c o m p a r e d with the r e s u l t s of a calculation made in the f r a m e w o r k of the F a b e r - Ziman method as g e n e r a l i z e d by Evans to alloys of transition metals. The e l e c t r i c a l r e s i s t i v i t y of m e t a l l i c alloys is d e t e r m i n e d by t h e i r atomic and e l e c t r o n i c s t r u c t u r e . P a r t i c u l a r i n t e r e s t is attached to alloys b a s e d on 3d t r a n s i t i o n m e t a l s in which the decisive s c a t t e r i n g m e c h a n i s m is resonant s c a t t e r i n g on 3d levels. T h e r e f o r e , in the i n t e r p r e t a t i o n of the concentration and t e m p e r a t u r e dependences of t h e i r r e s i s t i v i t y , one u s e s the p a r a m e t e r s of the 3d band in the m o s t d i r e c t f o r m . In the p r e s e n t p a p e r we give the r e s u l t s of the m e a s u r e m e n t of the t e m p e r a t u r e dependences of the r e s i s t i v i t y p of C o - Ga alloys between 800 and 17500C (i. e., at t ~ t m) and we c o m p a r e t h e m with calculated data obtained in the f r a m e w o r k of the method of [1, 2] and using the model of the electronic s t r u c t u r e of the C o - G a alloys p r o p o s e d in [3]. The s a m p l e s w e r e p r e p a r e d f o r investigation by melting m e t a l l i c cobalt (99.98% Co) and gallium (99.995% Ga) in an induction furnace in a p r o t e c t i v e a t m o s p h e r e of argon. The change in the weight of the m a t e r i a l s due to the m e l t i n g p r o c e s s did not exceed 0.5%. Metallographic and x - r a y a n a l y s i s showed that the phase c o m p o s i tion of the s a m p l e s c o r r e s p o n d e d to the phase d i a g r a m [4, 5]. The s a m p l e s p r e p a r e d in this way contain f r o m 0 to 100 at. % gallium. The r e s i s t i v i t y was m e a s u r e d by the " r o t a t i n g magnetic field" method [6]. Typical p o l y t e r m s of the r e s i s t i v i t y a r e shown in Fig. 1. They show that the p o l y t e r m s of the r e s i s t i v i t y of the solid alloys can be a p p r o x i m a t e d by l i n e a r dependences, the d i s p e r s i o n of the points not exceeding 2.5%. The concentration dependences of p and Op/3t f o r the solid C o - Ga alloys a r e shown in Fig. 2. Unfortunately, however, information is given f o r t h e m only f o r a limited n u m b e r of compositions. This is b e c a u s e the employed method cannot be applied to f e r r o m a g n e t i c alloys, and the Curie points of alloys with s m a l l gallium concentration a r e high and close to the solidus curve. In addition, in the investigated t e m p e r a t u r e range alloys with high gallium concent r a t i o n and low Curie points a r e a l r e a d y in the liquid phase. Data on the r e s i s t i v i t y of pure f e r r o m a g n e t i c cobalt, which w e r e needed to c o n s t r u c t the i s o t h e r m at t = 950~ w e r e taken f r o m [7]. As follows f r o m the data given in Fig. 2, the i s o t h e r m p(x) has an e x t r e m a l nature with m a x i m u m at x ~ 0.4. The dependence Op/St(x) is distinctive. Initially, with i n c r e a s i n g gallium concentration (up to x ~ 0.1) the t e m p e r a t u r e coefficient changes v e r y little with the composition; it then d e c r e a s e s rapidly, b e c o m i n g negative for x > 0.25 and r e a c h i n g a m i n i m u m at x ~ 0.4. This f e a t u r e of the dependence 8p/3t(x) indicates fulfillment of the Matthiessen rule f o r the solid solutions of gallium in cobalt (0 -< x -< 0.1 at 950~ which can be attributed to the o r d i n a r y m e c h a n i s m of impurity s c a t t e r i n g . Note that this a l s o a g r e e s with the data obtained [3] in an investigation of the magnetic susceptibility of C o - Ga alloys for 0 - x - 0.1. As was noted in [3], in the solid C o - Ga alloys the eg o r b i t a l s of the cobalt a t o m s have a s y m m e t r y close to that of the sP3 o r b i t a l s of the Ga a t o m s , so that they can overlap, f o r m i n g an ( s P 3 - e g ) bond. The r e s u l t i n g h y b r i d i z e d ( s P 3 - e g ) valence band has a n u m b e r of s t a t e s (per a t o m of the alloy) equal to
l(x)
=
[8x+4(1--x)].
(1)
Here allowance has been made for the fact that the number of sp~ gallium states is eight and the number of eg cobalt states, with allowance for the spin, is four. The nucleating centers of the CoGa phase that then forms become stable for x >- 0.I.
S. M. Kirov Ural Polytechnic Institute, Sverdlovsk. T r a n s l a t e d f r o m I z v e s t i y a Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 61-67, D e c e m b e r , 1978. Original a r t i c l e submitted J a n u a r y 9, 1978.
1572
0038-5697/78/2112-1572 $07.50 9
P l e n u m Publishing C o r p o r a t i o n
p.10~ (~.rn)
p .10 s
(~.m)
150 ~
I
~
"
mo112% . _ ~ ~ ~ ' ~
,
,-r
-/
12F-
YO
I~ "I0"
")',
r \
.
Liquid ','lpb.a--~ 2
9
7,T~.
100 I
5g ?0,~
1008
12#0
1400
I
l
I
)
I
m
J
g..2g gag g.60 0,00 ~Ga
lggO ~"o~
Fig. 2
Fig. 1
Fig. 1. T e m p e r a t u r e dependences of the r e s i s t i v i t y of Co - Ga alloys (the n u m b e r s next to the c u r v e s give the gallium concentration at. %). Fig. 2. Concentration dependences of the r e s i s t i v i t y and its t e m p e r a t u r e d e r i v a t i v e f o r C o - G a alloys at 950~ The o b s e r v e d f e a t u r e s in the dependences p(x) and Op/Ot(x) a r e evidently related to the f o r m a t i o n and filling of the (sp 3 - eg) band, and the e x t r e m a in the dependences at x ~ 0.4 c o r r e s p o n d to m a x i m a l filling of the band. The t e m p e r a t u r e dependences p(t) f o r liquid alloys with x -< 0.6 a r e linear, but f o r x > 0.6 a r e m o r e c o m p l i c a t e d (see Fig. 1), though in this l a s t c a s e they a r e n e a r l y l i n e a r for l a r g e t e m p e r a t u r e intervals f a r f r o m the liquidus c u r v e . F i g u r e 3 shows the p and 8p/St i s o t h e r m s f o r liquid alloys at 1500~ Qualitatively, they have the s a m e nature as the i s o t h e r m s f o r the solid s a m p l e s . However, the e x t r e m a in t h e m a r e shifted to l a r g e r gallium c o n c e n t r a t i o n s . F o r example, on the p i s o t h e r m the m a x i m u m c o r r e s p o n d s to x ~ 0.47. In addition, f o r the p 9 108
(a -m)
2gO
(~ 9 m 9 ~
I 2~,_~
168 Fig. 3. I s o t h e r m s (1500~ of the r e s i s t i v i t y and its t e m p e r a t u r e d e r i v a t i v e : 1) e x p e r i m e n t a l data; 2, 3) r e s u l t s of calculations in the f r a m e w o r k of models that do and do not take into account f o r m a t i o n of the ( s P s - eg) band, r e s p e c t i v e l y ,
12Q
8g
o,20
o/o
~8o
~o xoa 1573
liquid alloys the concentration coefficient of the r e s i s t i v i t y is a p p r e c i a b l y s m a l l e r than for the solid s a m p l e s . The c h a r a c t e r i s t i c f e a t u r e of the ap/St i s o t h e r m is the p r e s e n c e of the shallow m i n i m u m in the region of its negative values. T h e s e f e a t u r e s of the p(x) and ap/0t(x) i s o t h e r m s a r e typical f o r alloys of 3d t r a n s i t i o n m e t a l s with polyv a l e n t m e t a l s [8, 9]. The ideas developed by Ziman and F a b e r [10] f o r alloys of simple m e t a l s and g e n e r a l i z e d f o r alloys of t r a n s i t i o n m e t a l s [1, 2] m a k e it possible to u n d e r s t a n d the e l e c t r i c a l p r o p e r t i e s of such alloys. The r e s u l t s of the t h e o r e t i c a l calculations f o r alloys of t r a n s i t i o n m e t a l s with polyvalent m e t a l s [1, 2] a g r e e qualitatively with the e x p e r i m e n t a l l y e s t a b l i s h e d p(x) i s o t h e r m s . In [1, 2] the need to take into account the filling of the d band is a l s o pointed out. We made an analogous calculation using the r e l a t i o n s 1
3~:~~ e2hv~
p
4 ~ 3 d ~ w (q),
(2)
0
w h e r e ~20 is the atomic volume, VF is the c a r r i e r velocity on the F e r m i s u r f a c e , 0 q ~ sin - - ---- - -
2
(3)
2KF'
where 0 is the s c a t t e r i n g angle of the e l e c t r o n s , q = Iql = IK' - K I t o r s b e f o r e and a f t e r s c a t t e r i n g ,
is the difference of the e l e c t r o n wave v e c -
w(q)=(l--x)[t(q)[2[x+(1--x)a,,] +xlU(q)[2(1 -x+x.a~2)+2x(1-x)lt(q)llU(q)l(a.,,--1),
(4)
w h e r e t(q) is the t m a t r i x f o r the cobalt a t o m s defined by 2~ 17
t(q) =
1
m(2mEv) '~2 ~o ~ ( 2 l + 1)sin ~,rei~t P:(cos0),
(5)
and in the r e s o n a n c e approximation, taking into account only the contribution of the phase shift ~l with l = 2, is equal to ] t (q) I =
lO~h 3
1 9 - - sin "~2P., (cos 0).
m (2mEe)il2 P'o
(6)
H e r e E F is the F e r m i e n e r g y of the alloy and ~2 is the phase shift c o r r e s p o n d i n g to orbital quantum number, l = 2. To calculate the s c a t t e r i n g on the gallium a t o m s we u s e d the s c r e e n e d Coulomb potential [11]
4r~e2z
U(q) = - -
q-~_(20
cos q____RR ~ (q) '
w h e r e z is the n u m b e r of v a l e n c e e l e c t r o n s of the gallium a t o m s ; tivity of the e l e c t r o n gas, (q)
=
1
-
8 r,e' I1
q~ ~o 1_
Z(q) = -- -~
EF
-
(7)
R, radius of the gallium ion; e(q), p e r m i t -
q2
]
2 (qZ - Kb + K } ) / +
8ffFq
.ln
7. ( q ) ;
(8)
2Ke- q J ,
(9)
and K s = (2KF/~) 1/2 is the s c r e e n i n g p a r a m e t e r . The values of the p a r t i a l s t r u c t u r e f a c t o r s used for the calculation a r e given in Fig. 4. The values of all a r e taken f r o m [12]; a22 is taken equal to the values of the s t r u c t u r e f a c t o r f o r pure gallium obtained by x - r a y diffraction at 1450~ * We e s t a b l i s h e d a21 by g e o m e t r i c a v e r a g i n g of the p a r t i a l s t r u c t u r e f a c t o r s all and a22 [13]. The quantities 772 and R s e r v e d as fitting p a r a m e t e r s in the calculation of p for the p a r e m e t a l s . Using the above data, we m a d e two v a r i a n t s of the calculation. In the f i r s t , we did not take into account s e p a r a t e l y the filling of the d band, and the value of K F in the alloy was d e t e r m i n e d by m e a n s of the relation
KF= { 3~:'[O.9(l-. x)+ 3x] } ~J3,. ~o
*We a r e grateful to Yu. A. Bazin f o r providing it.
1574
(10)
I
I
I
2,~~ 22 2,o ~8 IA 1,2 1,g o,8 @ o,4 0,2 g
R ~2I
Fig. 4. Values of the p a r t i a l s t r u c t u r e f a c t o r s of liquid C o - G a alloys. The region of p o s s i b l e negative t e m p e r a t u r e coefficients of the r e s i s tivity is hatched.
o. 1 t
~5 I
I
1,5 2 2,5
g,5 4
where 0.9 is the n u m b e r of s e l e c t r o n s p e r cobalt a t o m and 3 is the n u m b e r of e l e c t r o n s p e r gallium a t o m that e n t e r the conduction band of the alloy. C o m p a r i s o n of the calculated dependence p(x) with the e x p e r i m e n t a l dependence shows (see Fig. 3) that, despite the qualitative s i m i l a r i t y of the r e s u l t s , the m a x i m u m of the ealeu ~ lated i s o t h e r m is displaced significantly to l a r g e r cobalt c o n c e n t r a t i o n s in the alloys; in addition, the value of Pmax/PCo exceeds the e x p e r i m e n t a l value by 1.3 t i m e s . This r e s u l t is s i m i l a r to the one obtained e a r l i e r [2, 8] f o r Co - Ge and Co - Sn alloys. In the second v a r i a n t the calculation was made u n d e r the a s s u m p t i o n that an ( s p 3 - e g ) hybrid band is f o r m e d and steadily filled in the interval 0 - x - 0.47. In this c a s e , the composition dependence of KF was calculated in a c c o r d a n c e with
Ke= I 25"lT + ll'25 x ]~/3 0 ~ x (11) KF = [ 110.1x--21.301I"3 0.47 < x ~ 1.
~o
Compa rison of the data p r e s e n t e d in Fig. 3 shows that this method of calculation m a k e s it p o s s ~ l e to obtain data that a g r e e b e t t e r with the e x p e r i m e n t a l data. It can be a s s u m e d that the somewhat too l a r g e calculated values of p a r i s e f r o m neglect of the concentration dependence of the phase shift 712 in the e s t i m a t e s made of the p values. As we have a l r e a d y noted, the t e m p e r a t u r e coefficient of the r e s i s t i v i t y is negative in an a p p r e c i a b l e range of concentrations. This has a l r e a d y been o b s e r v e d in alloys of transition m e t a l s with polyvalent e l e m e n t s [8, 9] and finds a qualitative explanation in the f r a m e w o r k of the Z i m a n - F a b e r theory; f o r in a c c o r d a n c e with this theory, ap/at can be l e s s than z e r o if the value of 2KF lies n e a r the m a x i m a of the partial s t r u c t u r e f a c t o r s . As follows f r o m the data given in Fig. 4, f o r the C o - Ga ailoys this can o c c u r f o r 2.5 ~ q ~ 3.05. It follows f r o m the concentration dependences of KF that in the f i r s t v a r i a n t of the calculations, which do not take into account the f o r m a t i o n of the (sp 3 - e g ) band, this c o r r e s p o n d s to the concentration r a n g e 0 _ x _< 0.4, while i n the second it c o r r e s p o n d s to 0.45 -< x -< 0.75. It follows that the qualitative c o r r e s p o n d e n c e between the r e sults of the calculation and the e x p e r i m e n t a l d a t a e s t a b l i s h e d in the f i r s t v a r i a n t is fortuitous. H e r e , the m a x i m u m of the r e s i s t i v i t y is produced b e c a u s e the m a x i m a l values of the s t r u c t u r e f a c t o r s a r e attained in the r a n g e 0 -< x -< 0.4, w h e r e a s this m u s t , a c c o r d i n g to the e x p e r i m e n t , be o b s e r v e d at 0.45 - x -< 0~ The second v a r i a n t d e t e r m i n e s the position of this region f a i r l y r e l i a b l y . Of c o u r s e , the final justification f o r the second v a r i a n t of the calculation can only be provided by n u m e r i c a l e s t i m a t e s of Op/at, and at the p r e s e n t t i m e t h e s e a r e difficult to m a k e b e c a u s e of the a b s e n c e of data on the t e m p e r a t u r e dependence of the p a r t i a l s t r u c t u r e factors. CONCLUSIONS 1. It has been shown that use of the method previously developed by Faber, Ziman, and Evans for transition m e t a l s m a k e s it possible to d e s c r i b e f a i r l y well the concentration dependences of the e l e c t r i c a l r e s i s t i v i t y and its t e m p e r a t u r e coefficient f o r liquid C o - Ga alloys.
2. It has been e s t a b l i s h e d that m o r e reliable r e s u l t s can be obtained using a v a r i a n t of calculation that t a k e s into account the f o r m a t i o n and gradual filling of the hybridized (Sl~ - eg) band of the Co - Ga alloys. 1575
LITERATURE 1. 2. 3. 4. 5. 6. 7.
8. 9.
10. 11. 12. 13.
CITED
R. Evans, H. J. Guntherodt, H. U. K//nzi, and A. Z i m m e r m a n n , P h y s . Lett., 38A, 151 (1972). K. Hirata, Y. Waseda, A. Jain, and R. S r i v a s t a v a , J. P h y s . F: Met. Phys., 7, 419 (1977). P . V . Gel'd, S. P. Dovgopol, V. A. Antropov, and L Z. Radovskii, High T e m p . - H i g h P r e s s u r e s , 8, 529 (1976). M . S . K . Hansen, Constitution of Binary Alloys, M c G r a w - H i l l , New York (1958). R . P . Elliott, Constitution of B i n a r y Alloys, McGraw-Hill, New York (1965). B . S . Vorontsov, V. A. Antropov, and I. Z. Radovskii, in: P h y s i c s of Metals and T h e i r Compounds [in Russian], Vol. 74, S v e r d l o v s k (1977). I. A. T s o u k a l a s , Phys. Stat. Sol., A23, k41 (1974). G. Busch, H. J. Gffntherodt, and H. U~ Kffnzi, P h y s . Lett., 29A, 608 (1969). O. D r e i r a c h , R. Evans, H. U. G//ntherodt, and H. U. K//nzi, J. Phys. F: Met. Fiz., 2, 709 (1972). T. E. F a b e r and J. M. Ziman, Phil. Mag., 1_!1, 153 (1965). V. Heine, Solid-State Phys., 2_~4, 1 (1970); M. L. Cohen and V. Heine, Solid-State Phys., 24, 37 (1970); V. Heine and D. Weaire, Solid-State Phys., 2__4, 249 (1970). Y. Waseda and S. T a m a k i , Phil. Mag., 322, 273 (1975). L. Z. Rudman, A u t h o r ' s A b s t r a c t of Candidate's Dissertation, Moscow (1976).
INTERPRETATION BY A LEPTON I.
G.
OF MODEL
RESONANCE OF
STATES
OF
PARTICLES
HADRONS
Kesaev
UDC 539.1.01
The possibility is b r i e f l y c o n s i d e r e d of using a lepton model to explain the l a r g e m a s s e s of r e s o n a n c e s and t h e i r widths, decays, and spin and c h a r g e p r o p e r t i e s as well as give a quantitative d e s c r i p t i o n of the laws governing t h e i r b e h a v i o r . In [1, 2] the p r e s e n t author b r i e f l y c o n s i d e r e d a lepton model of h a d r o n s and an a p p r o a c h b a s e d on it to the d e s c r i p t i o n of the p r o p e r t i e s of the s o - c a l l e d stable p a r t i c l e s , which in this model a r e a s s o c i a t e d with e n s e m b l e s of i n t e r a c t i n g muons c h a r a c t e r i z e d by a l a r g e positive binding energy. In this p a p e r we c o n s i d e r the possibility of using the s a m e model to d e s c r i b e the p r o p e r t i e s of r e s o n a n c e s , which a r e i n t e r p r e t e d as d y n a m i c a l l y unstable e n s e m b l e s of muons o r of stable h a d r o a s consisting of muons with sufficiently l a r g e negative binding e n e r g y of the individual f r a g m e n t s . T h r e e main f e a t u r e s of r e s o n a n c e s a r e c o n s i d e r e d : 1) the l a r g e m a s s e s c o m p a r e d to those of the stable h a d r o n s ; 2) t h e i r s h o r t l i f e t i m e s ; 3) the decays, whose products a r e predominantly the stable hadrons ~, K, and N. The l a r g e m a s s e s follow d i r e c t l y f r o m the relation m ~ ~ M m ~ - - U~H,
(1)
which e x p r e s s e s the hadron m a s s m H in e n e r g y units in t e r m s of its s t r u c t u r e n u m b e r M (the n u m b e r of muons in the hadron), the muon m a s s m/z, and the binding e n e r g y U/~H of the muons in the given hadron. To the stable hadrons t h e r e c o r r e s p o n d s on the s c a l e Of muon binding e n e r g i e s the region of positive values f r o m 76 to 850 MeV or, on the s c a l e of the specific binding e n e r g i e s U/~H/M, the region f r o m 7.5 to 50 MeV [1]. A s s u m i n g that the region of the r e s o n a n c e s is strongly displaced to negative values of U/~H, we obtain in a c c o r d a n c e with Eq. (1) a s i m p l e possibility f o r explaining the o b s e r v e d l a r g e m a s s e s of these p a r t i c l e s . On the o t h e r hand, the negative binding e n e r g i e s (as calculated f r o m the m a s s e s of the r e s o n a n c e s and t h e i r decay products) of the individual densely packed blocks of muons in the r e s o n a n c e s t r u c t u r e s in the f o r m of pions, kaons, and nucleons c o r r e s p o n d to r e l a t i v i s t i c v e l o c i t i e s of expansion. A s s u m i n g that the lifetime T R of the r e s o n a n c e is twice the t i m e taken f o r these f r a g m e n t s to expand o v e r the c h a r a c t e r i s t i c distance V. I. Lenin All-Union E l e c t r o t e c h n i e a l Institute. T r a n s l a t e d f r o m I z v e s t i y a Vysshikh Uchebnykh Z a v e denii, Fizika, No. 12, pp. 67-73, D e c e m b e r , 1978. Original a r t i c l e s u b m i t t e d J a n u a r y 11, 1978.
1576
0038-5697/78/2112-1576 $ 07.50 9 1979 Plenum Publishing C o r p o r a t i o n