INFLUENCE OF THE ANISOTROPY OF ELECTRON-PHONON SCATTERING ON THE LOW TEMPERATURE RESISTIVITY OF NORMAL METALS

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INFLUENCE OF THE ANISOTROPY OF

ELECTRON-PHONON SCATTERING ON THE LOW

TEMPERATURE RESISTIVITY OF NORMAL

METALS

R. Krsnik, E. Babić

To cite this version:

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JOURNAL DE PHYSIQUE

Cofloque

C6, suppfPmenr au no 8, Tome 39, aolit 1978, page

_C6-1052

INFLUENCE OF THE ANISOTROPY OF ELECTRON-PHONON SCATTERING ON THE LOW TEMPERATURE RESIS-

TIVITY OF NORMAL METALS

R. Krsnik and

E.

~ a b i 6

Institute of Physics of the University, P. 0. B. 304, Zagreb, YugosZavia

RBsum6.- Nous tentons de corrgler les diffdrences dans la r6sistivitL 2 basse tempdrature de

diffLrents mgtaux normaux P l'anisotropie deladiffusiondlectron-phonon. Auxerreurs expdrimentales

prhs, nous obtenons un schgma assez ggnsral qui montre que Zn et Sn sont les mdtaux P l'anisotropie

la plus forte.

Abstract.- An attempt is made to correlate the differences in the low temperature phonon resisti-

vities of various normal metals to the anisotropy of electron-phonon scattering. Within the uncertainty of experimental data a rather general picture is obtained which shows that Zn and Sn are the metals with the strongest anisotropy.

Our earlier measurements of A1 /l/ and Zn

/ 2 / based alloys showed that the low temperature resistivities of these systems have very different

temperature dependence. While in AI-alloys for T <

0.060D (gD is the Debye temperature) a T~ variation

predominates, in Zn system a pure T' dependence is

found for T < 0.049,,. We note that Zn system has the same temperature dependence both in a "pure limit" and in "impure region" as well as in polycrystals, quenched samples and monocrystals along either of crystallographic axes. Other normal metals (except for alkali metals which we will not discuss) in a sence show intermediate behaviour between AI and Zn. Namely for no normal metal (other than Zn) a pure T' dependence is observed in the "impure region".

In order to account for the observed behaviour the most promising seem to be those theoretical approaches based on the anisotropy of intrinsic electron-phonon (e1.-ph.) scattering due to anisotropic Fermi surfaces; taking into account strong umklapp (U) processes even at low temperatures. So far however the theoretical calculations agree only approximately with the experimental data which may be because the Fermi surfaces used in these calcu-

lations are not realistic enough.

Here we wish to show that the resistivity data for several normal metals can be related to the anisotropy of their Fermi surfaces in agreement

with the calculations of Lawrence and Wilkins 131.

Namely they predict a T' resistivity variation at

the lowest temperatures (T < 0.050D) and a somewhat

slower (T2,T') variation at higher temperatures.

The experimental data show a T5 variation at the lowest temperatures only for the metals with the most anisotropic Fermi surfaces like Zn and Sn (in the "pure limit") while for A1 and noble metals

(Au, Ag, Cu) no T' dependence is found at the lowest

temperatures /4,5/. In order to classify different systems according to their strength of the intrinsic anisotropy of e1.-ph. scattering we propose two methods.

In table I we compare for several systems thb

calculated coefficients (BN) of the

loch-~riineis-

sen resistivity term 161, (which does not include U

processes) with the coefficients (B ) of the mea-

p.1.

sured T' resistivity term /7/ in "pure limit". In the systems with a strong intrinsic anisotropy of e1.-ph. scattering the contribution of U processes

in B will be large and therefore we expect B

p.1. P. l

>> BN. (In calculating BN we used the values of 0 D

at T = 0 K and not ORbecause the latter is obtai-

ned by assuming only normal e1.-ph. scattering). In spite of a considerable uncertainty in the phonon resistivities,

( '

T

-

Po)p.l

,

and OD values

B

the ratio

*

is the biggest for Zn and Sn i.e.

B~

for the systems which show a pure T' dependence and

also have the most anisot opic Fermi surfaces. For

_B

the noble metals and A1 is much smaller al-

B~

though still greater than 1 which indicates that

also in these metals the contribution of the U-processes does not vanish even at the lowest temperatures.

A similar result is obtained by investigating pi the "breakdown" resistivities /7/ which mark the

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end of a "pure l i m i t " . "impure region" and t h a t Zn and Sn would be t h e b e s t systems i n o r d e r t o reach t h e " d i r t y " l i m i t .

Table I : Data r e l e v a n t t o t h e i n v e s t i g a t e d systems.

BD i s t h e Debye temperature a t T = 0 K , B is t h e c o e f f i c i e n t of t h e Bloch-Grcneissen r e s i s t i v i t y term B i s t h e c o e f f i c i e n t of a T' term i n t h e r e s i s - t ? i r l t y ( i n "pure l i n i i t " ) and T i s 'the temperature a t i q h i c h B i s taken.

p.1

C l e a r l y , although a s u b j e c t of a c o n s i d e r a b l e un- c e r t a i n t y , p i i s an important parameter because i t shows where t h e a n i s o t r o p y of t h e e1.-ph. s c a t t e r i n g begins t o be washed o u t by i m p u r i t i e s . However a d i r e c t comparison of p' v e r s u s T/OD f o r d i f f e r e n t

0

systems' / 7 / i s n o t a p p r o p r i a t e because i t does n o t take i n t o account t h e s t r e n g t h of the i n t r i n s i c e1.- ph. s c a t t e r i n g f o r t h e p a r t i c u l a r system. (A d i r e c t comparison gave 171 Zn and Sn a s t h e two l i m i t i n g c a s e s while t h e o t h e r m e t a l s were s i t u a t e d i n b e t - ween !). A b e t t e r way of comparing t h e d a t a i s t o p l o t

-

'0)p.l. v e r s u s

-

T because i n the more

p i OD a n i s o t r o p i c system ( b i g g e r (pT

-

p o ) p . l . ' a l s o t h e e f f e c t of i m p u r i t i e s ( p # ) w i l l show up e a r l i e r . Thus 0

-

P O ) ~ . ~ . ( t h e r a t i o of t h e e f f e c t i v e f r a c t i o n of P, of i n t r i n s i c e1.-ph. and e l e c t r o n - i m p u r i t y s c a t t e - r i n g ) should b e b i g g e r i n t h e c a s e of l a r g e r i n t r i n - s i c a n i s o t r o p y of t h e e1.-ph. s c a t t e r i n g . Although both p; and (pT

-

po)p.l. a r e s u b j e c t t o a conside- r a b l e u n c e r t a i n t y f i g u r e 1 shows t h a t t h e experimen- t a l d a t a f o l l o w r a t h e r w @ l l the expected behaviour. The d a t a i n d i c a t e t h a t t h e i n t r i n s i c a n i s o t r o p i e s of e1.-ph. s c a t t e r i n g i n Sn and Zn l a r g e l y exceed t h a t of A l ( n o t e l o g a r i t h m i c s c a l e ) which i n t u r n exceeds t h o s e of t h e noble m e t a l s . Thus we would e x p e c t a pure T5 r e s i s t i v i t y dependence i n Sn a l s o f o r

F i n a l l y , a s i m i l a r conclusion i s reached when comparing t h e i n c r e a s e s i n t h e phonon r e s h t i v i t y

(pT

-

p,) i n t h e "impure region" p e r decade of p. f o r d i f f e r e n t systems ( i n s e t t o f i g u r e I ) . Again we ob- t a i n t h e b i g g e s t i n c r e a s e f o r Zn and Sn, considera- b l y s m a l l e r f o r A 1 and even s m a l l e r f o r n o b l e m e t a l s . Apparently t h e e f f e c t of " i s o t r o p i s a t i o n " on t h e low temperature r e s i s t i v i t y i s t h e b i g g e s t i n t h e c a s e of a s t r o n g e r i n t r i n s i c a n i s o t r o p y .

[ ' T '

"

I

io2 T/eD F i g . 1 : R a t i o of i n t r i n s i c r e s i s t i v i t y

pi=(pT-po)p. l and "breakdown" r e s i s t i v i t i e s p; v s . l o g T/OD f o r s e v e r a l a l l o y systems : 0 Sn, Zn, V Al,WCu,nAu and

O A ~ .

I n t h e i n s e t t h e i n c r e a s e i n t h e phonon r e s i s t i v i t y (pT-p,) i n t h e impure r e - gion p e r decade i n t h e r e s i d u a l r e s i s t i v i t y (po) v s l o g T/€JD.

References

/ l / ~ a b i c ' , ~ . , Krsnik,R. and Ofko,M., J . Phys (1976) 73

121 Salvadori,E., BabiC,E., Krsnik,R. and Rizzuto, C., J , Phys. (1973) L195

/ 3 / Lawrence,W.E. and Wilkins,,J.W., Phys. Rev. (1972) 4466

/4/ Rumbo,E.R., J. Phys. (1976) 84

/ 5 / Barber,A.J. and Caplin,A.D., J. Phys.

2

(1975) 679

/6/ Ziman,J.M., E l e c t r o n s and Phonons (1960), Ox- f o r d U.P.