Electron transport process in simple amorphous metals at moderately low temperatures

journal of MOLECULAR

LIQUIDS ELSEVIER

Journal of Molecular Liquids 93 (2001) 207-218 www.elsevier.com/locate/molliq

ELECTRON TRANSPORT PROCESS IN SIMPLE AMORPHOUS METALS AT MODERATELY LOW TEMPERATURES Yu.P. Krasny 1, N.P. Kovalenko 2, V.T. Shvets 3, J. Krawczyk I

tOpole University, Oleska 48, 45-052 Opole, Poland. 21. 1. Mechnikov State Universi(y, Dvorianskaya 2, 2 70100, Odesa, ~/kraUle 30desa State Academy of Refrigeration, Dworianskaya 1/3, 270100 Odesa, Ukraine. The dependence of electroresistivity p(T)and electron contribution to thermoconductiviv,, x-(T) of s|mple amorphous metals is investigated. Calculation of kinetic coefficients is carried out in the nearly free electron approximation (Ziman theory). The form-factor was calculated in the quasiphonon model. In this approximation on the short wavelength part of.,dispersion curve" a minimum exists (..roton-like minimum"). It is shown that at moderately low temperatures 10K <_ T < 100K the ratio p(T) - p(0) has a lnaximum

T"

and

theT2[r(T)-Ic(0)]is

minimal in this temperature region. Such ,,anomaly" in the temperature

dependence of the kinetic coefficient is due to additional scattering of electrons on ,¢otons". Numeric ealeulatiens for amorphous Mg and Zn are presented. KeDvords: electrical resistivity, thermoconductivity, temperature dependence, electron scatering, amorphous metal. © 2001 Elsevier Science B.V. All rights reserved.

1. Introduction. In amorphous simple metals temperature dependence o f resistivity p(T) displays a sigmalike form at moderately low temperatures (10K _< T < 100K), see Fig. l and [1-3]. Since p~T) behaves like p (T) ~ T ~

at T < 1OK [1,4], then the quantity p(T) - p(0) T:

has a maximum

at moderately low temperatures. The ultrasound attenuation 7(T) and heat capacity ( ' ( 7 ' ) possess a similar functional dependence on temperature. The quantities ~

and ~

show

maxima at the same moderately low temperatures Such a behaviour of these quantities upon 0167-7322/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII S0167-7322(01) 00232-X

208

P i.

111.6

[~ / , c u

912

\

66.8

! /

65.4

i

~-J ~ j T(K)

0

40 80

120 160

Fig. 1. Dependence of the electrical resistivity p ( l ~ . cm)on temperature in the amorphous alloys Sn-Cu

temperature is explained by a minimum on the phonon dispersion curves (,,roton" minimum) observed in the small wave length region [5,6]. As it is shown in the present paper the electrical resistivity and deetron contribution to thermoconductivity behaviour in amorphous metals could be explained also by an additional scattering of electrons on ,,rotons". One should regard the approach considered as supplementary to other theories that explain the unusual (in comparison with crystals) dependence of the amorphous metal resistivity and thermoconductivity upon temperature [1-4] 2. Analysis of p(T) behaviour within the Ziman theory

So long as the ion and electron properties of liquid and amorphous simple metals are similar very much to each other, the Ziman theory for liquid metal resistivity can be applied to the amorphous metal case. The electrical resistivity and electron contribution to thermoconductivity of pure metal is defined by the following equation according to this theory [1,7]

209 ~,,

.

h¢_o

mp(T) = e2n ~2-~'h'Z ! dq. q ' IW(q)l2 .t'---:-:--. , , , -'exp

3

O)

~gel)

e~ p(r)

Here m and e are the electron mass and charge, respectively, n, is the number of electrons in a volume unit, Z is an ion valency, W(q), the screened pseudopotential form -factor describing the effective electron-ion interaction and ~(q, co)dynamic structure factor [8]

•~(q, oJ)=-~-~Idt..~(q,t)e -'~ l .V .V

~(q,t)=~ x x ~t" 11:112:1

I

" t

(21

(,)-,,,(o))1>

The following notations are accepted in (2): N is the number of ions in the volume V of the metal, R j(/) vector, defining the position ofthej-th ion at the moment t. Symbol <.....> in (2) denotes the thermodynamic averaging over vibrational degrees of freedom and the upper line in (2) (( ........ )) means the configurational averaging over ion equilibrium positions. The choice of the distribution function for the configurational averaging depends on disorder model features. One of the most realistic models for the topological disorder is a model of frozen liquid, which is exploited in the present paper. That is why the structure of the amorphous system under consideration is described, like in the theory for liquids, by the correlation function formalism. A hard sphet: model is used for the numerical ~aleulations. The ions oscillate with resl~ct to their equilibrium positions R0j at T ~ 0, that is Ry(t) = Ro.I + ~j(t) We restrict ourselves in (1) to the terms quadratic in ~ (the harmonic approximation) supposing that the ion displacements ~ t ) from the equilibrium positions are small ones in comparison with the mean distance between ions, that is

o+

{l+,, ,j,,o,]++(,j,,,

}

210 where

(4)

The Hamiltonian of vibrating ions is equal in the harmonic approximation = •~

SM2

Go + i=1 Z y ¢~

+ 1 N N

^ a ^P

0 2 U0

x~ j,=ls, Z Y.o,-,p Z ~,., L.~ a P ~ , CRop.~ ,

where M is the ion mass and U o = U(Ro~

..... Rou )

(5)

potential energy of N ions placed in

equilibrium positions. Since

z0 (

~ kBrJJ ,

(6)

then

and the dynamic structure factor takes the following form ,~(q,t) = {N "6,.0 +a(q)} +

_.oj,)j) ,

"

where

is the static structure factor. The Hamiltonian , ~ could be written in the form [5,6] ^

,Xf,*- ~ + ~

(9)

The first term in (9) represents the operator .~' averaged over all the possible quasiequilibrium positions of ions and the second term A~desefibes the deviation of the Hamiltonian of the considered ionic configuration for the amorphous system from the average

211

,~. For the calculation of the averaged thermodynamic potentials the addend A,~¢'can be ignored, as shown already in [5, 6]. The Hamiltonian ,Yf,~and the displacements ~jtake the form presented below in the second quantum presentation [5,6]

~ _ - ~ =Vo+ZZh~(k).

+

(1o)

k .i=l

Here J ( k , 2)is a polarization vector of phonon, which is characterized by momentum hk and energy hco~ (k), where A is the index of polarization and ('~, (~. are the operators of creation and annihilation, respectively. The frequencies cox(k) are defined by expressions [5,6]

(o~(k) = co~(k) = k2 +(k]+ M

""

co-',(k) = co~(k) = co~(k) = -

fd3k, .k~z+(k,).[a(#,. , . k,r. k,z + k ) - a(k,)]

,9,,

(12)

(2~r)~M J

~9°3

fd3k,.k~,*(k,).[a(kLr,

k,r,k,z + k ) - a ( k , ) ]

(13)

(21r) M a

Indexes I and t correspond to the longitudinal and transverse phonons, respectively, ~(k) is the Fourier transform of the interparticle interaction potential and ,90 = V is the volume per N atom. Numerical calculations indicate retort-like minima in the short wave region on the dispers!on curves. It is seen in Figure 2, where the dispersion curves for hard-sphere models of amorphous Mg and Zn are presented [5,6]. Let us substitute (10) and (11) into equation (7) and use the commutator relations for C~ and C'~. One can easily find that

212

go(lOl~sq)

COl

~l

i 0.2

0.4

gOt

0.6

0.8

2.77K(Aq)

Fig. 2a o~(lO~3s'J)

~0t 0.2

0.4

0.6

0.8

2.65K(A"t)

Fig. 2b Figure 2 Dispersion laws for hard-sphere models of amorphous Mg (a) and Zn (b) ro~ longitudinal oscillations: co, " transverse oscillations.

where

exp

xBI J

Let us direct axis z along vector q and perform integration in (1), taking into account that

q.gg(k, 1)-qcosO, q . # ( k , 2 ) : q s i n O c o s r p ,

q.#(k,3)=qsinOsin~o

Then, in the thermodynamical limit (N ---}oo, V --~ oo, but %

= ,90 = ¢onst ) we get the

following equation for the p(T)

p(v) = p(o) + Ap

c~ 6)

213 where m2

2kF

p(O)-12rc3eZnj~3Z !

,~2

2k~,

•

1

(17)

dq.q31W(q)12a(q) o

2 2f

~

[

kv

-

"(2,~) ~ 3 ~o

knT )

hint(k) I _ exp(. hm,(k))

k--?¢ j

~

,----r xpL h--g~,(k)~i JJ~ k.r(2~)~ o knT ) +

J"sin ~O.a(q- k)-

1 1

"~, ktjT J

,[. (18)

3. Discussion The obtained expression for Ap is correct for both crystalline and amorphous metals. The structure factor of an ideal crystal has the following form

(2x) 3 a(q) : ~ k_:,Z,6 ( q - k " ) ' '

n-

where k,, is the reciprocal lattice vector. The first two terms of (18) are equal to zero if integration is performed over the volume of the first Brillouin zone. It means that the transfer processes are neglected. The last term in (18) does not equal zero and it is proportional to 7'" at low temperatures. Thus we get the well known Bloch- Grfineisen result [9]. In the case of amorphous metals the first and the second addends give already nonzero contributions in (16). Let us use the Debye approximation for the phonon spectrum that is oJ~.(k) =

C:k

first two terms of(18) behave like T'- which corresponds to the results of[l,4].

Then. the

214 The temperature dependence of Ap(T) becomes more complicated if the phonon spectrum is defined by equations (12) and (13), see Figures 3 The numerical integration in (18) indicates the quantity of AP///T2 has such a maximum at the moderately low temperatures, as in Fig. 4, where the results of numerical calculations for the amorphous Zn and Mg are presented.

2.5 t Ap

I

1.5t-

0.5T~ 50

100

150

200

250

300

Fig.3a

JO

100

IJO

200

250

300

Fig. 3b

Figure 3. Dependence of the electrical resistivity Ap=[p(T)-p(O)](pf2.cm)on temperature T in hard-sphere models of amorphous Zn (a) and M g (b)

215

g

0.25 l 0.2"-

0.15 t 0.1~

i I

O.O5

7 ~ 50

0

~ 100

~ 150

¢ 200

I 250

T(~)

Fig. 4a g

', 0

b 50

---

, I00

~ 150

I 200

t 250

b 300

T(K)

Fig. 4b

~.cm']

Figure 4. Dependence of the quantity g= [P(7") - p(~VT 20~]/ (\ K 2 j'104°ntemp erature 7in a hard-sphere model of amorphous Zn (a) and M g (b)

216 The following qualitative estimates of the integrals over the k-region in (18) facilitate the understanding of the physical reason for the origin of such a maximum. Let us divide the integration region into two parts: from zero to ktx and from k,x to k o where k~ defines the maximum of the cox(k)function. One can put cox(k ) = Cxk and tend the upper limit to the infinity integrating in the (0, k~x) region, because the integrand decreases abruptly at/c - ,

o0.

Then, the value of the integral is proportional to T 2 and the final result for all the terms, containing this kind of integral, has the form A. T ~. The function of co~(k) can be approximated by the expression

hco (k)

+

2m~

at integration in the ((k~x, ko)) interval. Here Ax, k2x and mx are parameters defining the minimum of the dispersion curve cox(k) If the upper limit and the lower one of the integration are tended to +or and --~, respectively, then the sum of all the terms containing this kind of integrals takes the following form expVwhere the coefficients Bx and Cx are determined by parameters Ax, mx , and k,.x. Finally, the quantity of A'°///T2 takes the form

We have taken into account here that The first term in (19)

arises due

A t <

A t .

to the electron-acoustic phonons (co x (k) = Cxk ) interaction,

and the second term is caused by the electron scattering on rotons. The function A~/~//.: has a maximum at

A, I%T,, =

(5(~-B:3+I(5C'-Bt)"+eC'B' 2(".,

217 One can show that T, = 40K and T,, ~ 70K for Mg and Zn, respectively, making use of values for the At, k,.t and m~ pararne~ers, determined for hard-sphere models of amorphous Mg and Zn. These results correspond to the above numerical estimates, see Figures 4. One can conclude fxom the discussion, see also equation (I), that thermoconductivity of simple amorphous metals decreases at moderately low temperatures. Such an decrease is

caused by the additional scattering of electrons on ,,rotons".

References

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