Effect of the broadening of the spectral distribution on the Hall coefficient of amorphous metals

80

Materials Science and Engineering, A133 ( 1991 ) 80-84

Effect of the broadening of the spectral distribution on the Hall coefficient of amorphous metals M. Itoh, A. Ishida and T. Nagata Department of Physics, Faculty of Science, Shimane University, Matsue 690 (Japan)

Abstract The Hall effect of amorphous sp metals is studied theoretically for a hypothetical single-component system. Both the d.c. and the Hall conductivities are calculated in the framework of Edwards' theory by including the full broadening effect of the spectral densities. The electronic density of states (DOS) is also calculated in the same formalism. A clear correlation is observed between the structure-induced DOS minimum and the deviation of the Hall coefficient from its free-electron values.

1. Introduction Theoretical studies of electron transport in amorphous sp-electron metals have been made mostly based on the Faber-Ziman type of approach. Although the theory is a reasonable starting point and, indeed, it has shown its capability in describing various aspects of electron transport properties qualitatively or even quantitatively, the problem of the Hall effect lies beyond the scope of this formalism. There is a fundamental limitation in the Faber-Ziman approach in this respect because its basic assumption is the existence of the free-electron Fermi sphere, from which one can obtain nothing but the freeelectron value of the Hall constant. The calculation of the electronic density of states (DOS) is also outside the compass since no dispersion effect is incorporated in the formalism. However, it is indeed an outstanding problem of sp-electron amorphous metals that the Hall constant anomalies and the structure-induced minimum of the DOS appear to be correlated to each other [1, 2]. One is therefore forced to abandon the Faber-Ziman approach and to seek a more consistent formalism to incorporate the interplay between the scattering processes and the structure. A consistent formalism to deal with both the electronic density of states and the transport was in fact introduced as early as 1958 by Edwards [3], prior to Ziman's historical paper in 1961 [4]. 0921-5093/91/$3.50

Edwards' formalism can be suitably called the self-consistent second-order perturbation theory, in the sense that in applying the second-order perturbation theory all the intermediate states are renormalized in a self-consistent way. This is done by using the Green function plus the diagram technique, which results in a non-linear integral equation for the electron self-energy. The electronic density of states is then calculated from the diagonal matrix element of the Green function. Edwards also gave the prescription to evaluate the Kubo-Greenwood formula for the d.c. conductivity consistent with the DOS calculation. Again this results in a set of integral equations for the vertex functions. A number of applications have been made to calculate the DOS using Edwards' formalism [5], but few attempts have been reported so far for the electron transport (see, however, refs. 6 and 7). Recently the application of the above formalism to the Hall coefficient has been formulated by Itoh [8], and its calculation has been made possible in a consistent way with the resistivity and the DOS. In this paper we aim to apply this formalism to a model amorphous metal as a first attempt at the microscopic treatment of the Hall coefficient. As possible mechanisms to yield the Hall constant anomaly, we can raise (1) the deformation of the dispersion relation from the freeelectron parabola, and (2) the broadening of the spectral distribution. At first sight it may appear

81 that the deformation of the dispersion relation will have some effect on the Hall constant. Earlier work by the same author [9] has however shown that when the broadening is neglected the Hall coefficient is given by the free-electron expression R n = 1/nec, with the carrier number n being given by the volume of the shifted Fermi sphere. Since the shift of the Fermi surface is expected to be small [6], it is therefore interesting to perform a calculation including the full broadening effect. In this paper we attempt this by solving the integral equations numerically. The DOS, the d.c. conductivity ~lxx and the Hall coefficient R H are calculated as functions of the carrier number per atom (e/a) for a simplified model of a singlecomponent amorphous metal. The calculation shows that the Hall constant deviates from its flee-electron values when the Fermi energy lies around the minimum of the DOS. The position of the largest deviation is fairly close to e/a = 1.8, a magic number found by H~iussler and Baumann for noble-metal-based amorphous alloys [1]. Its direction is however opposite to their experimental data, rather along with the tendency found in Mg-Zn-based alloys [2]. 2. Formalism

The calculation of the Hall coefficient R u requires both the d.c. and the Hall conductivity RH=

(7xy/(Ixx2H

(1)

For the d.c. conductivity, Edwards' theory expands the Kubo-Greenwood formula into the perturbation series and sums up the ladder diagrams ([3]; see also ref. 7). The same level of approximation can also be applied to the Hall conductivity by using the formula found by Itoh [8, 10]

H

c

\OE]

x (Tr vxd(E - H) vyG +(E)

w d ( E - H) vyG+(E))

(2)

The details of the evaluation are reported elsewhere [8, 10] and we quote the result

Oxy_ f

- ~ - 3 e r c . , dE

( - ~of) f ( ~dk) 3 ~:1 )] (3)

× im[(vk + - )2Vk+ +(Gk+ )2 Gk- - ~(vk + + )3( Gk+

Here v ÷- and v ÷ ÷ are exactly the same vertex functions as appearing in the d.c. conductivity calculation [3, 7]. These are obtained as solutions of the integral equations for the current vertex parts. The retarded and the advanced Green functions Gk + and G k- are obtained from the self-energy, which is also obtained by solving an integral equation prior to those for the vertex functions. It must be noted that eqn. (2) is a rigorous formula and that the Hall conductivity is therefore determined only by the states in the vicinity of the Fermi level regardless of the approximation. This, of course, applies to the d.c. conductivity as well, so that the Hall coefficient is a purely Fermi surface quantity. 3. Models

We use Ashcroft's empty-core potential and the hard-sphere structure factor in the Percus-Yevic approximation to model the system. The single component system is considered for simplicity. In this paper we do not attempt quantitative calculations for the individual systems but rather try to extract general trends by varying the physical parameters. The packing fraction r/ of the system has been varied from 0.60 to 0.65. These high values of the packing fraction compared to the liquid state are expected to describe the strong short-range order in the amorphous state. We have investigated three possible cases for the position of the node q0 of the form factor relative to the position Kp of the first peak of the structure factor, according to q0 > Kp, q0 < KP and q0----Kp. For the first two cases we have chosen R c = 0.8 and 2.0 respectively in atomic units, where R c is the empty core radius. These values roughly correspond to the two possible values to reproduce the observed resistivity of liquid noble metals near the melting points [11 ]. Without using the small-core approximation to the d-states below the Fermi level, Moriarty [12] showed that the nodes fall in between these two values, but the slope of the potential near the node is much steeper than the empty core model. To simulate the situation suggested by Moriarty, we also study the case when q0 is very close to Kp, by adopting the empty core model and by multiplying the potential by the factor of 3. As for the effective valency of the atom we set Z = 2. In terms of the relative positions of q0 and Kp, the Mg-Zn-based metallic alloys are classified into the second case (i.e. qo
82 dart-Vosko's dielectric screening [14], for the evaluation of which we have fixed the electron density at e/a = 2. Here we report the results of the case of r/-- 0.65.

4. Numerical results In Figs. l(a)-3(a) we have plotted the calculated d.c. and Hall conductivity as functions of e/a, together with their values in the quasi-particle

approximation. The ratio of the Hall coefficient to its free electron value is also shown in Figs. l(b)-3(b). For comparison, also plotted in Figs. 4-6 are the electronic DOS (Figs. 4(b)-6(b)) and the resistivities calculated both in the Edwards and in the Faber-Ziman theory (Figs. 4(a)-6(a)). The structure factor and the pseudopotential form factor are also shown in the same figures 0.5

(a)

/x.

0.4

0

2 (a)

-2OO et /~\

__

/' ~, /,/

,01

\\

....

quasipaaicle

,

i

1

2

•

\~"'J i

0.2

Edwards

400 ~ P

0.1

-600

0.C

g

--

Edwards

....

quasiparticle

\Nil

-20

. . . . . . . 1 2 3 e/a

0

F

-30

'-800 1.02

e/a 1.10

-10 Ct

0.3

f

1.01

(b)

1.08 1.06

~

1.00

1.04 1.02

0.9~

1.00 0.98

i

0

' 1

0

' 2 e/a

' 3

4

i

1

Fig. 3. T h e same plots as in Fig. 1 for r/=0.65 Rc = 1.23.

2

da

Fig. 1. (a) T h e calculated d.c. and the Hall conductivity in the E d w a r d s theory and their quasiparticle approximations, and (b) the ratio of the Hall coefficient to its free-electron value, for r / = 0 . 6 5 and Rc = 0.8.

(a)

/

100

and

\ \\\\

0.5

a)~ / \ //

0.4

/

/ /I~.//,\\

o

//

0.1

-lO

\//

,~d 0'3 0.2

0

- -

Edwards

-20 cl

....

quasiparticle

-30

\

1.4

'

' 1

' 2 e&

1.0 0

1

2

e/a

O~ 0.1

1.1

OS

Zimart

t

-60 4

' 3

1.2 ~

u

-50

(b) ~

1.3

~ d w a r d s . . _ ~ S ~ j" ....

"0

-no

/

0.0 ~ 0

v

'

1

'

'

2 3 4 e/a Fig. 2. T h e same plots as in Fig. 1 for r/= 0.65 and R c = 2.0.

'

r

in

,V

0.2 Energy (a.m)

J

0.3

0

!

k(~.u.)

2

Fig. 4. (a) Resistivities as functions of'e/a calculated in both the E d w a r d s and the Ziman theories, (b) the electronic density of states, and (c) input structure factor and pseudopotentials, for r/=0.65 and R c = 0 . 8 . T h e scattering kernel is shown by the hatched area. T h e vertical lines labelled A and B indicate the positions of the maximum and the minimum, respectively, and C the Fermi level for e/a = 2.0. At A and B, e/a = 1.40 and 1.70, respectively.

83

~

500

2.0

(a)

•

Edwards

o~ 400

F\\

I I

3OO

zim~n

. . . .

\

1.8

"

1.6

"

1.4

A ~,2z.s~

•

//---...\ "-\..

.~ 20o

AgC~

~Cut~

o

r/

lOO ---m-~

"l

1

1

0

T

2 e/a

3

1.2

o

AA

~Gaal

[] c~

. &

1.0

_o

•

--

o oo

.l

•

0

~

°CO

o

o

N •

N+

•

• •

•

mO

o

]51 (b)

o

[]

0.8

[]

o o

o

0 0

0.6 0,4 0.0

0.1

0.2

0.3

0.4

1

0

3

2

Energy(a.u.)

k (a.u_)

Fig. 5. T h e s a m e plots as in Fig. 4 for r / = 0.65 a n d R c = 2.0. At A and B, e/a = 1.2 a n d 1.75, respectively.

~ 100

(a)

80 g.

,~ 60

/

m

•~ 40

\\

-

/ ~

j/

Edwards

-

Ziman

. . . .

20 0

0

i

i

a

1

2

3

4

e/a

-0.1

0.0 Energy

0.1 (a.u.)

0.2

0

1

2

3

k (a.tt)

Fig. 6. T h e same plots as in Fig. 4 for r / = 0 . 6 5 R c = 1.23. A t A and B, e/a = 1.4 a n d 1.8, respectively.

and

(Figs. 4(c)-6(c)), together with the scattering kernel. In the DOS figures the vertical lines labelled by C show the positions of the Fermi energies for e/a = 2.0. The difference between the full Edwards theory and its quasi-particle approximation shows the magnitude of the broadening effect. The deviation of the Hall coefficient from the free-electron result is rather small in spite of the large broadening effect found for both ax~ and axy. It seems that the broadening effects in oxx and axy tend to cancel each other in the Hall effect.

.

.

.

.

.

.

.

.

.

'

2

.

.

.

.

.

.

.

.

.

e/a Fig. 7. T h e plot of experimental Hall coefficients m a d e c o m parable to Figs. l(b)-3(b). T h e data for C u S n are taken from ref. 1, while all other data are taken f r o m ref. 2.

This is particularly so for the case of R c = 1.23, the case in which the first peak of the structure factor is almost killed by the node of the potential and the main scattering comes from the region around the second peak. In this case the broadening correction to the quasiparticle description amounts to 15-30% for axx and axr at maximum, whereas the Hall constant deviation is only 0.5% there. For R c = 2.0 the Hall constant deviation is substantial, but the calculated resistivity is pretty high compared with the typical sp-electron amorphous metals. It should be noted that the maximum resistivity, the DOS minimum and the largest deviation of the Hall coefficient occur almost precisely at the same carrier number, approximately at e/a = 1.75, a value very close to the one found by H~iussler and Baumann [1 ]. Concerning the direction of the deviation from the free electron, however, our result is opposite to the tendency reported by these authors for the noble-metal-based amorphous alloys. On the other hand, many of the amorphous metals including magnesium, zinc or gallium as their major components show the tendency shown in the present calculation [2]. We have made a tentative plot in Fig. 7 of the observed values of RH/R o taken from the literature [2], with the freeelectron value R 0--- 1/nec being calculated from the measured densities. The data for Cu-Sn are also plotted as a typical example of the noblemetal-based alloys [1]. It is tempting to talk about the resemblance between Figs. l(b)-3(b) and the

84

plot of the first group of the materials in Fig. 7, although the large deviation from the f l e e electron model occurs at much larger values of

anomaly found in noble-metal-based alloys to the d-states below the Fermi level, at least in a direct way, because the anomaly occurs at e/a ~>1.

e/a. Acknowledgments 5. Conclusion Although the present calculation is for a simplified model system, it has been shown that there generally exists a sufficient driving force in liquid or amorphous metals to yield the anomaly of the Hall constant. Most interestingly, we have recovered the magic number found by H~iussler and Baumann. This demonstrates clearly that the structure of the DOS and the transport anomalies come from the same physical origin, i.e. from the broadening. As for the direction of the Hall constant deviation, however, we should further investigate the multi-component systems. Experiment shows that the effect of the multi-componency is very important; the different behaviours of the Hall coefficient between Mgl_x(GaA1)x and Mgl_x(Ga2Al)x shown in Fig. 7 strongly indicate its importance. In particular, for the noble-metalbased alloys, such as Cu-Sn, we should take into account the large difference in the valencies of the constituent atoms, which might break the cancellation between the two broadening effects in axx and axy. As we have pointed out earlier, the Hall coefficient is determined solely by the states at the Fermi energy, as is seen from eqn. (2). Therefore we cannot attribute the Hall constant

The authors express their thanks to Professors U. Mizutani, H. Sato and T. Fuktmaga for useful discussions and encouragement.

References 1 P. H~iussler and E Baumann, Physica B, 108 (1981) 909; R H/iussler, Z. Phys. B, 53 (1983) 15; for a review, see P. Hgussler, in Glassy Metals Ili, Topics in Applied Physics, Springer, 1990. 2 U. Mizutani and T. Matsuda, J. Phys. F, 14 (1984) 2995; for a review, see U. Mizutani, Progress in Materials Science 2, vol. 28, 1983. 3 S. F. Edwards, Phil, Mag., 3 (1958) 1020. 4 J.M. Ziman, Phil. Mag., 6(1961) 1013. 5 L. E. BaUentine, Can. J. Phys., 44 (1966) 2533; Adv. Chem. Phys., 31 (1975) 263; N. C. Halder and K. C. Phillips, Phys. Status Solidi, B, 115 (1983) 9. 6 A. B. van Oosten and W. Geertsma, Physica B, 133 (1985) 55. 7 M. Itoh etal., Mater. Sci. Eng., A128(1991). 8 M. Itoh, J. Non-Crystalline Solids, 117(1990) 409. 9 M. Itoh, J. Phys. F, 14(1984) L179. 10 M. Itoh, J. Phys. Condens. Matter, to be published. 11 N. W. Ashcroft and D. C. Langreth, Phys. Rev., 159 (1967) 500. 12 J.A. Moriarty, Phys. Rev. B, 1 (4) (1970) 1363. 13 J. Hafner, J. Non-Crystalline Solids, 69 (1985) 325. 14 D.J.W. Geldart and S. H. Vosko, Can. J. Phys., 44 (1966) 2137.