Electronic transport properties of LaxY1−xAl2 alloys

Journal of Alloys and Compounds, 198 (1993) 117-126 JALCOM 657

117

Electronic transport properties of LaxYl_xA12 alloys A . T. B u r k o v A. F. Ioffe Physico-Technical Institute, 194021 St. Petersburg (Russian Federation)

E. G r a t z , E . B a u e r * a n d R . R e s e l Institute for Experimental Physics, Technical University of Idenna, g'Tedner Hauptstrasse 8-10, A-1040 ~enna (Austria)

(Received October 20, 1992; in final form January 25, 1992)

Abstract Transport properties of quasi-binary LaxYl_xAl2 alloys are presented over a wide temperature range from 4.2 to 1000 K. The observed non-linearity of the temperature-dependent electrical resistivity p and thermopower S is interpreted as originating from density-of-states (DOS) structures near the Fermi energy Er. On the basis of the linear Boltzmann transport equation, an energy-dependent function ~b(E) is calculated from the experimental p(T) and S(T) behaviour. Characteristics of the DOS can be compared directly with ~b(e); such a comparison with available theoretical DOS calculations is given. The change in DOS with alloy composition and the effect of impurity scattering on the electrical resistivity and thermopower are discussed.

1. Introduction In a previous paper [1] we presented experimental results on electronic transport properties of REAl2 (RE, rare earth) intermetallic compounds. It was shown that the observed temperature dependences of the thermoelectric power S(T) and electrical resistivity p(T) in the high temperature range are closely related to the variation in the electronic density of states (DOS) near the Fermi level. The microscopic function of spectral conductivity, ~o(e, T), was introduced on the basis of the well-known Boltzmann transport equation. The systematic variation in S(T) throughout the REAl2 series was interpreted on the basis of the rigid band approximation, taking into account the DOS features in the vicinity of the Fermi energy. The aim of this paper is to present experimental data on the thermopower and electrical resistivity of quasi-binary LaxYI_,~A12 alloys as well as pure LaAI 2 and YAI2 compounds over an extended temperature range from 4.2 K up to about 1000 K. The alloys are based on YAlz and LaA12, where the R E ions do not bear any magnetic moment [2]. However, transport properties of both compounds, in particular the thermopower, reveal rather different features [1]. This was interpreted by assuming a different position of the Fermi level within the DOS function, which has much *Author to whom correspondence should be addressed. 0925-8388/93/$6.00

in common in the two compounds. In the case of alloys, two phenomena must be taken into consideration: (i) the change in DOS with alloy composition; (ii) impurity scattering, which influences both the electrical resistivity and thermopower.

2. Experimental procedures Samples of the LaxY 1_~xl 2 compounds were prepared by means of high frequency melting in a water-cooled copper boat under a protective argon atmosphere and subsequent annealing under high vacuum at 800 °C for 80 h. The phase purity was checked by Debye-Scherrer photographs. Electrical resistivity and thermopower data in the temperature range 100-1000 K were obtained simultaneously using the same sample holder. Different sample holders were used at low temperatures. A fourprobe d.c. method was used for electrical resistivity measurements, while for the thermopower a differential method was applied. The size of the sample was typically about 0.5 x 3 X 10 mm 3. In order to check the stability of the sample at high temperatures, all measurements were made during the heating and cooling procedure. The estimated error in the electrical resistivity is about ___20%. This is mainly due to the uncertainty in the geometric factor of the sample, which is closely related to the well-known brittleness of cubic Laves phases, preventing the preparation of samples with good me© 1993 -Elsevier Sequoia. All rights reserved

A . T . Burkov et al. / Transport properties of LaxYx_xAl2 alloys

118

chanical quality. The uncertainty in the thermopower measurements is within +0.5 ~V K-1.

3. Results and discussion

From the linearized Boltzmann equation the following expressions for the thermopower and electrical resistivity can be obtained [3, 4]: 00

0

J e, r)(e - . ) ( S(T) =

/oe) de

1 "o

lelr

(2)

J ,o(e, 73(- /oe) de 0

where

e2

~ I(T, k)v

o,(e, 73- 12 3 h _

a4

(3)

f0 is the Fermi-Dirac distribution function and l(T, k), v and ~ are the mean free path, velocity and electrochemical potential of the electrons respectively. The integral in eqn. (3) is taken over a surface of constant energy, e(k) = constant in the k space. Equations (1)-(3) are of a general nature and can be applied to a description of the electrical conductivity and thermopower of conductors irrespective of the degeneracy of the electron gas. The area of their validity is limited by the boundaries of applicability of the linearized Boltzmann equation. For metals this approximation may be considered to be fully substantiated since, for example, it leads to Ohm's law for electrical conductivity which is confirmed experimentally with high precision. For metals one usually employs a further approximation associated with the fact that the electron gas in metals is strongly degenerate. In this case the function -af°/ ae has a sharp peak at e=/~, the integrand in (1) and (2) being non-zero in a narrow region (compared with Ev) near the Fermi energy. Expanding the function o~(e, T) into a Taylor series in the vicinity of e = ~ and restricting oneself to the first non-vanishing terms, one can readily obtain the following expressions for the electrical conductivity and thermopower [3, 4]: o(T) = ¢o(1*, T) S(T)---

31el T

(1 o) ~

,=~,

(4) (5)

Equation (4) leads to the well-known Bloch-Griineisen law describing the electrical resistivity

due to interactions of conduction electrons with thermally excited phonons. The Bloch-Griineisen law predicts a linear p(73 dependence for temperatures above the Debye temperature 0D. Equation (5) is called Mott's expression for the thermopower. According to this formula the thermopower at high temperatures should also be a linear function of temperature. However, the experimental data suggest that neither the electrical resistivity nor the thermopower follow the above-mentioned simple laws. We believe that the complex behaviour of these properties at high temperatures is intimately related to the structure of the DOS in the vicinity of the Fermi energy with a characteristic energy scale of 0.1-0.5 eV. The existence of such features is supported by the results of band structure calculations for metals [5]. In this case the approximation of strong degeneracy which is used to derive eqns. (4) and (5) cannot be considered as substantiated, particularly at high temperatures where the thermal energies of the carriers are comparable with the scale of the energy spectrum. This implies that the high temperature behaviour of the thermopower and electrical resistivity should be analysed by considering the more general equations (1)-(3). However, to use eqns. (1)-(3) for the interpretation of the data, one has to know the function ~o(e, 73, which includes all details of the DOS and conduction electron interactions with other excitations in a metal. Obviously, one can hardly calculate this function from first principles in a realistic model of the electronic spectrum and scattering mechanisms. This is why it seems to us to be sufficiently promising to use a phenomenological approach, by means of which it is possible to get interesting qualitative results. Let us assume that the function ~o(e, 7) can be represented by ~o(E, 73= A(T)~b(e)

(6)

This approximation is based on two assumptions: (i) the electronic structure does not vary substantially in the temperature range under consideration; (ii) the nature of the scattering processes does not change with temperature. There are different ways in which the temperature can influence the electronic structure. Thermal expansion of the lattice leads to a decrease in the bandwidth. On the other hand, thermal vibrations have in some sense an opposite effect, smoothing the structure as a result of thermal disorder. As is known from ref. 6, both effects are small and to some extent they compensate each other. A further effect is the change in the Fermi level position with temperature. To keep the conduction electron concentration constant, the following equation must be fulfilled:

A. T. Burkov et al. I Transport properties

n=

s 0

N(e)fO(e, 7’) de=constant

(7)

This last effect depends on the precise structure of the DOS near E, and, as will be shown later, is negligible in our alloys. The second assumption implies that no new scattering mechanism arises in the considered temperature range. For electron-phonon scattering this assumption is fulfilled above the Debye temperature. In this case the scattering probability Pkrk can be represented in the following way [7]:

where Qk and J/k’ are the initial and final conduction electron wavefunctions respectively, V is the ionic potential and p is the mean value of the square of the ionic displacements. It is clear that 3 is a function of temperature while the quantity in parentheses depends only on electronic energies. Within the scope of this discussion we conclude that eqn. (6) is valid for non-magnetic materials at least above the Debye temperature 0,. Under these conditions A(Y) depends only on the strength of the conduction electron scattering, thus determining the magnitude of the electrical resistivity. In contrast A(7) drops out of the temperature-dependent thermopower (eqn (2)). We assume that the function A(T) leads to a linear increase in the electrical resistivity with temperature (Bloch-Griineisen law). The function 4(e) reflects the structure of the DOS and the energy dependence of the scattering processes. Using this form of representation of the function W(E,T), one readily obtains from (1) and (2)

(8)

s S(T),-$0 s

of La,Y,_,AI,

alloys

119

One of the possible ways to solve eqns. (8) and (9) consists of constructing an appropriate function +(E) with a certain number of fitting parameters to be chosen such that eqns. (8) and (9) provide the best fit to the experimental temperature dependences of the thermopower and resistivity. One naturally has to use some physical considerations in order to chaos the most suitable function from the infinite multiplicity of possible functions. We have used this approach in previous papers [l, 81. Here we will consider another method which in some respects is more convenient. Let us assume that the function 4(e) can be approximated by a polynomial series

Substituting this representation and (9), we obtain

of 4(e) into eqns. (8)

(11) with

E--P

x=k,T In the same way one gets

m

(12)

4(4(~-~)( -?f’-‘N de

(9)

m

4(4( -VW

with the integral

de

0

Equations (8) and (9) can be considered as coupled integral equations for the function 4(e)_ Knowing A(r), one can obtain from the temperature behaviour of the thermopower and electrical conductivity an electronenergy-dependent function 4(e) by solving these equations; 4(e) can then be compared with the characteristics of the electronic structure of the respective metal.

if i is odd (13) if i is even (i= 2j) where C .= 2(2”-l- 1)~ Bj from ref. 9 and the Bj are Bernoull? numbers. Thuy

120

A. T. Burkov et al. / Transport properties of LaxYl_xAl 2 alloys ¢o

(14) y~O t~

(15) On the other hand, the integrals in eqns. (14) and (15) can be represented as a combination of experimental data from S(T) and O(T) in the following way: fo ~b(e)( - - ~ e ) d e = o(T0)~T)A(TO)A(T)=r(t)

~b(e)(e- ~) ( 0-~ _ ~ ) k ~ T f~

de=-£~Br(t)S(t ) e

(16)

(17)

where r(t) is the resistivity normalized to To and t = T/To. In order to calculate the temperature variation in the resistivity, we use the following expression for the function 1/A(T) [10]: OD/T

1 = 4R T ' f o z5 dz A(T) 0D5 (e ~-- 1)(1 -- e -z)

(18)

where 0D is the Debye temperature and R includes the temperature-independent electron-phonon coupling constant. Now we can also represent the righthand side of eqns. (16) and (17) as polynomial series: 1

e r(t)S(t)= ~,bit' kB i- o

(19)

m

r(t) = ~,d,t'

(20)

i--O

Combining eqns. (14) and (15) with eqns. (19) and (20) yields g'b t*

Z a t * 2 j + 1 [b" "rch2j+l# "~ ~,r~B"t / "-'2(j + 1) -- -/ - ~ i j--O i~O

(21)

m

~azj(kB T)2/Cz~= Zdit i j=o

(22)

i=o

It follows from eqns. (21) and (22) that

d2j a2~= (kBToyJC~j bE/" + a

(23) (24)

azi +1= (kBTo)ZJ+aC2o+l) It can be seen from eqns. (21) and (22) that r(t) should be expanded only in an even polynomial series

of temperature while the odd series should represent the (e/kB)S(t)r(t) dependence. The even and odd coefficients for the expansion of ~b(e) are given by eqns. (23) and (24) respectively. In this sense S(T) and p(T) form a minimal set of transport coefficients allowing us to determine the function ~b(e). Thus, starting from the general expressions for the thermopower and electrical conductivity (eqns. (1) and (2)), which follow from the linearized Boltzmann equation, transformations (23) and (24) are obtained. By means of these transformations the microscopic function ~b(e) (eqn. (10)) can be derived from the experimental temperature dependences of the macroscopic properties thermopower and electrical conductivity. In the following, the reliability and sensitivity of this procedure are demonstrated by considering few simple examples. Figure 1(a) shows linear dependences of both the electrical resistivity and thermopower as a function of temperature. Applying the above-described procedure, a linear function ~b(e) is obtained as presented in Fig. l(b). While the electrical resistivity is proportional to the mean value of the function ~b(e) in the vicinity of the Fermi level, the slope of this function determines both the sign and absolute value of the temperaturedependent thermopower. For this particular case Mott's formula gives a correct description of S(T); p(T) is in agreement with the Bloch-Griineisen law. Figure 2(a) shows the case where the p(T) dependence has a small curvature. Depending on the sign of the p(T) curvature, the function cb(e) has a minimum or maximum in the vicinity of the Fermi energy (Fig. 2(b)). The former case is thus responsible for a resistivity saturation at high temperatures. However, there exist a variety of models which also account for the saturation tendencies observed in most metallic materials. In particular, this resistivity saturation becomes more pronouced for highly resistive materials; this is known as Mooij's rule [11]. In spite of much theoretical effort (see e.g. refs. 12-14), there is no generally accepted theory which explains the saturation. Figure 3(a) represents a situation with a linear p(T) dependence but a non-linear S(T) behaviour. The ~b(e) function (Fig. 3(b)) also reveals a strong nonlinearity. Despite the strong non-linearity of ~b(e), it follows from eqn. (8) that all odd ~b(e) functions yield a linear behaviour of the electrical resistivity. The model examples presented indicate that ~b(e) can be deduced in a reliable manner in the approximate energy range +(3-5) kBTm~x around the Fermi energy, where Tmax is the upper boundary of the temperature range covered by measurements. Now we can apply this procedure to experimental thermopower and electrical resistivity results. The temperature-dependent resistivity and thermopower for various LaxYl_~kl2 alloys are displayed in Figs. 4 and

A . T. B u r k o v e t al. / Transport properties o f L a . Y ~ _ . A I 2 alloys

120

I

I

I

1.4

I

10

(a)

I

I

(b)

. o ...........

100

.(:1............(I ..0 .......... .....0"" "'" ,.0" ..... ..0 .........

80

1.2

j l

%"

/ • ! 1 . /

V

%.

E o

I

121

0

60

fi-<

1.(:]

/. ~

.....-.....

>

40

, / / ""°"'°"-O-o

0.8

o9

20

>

-10

0 0

t 200

I 400

I 600

I 800

0.6 -0.2

1000

0.0

0.2

[eV]

t e m p e r a t u r e T [K]

Fig. 1. (a) Schematic temperature-dependent resistivity P and thermopower S with (b) resulting q~(e) function. 120

I

I

I

0

I

(a) 100

O~

1.4

I

%-

..............

(b)

//•:"

0~0

1.2

//O/" ..m''"

f

80

I

~

•

.,.,.O"". i ~" -5

0

60

~-"

40

..,I/~~=;'" •

1.0

"~'o~__~

i 2oo

i 4oo

I 6oo

a soo

0.6

1000

t e m p e r a t u r e T [K]

//' .. J""

-10

0

',.....

//

CO 0 . 8

20

0

. ,./""" /." //"'

I -0.2

t 0.0

E 0.2

[eV]

Fig. 2. (a) Schematic temperature-dependent resistivity p and thermopower S with (b) resulting ~b(E) function.

5. Both p(T) and, in particular, S(T) reveal a systematic variation with respect to the composition of the alloys. The variation in S(T) has much in common with the data throughout the REAl2 series [1]. This implies that the impurity contribution to the thermopower has no specific temperature dependence or is comparatively small. More detailed analysis of the data is started with pure LaAlz and YAlz compounds. Applying the abovedescribed procedure, the experimental p(T) and S(T) data were used to obtain ~b(e) functions for both compounds as shown in Figs. 6 and 7. The comparatively

small difference in the 4,(e) functions results in a qualitatively different behaviour of the thermopower (Figs. 4 and 5). The function qS(e) can be compared with available data on the electronic structure of YA12 and LaAI2. It follows from density-of-states calculations [15-17] that the DOS has mostly d-character near the Fermi energy. Therefore it is reasonable to use Mott's s-d model, which suggests that the scattering rate of conduction electrons is proportional to Nd(e), the dband density of states near the Fermi energy [18]. In terms of our model the 1/¢(E) function should be proportional to Figure 6 shows the DOS of LaAlz

Nd(e).

A. T. Burkov et a L / Transport properties of LaxYl_xAl2 alloys

122

I

I

I

I

(a)

1.2

jD

120

(b)

10

j°-

Oj °

oJ

8

oY 80

o

y"

o/ y°

y

Y

1.0

oY

Y

o 40

Y

/

i ~Q

v

<

0

~

Y v

v~

I

0

I

200

I

400

I

600

800

0.8 -0.6

1000

0.0

T [K]

temperature

O.

e - l z [eV]

Fig. 3. (a) Schematic temperature-dependent resistivity p and thermopower S with (b) resulting qb(E) function. 15

~

~

v

10

~

'

v,vvvv v vvVV

vvV

u [] []

.

[]

,, •

•

u

o [3

>.

.* **

A" "

000

o°

oo°°

~oo-

o°

oO~

O00 00

<~oo

oooo

oOO°

000

0 0O0

oo

O

o°

5

oo°

0

0 •

000~

I 200

i

I

I

400

600

800

temperature

1000

T [K]

e22,~- .vvouou

,.o222x ...:~uuu

,~- . . - ~

50

^O,:A v ~

v "l'J

.<

:

100

o?t~,,-..'E,o'~

::- .-y

0~)

^~0000(~ AO00 V - VVVV

o?~x". - ~

OAA vV~r~

0'3 0

oo'~

oS

oo °

oo

AA

oo'.-,

100

^o

,,,,-

r~O0000

AA~'

...

,#

Sy°

150

000 ~

..."

.,"

~

O00no vv,V'~vvv'v vvv'~l~wv

v

u°

0 O

~

0 ~rl

I

0

200

I 400

[] • • o o I 600

temperature

LaM e Lao.9Yo.iA12 Lao.TYo.aA12 Lao.syo.4A1e Lao.sYo.6A1 e I 800

0

1000

T [K]

Fig. 4. Temperature-dependent electrical resistivity p and thermopower S data for various La~YI_~AI2 compounds (x= 1, 0.9, 0.7, 0.6 and 0.5). Solid curves represent calculated p(T) and S(T) dependences (see text).

as obtained in ref. 15 and the 1/4,(e) function, which for convenience is normalized to the value of the DOS at the Fermi level. Rather good agreement is seen with respect to the general shape of the curves and to the position of the maxima relative to the Fermi level. The DOS and 1/4b(e) function for YA12 are compared in Fig. 7. The agreement between 1/~b(E) and N ( e ) taken from ref. 16 is rather poor, but the DOS obtained by Nowotny [17] fits quite well to 1/qb(e). Moreover, assuming that the total number of states under the peaks in the DOS is the same for LaA12 and YA12, the relative

magnitude of the lAb(e) functions can be estimated and compared with the ratio of the experimental 3' values obtained from low temperature specific heat measurements [19]. Experimentally, y L ~ 1 2 / y v , , ~ = 2 , while a value of 1.8 follows from a comparison of the ~b(E) functions. In this treatment for getting the ~b(E) function, one effect is neglected which can have a considerable influence on the temperature variation in transport properties: the chemical potential changes its value, since the Fermi distribution function is smeared out with

A. T. Burkov et at / Transport properties of LGY~_~Alz alloys [

l

I

120

I

I

123

I

I

•V

'10

• Lao.4Yo.6A1 e •

•• •v • •

•V•

u___a

v vv

v v v~

u D []

v•

Vv

Vv

D~ 80

uuu

O[3[]

[3[3 AA

[3[3[3 A A

~

I

I

200

•v

DDADA A 0[~A A 013A 0AA 122~"

12~

A~

40

[3[3[3 " od~ . A A ^o"~-(a XJ

4O

E ¢9

-

i ~ mm

0 0 / OO.~

LaxY 1_xAl2

•"

::pc'Od'~r" I

400

VvV v

0[]A A

¢:~j~

,co

-5

.-"

,<>~

.-"

2N[31313[]oo[3[3°°[]

_

80

oDD °

VV V v

\

~,. •, ~\v

DC30

Vv v

Lao.aYo.7A1 e • Lao.eYo.sAI2 o Lao. IYo.oA1 e o YA1 e []

M

VV~

V• v

600

I

800

I

1000

0

200

I

I

400

600

temperature

temperature T [K]

I

800

1000

T [K]

Fig. 5. T e m p e r a t u r e - d e p e n d e n t electrical resistivity p and t h e r m o p o w e r S data for various LaxYl_xA12 c o m p o u n d s (x=0.4, 0.3, 0.2, 0.1 and 0). Solid curves represent calculated p(T) and S(T) d e p e n d e n c e s (see text). 35

80

I

I

I

I

........ c a l c u l a t e d DOS [12] ..... c a l c u l a t e d DOS [11] ~, DOS prop. 1 / ~ ( e ) 6 0

;>

/

,

Q)

0'q)]

203250

/'//////.////////,./,.,/" \.....

>

~Al

2

4O

In t__.a

%v

z

¢~ r./] i---i

15

v~~

10

.. ,' ..""

".,\

'

20

Z

...... , :. . . . . . . .

5

I

I

I

I

I

-0.2

-0.1

0.0

0.1

0.2

[eV] Fig. 6. 1/4~(e) function for LaAI2 and calculated density of states

0

I

-0.2

I

-0.1

-........

:-.:~: ...........:=.::.:-.:?.:~[.~

I

I

I

0.0

0.1

0.2

e - E F [eV]

[111.

Fig. 7. 1/~b(E) function for YAI 2 and calculated density of states [12, 13].

increasing temperature. To check the importance of this effect, calculations of p(7) and S(T) have been performed according to eqns. (8) and (9) using ~b(e) functions deduced from experimental data. Assuming that the DOS near the Fermi energy is proportional to 1/~b(e), the position of the Fermi energy can be determined at any given temperature by eqn. (7). The results of the calculations for LaA12 and YAI2 are shown in Figs. 4 and 5 as solid curves. The shift in the chemical potential does not exceed a few per cent of the DOS

peak width and thus has only a minor effect on the temperature-dependent thermopower and resistivity. In considering alloys, we have to include the impurity scattering of the conduction electrons. The total "spectral resistivity" of an alloy, p(e, T ) = 1/o)(e, T), can be represented as a sum of the phonon resistivity Pph(e-,T)=l/O)ph(G T) and the impurity contribution Pimp(E) = 1/09imp(e), which should be independent of temperature:

A. I". Burkov et al. / Transport properties of LaxYI_~AI2 alloys

124

1

1

T)

1

1

- a~ro(T) +Pph(7)

7) = O)ph(E,

-I- O)irnp(E)

7)= =

A(T)~b(ff)e°imP(ff)

+

(26)

It is difficult to analyse this expression in the general case. Although standard transport analysis of metals yields quite different energy dependences for phonon and impurity scattering, we consider an approximation with the same energy dependence for both processes. This assumption is supported by the fact that the DOS near the Fermi energy has mostly dcharacter, exhibiting a strong energy dependence. Since the transition probability for any type of scattering processes is proportional to the density of the final states, i.e. d-states for s-d scattering, it is natural to assume that the d-DOS variation with energy mainly determines the energy dependence of each scattering process. The spectral conductivity caused by impurity scattering is therefore given by

6Oimp(E) =a¢(e)

(27)

where, according to the definition of ¢(e) (eqn. (6)), 1/a is the residual resistivity of the alloy. The total spectral conductivity follows as

A(7)a ~o(e,T) ~b(e) A(7) +a

1

(25)

(28)

= Oo

+ o

(3o)

h(n

Equation (30) demonstrates that the impurity contribution to the total resistivity becomes temperature dependent. Figure 8 presents the residual resistivity and the phonon contribution to the total resistivity as functions of the alloy composition. The residual resistivity shows a typical behaviour for alloys: in a first approximation it follows the well-known Kurnakov-Nordheim law. In the low temperature region the phonon contribution to the resistivity is nearly a linear function of the composition, while at higher temperatures noticeable deviations from a linear dependence appear. This is in agreement with the conclusion that the impurity resistivity depends on the temperature. The composition dependences of the thermopower at 100, 300 and 650 K are presented in Fig. 9. At 650 K an almost linear dependence on the composition is found, implying that the impurity scattering has only a minor effect on the thermopower in this temperature region. There is, however, a noticeable deviation from linearity in the composition dependence of the thermopower at 100 K, where the impurity contribution dominates the total resistivity for the alloys. Figure 10 shows the composition dependences of the peak width and maximum position of the 1/¢(e) function. Both quantities have been evaluated by approximating ~b(e) as a lorentzian function. It is seen from this figure that the dependences of both parameters are non-

=

I

The factor A(7)a/[A(7)+a] vanishes in eqn. (9) for the thermopower. This indicates that impurity scattering does not contribute directly to the high temperature thermopower. From eqn. (28) the electrical conductivity follows as

v

100

I

I

I

p(650K)-p(4.2K)

• p(300K)-p(4.2K) A P0(4.ZK)

t /

i

1 t

I

8O / I O

a(7)= A(7)+a

~b(e) - -~e de

~ /

~60

~ - I

v

x

LaxY l_xAlz

0

Ck

_

A(T)__________~ao(7) a

(29)

40

.~IAr

•

A

with 20

.I / \

0 0

In terms of the resistivity p(7) we have A(T) + a

p(T)= A(7)a~o(7)

I

0.0

0.2

I

0.4

fraction

I

0.6

I

0.8

.0

of La x

Fig. 8. Composition-dependent residual resistivity P0 and phonon resistivity Pph at 300 and 650 K for LaxYl_xAl2.

A. T. Burkov et al. / Transport properties of La,¥1_~AI2 alloys 15

I

[

I

10

/ 0 / /

counted for completely by the change in the electronic structure due to alloying. Rather, impurity scattering becomes important.

/

4. Summary

/

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The thermopower and electrical resistivity of quasibinary alloys La, YI_,~A12 have been measured in the temperature range from 4.2 up to about 1000 K. Both the thermopower and electrical resistivity are non-linear in the temperature region above room temperature, which is in contradiction with Mott's formula for the thermopower and with the Bloch-GriJneisen law for the electrical resistivity. The non-linearity in S(T) and p(T) is attributed to the influence of the electronic band structure. The discussion is based on the linearized Boltzmann transport equation and takes into account features of the density-of-states function near the Fermi level. An energy-dependent function ~b(e) is calculated from experimental S(T) and p(T) data. For LaA12 and YA12 the function ~b(e) has been compared with available results of DOS calculations. It follows from this comparison that the DOS and 1/4,(E) show similar energy dependences. This agrees with the assumption that the observed non-linearity in S(T) and p(T) is closely related to density-of-states features near the Fermi level. The effects of impurity scattering and the change in the electronic structure due to alloying have also been discussed. The impurity contribution to the electrical resistivity is temperature dependent at high temperatures. This can be interpreted in the scope of the above-mentioned model by assuming that the electronimpurity and electron-phonon scattering processes have similar energy dependences

Acknowledgments

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of La x

Fig. 10. Position of maximum of N(E) = 1/~b(¢) and its width for La, Y t _ ~ l 2 alloys. A positive or negative shift corresponds to the position of the maximum of 1/~b(E) above or below the chemical potential respectively.

This work was supported by the Austrian Science Foundation under project 7994-TEC. We also thank A. Yu. Zyuzin for helpful discussions.

References linear, with a stronger variation on the LaAI2 side. This clearly demonstrates that simple approximations such as the rigid band model are barely applicable for description of the electronic structure of these alloys. On the other hand, the variation in the compositiondependent thermopower at 100 K (Fig. 9) appears to be very symmetrical and has much in common with the composition-dependent behaviour of the residual resistivity. This particular behaviour cannot be ac-

1 A. T. Burkov, E. 176 ( 1 9 9 2 ) 2 6 3 . 2 H . R . Kirchmayr

Gratz and M . V . Vedernikov, Physica B,

and C. A . P o l d y , in K. A . G s c h n e i d n e r J r . and L. Eyring (eds.), Handbook on the Physics and Chemistry of Rare Earths, North-Holland, Amsterdam, 1 9 7 9 , V o l . 2, p.

69. 3 J . M . Ziman, Electrons and Phonons, Clarendon, Oxford, 1 9 7 2 . 4 R . D . Barnard, Thermoelectricity in Metals and Alloys, Taylor and Francis, London, 1 9 7 2 .

126

A. T. Burkov et al. / Transport properties of La~YI-xAI2 alloys

5 D. A. Papaconstantopoulos, Handbook of the Band Structure of Elemental Solids, Plenum, London, 1986. 6 W. Jank, Ch. Hausleitner and J. Hafner, J. Phys. Condens. Matter, 3 (1991) 4477. 7 J. S. Dugdale, The Electrical Properties of Metals and Alloys, Edward Arnold, London, 1977. 8 A. T. Burkov, M. V. Vedernikov and E. Gratz, Solid State Commun., 67 (1988) 1109. 9 H. B. Dwight, Tables of Integrals and Other Mathematical Data, MacMillan, New York, 1961. 10 P. L. Rossiter, The Electrical Resistivity of Metals and Alloys, Cambridge University Press, Cambridge, 1987. 11 J. H. Mooij, Phys. Status Solidi A, 17 (1973) 521.

12 F. Brouers and M. Brauwers, J. Phys. (Lett.), 36 (1975) L17. 13 P.B. Allen, in H. Suhl and M. B. Maple (eds.), Superconductivity in d- and f-Band Metals, Academic, New York, 1980, p. 291. 14 K. Elk, J. Richter and V. Cristoph, Phys. Status Solidi B, 89 (1978) 537. 15 A. Hasegawa and A. Yanase, J. Phys. F: Met. Phys., 10 (1980) 847. 16 A. Hasegawa and A. Yanase, J. Phys. F: Met. Phys., 10 (1980) 2207. 17 H. Nowotny, personal communication, 1991. 18 N. F. Mott and H. Jones, The Theory of the Properties of Metals and Alloys, Oxford University Press, Oxford, 1936. 19 R. E. Hungsberg and K. A. Gschneidner Jr., J. Phys. Chem. Solids, 33 (1972) 401.