Chemical short-range order and the Meyer-Neldel rule for liquid alloys: AlCa and GaAlCa

Article abstract of DOI:10.1088/0953-8984/9/7/006

We have measured the electrical resistivity, ρ, its specific temperature dependence, α = (1/ρ)dρ/dT, and the thermopower, S, of two series of ternary liquid alloys: Ga_xAl_67-xCa₃₃ and Ga_xAl_50-xCa₅₀. We also provide new analysis for the binary liquid alloy AlCa. We do not see the unusually large values for S that were found earlier for amorphous solid ternary alloys of the approximate composition Ga_xAl_yCa₆₀. We do find that, while chemical short-range order (CSRO) appears to occur in the liquid binary alloy Al₂Ca, CSRO is apparently destroyed by substitution of one Ga atom for one Al per complex: Ga₁Al₁Ca₁. CSRO may exist in the liquid alloy Ga₂Ca. And we find that the activated conductivities of these ternary liquid alloys (and also of liquid AlCa) are consistent with the Meyer-Neldel rule (MNR), extending the range of applicability of the MNR to systems with activation energies about an order of magnitude smaller than previously observed. These results appear to rule out two physical models as universal bases for the MNR, but are consistent with one based on a hopping conductivity whose characteristic energy is that of a polaron shift.

Full text of DOI:10.1088/0953-8984/9/7/006

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J. Phys.: Condens. Matter 9 (1997) 1407–1415. Printed in the UK
PII: S0953-8984(97)75702-3
Chemical short-range order and the Meyer–Neldel rule for
liquid alloys: AlCa and GaAlCa
D You, H S Schnyders† and J B Van Zytveld‡
Physics Department, Calvin College, Grand Rapids, MI 49546, USA
Received 25 June 1996, in final form 7 October 1996
Abstract. We have measured the electrical resistivity, ρ, its specific temperature depend-
ence, α ≡ (1/ρ) dρ/dT , and the thermopower, S, of two series of ternary liquid alloys:
GaxAl67−xCa33 and GaxAl50−xCa50. We also provide new analysis for the binary liquid alloy
AlCa. We do not see the unusually large values for S that were found earlier for amorphous solid
ternary alloys of the approximate composition GaxAlyCa60. We do find that, while chemical
short-range order (CSRO) appears to occur in the liquid binary alloy Al2Ca, CSRO is apparently
destroyed by substitution of one Ga atom for one Al per complex: Ga1Al1Ca1. CSRO may
exist in the liquid alloy Ga2Ca. And we find that the activated conductivities of these ternary
liquid alloys (and also of liquid AlCa) are consistent with the Meyer–Neldel rule (MNR),
extending the range of applicability of the MNR to systems with activation energies about
an order of magnitude smaller than previously observed. These results appear to rule out two
physical models as universal bases for the MNR, but are consistent with one based on a hopping
conductivity whose characteristic energy is that of a polaron shift.
1. Introduction and theory
Recent measurements [1] of the electrical resistivity, ρ, its specific temperature dependence,
α (α ≡ (1/ρ) dρ/dT ), and the thermoelectric power, S, of the liquid alloys AlCa and
AlBa provide strong evidence for chemical short-range order (CSRO) in these alloys at the
composition Al2X, where X = Ca, Ba. This observation is unusual in that the constituent
elements in these alloys are all non-transition metals; theoretical calculations, on amorphous
AlCa [2, 3], however, imply that the electronic structure of this alloy becomes quite non-
free-electron-like near EF , and that there is evidence of strong Al–Ca bond formation of
electronic character at the concentration Al2Ca [3]. It is not surprising that liquid AlCa
should have similar characteristics. The rather sharp negative minimum observed in the
concentration dependence of α for liquid AlCa at the composition Al2Ca is therefore a
result of thermal activation of electrons from these bonds into the conduction band.
It is interesting to consider what would happen to the electronic properties of liquid
Al2Ca if isoelectronic Ga were substituted for some or all of the Al in this alloy. For this
purpose, we note that both pure liquid Al and pure liquid Ga are trivalent and that their
atomic diameters are very nearly the same. (The nearest-neighbour distances in the pure
˚
liquids, as obtained by Fourier inversion of their x-ray structure factors, are: r(Al) = 2.79 A
˚
and r(Ga) = 2.77 A [5].) The free-electron Fermi energies, EF , of these pure liquids are also
approximately equal: EF (Al) = 0.381 au and EF (Ga) = 0.365 au. Pure liquid Ca provides
† Also at: Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA.
‡ Present address: M J Murdock Charitable Trust, PO Box 1618, Vancouver, WA 98668, USA.
c
0953-8984/97/071407+09$19.50 ꢀ 1997 IOP Publishing Ltd
1407

1408
D You et al
˚
a contrast to these two very similar metals: r(Ca) = 3.75 A, and EF (Ca) = 0.160 au.
We expect that, if Ga is substituted for all of the Al in the liquid alloy Al2Ca, relatively
little change may occur either in the physical structure or the electronic properties of the
alloy. In particular, since liquid Al2Ca shows characteristics of CSRO, perhaps Ga2Ca will
as well. It is not so clear, however, what to expect when perhaps half of the Al atoms have
on average been replaced by Ga atoms, as in GaAlCa. We report these measurements in
the present paper.
It has also been reported [4] that, for amorphous Ga20Al20Ca60 (and, to a somewhat
lesser extent for Ga10Al30Ca60), the measured thermopower, S, is unusually large, having
a magnitude of about 10 µV ◦C−1; this was hard to understand considering that the
components are all simple non-transition metals. In the present study we look for an
unusual magnitude for S in the liquid ternary alloy at nearly the same composition.
In the last decade, in a different but yet related study, there has been an interesting
discussion of the possible physical origins of the so-called Meyer–Neldel rule (MNR). This
was first identified in 1937 by Meyer and Neldel [6]; it is now clear that this empirical rule
applies to many properties that show thermal activation in quite different systems; these
include crystalline, amorphous, and liquid semiconductors. Applied to a thermally activated
electrical conductivity, σ, where
ꢀ
ꢁ
E
σ = σ0 exp −
(1)
kT
the MNR notes that the following relation exists between σ0 and E for a class of similar
materials:
ꢀ
ꢁ
E
ε
σ0 = σ00 exp
.
(2)
Here E is a measure of the activation energy, T is absolute temperature, and σ00 and ε are
constants for the given class of materials. (We note with Fortner et al [7] that the value
of E obtained from fitting experimental data to equation (1) is not necessarily equal to the
true activation energy.) Typi−ca1lly, for solid or liquid semiconductors, with dσ/dT > 0
(or dρ/dT < 0), σ < 500  cm−1. Interestingly, for liquid semiconductors, values of
ε fall into three ranges for eight different alloy systems: (1) 30 meV < ε < 40 meV;
(2) 90 meV < ε < 125 meV; and (3) ε ' 180 meV [7].
One rather common suggestion for a physical basis of the MNR involves the statistical
shift of the Fermi energy, EF (T ), with temperature [8]. As noted by Fortner et al [7], this
explanation requires that kT /ε < 1 to avoid the non-physical result of a negative activation
energy, E. Fortner et al [7] show that several liquid semiconductors have kT /ε > 1, ruling
this out as a universal explanation of the MNR.
Another possible explanation of the MNR is based on multiphonon hopping, for which,
as the number, N, increases (where N is the number of phonons that are involved in
the promotion of a single electron to higher energy, E), the number of excitation paths
increases exponentially, and the MNR results [9, 10]. For this, in terms of our parameters,
we have Nε ' E, with N ꢁ 1. This implies that ε ꢂ E for MNR to be observed for
a given system. Fortner et al [7] also assume a variant of this hopping conductivity, but
one for which the characteristic energy, w, is that of a polaron; a system that exhibits a
strong polaron effect, appropriate for a flexible structure, may also exhibit MNR effects (for
w > E/2) even at high temperatures. Interestingly, the characteristics of σ(T ) for three of
the liquid alloy systems reported here allow us to test these explanations for the MNR in a
region of parameter space not previously explored.

Chemical short-range order in AlCa and GaAlCa
1409
Figure 1. Measurements of ρ, α (α ≡ (1/ρ) dρ/dT ), and S for the liquid alloy AlCa as
a function of Ca concentration on the isotherm T = 1079 ◦C (from reference [1]). The
vertical arrows indicate the range of values taken by these parameters for the ternary alloy
GaxAl67−xCa33
.
2. Experimental method
The base material of which the alloys were made was obtained from the Atomergic
Chemetals Corporation (99.999% Al), and from Aesar/Johnson–Matthey (99.99% Ca and
99.999% Ga). Before the Ca was used, however, it was encapsulated in Ta and baked at
high temperature and in high vacuum for three days to remove the hydrogen that normally
contaminates the alkaline earths. The material was then cut and the samples assembled
under an argon atmosphere in an evacuable glove box. Measurements were then made on
alloys held in high-density Al2O3 tubes; contact to the liquid sample was made via holes

1410
D You et al
drilled into the tubes and sealed with graphite plugs. A molybdenum stirring wire permitted
vigorous agitation of each alloy to ensure homogeneity. In other respects, the measurement
techniques were similar to those reported earlier [11].
3. Results
Experimental results for ρ, α, and S for the liquid binary alloy AlCa are shown in figure 1.
These were reported and discussed earlier [1], but the figure is repeated here because this
forms the base system on which the ternary alloys are formed. (These data are also used
in a test of the MNR.) The deep and sharp minimum in α at about 33 at.% Ca provides
partial evidence for the CSRO at the concentration Al2Ca which has also been described
[1]. The vertical arrows in the figure show the range of values which the parameters take
on as Ga replaces Al in the alloy while the Ca content is held at 33 at.%. (Some data for
GaxAl50−xCa50 have also been obtained, but, because the changes are small, the range of
values is not shown in figure 1.)
Measurements of S and ρ for the ternary liquid alloy systems GaxAl67−xCa33 and
◦
GaxAl50−xCa50 are shown in figure 2. All data are for the 1079 C (1352 K) isotherm, as
are those in figures 1 and 3. The variation of S with the Ga concentration, x, is monotonic
and featureless; dρ/dx, however, is negative for small x for both systems, and appears to
rise to zero for GaxAl67−xCa33 when about one-third of the Al atoms have been replaced
by Ga.
Measurements of α(x) for the same two liquid ternary alloy systems are shown in
figure 3. Again, the variation of α(x) for the alloy GaxAl50−xCa50 is probably small,
but α(x) for GaxAl67−xCa33 shows interesting structure, with α(x) becoming less negative
rapidly for small x, displaying a maximum at about 45 at.% Ga, followed by a shallow
minimum at about 55 at.% Ga. In each case in figures 2 and 3, vertical broken lines indicate
concentrations at which all Al atoms in that alloy have been replaced by Ga.
4. Discussion
We note immediately that we see no unusual magnitude for S at any concentration for either
∼
liquid ternary alloy system; in particular, S 0 for the alloy Ga Al Ca , a concentration
=
25 25
50
close to that for which the amorphous alloy showed an unusually large magnitude [4] (see
figure 2). Moreover, we also note that while dS/dx has the same sign for the ternary liquid
and the amorphous solid alloys, its magnitude for the liquid alloys is about an order of
magnitude smaller. dρ/dx, for small x, differs between these solid and liquid alloys both in
∼
∼
sign and in magnitude: dρ/dx −0.3 for liquid Ga Al
Ca , whereas dρ/dx +5.5
=
=
x
50−x
50
for amorphous GaxAl40−xCa60, where the units are µ cm/at.% [4]. And, significantly,
although α and dα/dx are of about the same magnitude for these same solid and liquid
systems, the dα/dx are of opposite sign. We offer no explanation for the difference between
our results and those for the amorphous alloy other than to note that the electronic properties
of amorphous solids often reflect the fabrication methods employed. (Naugle et al [4] also
comment on expected intrinsic differences in short-range order between liquid and glassy
alloys.)
Perhaps of greater interest is what appears to happen to the CSRO in liquid Al2Ca
as Ga replaces the Al in this alloy. ρ(x) initially decreases with increasing Ga content,
and α(x) becomes less negative rapidly as Ga is first added. Both of these (especially
the α(x) dependence) suggest that the Al2Ca CSRO association is disrupted rapidly as Ga

Chemical short-range order in AlCa and GaAlCa
1411
Figure 2. Measured values of S and ρ for two series of ternary liquid alloys (at T = 1079 ◦C):
◦
: GaxAl67−xCa33; and ꢀ: GaxAl50−xGa50. (The dashed lines show possible extrapolations
beyond the data.) M: from reference [1]. The vertical lines show the Ga concentration, for each
alloy, at which the Al concentration goes to zero.
is added to the binary alloy. This implies that no complexes of the type Ga1Al1Ca1 are
electronically preferred, and none form, even temporarily. We also note that, if one does a
simple statistical calculation in which Ga atoms randomly replace Al atoms in an assemblage
of Al2Ca ‘molecular’ complexes, 50% of these Al2Ca complexes will have at least one Al
replaced by Ga when the Ga concentration reaches 19 at.% (Ga19Al48Ca33), and 80% will

1412
D You et al
Figure 3. Measured values of α (α ≡ (1/ρ) dρ/dT ) for the two series of ternary liquid
alloys (at T = 1079 ◦C):
◦
: GaxAl67−xCa33; ꢀ: GaxAl50−xCa50. (Dashed lines are poss-
ible extrapolations—see the text.) M: from reference [1]. The vertical lines show the Ga
concentration, for each alloy, at which the Al concentration goes to zero.
have at least one Al atom replaced by Ga at 33 at.% Ga (Ga33Al33Ca33). This calculation,
coupled with the observation that α(x) has increased to nearly zero by x = 33 at.% Ga
(see figure 3), also suggests that no complex of the type Ga1Al1Ca1 is favoured. On the
other hand, however, the ultimate decrease in α(x) and subsequent minimum (for large x)
at least suggests that CSRO complexes of the type Ga2Ca do form, as do those of the type
Al2Ca. The formation of these Ga2Ca complexes would hold α(x) low for large x (figure 3
shows α(x) well below the extrapolation from the low-x region (dashed line) up to large x);
moreover, the measured value of α for the binary liquid alloy Ga2Ca (about 2×10−4 ◦
C
−1),
shown in figure 3, is far below the value of perhaps 8 × 10−4 ◦C−1 that one expects for
pure Ga, based on measurements made at lower temperatures [12]. It appears, therefore,
that α(x) may perhaps also be at a minimum at Ga2Ca for the liquid alloy GaxCa100−x, as

Chemical short-range order in AlCa and GaAlCa
1413
Figure 4. A semilogarithmic graph of σ0 versus E for parameters obtained from a fit of the
measured data for the three alloy systems to equation (1).
is the case for Al2Ca; this is also consistent with Ga2Ca complex formation (see figure 1).
It would be interesting to extend the calculations of Hafner and Jaswal [3] to both liquid
Ga2Ca and liquid Ga1Al1Ca1 to look for evidence for bond formation in the former and

1414
D You et al
Table 1. Values of the parameters obtained from a fit of our σ(T ) to the MNR.
Liquid
alloy
E-range
σ0-range
−1cm−1
ε
σ00
(104
−1
(10−2 eV) (104

)
(meV)
ε/E

cm−1
)
kT/ε
AlxCa100−x
1.1–17
GaxAl67−xCa33 0.6–6.2
GaxAl50−xCa50 2.8–4.8
0.6–3.1
0.8–1.5
0.75–0.88
108 ± 13 0.6–10 0.63
119 ± 9 1.9–20 0.74
1.06
0.96
0.90
127
2.6–4.5 0.60
lack of it in the latter.
Finally, as noted in the introduction, our measurements on the liquid alloys AlxCa100−x
,
GaxAl67−xCa33 and GaxAl50−xCa50 allow us to make an interesting new test of the physical
origins of the MNR. We recognize that, because our data for σ(T ) span a range of only
100–200 ◦C at about 1100 ◦C (1373 K), a variety of functional forms could be assumed for
σ(T ); we nonetheless can fit our data for these three alloys to equation (1), and we do this.
A least-squares analysis of ln σ versus 1/T yields a set of straight lines with correlation
coefficients ranging from about 0.6 to greater than 0.95. These coefficients are reflected in
the error bars displayed in figure 4. All of the data for which α < 0 were used. We plot
the values of σ0 versus E that were obtained from these fits on a semi-logarithmic graph,
figure 4, to test for adherence to the MNR. We note from figure 4 that the fit of these data
to the form of equation (2) is reasonably good, and is certainly not worse than it is for
several other systems (see reference [7]). Numerical values of ε and σ00 obtained from this
fitting procedure are shown in table 1. Note that we have only two data points for the alloy
GaxAl50−xCa50, and that the range of values of σ0 and E for this alloy is also small; the
results of the analysis for this alloy are therefore not of the statistical quality that they are
for the other two.
We see (table 1 and figure 4) that not only are these data consistent with the MNR,
but the parameter ε places all three alloys into the same class of materials: those with ε in
the range 90 meV 6 ε 6 125 meV. It is interesting that these liquid alloys yield the same
value of ε as do a class of solid and liquid semiconductors which have values of σ0 and
σ00 one to two orders of magnitude smaller. Moreover, the range of values that we find for
E (a measure of activation energy) is in many cases at least an order of magnitude smaller
that it is for the liquid semiconductors [7], and for other systems that have shown MNR
behaviour. Thus we have apparently extended the range of applicability of the MNR for
the first time to systems that have activated conductivities that are in the metallic range.
(We note that these alloy systems, though metallic, have conductivities that are too low for
the free-electron model to apply.)
The magnitudes which we obtain for ε and E appear to rule out the multiphonon hopping
model of Yelon et al [9, 10] as a universal mechanism, since ε is not substantially less than
E for these systems (see table 1). As was the case for the liquid semiconductors [7], we
find kT/ε ∼ 1 (and perhaps >1 in one case), making it unlikely that the statistical shift
in EF with temperature can be a universal mechanism for the MNR. Only the model of
Fortner et al [7], which posits a hopping conductivity, but whose characteristic energy is
that of a polaron shift, appears to be consistent with our results. And, as also noted by these
authors, the flexibility of a temporary CSRO complex in a liquid such as we have examined
is expected to be quite high, making this mode reasonable. (Fortner et al [7] draw on the
earlier theoretical work of Mott and Davis [13] and Emin [14] in their development of the
MNR via a hopping conductivity.) It is difficult, however, to understand how a hopping
conductivity mechanism may be active in the conductivity regime of our measurements.

Chemical short-range order in AlCa and GaAlCa
Acknowledgment
1415
We are happy to acknowledge a grant from the National Science Foundation (DMR-
9406812) in support of this research.
References
[1] Zuo D, Vos T, Nymeyer H, Reynolds L, Schnyders H S and Van Zytveld J B 1997 J. Non-Cryst. Solids at
press
[2] Nagel S R, Gubler U M, Hague C F, Krieg J, Lapka R, Oelhafen P, Gu¨ntherodt H-J, Evers J, Weiss A,
Moruzzi V L and Williams A R 1982 Phys. Rev. Lett. 49 575
[3] Hafner J and Jaswal S S 1988 Phys. Rev. B 38 7320
[4] Naugle D G, Delgado R, Armbruster H, Tsai C L, Callaway T O, Reynolds D and Moruzzi V L 1986 Phys.
Rev. B 34 8279
[5] Waseda Y 1980 The Structure of Non-Crystalline Materials (New York: McGraw-Hill)
[6] Meyer W and Neldel H 1937 Z. Tech. Phys. 12 588
[7] Fortner J, Karpov V G and Saboungi M-L 1995 Appl. Phys. Lett. 66 997
[8] Popescu C and Stoica T 1992 Phys. Rev. B 46 15 063
[9] Yelon A and Movaghar B 1990 Phys. Rev. Lett. 65 618
[10] Yelon A, Movaghar B and Branz H M 1992 Phys. Rev. B 46 12 244
[11] Walhout M, Haarsma L and Van Zytveld J B 1989 J. Phys.: Condens. Matter 1 2923
[12] Lyon R N (ed) 1952 Liquid Metals Handbook (Washington, DC: US Government Printing Office)
[13] Mott N F and Davis E A 1979 Electronic Processes in Non-Crystalline Materials 2nd edn (Oxford: Clarendon)
[14] Emin D 1975 Adv. Phys. 29 305