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This is to certify that the
dissertation entitled

AN EXPERIMENTAL INVESTIGATION OF THE ELECTRONIC
TRANSPORT PROPERTIES OF MOLTEN PURE METALS AND ALLOYS

presented by

Harold Scot SChnyders

has been accepted towards fulfillment
of the requirements for

Eh. D. degree in _Bh¥sics_

a Major professor

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1m mumps-m4

 

AN EXPERIMENTAL INVESTIGATION OF THE ELECTRONIC TRANSPORT
PROPERTIES OF MOLTEN PURE METALS AND ALLOYS

By

Harold Scot Schnyders

A DISSERTATION

Submitted to
Michigan State University‘
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1997

ABSTRACT

AN EXPERIMENTAL INVESTIGATION OF THE ELECTRONIC TRANSPORT
PROPERTIES OF MOLTEN PURE METALS AND ALLOYS

By

Harold S.Schnyders

A complete description of the nature of chemical ordering and it’s effects on the
conductivity and therrnopower in molten metals has not been developed, in part due to
an incomplete body of experimental evidence. In our studies, we seek to augment and
verify existing electronic transport property data for molten pure metals and alloys, and
to examine our results in light of recent experimental results and theoretical
considerations. Our experiments have revealed a previously unseen peak in the
conductivity in the liquid semiconductor Ag-Te near Ange, comparable to similar
features seen in AgZSe and AgZS, and also possibly associated with the persistence of
local ordering that enhances electronic conduction. We have also measured the metallic
resistivity and therrnopower of the weakly covalent pure liquids silicon and
germanium, and have developed a successful containment technique for the former,
highly corrosive liquid. Finally, we have found low resistivity, but negative temperature
coefficient, in liquid Cu-Ce near the congruently melting solid compound Cu6Ce; further
interpretation with neutron diffraction suggests that crystalline-like short range order
decomposing with temperature leads to this effect. The latter two studies are especially
important in that they demonstrate that weak ordering is not incompatible with metallic

conduction.

ACKNOWLEDGEMENTS

It is a pleasure to recognize the invaluable guidance of Professor John
VanZytveld during the conception and carrying out of these experiments; his persistent
attention, even through great distance, was absolutely essential to the success of our
work. The strong encouragement of the faculty members of Calvin College Physics
Department D. Van Baak, S. Steenwyk, J. Jadrich, S. Haan, R. Griffioen, A.
Kromminga, H. VanTil, M. Walhout, and J. Wolinski also carried this work to
completion. Thanks are also due Professor J. Bass for his leadership of the committee.

The actual experiments would have been impossible without the vast talents of
our machinist, John DeVries. Thanks are also due to Sue Sweetman, science division

secretary, and to the staff of the Calvin Computer Center.

iii

TABLE OF CONTENTS

LIST OF TABLES .................................................... v
LIST OF FIGURES .................................................. vi
INTRODUCTION .................................................... 1
CHAPTER 1
THE ELECTRONIC TRANSPORT PROPERTIES OF SILVER TELLURIDE
A. Review of the Silver Chalcogenides ............................. 7
B. Experimental Details ........................................ 12
C. Results ................................................... 17
D. Discussion ................................................ 23
CHAPTER 2
LIQUID SILICON AND GERMANIUM
A. Introduction ............................................... 29
B. Procedure ................................................. 31
C. Experimental Results and Discussion ........................... 37
CHAPTER 3
LIQUID Cu-RARE EARTH AND Ag-RARE EARTH ALLOYS
A. Introduction ............................................... 47
B. Electronic Transport Results .................................. 50
C. Discussion of Transport Measurements .......................... 54
D. Neutron Diffraction Procedure and Results ....................... 57
E. Reverse Monte Carlo Procedure and Results ..................... 65
F. Discussion of Evidence for Weak Ordering in Liquid Cu-Ce and Ag-Ce 71
CHAPTER4
CONCLUSION ..................................................... 8O
BIBLIOGRAPHY ................................................... 81
APPENDICES ...................................................... 85

iv

LIST OF TABLES

Table 1 - RMC Nearest Neigbor Distributions .............................................................................. 7O

LIST OF FIGURES

Figure 1 - The phase diagrams for Ag-Te and Ag-Se alloys, from [59] ............ 8
Figure 2 - The conductivity of liquid Ag-Se alloys, from Ohno et. a1. [8] .......... 10
Figure 3 - A diagram of the furnace and sample tube ......................... 14

Figure 4 - Current data for the conductivity of liquid Ag-Te at 950 C° - 0;
recent data for liquid Ag-Te at 1000 C°- A , from Ohno eta]. [14];
data for liquid Ag-Se from Ohno et. al. [1 l]- I ...................... 18

Present data (0) and data of Ohno et.al. (A) for the conductivity

of Ag-Te in a narrow range of concentrations around stoichiometry.

An extrapolation of our data and that of Ohno et. al. from outside the
stoichiometric region drawn toward Ange demonstrates the presence

of a peak in conductivity. In the lower half of the figure, the current

peak data is compared to the peak seen by Ohno et.al. [11] in

Ag—Se (V ). The boundary of the immiscible region is clearly marked

to demonstrate that the peak is not a result of phase separation ......... 19

Figure 5

Conductivity of two samples of Ag_-,0Te30 as the temperature boundary
(arrow) of the miscible- immiscible region is crossed (current data for

two independent measurements on two samples of Angejo- . v ), as
compared to the results of Ohno et.al. [10] for Ag.,,,s,,,- I . (The
comparable miscible-immiscible boundary for Angzg, is at 1050°C.)

This demonstrates the apparent stability of liquid Ag-Te at temperatures
and concentrations within the immiscible region. ................... 21

Figure 6

Current results for the thermopower of liquid Ag-Te -. evaluated at

960°C, compared to the data of Ohno et.al [14]-A at 950° C.

Significant differences greater than either experimental error exist in

the concentration range where the peak in the conductivity is seen. . . . 22

Figure 7

The thermopower and conductivity calculated using the Kubo

Figure 8
Greenwood formulas with a, =2ay =1300 0" cm" eV'l ............... 26

Figure 9 -

Figure 10-

Figure 11 -

Figure 12 -

Figure 13 -

Figure 14 -

Figure 15 -

Figure 16 -

Figure 17-

The electronic characteristics of high density graphite. The various

symbols indicate separate experiments done on samples of different

material manufacturer batch number, and on samples of the same batch
number with different thermal histories ........................... 32

Diagram of graphite sample container, support tube, and gradient coil. . . 35

The resistivities of liquids Si (t0p) and Ge. All of the data shown

are our own, the different symbols being separate experiments on

different samples. A random selection of 1/3 of the data accumulated

is displayed, for graphical clarity. The solid line is a fit to all of the

current data, including data points not shown, and the dashed lines are

the results of Sasaki et. al.[39] and Koubaa eta]. [3 8] for Si and Ge,
respectively. ................ _ ............................... 38

The thermopower, S, of liquids Si (tOp) and Ge. The lines and data points
are as described previously in Figure 11 ........................... 39

(Top) The l-dependent parts of the BHS pseudopotential for
Germanium. me and V pm are both pictured, but lie nearly
on top of one another. ......................................... 45

(Bottom) The various parts of the integrand of the Ziman formula

for liquid Si. S(Q) is the structure factor, fi'om Waseda [46], Q is the
scattering vector, k, is the Fermi energy, and Vm and Vvs are the

screened potentials, using the screening function of Ichumaru and Utsumi
[45] and Vashishta and Singwi [44], respectively. ................. ;. . 45

(TOp) Current data for resistivity, p, of Cu-Rare Earth alloys,

evaluated at 950°C for Cu-La (I ) and Cu-Ce ( o ) , and at

1050°C for Ag-Ce ( V ) . Cu-Nd (A ) (Busch et.al. [48]) is

at 1000°C. ............................................... 52

(Bottom) The thermopower, S, of Cu-Ce, Cu-La, and Ag-Ce.

The symbols and temperatures are the same as for Figure 15.

Error bars are calculated from a linear fit to all of the data‘for

each concentration, and unless shown are smaller than the

extent of the point. ........................................... 52

a= 1/p dp/dT for Cu-Ce, Cu-La, and Ag-Ce. Temperatures and
symbols are the same as for Figure 15. ........................... 53

vii

Figure 18 -

Figure 19 -

Figure 20 -

Figure 21 -

Figure 22 -

Figure 23 -

Figure 24 -

Figure 25 -

Figure 26 -

Figure 27 -

Figure 28 -

The phase diagrams of Cu-Ce and Ag-Ce. All of the Cu-RE
compounds considered here have a congruently melting crystalline
compound at Cu6RE (From [59]) . .............................. 5 8

(Top) The structure factor S(Q) -1 of CusCe, measured by neutron
diffraction. The solid line shows the experimental results, the
dashed line is the result of RMC analysis of the experimental data ..... 62

T(r) = r * g(r) of Cu6Ce. The pair correlation fimction, g(r)-1,

is the fourier transform of S(Q)-1 ............................... 64
(Bottom) The RMC partial structure factor, SCuCu(Q). Here, the solid line is
the simulation started from the crystalline arrangement, and the

dashed line is that started from the random arrangement ............. 60

The partial pair correlation function, gCuCu(r). Solid and dashed
lines are the same as for Figure 21 .............................. 69

The RMC bond angle distribution for random (top), post simulation
(middle), and crystalline (bottom) Cu6Ce. ........................ 69

The crystal structure of CuéCe (From [62]) ........................ 72

The resistivity, thermopower, and temperature coefficient
of the resistivity for Al-Ca alloy from Zuo et.al. [60], evaluated at
T=1079°C. ................................................ 74

The phase diagram of Al-Ca and Ga-Ca (From [59]). ............... 75
The temperature coefficient of the resistivity of Al,,,Ga,,Ca as

a function of concentration of Ga from You et.al. [61], evaluated at
T=1079°C. . ............................................... 77

The resistivity and thermopower of All,,GaxCa. .................... 78

viii

INTRODUCTION

This dissertation will explain our experimental determination of the resistivity and
therm0power of several pure liquid metals and alloys of current interest, and will also
explain our atomic structural study of one of the alloys (liquid Cu6Ce). The last several
decades of research have revealed that the liquid is a far more ordered environment than
thought earlier. Ionic bonding has in some cases been shown to result in atomic clusters,
some as large as 10 A. Such strong ordering powerfully influences the electronic
behavior, often resulting in behavior commonly associated with conduction near an
energy gap in the density of states (see Enderby and Barnes [1] for a complete review).
The liquids chosen for study in this work are meant to elucidate the effects of weak
ordering on the electronic prOperties; the lifetimes of the associations that we describe as
weakly bonded are perhaps shorter, and fewer atoms at any particular time participate.
The bonding in the solid compounds that melt to these liquids is more covalent than
ionic, in most cases. It is eXpected that the effects on the electronic or atomic structural
properties will be weaker and harder to detect in these systems, and therefore represent a
. class of liquids perhaps inadequately experimentally characterized, and not at all I
completely explained theoretically. Although the materials we’ve studied come from
various areas of the periodic table, and as a whole do not comprise a systematic study of
one narrow family of compounds, the various metals and alloys in this thesis have all

1

2
proven to be at the fringes of our understanding of liquids.

Some of the earliest attempts to understand liquid metals had at their roots Nearly
Free Electron theory, and the associated assumption that the potentials felt by plane wave
like electrons in metals are weak enough to be treated by perturbation theory. In this case,
the plane waves are disrupted by the disorder in a material, and the resistivity and
thermopower are directly related to the structure. In fact this formalism, first proposed
by Ziman [2], has successfully described many simple pure liquid metals, and the
extension of the theory by Evans [3] has been adequate in some cases presumed far more
difficult, such as the pure liquid transition metals and rare earths, whose semi full shells
of d and f electrons made it far from apparent that weak scattering theory was appropriate.
The success of the theory in practice, however, is certainly influenced by the application
of accurate structural data and suitable pseudopotentials. Binary alloys of the simple
metals with one another have largely been correctly described by the theory as well, by
expanding the formula to involve the three partial structure factors in such cases. The
theory has even correctly predicted the observed negative temperature coefficient in
divalent liquid metals and alloys.

Another class of liquids exists, however, for which weak scattering is clearly
unsuitable, namely liquid semiconductors and liquid semi-metals, of which the ionic
alloys are a part; for these one must retreat to the Kubo Greenwood (KG)[4] formalism.
Such formalism dates to approximately the time of Ziman’s formulation, but is far more
general, and may be used carefully in cases where the large resistivity would suggest that
the Fermi Energy is near or in an energy gap in the density of states. This theory

incorporates an energy dependant conductivity, 0(E), and the energy derivative of the

3

Fermi function (a more complete description is given in the Ag-Te section ofthis thesis).
The shape of o(E) is subject to some debate, but the metallic approximation 0(E)~x NtE)2
is often used, and the density of states at the edge of a band is often assumed to be free
electron like, and proportional to E”. Some early attempts at the conductivity and
thermOpower integrals of this theory result in analytic expressions for conductivity and
thermOpower, but these use the Boltzmann approximation for the electron energy
distribution, and render these results of limited validity (for example, they are not to be
used in cases where the Fermi energy is near the band edge). Enderby and Barnes have
recently relied on numerical integration to evaluate the integrals using the full Fermi
distribution and a linear dependance of 0(E) near the band edges. This has led to a highly
successful and much referenced method of wide applicability for calculating the transport
properties of liquid semiconductors.

As suggested earlier, there remain liquid metals that fall in between these
explanations, and several of the subject materials of this thesis are a sampling of this
group. However, AgZSe and Ang are semiconducting solids that melt into liquidswith
even poorer conductivity, and one might expect that the Kubo Greenwood equations
could adequately characterize these liquids. Surprisingly, at the stoichiometric
concentration in these liquids, ’the conductivity is enhanced with respect to surrounding
concentrations of Ag and chalcogenide, and a metallic (negative) temperature dependance
of the conductivity is observed. This anomalous behavior is confined to a narrow region
around the stoichiometric compound; it has never been observed in the chemically very
similar Ange, which we systematically examine in a similar concentration range. we

add our results to the considerable recent experimental work that exists on the

-4

connection between ionic mobility and the enhanced electronic conductivity in these
materials.

The transition in liquid Si and Ge from the crystalline semiconducting state to
metallic liquid is a poorly understood tOpic of great current interest. During this
transition, the number of nearest neighbors for an average Si only changes from 4 to
roughly 5.8, and the liquid structure as measured by neutron [5] and X-ray [6 ] scattering
reveals evidence for ordering strongly reminiscent of the tetrahedral networks in the
crystalline materials. In the liquids context, ordering does not mean long lived molecules,
but rather short lifetime associations of atoms, where lifetime is measured in terms of
the frequency of random thermal motions. Significant progress has been made in the area
of ab-initio calculations; the bond lifetimes in liquid Si have been estimated in computer
simulations to be about 0.1 ps, and approximately 30% of Si atoms are bonded at any
one time. In spite of recent computational insight into these liquids, considerable
difficulty surrounds the containment of liquid Si, which has hampered actual physical
measurements. We have successfully deveIOped a new containment technique for this
material, and utilize it to measure accurately the resistivity and therrnOpower.

Finally, liquid CuéCe shows signs of ordering in neutron diffraction experiments,
with a diffraction peak at low momentum transfer corresponding to correlations on a
length scale as large as 4.4 A; yet the resistivity is curiously featureless, except for the
temperature coefficient of the resistivity, which issharply negative at the concentration
Cu6Ce, a concentration corresponding to a crystalline compound which exists up to the
melting point. Cu-Ce and Ag-Ce appear to be alloys in which Chemical Short Range

Order (CSRO) and metallic conduction coexist, and in this way are members of a still

5
very small, largely unexplored class ofalloys.

Early characterization of the pure metals and alloys in this thesis has often been
limited to the conductivity, and our studies improve the accuracy and quantity ofthe data
for this important prOperty. However, in many cases, the thermopower has not been
measured, because of the inherent difficulty. High temperature thermOpower standards
for measurements have been in existence for decades, but were often extrapolated from
lower temperature data. Recent measurements at high temperature (1500 C°) of the
thermOpower of Pt by Roberts [7], utilizing the measurement of Thompson heat, has
established a reliable standard. The thermOpower is very important in liquid .
semiconducting systems, as theory indicates that the positive-negative transition in the
therm0power seen in many such systems provides a direct way to measure the width of
the energy gap. ThermOpower is also readily calculated for weak scattering pure metals
and alloys from the Ziman theory. We present new experimental results for
thermopower for the liquid c0pper lanthanides and for silver cerium, as well as for pure
liquid silicon. We also present data for liquid Ag-Te, including thermopower in the
immiscible region below 31 at. % Te.

All of the results presented in this thesis are discussed in light of the current
theories, and where possible, calculations have been done to test the applicable theories.
Liquid Si and Ge have been treated with Ziman formalism, using a recent pseudopotential
to calculate resistivity and thermopower; Ag-Te has been modeled with the Kubo
Greenwood formula. Cu-RE and Ag-Ce are discussed in light of recent experimental
evidence of the subtle effects of CSRO on the transport prOperties. It will be seen

throughout this work that significant gaps exist in the theoretical description ofliquids; it

6
is our hope that this work will provoke fresh thought on these subjects.
This thesis includes both a descriptive section for each of the studies undertaken,
and a copy ofthe publication or manuscript (as accepted for publication) ofthe same
study. In each case, the descriptive section is provided to augment the information

available in the publication. Both should be read, however, to provide the full picture of

each study.

v' w i v al

Liquid chalcogens have been of long term interest because of the unusual
properties they possess. Se and S are semiconducting liquids, while liquid Te has quite
metallic conduction. Scattering experiments on the pure liquids have revealed evidence
that may be interpreted as short lived structural units like short chains (Se, Te) [8]; the
presence of these units is intimately tied to the rich behavior of the electronic transport
and atomic structural prOperties as the chalcogenides are alloyed with each other, and
with other elements of the periodic table. Of particular recent interest are the electronic
and atomic structural properties of the chalcogens alloyed with the noble metals; the pure
noble metals are well described structurally by hard sphere packing of atoms, and are
very fine metallic conductors, and are considered in pure form to be monovalent. Ab
initio studies of liquid Ag-Se alloys [9], for example, have demonstrated the gradual
change from non bonding liquid to bonded networks as Se content is increased from 33
to 65 at. %, and have provided new understanding of the transition from metallic
conduction to semiconducting behavior as chalcogenide is added.

The electronic transport properties of liquid alloys of Ag with the chalcogens
around the composition Ag2X, where X = S, Se, are particularly surprising, and
significantly different than, for example, the compounds around Cu2X . In the latter
compounds the conductivity, 0, is clearly semiconducting (o < 500 Q " cm“), with o=40
Q " cm" and 160 Q " cm‘1 for S and Se compounds, respectively, and is at a minimum at
stoichiometry [1]. This behavior, as well as a positive temperature coefficient of the

conductivity, is rather typical of a liquid semiconducting alloy. In contrast, systematic

 

 

 

 

 

 

 

 

 

Weight Percent Selenium
MM! 19 29 35) a L 70 up 90 100
mm: "1 L1 + [,2 .l.
I 1,,
r2
540°C ‘ ‘ ~
800- L2 '1' L2 + L3 ‘i‘ L3 I:
l \
I ‘.
I I
‘ ‘.
o ' ero‘t: '.
a m'-(A‘) “.5 'u '-
L ‘t
3 '.
“ '.
‘6 '.
°' {I '.
E ‘00- < 1-
U Q .I»
F I
~zer‘c ar'c
ano- _
(Se)-
rzna'fc
In
<
.. u
10 2'0 3'0 do so so 70 so 90 mo
Ag Atomic Percent Selenium Se

Weight Percent Tellurium

Temperature °C

 

A!

Atomic Percent Tellurium Te

Figure 1 - The phase diagrams for Ag-Te and Ag-Se alloys, from [59].

9
studies in the neighborhood of Ag2X [10,1 1] (The phase diagram shows a peak in the

melting curve and a congruently melting solid compound at this concentration (Figure
1).) have revealed an unexpected rise in the conductivity to a sharp peak that is about
200 Q'cm'l in height at 900°C and approximately 4 at. % wide, and centered on
stoichiometry (Figure 2); this peak decreases to 100 Q'cm" in height by 1000° C. The
peak in conductivity is accompanied in this narrow concentration range by a negative
temperature coefficient, which becomes positive further from stoichiometry on either
side. According to earlier electronic transport studies on Ag-Te [12,13,14] , Ange
behaves in the former more common fashion. We note, however, that in the earliest
study of liquid Ag-Te, Dancy [12] lacked the ability to stir the alloy, and fiarther was
unable to probe the presumed immiscible region immediately below stoichiometry
(Figure 2 ). The more recent re-examination of Ag-Te by Okada et.al. [13] was conducted
over a broad and widely spaced range of concentrations, even in the critical neighborhood
of stoichiometry. The data of the final study by Ohno et.al.[14], while covering a finer
grid of points than in [13], is still sufficiently sparse to have missed narrow and small
features; furthermore, no details are given as to the measurement of properties within the
previously mentioned immiscible region.

Recent structural studies by Price et a1. [15] on liquids AgZSe and Ange revealed
a remarkable similarity between the structure factors of the liquids, although a prepeak at
about 1.8 A" in the structure factor of AgZSe was not seen in the Te compound. (This.
however, was attributed to the similarities in the scattering lengths of Ag and Te. )
There was no noticeable change in the positions or widths of the first coordination peak

in Ag-Se as temperature was varied, or as Se content was varied from 30-36 at. %,

10

 

I0 I I . I ' l I l 1 fl

:2 quuld Ag-Se Alloys .

(b) ~

—+—l373 K

'7 " —o-—1273 K ~

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5 L -
O I r I u l . l r I r
1 0 0.2 O. L 0.6 0.8 1.0

Ag Concentration of Selenium Se

Figure 2 - The conductivity of liquid Ag-Se alloys, from Ohno et.al [11]

11
although a significant change was noted in coordination of Se around Ag. According to

the x-ray attenuation density measurements by Tsuchiya [16], however, it is possible
that an incorrect number density was used to calculate coordination. It has been
concluded that no sudden structural change with temperature or concentration causes
the anomalous electrical conductivity behavior.

Stoichiometric compounds of Ag with S, Se, and Te are all fast ion conductors in
the high temperature solid phases of the alloys; the ionic conductivity reaches values of
5-7 Q'cm'l for these solids [l7]. Ionic conduction is by way of the Ag ions, which are
proposed to move by jumping between tetrahedral sites within a chalcogen lattice. A
picture of Ag jumping between chalcogen ion cages is used to describe the liquids as
well, since the ionic conductivity decreases by only ~30% upon melting [17], although Te
cages in Ag-Te had been thought to break up quickly with increasing temperature due to
the negative temperature dependence of the ionic conductivity in liquid Ange. Only very
recently has a full partial structure factor analysis of liquid Ange been completed by
Barnes et.al. [18] by the method of isotopic substitution and neutron diffraction. The
results show what was earlier postulated; that the Se-Se correlations are largely
responsible for the prepeak, and further that these distances are really quite similar to
those in the b.c.c. crystal from which the alloy melts.

The extreme asymmetry of the p-n transition of the thermopower revealed in the
data of Ohno et.al., and the variation of the conductivity with concentration make it clear
that the rigid band model is unsuitable for this system, as was similarly shown for Ag-Se
and Ag-S. In light of the structural similarities, and the comparable ionic conductivities

between solid and liquid AgZSe and Ange it is a serious question why AgTe should

12
behave so differently electronically near stoichiometry from the other silver

chalcogenides. We have therefore reexamined the well stirred liquid in fine concentration

steps.

B. Experimental Details

Our samples were contained in alumina (A1203) tubes, 24" long and closed on
one end. The tube is 0.315" outer diameter, and between 0.180" and 0.220" inner
diameter, the variation due to the nature of the casting process. Four holes were drilled
with a diamond drill bit into each tube, the first 7/8" above the closed end, and the others
spaced evenly from this hole 1 7/8" apart. The tubes were cleaned after drilling with a
brush in water to remove debris from the drilling process, and were then rinsed with
acetone. They were then calibrated at room temperature by tightly binding 0.002"
molybdenum foil over the holes, fastening four wires, one to each hole, and filling the
tube to a depth of ~6" with double distilled Hg. Current for the measurements is supplied
by an Hewlett Packard 6267B constant current supply, and to measure the current, the
voltage across a 0.010 standard resistor is measured. All voltages are read by a Kiethley
181 nanovoltmeter, with channel selection by a Kiethley 705 scanner and an external
computer. The resistance, R, of the mercury sample between the center holes was
measured using the conventional 4 probe DC resistance technique. The room
temperature resistivity of pure Hg is well known, and the tube constant is readily
calculated from the formula
R= pL/A

where p is the resistivity of the sample, and L and A are the effective length and cross

l3
sectional area of the volume between the center holes, respectively. The tube constant

at high temperature was later computed accurately knowing the coefficient of thermal
expansion of alumina, K=8.l * l 0‘6 /C°.

After calibration, the holes were plugged with machined cylinders ofhigh density
Aeromet grade graphite, supplied by Electronic Space Products International (ESPI).
Typical impurities for this grade graphite are as follows, in ppm: F e-100, V-lOO, Al-20,
Cu-S, Si-80, Ti-15, Ni-lO. The plugs are held in place only by tight fit; however, we
have had no difficulty with cracking of the tubes or leaking due to the difference in
thermal expansion of the materials (Aeromet grade graphite K=IO.34 *10'°/°C). Typical
tube preparation involves the sanding down of the fitted plugs until they stand < 0.004"
proud of the alumina, and encircling the tube over the plugs with molybdenum bands,
which serve as additional protection against leakage. The bands are made of 0.002" Mo,
and are secured by stainless steel screws, nuts, and clamps. The wires are held against
the M0 by one further band of 0.005" stainless steel, held by the same clamps, and chosen
for it’s strength and resistance to corrosion.

The sample tube, and it’s placement with respect to the various furnace coils, is
shown in Figure 3. The lifetime of the coils is affected by the surface loading
limitations (in watts/cmz) of the heater wire (0.032" Kanthalo A-l wire), and this
ultimately limits the highest temperature this furnace may achieve, about 1200°C, even
though the melting point of Kanthal is 1510°C. Kanthal is well suited to application in
air, as heating in this environment creates a protective oxide layer. Higher temperatures
can be attained in the same furnace with Mo wire, although much greater care with

regards to oxidation of the coils must be exercised. A 2" id. alumina tube surrounds the

l4

PUMP

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

wfifi r '
f
I 5.;
a at

 

Figure 3 - A diagram of the fumace and sample tube.

15
sample tube, and is held in place by a water cooled brass collar and an O-ring. This tube

has been threaded on the outside lower portion for easy and consistent placement of the
supplementary heating coils, which are used to control the temperature during
thermopower measurements. An IBM computer controls the duty cycles of the power
supplies for the upper and lower coils, and can control the temperature of the bottom and
top of the sample to ~01 C°. The bulk of the power delivered to the sample is from the
main furnace, also of Kanthal, and divided into upper and lower coils. This is raised to
surround the fiirnace tube during Operation.

The resistivity measurements were done using the four probe DC technique.
The upper and lower thermocouples are under normal circumstances 0.010" chromel-
alumel from Omega Engineering, Inc. Under especially arduous experimental
conditions, diameters as high as 0.020" have been used for greater temperature and
corrosion resistance. The temperature of the upper and lower thermocouples are
controlled to be the same during the resistivity measurement, but slight variations in the
thermocouple material result in a small error in the temperature measurement ( Omega,
the manufacturer of the chromel and alumel wire, specifies that the temperature measured
by these thermocouples is good to i0.75%). A difference in temperature between the
electrodes during a resistivity measurement would introduce an additional thermal emf to
the measured voltage; contributions to the resistivity measurement are eliminated by
reversing the current during the measurement, measuring the voltage in both directions,
and taking the appropriate difference, thereby eliminating the thermoelectric emf.

Thermopower measurements are conducted in the following manner. The upper

and lower portions of the sample are equilibrated in temperature , and the thermal voltage

16
is measured using chromel electrodes of the thermocouples . The temperature of the

upper portion of the sample is then ramped upward at a constant rate (~ 1°C/min) while
the lower temperature is kept constant. At periods throughout the ramping, the thermal
voltage is measured and recorded. When 10-15 points have been taken, during which
time the temperature difference aT has reached ~30°C, a quadratic fit of the thermal
voltage as a function of aT is conducted, from which SR,,,,,ve=dV/dT is extracted. The
thermOpower of the sample may be calculated using the formula

Sump“; Schmml :t dV/dT

where the i is determined only by the arrangement of the counterelectrodes at the input of
the voltmeter. The thermopower of Chromel has been measured in our lab [19] against
Cu and Pt, and is in excellent agreement with previous measurements [20]. This method
of thermopower measurement yields accurate results because of the smoothing effect of
the fit over numerous points, and allows the measurement of thermopower very near to
phase transitions.

The high vapor pressure of Tellurium did not appear to affect our measurements;
no condensed Te was found in the cooler regions of the sample tube. Samples are usually
stirred with 0.040" Mo wire; however, significant attack was noted on Mo stirring wires
for Te concentrations >34 at. %. For these concentrations, we changed to alumina stirring
rods. Ag and Te were of 99.99% and 99.999% purity, respectively ( metals basis), and
were supplied by Aesar-Johnson Matthey Co. The sample concentration was not varied
continuously during the experiment to avoid even the possibility of progressive
contamination. The majority of the samples were independently prepared in separate

tubes from the pure materials, making most data points completely independent of all

17
others.

£2.__R_e_sults

Our results for the conductivity of Ag-Te are shown in Figure 4 , including for
comparison the previous results for the higher Te concentration region by Ohno et al.
[14]. Our data are in excellent agreement with those of Ohno et.al., except as
stoichiometry is approached from the Te rich side, where our data fall noticeably below
his for concentrations between roughly 36 and 45 at. % . An extrapolation of the curve fit
to this data and the higher Te concentration data of Ohno et.al. reveals a peak in the
conductivity in the neighborhood of stoichiometry; this peak is shown in Figure 5, and is
about 60 (I'l-cm'l in height at 960°C, which is comparable to but less than the ~100 Q"-
cm" at 1000°C seen in AgZSe. As with the other silver chalcogenides, do/dc is
significantly steeper on the Ag rich side than on the Te rich side; however, the
temperature coefficient of the conductivity is positive at stoichiometry, in contrast to the
unusual negative temperature coefficient seen in AgZS and AgZSe. A positive
temperature coefficient of the conductivity is expected in liquid semiconducting alloys,
because the broadening of the Fermi distribution with temperature introduces more
electrons into the conduction process.

We have extended the resistivity measurements into the immiscible region; an
examination of the phase diagram marks 31 at. % to 9 at. % Te as the boundaries of the
phase separation, with a sharply concentration dependent temperature boundary. One
concentration, the liquid 30 at. % Te alloy, proved amenable to examination above and

below the miscible-immiscible phase boundary. No time dependance of the conductivity

18

 

 

     

 

 

 

=5 20 .—
U
,
L9;
7‘; l0 ._
‘3 .
O --
4O .1:
4000--
3 .
-°2ooo--
1000--
O M/IISCIBIE MISCIBLE ‘ .
Ag ‘ 20 4o 60 80 Ta

alto/0'5.

Figure 4 - Current data for the conductivity of liquid Ag-Te at 950 C° - 0;
recent data for liquid Ag-Te at 1000 C°- A , from Ohno et.al. [14];
data for liquid Ag-Se from Ohno et. al. [1 l] - I .

19

 

 

 

 

’8
o

 

 

. MlSClBLE

 

 

-.--—--‘-------- DOC-0-.---
.

 

 

I
r

25 so 35 ' 4o 45 so

<51. ‘70 Te

 

Figure 5 - Present data (0) and data of Ohno et. al. (A) for the conductivity of
Ag-Te in a narrow range of concentrations around stoichiometry. An
extrapolation of our data and that of Ohno et.al. from outside the
stoichiometric region drawn toward Ange demonstrates the
presence of a peak in conductivity. In the lower half of the figure,
the current peak data is compared to the peak seen by Ohno et.al.
[11] in Ag-Se (v ). The boundary of the immiscible region is
clearly marked to demonstrate that the peak is not a result of phase
separation. '

20
was seen in measurements taken at temperatures below the boundary of the immiscible

region during a time period of up to 1 hr after the sample had been vigorously stirred and
allowed to settle. In addition, no evidence for a sharp change in conductivity occurs at
~1075 °C , or at any temperature up to 1150°C; such a change was taken as an indication
of the immiscible-miscible transition in Ag-S and Ag-Se alloys (Figure 6). Nevertheless,
we stir all of the samples at intervals of less than 1 hr to ensure homogeneity, although we
believe that Ag-Te alloys are perhaps more stable than Ag-Se or Ag-S.

We also have measured the thermopower, including in the immiscible region
(Figure 7). In Ag-Te, it is found that the thermopower never descends below -13
uV/C° on the silver rich side of stoichiometry, in contrast to the highest extent that it
reaches on the Te rich side, +120 uV/C°. In addition, the dramatic dip of the
thermopower in Ag-S and Ag-Se that precedes the rapid change from negative to positive
S is absent in Ag-Te. The steeply sloped, rapidly increasing conductivity 0(c) on the Ag
rich side of stoichiometry and the shallow dS/d‘c and small IS] in the same range of
concentration are very similar to the combination seen below 28 at.% Ag in Ag-S and
Ag-Se. The temperature coefficient of the thermopower is positive in the silver rich
region and at stoichiometry , as it is in AgZSe and AgZS, and it changes to negative as
concentration passed to the Te rich side of stoichiometry.

Our thermopower data agree well with Ohno et.al. , except at concentrations in the
neighborhood of the peak in the conductivity. Whereas in other regions the data of Ohno
et.al. [14] fall consistently below ours by 4 uV/C°, in this region they exceed ours by as
much as 25 uV/C°, a far greater difference than the estimated error of Ohno et.al., about

10 uV/C°, and also greater than the 3-5 uV/C° we estimate for our data. (This estimate

GOO

0' (DJ-Err?)

4OO -

350 -- _.-..

BOO -- ,

 

21

 

 

 

 

 

 

250

Figure 6 -

900 9550 iooo 10230 Moo ”so

TlC°l

Conductivity of two samples of Ange.30 as the temperature boundary
(arrow) of the miscible- immiscible region is crossed (current data for

two independent measurements on two samples of AngeJo- 0 V ), as
compared to the results of Ohno et.al. [10] for Ag,,,,s,,,- I . (The
comparable miscible- immiscible boundary for Ag 705$ 29, is at 1050°C. )

This demonstrates the apparent stability of liquid Ag-Te at temperatures
and concentrations within the immiscible region. ‘ '

22

 

 

6‘63

, l
(‘3’

 

 

 

 

'30 : 'r t 'r‘ r i i,
IO 15 20 25 SO 35 4O 45 50

at. °Io Te

Figure 7 - Current results for the thermopower of liquid Ag-Te -0 evaluated at
960°C, compared to the data of Ohno et.al [l4]-Aat 950° C.
Significant differences greater than either experimental error exist in
the concentration range where the peak in the conductivity is seen.

23
is based on our experience with repeatability of measurements on separate samples of the

same concentration, and also on the scatter seen in a set of measurements on a given
sample.) The conductivity data in this concentration region are crucial to the
identification of the peak in the conductivity.

[2. Discussion

For the majority of liquid semiconductors, numerical integration of the Kubo
Greenwood equations using the full Fermi function, as demonstrated by Enderby and
Barnes [1], can yield a good qualitative agreement with experimental data. The

equations are

o=fo(E)-5—'gf—)dE

0

 

a E-E
S=£fO(E) ffldg
e o O kBT 5E

In these equations, 0(E) is the energy dependent conductivity , kB is Boltzmann’s

constant, and f is the Fermi function. The Fermi function is

l
’(5 'Ef)

kT
exp ° +1

 

f(E) =

 

with Ef being the Fermi energy and T the absolute temperature of the material. It is
assumed that the shape of the bands is constant even as the concentration of the alloy is
changed, or in other words, that the bands are rigid. Early treatments using these basic

equations make two assumptions. the first of which is that near the energy gap 0(E)o<

24
00, a constant, but also that o(E) drops sharply to zero at the gap. A second assumption

is that the Boltzmann approximation is valid ,
"(E 'EF)

E a
f() exn kT

Both of these assumptions together lead to the recognized result

 

where Ev is the energy of the valence band edge. While the Boltzmann approximation is
justifiable in situations where E,»EV » kBT, it is not good for calculations near a
conductivity edge, where the full Fermi function should be used, and results gained in this
manner are of limited applicability.

Enderby et.al.[l] differ from these early attempts by making the reasonable
assumption that N(E) o< E"2 near the edge of the band, and that o(E) oc (N(E))2 «X E. An

appropriate a choice of a simplified o(E) for valence and conduction bands is of the form

OC(E) = ac[E -Ec] V

0,,(E) =a,,[15:—15,,]v

with E, and Ec the valence and conduction band edges, respectively, and v = 1 in this
case. Here, we use the following relation for band gap, AE=Ec-EV , and the variables ac
and aV may be calculated using the band width, the atomic separation, and the total

number of electron states per unit volume in the band [21].

25
Enderby and Barnes have shown that in liquid semiconductors where rigid band

formalism is applicable, the magnitude of AS=Sm,-Sm,n , where Smax and Smin are the
maximum and minimum observed thermopower, is directly related to the width, AE, of
the energy gap. Our data indicate AS=l30 uV/K, which at 1233 K corresponds to AE~0
eV. Using the Kubo Greenwood formulas with AE~0 eV as a starting point, we have
done several tests to gauge the effects on the conductivity and thermopower of varying
the slope of the (linear) valence and conduction band edges, ac and a,,. Some results of
numerical integration of the Kubo Greenwood formulas are shown in Figure 8, along
with the energy dependant conductivity used for each calculation. The conductivity is
given by

0 = CC + oh

where 0e and 0,, are the contributions from electrons and holes, respectively. The
thermOpower is calculated by a conductivity weighted average

S : oese + OhSh

 

o + o
e h .
with Sc and Sh the thermopower from electrons and holes. Quite apart from

reproducing the peak in the conductivity, we have found that there is no choice of ac and
av that can simultaneously reproduce even the general features of conductivity and
thermopower together . Our tests, using a value of ac ~1300 Q" cm'l eV", and a choice of
ac=2aV (This choice of asymmetry between conduction and valence bands is comparable
to that chosen for a similar calculation on Ag-Te [14].) can indeed roughly reproduce the
magnitude and asymmetry of o(c) in the concentration range around stoichiometry, and a
thermopower that rises to a maximum of roughly 150 uV/C°on the Te rich side.

However, with this choice of o(E), the minimum of o(c) and the p-n transition of S(c) is

26

 

Olff'crfi' I
CH
O
O

 

ES
0

 

 

 

-04 - -O.Z O 0.2 0.4
excess Te excess Ag

E - E:F (eVl

Figure 8 - The thermOpower and conductivity calculated using the Kubo
Greenwood formulas with ac =2a,, =1300 0" cm" eV'l

27
shifted several atomic percent to the tellurium rich side of stoichiometry; in addition, the

p-n transition is broadened, and S(c) also descends far below 13 uV/C°.

Although the peak in the conductivity appears to be unique to the silver
chalcogenides, there are other liquids that are also poorly modeled by this formalism . In
particular, in the concentration neighborhood of the fast ion conducting system liquid
Mg3Bi2, o experiences a steep and sudden drop as-stoichiometry is approached from
either side [22]. However, the thermopower maintains very small values (lS|<10 uV/C°),
even through stoichiometry. It has been suggested that this is a result of a rapid Opening
of an energy gap as a result of rapidly changing mobility within 1 at. °/o of stoichiometry,
where previously there had been perhaps a pseudogap. Such an explanation is not
unlike that advanced by Ohno et. al. [1 l] to explain AgZSe, where it is supposed that
do(E)/dE changes very rapidly at concentrations near the peak in conductivity.

Significant experimental evidence for a tie between ionic and electronic mobility
has now been accumulated, although theoretical considerations are still sparse (the
possibility of a connection between ionic conduction and the energy gap has been
explored by Ramasesha [23]). In crystalline NaSn and Cst [24], for example, the
electronic conductivity jumps when a phase boundary is crossed from the higher
symmetry, lower temperature [5 phase, to the lower symmetry, high temperature ionic
conducting a phase. The lowering of the symmetry would typically result in the
narrowing of the energy bands, and perhaps the widening of the band gap; the increase in
the conductivity is contrary to this; Indeed, a similar transition in the ionic and
electronic conductivity is seen in the solid AgZSe and Ang compounds as the [3 to a

phase boundary is crossed [25], and apparently also in crystalline Mg3Bi2 [26]. (We have

28
already noted that these are ionic conductors in the liquid as well as the high temperature

solid.)

It is significant, however, that recent ab-initio calculations on liquid Ag-Se [9]
have revealed that an energy gap in N(E) does exist at stoichiometry; further muffin tin
[27] calculations have verified the presence of this gap, and have shown the presence of a
gap in Te and S compounds at stoichiometry as well. The ab-initio calculations have
shown that the gap disappears as Se concentration is increased from 33.3 at. °/o to 65 at.
%, and it may not be ruled out that at least for these systems, the anomalous conductivity
may depend as significantly on changes in N(E) as changes in the mobility u(E). It is
unfortunate that no ab-initio results exist for Te or S alloys with silver, or for any such
system in fine concentrations steps around stoichiometry; it is clear that the present

results warrant all of these calculations.

mum

AJnteruction

Pure liquid Ge and Si have for decades been the subject of theoretical and
experimental inquiry because of the similarities and differences between their solid and
liquid phases. Both are semiconductors in the solid, but in each the resistivity drops
enormously upon melting to metallic values (pQOOuQ-cm); at least for liquid Ge,
experiment has shown that this decrease in the resistivity is associated with the
disappearance of the energy gap in the density of states [28]. Remarkably, the structure
apparently remains open and far from the hard sphere packing normally associated with
metallic conduction, which often correspond to coordination numbers of 10-12. The
shoulder observed in numerous x-ray and neutron scattering experiments on the high Q
side of the main peak of the structure factor is unreproducible by even more SOphisticated
hard sphere modeling [29]. The fourier transformed structure factor has an ill defined
first minimum, which makes estimation of coordination number from the area under the
first peak of g(r) difficult; coordination has been .found to be as small as 5.8 and 6.1 for
liquid Si and Ge, respectively [30]. Estimates range as high as 9.5 for both liquids
depending on the definition of the upper boundary of the first peak.

The shoulder in the experimental structure factor S(Q) is excellently reproduced
by Reverse Monte Carlo (RMC) modeling [31]. (The RMC method has been used to
analyze our liquid Cu-Ce data, and a full account of this technique will be delivered in
that section.) It is interesting that the same method has demonstrated that requiring

progressively more atoms within the simulation to be coordinated as they are in the

29

30

covalently bonded crystal results in the splitting of the first peak in the structure factor,
and that even weak covalency can result in a shoulder like the one seen in liquids Ge and
Si. In addition to this, this technique can yield the experimentally unobtainable bond
angle distribution, although it must be emphasized that the configuration achieved by the
RMC method is not the only configuration that can yield a particular S(Q). ( It is not
unique.) For Ge, the bond angle distribution shows a peak at around cos 6 = 0.65 ,which
suggests that the structure is weakly diamond like, with significant overlap of the first and
second coordination shells.

Evidence for remaining covalent bonding in the liquid has also been noted
through the recent development of ab initio molecular dynamics. These simulations, in a
series of steps, calculate the interatomic forces through full quantum mechanical
considerations, and then allow the ions to move according to classical equations of
motion. (See, for example, Stich et.al. [32].) The method has been used with success for
several pure liquid metals [33] and alloys [9]. A consideration of the electronic wave
functions of liquid Si by this method has shown a transient accumulation of charge
between neighboring atoms, which amounts to more than effects due merely to the
proximity of the atoms. The excess charge is in fact interpreted as covalent bonds in the
liquid. These experimentally inaccessible results are grounded in experiment; good
agreement exists between simulated and experimental atomic structure, density of states,
and bulk modulus for liquid Si. However, the experimental resistivity, also valuable for
comparison to simulation, varies by as much as 20 % from one experiment to another,

and an improvement in the accuracy of these data is clearly necessary.

31

B,Erocedure and results.

Liquid Ge is completely compatible with Alumina; all measurements were done
in the same way as the Ag-Te alloys, and no special care was needed to isolate the
material from the air during preparation. The experiment, however, was still done in an
argon atmosphere with 1-2 psi overpressure after baking the system under vacuum. Ge of
purity 99.9999% was supplied by Aesar-Johnson Matthey Co.

Si was also supplied by Aesar-Johnson Matthey , and was of 99.99% purity.
Liquid Si presents unique containment problems, as our own tests showed that it
vigorously attacks alumina and quartz. A survey of the literature showed that a number
of previous measurements had been conducted in containers of this sort, and were
therefore to be treated with caution.

Earlier workers [34] had shown that high density graphite could contain liquid Si
with minimal corrosion, although none had characterized graphite electronically for the
purpose of transport measurements; this early work had also shown that conventional
machined graphite rods immersed in l-Si experience a change in radius due to corrosion
of only about 1.02 X 10‘3 cm hr“. In the same tests, neither x-ray diffraction nor light
microscope inspection of the boundary could reveal evidence of the formation of silicon
carbide, an electric insulator. We have performed our own test of this by measuring the
resistance of a graphite-(liquid-Si)-graphite junction over a time period as long as 12
hours, with the result that no increase of resistance was measured, suggesting that no
appreciable amount of SiC was formed. Our tests were performed on Aeromet grade

graphite supplied by ESPI, their hardest, least porous graphite. ESPI has confirmed that

RITIIR125°CI°

SIN/C°)

32

 

0.9 ~-

67... . . . , A

0.6 --' -
0.5 a: 3'

 

 

 

l I ‘
_:2 3 l 'r 1 f L I I

 

 

1260' Isoo I400 1500
T(C°l

900 1000 I loo

Figure 9 - The electronic characteristics of high density graphite. The various
symbols indicate separate experiments done on samples of different
material manufacturer batch number, and on samples of the same
batch number with different thermal histories.

33
additional ingredients are not responsible for the different grades of graphite, but that the
difference lies only in the individual steps of the preparation process.

For measurements to be made using a conductive graphite container, a complete
knowledge of the electronic characteristics of the high density graphite is necessary for
the extraction of the sample resistivity and thermopower. We have characterized
numerous samples of the graphite through the full range of temperatures necessary for the.
liquid measurement, and have further examined several samples through numerous
temperature cycles. The results of our calibration are shown in Figure 9 . Although the
resistivity of the graphite may vary from one rod to the next by as much as 10%, the ratio
of R(T)/R(room temperature) was highly reproducible. The thermopower of the rods was
measured against W-Re 26% electrodes, and was highly consistent from one rod to the
next, being within i1uV/C° on all the rods tested, and also independent of the thermal
history of the material.

The graphite containers for our measurements were not available commercially,
but were prepared from .3 75" dia. x 6 “ rods of graphite by drilling with a special hollow
core drill that allowed compressed air to expel the graphite powder, thereby maintaining a
smooth machined inner surface . The wall thickness was about 0.76 mm. Triangular
grooves were machined on the outside for accurate and consistent placement of contact
wires. Voltage leads for the resistivity measurements were Mo wires nominally of 0.102
mm diameter; these were placed in the grooves and held in place by semi-circular A1203
clamps. The uncertainty introduced to the measurement of sample resistivity by the

effect of the grooves on the cross sectional area of the tube was estimated at below 0.1%.

34

The thermoelectric voltage was measured sufficiently far away from the ends of the tube
so that end effects in the therrnOpower were negligible [35]. The counter electrodes were
too thick (0.010") and inflexible to be bound under the alumina clamps, and were
therefore affixed with 0.015" molybdenum wire in such a way as to minimize the
longitudinal area in contact with the tube, to avoid as much as possible the addition of a
third parallel component to the thermopower measurement. Figure 10 is a diagram of the
arrangement.

The tubes were calibrated in a slightly different fashion than the method for the
alumina tubes used in the Ag-Te experiment. The first step was to measure the resistance
of the empty tube, wired as it would be for the actual experiment. The tube was then
filled with Hg, and the resistance of the parallel arrangement measured. After calculating
by the parallel resistors formula the resistance of the Hg, R=pL/A was used to extract the
tube constant, A/L, using the well known resistivity of Hg at room temperature, 95.7
it!) cm. With the room temperature resistance of each tube known, the resistance of the
tube at high temperature was calculated using the calibration curve for graphite
described earlier. Resistivity of the sample was calculated by considering the graphite
tube and the sample to be resistors in parallel. As in all of our resistivity measurements,
the current is reversed midway through the measurement to eliminate thermal voltages.

The thermocouples used for temperature measurement were of W-5% Re and W-
26% Re. These component wires were supplied in matched lengths by Omega
Engineering Inc., and thermocouples formed of these matched lengths were quoted to be

accurate to i=1%. The W-26% Re leg also served as the counterelectrode for

35

 

Figure 10 - Diagram of graphite sample container, support tube, and gradient coil.

36

measurement of S, and its thermopower was measured to above 1500°C against Pt.
(The thermopower of pure platinum by Roberts et.al. [7] was measured using the
Thompson heat method.) In addition, a calibration up to 1200°C was conducted against
chromel; the Pt calibration data was used in conjunction with our lower temperature
chromel calibration data to extract the absolute thermopower of W—26%Re. We
estimate the uncertainty in S for W-26%Re to be 3:05 uV/C°. It is worth noting that
another calibration of W-26% Re against molybdenum, using the early calibration of
Cusack and Kendall (195 8) [3 6] , differed from results gained from Pt and Chromel in
temperature region above 1250°C; we expect the most recent high temperature standards
to be more accurate.

The furnace is different than the one described for Ag-Te alloy, and capable of
much higher temperatures. Kanthal Super elements (made of MoSi2 sheathed in a thick
layer of SiOz) were used; this material has a maximum operating temperature of 1700°C
in air. The maximum surface load, 12.5 W/cmz, is again the limiting factor for
temperature, and this is achieved with 140A at 1500C°. At these high temperatures the
efficient radiative transfer of heat makes the delicate maintainance of the constant lower
temperature during thermopower measurements described earlier difficult. Instead, a
supplemental coil with Molybdenum windings supplying ~300 W was hung around the
uppermost part of the sample within the external fumace tube. A temperature gradient
was established by lowering the main fumace current, and by increasing the supplemental
coil current. When conducted properly, the lower temperature dropped steadily, and the

upper temperature rose steadily, such that the average temperature would remain the

37

same. The thermoelectric voltage was measured for numerous temperature differences
ATi, all with the same average temperature, and the “small thermal gradient method” [3 7]
was used to calculate the relative thermopower using the following formula,
Sm: (e,-ej )/ (ATi -ATj )
where ei is a thermoelectric voltage and ATi is the associated temperature difference.
This yields a relative thermopower free from errors in the thermocouple calibration, with
the assumption that the parasitic voltages remain constant over small temperature
intervals of the same average temperature, and may be subtracted out. The absolute
thermopower of the sample is extracted using the W-Re 26 % thermopower calibration
data 8,, the thermopower of the graphite SW, and the following formula
Sliquid=(1+B)(Sx + dV/dT) ‘ BStube
where
B=pliquidAtubelptubeAliquid
with p being the material resistivity and A the cross sectional area of the material. The
dimensions chosen for the tube, and it’s nearly ideal electronic properties, render B of
minimal importance to the calculation; the value of B in these experiments was only
about 0.02, and the contribution of the BSmb, term to the measured thermopower only
about 0.12 uV/C°. Therefore, uncertainties in the tube parameters contributed minimally
to the final values obtained for S for liquid Si.

Q. Expeg'mental Results and Discussion

Liquid Si has been seen to melt very slowly [3 8], passing through a semi solid

phase before becoming completely liquid; the slow destruction of order could result in

38

 

A A A
A ‘ I A

 

85
75 __"-.¢“._-_'__'__.3_’:_;_:.-- _'J-!3--—_
65

35 -.
25 . A
1420

 

14-410 1460 1480 1500 1520
TIC”)

 

 

 

 

 

 

 

50-=:::::::.:::.:.
940 960 980 100010201040106010801100

T1C°1

Figure 11 - The resistivities of liquids Si (top) and Ge. All of the data shown
are our own, the different symbols being separate experiments on
different samples. A random selection of 1/3 of the data accumulated
is displayed, for graphical clarity. The solid line is a fit to all of the
current data, including data points not shown, and the dashed lines are
the results of Sasaki et. al.[39] and Koubaa et. a1. [3 8] for Si and Ge,

respectively.

 

4301
‘H‘I

3

2
.11
“\10
%-II I l .' I
-ZW
-31

Fri

 

 

 

 

“SI-Ir.:;:;.,...
1460 I440 1450 1480 1470 1480 1490 I5CO 1510
T1C°1
5
4
3
2
6‘1
09-, “““:“----'--——--'---—_--__-__
-2
x7)
.4
960 980 1000 1020 1040 1060 1080 ICC
T1C°l

Figure 12 - The thermopower, S, of liquids Si (top) and Ge. The lines and data points
are as described previously in Figure l 1.

40

time dependent electronic properties before the material is fully melted. To investigate
this possibility, we have measured the resistivity at constant temperature for periods of
up to 30 minutes at Tmcl,+50°C, and have seen no discernible change. Furthermore,
results at the beginning and the end of an experiment were consistent, and further suggest
that the liquid state is achieved quite immediately at temperatures above the melting
point.

The results for the resistivity and thermopower of liquids Si and Ge are shown in
Figures 11 and 12. We find that our results for the 3 separate samples of Ge are very
consistent, and further agree very well with the recent results of Koubaa et. al. [39], who
also contained their sample in alumina. Our measured thermopower of Ge also agrees
well with the same workers. Our resistivity results for liquid Si are very similar to those
of Sasaki et. al. [40], who contained their sample in boron nitride containers and used a 4
probe measurement technique. Unfortunately, these workers have made no attempt to
measure thermopower. Our results for thermopower differ significantly from those of
Glazov et.al [41] , and we attribute this to our vastly improved containment technique.
(The containment for the earlier measurement was A1203.)

Because the resistivities of liquid Si and Ge are comparatively low, one might
expect that an analysis according to Ziman formalism is appropriate. Ziman theory
emerges from perturbation theory, and is often used to explain the so called weak
scattering metals, that is, cases when the electron interaction with the ionic cores is not

strong. The Ziman formulas for resistivity and thermopower are shown here

41

3n if 1
ezhv: V4k

 

2b,,
P= , fQ’.S<Q)1th>PdQ
F 0

5: 1t’kaT[e5(1r1[t>(r:)1)

3|e| 55 ]

 

where S(Q) is the structure factor of the material, u(Q) is the fourier transform of the
scattering potential, and vf and kf are the Fermi velocity and Fermi wavevector,
respectively. The Fermi wavevector may be calculated using Nearly Free Electron
techniques and the conduction electron density, n, using the formula

kF = (31t2n)“3

The structure factor, which will be seen again in the following section, is defined for pure
liquids as

S(Q)=1/N (2121‘ €pr -iQ.(rj- 1' r )1)

where N is the number of atoms, and rj is the position of an atom. Also, Q = ki-kf ,
where hk is the momentum of the scattered particle, and (llgl=|k,1) for elastic scattering.
The isotropy of the liquid allows Q in S(Q) to be a scalar quantity.

The earliest attempts to calculate the electronic properties of l-Si and l-Ge using
this theory used by necessity hard sphere structure factors for the materials. Hard sphere
packing routines typically have as input only the density, packing fraction, and
concentration of constituents (for binary alloys). No hard sphere structure factor,
including the more sophisticated softened hard sphere structure factor employed by
Rahman [29], can adequately model the shoulder on the high Q side of the main peak of

the experimental S(Q) of both liquids Ge and Si. It is interesting to test this weak

42
scattering theory using the experimental structure factors against our current, accurate
experimental results, to examine its success with regards to weakly ordered liquids.

Ziman theory also involves a sufficiently weak scattering potential to satisfy the
assumptions of perturbation theory. Although the full, all electron potential can in
principle be calculated with the Dirac equations, we are left with potentials for the
conduction electrons that are strong in the core region of the atom. A solution is the
creation of a pseudopotential, a potential that is not strong in the core region, but
maintains the same energy dependent scattering pr0perties to first order as the true
potential. This potential may be used in weak scattering Ziman theory.

For this purpose we have employed the highly versatile Bachelet, Hamann, and
Schluter (BHS) pseudopotential [42]; versions of this potential suitable to various
computing situations are regularly seen in current molecular dynamics studies. This
pseudOpotential is constructed in a series of steps from the fully relativistic atom, but at
distances outside the core region may be used in normal Schroedinger equations. The
accuracy of the BHS pseudopotential has been demonstrated by agreement of calculations
to within several percent of experiment for lattice constants, cohesive energies, bulk
moduli, and phonon fi'equencies. It has also been used in several recent molecular
dynamics simulations [31,43].

The total bare ion average pseudopotential is assembled from the expanded parts

as follows

vpsiona.) = 2‘ <1 I [V‘ion(r) + L's V‘spin orbit(r)] I I >

43

with
L.S= ‘/2 P- 1/2 (L2+Sz)
where V,’°"(r) is the additional potential felt by electrons with different angular momenta.
Each 1 component is then weighted according to the fraction of total electrons present in
each 1 orbital. For example, for both Si and Ge, there are 2 s-electrons and 2 p-electrons,
and the potentials would be added
vion(r)= .5 V ps.!=0 ion + _5 V ps‘w ion

For ease of tabulation, the BHS pseudopotential is separated into along range 1-
independent Coulomb part, and into short range l-dependent parts

(ion = Vcore(r)+ Avtion

All of the components added together merge to the all electron coulombic potential
outside of a radius Rm, . Both l-dependant and l-independent parts are expanded in terms
of Gaussian like functions
Avtion = Zr (Ar + rzA,+3)exp (“'0‘ i 1' 2)
yielding a set of expansion coefficients A, and decay constants or,. An additional set of
coefficients is created by a similar expansion to describe the spin orbit interaction. Some
of the values for the coefficients are quite large and have many significant figures, and
are not easily tabulated. Bachelet et.al. have published the coefficients A, in terms of
different, less numerically ponderous coefficients C, , related by a matrix multiplication to

A,. The 6X 6 matrix was required to be generated to 100 bit accuracy by integrals of the

44

form

Q

Sm=fr2q>i(r)q>m(r)dr

with
(I), (r) = exp(-a, r2) for i = 1,2,3
(I), (r) = r2 exp(-a, r2) for i = 4,5,6
The vectors C, were multiplied with this matrix to the same accuracy to attain the
original A,’s with the intended number of significant figures. We were able to
accomplish this conversion using Mathematica. As a check that our matrix inversion
program was correct, we compared our calculated A,’s to the one set provided in the
BHS paper for this purpose (for Si), and found complete agreement.

The 1 dependant potentials for Ge are shown in Figure 13. We notice that in
each, the singularity at r=0 has been removed. The fourier transform to generate V(Q)
was accomplished within a program by successively fitting portions of the curve with a
least squares method, and numerically integrating the fourier integral. V(Q) was then
screened,
V(Q)=V’(Q)/€(Q)
where V’(Q) is the transform of the (unscreened) BHS potential, and €(Q) the calculated
screening function; the resulting screened potential is suitable for input into the Ziman
integral. The screening of Vashishta and Singwi (VS) [44] and of Itchumaru and Utsumi
(IU) [45] were used; Figure 14 is a comparison of the squares of the screened potentials

for liquid Si, and a fairly constant difference between them is seen along the full range

Figure 13 -

Figure 14 -

 

45

 

 

 

 

 

 

 

 

 

 

 

 

cal/5.“)

(TOp) The l-dependent parts of the BHS pseudopotential for
Germanium. me and V W2 are both pictured, but lie nearly
on top of one another.

(Bottom) The various parts of the integrand of the Ziman formula

for liquid Si. S(Q) is the structure factor, from Waseda [46], Q is the
scattering vector, k, is the Fermi energy, and VIU and VVS are the
screened potentials, using the screening function of Ichumaru and Utsumi
[45] and Vashishta and Singwi [44], respectively.

46

of Q values considered by the integral. The structure factor of the liquids, measured by
Waseda [46] by X-ray diffraction, was also entered; it should be noted that the data of
Waseda for liquid Si agrees well with the recent .x-ray results of Kita et. al. [30] and with
the neutron results by Gabathuler at. al. [5].

The computation of V(r), the fourier transformation of V(r) to V(Q), the
calculation of the screening function, and finally the numerical integration of Ziman’s
formula to yield resistivity were all done with BASIC programs that were written for
these express purposes.

The value that we calculate for p for liquid Si using VS screening, 66.6 ufl cm, is
rather close to the experimental value, but the result for liquid Ge using similar screening
is not nearly so good (41.2 p0 cm). Furthermore, we found that if we used IU screening
in this calculation, we obtained a value for p for liquid Ge that was even lower by about
15 pi) cm. These screening functions are very similar, but for both liquids the small
differences between them occur in the Q range where the contribution of S(Q) to the
resistivity integral is significant (Figure 14).

It is interesting that a calculation that completely ignores bonding, such as Ziman
theory, produces a value of the resistivity close to what is actually observed. The results
of the computationally far more complex ab-initio simulations of liquid Si [32] are
marginally worse than the results of Ziman theory. It seems to be more than good
fortune, however, that the recent ab-initio results for liquid Ge by Kulkarni et. al.[47],
who have extended the simulation to several temperatures and have extracted the

temperature coefficient of the resistivity, are in excellent agreement for both p and dp/dT

47

with our experimental results. It is our hope that our new, reliable data for p of liquid
Si will provoke similar simulations, and that perhaps thermopower calculations will be
considered by these methods. We also look to the very recent development of more
effective containerless methods to explore the structure and transport of this corrosive
liquid material at extremely high temperature and in the supercooled state. Preliminary
results on levitated liquid Si [48] shown good agreement with our results for the

resistivity.

IV. Liquid Cu-Rars Earth and Ag-Rare Earth Alloys

A. Introduction

The electrical resistivities of the alloys of copper with the light rare earths (RE:
La, Ce, Nd, Pr) were experimentally studied about 25 years ago by Busch et.al. [49];
there was no attempt made to measure the thermopower. At the time, the transport
properties of the pure liquid rare earths were of significant interest because of the quite
linear variation of the resistivity across the series-of pure liquid metals from La to Er ,
which defied explanation in terms of a continuously varying spin disorder resistivity
across the series as a result of the continuous filling of the f shell. Furthermore, the
resistivities of dilute (<20 at. %) RE alloys with Cu were nearly identical, and the
temperature coefficient always small and positive, further suggesting that the f-electrons
were of little importance , and were likely significantly below the Fermi level, and close
in energy to the core electrons of the RE atoms. The resistivity data of the alloys of the
rare earths with monovalent (Cu) elements, and even polyvalent (Sn) elements, were
satisfactorily described by considering the trivalent RE’s to have 2 electrons in a virtual d
band, and the 3rd in an 3 like state, and free to conduct. Alloying of RE with monovalent
copper resulted in a liquid that had positive temperature coefficient; the alloy with
tetravalent Sn also had positive TC, except near the equiatomic concentration, where the
valence of the alloy was near 2, and the temperature coefficient negative. A theory
describing these alloys would build on the modification of Ziman theory by Evans [3]
which extended coverage to the more complicated, partially full d-shell pure transition

metals, by the incorporation of a t matrix in the place of the potentia. (Recall a

48

49

pseudopotential was used in the calculations done in the present work for liquid Si and
Ge.) A t-matrix formulation has successfully described stronger scattering liquid Ba
(p=320 uohm cm [50]). In addition, in the place of a total structure factor, there would be
a weighted sum of the partial structure factors of the constituents in the alloys. We are
not aware that any calculations were ever done by this method for comparison to
experiment.

Recent interest in Cu-Ce has focused on the stoichiometric solid compound
Cu6Ce, in both crystalline and amorphous forms; crystalline CuéCe was one of the first
Heavy F ermion (HF) compounds discovered [51]. Perhaps an even more compelling
reason for a reexamination of the liquid is the recent discovery of HF behavior in the
amorphous solid [52]. This is evidenced by a -log(T) dependance of the resistivity in the
disordered solid at very low temperatures (<10K), which is also seen in crystalline Cu6Ce
in the same temperature range, as well as in other HF compounds, and is rather a
signature of HF behavior. The amorphous compound also shows continued negative
temperature coefficient of the resistivity to 300K, the highest temperature surveyed. No
such effect is seen in amorphous CufiLa; crystalline Cu6La has not been reported as HF
compound, as the source of the HF behavior is almost certainly the Ce-contributed
localized f electron in the alloy, and it’s detailed interaction with the itinerant electrons.

With the evidence that disorder does not destroy the interesting electronic effects,
we have therefore measured the electronic properties of liquid Cu-La and Cu-Ce alloys
in a range of concentrations that encompass the stoichiometric composition CugRE. For

comparison, we have also examined the Ag-Ce system; to our knowledge no liquid alloy

50
Ag-RE has been systematically studied for either p or S, and a comparison to Cu-Ce
would be of certain interest. The measurements on Cu-RE and Ag-Ce were done in
aluminum oxide containers in a fashion very similar to the measurements on Ag-Te .
Owing to the reactivity of the rare earth materials with oxygen and water vapor, the
samples were prepared in an evacuable glovebox which was backfilled with argon to
several psi overpressure. Sample materials entered the glovebox through an airlock; both
the box and the airlock are evacuable to 20 microns. The inert atmosphere over the
sample was preserved for the short trip to the fumace by simply fastening a clamped
rubber fitting over the end of the tube before exiting the glovebox; the clamp was
loosened by a feedthrough in the furnace vacuum system after the system had been
evacuated. The rubber fitting was several inches above the water cooled O-ring (see
Figure 3) , in a relatively cool portion of the furnace. Pressures of ~10“ torr were
achieved in the furnace system before backfilling with Argon. The sample was stirred
with 0.040" Mo wire to ensure homogeneity, and to eliminate bubbles. No attack was
noted on the Mo for any Cu-RE or Ag-RE alloys. Neither was any attack noted on the
alumina or the graphite plugs; the post experiment solidified samples were seen to be
uniform and well stirred, devoid of bubbles, and were separated from the alumina tube
without difficulty. Materials were supplied by AESAR/ Johnson Matthey; lot analyses
of La and Ce yielded impurity levels of 60 and 200 ppm (wt), respectively; Ag and Cu,
from the same supplier, were of 99.99% and 99.999% purity, respectively.

mm

The results for the resistivity measurements on liquid Cu-RE and Ag-Ce alloys

51

are shown in Figure 15. In all of alloys, the resistivity, p, rises smoothly and quickly as
RE is added, with the p(c) curves of the different alloys being essentially
indistinguishable until ~30 at. % RE. At no time does the resistivity exceed ~200 ufl-cm,
even in the neighborhood of the compounds CuORE or CquE, which correspond to solid
crystalline compounds that exist up until the melting point. Our results for p are in
excellent general agreement with the previous results of Busch et.al. [49] for lower RE
concentrations, being uniformly increasing with increasing RE content, although they
appear to be consistently ~5% above his values. We do note, however, that the unnamed
ceramic material that Busch et.al. used for containment was subject to some corrosion,
and that the RE materials used in the study were of lower purity than our own.

The thermopower (Figure 16) is as featureless as the resistivity, and remains
small and metallic in magnitude at all concentrations studied. The temperature
coefficient of the resistivity, however, displays sharp structure in the neighborhood of
Cu6RE, dropping in both Ce and La alloys to a narrow minimum, which furthermore is
negative in CufiCe (Figure 17). It is highly unusual to see such a small resistivity ( 95
uQ-cm) along with a Negative Temperature coefficient (NTC). NTC was also seen in a
comparatively broad range of concentrations surrounding the congruently melting
compound Ag3Ce, although the same magnitude of NTC as is seen in CugCe was only
reached when p had exceeded 140 uQ-cm (in comparison to 95 tin-cm for CufiCe).

For both CugLa and Cu6Ce, an attempt was made to measure the properties of the
high temperature solid material as well as the liquid, as current data for the solid extends

to only roughly 300°K. Visual inspection of the solidified sample revealed that it was a

Figure 15 -

Figure 16 -

52

 

 

 

 

 

 

 

 

 

0.8 1
E.
U)
1:?"
- 10 - : - t - : . : .
0 0.2 0.4 0.6 0.8 1

(Top) Current data for resistivity, p, of Cu-Rare Earth alloys,
evaluated at 950°C for Cu-La (I ) and Cu-Ce ( o ) , and at
1050°C for Ag-Ce ( v ) . Cu-Nd ( A ) (Busch et.al. [48]) is
at 1000°C.

(Bottom) The thermOpower, S, of Cu-Ce, Cu-La, and Ag-Ce.
The symbols and temperatures are the same as for Figure 15.

Error bars are calculated from a linear fit to all of the data for
each concentration, and unless shown are smaller than the

extent of the point.

53

 

 

 

 

 

 

Figure 17 - a= l/p dp/dT for Cu-Ce, Cu-La, and Ag-Ce. Temperatures and
symbols are the same as for Figure 15.

54

highly uniform polycrystalline sample. Because of differing expansion coefficients, we
were unable to establish contact with the entubed solidified sample. In several instances,
we were able to remove the sample from the tube in one ~ 6" cylindrical rod, and directly
fasten wires to it, but our measurements were not reproducible for either Ce or La
samples. Interestingly, amorphous Cu6La was noted by Suzuki et.al. [52] to be extremely
brittle and prone to cracking; in fact, microcracks were blamed for unusual electrical
properties in samples of this material.
i i rt Me urement

The NTC’s in Ag-Ce and Cu-Ce are in both cases of the same order of magnitude
seen in the pure divalent liquids Sr and Zn, and in the divalent alloy SnCe. One of the
great successes of Ziman theory [53] is the explanation of NTC in divalent liquid metals
and alloys, which occurs as a result of the temperature dependance of the main peak in
the structure factor, S(Q), and it’s proximity to 2k, (where kf is the Fermi wave vector).
In pure metals and in alloys, the structure factor generally exhibits temperature
dependence in the form of both lowering and broadening of the main peak, a result of
modification of the interatomic distances with temperature. In divalent metals, the upper
limit of the resistivity integral, 2k,, is located in such a position that the contribution of
the lowered and broadened main peak is lessened as the temperature increases, leading to
NTC. The valence of an element is often inferred from the oxidation state of a material;
for example, the rare earths, except for Eu and Yb, are thought to be trivalent. In Cu6Ce
and Ag3Ce, however, a concentration weighted average of the valence of the constituents,

using 3 for Ce and l for Cu and Ag, yields a valence of 1.3 and 1.5 respectively, rather

55

too far from divalence to be explained by Ziman formula. (It should be noted that
divalent metals considered with t-matrix formalism are affected in the same way by the
location of 2kf with regards to the main peak as materials able to be considered in the
original theory using pseudopotentials.) The early suggestion of t-matrix treatment by
Busch et.al. appears inadequate for this situation.

Quite recently, several studies of the dilute transition metals with the noble
metals, (for example, liquid Au-F e [54] and Cu-F e [55]) have revealed an overall
concentration dependance of the resistivity very similar to that seen for Cu-RE, and have
for Au—Fe revealed that the temperature coefficient becomes negative at roughly 8 at.%
Fe, reaching a minimum at about 15 at. %. Here, as in Ce-Cu, the minimum temperature
coefficient is associated with an unusually small resistivity, about 90 ufl-cm, although the
minimum seen in both noble metal-iron compounds appears to be significantly wider in
concentration than in Cu-Ce. A calculation has been done using t-matrices, assuming
the adequacy of hard sphere structure factors of Ashcroft and Langreth , and making the
assumption that Fe is monovalent , and has found that Evans theory has significantly
underestimated the resistivity at all concentrations, and was unable to reproduce the
observed NTC. Further analysis revealed that the NTC may be tied to the spin disorder
resistivity, as the transition metal atoms appear from magnetic susceptibility
measurements to retain a significant moment due to their d electrons even in the alloy.
Susceptibility measurements on liquid alloys of Ce with Cu, and with La and Pr have
indeed shown that the magnetic moment changes nearly linearly with concentration, and

that the source of the moment is f-electrons [56]. However, if spin disorder effects from

56

the filling f band were significant to transport in the Cu-RE systems, the resistivity of
dilute Cu-RE alloys would vary greatly with differing RE; for our data, the concentration
curves of the various RE alloys are nearly indistinguishable in this regime.

NTC’s are also expected and observed for liquid and amorphous metals and
alloys for which Ef falls approximately at a fairly deep minimum in the Density of States
(DOS). Fresard, Beck, and Itoh [57] have considered an elastic scattering model, and
have used the full Fermi Dirac distribution to calculate the temperature dependance of the
resistivity. They have found that their model does indeed reproduce NTC at high
temperature, but only in cases where p >400 uQ-cm. They also note an extreme
sensitivity in the thermopower to the features in the DOS. Neither property describes our
experimental data.

NTC may also result for liquids fi'om the release of electrons from ionic or
covalent bonds into the conduction band as temperature increases. These NTC’s tend to
be quite sharply concentration dependent and occur most frequently at maxima in the
melting curves where congruently melting solid crystal structures exist. In fact, extrema
in p(c) and evidence for bonding in the structure factor S(Q) are so often seen together,
that some workers have defined new stoichiometric compounds by the concentration
positions of these extrema in cases where diffraction studies are difficult, and have even
qualified the strength of the bonding by the relative heights of the peaks in the resistivity
[58]. No feature is apparent in p(c) at stoichiometry in either Cu-Ce or Ag-Ce, and the
resistivity in the neighborhood of these compositions is quite low (~100 [IO-cm) to be

associated with strongly bOund electrons.

57

The nature of the phase diagram of Cu-RE and Ag-RE alloys (Figure 18),
especially the presence of a congruently melting solid compound at CugRE, suggest that
it is possible that short range order exists in the liquid. According to Massalski [59], no
compound Ag3Ce has been identified, although the highest point of the melting curve is
centered precisely at this concentration; an interrnetallic compound Ag5,Ce,4 is located in
this area. It is worth noting, however, that phase diagrams have incompletelyiidentified
solid compounds before, so much so as to make Xu [58] comment, “Evidently the results
of thermodynamic and crystallographic investigations are almost incongruous.” Many of
the liquid compounds identified in the work of Xu were not signaled by recorded
congruently melting solids, but were discovered in part by extrema in the electronic
properties of the liquid.

WWW

To pursue the notion of CSRO further, we have undertaken a neutron scattering
study of liquid CuGCe at the Glass, Liquid, and Amorphous Diffractometer (GLAD) at
the Intense Pulsed Neutron Source at Argonne National Lab. Sample preparation was
conducted in our lab, the pure ingredients being melted and stirred in an alumina tube,
following glovebox preparation similar to the samples used in the electrical
measurements. The mixed sample was then transferred in the glovebox from alumina to a
vanadium tube, and the tube was sealed under argon atmosphere by TIG welding
vanadium plugs into the ends. There was concern of the solubility of the vanadium
container in Ce, as a thin band of concentrations with comparatively low melting

temperature exists for dilute V in Ce. However, immersion of a known mass of V in

Temperature ‘C

Temperature 0C

58

Weight Percent Cerium

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

to 20 no to so so 70 so 90 10
:29!) r '1 r I r v 1 II I I y L '1 I
1113
roum‘c ’"
rooo- .
a: 'c
° 3
‘ 414
500- _.
x \/
/’ an
MC
0
son 9'
' 5
o‘\
400-
8 8 8
*(Cu) 3' :7 5'
U U U
200- ,
er'c
,. (flCe)-
1'0 20 3‘0 [0 50 6'0 7‘0 5'0 9'0 it 0
Cu Atomic Percent Cerium Ce
Weight Percent Cerium
( 10 an 30 40 so so 70 100
:m I. ' . 1 . 1 . ' . ' ‘ . .
noo- _
.c' ‘
:"1 i
c ,1 II r
.'
_ l _
90° ' ufc
m 1
F ni'c
~ono .
rn :- 4a (a)
“rar’c
7w« 0 _
z 8
5- a
3 .
o < safe
0 513°C
50°“ .3 m '1 A "
< o ‘.
‘3: 8 I.
‘1“) a g (“Kiel—'7'
M ii) 20 ab 4'0 50 ab 7‘0 so 90 mo
Ag Atomic Percent Cerium I Ce

Figure 18 - The phase diagrams of Cu-Ce'and Ag-Ce. All of the Cu-RE
compounds considered here have a congruently melting crystalline
compound at Cu6RE (From [59]) . ‘

59

liquid Cu6Ce for an extended time period, and dissolving away of the post experiment
solidified Cu-Ce with aqua regia (50% HCl, 50% H2N03; vanadium is immune to this)
and subsequent re-weighing, revealed no significant dissolving of the V by the liquid.
Another empty tube of the same dimensions, also sealed on both ends, was prepared so
that a separate scattering experiment might be done on it, and the tube scattering might be
subtracted from the total sample plus tube signal. On each tube, one of the end pieces had
a threaded hole to support the tube in the hanging position when it was in the neutron
beam. This also allowed for the very precise positioning of both the empty and full tubes,
which is essential for proper subtraction of the tube contribution. The tube was
sufficiently long that the neutron beam, which at position of incidence measures roughly
lcm wide by 5 cm high, would impinge only on the smooth length of the cylindrical
container, and would not strike the welded ends, which were not uniform from one tube
to the next. The temperature was controlled to within 3:1 C° by a furnace consisting of
several concentric, cylindrical vanadium sheets; the container holding the furnace was
evacuated to ~10'3 torr while the measurements were being taken. Finally, the scattering
of the empty furnace had to be measured, as it also is subtracted in the analysis. All
scattering experiments were conducted in the so [called “downstream position”, meaning
the sample is several meters closer to the detector than in the upstream position. In the
downstream position, the detectors subtend an angular range 45 °< 2B <115 °.

The sample, empty tube, and furnace data provide a portion of the information
required to extract the structure factor from the observed scattered intensity. In addition ,

another scattering experiment was done on a solid vanadium piece of the same

60

dimensions as the above sample tubes. Vanadium is a nearly completely incoherent
scatterer, so the data obtained by scattering from a standard sample of the same geometry
as the actual sample may be used to normalize the total scattered intensity. A standard
data analysis package was used to treat the results. Corrections for the absorption of the
furnace, tube and sample require input of the radii of the various annuli of the concentric
cylindrical sample, container, and firmace, as well as the number density of scatterers in
each material in order to calculate the multiple scattering and attenuation in these
elements. A concentration weighted density of p=0.063 scatterers/A3 for liquid Cu6Ce
was used, as no experimental liquid density was available; this is slightly less dense than
the crystalline Cu6Ce density, about 0.065 scatterers/A3. A final portion of the analysis
is to correct for inelastic scattering using Placzek corrections.

We choose to define the structure factor in terms of the Aschroft Langreth partial

structure factors S,,(Q)

2
S(Q) = Z c..c,.b,.b,S,.,.(Q>

r1=r

where c, is the atomic percentage of a given constituent, bi is the neutron scattering

length of the same constituent, Q is the momentum transfer, and

N:

N
sum) #111,115)"2 (2 X: exp(-iQ. (ru-rjm))) - (NIN )t‘aQ.0

[=1 m=1

In this expression, N, is the number density of scattering centers, and r,, is the vector

61

between adjacent atoms, and again the vector nature of Q in S(Q) has been removed
because of the isotropy of the liquid. Figure 19 is the experimental structure factor
obtained from the measured cross section by
do/dQ = (b)2 [S(Q)-1] + (b2)
where <b> = c,bl +c2b2 .

A fairly strong prepeak exists at Q ~ 1.6 A“, and is followed by the main peak at
2.5 A". Although our scattering experiment involved only the total structure factor of
Cu6Ce, the x-ray diffraction results of Waseda [46] for liquid Cu-Ce alloys display no
evidence of such a prepeak in any of the partial structure factors S,,(Q) even at the nearby
concentration 25 at. % Ce; this is significant in that by 25 at. % Ce, the temperature
coefficient of the resistivity has risen to positive values. In all of the partial S(Q)’s from
the work of Waseda, the main peak stays at essentially the same Q value for 20 at.%<Ce
content<80 at. %. It appears then that the intermediate range ordering indicated by the
prepeak at low momentum transfer is confined only to compounds around the highest
peak of the melting curve.

The pair correlation function may be obtained by fourier transform of the

experimental structure factor

 

g(r)=1+ ‘ f(S(Q)-1)flg(£r-)-de9
r

2
2“"00

Here p() is the density in scatterers/ A3. In this case the integral is over all Q, but a real

experiment may have data only up to Q~40A“. We have used a Lorch function to de-

0.5 --

 

 

 

8(01-!
o

 

 

 

 

 

_
I

 

 

 

 

 

.0 5 IO . 15 20 25
Q(A")

Figure 19 - (Top) The structure factor S(Q) -l of Cu6Ce, measured by neutron
diffraction. The solid line shows the experimental results, the
dashed line is the result of RMC analysis of the experimental data

Figure 21 - (Bottom) The RMC partial structure factor, SCuC,(Q). Here, the solid line is
the simulation started from the crystalline arrangement, and the
dashed line is that started fi'om the random arrangement

63

emphasize the structure factor at high Q and to minimize the truncation errors in the
integral. The pair correlation function may also be expressed as a sum of the partial pair
correlations

2
g(r) = Z cicjbibj gU.(r)

i,j=1

with c and b the same as before.

The total correlation function, T(r) = r*g(r) is shown in Figure 20. An inspection
of the high r side of the main peak in T(r) reveals a shoulder at roughly 3 A, which is
explained as follows. We have fit gaussians beneath T(r); the difference curve between
the fitted distributions and the experimental T(r) is also shown in the figure, and
illustrates that in the region below ~3.5 A , 2 gaussians yield a very satisfactory fit. The

shape of the gaussians indicates the distribution of interatomic separations. The area

rl ll

C=41tp0 fr2g(r)dr

rain

with p0 being the density, and the limits of integration defined by the width of the
gaussian, is related to the average number of atoms within rmin < r < rmax around another

atom by the following formula

C:

 

1 Z —0_ _
_ c b, bC,(J)
(b)2 :31 i I

64

.1on Co enema: age as a
.7va 528:3 cone—oboe :3 BE. .0050 .3 3m _. .._ .1. 3H - om 233m

_ gm .
on one 0m 0m 9

 

0w

9 8 .
. . xv. 111 >1>

 

b.\o’Lo 5o '. o
’1/ .\ . .A.... la...\
I \\
. I a \\
. / .
vi \
c. ., ..
a s
.c , “a
\ x .
c x / .
s .
.
.a ..
C

 

 

 

65
where C, is the coordination of the ith type of atom.

The first gaussian in our data is certainly due to Cu-Cu correlations exclusively,
as in pure liquid copper the main peak in g(r) is at 2.5 A. The second gaussian, fitted
beneath the shoulder, probably contains both Cu-Cu and Cu-Ce correlations, as both
types of pairs exist at distances of 3-3.4A in the crystal; however, the scattering length of
Cu is roughly twice that of Ce , and Cu is present in far greater numbers. If we consider
the weight of the CuCe correlations in the second peak to be quite small, and the peak to
consist solely of CuCu correlations, we calculate a CuCu coordination from the area of
both of the gaussians to be 10.7, which compares favorably with the average coordination
of Cu to Cu in the crystal, about 9.3. Proper accounting of the CuCe coordination
represented in the second gaussian would of course bring the liquid coordination even
closer to that seen in the crystal, and further from the 10-12 nearest neighbors seen in a
typical close packed liquid. Certainly, varying the weighting of the partials is possible by
either isotopic substitution and neutron scattering, or by combining our present neutron
data with X-ray data. Neither option was viable at the time of this experiment. We
have instead done a Reverse Monte Carlo (RMC) simulation on the total structure factor,
which in previous studies on alloys has yielded a fair appraisal of the structure even from
one set of scattering data [20].

d re ult
RMC is a program which considers the structure factor, as defined above, of a
representative model ensemble of atoms. These atoms are moved singly in accordance

with user specified constraints such as closest distance of approach and coordination

66

number, and after each move a comparison of the ensemble structure to the experimental
structure is made, with the move accepted if it results in a lessening of the difference
between the two, as measured by the quality of fit parameter

r [SC(Q,-) -SE(Q,)1
2:2 2
i=1 0(9)

 

X

It is also accepted with a probability exp[-(x2-x’2)/2] if x2 increases (x ’ is the new x) ,
but rejected if no change occurs. Here SC(Q ,) and SE(Q ,) are the computed and
experimental structure factors, respectively. The simulation is continued from the
modified ensemble, until x2 varies randomly about a constant value. The ensemble is
then in a configuration that will yield the experimental structure factor, although this
ensemble is by no means the only configuration that can yield a particular structure
factor. The partial pair correlation functions g,,(r) are readily calculated fiom this
configuration of atoms. RMC has proven in some cases to yield results for partial pair
correlation that are nearly as good as those obtained by a proper isotopic substitution
experiment, but experience varies according to the quality of the data, and according to
the relative weighting of the partial components of the total structure factor. We expect
that the weighting factors will make the CuCu correlations most readily modeled, and
CuCe passably so; the weightings of the partials are as follows: Cu-Cu=.4375, Cu-
Ce=.0916, and Ce-Ce=.00479.

Two different initial configurations were selected for our simulations; these will

be referred to in what follows as case 1 and case 2. l) A random arrangement of 1050

67

atoms of the stoichiometric concentration, with minimtun distances of approach
rCuCu=2A, rCuCe=2'55A9 and rC,C,=2.85A, which are very similar to the first peak positions
of liquid CuJSCeZ, according to the results of Waseda [45], but are less than the distances
rCuCu=2.5 A, rCuCC=2.9A, and rCeCe=3'1A seen in the crystalline solid. 2) A crystalline
configuration of CuéCe [59], with 3500 atoms and with the minimum distances of
approach the same as above. The program fitted the configurations to the experimental
structure factor data; the results have been shown in Figure 19 by the dashed line. The
two simulations ultimately converged to x2 values within 1% of one another, and both
simulations were successful in reproducing the prepeak.

The main peak of the total S(Q) was well reproduced in height, width, and
position for both starting configurations, although the following peaks were not modeled
as effectively. In both simulations, the prepeak is placed at the correct Q value, but as a
whole is rather too high, when compared to the main peak. The height of the pre peak in
the total structure factor agrees well with experiment, although is perhaps minimally
better modeled by case 2; the use of a larger configuration in an attempt to minimize the
ripples had no effect, so for the random arrangement a smaller configuration was used for
rapidity of convergence. In both cases, however, the prepeak is unmistakable in ScuCu(Q)
(Figure 21) , and substantially larger, relatively speaking, than in the total S(Q), because
of a very deep minimum in SCuC,(Q) at the same Q. This serves to diminish the height of
the pre peak in the total structure factor.

The depth of this minimum in SCuC,(Q) is not significantly affected by varying the

minimum approach reuce between 2.55A and 2.9A, although the post simulation partial

68

gCuC,(r) looks rather unnatural and non physical for r<2.9A; by this we mean that
although the distribution shows that a significant portion of the Cu-Ce pairs remain at
distances 2.9A in a comparatively broad peak, a spike appears in the distribution at the
minimum distance of approach. The CuCe partial appears to be weighted just heavily
enough to be adequately modeled. The Ce atoms collect artificially at the lowest allowed
distance of approach because of the extremely small weighting of SCeC,(Q), and these
correlations will not be discussed further. It is interesting that the variation of CeCe and
CeCu minimmn distances of approach had no noticeable effect on the position of the
prepeak or the main peak in SCuCu(Q). To check whether this was plausible, we have
examined a simple hard sphere, close packed arrangement according to the Percus-Yevick
approach [ see, for example, Waseda [46]). This calculation, like RMC, showed that the
CuCu structure in a close packing scheme was unaffected by changes in the sphere
diameter of Ce, but further was unable to reproduce the prepeak.

It is worth noting that for both case 1 and case 2, RMC gc,,c,,(r) (Figure 22) has a
peak at 2.5 A, and that the high r side of this peak is significantly less sharply inclined
than the low r side, which is rather suggestive of a distribution of CuCu nearest neighbor
distances with significant numbers at about 2.9A, in agreement with the shoulder seen in
the experimental g(r). We have calculated the average CuCu coordination number from
RMC, using the same rmax for the limit of the integral in each case, and have found each
copper to have 12.8 neighbors for the simulation starting from case 1, and 12.1 for case 2.
Table 1 shows that although the average coordination of these cases after simulation are

very similar, careful consideration of the coordination of each individual Cu atom will

69

 

 

3.5

2.5 f.

g(r)

0
15..
O
D

 

0.5 I

 

 

 

F (A)
Figure 22 - The partial pair correlation function, gCuC,(r). Solid and dashed
lines are the same as for Figure 21.

1.5' l u...
" I

 

 

 

 

 

-05 "- iii/l illli I” 1111111 Iii/L

150 200

 

O 50 100
Angle (deg)

Figure 23 - The RMC bond angle distribution for random (top), post simulation
(middle), and crystalline (bottom) Cu6Ce.

70

Table 1 - The RMC nearest neighbor distribution for Cu-Cu coordinations, including the
results of the simulations started from a random (column 2) and a crystalline (column 3)
arrangement. The first column is the coordination number, the remaining columns express
the percentage of Cu atoms with the associated coordination. The coordination is
calculated by counting Cu atoms within a spherical shell centered on a Cu atom, of outer
radius 4 A, and inner radius 2 A. It is seen that there is less spread in the coordination
number for the simulation started from crystalline arrangement, and that the average
coordination is lower, although the quality of fit parameter for both simulations was

nearly equal.

 

Won PercentofTotaLCasel e ce t t tal ase2
5 0.1 0

6 0.3 0

7 1.0 0

8 3.0 0

9 4.9 1.5
10 9.3 10.1
11 11.4 23.8
12 13.3 26.5
13 16.6 21.4
14 14.3 11.6
15 12.0 . 4.5
16 6.7 0.5
17 4.7 0.1
18 1.2 0

19 0.7 0
Average 12.8 12.1

Coordination

71

reveal that a greater percentage of the case 2 Cu atoms are lower coordinated than case 1
Cu atoms. This is a possible reason for the slightly stronger prepeak in the case 2
Scucu(Q)-

RMC may also yield information on experimentally unobtainable bond angle
distributions in the liquid, and herein lies one of the strongest pieces of evidence for
CSRO in liquid CuCe. Figure 23 shows the CuCuCu bond angle distributions, measured
for atoms up to 4A away from one another to include all of the atoms presumed held in
the second gaussian in T(r). Shown are the initial crystalline arrangement, the post
simulation ensembles of case 1 and 2, and the initial random arrangement subjected
only to moveout of atoms to minimum distances of approach . A completely random
arrangement of atoms will have a flat distribution, and this is what we see for the initial
random arrangement. After the simulation, for both cases 1 and 2, there are obviously
preferred angles in the liquid; the bond angle distribution has rather broad peaks of
roughly the same magnitude at 55° and at 105 °, which correspond nicely to the sharper
distribution of peaks shown for the crystal. The crystalline CuGCe arrangement is shown
in Figure 24; it is apparent from this that 105 ° is not merely ~2*55 °, but rather that the
angles are independent of one another, and serve to define the corners of a parallelogram
of 4 coppers, with the sides of the parallelogram of the lengths seen in the CuCu g(r).

E, Discussion

A first sharp diffraction peak in S(Q) at low momentum transfer (0.7-1.0 A") has
in many studies of molten materials been associated with the presence of clustering, and

is very often associated with extrema in the electronic properties, the nuclear magnetic

 

 

 

 

 

CEDG G) ®®©® @Q

.4

 

’J‘fi

 

 

Cell content

Number Atom Coordinates
x y x
1 Ce .7602 1I4 .0648
.2602 1/4 .4354
.7398 .'4 .5648
.2398 3/4 .9354
2 Cut .1467 1/4 .1418
.6467 1I4 .3582
.3533 314 .8418
.8533 314 .8582
3 Cu2 .1821 314 .2450
.6821 3/4 .2550
.3179 1/4 .7450
.8179 1/4 .7550
4 Cu3 .9381 3/4 .0976
.4381 - 3/4 .4024
.5619 114 .5976
.0619 1/4 .9024
5 CM .5986 3M .0154
.0986 3/4 .4848
.9014 1/4 .5154
.4014 1/4 .9848
6 C05 .4354 .0042 .1908
.4354 .4958 .1908
.9354 .4958 .3092
.9354 .0042 .3092
.0648 .5042 .6908
.0646 .9958 .6908
.5646 .5042 .8092
.5648 .9958 .8092

Figure 24 - The crystal structure of CugCe (From [62])

73

resonance spectrum, and even in the density . For example, resistivity measurements
coupled with neutron diffraction by Xu [5 8] has revealed what is thought to be the
presence of clustering in equiatomic alkali-Sn and alkali-Pb compounds, so called Zintl
alloys. In these cases, it is proposed that covalently bonded tetrahedral clusters of (Pb,)“‘
form, which are separated by the alkali ions. The peak at low Q in S(Q) is a result of
the inter-cluster distance; the other peaks are interpreted as the intra-cluster distances.
There are also documented cases where a smaller peak is located on the low Q
side of the main peak at perhaps slightly higher Q values than the clustering features
defined previously, and these compounds are often thought to display chemical short
range order. For example, the liquid compound Mg-Bi, discussed briefly in the Ag-Te
section, displays in the structure factor of the stoichiometric compound Mg3Biz a prepeak
at 1.8 A", very similar to the position of our prepeak (1.6 A"). Guo et.al. [20] have
shown that RMC analysis of a single total structure factor (in this case as well as ours,
an isotopic substitution experiment wasn’t possible), with a starting configuration
modeled after the fast ion conducting high temperature [3 phase of the material, shows
that the prepeak is present in the Mg-Mg partial structure factor, and is significantly
decreased in height in the total structure factor by a deep minimum in SMgB, (Q). These
precise structural characteristics have been identified by Enderby and Barnes [1] to signal
heterocoordination in numerous ionic liquids, and are similar to the outcome of our own
RMC analysis. It is quite difficult to consider Cu-Ce ionic; we suggest that the
proposed ordering in our system is probably due to covalent bonding that is not

sufficiently strong to open an energy gap or pseudo gap in the density of states, and thus

74

 

 

 

 

 

 

 

 

 

 

Figure 25 - The resistivity, thermopower, and temperature coefficient
of the resistivity for Al-Ca alloy from Zuo et.al. [60], evaluated at
T=1079°C. '

75

Al-Ce Phase Diagram

Weight Percent Calcium

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

”M 1,0 2,0 :10 4'0 510 so 7P 6‘0 03 mo
noo-
rooo-
o
o 900.
U
L.
3
u we
I-
U
D.
E
o voo-
[—
500.
m.
'"r'ozoa'ot'os'oe'or'oe'ooorrI-o
Al Atomic Percent Calcium Ca
Ca-Ga PhaseDiagnm
Weight Percent Gallium
(10439405969798? 9.0 mo
tooth 990°C _
are
900- L first _
m
o
O
a / n "
eoo- I .3“ I ‘ v _
§ ‘0'“) ” °- -—3 a
E. 512': 6 <5
E “a“
g 400- _
a I“ a ‘ g
tac- ‘%‘-’n ‘4. o .—
m- ‘ ’ a a a a 9 -
'eernu’c
c . . . . , c.)
i 10 an no go an en 70 no so no
Ca Atomic Percent Gallium Ca

Figure 26 - The phase diagram of Al-Ca and Ga-Ca (From [59]).

76

measurably effect either p or S. Evidence for liquid compounds is often seen near
melting curve peaks; the phase diagram for Cu compounds with La, Ce, Nd, and Pr are
very similar to one another, and all do form congruently melting solid compounds at
Cu6RE.

We suggest that or is perhaps a more sensitive indicator of CSRO in liquid alloys
than either p or S, and in alloys where short lived temperature dependant covalent bonds
are suspected, inspection of a for extrema may be the preferred method for identification
of liquid covalent compounds. We present as further evidence of the effectiveness of or
as an indicator of CSRO, an earlier study by Zuo et.al. [61] on liquid AIZCa. Al-Ca also
possesses small p (150 p!) cm) and NTC (-9 xlO“ C°") at stoichiometry (Figure 25)
(The stoichiometric alloy is a congruently melting solid with a high melting point (Figure
26).), but no visible extrema in p or S. A study of the substitutional alloy GaxA167,,Ca
by You et.al. [62] shows that [or] is fairly rapidly diminished as x is increased (Figure 27),

suggesting that the AlzCa association is broken quickly as Ga is added, and that
complexes of AlGaCa do not form. The percentage change in or is far larger as CSRO is
disrupted than the change in either p or S, although in this case it does change by 15 % as
the AlzCa complexes are disrupted by the additional Ga (Figure 28).

A further result of You’s study is that CSRO does not necessarily result in NTC,
but can certainly lead to depressed temperature coefficient, when compared to values at
surrounding concentrations . A systematic study of liquid Ga-Ca alloys has not been
done, but we do notice that as Ga is substituted for Al in the ternary alloy, da/dc displays

a marked change from linearity several atomic percent Ga before all of the Al had been

77

 

 

 

 

 

 

 

 

 

 

-6 1—

...8 .... '—
-10 T
1 L 1 I t l
0 15 30 45 60 75

5‘1' 0 :3 Ga

Figure 27 - The temperature coefficient of the resistivity of Al,,,Ga,,Ca as
a function of concentration of Ga from You et.al. [61], evaluated at

T=1079°C.

78

 

 

 

 

 

 

 

s (,rv °c"')

 

 

 

 

 

 

 

 

 

 

 

O
6‘
()1
O
N
.(s
[3 u.
(D
O
\l
U!

5T

Figure 28 - The resistivity and therrn0power of Al,.,Ga,Ca.

79

replaced. Furthermore, or for GaZCa is significantly lower than an extension of this line
would suggest. We compare this case of depressed temperature coefficient to our data for
liquid CuéLa, and even to the data of Busch for Cu6Nd; for both, or is positive, but clearly
at a minimum.

We have presented significant evidence that CSRO is responsible for the NTC in
liquid Cu-Ce; although structural studies have not been carried out on Ag-Ce, we expect
the same explanation to be valid in that system. Perhaps the strongest argument in favor
of CSRO in liquid CugCe is the close similarity of the CuCuCu bond angle distributions
for the RMC generated data and the crystalline material, as they emerge without special

provocation from the simulation.

V. . L IONS

It is apparent from our work that one must move only slightly away from simple
liquids or ionic liquids, the end points ofthe bonding spectrum, before theoretical
inadequacy results. Cu-RE, Ag-RE, Si, and Ge are several ofa growing set of liquids that
demonstrate weak covalency, and are only marginally described by Ziman theory. At the
other extreme, some authors have suggested that Ag-Te will only be adequately described
using covalent corrections to complete ionicity. Unfortunately, it is obvious that such
covalent considerations lend considerable difficulty to the theoretical description.

We may look with cautious Optimism to the relatively new ab-initio techniques that
consider the electronic and ionic motion on an equal basis, and have thus achieved some
success in modeling the atomic structure, density of states, and conductivity. The method is
vulnerable, however, in that it uses the local density approximation, which may be invalid at
critical concentrations like Ange. It is also tremendously computationally expensive,
especially for situations where higher shell electrons must be considered; f-electron
compounds are currently beyond reach, and d-electrons have only recently been considered
for the first time.

We expect that more weakly bonded liquids will be revealed as experimental
methods and material purity improve, and make visible the somewhat subtle electronic
effects. The increasingly intense X-ray and neutron sources, when used in conjunction, are
capable of revealing the full structure ofthe liquids. It is apparent even from the data in this
dissertation that any description of weak bonding in liquids will hinge on both ofthese

important experimental data sets.

80

A’A‘. 9n"

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Eur. nwisug‘ ~
. ..a
I

Ix)

10.

ll.

12.

l3.

l4.

l6.

l7.

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D. Zuo, T. Vos, H. Nymeyer, L. Reynolds, H.S. Schnyders, J. B. VanZytveld,
1996, J. Non-Cryst. Solids 205-207.

D. You, HS. Schnyders, J.B. VanZytveld, 1997, J. Phys. Condens. Matter 9, 407.

Atlas gf Crystal Structure TVpes for Intennetallic Phases, ASM International,

Materials Park, OH, 1991.

~

.'.MIA$.49o 1.1

.2
', blufll’l‘luf.

 

APPENDICES

J. Phys: Condens. Matter 3 (mo) roars-loss). Primed in the UK

Electrical resistivity and thermopower of liquid Ge and Si

H S Schnyders? and J B Van Zytveldi‘.
Physics Depanment. Calvin College. Grand Rapids. MI 49546. USA

Received 2 July 1996

Abstract. The electrical resistivity p and thennopower S of pure liquid silicon and pure liquid
germanium have been carefully measured. For silicon..a new containment material was used.
namely high-density graphite. This graphite has a low thermopower (6 “V ’C'" at 1500 ’C)
a high resistivity (1000 an cm at ISOO‘C). and little or no reaction with Si. making it an
ideal containment material. The results for each liquid show a metallic value of resistivity.
a small but positive temperature coefficient of the resistiVity and a small thermopower. In
particular. for liquid 51. p .. 75.2 4.- 0.5 an em. dp/dT - (1.7 .4.- o.9) x 10‘: an cm’C"
and s - -2.l s.- 0.6 a we“ and. for liquid Ge. p - 66.8 2 0.2 an em. dp/dT -
(17:02)): 10-: an cin=c-' and s - -o.a 205 a v=c'; all values are for the respccdve
melting temperatures of Si and Ge. We also report a calculation of the resistivity of each liquid.
using the Ziman fomialism. with a recent pseudopotential and an experimental strucrure factor.
Both our experimental and our calculated results are compared with other work.

1. Introduction

The properties of liquid Si (l-Si and liquid Ge (t-Ge) have recently become the focus of new
attention with the advent of successful new computing techniques which are yielding results
previously unavailable. Among the unusual characteristics of these liquids which the new
theoretical methods address are neutron and x-ray structure factors A(Q). with a substantial
shoulder on the high-Q side of the main peak. yielding a radial distribution function with a
first-nearestoneighbour coordination as low as 5 [I]; a value 0f approximately 12 is normally
associated with a close~packed simple liquid structure. This low co—ordination implies an
open structure and perhaps indicates that some remants of the crystalline tetrahedral network
remain. Early attempts to explain the structure facror of these liqids with multiple-diameter
hard-sphere modelling [2] reproduced the first peak of A( Q) fairly well. but became worse at
higher Q. Other attempts suggests two kinds of coordination. with some of the atoms being
fourfold co—ordinated and some being close packed [3]. More recently. Jank and Hafner
[4] calculated the structure of these liquids. using pseudopOtential perturbation theory (PPT)
with a simple local Ashcroft pseudopotential; they concluded that the unusual coordination
numbers of (Si and l-Ge are a result of Freidel oscillations superimposed on the repulsive
hump of the interatomic potential, and that the balance of volume energy and this porential
leads to the splitting of the firsr coordination shell. The results for this calculation of the
structure factor are in good agreement with the especially interesting new work of Stich
er a! [5], who have followed an air-initio approach. working from local density functional

1 Also at Physics and Astonomy. Michigan State University. East Lansing. MI 43324. USA.
t Present address: M l Murdock Charitable Trust. PO Box 1618. Vancouver. WA 98668. USA.

oess-a9amasoroavsw9swso on 199610? Publishing Ltd 10875

85

 

86

10376 H S Schnyders and J 8 Van Zyrveld

(LDF) theory and using Standard equations of motion for the ionic and for the electronic
degrees of freedom. In addition to A(Q). their model yields information on the elecuonic
charge density dynamics; this displays a build-up of charge between neighbouring atoms
in t-Si that is interpreted with the aid of the triplet coordination function as the remnants
of tetrahedral bonding in the liquid. Further Study of (Ce by Kresse and Hafner [6] using
the LDF has validated the earlier PPT study [4] and has also produced a self-diffusion

coefficient that agrees well with experiment.
These computer models have also shown that the density-ofostates (DOS) function

of silicon becomes metallic and nearly free electron like upon melting in spite of the
unusual coordination [4,7]. losing the sharp features associated with 5pJ bonding in the
solid. Although measured ultraviolet photoemission spectra have shown that the DOS for
l-Ge has a minimum between the s and p states [8]. the minimum is well away from E;
(calculated from nearly-free-electron theory) and is not expected to affect the electronic
transport properties.

Evidence such as this would suggest that the Ziman [9] theory, with an adequate
pseudopotential and an accurate Structure factor. could provide a relatively good calculation
of the electronic transport properties for these liquids. In the Ziman formulation. the

resistivity p for a pure liquid metal may be written

321 N l a" 3 2
= ——--—«r A u d (I)
p elm} V m [a Q (0)1 (0)1 Q
where u(Q) is the screened pseudopotential. A(Q) is the structure factor and Q is the
magnitude of the scattering vector. The thermopower is related to the logarithmic derivative

of the resistivity:
_ +“zkaT (allnlp(E)ll) (2)
314 at: m;

Improving theoretical (including molecular dynamics) techniques and more satisfactory
pseudopotentials have created a demand for higher-quality measurements of the electronic
properties of l-Si and t-Ge. Differences of about 20% between measured values of these
properties by various workers make new and accurate measurements desirable. Here we
present new measurements of p and S for these liquids that carry uncertainties of less than
1% for p and less than 0.6 MI °C" for S.

2. Experimental method

The experimental method used to study z-Ge has been described in detail elsewhere [10].
The only modification to this is the addition of graphite plugs. used to seal the contact
holes in the containment tubes; these more securely contain the liquid and allow electrical
contact to be made to the sample. t-Ge can be contained easily in high-density Al203 tubes
with no evidence of attack. as is apparent from visual inspection. from the stability of the
meauremean and from the ease of separation of the solid sample from the post-experiment
tube. The resistivity was measured by the four-probe DC technique. The thermopower of
germanium was measured relative to chromel counterelectrodes; S(T) for chromel is from
[10].
Numerous attempts have been made to measurethe resistivity of l-Si; we attribute
some of the great disparity in the measurements to contamination of the samples due to
containment in A130; crucibles: we consider that all such measurements should be treated
with caution because of serious corrosion that we have noted in our tests of t-Si in A1203.

87

p and s oft-Ge and e51 10377

One recent measurement ofp was done on a sample held in a boron nitride container [I l];
we expect that this should provide a much better result. We find only one attempt to measure
the thermopower of {-51 [12]; this measurement was conducted in an A1303 container and
must therefore also be treated with care.

Because of the great difficulty usually experienced in containing t-Si. we shall describe

our method in some detail. We found that we were able to contain this liquid very well
within high-density graphite. which allowed us to sidestep the oxidation problems that
plague eitperiments done in quartz or alumina containers. Other workers have found that
conventional machined graphite rods immersed in gl-Si experience a change in radius due to
corrosion of only about 1.02 x 10‘J cm h“ [131:‘for our experimental containers this would
result in about a 2% change in the cross-seetional area of the sample over the course of an
average experiment. Further. these workers discovered that x-ray diffraction measurements
on the boundary between {-51 and graphite display no evidence of diffraction peaks that
would indicate the presence of silicon carbide [13]. (If an insulating layer of SiC were
to form at this interface. it would prevent measurement of the transport properties.) In a
separate tcSt. we have menured the resistance of a graphite—C-Si-graphite junction for as
long as 12 h and have seen no increase in the resistance. implying no measurable formation
of SiC.
‘Ihe previous tests done by Other workers (noted above) were on graphite prepared by
an unknown method and characterized only by some indication of its hardness (6—7 on the
Mohs scale) and low porosity. Our graphite was supplied by ESPI, and was their Aeromet
grade. This is the hardest. least porous graphite that they make and appears to be at least
as good for containing {-81 as the previously tested material.

Not only are the containment properties of this graphite very good. but its electronic
properties make it a nearly ideal sample holder at these temperatures. In particular. this
material has a low thermopower (about 6 pV'C“ at 1500 ‘C) and a high resistivity (about
1000 p.32 cm at 1500 °C). Tests of the resistivity and thermopower of Aeromet-grade graphite
from room temperatures 1600°C were conducted on numerous rods of difl'erent batch
numbers. and several individual graphite'rods were tested throughout the full range of
temperature several times. Although the resistivity varied by about 10% from one rod to
the next. the ratio p(T)/p (room temperature) was highly reproducible. The thermopower
of the graphite. however. was consistent to within i1 uV 'C'I on all rods tested. Standard
curves were established on the basis of these tests so that a calibration for each tube constant
could be done with mercury at room temperature. and the parameters of the tube at high
temperatures could be computed. Plots of the calibration data are presented in figure 1.

To form each graphite tube, a graphite rod was drilled with a special hollow core drill.
leaving a wall thickness of about 0.76 mm; triangular grooves for placement of contact
wires were machined around its surface to a depth of about 0.20 mm. Voltage leads for
resistivity measurements were made with Mo wires of nominal 0.102 mm diameter placed
in these grooves and held in place with alumina clamps. The uncertainty introduced into
the measurement by the machining of the grooves was estimated at below 0.1%. The
thermocouples used for temperature measurement were W-5% Re and W—26% Re. were
supplied in matched lengths by Omega Engineering Inc. and were quoted to be accurate
to i-l%. The W—26% Re leg also served as the counterelectrode for measurement of S.
and its therm0power was measured to 1500 °C against Pt. using the results of Roberts er
a! [14] for Pt. Further calibration of W—26% Re below 1200 °C was conducted against
chromel. with the thermopower of chromel taken from [10]. Moreover. these calibrations
were also compared with data obtained from the thermocouple voltage curve for W versus
W-26% Re published by Omega Engineering Inc. using data from Roberts et a! [14] for

 

88

10373 H S Schnyders and J B Van Zyrveld

 

 

 

 

 

 

 

 

 

9C0 lOCO IICO IZOO 13m |4C0 ECO
TC‘l

Figure l. R(T)/R (room temperature) and therrnOpower of ESP! Aeromet-grade high-density
graphite. The therrrwpower was measured with Vii-26% Re counterelectrodes. The various

symbols signify different graphite rods tested in separate experiments.

S(T) for W. Our best fit to these data for W—26% Re is
Sw-x.(T) = -3.985 32 + 6.91069 x 1041 - 2.932 94 x ro-“rz awe-l (3)

for .the range of temperatures from 1000 to 1500 °C. We estimate the uncertainty in S to be
:l:0.5 uV °C“ for this material.

Values for p and S for l-Si were extracted from measured data by considering the l-Si
and the graphite tube as parallel conductors. A 'small-thermal-gradient method’ was used
to obtain the relative therm0power dV/dT of the parallel combination. The thermopower

S of the liquid was then obtained from the relation
3: = (1+ B)(S, + dV/dT) - BS, - (4)

where S is the thermopower of the counterlectrode (here W—26% Re). 5, is the thermopower
of the graphite tube and

B =PlAr/PrAl (5)

where A, and A: are the cross-sectional areas of the tube and the sample. respectively. and
p, and pr are their resistivities. Because of the dimensions chosen for the tube and because
of its electronic properties. the value of B in these experiments was only about 0.02; the
contribution of B to the measured thermopower of the parallel system was therefore also
very small. only about 0.12 uV‘C". As a result. uncertainties in the tube parameters
contributed minimally to the final values obtained for S for l-Si.

Germanium and silicon sample materials were supplied by Aesar-Iohnson Matthey Co.
and were of 99.999993: and 99.99% purity. respectively. All measuremean were done in a
system baked under vacuum and backfilled with argon.

89

p and 5 oft-Ge and l-St' 10379

3. Results

Our experimental results for f-Ge are shown in figures 2 and 3 and those for C-Si are
shown in figures 4 and 5; these are also summarized in table 1. with some previous results.
Both l-Si and (Ge have the characteristics of metallic behaviour: fairly small values of
resistivity. positive temperature slope of the resiStivity and small thermopower. We made
measurements on three separate samples of C-Ge; as can be seen in the figures. all the
results are very consistent with one another. By making a least-squares linear fit to all
our data we find p(LGe) at the melting point (938°C) to be 66.8 i 0.2 ufl cm and
dp/dT = (2.7 i 0.2) x 10"2 11.9 cm °C". These values agree exceptionally well with the
most recent results reported by Koubaa er a! [15] (see table 1); we note also that these
workers contained their sample in alumina and measured resistivity with the direct-contact
fouroprobe technique. In addition. our results for the therm0power. (—0.3 i- 0.5) uV °C".
are also in good agreement with those of the same workers. as well a other recent work

(see table 1).

 

 

 

 

 

50 :L;~! : 5 ##i
940960_ tooomomoaoeooeouoo

Tt'l

Figure 2. The electrical resistivity of (Ge. Only a fraction of our resistivity points are shown for
better graphical clarity. and different symbols signify separate samples. The results of Koubaa
and Gasser [15) are shown a a broken line.

Our value of p(75.2 $0.6 ufl cm at T", = 1410°C) for (Si. also obtained by a least-
squares fit to all data. is very close to the recent value of Sasaki er al [11]. who also used a
four-probe measurement technique with boron nitride containment. As is evident in figure 4.
one of our sets of measurements of resistivity of l-Si was somewhat higher than the others;
the uncertainty and final value quoted include all the data. Using all our measurements of
(Si. the slope of the resisdvity is (18:09) x 10'2 p.32 “C". The measured thermopower
of l-Si was very consistent through all experiments (three different samples): our value of
-2.1i0.6 uV °C" differs substantially from the earlier work of Glazov er a! [12]. however.
We consider this difference a result of our improved containment method. (Containment

for this calier mearurement was A1203.)

4. Discussion

It is clear that our primary goal in this research. that of obtaining highoquality measurements
of p and S for the C-Ge and (Si. has been met. It will now be interesting to discover

 

90

 

 

 

 

 

10880 H S Schnyders and J B Van Zyrvcld

5

4

3..

2..
5'" .
(0.“ “““""‘"““--~-------
.2..

5..

4..

_5 g ‘ : s t r
960 geotOCOIOZOIOdOIOGOKBOIDO

TlC‘l

Figure 3. The thermopower of l-Ge. The results for the three samples are shown with different
symbols. The experimental thermopower of Koubaa and Gasser [15] is shown as a broken line.

 

125
“5 <

'E

3.

Q
.‘ . ‘-' ‘ l-“ “
V v V
-«.o.‘obh--'¢ ‘3- -J.~.- ..,- ’..--

 

 

 

l

 

A
v

1420 1440 I460 148) 1500 1520
TC)

lh’ifit‘fitilfllalfiihig

Figuret The electrical resistivity oft-Si. Again. onlynfraction of the resistivity points uken
areshomandresults fromfotusamplesmsuredbythesamemethodareshownadifl’erent
symbols. Foreompaisoathererults ofSasakicral [11] areshownas nbrnken line.

how successfully various theoretical methods can reproduce these values for the electronic
transport parameters. (Numerical results of these calculations are shown in table 2.) .

Some early calculations of p and S for l-Ge and t-Si [16. 17] were relatively successful.
utilizing the Ziman method and either a modified hard-sphere structure factor [17] or an
experimental structure factor [16]. (In [16] the Ge A(Q) was used in the calculations for
Si: the results for this calculation of p and S were not nearly so good.) Their success
is somewhat surprising. especially in the case of Rahman [17]. since the softened hard-
sphere structure factor used in that work inadequately describes the shoulder seen in the
experimental scattering data [18.19]. Other workers [11.18.20] have used the extended
Ziman formulae. a t-mauix method developed especially to address the transition metals.
and an experimentally determined A( Q). The results of these calculations come very close
to the measured values of these parameters in each case (see table 2).

Koubaa and Gasser [15] have recently attempted a calculation for l-Ge. again utilizing
a pseudopotential in the Ziman formalism. but have chosen the Bachelet-Haman-Schluter

 

91

 

 

 

 

p and S off-Ge and C-St' 10881
5
4 ..
3 ..
2 ..
a“ ' "
t2 0 -
a e
0')" " I U . ' A
-2 MT - " '
.3 ..
-4 .

 

1430 14:30 1460 I460 1470 143) 1490 1500 1510
T‘C'l

Figure 5. The thermopower of («Si. The different symbols denote different samples.

Table 1. Experimental values for p. dp/dT. S and dS/dT for t-Ge and C-Si at their respective
melting temperatures. (The numbers in the square brackets identify the references.)

 

 

p dp/dT s dS/dT
Metal {an em) (to-2 no cm'C") (“vac-l) (10" awe")
Ge 66.8 i 0.2 [present] 2.7 2 01 [present] -0.3 :l: 0.5 [present] 0 2 5 [present]

67.8 [15] 1.3 [15] -o.4 [is] . . .4 [15)

66 [26] :15 [as] -05 [20] -3.3 [20]

85 [27] 0 (27)

so [28]
Si 752 e 0.6 [present] 1.7 a; 0.9 [present] ~11 a: 0.6 [present] 9.1 e 5.3 [present]

72 [11] (nonolinenr) [11] -14 [12]

60 [27]

so [26] «.- 6 [12)

84 (12]

 

(BHS) [21] non-local pseudopotential. one that is widely recognized currently for its broad .
utility. They also use the screening function of Vashishta and Singwi [22]. In separate
calculations. with either a hard-sphere or an experimental structure factor. they have noted
that the use of the experimental structure factor improves their results by 8 an cm over
the calculation employing hard spheres; their best calculation. however. is still 20% below
our'experimental value for p for t-Ge (see table 2). Their calculated value for S for
t-Ge is relatively good (-2.2 uV °C". compared with an experimental value of about
-0.3 uV°C").

In order to understand better the limitations of this pseudopotential method. and to
extend its application to Z-Si. we have also calculated p for l-Ge (and for l-Si) using
the BHS pseudopotential [21]. the experimental Structure factor of Waseda [23]. and the
screening function of Vashishta and Singwi [22]. (We did not pursue a thermopower
calculation because of the sensitivity that we discovered in the calculation of P ‘0 the
imput parameters.) In the range of Q required for calculation of transport properties. the
x-ray diffraction structure factor reported by Waseda [23] compares well with other work.
including the recent x—ray data of Kita er a! [l]. and the neutron data of Gabathuler and Steeb

 

92

10882 H s Schnyders and J a Van zyrveid

Table 2. Calculated values of p and S for 06¢ and (Si near their respective melting
temperatures. (The numbers in the sequence brackets identify references.)

 

 

p“, Pale Sup Scale
Metal (an cm) (at: cm) (“WC-l) (“VT")
Ge 66.8 [present] 41.2 [present] —0.3 [present] -2.2 [15]
55 [15] -3.3 [20]
66.2 [18] -6.4 [16]
45.5 [20] -l.4 [17]
75.9 [16] -3.2 [18]
57.4 [17]
Si 75.2 [present] 66.6 [present] -2.1 [present] -3.6 [18]
57 [5] -l.1 [17]
69 [ll] -lO.l [16]
67-3 [18} (Ge A(Q)l
46.3 [17]

180115110: A(Q)l

[19]. The BHS pseudopotential also includes relativistic eEects on the valence electrons
by the atomic core region. This treatment has been validated by close agreement between
calculations [21] and experiment for such properties as bulk moduli and phonon frequencies.
Si and Ge have been specifically noted as two elements well described by this formulation
[21]. (Stich er a! [5] and Kresse and Hafner [24] have also used BHS patentials in their
calculations recently. These potentials were modified slightly to lessen the number of high-
‘frequency components in the potential and to case its use in LDF calculations. but these
modifications appear mostly above Q = 2k; and are outside the area of importance for
calculation of transport properties by the Ziman formula.) For our purposes the non-local
BHS pseudopotential should be fully adequate.

The value that we calculate for p for l-Si. 66.6 ufl cm. is rather close to the experimental
value. but the result for t-Ge is not nearly so good. (41.2 p.82 cm). We found that. if we
used the screening function of Itchumaru and Utsumi (TU) [7.5] in this calculation. even
though it differed rather little from that of Vashishta and Singwi. we obtained a value‘for p
for l-Ge that was even lower by about 15 p.52 cm. (The small difl'erence between the two
screening functions occurs mainly in the range of Q where A(Q) is large and contributes
substantially to the calculation of p.) The calculation of p for t-Si is affected much less by
use of the IU screening function. lowering p‘ by only 4 [1.52 cm. Nonetheless. because of
these sensitivities. we do not report calculations of thermopowers for these liquids.

It would now be especially interesting to see the results of applying the recently
developed and highly suecessfirl rib-initio molecular dynamics methods of Car and Parrinello
[7] to a calculation of the transport properties of l-Ge and l-Si. Some work along these lines
has begun. As nated above, Stich er a! [5] have successfully used optimally smooth BHS
pseudopotentials to calculate numerous properties of t-Si. including the structure factor and
DOS; additionally. an extrapolation of AC conductivity to the DC limit for (-Si gives a
resistivity of 57 p.52 cm. in reasonable agreement with the measured value. Kresse and
Hafner [6] have calculated the DOS for t-Ge With the same method and this is in very
good agreement with experiment. although no estimate of resistivity was given. Now that
good experimental data are available for p and S for C-Ge and l-Si. it would be especially
interesting to apply these new methods to the calculation of these parameters.

5‘

93

p and 3 off-Ce and C-Si 10883

Acknowledgment

We are happy to acknowledge a grant from the National Science Foundation (DMR-
99106812.) in support of this research.

References

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94

J. Phys.: Condens. Matter 9 (1997) 10121-40123. Printed in the UK P11: 50953-8984(97)83636-J

The electronic transport properties of liquid Ag—Te

H S Schnyders”. I Hahnf. D Streutker‘l’ and I B Van Zytveldf§

1 Physics Department. Calvin College. Grand Rapids. Ml 49546. USA
1 Department of Physics and Astronomy. Michigan State University. East Lansing. Ml 43824,

USA
Received 22 April 1997. in final form 21 July 1997

Abstract. We have systematically measured the conductivity and the thermopower of the
Ag-Te system {or concentrations in the neighbourhood of Agz'l'e. Our ability to stir the
sarnple while taking measurements has allowed us to accurately study the immiscible region
below 31 at.% Te and to measure the complete extent of the p—n transition in the thermopower
as stoichiometry is crossed. Our results are compared to the recent experimental results of
Ohno er al [1]. and are found to be largely in excellent agreement for thermopower and
conductivity: however. in our data approximately at stoichiometry. a previously unseen feature
in the conductivity is observed. and a significant deviation from the therrnOpower dna ot‘ Ohno
er a! is also seen. This conductivity feature is consistent with. but much smaller than. peaks that
have been observed in the conductivities in liquid Ag18e and Agzs. We also include data for
da/dT and dS/dT that have not been previously available. Our results are discussed in terms
of the Kubo-Greenwood equations. and are compared to the highly unusual systems Agée and

A14.

1. Introduction

Silver chalcogenide systems have been the subject of intense scrutiny in recent years
because of the observation in AgZSe [2] and Ang [3] of an unexpected rise in the
conductivity, 0’. to a sharp peak at stoichiometry. At temperatures of about 1000°C. the
peak has been observed to be about 100 {2“ cm" in height. approximately 6 at.% wide.
and centred on stoichiometry. The pealt in conductivity is accompanied in this narrow
concentration range by a negative temperature coefficient; further from stoichiometry on
either side. the temperature coefficient is positive. In the same range the thermOpower. S,
undergoes a p—n transition; for example. the thermopower of AgZSe changes from —60 to
+120 1.1V °C“. Although a p-n transition is seen in a multitude of liquid semiconducting
systems (a < 500 (2“ cm"). the conductivity in these two unusual systems departs
from the more regularly observed sharp minimum and positive temperature coefficient at
stoichiometry. In this way. Agée and Ag—S are unique among all liquid semiconductors.
It is also interesting to consider whethi:r these phenomena might be related to the fast-ion
conducting character of AgZS and Ange in the high-temperature solid and in the liquid [4].

Perhaps just as curious as the feature in liquid AgZSc and AgIS is the apparent lack
of such a phenomenon in the similar liquid alloy Agz'l'c. Early measurements of the
conductivity of Ag—Te by Dancy [5]. relying on convection alone to stir the sample.
showed no evidence of the Ag—Se type conductivity behaviour. but rather displayed the

5 Present address: M l Murdock Charitable Trust. PO Box 1618. Vancouver. WA 98668. USA.

0953-8984/97/4610121+08519.50 © 1997 10? Publishing Ltd, 10121

95

10122 H S Schnyders er a!

more conventional sharp minimum at stoichiometry. associated with a positive temperature
coefficient. The immiscible region below 31 at.% Te prevented the complete measurement
of the conductivity and the p—n transition in the thermopower in previous experiments. Later
results by Okada er a! [6] were also measured only above 31 at.% Te. and the 31-40 at.%
region was largely untouched.

Following these results, the apparent difference in behaviour between Ange and AgZSe
was addressed at a structural level via neutron diffraction by Price er a! [7]. who found
that in both liquid alloys the structure was very similar. AgISe displayed a prepeak at
Q ~ 1.8 A". which was found in a later analysis by ab initio simulations to be the result
of Se-Se coordinations not unlike those seen in the fast-ion conducting solid. It was nOted
that the prepeak was absent in Ange. but this was expected to be a result of the similarity of
scattering lengths of the two materials. Most importantly. there was no noticeable structural
change as the concentration of selenium was varied in a narrow range about stoichiometry.
suggesting that structure was not responsible for the anomalous peak in the conductivity.

Very recent results by Ohno er al [1] have sampled the neighbourhood of stoichiometry
in more closely spaced concentration steps. and have again found no Special features. No
mention is made. however. of the stability of the liquid at temperatures below the miscible-
immiscible boundary at Te concentrations between 9 and 31 at.%. although measurements
appear to have been made at these temperatures. Such a boundary is seen in roughly the
same concentration range in both Ag-S and Agée, and the earlier studies have considered
only data taken at higher temperatures. in the miscible region.

Additionally. the experimental thermopower data taken at concentrations and
temperatures within the immiscible portion of the phase diagram suggest by the asymmetry
of the p-n transition that the slopes of the valence and conduction bands are highly
asymmetric. following the formalism of Enderby er a! [8]. But rigid band modelling.
comparing experimental data to those generated by numerical integration of the Kubo-
Greenwood equations. is incapable of reproducing even the general features of the
conductivity and the thermopower. We have therefore studied Ag-Te once more in
the neighbourhood of stoichiometry. proceeding in even finer concentration steps. and

examining only the well stirred liquid.

2. Experimental details

The experimental apparatus used to study liquid Ag—Te has been described in detail
elsewhere [9]. The only modification is the addition of graphite plugs. used to seal the
contact holes in the alumina containment tubes; these more securely contain the liquid and
allow electrical contact to be made with the sample. The alloy showed no evidence of
attack on the A1203. and the solidified sample separated easily from the post-experiment
tube. The tubes were Open at the top to allow stirring. and all of the samples were agitated
repeatedly during the experiments. Even in Te rich samples. the cool upper inside of the
sample tube was free from condensed Te after the experiments. leading us to believe our
concentrations were largely unaffected by evaporation. Conductivity was measured with the
four-probe DC technique. and therrnOpower was measured with a chromel counterelectrode.
with S (T) for chromel from [10]. Our calibration for chromel is in excellent agreement with
that of Cook and Laubitz [l l]. and we estimate the uncertainty in the absolute thermopower
of chromel to be :h0.2 11V °C". The sample concentration was not varied continuously
during the experiment to avoid any possibility of progressive contamination; the majority
of the samples were independently prepared in different alumina tubes from pure Ag and

Te. making most data points completely independent of all others.

96
Electronic transport properties of liquid Ag—Te 10123

 

30

110 mhrlé‘c°"l

 

 

 

 

 

Figure 1. Measurements of a and Us: da/dT fr .‘ liquid Ag—Te. 0. current results as a function
of Te concentration on the isotherm 1' :3 960’C: A. results of Ohno er al [1] for Ag-Te.
T a 1000 °C; I. results for Ohno er a! [2] for Ag-Se. The upper concentration boundary of
theimmiscibleregion. 31 atfi.Te.isdenotedbyadashedvertieal line;theotherboundaryis
at9at.%Te. Theuncertaintyina issmallerthantheextentofthepoints.

Silver and tellurium sample materials were supplied by Aesar-lohnson Matthey Co. and
were of 99.99% and 99.999+% purity. respectively. The samples were prepared in air. but
all measurements were made in a system that was baked under vacuum. and maintained
during the experiment under 1 atmosphere of argon.

3. Results

Our results for the conductivity of Ag—Te as a function of Te content are shown in figure 1.
and are seen to be in good general agreement with the results of Ohno et a! [1]. Our points
result from the evaluation at 960 °C of a linear fit to all of the data at each concentration. We
also provide our data for the temperature coefficient of the conductivity. which also agree
well with the limited data of Ohno er a! [1]. and appear to be rather asymmetric. Also
shown on this graph are the boundaries of the immiscible region. according to Massalski
[12]; the immiscible region extends from 31 at.% Te to 9 at.% Te. with a very sharply
concentration dependent temperature boundary.

Figure 2 shows a curve fit to our data outside the immediate neighbourhood of
stoichiometry. This curve falls below our data close to stoichiometry. and would suggest
that our conductivity data do form a peak approximately at stoichiometry. extending from

97
10124 H S Schnyders et a!

 

1200 E

015.6511

093.0???“

 

minim" 1

 

 

 

25 50 55 40 45 50
more

Figure 2. The conductivity of Ag-Te for a narrow range of Te concentrations. shown with a
line drawn as a guide to the eye. Symbols are the same as for figure 1. and the irruniscible
boundary is again marked by a dashed line. The lower portion of the graph is the difference
between the curve extrapolated from the experimental data at higher Te concentrations and our
data near stoichiometry. 0. present data; a. data for Ag—Te from Ohno er al [1]; V. similar
difference curve for Ange. from Ohno er a! [2].

roughly 32 at.% to 36 at.% Te, with a maximum height of about 65 (2" cm". Also
included on this graph is the difference between this curve and our experimental data; for
comparison. similar difference data at 1000°C for the peak in the Ag—Se system have been
plotted as well (data from Ohno et a! [2]). The peak in the conductivity of Ag-Te begins
at roughly 32 at.% Te. and extends to roughly 36 at.% Te. It is quite similar to the peak in
Agée. which begins at roughly 31 at.% chalcogenide, and extends to 37 at.%, and which
shows a significant broadening toward higher Se content as temperature is increased.

Figure 3 shows a graph of the conductivity at 30 at.% Te. which we have included
to demonstrate the apparent lack of any mixing difficulties within the immiscible region.
Our graph includes data for two different 30 at.% samples. both of which were measured
during repeated stirring at temperatures above and below the boundary of the immiscible
region. The Ag-Te phase diagram indicates the boundary is at roughly 1075 °C for this
concentration. indicated with an arrow in the figure; the conductivity varies smoothly through
this temperature. At no temperature in any of the Te alloys in this region is there a dramatic
change of the slope or the magnitude of the conductivity which would indicate the presence
of a mixed—unmixed boundary. In particular. the temperature coefficient is quite small
and positive over all temperatures surveyed. in contrast to the very large positive slopes
displayed by the unmixed Agée and Ag—S alloys, which change to small positive slopes
when the miscibility boundary is crossed. Furthermore. at various temperatures within
the immiscible region. the sample was stirred, measured, and allowed to settle for several
hours. at which time more measurements were taken. There was no time dependence of
the electronic properties, which makes us believe that these components separate far more
slowly in the immiscible region than in the other silver chalcogenide alloys.

98
Electronic transport properties of liquid Ag-Te 10125

 

 

 

 

#

900 960 1000 1100 1150
Tlc°l

 

Figure 3. The conductivity for 30 at.% Te as a function of temperature. All of the data are
current data. but different symbols denote separate experiments. Only a fraction of the data
is shown of graphical clarity. The arrow denotes the temperature of the immbciblmxible
boundary of the liquid.

Our thermopower data (figure 4) we are generally in good agreement with Ohno et al [1].
Our study includes thermopower measurements in the immiscible region. and has verified
that the thermopower extends at its lowest point to only —13 “V °C". The dramatic dip
in the thermopower seen in Ag-Se or Ag-S between 29% and 32% chalcogenide is absent
in our Ag—T‘e data. Interestingly, the concentration dependence of our thermopower data
through stoichiometry is very close to that in AgZSe. about 20 11V °C at.%". (Measurements
on Ag—S were not completed in this region because of miscibility problems.) The zero of
the thermopower is at the same concentration as in the data of Ohno et al [1]. 31.5 at.% Te,
and less than the 36 at.% zero seen in Ange; it is not surprising that this is at a lower
chalcogen concentration than in Ag-Se. as the deeply negative thermopower is not seen in
our data. Additionally, our data recognize a small positive temperature coefficient of the
thermopower at stoichiometry. in contradiction to the earlier study of Endo er al [4]. A
positive temperature coefficient 13 also seen in AgZSe and Ang.

It 1s interesting to note that while the data of Ohno et al [1] lie consistently 3—4 1.1V °C"
below ours outside the neighbourhood of stoichiometry. perhaps due to their different
counterelectrode material. in the area from ~34 at.% To to 40 at.% Te their thermopower is
above ours by as much as 25 11V °C"'. The difference between these measurements is well
beyond the error quoted for their measurements (~10 12V °C"), and far greater than our
estimated errors. about 3—5 12V °CT‘. These concentrations are crucial to the definition of
the curvature on the conductivity graph as stoichiometry is approached from the tellurium
rich side, and ultimately to the identification of the peak in a at stoichiometry.

4. Discussion
For the majority of liquid semiconductors. an analysis according to the method of Enderby

et al [8] can yield a good qualitative agreement with experimental data. This method
involves the numerical integration of the Kubo-Greenwood equations

_ °° df
cr—-/o o(E)a-EdE

99
10126 H S Schnyders et al

 

Q
01

aslarlo‘tMc'?)
O

 

 

    

 

-30 . A ‘l-\"i -
IO 15 20 25 30 35 4O 45 50
«It'h‘l’e

 

 

Figure 4. The thermopower. S. and temperature coefficient of the thermopower. dS/dT. for
liquid Ag-Te. Again. 0, current results at T - 960°C; A. results of Ohno er a! (1] for
Ag—Te. r . 950°C. Estimated uncertainties in s are 3-5 awe-l. Again. the boundary ofthe
immiscible region is marked by a vertical dashed line.

__ k5 °°a(E)E-E, 5f
- -T€-l o 0’ ka E

where f is the Fermi function. k3 is Boltzmann’s constant. and E I is the Fermi

energy. o(E) is the energy dependent conductivity. and the valence and conduction band

contributions may be expressed generally as
”C(E) = 0.1-(E - Er) (E 2 EC)
031(5) = av(£u - E) (E g Ev)

where E: and E. are the energies that denote the edges of the bands. The variables :1c and
a, may be calculated using the band width, the atomic separation. and the total number of
electron states per unit volume in the band.

Our new data would suggest that the behaviour of AgTe around stoichiometry is rather
more complex than a simple o(E) could explain. A consideration of the overall shape
of o(c) for Ag-Te. neglecting the feature near stoichiometry, and of the behaviour of
S(c) on the Te rich side of stoichiometry would suggest that o(E) falls rather steeply to
a gap or pseudogap within a few atomic per cent of stoichiometry. Using the random
phase approximation and a model o(E) according to Barnes [13]. with bandwidths from a
calculation of the density of states for liquid Agz’l'e by Koslowsld [14], we have calculated
0: ~ 1300 52" cm" eV“, in reasonable agreement with the choice of Ohno er al of
2000 9" cm" eV" [1]. A choice of ac = 20., for Ag—Te. similar to that chosen for Ag-Te
[I]. does indeed reproduce the asymmetry and magnitude of o(c) about stoichiometry. and

 

100
Electronic transport properties of liquid Ag-Te 10127

the thermopower rises to a maximum of roughly 100 uV°C". However. no choice of
ac. a". or band gap AE could reproduce the shallow S (c) and steep o(c) in this concentration
range Simultaneously. A small feature can indeed be reproduced in the conductivity and
the thermopower on the tellurium rich side of stoichiometry with a more detailed o(E)
similar to that suggested by Enderby et a! [31 for CuzTe, bl“ only With ll“: 58““: inability
to reproduce even the general shape of both o(c) and S (c). We conclude that a rigid band
model is not appropriate for liquid Ag-Te.

A neutron diffraction study by Price era! [7] indicates a prepeak exists in liquid AgZSe at
Q ~ 1.8 A“, which has been determined experimentally by partial structure factor analysis
to be most substantially composed of a contribution from the comparatively immobile Se
ions. There is no apparent change in structure as concentration of Se is scanned through
stoichiometry. making it unlikely that a structural change is the source of the unusual
conductivity peak. Price er al also note the strong similarity of liquid Ag28e structure to
liquid Ange; an absence of the prepeak in Agz'l‘e is attributed to the similarity in scattering
lengths of Ag and Te. It should be noted that all three silver chalcogenides have been shown
to be fast~ion conductors.

It is therefore interesting that a recent neutron diffraction study of liquid Mg,Bi2 by Guo
et al [15] indicates an ordering in the liquid similar to the fast-ion conducting solid phase
of the same material. Furthermore. the structure shows broad similarity to that of AgZSe.
including the presence of a prepeak at low wavenumber. Reverse Monte Carlo analysis
suggests that the Bi-Bi distances generate the prepeak in Mg3Bi2, and that a shallow
first minimum in the Mg—Mg real space distribution indicates that Mg ions are indeed
quite mobile. Additionally. transport studies of Mg—Bi have shown that the inadequacy
of a rigid o(E) is not limited to the silver chalcogenides. Guo et al find that fast-ion
conducting liquid Mg3Bi2 is also difficult to explain using the Kubo—Greenwood formalism
with any simple model o(E). because the conductivity drOps to a sharp minimum while
IS I < 10 uV °C“ around stoichiometry. By changing the concentration by only 1.5 at.%
on either side of stoichiometry, the conductivity rises to values expected of a material with
a shallow pseudogap.

In the absence of a structural explanation for the conductivity anomaly in Ag-Se. Ohno
et al [3] suggested that the density of states in AgZSe may be modelled by rigid band
behaviour, but that a sudden change in do (E) /dE may be responsible for the unusual peak
in the conductivity which spans the concentration range 31-37 at.% Se. The change in o(E)
is believed to occur as a result of changes in mobility; magnetic susceptibility measurements
indicated N (E) was at a minimum at stoichiometry. The susceptibility measurements
are in agreement with the conclusions of Koslowski [14]. who calculated the density of
states for all three silver chalcogenides using a simple tight-binding method. and found
that with sufficient introduction of chemical bonds, the Fermi energy was indeed located
in a pseudogap located between the silver 3 and tellurium p bands. Guo also postulated
a change in N (E) and the mobility to explain the small dS/dc through stoichiometry
and shallow dH/dc away from stoichiometry in Mg3Biz, and further pointed to the work
of Fortner et al [16] which suggests a possible link between mobile ions and enhanced
mobility. Our new data. and the similarity in structure and transport to other similar alloys.
suggest that a similar dramatic change in mobility or density of states occurs in Ange
around stoichiometry.

At least four liquid fast-ion conductors have now been identified as being inadequately
explained by the Kubo—Greenwood equations. Our study has revealed the presence of a peak
in the conductivity near stoichiometry similar to that seen in the other silver chalcogenides,
though smaller in magnitude. Ag-Te has also been studied at temperatures above and below

1
10128 H S Schnyders et a} O

the immiscible boundary. and found at temperatures inside this region to be a surprisingly
stable liquid.

Acknowledgment

We are happy to acknowledge a grant from the National Science Foundation (DMR-
9406812) in support of this research.

References

[1] Ohno S. Mizushina Y. Okada T and Togashi M 1996 J. Phys. Soc. Japan 65 2182
[2] Ohno S. Barnes A C and Enderby l E 1990 J. Phys: Condens. Matter 2 7707
[3] Ohno S. Barnes A C and Enderby J E 1994 J. Phys: Condens. Matter 6 5335
[4] Endo H. Yao M and lshida K 1980 J. Phys. Soc. Japan 48 235
[5] Dancy E 1965 Trans. Metall. Soc. AIME 233 270
[6] Okada T. Kakinuma F and Ohno S 1983 J. Phys. Soc. Japan 52 3526
[7] Price D. Saboungi M-L. Susman S. Volin K. Enderby .1 and Barnes A 1993 J. Phys: Condens. Matter 5
3087
[8] Enderby J E etal 1990 Rep. Prog. Phys. 53 85
[9] Walhout M. Haarsma 1.. and Van Zyrveld 1 B 1989 J. Phys: Condens. Matter 1 2923
[10] Kooistra J. Dreyer K and Van Zytveld J B 1988 J. Phys. F: Met. Phys. 18 1225
[11] Cook S G and Laubitz M S 1976 Can. J. Phys. 54 928
[12] Massalski T 1986 Binary Alloy Phase Diagram (Metals Park. OH: American Society for Metals)
[13] Barnes A C 1993 J. Non-Cryst. Solids 156-158 675
[14] Koslowski T 1996 J. Phys: Condens. Matter 8 7031
[15] Guo C. Barnes A C and Howells W S 1996 J. Phys: Condens Matter 8 10823
[16] Former J. Saboungi MoL and Enderby J E 1994 Phys. Rev. Lett. 74 1415

102

Diffraction Electronic Transport and Nerflm Evidence for Chemical Short Range
Ordetfll Liquid Cque.

Harold S. Schnyders*
J.B. VanZytveld**

Physics Department, Calvin College, Grand Rapids, MI 49546 USA

 

*Also at: Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
"Also at: M J Murdock Charitable Trust, PO Box 1618 ,Vancouver WA 98668

103

Abstract

We have measured the electronic resistivity and thermopower of liquid Cu1_cCeC, Cu1-cLac,
and Ag1.cCec alloys for c<0.4. In all of the alloys the resistivity is very similar, rising

smoothly and steeply as x increases, but at all times remaining less than 200 ufl-cm.. The
thermOpower is also featureless and metallic. However, the temperature coefficient, a=1/p
dp/dt, displays in the Cu alloys a minimum at Cu6RE (RE=Rare Earth), which furthermore

is negative in CugCe; a negative temperature coefficient is also Seen at Ag3Ce. We also
present the results of neutron diffraction scattering from liquid CugCe, and of a Reverse

Monte Carlo (RMC) analysis of the structure data, which shows evidence for ordering in
the liquid, with characteristics similar to crystalline CugCe. We discuss several possible

explanations for the negative temperature coefficient in both Ag and Cu alloys, and
conclude that it’s origin the presence of Chemical Short Range Order (CSRO).

Introduction
The resistivities of liquid copper alloys with Rare Earths (RE) were first studied by

Guntherodt et.al. [1]. All of the alloys studied, including those with La, Ce, Nd, and Pr,
show remarkably similar behavior up to 40 at. % RE. Furthermore, the highest resistivity
of the alloys, approximately at the equiatomic concentration, is still metallic in magnitude.
In addition, the temperature coefficient of the resistivity, a=l/p dp/dT , was in all alloys at
all concentrations determined to be small and positive. A simple description seemed in
order, due to the apparent lack of effect on p or a of the additional f electrons contributed
by the RE’s; weak scattering Faber-Ziman theory seemed appropriate because magnitude of
the resistivity was far less than in divalent liquid Ba (p=338 tin-cm [2]), which was
successfully modelled by the theory.

In later structure measurements by anomalous X-ray diffraction, Cu-Ce was
determined by Waseda [3] to be unremarkable across a wide range of rare earth
concentrations; these authors suggested that the alloy was most likely a simple random
mixture of Ce and Cu atoms, based on the apparent invariability with respect to Ce content
of the peak positions in the partial structure factors. However no structure measurements
were made below 20 at. % Ce. . '

Additional recent interest in some RE-alloys stems from the fact that they are Heavy
Fermion compounds. This is also true for Cu6Ce. Sputter deposited amorphous Cu6Ce

was discovered comparatively recently by Suzuki et.al. [4] to display at low temperatures

104

the -log(T) dependance of the resistivity characteristic of Heavy Fermion behavior, and
similar to the behavior recorded previously in CuGCe crystal [5]. The amorphous alloy

shows a continued linear temperature dependance of p with negative a up to the highest
temperature surveyed, about 300 K. The absence of thermopower data for liquid Cu-RE
compounds, the availability of high purity materials, and the unusual negative temperature
coefficient (NT C) in the amorphous solid have prompted us to systematically study these
liquid materials. We have for comparison measured the electronic transport properties of
liquid Ag-Ce; we believe these are the first electronic transport measurements on this liquid
system. In addition, we have conducted a neutron diffraction study of liquid Cu6Ce, and

have also done a Reverse Monte Carlo (RMC) analysis. The results of the transport
measurements and the structure measurements are discussed in separate sections below.

Trans ort erties: Ex erimental detail

We obtained sample materials from AESAR/Johnson Matthey: lot analyses of La and Ce
yielded impurity levels of 60 and 200 ppm (wt), respectively; Ag and Cu, from the same
supplier, were of 99.99% and 99.999% purity. Samples were prepared and handled under
an argon atmosphere in an evacuable glove box to avoid contamination. During
measurements, our alloys were held in high density A1203 tubes, and were stirred

vigorously to ensure uniformity. Sample material separated easily from the wall of the post
experiment tube, and no reaction is thought to have occurred. (For a more detailed
description of the sample containment see Walhout et.al. [6 D. The resistivity was
measured using a standard 4 probe technique, and the thermopower was measured against
chromel counterelectrodes [7]; the uncertainty in the absolute thermopower of chromel is
estimated to be + 0.2 uV/°C. The error in the resistivity and the thermopower are
calculated from a least squares fit to all of the data, and in the majority of cases do not
exceed 0.5 14.0 cm and 0.3 uV/°C, respectively.

Results

Our results for the resitivity of liquid Cu-Ce are shown in figure 1, along with our data for
Cu-La and Ag-Ce, and Cu-Nd data from Guntherodt and Zimmerman[8]. All data points
shown are from a linear fit to all of the data, evaluated at 950° C, except for the Ag alloys,
which were evaluated at 1050° C . In all cases, the resistivity rises smoothly from pure

105

Cu(Ag) as rare earth content is increased. The resistivity of the different alloys is
remarkably similar up to 35 at. % RE; none of the alloys show any features in the
resistivity, even at CuGRE or CquE, or at Ag3Ce, which are the concentrations of
congruently melting crystalline compounds, and correspond to peaks in the melting curve.
Even near the equiatomic concentration and the highest resistivity in the alloys, all of the
materials hold values of the resistivity which are well within the bounds of metallic
conduction. _ ,

Figure 2 displays the concentration dependance of the thermopower; in all cases,
dS/dc is quite steep from pure Cu(Ag) to ~30 at. % rare earth; again, S is featureless and
smooth, and at all concentrations examined holds a value which is decidedly metallic and
between the values of S for the pure constituents.

Figure 3 shows the concentration dependance of the temperature coefficient of the
resistivity, a. In Cu—Nd , a is quite small and positive over a wide range of RE
concentration, but appears to have a very shallow minimum at CuéNd. In both Cu6Ce and
Cu6La, a is markedly depressed compared to surrounding concentrations, and
furthermore is negative in Cu6Ce. It is highly unusual to see structure in a and not in the
resistivity or thermopower; furthermore, to our knowledge, this is the lowest resistivity in
a liquid alloy, about 95 tin-cm, associated with NTC. Our value of a= -1.8"‘10‘4

C"‘1 is comparable to that seen in the divalent pure liquid elements like Sr and Zn, and in

the divalent alloy SnCe. In Ag3Ce, a = -1.6*10’4 C ’1, but again, no features are seen in

p or S .
In all of the studied materials, the rninima in a correspond to peaks in the melting
curve. Cu6RE, for RE: La, Ce, Pr, and Nd all possess the same crystal structure and melt

congruently from an orthorombic structure; Ag3Ce melts from a hexagonal crystal

structure. It is interesting that the minimum in a around stoichiometry is most pronounced
in Cu6Ce, which has the highest melting point of the Cu-RE compounds studied, and is

least pronounced in CuéNd, the lowest melting point compound. With this in mind, we

have conducted a neutron diffraction study of liquid Cu6Ce.

Neutron Diflractigg: Procedure

The neutron diffraction study was performed on the GLAD (Glass, Liquid, and

 

106

Amorphous Diffractometer) at the Intense Pulsed Neutron Source (IPNS) at Argonne
National Laboratory. Sample material of the purity noted earlier was prepared in an argon
atmosphere, as above, and was initially melted and stirred in an alumina tube under argon,
prior to being sealed under argon in a vanadium tube for the scattering experiment. The
alloy sample was not at any time exposed to the atmosphere. The experiment was
conducted at 1000 C within a cylindrical vanadium furnace in a vacuum. Data for the
empty tube, and for the furnace, were subtracted from the experimental spectra before

analysis.
Diffraction results

The experimental structure factor , S(Q), is shown in figure 4. A significantly
robust prepeak is seen at 1.6 A4; in other alloys, a prepeak in the structure factor at low Q
value has been interpreted [9] as evidence of CSRO .

Fourier transform of the data up to Q=25A'1, using a Lorch window to truncate,
leads to the pair correlation function g(r). The total correlation function, T(r)=r*g(r), is
shown in figure 5. The main peak in T(r) is at 2.5 A, and a shoulder is seen on the high r
side at 3.05 A. Because of the relative concentrations of Cu and Ce, and because the
scattering length of Cu is nearly twice that of Ce, the structure seen in T(r) is most
significantly composed of Cu-Cu correlations. The total structure factor may be calculated
according to the following formula
S(Q) = zij Ci Cj bi bj (Sij(Q)-1)
where C, is the concentration of constituent i, bi is the scattering length, and Sij(Q) is the
partial structure factor. For our alloy, the weightings of the partials are as follows: Cu-
Cu=.4375, Cu-Ce=.0916, Ce-Ce=.0048.

The coordinations may be calculated from the area of gaussians fitted beneath the
T (r) curve. The first gaussian is certainly Cu-Cu correlation exclusively; the second,
however, is probably a combination of CuCu and CuCe coordinations, mostly CuCu. If
we take the total area under both Gaussians as CuCu, we calculate Cu coordination of
10.7 , which compares favorably with the average coordination of Cu to Cu in crystalline
Cu6Ce calculated out to 4 A, about 9.3 . However, CuCe distances of ~3-3.4 A also exist

in the crystal, and almost certainly comprise a portion of the second gaussian. To continue
this analysis, we have utilized RMC to further examine the structure factor, and to shed

 

107

light on the origin of the prepeak.
Two different initial configurations were selected for our simulations; 1. A
random arrangement of 1050 atoms of the stoichiometric concentration. The following

minimum distances of approach were used: role“: 2A, roles: 2.55A, and rCeCc=2.85A,
very similar to the first peak positions of liquid Cu.75Ce_25, according to the anomalous x-
ray diffraction results of Waseda [3]. 2) A crystalline configuration of Cu6Ce[10], with

3500 atoms, and with the minimum distances of approach the same as above. The
program fitted the calculated S(Q) to the experimental structure factor data; the results are
shown in figure 4. The quality of fit parameter, c2 , of the simulation started from the
initial crystalline configuration, and of the simulation started from a random initial
arrangement ultimately converged to within 2% of one another; both simulations were
successful at reproducing the prepeak.

The main peak in S(Q) is very well reproduced in height, width, and position for
both starting configurations, but for the second and following peaks the RMC fits were
less effective. The baseline against which the height of the prepeak may be measured
appears to have been set rather too high by the simulation. However, for both
simulations, the height of the peak in S(Q) with regards to this baseline is in good
agreement with the height of the experimental peak, but it is only as pronounced as the
experimental prepeak in the simulation starting from the crystalline arrangement. The
prepeak is found in both cases in the CuCu partial structure factor; and is centered at 1.6
A'1 (figure 6). At the same momentum transfer, there is a deep minimum in the CeCu
partial, which in spite of it’s significantly lower weighting in the total structure factor
serves to diminish the height of the prepeak . We find that essentially the same quality of
overall fit as measured by c2 is achieved for minimum approach values 2.55A<
rCuCe<2.9A, although inspection of the post simulation CuCe partial g(r) shows that for

rCuCe<2.9A, the partial looks rather unnatural and probably non-physical; we believe that
the weighting of the CuCe partial is just large enough to yield an adequate fit. SCeCc(Q),

on the other hand, was not well modelled by RMC due to the relative lack of information
about it in the experimental S(Q), and will not be discussed further . The main peak of the
CuCe partial S(Q) is at 3 A—I, and the CuCu peak is at 2.5A-1.

Fourier transformation of the RMC partial structure factor of CuCu yields gCuCu(r)

with a main peak at 2.5A (figure 7); the high r side of this peak is significantly less sharply
inclined than the low r side, and perhaps indicates the presence of a shoulder, or nearby

 

 

108

smaller peak at a value of about 2.9A, in agreement with the gaussians fitted to the
experimental g(r). The next peak at 4.4A is still quite strong, and is precisely a CuCu
distance in the crystalline solid. The RMC partial gCuCc(r) (not displayed) is peaked at 2.9

A, close to the CuCe distances of 3A-3.4 A seen in the crystalline solid.

RMC analysis has also allowed an examination of the bond angle distributions; we
expect the simulated CuCuCu bond angles to be most reliable because of the exceptional
weight of CuCu correlations in S(Q). An examination of the pre-simulation bond angle
distribution for the initial random configuration subjected only to moving out of the
particles to minimum distances of approach, and using an atom-atom separation cutoff of
4A, showed a rather flat distribution (figure 8), indicative of a random arrangement. The
post simulation angle distribution with a similar cutoff shows a peak at about 55°, and a
somewhat more broad peak at around 105°; interestingly, crystalline Cu6Ce has bond

angles in a range of angles around 60° and 110°, but a lack of bond angles in between; the
post simulation bond angle distribution reflects a lesser number of bonds in this same
region. There is, however, no evidence in the liquid of the much less common 140°

angles also seen in the crystal.
Diagram.

Several possible explanations for NTC in alloy systems exist, but the results for these
systems appear to provide an unusual challenge. Divalent pure liquid metals and liquid
alloys often display a NT C. This behavior for pure metals occurs as a result of the
temperature dependance of the main peak in the structure factor, S(Q), and is seen uniquely
for divalent liquids because of the relative positions of 2kf (where kf is the Fermi wave

vector) and this primary peak in S(Q) [11]. A similar situation occurs for liquid alloys.
This characteristic NTC has been seen for liquids with rather low values of p, as in our .
alloys, but is seen only if the effective valence of the liquid is close to 2 [12]. If we
assume that La and Ce each contribute 3 electrons per atom to the conduction band, and Cu
and Ag contribute 1, the Zeff = 1.3 for Cu6Ce and Cu6La and 1.5 for CeAg3, rather too far

from 2 to show divalent character.
NTC’s are also expected and observed for liquid and amorphous metals and alloys
for which E falls approximately at a fairly deep minimum in the Density of States (DOS).

Fresard, Beck, and Itoh [13 ] examine this model and further assume non-degeneracy of

109

the conduction electrons (ie: that dfFD/dE evaluated at Ef cannot be approximated by a d-
function, an assumption that is particularly important at high temperatures, where fFD is the

Femri-Dirac distribution function) but take only elastic scattering into account, and find that
they obtain NTC at high temperatures even in this case, but only for p > 400 tin-cm.
They take this to be a possible explanation (at high temperatures) of the behavior that has
come to be called the Mooij Correlation: NTC associated with large p. However, they
also found that in such cases the thermopower is especially sensitive to features in the
DOS. Our experimental thermopower and resistivity show no signs of the behavior
predicted by this model.

If NT C results from the release of electrons into the conduction band from covalent
or ionic bonds as the temperature increases, one often observes a negative minimum in
a(c) accompanying a maximum in p(c). These NT C’s also tend to be quite sharply
concentration dependant and to occur at compositions corresponding to maxima in the
respective melting curves, and to rather strongly bound crystal structures in the solid alloy.
While the NTC’s that we see in liquid Cu-Ce and Ag-Ce do occur at maxima in their
melting curves, and these also do correspond to preferred solid crystal structures, no
feature is apparent in p(c) at these compositions that would reflect evidence for bound
clusters in these liquids. Moreover, p at these compositions is quite low (~100 an -crn),
and reaches it’s peak at much higher RE-concentrations.

Salmon [9] has discussed the importance of the position, width, and height of the
first sharp diffraction peak in liquids. Significantly, the width of the peak has proved in
several cases to be directly related to the r space distance over which CSRO is present, and
the height of the peak to the amplitude of the density fluctuations in g(r). The height of the
prepeak in the RMC SCuCu(Q) suggest that the concentration fluctuations are not large.
Also, the width implies that the order tapers quickly in the liquid. Such weak order is quite 1
consistent with the absence of features in p or S. We suggest that a is perhaps a more
sensitive indicator of CSRO than p or S, and that the NTC in Cu-Ce and Ag-Ce is
indicative of ordering in those systems. We turn for an example of this sensitivity to the
liquid AlzCa system [14], which also possesses small p (< ISOW-cm) and a NTC (a= -9

x 10'4 C ‘1). The resistivity in this case is also featureless and metallic, as is the
thermopower; however, the substitution of Ga for Al gives strong evidence that clusters of
A12Ca do exist in this system.

Perhaps the strongest single argument in favor of CSRO in liquid Cu6Ce is the

110

close similarity of the CuCuCu bond angle distributions for the RMC generated data and
the crystalline material. (Figure 8). The cluster of bond angles in the liquid near 55°
compares well with the crystalline configuration, which is peaked slightly closer to 60° . It
is unlikely that these angles result merely from close packing in the liquid because of the
substantial size difference between Cu and Ce atoms (about 50%). It is also clear from
examination of the arrangement of atoms in the crystal that bond'angles near 115°-120° at
least for the crystal are quite independent of those at 60° (not necessarily 2 x 60°).
Perhaps those near 110° in the liquid are similarly independent.

We have argued that CSRO does occur in liquid Cu6Ce. This implies that CSRO

can be seen even in liquid alloys with p <100 11.0 -cm . The magnitude of the NTC and the
height and width of the prepeak in the structure factor indicate that perhaps only a small
fraction of the atoms participate in clustering on average, which could explain the lack of
discernible features in p or S. It is significant that no signs of a prepeak are seen in
SCuCu(Q) for liquid Cu.75Ce.25 [3], even at this concentration comparatively close to

Cu6Ce. A shallow minimum can be seen in Waseda’s partial Sw(Q) at 1.6 A‘l, but it

cannot compare in depth to the minimum modelled in our RMC calculations. It might be
reasoned that weak ordering occurs in a reasonably narrow concentration range around
stoichiometry; this is consistent with the narrowness of the minimum in a. The chemical
similarity with other Cu alloys with the light rare earths, and similarity of a for liquid Cu-
La, invite further study into possible ordering in these alloys, and possibly RE alloys with
other noble metals.

We are happy to acknowledge a grant from the National Science Foundation
(DMR-9406812). We appreciate the use of the facilities at IPN S, and thank in particular
David Price and Ken Volin for their help in the neutron data acquisition and analysis.
Thanks are also due Robert McGreevy, for supplying the RMC source code.

[ll

[31
[31

[41

[5]
[7]
[31
[9]
[10]

[11]
112]
[13]
[141

[15]

111

G. Busch, HJ. Guntherodt, H.U. Kunzi and HA. Meier, Electronic Structure of Liquid
Transition and Rare Earth Metals and their Alloys, in: The Properties of Liquid Metals.
edited by S. Takeuchi (Taylor and Francis, London, 1973), p. 263-276.

J.B. Van Zytveld, 1977, Inst. Phys. Conf. Ser. No. 30, p. 212.

Y. Waseda, The Structure of Non~Crystalline Materials; Liquids and Amorphous Solids.
1930, McGraw-Hill Inc., USA.

K- SUZUKI. K. Sumiyama, Y. Homma, H. Ameno, and T. Hihara, 1993, J. Non-Cryst.
Solids, 156-153, 323.

F. Steglich, J. Aarts, C.D. Bredl, W. Lieke, D. Meschede, W. Franz, and H. Schafer,
1979, Phys. Rev. Lett. 43, 1892.

M. Walhout, L. Haarsma, and J.B. Van Zyrveld, 1989, J. Phys.: Condens. Matter 1, 2923.
I. Kooistra, K. Dreyer, and J.B. Van Zytveld, 1988, J. Phys. F.: Met. Phys. 18, 1225.
H.-J. Guntherodt and A. Zimmermann, 1973, Phys. Kondens. Materie 16, 327.

P. Salmon, 1994, Proc. R. Soc. Land. A, 445.

“Atlas of Crystal Structure Types for Intennetallic Phases”, ASM International, Materials
Park, OH, 1991

L. Rottman, J.B. Van Zytveld, 1979, J. Phys. F: Met. Phys. 9, 2049-56.

J.M. Ziman, 1970, Proc. R. Soc. Land. A 318, 401.
p. Fresard, H. Beck, and M. Itoh, 1990, J. Phys.: Condens. Matter 2, 8827.
D. You, I-LS. Schnyders, and J.B. Van Zytveld, 1997, J. Phys.: Condens. Matter 9,

1407- 15 .

J. Van Zyt'veld, “Liquid Metals and Alloys”, Handbook on the Physics and Chemistry of
Rare Earths, Vol. 12, edited by KA. Gschneidner Jr, and L. Eyring, Elsevrer Scrence
Publishers B.V., 1989.

112
E' C .

Emmi: p as a function of atomic fraction, c, of RE, for the liquid alloys: Cu-Ce, Cu-La. Cu-
Nd, Ag-Ce. Present results: 0 -Cu-Ce; I -Cu-La; V -Ag-Ce. A -Cu-Nd, reference [3].
In all cases, the error bars are encompassed by the point.

figure]; S as a function of atomic fraction, c, of RE, for the liquid alloys: Cu-Ce, Cu-La, Ag-
Ce. Present results: I -Cu-Ce; I ~Cu-La; V-AgoCe. Unless shown, error bars are to

be considered smaller than the extent of the point.

figure}; 1/ p d 'p/dT as a function of atomic fraction, c, of RE, for the liquid alloys: Cu-Ce, Cu-
La, Cu-Nd, Ag-Ce. Present results: a -Cu-Ce; I -Cu-La; V -Ag-Ce. A -Cu-Nd,
reference [8]; data for pure liquid Ce, La, Nd from [15].

Emmi; The experimental total structure factor, S(Q) ~ 1, for liquid Cu6Ce (solid line), shown
with the converged Reverse Monte Carlo fit started from the crystalline configuration of 3500

atoms (dashed line).

Eigurfi; The total correlation function, T(r) = r ‘ g(r); g(r) is obtained by fourier transform of
the experimental structure factor S(Q) for liquid Cu6Ce. The dashed lines result from a Gaussian

deconvolution of T(r).

Eigurgfi; The RMC partial structure factor, 3mm) - l, for liquid Cu5Ce: dashed line -
simulation results starting from a random initial configuration; solid line- simulation results
starting from the crystalline initial configuration.

Elmira]; The RMC partial pair correlation function, gawk), for liquid Cu6Ce: dashed line -
simulation results starting from a random initial configuration; solid line- simulation results
starting from the crystalline initial configuration.

figural; The RMC CuCuCu bond angle distribution for CU5Ce. Uppermost line: Bond angle

disuibution of the initial random arrangement Middle line: Bond angle distribution at the

‘ conclusion of RMC simulation for initially random arrangement (solid line), and initially
crystalline arrangement (dashed line). Bottom line: Bond angle distribution for initial crystalline

configuration

 

 

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