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Michigan State
University

 

TRANSPORT PROPERTIES OF DILUTE

TIN-INDIUM ALLOYS

BY

a)

Richard E Fryer

A THESIS

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

MASTER OF SCIENCE

Department of Physics and Astronomy
College of Natural Science

1964

.22 q
8 *\31

fa.
03.x
”WW“

ABSTRACT

TRANSPORT PROPERTIES OF DILUTE
TIN-INDIUM ALLOYS

by Richard E. Fryer

Five alpha phase tin—indium alloys, varying from
l/4% to 5% indium, were cast and drawn to .101" rods about
10cm. long. The residual resistivity and the thermal
conductivity from 80K to 300K was measured. The thermo-
electric power and the electrical resistivity of pure tin
and two of the above alloys was determined from 4.20K to
3000K. These samples for the thermopower and resistivity
measurements were obtained by extruding the thermal conductivi—
ty sample into two wires, each about a meter long.

The diffusion thermopower of all samples was
negative, and the pure tin sample had a large phonon drag
peak at 280K. The phonon drag term was effectively quenched
with 1% solute. It is suggested that a change in the Fermi
surface upon alloying is responsible for this effect.

Some of the measurements relating to thermal
conductivity, or the analysis, was apparently in error,
and a reliable estimate of the lattice conductivity was

not obtained.

ACKNOWLEDGMENTS

I wish to thank Dr. F. J. Blatt for suggesting
the problem, and for his assistance and guidance during
much of this work. I also wish to express my sincere
appreciation to Dr. Peter A. Schroeder for his encourage-
ment and suggestions concerning interpretation of the data.
Financial support by the Atomic Energy Commission

is gratefully acknowledged.

ii

TABLE OF CONTENTS

Chapter Page
I. INTRODUCTION . . . . . L . . . . . . . . . . . 1

II. THEORY . . . . . . . . . . . . . . . . . . . . 2
Thermoelectric Power 2

Electrical Conductivity 9

Thermal Conductivity 10

III. APPARATUS . . . . . . . . . . . . . . . . . . 12
IV. PREPARATION OF SAMPLES . . . . . . . . . . . . 22

V. DESCRIPTION AND ANALYSIS OF RESULTS . . . . . 25
Fermi Sufrace of Tin 25
Thermoelectric Power 25

Electrical Resistivity 30

Thermal Conductivity 35

REFERENCES . . . . . . . . . . . . . . . . . . . . . . 43

iii

Table

LIST OF TABLES

Page

9 values of Sn—In samples and size of
D

thermopower samples . . . . . . . . . . . 24
Residual resistivity and resistance ratios

of Sn—In alloys . . . . . . . . . . . . . 24
Temperature dependence of ideal

resistivity curves . . . . . . . . . . . . 34

iv

LIST OF FIGURES

Figure Page
1. The Pb-alloy thermocouple . . . . . . . . . . 3
2. Schematic of thermal conductivity

apparatus . . . . . . . . . . . . . . . . l3
3. Wiring diagram for apparatus A . . . . . . . . 15
4. Apparatus A, thermal and electrical
conductivity . . . . . . . . . . . . . . . l6
5. Apparatus B, thermoelectric power and
electrical conductivity . . . . . . . . . l9
6. Wiring diagrams for apparatus B . . . . . . . 21
7. Fermi surface for tin . . . . . . . . . . . . 26
8. Thermoelectric power of pure tin and
tin—indium alloys . . . . . . . . . . . . 27
9. Characteristic thermopower of Indium in
tin . . . . . . . . . . . . . . . . . . . 31
10. Resistivity of tin and tin-indium alloys . . . 32
11. Log (p — p ) against log T for tin and tin—
. . o
indium alloys . . . . . . . . . . . . . . 33
12. Thermal conductivity of tin-indium alloys . . 36
13. WT versus T3 for tin-indium alloys . . . . . . 38
14. K/T versus T for tin—indium alloys . . . . . . 39
15. Total and electronic thermal conductivity
of tin-indium alloys . . . . . . . . . . . 4l

I. INTRODUCTION

To gain an understanding of the properties of
conduction electrons in a metal, several methods of getting
detailed information about the Fermi surface have been
devised. Many of these methods are only useful in the cases
of extremely pure metals and low temperatures. Transport
effects are not restricted to these conditions, however.
Thermal conductivity, electrical conductivity, and thermo-
electric power are transport properties which are sensitive
to variations in the properties of conduction electrons, and
they are theoretically related.

The object of this thesis is to (1) describe the
two apparatus and the methods used to measure the above
transport properties, and (2) describe and analyze the

results of those measurements.

II. Theory

Thermoelectricgpower

 

If a temperature gradient is set up in a metal.
electrons will instantaneously flow towards the cold end.
A potential difference builds up between the ends of the
metal until the current produced by the temperature gradient
is balanced by an equal and opposite current produced by the

EMF. If the temperature at one, T is fixed, and the

1!

other. T2, is varied, the derivative of the EMF with

respect to T2 is called the thermopower of the metal at

temperature T2:

_ d EMF)
S — dT Eqn. 1.

If a thermocouple is made of two different metals,

such as in Fig. 1, then Jones (1956) shows:

T2

EMF

(5 - SA) dT Eqn. 2.

B

T1

where SA and SB are the characteristic thermopowers of

metals A and B respectively. If T1 is held constant, then:

d(EMF) _ _ _
—EEE——-— sAB — SB(T2) SA(T2) Eqn. 3.

where SAB is the thermopower of the two metal thermocouple
combination. If the wires (metal B) and the sample
(metal A) are the same material, Eqn. 3 shows that the EMF

will not vary with termperature. In fact, it will be zero.

EMF

Figure 1. Schematic of alloy - lead thermocouple assembly.

An experimental measure of the thermopower of the alloy,
wire A, is then possible if the thermopower of metal B is
known.

If the-thermopower is derived considering only that
some mechanism exists that can re—establish the electron
distribution once it has been disturbed, but ignoring any
effects due to the drift velocity of phonons, then the
result is called the diffusion thermopower, and Wilson

(1954) shows that this has the form:

 

E = g Eqn. 4.

Where K is the Boltzman constant, ”(E) is electrical
conductivity of the metal, f; is the Fermi energy, and
e the electron charge. This expression is valid when

T 5 5D (:D is the Debye temperature) or where impurity
scattering dominates (low temperature).

There is a second contribution to the thermopower
due to the transport of heat by lattice vibrations.
These vibrations are quantized in energy packets whose
net velocity is opposite the temperature gradient. If
there is an interaction between the phonons and the
electrons, the flow of phonons will tend to carry, or
"drag" the electrons with them from the hot to the cold
end. Thus, this contribution is called phonon drag, and
is denoted by Sg' The total thermoelectric power is the

sum of S and 89:

d
S = S + S Eqn. 5.

It is desirable to distinguish the terms in Eqn. 5.
Calculation of the phonon drag term involves a
knowledge of phonon-electron interactions. There are two
kinds of interactions, normal and Umklapp. These terms
give rise to a phonon drag thermopower of opposite sign.
Sondheimer (1956) shows that for pure metals at low

' I

temperatures (T‘g.

C
Sg = S—EIE Eqn. 6.

where Cg is lattice specific heat which varies as T3
and N is the density of conduction electrons. This
derivation neglects Umklapp process and phonon—impurity
scattering.

As the temperature increases, it is necessary to
look at the transfer of momentum between phonons and
electrons in detail. There are two kinds of electron—
phonon collisions, normal and Umklapp (U) processes.

—) .—_‘b
In the normal process, where K and K' are the electron
wave vectors before and after collision, and q’is the
phonon wave vector:
.a _A ,;
K' - K.: q = O Eqn. 7.
Since the phonons will be traveling from the hot to cold
end, they will tend to "drag" the electrons with them.
As the momentum of the phonons is much smaller than the
momentum of the electrons being scattered, there is a small

change in momentum, and a small, usually negative, contri—

bution to the thermopower.

In the Umklapp process, the reciprocal lattice

,_3
vector, g, must be included, and Eqn. 7 becomes:

K ‘ K'.i q =_; Eqn. 8.

This results in a large change in momentum of the electron,
and a correspondingly large change in Sg. In addition,
the momentum reversal (which we would expect from the
interpretation of a U process as a Bragg reflection) may
result in a change of sign in 89. Ziman (1961) has shown
that when an extended zone is considered, distinction be—
tween normal and U processes is not necessary. He further
shows that the contribution to S9 will be positive if the
.—A A
q vector lies in an unoccupied region of K space, and
negative if it lies in an occupied region.

For alloys, impurity atoms cause considerable

).

phonon scattering, and at high temperatures (T > 3D

all kinds of phonon-phonon and phonon—electron collisions
will become effective. These can be suitably accounted
for by an averaged relaxation time (MacDonald, 1954):

C9 7p
S _ +

g 3Ne T Eqn. 9.
P

(w-

L
pe
where Tpe is the average relaxation time for phonon-

electron scattering, and Tp is a relaxation time for

all other events. Thus, when phonon-electron scattering
dominates, Tpe/Tp << 1, and the low temperature limit
becomes Eqn. 6. At high temperatures, T /T >> 1, and

P8 P
the equation becomes:

C T
Sg N 3N8 Tpe Eqn. 10.

which is proportional to T— . At room temperature in noble
metals, Eqn. 10 predicts an appreciable Sg term. Instead,
it is very small, possibly because the normal and U pro-1
cesses more or less precisely cancel each other. A
detailed quantitative treatment is not possible except for
monovalent metals, and even there the agreement with
experiment is not complete. However, it is possible to

say that the peak in S9 should appear between (FD/5)

and éD/lO), and is very sensitive to temperature and to

the degree of distortion of the Fermi surface from a sphere.
In tin, the degree of distortion is striking (Gold and
Priestley, 1960).

For diffusion thermopower, Kohler (1949) has shown

that:

d — W Eqn. 11.

where Wj and Sj are the thermal resistivity and the
thermopower of the jth scattering mechanism if it alone
were present. If the Wiedemann—Franz law can be validly
applied to each of the thermal resistances (L0 is the

Lorenz number), that is, if:

LO =-W%— Eqn. 12.

which is true if impurity scattering dominates or if T > 5D'

then the Kohler relation can be written:

p. S.
S =-l—el~—l Eqn. 13.
d ‘1" pj
J
which is the Nordheim—Gorter relationship. If Matthiessen's
rule is obeyed:
2 p, = p Eqn. 14.
j J

If two scattering mechanisms (a and b) are present,

= + . .
de pa Sa pb Sb Eqn 15
where p is the total resistivity and Sd is the thermopower

of the alloy. In an alloy, Sa and Pa become the parameters

of the pure metal and are associated with lattice scattering.

In this case, Sb becomes the characteristic thermopower
of indium in tin. Substituting pb from Eqn. 14 into Eqn. 15

and dividing by p:

P S _ P
S = _2__e. +.ln____e1 Sb Eqn. 16.
d P P
collecting terms:
pa (S - S )
Sd = Sb +E_ a b Eqn. 17.

A plot of Sd against l/p, at a constant temperature,
for various samples will give a linear relationship if
b are

constants over the impurity range being considered. The

the assumptions are fulfilled and if Sa and S

intercept at 1/p = O is S Blatt (1957) shows that if

b'
the metal can be approximated as a single band of standard

form, then the characteristic thermopower of the impurity

in the parent metal should be proportional to T.

Electrical Conductivity

 

The Bloch-Gruneisen formula predicts the temper-
ature variation of resistivity in this manner:

K T5

3\
F =._:€_ J5 Ci} Eqn. 18.

where Jn(x) are transport integrals (Sondheimer, 1950).

In the low temperature limit (T << 2) Eqn. 18 reduces to:

5
P = 92‘ ('2) Eqn. 19.

And for high temperatures (T/q42.5):

_CT
P ’ 497.6 .2 Eqn' 20'

The validity of this formula is dependent on the constancy
of 5D. The best approximation to this is in a pure
metal, but even here there may be significant departure
from the predicted temperature dependences at the low
temperature limit, as it depends on the Debye temperature
to the sixth power.

Matthiessen's (Jones, 1956) rule states that:

P = pi + p0 Eqn. 21.
where pi is the ideal resistivity, arising from scatter—
ing of electrons by phonons in the pure metal, and p0 is
the residual resistivity, which is due to scattering of
electrons by imperfections and impurities.

According to this theory, p0 is independent of
temperature. This expression is based in part on the

assumptions that: (1) The addition of impurity atoms of

a different valence than the parent metal doesn't change

10

the effective number of electrons. (2) The impurity
atoms do not affect pi by changing the lattice vibrations.
Neither of these approximations are likely to be strictly

obeyed, so Eqn. 21 will not be exact..

Thermal Conductivity

Lattice vibrations carry heat, as can be verified
by maintaining a temperature gradient in an insulating
crystal. In a metal, however, conduction of heat by
electrons is the dominant process. These mechanisms are
essentially independent, and can be added to give the total

conductivity (Klemens, 1956):

K = K + K Eqn. 22.
e 9
where Kg represents lattice conductivity, and Ke is the
electronic conductivity.
In a pure metal, the thermal resistance is largely
due to scattering of electrons by lattice vibrations.

Wilson (1954) has shown that this thermal resistance,

called the ideal resistance, has the form:

5 . .

4 T ” l t
.314 (r) J <—) —-—J <—>
1 LOT ._ . 5 T 2__2 7 T

where p“ is the resistivity at the Debye temperature, K is
the Fermi radius, and Q is the Debye radius. In the
limit of high temperatures, this reduces to:

Eqn. 24.

11

which istflmaWiedemann—Franz law, and the low temperature

limit is:

N

w. _.... .5 (.0) (— _..__._ 25.

At low temperatures, the residual thermal resistance
can be calculated from the Wiedemann-Franz ratio. When
Matthiessen's rule, Eqn. 21, is valid, the thermal resis—

tance considering electrons only becomes:

po 2
W = W + W. = -—- + BT Eqn. 26.
e o 1 LT -

The coefficient of T2 in Eqn. 25 is represented by the
constant, B. In alloys, W0 is large, so Ke is small.
Then Kg, the lattice conductivity, and Ke are often of
comparable magnitude. If Ke can be calculated and K
experimentally determined, then K can be found from
Eqn. 22. It is not easy to measure Kg on a pure sample,
for it is difficult to measure K with enough accuracy to
deduce Kg reliably.

Klemens (1958) has also shown from consideration
of the mean free path that Kg should vary as T“1 at
relatively high temperatures, and as T2 at low. At high
temperatures, phonon-phonon Umklapp and phonon-impurity
scattering control the conductivity; at low, phonon-

electron and phonon-dislocation scattering dominates.

III. APPARATUS

Two pieces of apparatus were used in the measurements
described. On the first apparatus, hereafter referred to
as apparatus A, thermal conductivity measurements were
made. The other apparatus, referred to as apparatus B,
was used to measure thermopower and electrical resistivity.
The thermal conductivity apparatus was built by
Garber, Scott, and Blatt (1963). It was originally designed
with a form factor that would allow thermal conductivity,
resistivity, and thermopower to be measured at the same
time, using a potentiometer that was capable of measuring
.01 microvolt directly. The instrument used in these
measurements, a Cambridge Microstep, was limited to .l
microvolt direct measurement, thus increasing uncertainty
in temperature, resistance, and thermopower by nearly an

order of magnitude over previous measurements.

In Fig. 2, the heat flow, 6, down the sample is

given by: ~ \
K(T) = fi—g—f; Eqn. 27.

where AL is the length of the sample, AT the temperature
between the two ends after equilibrium has been reached,
and A the area. The temperature difference is maintained
at 20 to 30K, and D, the power, is calculated from the
voltage and current in the lower heater (T2). The power

in the upper heater (Tl) doesn't need to be measured,

since it raises the temperature of the whole sample.

12

13

//////////m

Heat Leak

 

Upper Heater

 

Sample

 

 

 

Lower Heater

 

Figure 2. Schematic of thermal conductivity apparatus.

14

A schematic of the measuring circuits and detailed
drawing of apparatus A appears in Figs. 3 and 4. The
carbon resistors were not used in these measurements. A
16 lead platinum seal brings all of the electrical connec-
tions out of the vacuum can, except for the Pb potential
leads, which were taken through the can directly to the
copper wires in the measuring circuits by a special glass—
Kovar seal. Number 36 wires were used from the seals to
the outside of the dewar to minimize heat influx. The
glastic rod which supports the sample, heaters, and
temperature measuring plates has a very small thermal
conductivity, and the upper heater block is connected
via a small pure copper wire to a copper plate which is
in good thermal contact with the bath. A11 contacts to
the sample are physical, and mylar film .00024" thick
used with nylon bolts provides thermal contact where
electrical insulation is required. The temperature
measuring plates were the location of thermocouple
junctions, and were also the potential measuring points.
These plates contacted the sample in a tapered opening of
width 1/64" around the sample.

The heaters were wound with about 1000 ohms of
manganin wire. The resistance of the leads to the
heater is much less than 1000 ohms, so the power loss
in the leads is much smaller than that dissipated in the
heater.

A gold-2.1 at .1% cobalt vs. silver~37 at .% gold

15

._L_12V. 40
{if VAC 110
~1K,, VAC
Upper Heater
a a Lower Heater
[:J;{:: Potentiometer #1

Heater Circuits

 

 

 

 

 

 

 

 

 

 

 

 

6V. [/ ;; ——_‘$\‘_____;Potentiometer #3

Sample

.. Z {N

Thermocouples

 

 

 

 

 

 

 

 

 

 

Potentiometer #2

 

 

Temperature and Residual Resistance Circuit

Figure 3. Wiring diagram for apparatus A.

l6
PUMPING.LINE

Z"

      

 

 

 

 

 

PLATINUM—
GLASS SEAL

l6 LEADS

CONTACT
WITH BATH

 

 

W
HEAT SINK AND TEMP. REFERENCE
l6 LEAD PLATINUM-GLASS SEAL
HEATER

TEMPERATURE MEASURING PLATE
CARBON RESISTOR
THERMOCOUPLE JUNCTION
NYLON SCREW

GLASSTIC SUPPORT

SAMPLE

NYLON SCREW

THERMOCOUPLE JUNCTION
CARBON RESISTOR

TEMPERATURE MEASURING PLATE

HEATER
VACUUM CAN

 

 

Figure 4. Apparatws A, thermal and electrical conductivity.

l7

thermocouple is connected between the copper heat sink and
the upper measuring plate, and this determines the temper-
ature of the upper end of the sample. Another is connected
between the plates, and determines the temperature dif-
ference across the sample. The thermocouple wire had been
previously calibrated.

The connection of current leads for the resistance
measurements is made to the copper heater block. Near
liquid helium temperature the conductivity of the sample
was greater than that of the heat leak. A gradient of
about 20K caused the temperature of the upper end, T1'
to increase 30K. This prevents an accurate measurement of
the conductivity below 80 or 90K.

The thermopower and resistivity cryostat, (apparatus
B) was built by Wold (1963) and was designed to keep one
end of the sample-Pb thermocouple at the bath temperature,
and the other at a higher variable temperature. At the
same time, a resistance sample was kept at this higher
temperature, which could be controlled between 4.20K and
3000K. The apparatus consisted of the sample holder, the
stainless steel support tube, and a low temperature junction
which was always in the liquid.

The sample holder was a copper cylinder with holes
machined in it parallel to the axis. Inserted in these
were the platinum resistance thermometer and the resistance
sample, mounted on a vycor rod. One end of the block was

machined to a small diameter projection, on which a

18

manganin heater of approximately 250 ohms resistance was
wound. Also, slots were turned around the block concentric
with its axis to facilitate wiring changes. The base was
drilled to let leads in, and then threaded to fit a copper
can which enclosed all wiring connections. The other end
of the stainless steel rod has a 21 pin connector, at

which all electrical connections (except the heater's) are
made. A styrofoam sleeve thermally insulates the can from-
cold gas. The apparatus is represented in Fig. 5.

To avoid discontinuities in measured voltages at low
temperatures, the bismuth~cadmium eutectic (non-super—
conducting) solder was used for all connections to the
samples. The resistivity sample and the platinum thermo-
meter had both voltage and current leads. The cold junction
of the Pb—sample thermocouple is connected to this plug, and
it is interesting to note that this eleminates the need of
one Pb wire, since both ends of it would be at the bath
temperature. The current used for the resistance sample
was 20 ma. for the alloys, 100 ma. for the pure, and 2 ma.
for the platinum resistance thermometer. These currents,
supplied by separate large capacity 6 volt storage batteries,
were regulated by series helipot resistors, and were monitored
using one ohm standards and a Leeds and Northrop type K3
potentiometer.

It was found that a stable temperature for the copper
block was much more easily reached at temperatures near the

bath if there was a heat leak from the copper block to the

l9

 

 

//

|
’ er collar

stainless steel
/ tubing

styrofoam

 

 

 

 

1:

 

 

 

 

copper can
//

wl”’,.copper block
3/

 

 

Bakelite disk

: Heater
r” -
K

P’,——- stainless steel tubing

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

F”——— 21 pin plug

 

Figure 5. Apparatus B, thermoelectric power and electrical
conductivity.

20

bath. To provide this heat leak, two number 20 wires of
7” and 11" length were connected to the block. When the
block is raised to a high temperature with respect to the
bath, the liquid level drops first below one, then the
other wire, thereby minimizing the heat leak when it is no
longer useful.

As illustrated in the wiring diagram, Fig. 6,
current reversing switches were used in the thermometer
and the sample circuits to eliminate voltages arising in
the circuit or at the connections. The dewar was filled
with liquid well up onto the block, so that the block
temperature was the same as the 21 pin connector, and a

preliminary check could be made of all measurements.

21

110 VA \ ® 250.0. Heater

_—_l \,

Heater Supply

 

 

Potentiometer #1

 

 

 

 

 

 

J
f '

15L T '
Batt. SUP- 1 Resistance Sample

‘Potentiometer #2

 

 

 

 

 

 

 

 

Resistance Sample Circuit

 

 

Potentiometer #1

 

 

 

 

 

 

 

1. 6V. 1&0— ‘1 Platinum
Thermometer

 

 

 

 

 

 

 

 

Potentiometer #2
Thermometer Circuit

Figure 6. Wiring diagrams for apparatus B.

IV. PREPARATION OF SAMPLES

The pure tin sample was made from high purity
(99.9999%) tin supplied by Cominco, and all alloys were
made of tin (supplied by Cominco) and indium (supplied by
ASARCO) of 99.999% purity. After etching in dilute HNO3.
the metals were flushed in distilled water and dried by a
heat gun. They were then weighed on a Mettler balance to a
precision of 10 micrograms and placed in a graphite crucible
which had previously been outgassed under vacuum at
temperatures above lOOOOC. Then the metals were melted
together, keeping the pressure below 10-3mm Hg, and the
crucible was agitated to facilitate mixing. The alloys
were then poured into a cool cylindrical copper mold with
length 2-1/2" and .150" diameter. The pour invariably
wetted the copper mold near the opening, thus making a check
of the loss of the metal due to evaporation impossible.

The diameter of the samples was small enough so
that they could be inserted directly into a die plate, and
be drawn to about .103” diameter. These samples were then
heavily etched in HNO3 to reduce them to the required
.101" diameter to fit the sample holder. Five alloys
(1/4%, l/2%, 1%, 2%, 5%) were made, 8 to 10 cm. long.

These were annealed in silocone oil at 1850C for about
10 hours and then slowly cooled. After the thermal conduc-

tivity measurements were made, some of these samples were

22

23

extruded through diamond dies to produce a thermopower and
a resistivity specimen for apparatus B. Resistivity
samples were 0.01" diameter and about 1 meter long. The
thermopower samples varied in diameter (see Table l), but
all were about 1 meter long. All samples were heavily
etched to remove impurities that might have entered the
surface during the extruding process. The resistivity
sample was wound on a 1/4" diameter by 2" vycor tube, and
was sealed under vacuum into a 1/2" pyrex tube; the
corresponding thermopower sample was sealed under vacuum in
a 3/8" diameter pyrex tube long enough to avoid bending
the sample. Then both samples were annealed for 30 hours

at 1850C and cooled slowly at room temperature.

24

Table 1. yo values of Sn-In samples and size of thermopower
samples.

 

 

 

 

 

 

 

Dia.
Sample (In in Sn) Thermopower 50

Sample

Pure Sn .0286" 1550K

0.99% .0236" 1450K

2.05% .0236" 1380K

Table 2. Residual resistivity and resistance ratios of
Sn—In alloys.

Sample (In in Sn) p273/p4.2 App. B p4.2, uflcm App. A
5.01% - 2.62 i .03
2.05% 10.55 1.06 i_.04
0.99% 20.10 0.43 i .08
0.56% — 0.24 i .04
0.25% - 0.13 i .01

Pure Sn 7150 0.00162

 

V. DESCRIPTION AND ANALYSIS OF RESULTS

Fermi Surface of Tin

 

The Fermi surface of tin has been the subject of
considerable interest recently. The zone structure using
a free—electron spherical Fermi surface, Fig. 7, has been
worked out by Gold and Priestley (1960). Experimental
resutls of several investigators agree substantially with
this geometry, but Gantmakher (1963) has shown that a
nearly—free electron model, based on Harrison's method,

results in some improvement with his data.

Thermoelectric Power

 

The thermal EMF of a pure Pb—alloy thermocouple was
measured as a function of the temperature (T2 in Fig. l)
of the copper block in apparatus B. This was plotted vs.
temperature and the slopes obtained by use of a mirror.

This slopeis S the thermopower for the alloy—Pb

AB'
combination. The thermopower of the alloy (or the pure
metal), plotted in Fig. 8, was then obtained from Eqn. 3,
using the data of Christian, et a1 (1958) for the thermo-
power of Pb.

The pure tin sample has a positive maximum of 280K,
which is probably caused by phonon drag. An interpretation

of the data as a positive peak superimposed on a negative

diffusion component is consistent with phonon drag theory,

25

26

' .fl / ‘
,,- ' :\_, ~ \
/- \ \.\. f” ‘.'l )\‘/‘(‘\.\; lope 6

zero 61
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, \"' / 3 .
:cnez: ....”" \-/
hoies/
r- . [3c
.......... . .
. L,
. X ,. a i
l .
H

 

 

.--_-----_- -. .-
Or .
.02
.. " l
\

<;—/

zone 4 (o) hoic'.

 

zone 3’ 9207c:

 

n 4: 2 a
to c (b) :19: for: tom 51¢IQC1'O'1!

Figure 7. Fermi surface for tin.

27

 

 

 

 

 

 

 

 

 

r 1 T IMT 1 1 l 1 1 1 1 1 1
1 .4
1 \
L» \ S ,uv 3 _
1 ‘ a.
1_ .2 __
L \ .1
i 20 40 \~\ 80 100 120 140 160 180 200 220 240 260 280
1 1 1 \ 1 1 1 1 1 J 1 1 1
T l I I I I T T l I l T I
T °K
,__V - 1 __.
i
,-—\ \. -.2 “—1
\\
—— \ — 3 is
\\ a
J” Pure Tin _.4 _1
l
1.— -.5 —I
l
1 i
P— 1% In —.6 ——+
E , 1
L
1
2
T- ‘\
__ 2% In
1 J J J I I L I l l | l l .
Figure Thermoelectric Power of pure tin and tin—indium alloys.

28

and the location of the peak, about 280K, is in the

expected range. The large size of the peak is not surprising
either. Inspection of the Fermi surface reveals several
surfaces, particularly in zone 4, that allow scattering of
electrons by phonons with q vectors lying in unoccupied
regions. These processes, as pointed out previously, give

a positive contribution to the phonon drag term. From 80

to 170K, the pure tin thermopower varies approximately as

T3.

Pullan (1953) found that from 3.80 to 4.3OK,the
thermopower of pure tin varied from -T0'9 to —T3°l,
depending on crystal orientation. That is, his data was
increasingly negative with temperature increase. Since
the thermopower of a metal is by definition zero at 00K,
our curve must reach a minimum and go to zero. Therefore,
these measurements agree with those of Pullan.

The 1% and 2% Sn-In samples had negative thermo-
powers at all temperatures. The addition of 1% In to Sn
reduces the phonon drag peak by a large factor, and in the
2% sample, it has nearly disappeared. There are four
recongized ways that the addition of a small amount of
impurity can reduce the phonon drag term in this manner.

In the first three, the addition of solute results in the
scattering of phonons which would otherwise have contributed
to the phonon drag process. In Eqn. 9, :pe becomes large

and S9 -——9 0. The observed reduction will occur if:

(1) The solute atoms differ appreciably in mass from the

29

solvent. (2) The solute atoms badly distort the lattice
locally. (3) The solute atoms locally distort the force
constants, between adjacent atoms, resulting in modified
elastic constants. (4) The Fermi surface is substantially
altered by the addition of the solute atoms. In order to
assess the importance of some of these factors, they are
compared with the equivalent factors in the Cu-Zn results
of Henry and Schroeder (1963). In the results, the phonon
drag term is largely unaltered by the addition of up to 35%

Zn to Cu.

ale
2

The differential mass term, , in Sn-In alloys
is .034, and in Cu-Zn is .029. This difference cannot
account for the effect. The change in lattice spacing for
Sn—In alloys in the a phase is less than the corresponding
change in Cu-Zn alloys (Hansen, 1958). The characteristic
temperature decreased with increasing impurity, varying
from 1550K for Sn to 1380K for 2%. The corresponding
change in the Cu—Zn measurements is from 3700K for pure Cu
to 3380K for 2.8% Cu—Zn; implies that a change in elastic
constants is not responsible for the effect. I suggest
then, that the drastic reduction in the phonon drag term may
be ascribed to a change in the Fermi surface.

The valence of Sn is (4) and In is (3). In the
periodic table, antimony, with valence (5), lies on the
opposite side of Sn from In. It would be instructive to

make similar measurements on tin—antimony alloys. In

particular, one would see if the effect of increasing the

30

number of electrons in the sample also affects the phonon
drag term.

The characteristic thermopower of In in Sn can be
found from Eqn. 17 by plotting Sd vs. l/p. Previous
measurements by Schroeder, Woolam and Wolf (to be published)
on Ag-Pd and Cu—Ni alloy systems show that the character-
istic thermopower of Solute in solvent is accurately
proportional to T, and passes through the origin. The

Sn—In results, Fig. 9, show that this alloy system cannot

be approximated by a single band of standard form.

Electrical Resistivity

 

In apparatus B, measurements were made on the
resistance samples at intervals from 4.20K to 3000K.
Table 2 lists the residual resistivity and resistivity ratios,
p273/f4.2, of the three samples. Kunzler and Renton,
(1957) show that for pure tin samples, the size effect
may be an important factor in determining the resistance
ratio. If their data can be applied to this pure tin
sample, the resistance ratio will be approximately 21,500,
an increase by a factor of 3.1. This value of
is in fairly good agreement with resistance ratios of
single crystals of zone refined tin. Fig. 10 shows the
variation of p with T for the three samples. The log
of ideal resistivity, log (p - p0) is plotted vs. log T
in Fig. 11 to determine the power of the temperature
dependence at low temperatures, and these powers are listed

in Table 3.

31

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32

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34

 

 

 

Table 3. Temperature dependence of ideal resistivity curves.
__. . -1
Sample (In in Sn) Slope (u cm deg )
Pure Sn T4.0 i 1
0.99% T3.6 i 1
2.05% T3.8 i l

 

At high temperatures (T > g) p varied as T, as
predicted by the Bloch-Grfineisen formula (Eqn. 20).
However, at the low temperature limit, the predicted T5
dependence (Eqn. 19) is not obeyed. Pure tin varies as
T4'O, 1% In as T3.6 and 2% In as T3°8. Thermal scattering
is responsible for the predicted T5 variation, and the
deviation of the alloys from this law is not surprising,
since the derivation is based on a number of assumptions.
For pure tin, however, the T4'0 dependence suggests that
in addition to thermal scattering, at least one more
temperature dependent mechanism is operating. Aleksandrov
and D'Yakov (1963) find that for pure tin, the resistivity
does vary as T5, and they suggest that derivations from
this in several other metals should approach the T5
law for conditions of decreasing temperature and increasing
purity. It may be that our sample would agree with the T5
dependence at a low enough temperature.

The best fit to a theoretical Bloch—Grfineisen

law was obtained by choosing various values of ID and

35

calculating the quantity p/pc from the data. The 5D
which best approximated the curve at high temperature

had to be reduced about 40% to fit the data near T/fi = .1.
This poor agreementis consistent with the T4 resistivity
variation. Some thermopower and electrical conductivity
measurements were made on apparatus A, but they have

considerably less precision than measurements taken with

apparatus B. The results do agree qualitatively.

Thermal Conductivity

 

The thermal conductivity of five Sn-In alloys was
measured between 80K and 300K at approximately 1—1/20K
intervals. The total thermal conductivity, K is plotted
against T in Fig. 12. The curves suggest that the con-
ductivity maximum, characteristic of the pure metal,
is shifted to higher temperature as the percentage of solute
is increased. Shiffman (1958) made similar measurements
on Sn—In alloys. Some of his results appear in several
figures, and are drawn in red.

If the thermal conductivity were entirely electronic,
the thermal resistivity would be given by Eqn. 26, and
a plot of WT against T3 would be a straight line with
“intercept po/2 and slope B. In a pure metal, the lattice
conductivity is usually two or three orders of magnitude
smaller than the electronic, and then the above approxi—
mation is valid. Going from a pure metal to an alloy, the

residual thermal resistance is substantially increased, and

36

   
    

V 1/4% In D 2% In

A 1/2% In 0 5% In

C) 1% In
100 T OK 200 300

111 I 111111-1__.l

Figure 12. Thermal conductivity of tin-indium alloys.

 

 

37

the lattice term is no longer negligible. Fig. 13 shows
this plot for the same data as in Fig. 12. The values of
pO/LO measured are marked on the axis at zero degrees, and
the extrapolation of the curves is also indicated. There
was no minimum in the measured conductivity. However, the
data suggests that such a minimum exists and that it shifts
to higher temperature as the percent of solute increases.
The dip in the curves at low temperatures («9100K) is at
least partially due to a lattice term, Kg. As the percent
of solute is increased, the K9 term becomes a larger part
of the total conductivity, and Eqn. 26 is no longer a good
approximation.

For high concentration alloys, the ideal part of the
electronic component, Wd’ should be small enough to neglect.
The conductivity will then be of the form of Eqn. 22,

and Ke becomes-£2

A

‘ 1

LT 2

K 7 + AT Eqn. 28.

A plot of K/T against T for these results is shown in
Fig. 14. At relatively high temperature and high concen—
trations, the measurements are approximately straight

lines. The slope of these lines is either negative or zero,

however, this corresponds to a negative or zero lattice

conductivity: a physically unreal situation.

If the ideal resistivity is included in Eqn. 28,

the result is:

%- q l + AT Eqn. 29.
L9_+ Bt
L

3

 

«~11

u

’1'" <40

 

 

 

 

L_

1

—er ',- , v

E

‘.__-- - ._,;.

 

lloys.

1n—indium a

for t

WT vs. T3

Figure 13.

 

unur [NU C I

”PI-IL . gnaw." ‘4‘

 

 

_
.
afi .
_
a

1 A”

1
L- ,, ._.

 

r—vr

1
‘1
,7-

 

 

lloys.

 

ium a

d

1n—1n

K/T vs. T for t

Figure 14.

 

 

 

 

 

--~

1‘ ~-

40

It is difficult to use this formula as a basis of analysis.
Shiffman has simplified this by expanding the Ke term in
Eqn. 29. This method cannot be used on our data, however,
because the necessary approximation is not accurate for
our samples. Another method of accounting for lattice,
residual, and electronic resistivity is the following.

The lattice thermal conductivity is obtained from Eqn. 22,
where Ke = %; in Eqn. 26.

The value of W0 is obtained from the resistivity

measurements. The ideal thermal resistance, W5, can be
obtained several ways. For the purpose of this analysis,
however, W; was calculated in the following way. Rosenberg

(1955) measured the thermal conductivity of pure tin, and
the purity of his sample was very nearly the same as ours.
From his thermal resistance data, the residual thermal

P

resistance, 3%, was subtracted at several points between

100

and 300K. The remainder was assumed to be Wi. Figure
15 shows the electronic thermal conductivity, Ke’ in
relation to the total conductivity. For one sample, K

would be positive. However, the fact that the calculated
electronic conductivity is larger than the total measured
conductivity in the other samples indicates that: (1) There
is a relatively large, unaccounted for error in some of the
data, or (2) Eqn. 26 doesn't accurately describe these
alloys.

The error in any quantity in Eqn. 27 is believed

to be less than 2%. However, there is an error introduced

 

 

1/231.

 

 

 

2%

/fl’ - $ 5%

 
    
   

 

 

__ ,2
l %
V /4/ D 2%
_—f;’,,_
.l o (3 5%
T OK 200 A 1/24’ 300
C) %
1 1 I l I 1 L I 1 l1 1 I
Figure 15. Total and electronic thermal conductivity of

tin—indium alloys.

42

by the fact that, as the specific heat increases, the

rate that the sample reaches thermal equilibrium decreases.
Small random variations in the thermocouple EMF made the
temperature change difficult to resolve when the sample
was near equilibrium, and true equilibrium was not always
attained. The estimated error from this source is
approximately 3%-5%. Errors in the electronic thermal

conductivity are indicated approximately in Fig. 15.

REFERENCES
Aleksandrov, V. N. and I. G. D'Yackov, Zh. eksper. theor.
Fiz. (USSR)_43, No. 3(9)(1962).
Blatt, F. J., "Solid State Physics," 4, 225(1957).

Christian, J. W., Jan, J. P., Pearson, W. B., and Templeton,
F. M., Proc. Roy. Soc. A 245, 213(1958).

Gantmakher, V. F., Soviet Phys. JETP 11, 549(1963).

Garber, M., Scott, B. W., and Blatt, F. J., Phys. Rev.
130, 2188(1963).

Gold, A. V., and Priestley, M. G., Phil. Mag._5, 1089(1960).

Hansen, M., "Constitution of Binary Alloys," McGraw—Hill
(1958).

Henry, W. G. and Schroeder, P. A., Can. J. Phys. 41, 1076
(1963).

Jones, H.,‘Encyclopedia of Physics," 28(1956).

Klemens, P. G., "Encyclopedia of Physics," 14_pt.4(1956).
Klemens, P. G., "Solid State Physics,” Z, 1(1958).
Kohler, M., Z. Phys., 126, 481(1949).

Kunzler, J. E. and Renton, C. A., Phys. Rev._108,’1397(l957).
MacDonald, D. K. C., Physica, 29, 996(1954).
Pullan, G. T., Proc. Roy. Soc. A 212, 280(1953).

Shiffman, C. A., Proc. Roy. Soc. 71, pt.4(l958).
2

Sondheimer, E. H., Proc. Roy. Soc. A 03, 75(1950).

—.—-—

Sondheimer, E. H., Can. J. Phys., 34, 1246(1956).

Wilson, A. W., "The Theory of Metals," 2nd Ed., Cambridge
Univ. Press (1954).

Wolf, R. A., Thesis, Michigan State University (1963).

Ziman, J. M., Phil. Mag. Supplement 10, No. 37(1961).

43

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