This dissertation has been microfilmed exactly as received 67-8925

MARTIN, Joel Jerome, 1939- THERMAL CONDUCTIVITY OF MAGNESIUM STANNIDE.

Iowa State University of Science and Technology, Ph.D., 1967 Physics, solid state

University Microfilms, Inc., Ann Arbor, Michigan THERMAL CONDUCTIVITY OF MAGNESIUM STANNlDE

by

Joel Jerome Mart in

A Dissertation Submitted to the

Graduate Faculty in Partial Fulfillment of

The Requirements for the Degree of

DOCTOR OF PHILOSOPHY

Major Subject: Physics

Approved:

Signature was redacted for privacy.

n Charg^^of Major Work

Signature was redacted for privacy.

Headf hf'jha j^r Department

Signature was redacted for privacy.

De^l of Grâbuate College

Iowa State University Of Science and Technology Ames, Iowa

1967 TABLE OF CONTENTS

Page

ABSTRACT vi

1. INTRODUCTION 1

A. Properties of Mg2Sn 1

B. Purpose of this Investigation 5

I I . THEORY 6

A. Theory of Thermal Conductivity of Semiconductors 6

B. Seebeck Effect 17 ill. EXPERIMENTAL PROCEDURE 20

A. Samples 20

8. Measurements 25

C. Errors 35

IV. RESULTS AND DISCUSSION _ 4l

A. Thermal Conductivity Results 41

B. Thermal Conductivity Discussion 41

C. Seebeck Coefficient Results and Discussion 59

D. Conclusions 66

E. Future Work 67

V. LITERATURE CITED 68

VI . ACKNOWLEDGEMENTS . 72

VII. APPENDIX 73 LIST OF TABLES

Page

Table 1 . Sample characteristics 25

Table 2. Relaxation time parameters 49

Table 3. Diffusion Seebeck coefficient parameters 62

Table 4. Sample experimental data 73

Table 5. Thermal conductivity and Seebeck coefficient results 74 I V

LIST OF FIGURES

Page

1 Crystal structure of Mg2Sn 2

2 Electrical resistivity, p, of the thermal conduc­ tivity samples 22

3 Hall coefficient, R, of the thermal conductivity samples 23

4 Hall coefficient, R, and electrical resistivity, P, of Mg2Sn samples K-11 and K-13 at low temper­ atures 24

5 Sample holder 27

6A Sample heat sink clamp 29

6B Thermometer clamp 29

7 Block diagram of the thermal conductivity appa­ ratus 33

8 Percentage correction of the measured thermal conductivity values calculated from the last two terms of Equation 66 40

9 Thermal conductivity results 42

10 The thermal resistance of Mg2Sn above 100° K 44

11 The thermal conductivity of Mg2Sn calculated from the Callaway theory with the size of the sample calculated from the Casimir theory 47

12 The figure shows the magnitude of the correction term in the Callaway theory 48

13 The thermal conductivity of Mg2Sn calculated from the Callaway theory with the size of sample K-13 adjusted to fit the data at 4.2° K 51

14 The thermal conductivity calculated with bound donor electron phonon scattering for samples K-13 and K-13 B is shown 56

15, The thermal conductivity calculated with bound donor electron phonon scattering for samples K-1 I and K-15 is shown 5 7 V

page

Figure 16. The absolute Seebeck coefficient for the diffusion range is shown 60

Figure 17. The phonon drag Seebeck coefficient is shown 64

Figure 18, The total Seebeck coefficient at low tempera­ tures is shown 65 V i

ABSTRACT

The thermal conductivity of several n-type MggS" samples was measured

from 4.2 to 300° K. The samples had uncompensated donor concentrations

on the order of 2 x 10^^ donors/cm^. Above 150° K, the thermal conduc­

tivity showed a T~^ temperature dependence which is characteristic of

lattice thermal conductivity. The theory for high temperature thermal

conductivity of Leibfried and Schloemann is In agreement with our results.

The bipolar electronic contribution was estimated to be U % of the total

thermal conductivity at 300° K. Below 100° K, the data were analyzed in

terms of the Callaway theory by combining the relaxation times for phonon- phonon scattering, isotope scattering, boundary scattering and bound donor electron-phonon scattering. Above the thermal conductivity maximum it was necessary to retain the exp (-0/aT) term in the phonon phonon scattering process to obtain the correct temperature dependence. The calculation in the neighborhood of the thermal conductivity maximum Indi­ cated that the only point defect scattering present in the samples was caused by the isotopes of Mg and Sn. At 4° K 'the data showed a smaller size dependence than the theory predicted if only boundary scattering was Included. in addition, the measured values were about half of the curves calculated with only boundary scattering. This result was ex­ plained in terms of an additional phonon scattering caused by the bound donor electrons. With this mechanism It was possible to account for the size difference and the difference In doping in the samples. As an aux­ iliary experiment the Seebeck coefficient was measured at the same time as the thermal conductivity. The phonon drag contribution to the Seebeck coefficient shows a T~^'^ temperature dependence from 30 to 100° K. I. INTRODUCTION

A. Properties of Mg2Sn

I . Crysta1 structure

Mg2Sn is a I I-1V compound semiconductor of the Mg2X family where X can

be Si, Ge, Sn or Pb. The Mg2X compounds crystallize in the Pluorite struc­

ture. Figure 1 shows the cubic unit cell of Mg2Sn. Mg2Sn has a lattice

parameter of 6.7625 A at 26° C (49). Shanks' found a value of 9.S x 10 ^

(deg)"' for the temperature coefficient of the lattice parameter at 300° K.

2. Energy gap

Winkler (53) determined an energy gap of (0.36 - 3 x 10~\) eV from his

resistivity, Hall effect and Seebeck effect measurements. Blunt _et_ a_L- (&)

found an energy gap of 0.33 eV from their Hall effect data. Nelson (40)

found a gap of 0.36 eV from his Hall coefficient measurements. An energy

gap of 0.34 eV was determined from Hall effect measurements by Lawson e_t

±1. (34)

The optica] measurements of Blunt e_t _a]_. (6) yielded an energy gap of

0.33 eV at 5° K; they found that the energy gap decreased as the tempera­

ture increased. The optical absorption measurements of Lawson _e^ aj_. (34)

gave an energy gap of 0.18 eV at 294° K. They found that if their measure­

ments were interpreted in terms of indirect transitions the energy gap in­

creased as the temperature increased. Lipson and Kahan (37) interpret

their optical absorption data in terms of an energy gap of about 0.18 eV at

'shanks, H. R., Iowa State University, Ames, Iowa. Private communica­ tion. 1 966. 2

O Sn ATOM

oMg ATOM

Figure 1. Crystal structure of Mg2Sn 3

0° K and a temperatu re dependence of -1.7 x 10"^ eV/deg.

At the present time, the energy gap of Mg2Sn is not understood. A gap

of about 0.36 eV seems to result from electrical measurements, but a gap of

about 0.18 eV seems to result from optical measurements.

3. Mob ility

Winkler (53), Ge i ck e_t _aj_. (21) and Lichter (36) found that the mobil-

- 2 5 ities of both holes and electrons have a temperature dependence of T in

the intrinsic range (T > 300° K). The temperature dependence was at­

tributed to optic mode scattering. The ratio of the electron mobility to

the hole mobility in the intrinsic range was found to be 1.20 - 1.25 by

Lipson and Kahan (37) and 1.23 by Blunt e_t_aj_. (6) The electron mobility

at 300° K is about 300 cm^\/"'sec"'.

Blunt et _a]_. (6) found that the mobilities of both electrons and holes

had a temperature dependence of T ^^ between 80 and 200° K. More recent­

ly, Ge i ck e_t a_l_. (21) and Lichter (36) found that the hole mobility varies

as T"'*^ for temperatures below 300° K. The T~''^ dependence was inter­

preted in terms of acoustic mode scattering.

4. Effect ive masses

Lipson and Kahan (37) have reported the most recent effective mass

values. They found that the conductivity effective masses are 0.15 m and

0.10 m for electrons and holes respectively; and that the density of state

effective masses are 1.2 m and 1.3 m for electrons and holes respectively where m is the free electron mass. k

5. Elect ronIc st ructure

Lipson and Kahan (37) have suggested that the band structure of Mg2Sn

consists of a triply degenerate valence band located at k = 0 and conduc­

tion band minima located away from k = 0. The magneto resis tance measure­ ments of Umeda (50) indicate that the conduction band of Mg2Sn consists of elliptical energy surfaces along the < 100 > direction of the reci proca1 lattice. Grossman and Temple (15) made piezo-resistance measurements which also indicated conduction band minima along the < 100 > directions.

6. Dielectric constant

McWilliams and Lynch (38) have determined the high frequency dielec­ tric constant of Mg2Sn; they found = 17.0. Their data indicate that

] 7 - 1 the transverse optic mode lattice frequency is 5.6 x 10 sec" . The far- infrared reflectivity data of Geick e_t aj_. (21 ) indicate that has a constant value of 7.^ below 300° K and increases to 8.2 at 500° K;

Eq is the static dielectric constant.

7. Elastic properti es

Davis e_t (17) have measured the sound velocities of Mg2Sn, They found for the < 100 > direction, Cg = 4.78 x lO^cmsec ' and c^ = 3.19 x

5 -1 10 cmsec where c^ is the longitudinal wave sound velocity and c^ is the transverse wave sound velocity. They have calculated the phonon dispersion curves on the basis of several models. They found that the dispersion curves calculated from a shell model gave a best fit to the Debye tempera­ ture versus temperature curve obtained from the heat capacity measurements of Jelinek et al. (29) 5

8. Therma1 propert ies

The heat capacity of Mg2Sn has been measured from 5 to 300° K by

Jelinek aj_. (29) They report a Debye temperature of 388.4° K at 1° K.

The Seebeck effect has been measured by Boltaks (7) and by Winkler

(53) from 80° K to 600° K. Several of Winkler's samples showed a sign

change in the Seebeck coefficient near 150° K which indicates that they f were p-type; all of his result showed a positive temperature coefficient

at low temperatures. Aigrain (I) has measured the thermomagnetoe1ectric

effect in Mg2Sn; he found that the effective mass of the holes is 0.14 m^.

Busch and Schneider (9) have measured the thermal conductivity of

Mg2Sn from 73 to 473° K. Their results did not indicate any electronic"

contribution to the thermal conductivity.

B. Purpose of this investigation

The purpose of this investigation was to measure the low temperature

thermal conductivity of Mg2Sn. The relaxation time for a number of phonon

scattering mechanisms that may be present have been worked out by a number

of authors. Comparison of the experimental data with the values calculated

from the theory of Callaway (10) allows one to test the magnitude of the

various phonon relaxation times.

A simple modification of the apparatus allowed the Seebeck coeffi­

cient to be measured at the same time as the thermal conductivity. The

Seebeck data can be interpreted in terms of the existing theories of trans­ port properties of semiconductors, in addition, the phonon-drag component of the Seebeck coefficient provides information on electron-phonon inter­ act ions . - - 6

I 1. THEORY

A. Theory of Thermal Conductivity of Semiconductors

1 . Sepa ration of lattice and e1ect ron i c components

in a semiconductor, heat may be carried by phonons (quantized lat­

tice vibrations) and by charge carriers (electrons and holes). The total

thermal conductivity, K, can be separated into an electronic component.

Kg, and a lattice component, Kp, as follows

K = Kg + Kp . (])

Equation 1 holds if the interaction between the electrons and the

phonons is small. The validity of Equation 1 has been discussed by Drab­

ble and Goldsmid (18). For the purposes of this discussion, we shall

assume that Equation 1 is valid.

2. Elect roni c the rma1 conduct ivity

Following Smith (4?) but using the method of averaging relaxation

times of Herring (26) the electron current density, J, in a semiconductor with an applied electric field,£., and temperature gradient, dT/dx, is

, ne r? _ ô ^Ef'^"] < v^t > ne dT < v^ET > J = —ce + T ^ n + n , , ^ mg ôx\ T / < v > Tmg dx < v^ > (2) where n is the number per unit volume of electrons, mg is the effective mass, e is the magnitude of the electronic charge, T is the absolute temperature, Ef is the Fermi level, E is the energy of the electron measured from the bottom of the conduction band, v is the electron veloc­ ity and T is the relaxation time. For a non-degenerate semiconductor, the quantity < ( ) > is given by < ( ) > = ( ) f°E2dE r f°E^dE :3a)

where f° = exp (-E/kT).

Often, T may be expressed as

-s T = AE (3b)

Then Equation 3a gives:

< v^T >/< > A Rl-

< v^Et >/< > = AkT - i!)

< v^E^T >/< > = Ak^T^P[|- (3c)

where ^(x) is the gamma function.

The electron heat current density, is

n dT < v^E^t > mg"- ôx \ T /-' < > Tmg dx < > (4)

The thermal conductivity is measured under the condition of zero electric

current. Therefore, Equation 1 is solved for ^with J = 0, and the

result is substituted into Equation 4 to give

n < v^E^t > < v^T > - < v^Et >2 dT meT < V^ > < V^T > dx (5) or

'dT < V^E^T > < V^T > - < V^ET >2 " mgT < v^ > < v^T > (6)

Since the electrical conductivity a = ILë£ , we have mp < V

Ke = (%) o T , (7) where 8

2 2 2 2 2 f _ < V E T > < y T > - < v ET > - k,2T2 ^< .Zry^t ^2 (8)

or

Ke'e -= l2fl - =/s (7)le' (T T . (9)

For the case of acoustic mode scattering, s = —, we find -£ = 2. For 2 a simple meta]. Equation 8 gives the Lorenz number, -C. =

I • S In a non-degenerate p-type semiconductor, if we assume T = AE

the hole current is given by

Ô /Efl . /5 ,\1 dT ^ (10)

where p is the number of holes per unit volume, |j.|^ is the hole mobility

< tv2 >/< v2 >) and Ef = A E - E^ where A E is the energy gap.

Similarily, the hole heat current density is given by

Q. = Pl^h ( 2 - ^ 'j kT l^e £ - [t — I ~^\ + k ( - - s ' ' ' — ^ ira"(i-••)!£}

(11)

The second term in Equation II represents the transport of the recombina­

tion energy of electrons and holes. It is zero for zero hole current.

Hence, the electronic contribution to the thermal conductivity of a p- type sample is

^ - (f - =j( • (12)

If we are dealing with an intrinsic semiconductor where both electrons and holes are present, we must write the total charge and thermal cur­ rents as 9

J (13a)

and

Q. = + Oh ; (]3b)

where is given by Equation 1, Jj-, by Equation 10, Q,g by Equation 6 and

Q.(^ by Equation 11. Setting J = 0 (note, this is not the same as =

0 and = 0) we find for the total electronic contribution to the

thermal conductivity

s') + AE/kTj^ np^e^h o T . (14)

The first term in Equation 14 represents the combination of the heat

currents given by Equation 9 and Equation 12, The second term represents

the transport of the recombination energy of the holes and electrons.

This term is often called the ambipolar or bipolar contribution to the

electronic thermal conductivity. The bipolar term is usually large com­

pared to the first term because A E/kT > > 1.

3. Lattice thermal conduct ivity

The original description of lattice thermal conductivity was writ­

ten in 1929 by Peierls (41). In that paper he quantized the lattice

vibrations into phonons and showed that in a perfect crystal the only

processes which give rise to a finite thermal conductivity are those pro­

cesses which do not conserve crystal momentum (U or Umklapp processes).

The theory of lattice thermal conductivity for various scattering mecha­ nisms, including imperfections, has been reviewed by Klemens (32, 33) and 10

Carruthers (13).

At high temperatures (T > O) Leibfried and Schloemann (35), using

a variational approach, have obtained the expression

5 4"' \h/ yZy ' (15)

where k is Boltzmann's constant, h is Planck's constant, M is the mean

atomic mass, 5 is the cube root of the atomic volume, 0 is the Debye

temperature and j is the Gruenei sen anharmonicity parameter. Equation

15 is a modification of Leibfried and Schloemann's original expression

that was pointed out by Steigmeier and Kudman (48).

Callaway (lO) has developed a phenomenologica1 theory which combines

the relaxation times for the different scattering processes. His theory

has had considerable success in fitting thermal conductivity data on a

number of materials from very low temperatures to about 100° K. Car­

ruthers (13) discusses Callaway's theory and a number of the phonon scat­

tering processes that occur at low temperatures. The relaxation time

approach of Callaway will be used to interpret the low temperature data

of this experiment.

The following is an outline of the Callaway theory. (10) The

Boltzmann transport equation for the phonon distribution , where q is

the phonon wave vector, is

W, " -VT ^. (16) where c^ is the phonon group velocity for polarization e, t is the time,

T is the absolute temperature and is the collision operator. Equa- \at/c tion 16 is valid only when Nq depends on position through the temperature. 11

n the relaxation time approximation

N° - N, 'la T (q) ' (17)

where is the equilibrium distribution and T (q) is the relaxation

time.

Peierls (42) and Klemens (32) have shown that normal (N) processes

cause the low momentum phonons to create higher momentum phonons; and,

therefore, the normal processes cause Nq to relax to a state of higher

momentum, Nq ( \ ), than the equilibrium distribution Nq° where Nq ( A. )

is given by

1 N„ (X) = + % . q e kT _ I (18)

In order to include N processes. Equation 16 is written as

Nq(^) - Nq Nq° - Ng

&t/c "^r (19) where is the relaxation time for N processes and is the relaxation

time for resistive processes. To first order

Nq (\) = Nq(0) + \ , or

Tim kT No (\) = Nq(0) kT / fg \2 '

( _ ij (20) where Ti is Planck's constant divided by 2jt, m is the phonon angular fre­ quency and k is Boltzmann's constant. Let x = Tim/kT.

Combining Equations 16, 19 and 20 gives 1 2

(cx'- ,)2 + Tj"q = 0 N 7 (21)

where Nq° - . Let

tiœ e "q - -VT i^j2 (gx _ 1)2 • (22)

Callaway (10) defines

_L - _L ^ _L Tc TN Tr ' _ (23)

Note that Equation 22 simplifies Equation 21 to

R T 1 Tico — \ - q [-' + T c -VT = 0 . (24)

Now in an isotropic medium X~vT, so we can define a parameter B, such

that

X = - TiBcVt/T . (25)

Using q = cm/c^, we get

X • q = - TiCjùBc * T/T . (26)

Therefore, Equation 22 gives

T = Tc [l + B/TJ, (27) and

r", „ / 1 — , , ^ e ^ Ticu "q = - -Tc L' + B/t,] c . VT-^prrrpiPi^ • (28)

Normal processes cannot change the total phonon momentum. Therefore,

^ ,0. J dt N ^ (29)

By using Equation 26 and getting rid of the constants. Equation 29 becomes

pe/T x^e* T - B (e* - 1)2 Tw ^ ° ' (30) 13

or solving for the constant

pP/T P/T x^e* dx dx (e* - 1) (e* - I)" T, N (31)

To get the heat current Q,, note that

q,p q,G q,e q,G

if we replace the / by an integration and note that Nq° makes no contri­

bution, we get

("hco) e^ ? ? K = 1 + — Cr cos a 2 G (2%) 3 kT (e* - 1) (32) where a is the angle between and T.

if we assume the isotropic case and let three Debye modes, m = cq, represent the three acoustic modes with c an average sound velocity then

'G/T xV e/T T K = dx + B ^ x^ dx 2rt c \ Ti (c"" - 1)^ (e* - ]) (33) where 9 is the Debye temperature. Equation 33 will be used to fit the low temperature thermal conductivity data of this experiment, is a combined relaxation time found by the reciprocal addition of the relax­ ation times for the different scattering mechanisms.

4. Phonon relaxat ion t imes

a. Boundary scattering At temperatures below the thermal con­ ductivity maximum only long wavelength phonons are present. These phonons will be scattered by the crystal boundaries. Casimir (14) first worked out the relaxation time, Tg, for boundary scattering given in Equation 34. 14

Tg" = c/L (34)

where c is the average sound velocity and L is the effective sample di-

i/o ameter. For a rectangular sample, L = 2tc" where 1 j I ^ is the

sample cross-section. Berman e_t (4) have determined corrections for

sample roughness and for finite sample length. At low temperatures bound­

ary scattering causes the thermal conductivity to be proportional to .

b. Defect scattering Klemens (32) has found the phonon relaxa­

tion time for several types of crystal defects. One of the more impor­

tant defects is a point defect caused by the different isotopes in the

crystal. For point defects Klemens found

= Am (35a) where GO is the phonon frequency. Slack (46) has modified Klemens expres­ sion for A for elemental materials to include compounds. For a compound

A/ByC;.

A = V p/4jrc^ ; (35b)

X + y + X + y + A (35c)

AMj f : . M, -A \ / (35d) where V is the molecular volume, c the average sound velocity; My^, Mg are the average masses of the atoms,

_ xMy^ + yMg + M = x + y + (35e) fj is the isotopic abundance of the ith isotope, and AM. is the difference between the mass of the ith isotope and the average mass, 15

c. Phonon-phonon scattering Herring (25) has investigated acous­

tic mode three phonon scattering of long wavelength phonons for different

crystal structures. He found for face centered cubic crystals at low

temperatures

T"' = , (36) where B is a constant. Callaway (10) used this expression for normal

processes and for Umklapp process, he wrote

Tu"' = ByGxp (-e/aT) , (37) where By and a are constants, and 0 is the Debye temperature. a should have a value of about 2. At high temperatures Klemens (32) finds

Ty-' = B'afl (38) where B' is a constant.

Pomeranchuk (44). has suggested the possibility of four-phonon scat-

__ 2 tering processes which would introduce a T~ dependence in the thermal conductivity at high temperatures. However, no conclusive experimental evidence of four-phonon processes has been found.

Blackman (5) has discussed thermal conductivity data on alkali- halide compounds in terms of optic mode scattering of the form: two acous­ tic mode phonons create one optic mode phonon. Steigmeier and Kudman (48) have also suggested optic mode scattering to explain their data on lll-V compounds.

d. Other relaxation times Several other relaxation times have been used to explain various anomalies that have appeared in thermal con­ ductivity data,

Resonance scattering, has been used by Pohl (43) and Walker and 16

Pohl (52) in the form

_1 _ Rco^TP - ^2)2 + (O/ajZufwoZ ' (3g)

to explain dips in thermal conductivity data, m is the resonant fre­

quency, fi describes the damping, p = 0 has been used for dips at tem­

peratures below the thermal conductivity maximum and p = 2 has been

used at temperatures above the maximum. Wagner (51) has derived on

the basis of scattering caused by localized modes introduced by impuri­ ties.

Bound donor electron - phonon scattering of the form

-1 ^ Gco^A^ [(haii)2 - (A)2j2 1^1 + (r^W/^c^jj 8

G = ^"^2 7 (~~f 34Kp2c/ ^ A y (40) has been used by Keyes (30, Griffin and Carruthers (23), Goff and

Pearlman (22) and Holland (28) to explain thermal conductivity data on semiconductors at low temperatures that is considerably lower than the theory of boundary scattering would predict. Equation 40 is the form derived by Griffin and Carruthers; A is the splitting of the ground state of the donor level and r^ is the effective electron radius.

"ex is the number of unionized uncompensated donors, p is the mass density, c is the sound velocity, is the shear deformation potential and F is a factor depending upon the phonon polarization branch and the electronic structure. F has a value around 0.2.

Ziman ($4) has calculated a relaxation time, for the scattering of phonons by degenerate electrons in a parabolic band. 17

-1 _ _r_ X ] ^îTh^p T5 (1 - e"*) X ' (41)

where C measures the strength of the interaction, m is the effective mass,

c is the sound velocity, p is the mass density, kT^ = 1/2 mc^ and w is

a complicated function of T, T^, x and the Fermi temperature. J. A. Car-

ruthers £t _aj_. (11, 12) have used to qualitatively explain a reduced

thermal conductivity at very low temperatures in p-type Ge and Si.

B. Seebeck Effect

The theory of the Seebeck effect in semiconductors has been dis­ cussed by Herring (26) and by Johnson. (30) The diffusion Seebeck coef­ ficient, Sof an n-type sample can be obtained from Equation 2 with

J = 0. The resu1t is

k rAet Ef

^ kT kT or

3 k Aet nh Sd In e L kT 2 (2rtmekT)3/2 (42) where

Act = < v^ET >/< V^T > .

The quantity is the average energy of the transported electrons rel­ ative to the band edge. For a p-type semiconductor the ~ sign in front of Equation 42 is replaced by a + sign. For mixed conduction, we require that the sum of Equation 2 and the equivalent equation for holes is zero; the result is 18

C Ast r_flLl ph' 1 L kT ' 2(2amekT)3/2j kT " 2(2nmhkT) Sd = - k y e PW-h (43)

where ' indicates holes. If we let b = Equation 4] becomes

3 1 r r A £ ' t ph Sd p n e nb + p kT 2(2jtm,^kT) 3/2

O Aet nh-' - nb kT " " 2(2%m_kT)3/2JJ (44)

There is an additional component to the Seebeck coefficient caused

by the scattering of the charge carriers by the phonons. in the absence of a temperature gradient this scattering is isotropic. However, when a

temperature gradient is present there is a phonon current from the hot end to the cold end; this phonon current causes the charge carrier scat­

tering to be anisotropic with the carriers scattered toward the cold end more often than toward the hot end. Herring (26) has discussed the phonon-drag Seebeck coefficient in semiconductors. He found that the phonon-drag Seebeck coefficient is given by

Sp = + ^ c(q) 2 f(q)T(q)/pJ . where c(q) is the sound velocity of mode q, T(q) is the phonon relaxation time for mode q, f(q) is the fraction of crystal momentum given up by the carriers to the acoustic mode q, is the mobility due to phonon scat­ tering and T is the absolute temperature. f(q) is a strongly peaked func­ tion for q of the order of the thermal electron wave vector. Therefore, only low energy phonons are important in phonon drag. The - sign is for electrons and the + sign is for holes. The summation is carried over all 19

acoustic modes. The phonon wave number q in Equation 4$ is an average

q of the order of the wave number of a carrier with energy i

Herring (26) obtains several expressions for the temperature depen­

dence of Sp. For longitudinal acoustic mode scattering he finds

Sp a T-lO/2/p . (46)

t f p. a Equation 46 becomes

Sp a 1-7/2

For a more general acoustic mode scattering, Herring finds

Sp a , (48) where

T(q) = 1/A q2 + S -P3 - s - AR (49)

I f p. GL acoustic mode scattering, we see that the phonon drag

Seebeck coefficient is proportional to T® where a > - ill.

At low temperatures T(q) = L/C where Lisa characteristic size of the crystal. Then we expect Sp a T^^^ 20

111. EXPERIMENTAL PROCEDURE

A. Samples

I . G rowth

The samples used in this experiment were grown by a modified Bridgman

technique. The method is a variation by Grossman' of the method described

by Morris e_t (39) to grow Mg2Si. in Grossman's method, the melt is passed slowly through the melting point twice. it is thought that the

first pass forms the compound, and the second pass forms the single cry­

stal. The method would also provide a zone-refining effect.

All three crystals were grown from Sn of 99.999 % purity obtained

from Vulcan Materials Gompany. Crystals SBI69 and SB202 were grown from

99-995 % purity Mg obtained from the Dow Chemical Gompany. Crystal SB239 was grown from Dow Chemical Mg that had been vacuum distilled in this

laboratory.

High purity samples are needed for thermal conductivity measurements so that the intrinsic properties of the material are not masked by the impu rit les.

2. Shapi ng

The thermal conductivity samples were cut from the crystal with a wire saw. An abrasive slurry of #600 SIC grit suspended in kerosene was applied to the wire of the saw to serve as the cutting agent. After the saw cuts were made the sample surfaces were trued by lapping. The samples

'crossman, Leon D., Iowa State University, Ames, Iowa. Private communication. 1966. 21

were rectangular parallelepipeds approximately 3 mm by 3 mm by 2 cm. The

samples were then stored in small bottles containing a desiccant.

3. Character!zati on

Small pieces of the original crystals were spectroscopical 1 y ana­

lyzed. The spectroscopic results are given in Table 2.

The electrical resistivity and Hall coefficient were measured from

77 to 300° K by a standard 5 probe technique. Figure 2 shows the electri­ cal resistivity, and Figure 3 shows the Hall coefficient for the thermal conductivity samples. Figure 4 shows the results of electrical resistiv­ ity and Hal I coefficient measurements made on samples K-11 and K-13 at lower temperatures. Sample K-3 was accidentally broken before electrical measurements were made.

For a saturated extrinsic semiconductor, the Hall coefficient, R, is g iven by

o _ 1 - - 8 (Ng - N^je ' (50)

where Nq is the number of donors per unit volume, Ny;^ is the number of acceptors per unit volume and e is the magnitude of the electronic charge.

Equation 50 was used to calculate Nq - for the thermal conductivity samples; the results are given in Table 1. Table 1 also contains the physical dimensions of the thermal conductivity samples.

Back reflection Laue X-ray patterns were made at a number of points along each sample. All of the patterns showed the same orientation, which indicates that the samples were single crystals. There were no grain boundaries visible. 22

°K 300 200 150 100 80 1.0

0.5

^0.2 8 o^ u I 8 o'o X £ 0. - flj - • ° QS> (L 0 4)

0.05l-o

• K-ll O K-13 0.02 A K-15

0.01 7 8 9 10 II 12 13 1000

Figure 2. Electrical resistivity, p, of the thermal conductivity samples 23

3 300 200 150 100 80 10

500 O O O ° A/0 a A • 200 — g

10^ cP

_j 50 3 O O • \ m 5 O O 20 E g 10 • K-ll 9 o K-13 A K- 15 o

7 8 10 11 12 13 1000 (°K)

Figure 3. Hall coefficient, R, of the thermal conductivity samples 24

100 50 20 '".o"

• • XI 10' o(A] o 0 o o c 2 C 10 10

•

l.Ok 2 • 0 1 5 • % cP O Q • K-ll X 0.1 O K-13

0.01 I I I L 10 20 30 40 50 60 70 80 90 100

(°K-I)

Figure 4. Hail coefficient, R, and electrical resistivity, p, of Mg2Sn samples K-1I and K-13 at low temperatures. (Both samples were n-type) 25

Table 1. Sample characteristics

di mens ions Spect roscopi c ND - Sample mmxmmxmm ResuIts^ numbe r/ cm-^

K-3b 5.856 X 5.394 X 16 Ca - F. T. Cu - T. - V. W. Fe - T. Si - T. Y - F. T.

K-Iic 2.706 X 4.405 X 20 Al - T. 2.5 X lo'G 0 1 X LO^G VuU 1 . 898 X 4. 368 X 19 Ca - T. 1 .8

K-13BC 2.076 X 1.866 X 19 Cr - F. T. 1 .8 X Cu - F. T. Fe - T. Si - T. V - T.

K-15^ 4.298 X 4.242 X 20 Ca - T. 2.5 X lO^G Cr - F. T. Fe - F. T. Si - T.

^V. W. = very weak, T. = trace, F. T, = faint trace, other elements were not detected.

'^Cut from crystal SBI69.

^Cut from crystal 58239.

*^Cut from crystal SB202.

B. Measurements

1 . Method

The thermal conductivity is given by

q = - K\7T , (50 where q is the energy per unit time per unit area, K is the thermal con­ 26

ductivity and T is the gradient of the temperature. In the case of a

long cylindrical sample with longitudinal heat flow and no heat losses,

Equation 51 becomes

A T a = - K A L , (52)

where Q. is the energy per unit time put into one end. A is the cross-

sectional area, L is the length of the sample and A T is the temperature

difference between the ends of the sample. Therefore,

K = - i -A- AAT ' (53)

The experimental determination of K waa based on Equation 53.

The Seebeck coefficient of the thermal conductivity sample can be

determined by a simple modification of the experiment. if the thermal

conductivity sample is of material a and if wires of material b are at­

tached to each end of the sample, then the Seebeck effect causes an e. m. f., E, to appear at the free ends of the wires. E is given by

E = Sab AT , (54) where is the Seebeck coefficient of a with respect to b. The absolute

Seebeck coefficient of a is given by

Sa = Sab - Sb , (55) where S^ is the absolute Seebeck coefficient of b.

2. Sample holder and sample mount inq

The thermal conductivity measurements were made with the sample holder shown in Figure 5 mounted in a liquid helium cryostat of standard design.

The sample holder was designed to allow operation with the sample holder immersed in the liquid helium or nitrogen bath so that the thermal re- 27

TRANSFER GAS TO VACUUM PUMPS

STAINLESS —= STEEL TUBING

COPPER WOODS METAL VACUUM SEAL STAINLESS VARIABLE HEAT STEEL TUBING LEAK CHAMBER AMBIENT HEATER

SAMPLE LEADS COPPER — HEAT SINK

SAMPLE CLAMP COPPER T- D COVER " THERMOMETER CLAMPS GRADIENT HEATER

I 1

Figure 5. Sample holder 28

si stance between the sample and the bath would be small. In addition,

transfer gas could be added or removed from the heat leak chamber to vary

the thermal resistance to the bath. The sample holder cover was soldered

to the sample holder with Wood's metal. When in operation, the sample

holder was evacuated to a pressure better than 5 x 10"^ Torr. Heat con­

duction to the sample was minimized by using small diameter (0.005 in. or

less) wires and by thermally anchoring the wires to the copper heat sink.

A 40 ohm heater wound on the heat sink was used to obtain ambient temper­

atures above the bath temperature.

The sample was connected to the sample holder with the sample heat

sink clamp shown in Figure 6A. The thermocouples and the germanium re­

sistance thermometer were attached to the sample with the thermometer

clamps shown in Figure 6B, The thermocouples were soldered to the small

copper tabs. The nylon screws and phosphor bronze springs were used to

compensate for the different thermal expansion coefficients of the sample

and the clamp parts. Good thermal contact was achieved by tightening the

clamps until the indium pad or knife edges deformed. The copper tab on

the thermometer clamp nearest the gradient heater was electrically insu­

lated by a layer of polyester film tape so that a differential thermo­

couple could be used to measure the temperature difference.

The temperature gradient in the sample was established by a ten ohm

heater wound on the free end of the sample.

The constantan wire of the thermocouple attached to the clamp on the

"cold" end of the sample and a constantan wire attached to the clamp on

the "hot" end were used to measure the Seebeck coefficient of the sample with respect to constantan. The absolute Seebeck coefficient of constantan 29

•INDIUM PAD , PHOSPHOR BRONZE

NYLON SCREWS

COPPER SCREW

Figure 6A. Sample heat sink cl amp

COPPER TAB FOR THERMOMETER COPPER INDIUM KNIFE EDGES COPPER PHOSPHOR BRONZE LEAF SPRING NYLON SCREW

Figure 6B. Thermometer clamps 30

was found by subtracting the Seebeck coefficient of Cu as tabulated by

Cusack and Kendall (16) and Borelius (8) from the sensitivity of Cu versus

constantan thermocouples as given in the tables published by Powell et al.

(45)

3. Thermometry

The problem of temperature measurement in a thermal conductivity

experiment can be divided into two parts; the determination of the temper­

ature gradient and the determination of the ambient sample temperature.

In addition, the wide range of temperatures to be covered in this experi­

ment required that the thermometry problem be divided into two ranges.

For the temperature range 4 to 45° K, the ambient temperature of the

sample was measured by a Texas Instruments model 340 type 108 germanium

resistance thermometer soldered to the copper tab on the cold end of the

sample. The resistance of the thermometer was measured by a 4-wire tech­ nique with the measuring current adjusted to maintain a power dissipation below one microwatt in the resistor. The resistance thermometer was cali­ brated by means of a comparison of its resistance with the resistance of a second Texas instruments germanium resistance thermometer which had been calibrated by the manufacturer. The calibration of the thermometer used in this experiment should be good to about 0.5 % of T.

For the temperature range 4 to 45° K, the temperature gradient in the sample was measured by a silver "normal" (Ag + 0.37 at. % Au) versus gold- iron (Au + 0.03 at. % Fe) versus silver "normal" differential thermocouple.

One junction of the thermocouple was soldered to the copper tab on the

"cold" end sample clamp and the other junction was soldered to the electri- 31

ca11 y insulated copper tab on the "hot" end clamp. This Lhermocouple has

been described by German ejL 21- (3) They have published a curve of the

sensitivity of the thermocouple from 1 to 300° K. Silver "normal" has a

Seebeck coefficient similar to the Seebeck coefficient of pure copper;

however, it does not have the high thermal conductivity maximum at low

temperatures that copper does. The differential thermocouple voltage was

measured with a Keithley model I48 nanovoltmeter.

For the temperature range 45 to 300° K the temperatures of the two

clamps were measured with silver "normal" versus constantan thermocouples.

The ambient temperature was determined by the average of the two thermo­ couple readings, and the temperature difference was determined by the difference between the two readings. A calibration of silver "normal" versus constantan thermocouples has been published by Powell e_t a_l_. (45)

The e. m. f. at liquid nitrogen temperature of the thermocouples used in this experiment differed by about 80 i^V from the published value. A new e. m. f,, E, versus temperature, T, curve between 77 and 300° K was pre­ pared by fitting the equation

E(T) = A(T - 273) + B(T - 273)^ + C(T - 273)^ , (56) to the measured e. m. f. value at liquid nitrogen temperature and the dE/dT at 273° K and the e. m. f. at 300° K of Powell ^ _a]_. (45) A thermo­ couple table giving the e. m. f. and dE/dT at 1° K intervals was calcu­ lated by evaluating Equation 56 and its first derivative. The table was extended to liquid helium temperatures by comparing the thermocouples against the germanium resistance thermometer. 32 .

4. Data taking

A block diagram of the complete apparatus is shown in Figure 7. The

Leeds and Northoup 1<-3 potentiometer was used to measure the thermocouple

E. M. F.'s, the Seebeck e, m. f., the gradient heater voltage and current

and the germanium resistance thermometer voltage and current. The dif­

ferential thermocouple e. m. f. was measured with the Keithley #148 nano-

voltmeter. The thermocouple leads were brought out to an ice bath. Copper

leads from the ice bath connected the thermocouples to a low-thermal ter­

minal strip. The other current and voltage leads were also connected to

the low-thermal terminal strip. A low-thermal switch connected the termi­

nal strip to the potentiometer and the nanovoltmeter. Power Designs model

5005R D. C. power supplies were used to supply both heaters. The gradient

heater supply was operated in the constant current mode. The ambient

heater supply was controlled by a servo-amplifier system. The germanium

resistance thermometer current source was a mercury battery in series

with a large resistance.

The data were taken in the following manner. From 4 to 45° K, a tem­

perature gradient was established by adjusting the gradient heater power

supply. When thermal equilibrium was reached: the differential thermo­

couple voltage, E, was read on the nanovoltmeter, the gradient heater cur­

rent, 1, and voltage, V, the germanium resistor current and voltage, and

the Seebeck voltage, V^, were read with the K-3 potentiometer. The gra­

dient heater power was reduced and the ambient heater power was increased

until the resistance thermometer indicated the same temperature, T. The measurement process was repeated for the lower gradient heater power set­

ting. The thermal conductivity, K, was calculated from GERMANIUM RESISTANCE AMBIENT THERMOMETER LEEDS NORTHRUP TEMPERATURE CURRENT SOURCE CONTROL POTENTIOMETER

AMBIENT HEATER SWITCHING CIRCUIT

THERMOCOUPLE ICE BATH T+ AT AT GRADIENT HEATER

GRADIENT HEATER POWER KEITHLEY**I48 SUPPLY NANOVOLTMETER

Figure 7. Block diagram of the thermal conductivity apparatus 34

L 1]V] - 12^2 dE = x-iTTir-' ,3,,

where the subscripts denote the two power settings, dE/dT is the sensiti­

vity of the differential thermocouple at temperature T, L is the thermo­

meter clamp spacing and A is the cross-sectional area. The Seebeck coef­

ficient, S, with respect to the constantan leads was calculated from

Vsi - Vc2 dE S(T) = - ^2 "T

The process was repeated at different ambient temperatures. The above

technique has the advantage that spurious thermal voltages in the thermo­

couple and Seebeck circuits cancelled. Data at the lowest temperatures

were taken by pumping on the liquid helium bath. Temperature gradients

of 0.1 to 0.5° K were used.

From 4$ to 300° K, the data were taken by setting the gradient heater

power supply for the desired power input. When thermal equilibrium was

reached; the thermocouple E. M. F.'s, E] and E2, at the two thermometer positions; the Seebeck voltage, Vg; the gradient heater current, 1, and voltage, V, were read on the K-3 potentiometer. The ambient temperature,

T, of the sample was determined, and the thermal conductivity was calcu­

lated from

, . _ L I V dE A E] - E2 dT ' (59) where dE/dT is the sensitivity of the thermocouples at temperature T.

The Seebeck coefficient with respect to constantan was calculated from

Vs dE = El - E2 cTf • (60)

Below 77° K, the ambient temperature of the sample was obtained by 35

immersing the sample holder in a liquid nitrogen bath in the cryostat

helium chamber and pumping on the bath. Above 77° K, transfer gas in

the helium chamber and its vacuum jacket was used to couple the sample

holder to the liquid nitrogen jacket. A servo-amplifier temperature con­

troller was used to control the temperature above 77° K. Temperature

gradients of 1 to 2° K were used.

Table 4 contains sample experimental data recorded for the two tem­ perature regions'.

C. Errors

1. Measurement errors

The uncertainties involved in this experiment entered through the measurement of the sample cross-section, the thermometer spacing, the ambient temperature, the temperature gradient and the power input to the sample.

The cross-sectional dimensions of the samples were measured with a micrometer that had a least count of 1 x 10"^ cm. Therefore, the dimen­ sions of the samples were accurate to about 2 x 10"^ cm or 1.0 %, and the uncertainty in the cross-sections are were less than 2.0 %.

The thermometer spacing was measured with a traveling microscope.

The distance between the centers of the thermometer clamps was taken to be the thermometer spacing because the screws on the clamps made it im­ possible to see the knife edges. The uncertainty in the thermometer spa­ cing was probably not more than 5 %. The uncertainties mentioned above remained constant during a data taking run and did not affect the shape of the thermal conductivity curve. 36

Between 4 and 45° K, the germanium resistance thermometer was used

to measure the ambient sample temperature. Thefe was aTT uncertainty of

about 0.5 % of T in this temperature range. Above 45° K, the ambient

temperature was determined by averaging the two thermocouple voltages.

The thermocouple voltages were measured with a Leeds and Northrup K-3

potentiometer. On the range used to measure the thermocouple voltages,

the K-3 potentiometer measured voltage to about 0.5 [iV. 0.5 p.V corre­

sponded to a maximum temperature uncertainty of 0.05° K at 50° K. The maximum uncertainty in the thermocouple calibration was less than 2° K

which occurred midway between 77 and 273° K.

Between 4 and 45° K, the temperature gradient was measured by a dif­

ferential thermocouple read out by a Keithley # 148 nanovoltmeter. The manufacture listed a maximum error of 2 % for this instrument. The ther­ mocouple sensitivity was assumed to be within 3 % of the values given by

Berman e_t ^1- (3) Therefore, the uncertainty in the temperature gradient was about 5 %. Above 45° K, the temperature gradient was determined by

dividing the difference between the two thermocouple voltages by the ther­ mocouple sensitivity. Therefore, the uncertainty in the temperature gra­ dient was about 5 % at 45° K and fell to about 2 % at 300° K.

The power input to the sample was determined by measuring the current

through the gradient heater and the voltage drop across the heater. The current (10 to 100 mA) was determined by measuring the voltage drop across a 100 ohm standard resistor with the K-3 potentiometer. The heater voltage

(0.1 to 1 M) was measured with the K-3 using voltage probes attached to the heater so that lead resistances could be neglected. The potentiometer made it possible to measure the voltage and current to about 5 significant figures. The supply was stable to about 4 figures. Hence, the heater

power input was accurate to about 0.1 %.

The uncertainties involved in the Seebeck coefficient enter through

the measurement of the Seebeck voltage and the temperature gradient. The

Seebeck voltage was measured with the l<-3 potentiometer which introduced

an uncertainty of about 1 % at temperatures below about 15° K. At higher

temperatures, ti'io uncertainty was much less. The uncertainties in the

temperature gradient and the ambient temperature were the same for the

Seebeck effect and the thermal conductivity. Some of the scatter observed

in the Seebeck data may be caused by the large Seebeck coefficient which was greater than 500 ^V/deg at most temperatures more accurately measuring

the true temperature gradient than the thermocouples.

2. Heat 1oss errors

Thermal conductivity experiments depend upon the complete knowledge of where all of the heat generated in the gradient heater goes. Ideally, when thermal conductivity is measured by longitudinal heat flow, none of the heat generated in the gradient heater should flow out the sides of the sample. There are several possible mechanisms for such lateral heat losses: radiation, conduction by the leads attached to the sample and convection. The measurements in this experiment were made at pressures less than 5 x 10~5 torr. Therefore, convection and conduction losses were neg1igible.

The radiation heat loss, can be approximated by

Gr = 4 aff£ T%T , (61) where a is the surface area of the sample, a is the Stefan-Boltzmann con- 38

s tant, £ is tile emissivity of the sample, T is the absolute temperature

and ÔT is the temperature difference between the sample and the sample

hoider.

The only possible conduction loss was through the wire leads con­

nected to the sample. Of these leads, only the silver "normal" and gold-

iron thermocouple leads, and the 0.002 inch diameter copper heater current

leads had thermal conductivities large enough to cause any problem. The

heat loss through the leads, Qj, was

where the summation is over all leads, K, is the thermal conductivity, Ij

the length, a| the area and &Tj is the temperature difference for the i th

lead. If we assume that ôTj for all leads equals ôT, the temperature difference between the sample and the sample holder, we find that

Cb = (6.3 X 10-46T) Watts. (63)

Therefore, the actual heat conducted through the sample, was

= Qm - G] - Or , (64) where is the measured power input. The thermal conductivity, K, is given by

(ia L * jYT A (65) where L is the thermometer separation, A is the sample cross-sectional area and AT is the temperature gradient.

Therefore, the final result for the thermal conductivity in terms of the measured power input when we take into account the corrections for the heat conducted away by the leads and the heat radiated, is 39

£m L K AT A (66)

We must now make a reasonable estimate of OT/AT and £, in order to determine an approximate correction to the thermal conductivity. The quantity ÔT/AT should be approximately constant, and an approximate value might be 2. We assume that £ = 0.5. With these values for OT/AT and , the correction terms have been evaluated for samples K-3, K-11, K-13 and

K-15. The percentage corrections calculated from the last two terms of

Equation 66 are shown in Figure 8. 40

20

X-5 "LE K-ll SA.VPLE K-i5 SAMPLE K-13

o

o y/ i— o u d: o o

100 200 300 T(°K)

Figure 8. Percentage correction of the measured thermal conductivity values calculated from the last two terms of Equation 66 41

IV. RESULTS AND DISCUSSION

A. Thermal Conductivity Results

The thermal conductivity data are shown in Figure 9. The correction

for heat losses, which was discussed in III. C. 2, has been applied to

the data. The corrected data for samples K-11, K-1 3 and K-15 are in good

agreement with the data of sample K-3 which required a much smaller cor­

rection owing to its small ratio of length to area. This agreement indi­

cates that the calculated correction has the correct magnitude. The

thermal conductivity data are tabulated in Table 5.

The thermal .conductivity data are in good agreement with the data

reported by Busch and Schneider (9) for the temperature range 73 to

473° K.

At 4° K the thermal conductivity of the samples should be propor­

tional to the effective size of the samples. T-o check the size effect, we made sample K-13B by cutting sample K-13 in half. The thermal conduc­

tivity of sample K-13B was about 10 % lower than the thermal conductivity of sample K-13 at 4.2° K, but the effective size of sample K-13B was 30 %

less than sample K-13.

B. Thermal Conductivity Discussion

1 . Elect roni c contribut ion

!n the intrinsic region, there exists the possibility of heat con­ duction by electrons. In the mixed and intrinsic regions the electronic thermal conductivity, Kg, is given by

(5 - 2s + AE/kT)^ bnp Ke a T , (bn + p)^ (67) 42

10,00 r TTl [ rfît^CP^D € % t • - Û (P 5 00H- oo L L_ s Î ° g

2 2.00i~ 0 5 i # a i S § • - '°°F & ^ 1— • SAMPLE K-3 cb > !Z " SAMPLE K-ll °'fl 0 I 0 SAMPLE K-13 Q 1 0 50*— . SAMPLE K-I3B g o r— A SAMPLE K-15 (J 9 BUSCH a SCHNEIDER < : § i A 0 20 0 A

010— %\

005-

003' I 3 5 10 20 50 100 200 500 TEMPERATURE CK)

Figure 9. Thermal conductivity results 43

if the electron distribution is assumed to be non-degeneraLe. The elec­

tron and hole relaxation times are given by T A E"^. The quantity b

is the mobility ratio, n is the number of electrons/cm^, p is the number

of holes/cm^, A E is the energy gap at temperature T and a is the elec­

trical conductivity. At 300° K, Mg2Sn with 2 x lo'^ donors/cm^ is not

completely intrinsic. However, for the purposes of estimating K , it

is reasonable to assume n = p. For acoustic mode scattering s = 1/2.

Lipson and Kahan (37) found a mobility ratio of 1.20 to 1.25. If we use

the energy gap at 0° K and its temperature dependence as found by Winkler,

(53) we obtain an energy gap of 0.27 eV at 300° K. The electronic con­

tribution to the thermal conductivity is then 0.003 watt/cm-deg or about

4 % of the measured thermal conductivity. For lower temperatures the

electronic contribution will be less than it is at 300° K. The electronic

component of the thermal conductivity is, therefore, comparable to the

experimental error and too small to separate from the total thermal con­

ductivity. For the rest of this discussion we shall neglect the electronic

contribut ion.

2. High temperatu re lattice thermal conduct ivity

At high temperatures the lattice thermal conductivity should be proportional to T~^. The thermal resistance of the four samples measured at high temperatures is shown in Figure 10. Leibfried and Schloemann (35) have obtained an expression for the thermal conductivity at high temper­ atures (T > 0), they found

K .i 41/3 5 \h/ 7 T (15) SAMPLE K-3 CO SAMPLE K-ll CO w SAMPLE K-13 SAMPLE K-I5

THEORY

UJ

100 120 140 160 180 200 220 240 260 200 500 TEMPERATURE ("K)

Figure 10. The thermal resistance of Mg2Sn above 100^ Kis approximately linear in T. (Tlie solid I in shows the thermal resistance calculated from the theory of Leibfricd and Sclil oc-.vr:) 45

M is Llir mean a Loin i c iiuibs, l-> is Line cube rooL of Liic -ilfjinic vo I u 'r , '• is

the Deby I' temperature and 7 is the Grueneisen anharmon icily p-i r- jLer.

Here P re|iresents the Lhree acoustic modes, so we Find ' ' = 7'.. ' K. 11

we fit Equation 15 to the data with 7 as an adjustable paramrJcr, we

find, with 7 = 1.4, the solid line given in Figure 10. The agreement

between thr theory and L he experiment is satisfactory. The Ihe rnia I re­

sistivity is linear in T for T> 120° K. In other words, the T~' depen­

dence of tlie thermal conductivity continues for a considerable distance

belP. Holland (27) his pointed out a similar result for Ge and SI.

3 . Low temperatu re lattice therma1 conduct ivity

In tlie range 4 to 80^ K, the theory developed by Callaway (10) will be used to interpret the measured thermal conductivity. Callaway found

k fkTl3 re/T ^ h/ (e^ - 1)^ ' (68) where x = Tico/kl and is a combined relaxation time found by the re­ ciprocal addition of the relaxation times for the resistive processes plus the relaxation time for the phonon-phonon normal process. The first

term of Equation 33 is given by Equation 68; the second term is a cor­

rection to the theory that takes into account the effect of the normal processes (non-resistive processes). Usually, the correction term is small and can be neglected. it will be evaluated to check its magnitude.

In Equation 68, 0 is the characteristic temperature of one Debye mode which represents an average acoustic mode. 0 is given by

Ti ^6it^ ® (69) where a is the lattice constant. For Mg2Sn 0- = 154° K. 46

Holland (27) indicates that the average sound velocity, c, in the

crystal is found from

c"' = (2ct"' + C]-')/3 , (70)

which gives c = 3-59 x 10^ cm/sec.

In pure material, the only phonon scattering processes present are

the following: boundary scattering, Tg"' = c/L; point defect scattering

caused by the isotopic mass difference, T|~' = Ao)^, with A given by

Equation 35; Umklapp phonon-phonon processes, ' = Byexp (-0/aT)

— Î 2 1 and normal phonon-phonon processes, = Bj^CD T^. Tg is effective

from very low temperatures to just above the thermal conductivity maximum.

T| is effective in the region near the maximum. The phonon-phonon pro­

cesses become important in the region of the thermal conductivity maximum

and for all higher temperatures. However, the phonon-phonon relaxation

times given above are correct only for low energy phonons so we can expect

the theory to break down at high temperatures.

Equation 68 will be numerically evaluated with

T^"' = c/L + Am^ + (Byexp (-e/aT) + B|^) . (y,)

L is given by the Casimir (l4) theory, L = IsC^ ^ ,2) where I^ 1 ^ is

the sample cross-sectional area. A is given by Equation 35. By, B^ and a are adjustable parameters. The value of a is approximately 2. Figure

11 shows the result of the calculation for the four samples. Table 2 lists the values of L, A, By, Bj^ and a that were found. L and A are values calculated from the theory.

The correction term given in Equation 33 was evaluated for the case of sample K-ll. The result is given in Figure 12. The correction term 47

10.0

- 5.0 0 LU h- d 1 5 O

>- > H 2.0 - o 3 û Z O o V THEORY < 1.0 L"2.2mm (K-I3B) 2 ù: L= 3.1 mm (K-13) LU X - L* 3.9 mm (K-l I) h- L = 4.8mm(K-l5) EXPERIMEMT: 0.5 V SAMPLE K-I3B L^Z.Zmm • SAMPLE K-I3B L=3.1 mm O SAMPLE K-ll L = 3.9mm 6 SAMPLE K-15 L»4.8mm 0.3

10 20 50 100 TEMPERATURE (°K)

Figure II. The curves represent the thermal conductivity calculated from the Callaway theory with the size of the sample calculated from the Casimir theory. (The measured thermal conductivity values show a smaller size effect and a steeper T dependence than the theory) 48

0.20 —

0.10-

0.05

0.02-

0.01 5 10 20 50 100 TEMPERATURE, (*K)

Figure 12. The figure shows the magnitude of the correction term in the Callaway theory. (The correction is less than 5 % below 25° K, about 7 % at 30° K and falls to around 5 % at 80° K) 49

Table 2. Relaxation time parameters

Sample L(mm) A(sec3) By(secdeg^) ^^(secdeg^) a

K-1I 3.9 5.,6 x 10-4'» 5.0 X 10-22 9.0 X 10-23 3

,0-44 K-13 3.1 5.,6 X 5.0 X 10-22 9.0 X 10-23 3 1 1 N)

K-13B 2.2 5.,6 X o 5.0 X 10-22 9.0 X O 3 1 O 1 ro

K-15 4.8 5..6 X 5.0 X 10-22 9.0 X O 3

has a maximum value at 30° K where it is 1 % of the measured thermal conductivity. It is less than 3 % for temperatures below 20° K. For

the sake of simplicity we shall not use the correction term in the rest of this discussion, even though it is not completely negligible.

For the temperatures from 20 to 50° K the values found for By, and a give a good fit to the experimental results. Callaway (10) was able to fit Ge, and Holland (27, 28) was able to fit Si and the Ill-V compounds over a similar temperature range by setting a = <» and making

(By + B|yj) an adjustable parameter. It was not possible to fit the MggSn data with a = oo. Callaway showed that a = <» gives a tempera­ ture dependence for temperatures just above the thermal conductivity maximum. The temperature dependence of Mg2Sn is T'^.S,

The point defect scattering strength predicted by the theory of isotope scattering seems to give a good fit in the region of the thermal conductivity maximum. The size effect carries over into this region so that the isotope scattering is slightly obscured.

The fit at lower temperatures is not correct. The data show a steeper 50

temperature dependence than that given by theory. Furthermore, the dif­

ference between the measured thermal conductivities of the four samples

at 4° K is smaller than the theory predicts. For example, the theory

indicates that at 4.2° K samples K-13 and K-13B should differ by 20 %;

the measured values differ by 10 %.

Perhaps boundary scattering is correct, but the correct size, L, of

the sample is not given by the Casimir (14) theory. L was reduced to

1.0 mm for sample K-13 to make the theory fit a 4,2° K, and the L - values

were scaled for the other samples. Figure 13 shows the result. The

situation has not been improved. The maximum has been reduced below the

measured value and the size dependence is still incorrect.

If the strength of the isotope scattering were reduced the maximum

would be increased, but the size dependence would still be incorrect.

The A calculated from the theory of Klemens (33) (Equation 35) has been

found by Callaway (10) and Holland (27) to give a good fit for Ge and Si.

The relaxation time that we have been using has included only those

scattering processes which are present in a pure, defect free crystal.

We must now consider the contributions to that are introduced by cry­

stal defects and impurities. The additional scattering should increase

the temperature dependence at low temperatures and have only a small effect at the thermal conductivity maximum.

Klemens (32) has shown that crystal defects such as dislocations produce at"' CC where 0

10.0

— 5.0 0 UJ I- Q 1 2 1u

>- t- 2.0 (_) =) zo o o V/THEORY; _J < 1.0 L"2.2mm (K-I3B) s cr L® 3.1 mm (K-13) UJ X L * 3.9mm (K- 11) I- L = 4.8mm( K ~15) EXPERIMENT; 0.5 V SAMPLE K-I3B L«2.2mm • SAMPLE K-13B L» 3.1mm O SAMPLE K-ll L»3.9mm A SAMPLE K-15 L=4.8mm 0.3

10 20 50 100 TEMPERATURE (°K)

Figure 13. The curves represent the thermal conductivity calculated from the Callaway theory with the size of the sample, L', for sam­ ple K-13 adjusted to fit the experimental data. The other L' were found by scaling L' = L'J^L/L]^, where L is the value found from the Casimir equation. (The measured thermal con­ ductivity values show a smaller size effect and a steeper T dependence than the theory predicts) 52

obtain the proper temperature dependence we need a T"' which decreases

as CO increases for the range of co important for thermal conduction.

Ordinary chemical impurities would introduce point defect scattering

with T~' OL This scattering would cause a small increase in A but

it would largely be masked by the large isotope effect. The resonance

scattering found by Pohl (43) for small amounts of KNO2 in KCl does not

introduce the correct temperature dependence.

4. Elect ron-phonon interact ion

Because we are dealing with a semiconductor we should consider the

possibility of electronic effects. The number of electrons present at

low temperatures is too small for any electronic thermal conduction.

However, electron-phonon scattering is possible. Several electron-phonon

scattering mechanisms have been proposed.

Ziman (54) has considered phonon scattering by a degenerate parabolic

electron band. J. A. Carruthers e_t (''» 12) have used Ziman's re­

sult to qualitatively explain their data on a number of p-type Ge and

Si samples. One of their samples, Ge7, had a carrier concentration

(2.3 X 10'^) similar to our Mg2Sn samples. Their sample Ge7 showed a

similar temperature dependence at low temperatures. However, it seems unlikely that the distribution function for the charge carriers can be degerate; since 2.5 x lo'^ carrier/cm^ would have a degenerate tempera­ ture of 3° K, and the Hall effect data on samples K-ll and K-13 indicate that there are approximately 5 x 10'5 electrons/cm^ in the conduction band at 10° K. This concentration is sufficiently low that degeneracy would not occur at the temperatures of interest, J. A. Carruthers et al. (12) suggested that the impurity levels formed a degenerate band for the!

sample Ge7. In the case of the Mg2Sn samples there is approximately one

(• — donor atom in 10 Mg2Sn molecules. Therefore, the donors should be sep­

arated by approximately 100 lattice units. The donor-electron orbit

should be 3 to 6 lattice units. Therefore, the overlap of the donor

electron wave function is small, and the donor levels would not form a

band. Furthermore, the large low temperature Seebeck coefficient of the

Mg2Sn samples indicates that the electrons are non-degenerate. Griffin and Carruthers (23) proposed an alternative explanation of

J. A. Carruthers' (12) data on sample Ge7. Griffin and Carruthers suggested that the result could be explained in terms of resonance scat­ tering by the acceptor states, in the method of Griffin and Carruthers, the impurity sites are isolated from one another. They did not discuss the problem for p-type Ge in detail.

Griffin and Carruthers (23) discussed the problem of phonon scat­ tering by bound donor electrons in Ge and Si. The degeneracy of the donor ground state in the effective mass approximation is partially re­ moved by the interaction with the conduction band structure. The 4-fold degenerate ground state in Ge is split into a singlet and triplet state.

The 6-fold degenerate state in Si is split into a singlet, a doublet and a triplet state. There are, of course, higher energy levels.

Griffin and Carruthers (23) considered the phonon scattering by vir­ tual transitions between the ground state levels; particularly, between the singlet and triplet levels in Ge. (For Si only the singlet-doublet transition is permitted.) For either Ge or Si, they found for the relaxa­ tion time, Tpg, 54

1 Gcu^A^ ^De" = (hm) ^ (1 + kc^)^

G = ,"e* F 3^jtP c? \ A / ' (40)

In this expression, A is the energy difference between the levels, r^

is the average radius of the donor electron orbit, n^^ is the number of

un-ionized uncompensated donors, p is the mass density of the crystal,

c is the sound velocity, is the shear deformation potential and F is

a factor depending upon the phonon polarization branch and the electronic

structure. F has a value of approximately 0.2. Goff and Pearlman (22)

found thermal conductivity results for As and Sb doped Ge similar to our

results for Mg2Sn. Griffin and Carruthers (23) were able to fit the data

of Goff and Pearlman (22) with no adjustable parameters.

The structure of the conduction band of Mg2Sn is similar to Si.

Therefore, the donor energy levels and wave-functions of the donors should

be similar to those of Si, and as given in Equation kO would apply

for Mg2Sn, Unfortunately, E^, A, and r^ are unknown. We shall, however,

attempt to make reasonable estimates of these quantities. E^ should not

be too different from the values of Ge and Si. (11 and 19 eV respectively)

The Hall effect data on sample K-11 indicate that the donor level is

1.3 X 10"^ eV below the conduction band. Hence, A should be less than

1,3 X 10"3 eV. r^ should be 15 to 50 A.

Equation 68 has been evaluated with T[)q~' included in The

theoretical values of L and A were used. G, A and r^ were used as ad­

justable parameters. The best fit to sample K-13 was obtained with

G = 1.70 X 10-44 sec3, A = 5.0 X 10"^ eV and r^ = 40 A. The 55

result of the calculation is shown in Figure 14. The same parameters were

used to calculate the thermal conductivity of sample K-13B, The result

of the calculation for sample K-13B is also shown in Figure 14. The

sizes calculated from Casimir's (14) theory were used in both cases.

The bound donor electron phonon scattering accounts for the size

difference between the two samples. The temperature dependence, however,

is not correct for temperatures between 6 and 11° K; but the fit is much

closer to the experimental result than the fit without the electron phonon

scattering. The curves calculated without the electron phonon scattering

in the last section for these samples are included in Figure 14.

In order to fit samples K-11 and K-15, it was necessary to increase

G to a value of 2.38 x 10~^^ sec^ . This increase in G corresponds to

the increase in uncompensated donors for samples K-11 and K-15 over sample

K-13. Samples K-1 1 and K-15 had 2,5 x lo'^ uncompensated donors/cm^ while sample K-13 had 1.8 x lo'^ uncompensated donors/cm^. The result of the calculation for samples K-11 and K-15 is shown in Figure 15. Again we are able to correctly account for the size dependence with the electron phonon scattering. o The electron orbit radius = 40 A is a reasonable value. r^ = o o 15 A in Si and about 40 A for Ge. The large dielectric constant of Mg2Sn would tend to make r^ large.

The ground state splitting, A = 5 x 10"^ eV, is less than the separation between the donor level and the conduction band. In Si, where the donors are about 4 x 10"^ eV below the conduction band, A is 1 x 10"^ eV.

With the values of G and A given above we can calculate E^. The 56

a h-a I Io2

> b S z o o < 0 K-13 L= 3.1 mm WITH ELECTRON- £C PHONON SCATTERING K-I3B L=Z.2mm ^ .5

L» I.Omm I L «0.67mm I NO ELECTRON - PHONON SCATTERING

.3

3 5 10 20 50 100 TEMPERATURE (°K)

Figure 14. The thermal conductivity calculated with bound donor electron phonon scattering for samples K-13 and K-I3B is shown. The theoretical sizes were used in the calculation. (The thermal conductivity calculated for these samples without electron phonon scattering is also shown) 57

10.0

^ 5.0 d UJ Û I 2 O œ I- g 2.0

>- H >

O .0 — n Q Z o o o K-11 L=3.9 mm WITH ELECTRON Ù K-15 L = 4.8mm PHONON i 0-5 SCATTERING LU X & 0.3 -A

10 20 50 100 TEMPERATURE (*K)

Figure 15. The thermal conductivity calculated with bound donor electron phonon scattering for samples K-11 and K-15 is shown. (The strength of the interaction was increased from its value for sample K-13 by the ratio of the uncompensated donors) 58

result is E^j = 10.3 eV, which seems reasonable. For Si, = II eV,

and for Ge, Ey = 19 eV.

The results for r^, A and E^ given above should not be interpreted

as the correct values needed to describe the donor states in Mg2Sn. They

are given only as an indication that the donor electron phonon scattering

mechanism is of the correct magnitude.

There are several other donor electron phonon scattering process

which might improve the temperature dependence of the calculated thermal

conductivity. Among these are scattering between the donor ground state

and the donor excited states. The possibility exists that some of the

donor's excited levels are filled by thermal excitation and we could have

scattering between this level and other donor levels. in addition, scat­

tering could take place between the filled donor level and the conduction

band. The relaxation times for these processes are not known. Griffin

and Carruthers (23) have argued that in Ge their effect is smaller than

the scattering between the ground state levels. Even if the relaxation

times were known their addition to the calculation would simply add more adjustable parameter unless considerably more information were available

about the band structure and impurity levels of Mg2Sn. However, their

relaxation times would still be proportional to the number of uncompensa­

ted donors, n^^. We have demonstrated that the additional relaxation

time needed to account for the size dependence of these samples is propor­

tional to Hgx- 59

C. Seebeck Coefficient Results and Discussion

Tine Seebeck coefficients of tine 4 n-type samples K-1 1 , K-13, K-13B

and K-]5 were measured at the same time as the thermal conductivity.

The impurity concentration in all of the samples was approximately the

same. In order to make a complete analysis of the Seebeck coefficient,

a number of different impurity concentrations should be measured, p-type

samples should be measured, and the measurements should be extended to

higher temperatures so that the samples become completely intrinsic.

The analysis of the Seebeck coefficient of Mg2Sn is complicated by the

fact that mixed conduction sets in at approximately the same temperature

that the phonon drag contribution dies out.

The absolute Seebeck coefficients of the four samples is shown in

Figure l6 for the temperature range where the diffusion term is dominant.

From liquid nitrogen temperature to 200° K the data for sample K-13 a re

considerably lower than the data for the other samples. This result is

not understood because sample K-13 contains fewer donors than the other

samples and, therefore, it should have a higher diffusion Seebeck coef­

ficient.

From Equation 42 we find that in the extrinsic region the diffusion

Seebeck coefficient, S^, can be expressed as

k An - In (7h3/2e |R| (2amkT) 3/2)1 , e (74) where

7 |R| ne and 1000 o SAMPLE K-ll • SAMPLE K-13 V SAMPLE K- 13 B 800 A SAMPLE K- 15 S Q 600 > 3 O—o

200

0 100 200 300 TEMPERATURE (*K)

Figure 16. The absolute Seebeck coefficient for the diffusion range is shown. (The curves represent the diffusion Seebeck coefficient calculated from Equation 74 and 75) 6l

Ae t 3 iTig

R is the Hall coefficient and 7 is a number which depends upon the scat­

tering mechanisms. For acoustic mode scattering and small magnetic fields

7 = %^/8.

In the intrinsic range, Sis given by

Sy = ^ I^P {Ap - In (ph3/2(2#mkT)3/2^j

- nb |An - In (nh^/Z (2TtmkT) j , (75)

where

Aet' 3 mh "p' ~ IT

and

Aet 3 mg ~ ^ Ï T •

n and p are respectively the electron and hole densities.

In the extrinsic range, the diffusion Seebeck coefficient will be

calculated from Equation 74 with the value of adjusted to make the

calculated value fit the experimental data at 150° K. in the mixed con­

duction range the diffusion Seebeck coefficient can be calculated from

Equation 75- In this region it is necessary to find n and p from the Hall

coefficient. The two equations

n = P + "ex ,

and 0 7 p - b n R = - 0 (p + bn)2 (76) with b = 1.23 were solved to find n and p as a function of temperature. 62

Since we are assuming that the mobility ratio, b, is independent of tem­

perature we have

3 mh "p =

Api has been treated as an adjustable parameter. The density of states

effective masses reported by Lipson and Kahan (37) nig = 1.2 m and

m^ = 1.3 m, were used to calculate Ap from A^, The curves in Figure

16 are the results of the calculation. Table 3 gives the parameters Ap

and Ap that were found. The diffusion Seebeck coefficient was not calcu­

lated in the mixed conduction range for sample K-13 because A^ would have

to be strongly temperature dependent.

Table 3. Diffusion Seebeck coefficient parameters

Sample A^^

K-ll 1.81 2.10 2.02

K-13 -1.03

K-15 1.95 2.30 2.20

^For Equation 74.

^For Equation 75.

For acoustic mode scattering, Ae^/kT = 2.0, and Ae^/kT > 2.0

for other types of scattering. With m^ = 1.2 m we find in the extrinsic

range that As ^./kT = 1.54 and 1.87 for samples K-ll and K-15. An ef­

fective mass less than m is required to make Ae^/kT greater than 2.0.

A more complete study of the Seebeck coefficient of both n and p type 63

MggSn should be made to check this result.

Figure 17 shows the phonon drag Seebeck coefficient calculated by

subtracting the diffusion Seebeck coefficient from the measured values.

The phonon drag Seebeck coefficient shows a T~^'^ temperature dependence

from 30 to 100° K. Above, 100° K, the phonon drag contribution is too

small to be accurately determined. The theory of Herring (26) predicts

a T~" temperature dependence. For longitudinal acoustic mode scattering

n = 3.5. For a more general type of scattering Herring finds n < 3.5.

Heller (24) found n = 3.0 for Mg2Si. Geballe and Hull (20) found

n = 2.4 for n-type Ge and 2.3 for n-type Si.

The maximum of the phonon drag Seebeck coefficient occurs near 18° K

while the thermal conductivity maximum occurs at 13° K. This result

demonstrates that different phonons are involved in the two phenomena.

The phonons which contribute to the phonon-drag have wavevectors of the same size as the wavevectors of thermal electrons. Therefore, at 10° K

the phonons that contribute to the phonon-drag have m 6 x lo"^ sec~'.

The phonons which contribute to the thermal conductivity have energies ] 9 on the order of 3 kT so that their angular frequency is near 4 x 10 sec"' at 10° K. Impurities, such as isotopes, scatter high frequency phonons more effectively than low frequency phonons so impurity scattering does not make a significant contribution to the phonon drag. Boundary scattering is independent of frequency so that boundary scattering affects both the thermal conductivity and the phonon drag Seebeck coefficient.

Heller (24) observed a large size effect in the low temperature

Seebeck coefficient of Mg2Si. The total Seebeck coefficient at low temper­ atures is shown in Figure 18 for the Mg2Sn samples. Sample K-13B is sam- 64

10

o O1x1 > 3

SAMPLE K-ll SAMPLE K-13 SAMPLE K-I3B SAMPLE K-15

1 0

AO

10 20 50 100 200 500

TEMPERATURE C K )

Figure 17. The phonon drag Seebeck coefficient is shown. (From 30 to 100° K the temperature dependence is T'^.S) 5x 10

O : y 2x10

Z3

(f) O SAMPLE K- II D SAMPLE K-13 V SAMPLE K-I3B A SAMPLE K-15

5x10 10 20 50 TEMPERATURE (°l<)

Figure 18. The to^al Seebeck coefficient at low temperatures is shown. (The Seebcck coefficient shows a T temperature dependence) 66

pie K-13 cut in half. In these samples, the Seebeck coefficient shows a

small size effect just the same as the thermal conductivity. This small

size effect is probably caused by the additional phonon scattering intro­

duced by the bound donor electrons. Herring's (26) theory of phonon drag

indicates that the phonon drag Seebeck coefficient should be proportional

to in the boundary scattering region. Our total Seebeck coefficient

shows a T^'5 temperature dependence at low temperatures. This higher

temperature dependence is probably an additional effect of the donor

electron phonon scattering.

D. Conclusions

The thermal conductivity of Mg2Sn single crystals has been measured

from 4 to 300° K. From 120° K to 300° K the thermal conductivity is pro­

portional to T~^. The theory of Leibfried and Schloemann (35) for lattice

thermal conductivity due to phonon-phonon scattering adequately repre­

sents the data in this temperature region.

At low temperatures the data have been fit with the Callaway (lO)

theory. In order to account for the small size dependence of the thermal

conductivity below 15° K it has been necessary to include phonon scat­

tering by the bound donor electrons.

The results in the neighborhood of the thermal conductivity maximum

indicate that the only significant point defect scattering is caused by

the mass differences of the isotopes of Mg and Sn.

It has been found necessary to retain the exp (-0/aT) term in the

Umklapp process relaxation time in order to fit the data at temperatures above 15° K. 67

The Seebeck coefficient also shows a small size dependence at very

low temperatures. Presumably this result is also caused by the donor

electron phonon scattering mechanism. At higher temperatures the phonon

drag Seebeck coefficient is proportional to T~ in agreement with the

theory of Herring. (26) The data at higher temperatures indicate that more measurements need to be made before the diffusion Seebeck coeffi­ cient can be understood.

E. Future Work

The measurements should be extended to both lower and higher tempera­ tures. At higher temperatures, the electronic thermal conductivity should become à significant part of the total thermal conductivity. At very low temperatures, the theory of donor electron phonon scattering indicates that there may be a change of slope in the K versus T curve. In addition, measurements on purer and on doped samples should be made in order to confirm the electron phonon scattering mechanism.

A more extensive study of the Seebeck coefficient should be made.

The measurements should be extended to higher temperatures and to p-type samples so that the diffusion Seebeck coefficient can be properly analyzed. 68

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VI. ACKNOWLEDGEMENTS

The author wishes to thank Dr. G. C. Daniel son for guiding this research. Thanks are also due the National Science Foundation for sup­ port during part of this study. Mr. Arthur Klein helped with the compu­ ter programs. Mr, 0. M. Sevde helped set up the experiment. The author also wishes to thank Mr. Howard Shanks and Mr, Paul Sidles for several helpful discussions.

V 73

Vil. APPENDIX

Table k. Sample experimental data

Sample K-13 L = 0.886 cm A = 0.0829 cm^

Data at 9.13° K

I, = 81.055 mA I2 = 52.610 mA V, = 0.56669 V V2 = 0.36341 V E] = -1.70 P.V Eg = -1 .00 \i\l Vsi = -194.4 |j,V Vs2 = -107.0 1J.V

K = 1'iVl - I2V2 dE A E] - E2 dT

_ _ in rn 8.1055 X 10-3 X 0.56669 - 5,2610 x 10-3 ^ 0.36341 , Lc\ ^ lu.oy -1.70+1.00 X

K = 6.30 ^ cmdeg

V.] - V52 dE -194.4 + 107.0 S = (-15.45) = -i860 [iV/deg E, - E2 dT -I.70 + ].00

Data at 97.8° K

1 = 80.884 mA V = 0.57294 V E] = 5.1215 mV Eg = 5.1539 mV V3 = -1033.2 ^iV

K = -10.69('.79X,0-2)

w K = 0.278 cmdeg

^ " Ej - Eg dT - 5.1215 -V1539 ('"79 ^ '0-2)

S = -581 (iV/deg 74

Table 5, Tinernia 1 conductivity and Seebeck coefficient resu1ts

K (measured) K (corrected)^ S (absolute/ Watt Watt T° K cmde 9 cmdeg de g

Samp] e K-3

80.2 .338 .336 80.7 .339 .337 82.7 .325 .323 83.5 .338 .336 83.9 .330 .328 91 .7 .314 .312 91.8 .314 .312 107.1 .234 .232 107.1 .234 .232 119.8 .202 .200 119.9 .201 .199 128.4 .190 .187 128.8 .184 .181 141.1 .165 .1 62 141.] .167 .164 158.7 .149 .146 158.8 .146 .143 159.5 .147 .144 ] 61 .142 .139 177.6 .127 .123 192.7 .117 .113 208.7 .110 .106 215.5 .108 .103 229.8 .101 .096 230.1 .101 .096 252.7 .094 .088 252.8 .094. .088 260.7 .092. .085 276.2 .086 .080 277.4 .088 .080 286.6 .085 .077 303.3 .083 .074 301 .0 .082 .073

^Corrected for heat loss according to Equation 66, 75

Table 5 (Continued)

K (measured) K (corrected)^ S (absolute) Watt Watt (iV T° K cmdeg cmdeg deg

Samp]e K-]1

4.26 1.50 1 .50 -763 6.52 4.17 4.17 -1427 7.35 4.37 4.37 -1478 9.04 6.42 6.42 -2088 10.52 7.39 7.39 -2378 12.73 7.35 7.35 -2708 14.52 6.56 6.56 -2792 16.72 6.58 6.58 -3066 18.83 5.65 5.65 -3051 20.80 5.24 5.24 -3120 23.34 4.51 4.51 -3055 25.80 3.26 3.26 -2244 30.14 2.58 2.58 -1988 33.70 1.84 1.84 -1473 49.7 0.834 0.829 -1090 51 .5 0.713 0.708 -880 67.1 0.446 0.441 -700 80.1 0.382 0.377 -750 80.9 0.373 0.368 -777 83.5 0.359 0.354 -719 107.1 0.28] 0.275 -765 131 .9 0.208 0.200 -715 155.3 0.173 0.164 -704 155.1 0.168 0.160 -692 197.3 0.127 0.116 -528 205 0.129 0.118 -442 274 0.0950 0.074 -94.2 289 0.0944 0.075 -97.5 303 0.0956 0.075 -86.3

Samp]e K-1 3

4.2 1.37 1.37 -81.8 5.16 2.32 2.32 -1019.0 5.65 2.74 2.74 -1088.5 6.76 4.43 4.43 -1407.5 7.38 5.0] 5.01 -1539.5 9.13 6.30 6.30 -1853.0 12.27 7.22 7.22 -2487.0 14.67 7.17 7.17 -2807.0 76

Table 5 (Cont inued)

K (measured) K (corrected)^ S (absolute) Watt Watt (iV T° K cmdeg cmdeg deg

16.80 6.30 6.30 -2867.0 18.42 5.58 5.58 -2726.0 20.58 5.07 5.07 -2695.5 21.52 4.84 4.84 -2665.0 21.6 4.36 4.36 -2445.0 26.38 3.18 3.18 -2084.0 26.76 3.47 3.47 -2234.0 29.8 2.55 2.55 -1773.5 32.7 2.045 2.05 -1513.0 39.0 1.429 1 .43 -1082.8 49.5 0.944 0.937 -1028.0 49.8 0.960 0.953 -1048.0 50.1 0.870 0.863 -944.0 50.2 0.826 0.826 -705.0 63.3 0.560 0.554 -742.0 79.9 0.392 0.386 -664.6 80.5 0.3575 0.351 -605.6 81.1 0.3575 0.351 -607.6 97.8 0.278 0.271 -562.4 97.8 0.2775 0.271 -564.4 129.0 0.196 0.188 -488.7 1 29.0 0.196 0.188 -487.7 151 .0 0.170 0.161 -471.2 151.0 0.1676 0.160 -469.0 173.9 0.1436 0.133 -438.4 174.3 0.1405 0.130 -436.4 189.8 0.1321 0.121 -371.3 21 1 .0 0.1223 0.110 -240.0 274.8 0.0974 0.0742 -86.9 274.9 0.0967 0.0735 -87.2 303.0 0.1069 0.6867 -74.0 303.7 0.1080 0.0878 -71 .9

Sampl e K'-13B

4.08 1.112 -587 4.25 1.260 -64l 7.94 4.710 -1556 8.94 6.300 -2098 10.2 7.110 -2438 10.92 7.150 -2517 13.22 6.860 -2727 77

Table 5 (Con ti nued)

K (measured) K (corrected)^ S (absolute) Watt Watt liV T° K cmdeg cmdeg deg

]6.08 6.560 -3116 20.92 4.940 -2870 26.75 3.390 -2474 34.74 1 .800 -1930 48.6 0.900 -1041 53.7 0.749 -931 58.0 0.651 -765 67.8 0.483 -728 80.3 0.400 -740 82.7 0.3665 -700 82.8 0.358 -659

Sample K-15

4.915 2.32 2.32 -955 6.35 3.42 3.42 -1278 6.48 3.69 3.69 -1436 6.73 4.40 4.40 -1972 7.75 5.70 5.70 -2143 8.87 6.60 6.60 -2363 1 1 .08 7.46 7.46 -2658 1 3.44 7.36 7.36 -2833 1 7.46 6.12 6.12 -3136 20.5 4.50 4.50 -2691 25.85 3.79 3.79 -2645 57.0 0.665 0.660 -911 68.0 0.509 0.504 -867 81.3 0.384 0.378 -741 82.5 0.369 0.364 -745 84.2 0.349 0.344 -730 96.3 0.307 0.302 -722 96.4 0.308 0.303 -728 118.1 0.232 0.226 -691 131.6 0.212 0.206 -690 146.0 0.193 0.187 -691 161.3 0.175 0.168 -676 275 0.100 0.081 -100 276 0.099 0.081 -97 302,2 0.100 0.080 -75 303.7 0.101 0.081 -73