Retrospective Theses and Dissertations

1967

Thermal conductivity of magnesium stannide Joel Jerome Martin Iowa State University

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MARTIN, Joel Jerome, 1939THERMAL 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 i n

A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY

Major Subject:

Physics

A p p r o v e d:

Signature was redacted for privacy. n Charg^^of Major Work

Signature was redacted for privacy. H e a d f hf'jha j ^ r D e p a r t m e n t

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 1.

II.

ill.

IV.

INTRODUCTION

1

A.

Properties o f Mg2Sn

1

B.

Purpose of this Investigation

5

THEORY

6

A.

Theory of Thermal Conductivity of Semiconductors

6

B.

Seebeck Effect

17

EXPERIMENTAL PROCEDURE

20

A.

Samples

20

8.

Measurements

25

C.

Errors

35

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

VI .

ACKNOWLEDGEMENTS

VII.

vi

APPENDIX

68 .

72 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

74

conductivity and Seebeck coefficient results

I V

LIST OF FIGURES Page 1

Crystal

2

Electrical resistivity, p, t i v i t y samples

3

4

structure o f Mg2Sn

2 of the thermal

conduc­ 22

Hall coefficient, R, of the thermal samples

conductivity

Hall c o e f f i c i e n t , R, and e l e c t r i c a l

resistivity,

23

P, o f M g 2 S n s a m p l e s K - 1 1 a n d K - 1 3 a t l o w t e m p e r ­ atures

24

Sample holder

27

6A

Sample heat sink clamp

29

6B

Thermometer clamp

29

5

7

8

9

Block diagram of ratus

the thermal

conductivity appa­ 33

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

40

Thermal

42

conductivity results

10

The thermal resistance o f Mg2Sn above 100° K

44

11

T h e t h e r m a l c o n d u c t i v i t y of M g 2 S n c a l c u l a t e d f r o m the Callaway theory with the size of the sample calculated from the Casimir theory

47

The figure shows the magnitude o f the correction term in the Callaway theory

48

The thermal conductivity o f Mg2Sn calculated from the Callaway theory with the size of sample K-13 adjusted to f i t the data at 4.2° K

51

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

56

The thermal conductivity calculated with bound donor electron phonon scattering for samples K-1 I and K-15 i s shown

57

12

13

14

15,

V

page Figure 16.

The absolute Seebeck coefficient for the diffusion range is shown

60

Figure 17.

The phonon drag Seebeck coefficient i s shown

64

Figure 18,

The total Seebeck coefficient a t tures i s shown

65

low tempera­

Vi

ABSTRACT

The thermal

c o n d u c t i v i t y o f s e v e r a l n - t y p e M g g S " s a m p l e s was m e a s u r e d

from 4.2 to 300° K.

The samples had uncompensated donor concentrations

on the order of 2 x 10^^ donors/cm^. t i v i t y showed a T~^ lattice thermal

Above 150° K,

the thermal

conduc­

temperature dependence which is characteristic of

conductivity.

The theory for high temperature thermal

conductivity of Leibfried and Schloemann i s In agreement with our results. The b i p o l a r e l e c t r o n i c c o n t r i b u t i o n was estimated t o be U % o f the t o t a l thermal

conductivity at 300° K.

Below 100° K, the data were analyzed i n

terms of the Callaway theory by combining the relaxation times for phononphonon scattering,

isotope scattering, boundary scattering and bound

donor electron-phonon scattering.

Above the thermal

i t was necessary t o retain the exp

(-0/aT)

conductivity maximum

term in the phonon phonon

scattering process to obtain the correct temperature

dependence.

The

calculation i n the neighborhood o f the thermal conductivity maximum I n d i ­ cated t h a t the only point defect scattering present i n the samples was caused by the isotopes o f Mg and Sn.

At 4° K 'the data showed a smaller

size dependence than the theory predicted i f only boundary scattering was Included.

in addition, the measured values were about half of the

curves calculated with only boundary scattering.

This r e s u l t was ex­

plained in terms of an additional phonon scattering caused by the bound donor electrons.

With t h i s mechanism I t was possible t o account f o r the

size difference and the difference In doping in the samples.

As an aux­

i l i a r y experiment the Seebeck c o e f f i c i e n t was measured a t the same time as the thermal conductivity.

The phonon drag contribution to the Seebeck

c o e f f i c i e n t shows a T~^'^ temperature dependence from 30 t o 100° K.

I.

A.

I.

Properties o f Mg2Sn

Crysta1 structure Mg2Sn i s a

I I -1V compound semiconductor o f the Mg2X

be S i , Ge, Sn or Pb. ture.

Figure 1 shows the cubic u n i t c e l l o f Mg2Sn.

(deg)"'

family where X can

The Mg2X compounds c r y s t a l l i z e i n the

parameter of 6.7625 A at 26° C

2.

INTRODUCTION

(49).

Shanks'

Pluorite struc­

Mg2Sn has a l a t t i c e

found a value of 9.S x 10 ^

for the temperature coefficient of the lattice parameter at 300° K.

Energy gap Winkler (53) determined an energy gap o f (0.36 - 3 x 10~\) eV from h i s

r e s i s t i v i t y , Hall effect and Seebeck effect measurements.

B l u n t _et_ a _ L -

found an energy gap o f 0.33 eV from t h e i r Hall e f f e c t data.

Nelson (40)

found a gap of 0.36 eV from his Hall coefficient measurements. gap o f 0.34 eV was determined from Hall

(&)

An energy

e f f e c t m e a s u r e m e n t s b y L a w s o n e_t

±1. (34) T h e o p t i c a ] m e a s u r e m e n t s o f B l u n t e_t _a]_. ( 6 ) y i e l d e d a n e n e r g y g a p o f 0.33 eV a t 5 ° K; they found that the energy gap decreased as the tempera­ ture increased.

The o p t i c a l a b s o r p t i o n measurements o f Lawson _e^ aj_.

gave an energy gap o f 0.18 eV a t 294° K.

(34)

They found that i f their measure­

ments were interpreted i n terms of indirect transitions the energy gap in­ creased as the temperature increased.

Lipson and Kahan (37)

interpret

t h e i r optical absorption data i n terms o f an energy gap o f about 0.18 eV a t

'shanks, H. R., tion. 1966.

Iowa State University, Ames,

Iowa.

Private communica­

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 o f Mg2Sn i s not understood.

A gap

o f about 0.36 eV seems t o r e s u l t from e l e c t r i c a l measurements, but a gap o f about 0.18 eV seems t o r e s u l t from o p t i c a l measurements.

3.

Mob i l i t y W i n k l e r ( 5 3 ) , Ge i c k e_t _aj_.

(21) and Lichter (36)

found that the mobil-2 5

i t i e s of both holes and electrons have a temperature dependence of T the intrinsic range (T > 300° K).

The

t r i b u t e d t o o p t i c mode s c a t t e r i n g .

in

temperature dependence was a t ­

The r a t i o o f the electron mobility to

the hole m o b i l i t y i n the i n t r i n s i c range was found t o be 1.20 - 1.25 by Lipson and Kahan

(37) and 1.23 by B l u n t e_t_aj_.

(6)

The electron mobility

at 300° K is about 300 cm^\/"'sec"'. B l u n t e t _a]_.

(6)

found that the mobilities of both electrons and holes

had a temperature dependence of T ^ ^ between 80 and 200° K. l y , G e i c k e _ t a_l_.

(21) and Lichter

(36)

as T"'*^ for temperatures below 300° K.

More recent­

found that the hole mobility varies The T ~ ' ' ^ dependence was

inter­

preted i n terms o f acoustic mode s c a t t e r i n g .

4.

Effect ive masses Lipson and Kahan (37) have reported the most recent e f f e c t i v e 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 f o r electrons and holes respectively where m is the free electron mass.

k 5.

Elect ronIc st ructure Lipson and Kahan

(37) have suggested t h a t the band s t r u c t u r e o f Mg2Sn

consists of a t r i p l y degenerate valence band located a t k = 0 and conduc­ t i o n band minima located away from k = 0. ments o f Umeda

(50)

The magneto resis tance measure­

indicate that the conduction band o f Mg2Sn consists o f

e l l i p t i c a l 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

t r i c constant of Mg2Sn; they found

= 17.0.

Their data indicate that

the transverse o p t i c mode l a t t i c e frequency i s 5 . 6 x 10 i n f r a r e d r e f l e c t i v i t y d a t a o f G e i c k e_t aj_.

(21 )

]7

-1

sec" .

The f a r -

indicate that

has a constant value of 7.^ below 300° K and increases t o 8.2 at 500° K;

Eq is the static dielectric constant.

7.

Elastic properti es D a v i s e_t

(17) have measured the sound velocities of Mg2Sn,

They

found f o r the < 100 > direction, Cg = 4.78 x lO^cmsec ' and c^ = 3.19 x 5 -1 10 cmsec where c^ i s the longitudinal wave sound velocity and c^ i s 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 f i t t o the Debye tempera­ ture versus

temperature curve obtained from the heat capacity measurements

of Jelinek et al.

(29)

5

8.

Therma1 propert i es The heat capacity o f Mg2Sn has been measured from 5 t o 300° K by

Jelinek

aj_.

(29)

They report a Debye temperature o f 388.4° K a t 1° K.

The Seebeck effect has been measured by Boltaks (7) and by Winkler (53)

from 80° K to 600° K.

Several o f Winkler's samples showed a sign

change i n the Seebeck coefficient near 150° K which indicates that they f were p-type; a l l o f his result showed a positive temperature c o e f f i c i e n t at low temperatures. effect

Aigrain ( I ) has measured the thermomagnetoe1ectric

i n Mg2Sn; he found that the e f f e c t i v e mass of the holes i s 0.14 m^.

Busch and Schneider (9) have measured the thermal Mg2Sn from 73 t o 473° K.

conductivity of

Their results did not indicate any electronic"

contribution to the thermal conductivity.

B.

Purpose of this investigation

The purpose o f t h i s investigation was t o measure the low temperature thermal conductivity o f Mg2Sn.

The relaxation time f o r a number of phonon

s c a t t e r i n g 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 t o test the magnitude of the various phonon relaxation times. A simple modification of the apparatus allowed the Seebeck c o e f f i ­ c i e n t t o be measured a t the same time as the thermal c o n d u c t i v i t y .

The

Seebeck data can be interpreted in terms of the existing theories of trans­ port properties of semiconductors,

in addition,

of the Seebeck coefficient provides

information on electron-phonon inter­

actions .

the phonon-drag component

- -

6

I 1.

A.

1 .

THEORY

Theory of Thermal Conductivity of Semiconductors

Sepa r a t i o n o f l a t t i c e and e1ect roni c components i n a semiconductor, heat may be carried by phonons (quantized l a t ­

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

The total

thermal conductivity, K, can be separated into an electronic component. Kg, and a l a t t i c e component, Kp, as follows K

=

Kg + Kp .

(])

Equation 1 holds i f the interaction between the electrons and the phonons is small.

The validity of Equation 1 has been discussed by Drab­

ble and Goldsmid (18). assume that Equation 1

2.

For the purposes o f t h i s discussion, we shall is valid.

Elect roni c the rma1 conduct i v i t y Following Smith

times of Herring (26)

(4?) but using the method of averaging relaxation the electron current density, J, in a semiconductor

with an applied electric field,£., and temperature gradient, , J

=

n e r? —ce mg

_ + T

dT/dx,

is

ô ^Ef'^"] < v ^ t > ne dT < v^ET > ^ n + n , ôx\ T / < v > Tmg dx < v^ >

, ^ (2)

where n i s t h e number per u n i t volume o f e l e c t r o n s , mg i s the e f f e c t i v e mass,

e is the magnitude of the electronic charge, T is the absolute

temperature, Ef i s the Fermi level, E is the energy of the electron measured from the bottom of the conduction band, v is the electron veloc­ i t y and T

is the relaxation time.

the quantity < (

) > is given by

For a non-degenerate semiconductor,



(

=

)

r

f°E2dE

f°E^dE

:3a)

exp (-E/kT).

Often, T may be expressed as T

=

AE

-s

(3b)

Then Equation 3a gives:

where

^(x)

< v^T > /


< v^ Et > /


=

AkT

< v^E^T > /


=

Ak^T

A

Rl-

i!)

-

^P[|-

(3c)

i s t h e gamma f u n c t i o n .

The electron heat current density,

is n

mg"-

ôx \ T/-'




dT

< v^ E^ t >




(4)

The thermal conductivity i s 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 < V^ > < V^T >

meT

dT dx

(5)

or 'dT "

Since the electrical

< V^E^T > < V^T > - < V^ET >2 mgT < v^ > < v^T >

conductivity a

=

IL ë £

mp

Ke where

=

(%)

(6)

, we have < y T > - < v ET >

-

2

k, 2 T 2 ^ < .yZ^ rt ^ 2

(8)

or

K' e e

= -

7 )' lf2l - =s / (l e

For the case o f acoustic mode scattering, s a simple meta].

Equation 8 gives

(T T . =

(9)

—, we f i n d -£ =

the Lorenz number,

-C. =

2.

For

2 I

In a non-degenerate p-type semiconductor,

i f we assume T

=

AE

•S

the hole current is given by Ô /Efl

.

/5

, \ 1 dT ^

(10) w h e r e p i s t h e n u m b e r o f h o l e s p e r u n i t v o l u m e , |j.|^ i s t h e h o l e m o b i l i t y < t v 2 > / < v 2 > ) a n d Ef

=

A E - E^ w h e r e 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 ' ' '— ^ i r a " ( i - • • ) ! £ }

(11) The second term in Equation I I represents the transport of the recombina­ tion energy of electrons and holes. Hence,

I t is zero for zero hole current.

the electronic contribution to the thermal conductivity of a p-

type sample is

^

- (f - =j(

•

(12)

I f we are dealing w i t h an i n t r i n s i c semiconductor where both electrons and holes are present, we must w r i t e the t o t a l charge and thermal cur­ rents as

9

J

(13a)

and Q. where

+ Oh ;

(]3b)

i s g i v e n b y E q u a t i o n 1 , Jj-, b y E q u a t i o n 1 0 , Q,g b y E q u a t i o n 6 a n d

Q.(^ b y E q u a t i o n 1 1 . 0 and

=

=

Setting J

=

0 (note,

this is not

the same as

=

0) we f i n d f o r the t o t a l e l e c t r o n i c c o n t r i b u t i o n t o the

thermal conductivity

s') + A E / k T j ^ np^e^h

o T .

(14)

The f i r s t term i n 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 i s usually large com­

pared to the f i r s t term because A E/kT > > 1.

3.

Lattice thermal

conduct i v i t y

The original description of l a t t i c e thermal ten i n 1929 by Peierls (41).

conductivity was w r i t ­

In that paper he quantized the l a t t i c e

vibrations into phonons and showed that i n a perfect crystal

the only

processes which give rise to a f i n i t e thermal conductivity are those pro­ cesses which do not conserve c r y s t a l momentum (U o r Umklapp processes). The theory of l a t t i c e thermal nisms,

conductivity for various scattering mecha­

including imperfections, has been reviewed by Klemens (32, 33) and

10

Carruthers (13). At high temperatures

(T > O) L e i b f r i e d and Schloemann (35), using

a variational approach, have obtained the expression

5 4"' \ h /

yZy

'

(15)

where k i s Boltzmann's constant, h i s Planck's constant, M i s 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.

15 is a modification o f Leibfried and Schloemann's original

Equation expression

that was pointed out by Steigmeier and Kudman ( 4 8 ) . Callaway (lO) has developed a phenomenologica1 theory which combines the relaxation times for the different scattering processes. has had considerable success in f i t t i n g thermal

His theory

conductivity data on a

number of materials from very low temperatures t o about 100° K.

Car­

ruthers (13) discusses Callaway's theory and a number of the phonon scat­ tering processes that occur at low temperatures. approach of Callaway w i l l

be used to interpret

The relaxation time

the low temperature data

of this experiment. The following is an outline of the Callaway theory. Boltzmann transport equation for the phonon distribution the phonon wave vector,

(10)

The

, where q is

is

W ,

"

- V T

^

.

where c^ i s the phonon group velocity for polarization e, T is the absolute temperature and

(16) t is the time,

is the collision operator.

Equa-

\ a t /c t i o n 16 i s v a l i d o n l y w h e n N q d e p e n d s o n p o s i t i o n t h r o u g h t h e t e m p e r a t u r e .

11

n the relaxation time approximation

N° - N,

'la

T (q) where

'

(17)

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 t o create higher momentum phonons; and, therefore,

the normal processes cause Nq t o relax t o a s t a t e o f higher

momentum, Nq ( \ ) ,

t h a n t h e e q u i l i b r i u m d i s t r i b u t i o n N q ° w h e r e N q ( A. )

is given by 1 N„ ( X ) =

+ % . q e

kT

_ I

(18)

In order t o include N processes. Equation 16 is written as N q ( ^ ) - Nq

Nq° - Ng

&t/c where

"^r

(19)

is the relaxation time for N processes and

time for resistive processes.

Nq ( \ )

is the relaxation

To f i r s t order

=

Nq(0) + \

,

or Tim kT No ( \ )

=

Nq(0)

kT

/ fg (

\2 ' _ ij

(20)

w h e r e Ti i s P l a n c k ' s c o n s t a n t d i v i d e d by 2jt, m i s t h e p h o n o n a n g u l a r f r e ­ quency and k is Boltzmann's Combining Equations 16,

constant.

Let x

19 and 20 gives

=

Tim/kT.

12

(cx'- ,)2

where Nq° -

.

N

+ Tj"q

=

7

0

(21)

Let

"q

tiœ e -VT i^j2 (gx _

-

1)2 •

(22)

Callaway (10) defines

_L - _L ^ _L Tc

TN

Tr '

_

(23)

Note that Equation 22 s i m p l i f i e s Equation 21 t o

R

T 1 Tico —

[-' +

\

- q

T c -VT

=

0 .

(24)

Now i n an i s o t r o p i c medium X ~ v T , so we can d e f i n e a parameter B,

such

that X Using q

=

=

- TiBcVt/T .

(25)

cm/c^, we get X

• q

=

- TiCjùBc

*

T/T .

(26)

Therefore, Equation 22 gives T

=

Tc [ l + B/TJ,

(27)

and r", "q

=

„/

1—

, ,^

e^

Ticu

- -Tc L ' + B / t , ] c . V T - ^ p r r r p i P i ^ •

Normal processes cannot change the t o t a l phonon momentum.

J

dt

N

^ ^

,

0

(28)

Therefore,

. (29)

By using Equation 26 and g e t t i n g r i d o f the constants. Equation 29 becomes

pe/T

x^e* (e* - 1)2

T - B Tw

^

° '

(30)

13

or solving for the constant pP/T

P/ T dx

x^e* dx

(e* - 1)

(e* - I ) " T,

N

(31)

T o g e t t h e h e a t c u r r e n t Q,, n o t e t h a t

q,p

q ,G q , e

q,G i f we replace the

/

by an integration and note that Nq° makes no c o n t r i ­

bution, we get

K

=

1

+ —

("hco)

(2%) 3

kT

e^ (e* - 1)

2

? ? Cr cos a G (32)

where a is the angle between

and

T.

i f we assume the i s o t r o p i c case and l e t three Debye modes, m

=

cq,

represent the three acoustic modes w i t h c an average sound velocity then

'G/T K

e/T T

xV

=

dx + B 2rt c \ Ti

^ x^

(c"" - 1)^

where 9 i s the Debye temperature.

dx (e* - ])

(33)

Equation 33 w i l l be used t o f i t the

low temperature thermal conductivity data of this experiment,

is a

combined relaxation time found by the reciprocal addition of the relax­ ation times

4.

for the different scattering mechanisms.

Phonon r e l a x a t i on t imes a.

Boundary scattering

At temperatures below the thermal

d u c t i v i t y maximum only long wavelength phonons are present. w i l l be scattered by the crystal boundaries. out the relaxation time, Tg,

Casimir (14)

con­

These phonons f i r s t worked

for boundary scattering given in Equation 34.

1 4

Tg"

=

c/L

(34)

where c is the average sound velocity and L is the effective sample dii

ameter.

For a rectangular sample,

sample cross-section.

L

B e r m a n e_t

=

/

o

2tc"

where 1 j I ^ is the

(4) have determined corrections for

sample roughness and for f i n i t e sample length.

At low temperatures bound­

ary scattering causes the thermal conductivity to be proportional

b.

Defect scattering

to

.

Klemens (32) has found the phonon relaxa­

tion time for several types of crystal defects.

One o f the more impor­

tant defects is a point defect caused by the different isotopes i n the crystal.

For point defects Klemens found

= w h e r e GO i s t h e p h o n o n f r e q u e n c y .

Am

(35a)

Slack (46) has modified Klemens expres­

sion for A for elemental materials t o include compounds.

For a compound

A/ByC;. A

=

X + y +

V p/4jrc^ ;

(35b)

X + y +

A

(35c)

AMj f:

-A

. M, \ /

(35d)

w h e r e V i s t h e m o l e c u l a r v o l u m e , c t h e a v e r a g e s o u n d v e l o c i t y ; My^, Mg a r e the average masses o f the atoms, _ M

=

xMy^ + yMg + x + y +

f j is the isotopic abundance of the i t h isotope, and

(35e) AM.

between the mass o f the i t h isotope and the average mass,

is the difference

15

c.

Phonon-phonon scattering

Herring (25) has investigated acous­

t i c mode three phonon scattering o f crystal structures.

long wavelength phonons

for different

He found f o r face centered cubic c r y s t a l s a t

low

temperatures

T" ' where B is a constant.

=

,

(36)

Callaway (10) used this expression for normal

processes and for Umklapp process, he wrote Tu"'

=

where By and a are constants, have a value of about 2.

ByGxp ( - e / a T )

(37)

and 0 i s the Debye temperature.

At high temperatures Klemens (32) Ty-'

where B'

,

=

a should

finds

B'afl

(38)

is a constant.

Pomeranchuk (44). has suggested the p o s s i b i l i t y of four-phonon scat__ 2 t e r i n g 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 a l k a l i halide compounds

i n terms o f o p t i c mode scattering o f the form: two acous­

t i c mode phonons create one o p t i c mode phonon.

Steigmeier and Kudman (48)

have also suggested o p t i c mode s c a t t e r i n g t o explain t h e i r data on l l l - V compounds.

d.

Other relaxation times

Several other relaxation times have

been used to explain various anomalies that have appeared i n 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 '

to explain dips quency, fi

in thermal

conductivity data,

describes the damping, p

=

m

is the resonant fre­

0 has been used f o r dips a t tem­

peratures below the thermal conductivity maximum and p used at

(3g)

temperatures above the maximum.

=

2 has been

Wagner (51) has derived

the basis of s c a t t e r i n g caused by localized modes introduced by

on impuri­

ties. Bound donor electron - phonon scattering of the form -1

^

Gco^A^ [(haii)2 - ( A ) 2 j 2

G

has been used by Keyes

=

(30,

Pearlman (22) and Holland (28)

1^1 + ( r ^ W / ^ c ^ j j 8

^"^2 7 ( ~ ~ f 34Kp2c/ ^ A y

(40)

Griffin and Carruthers

( 2 3 ) , G o f f and

to explain thermal conductivity data on

semiconductors at low temperatures that is considerably lower than the theory of boundary scattering would predict. derived by G r i f f i n and Carruthers;

A

Equation 40 is the form

is the splitting of the ground state

of the donor level and r^ is the effective electron radius. "ex i s the number of unionized uncompensated donors, p i s 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, of phonons by degenerate electrons i n a parabolic band.

for the scattering

17

-1

_

_r_

X

]

^îTh^p T5 (1 - e " * ) where C measures the strength of

X

(41)

the interaction, m is the effective mass,

c i s the sound v e l o c i t y , p i s the mass density, kT^ a complicated function of T, T^, x and the Fermi r u t h e r s £ t _aj_. ( 1 1 , 1 2 ) h a v e u s e d thermal conductivity at

'

=

1/2 mc^ and w i s

temperature.

J. A. Car-

to qualitatively explain a reduced

very low temperatures i n p - t y p e Ge and S i .

B.

Seebeck Effect

The theory of the Seebeck effect i n semiconductors has been dis­ cussed by Herring (26) and by Johnson.

(30)

The diffusion Seebeck coef­

f i c i e n t , S o f an n-type sample can be obtained from Equation 2 with J

=

0.

The resu1t i s k

rAet

Ef

^ kT

kT

or k Sd

Aet L kT

e

nh In

3

2 (2rtmekT)3/2

(42)

where A c t The quantity

=

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

is the average energy of the transported electrons

ative t o the band edge.

For a p-type semiconductor

of Equation 42 is replaced by a + sign.

the ~ sign in front

For mixed conduction, we require

t h a t the sum o f Equation 2 and the equivalent equation f o r holes the result is

rel­

is zero;

18

C Ast Sd

=

-

where '

L kT

k e

r_flLl '

2(2amekT)3/2j

ph'

kT

"

1

2(2nmhkT) y

PW-h

indicates holes.

I f we l e t b 1

Sd

e

r

nb + p

=

ph n

kT

There is an additional by the scattering of

kT

3

2(2jtm,^kT) 3/2

O nh-'

Aet - nb

Equation 4 ] becomes

r A £' t

p

(43)

"

" 2(2%m_kT)3/2JJ

(44)

component t o the Seebeck coefficient caused

the charge carriers by the phonons.

of a temperature gradient this scattering is isotropic.

in the absence 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 i n semiconductors.

He found t h a t the

phonon-drag Seebeck coefficient is given by Sp

where c(q)

=

+ ^

c(q) 2 f(q)T(q)/pJ .

i s the sound v e l o c i t y o f mode q, T(q)

time f o r mode q, f ( q )

is the phonon relaxation

i s the f r a c t i o n o f crystal momentum given up by the

c a r r i e r s t o the acoustic mode q,

i s the mobility due to phonon scat­

tering and T is the absolute temperature.

f(q)

is a strongly peaked func­

t i o n for q of the order of the thermal electron wave vector. only low energy phonons are important i n phonon drag. electrons and the + sign is for holes.

Therefore,

The - sign is for

The summation i s carried over a l l

19

acoustic modes.

The phonon wave number q i n Equation 4$

is an average

q o f t h e o r d e r o f t h e wave number o f a c a r r i e r w i t h energy i - ill.

L/C where L i s a characteristic size a

T^^^

(49)

20

111.

EXPERIMENTAL PROCEDURE

A.

I.

Samples

Growth The samples used i n this experiment were grown by a modified Bridgman

technique. by Morris

The method i s a variation by Grossman' e_t

(39)

to grow Mg2Si.

of the method described

in Grossman's method,

passed slowly through the melting point twice.

the melt

is

i t is thought that the

f i r s t pass forms the compound, and the second pass forms the single cry­ stal.

The method would also provide a zone-refining e f f e c t . All

three crystals were grown from Sn of 99.999 % p u r i t y obtained

from Vulcan Materials Gompany.

Crystals SBI69 and SB202 were grown from

99-995 % p u r i t y Mg o b t a i n e d f r o m t h e Dow Chemical Gompany.

Crystal SB239

was grown from Dow Chemical Mg t h a t had been vacuum d i s t i l l e d i n t h i s laboratory. High purity samples are needed for thermal conductivity measurements so that the i n t r i n s i c properties of the material are not masked by the impur i t l e s .

2.

Shapi ng The thermal conductivity samples were cut from the crystal

wire saw.

with a

An abrasive s l u r r y of #600 SIC g r i t suspended i n kerosene was

applied t o the wire o f the saw t o serve as the cutting agent.

After the

saw cuts were made the sample surfaces were trued by lapping.

The samples

'crossman, Leon D., communication. 1966.

Iowa State University, Ames,

Iowa.

Private

21

w e r e r e c t a n g u l a r p a r a l l e l e p i p e d s a p p r o x i m a t e l y 3 mm b y 3 mm b y 2 cm. samples were then stored i n small

3.

bottles containing a desiccant.

Character!zati on Small pieces of the original

lyzed.

crystals were spectroscopical 1 y ana­

The spectroscopic results are given i n Table 2.

The electrical

resistivity and Hall

coefficient were measured from

77 to 300° K by a standard 5 probe technique. cal

The

r e s i s t i v i t y , and Figure 3 shows the Hall

conductivity samples.

Figure 2 shows the e l e c t r i ­ coefficient for the thermal

Figure 4 shows the results o f e l e c t r i c a l

resistiv­

i t y and Hal I c o e f f i c i e n t measurements made on samples K-11 and K-13 a t lower temperatures.

Sample K-3 was accidentally broken before e l e c t r i c a l

measurements were made. For a saturated extrinsic semiconductor,

the Hall

coefficient, R,

is

gi ven by o

_ -

-

8

1 (Ng - N^je '

(50)

w h e r e N q i s t h e n u m b e r o f d o n o r s p e r u n i t v o l u m e , Ny;^ i s t h e n u m b e r o f acceptors per unit volume and e is the magnitude of the electronic charge. Equation 50 was used t o calculate Nq samples; the results are given in Table 1.

for the thermal conductivity Table 1 also contains

the

physical dimensions of the thermal conductivity samples. Back r e f l e c t i o n Laue X-ray patterns were made a t a number o f points along each sample.

A l l o f the patterns showed the same o r i e n t a t i o n ,

which indicates that the samples were single crystals. grain boundaries visible.

There were no

22

°K

1.0

300

200

150

100

80

0.5

^0.2

8

u I

o^

8

o'o

X

£ 0. - f l j

-

• °

QS>

(L

0 4) 0 . 0 5 l-o

• K-ll O K-13 A K-15

0.02

0.01

7

8

9

10

II

12

13

1000

Figure 2.

Electrical r e s i s t i v i t y , p, of the thermal conductivity samples

23

3 300 10

200

80

100

150

500 O

O a A

O

°

A/0

•

200 — g 10^

_j 3 O O \ m 5 O

cP

50 • O

20

g

E

•

10

9

o K-13 A

o

Figure 3.

K-ll

K - 15

7

8

1000

(°K)

10

11

12

13

Hall c o e f f i c i e n t , R, o f the thermal conductivity samples

24

20

100 50

'".o"

• XI

• 10'

o(A]

o

0 o o

c 10

10

2

C

• l.Ok 2 0 1 5 % O X 0.1

0.01

10

• cP

•

Q

20

30

40

• K-ll O K-13

50

I 60

I 70

I 80

L 90

100

(°K-I)

Figure

4.

H a i l c o e f f i c i e n t , R, and e l e c t r i c a l r e s i s t i v i t y , p, o f Mg2Sn s a m p l e s K - 1I a n d K - 1 3 a t l o w t e m p e r a t u r e s . ( B o t h s a m p l e s w e r e n-type)

25

Table 1.

dimens i ons mmxmmxmm

Spect roscopi c ResuIts^

ND numbe r / cm-^

K-3b

5.856 X 5.394 X

16

Ca Cu Fe Si Y

K-Iic

2.706 X

4.405 X

20

Al - T.

2.5 X l o ' G

1 VuU 0

Sample

Sample characteristics

1 . 898 X 4. 368 X 19

Ca - T .

1 . 8 X L O ^G

K-13BC

2.076 X 1.866 X 19

Cr Cu Fe Si V

-

F. T. F. T. T. T. T.

1 .8 X

K-15^

4.298 X 4.242 X 20

Ca Cr Fe Si

-

T. F. T. F. T. T.

2.5 X lO^ G

^V. W. = very weak, T. = t r a c e , were not detected. '^Cut from c r y s t a l

-

F. T. T . - V. W. T. T. F. T.

F. T, = faint trace, other elements

SBI69.

^Cut from crystal 58239. *^Cut from c r y s t a l SB202.

B. 1.

Measurements

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 long cylindrical

T is the gradient of the temperature.

In the case of a

sample with longitudinal heat flow and no heat losses,

Equation 51 becomes

A T a w h e r e Q. i s t h e e n e r g y p e r u n i t

=

- K A

L

(52)

time put into one end.

sectional area, L is the length of

=

A is the cross-

the sample and A T is the temperature

difference between the ends of the sample.

K

,

Therefore,

- i -AA A T '

(53)

The experimental determination o f K waa based on Equation 53. The Seebeck coefficient of the thermal

conductivity sample can be

determined by a simple modification of the experiment.

i f the thermal

conductivity sample is of material a and i f wires of material b are a t ­ tached t o each end of the sample, f . , E,

t o appear a t the free ends of the wires. E

where

then the Seebeck effect causes an e . m.

=

E is given by

Sab A T ,

i s the Seebeck coefficient of a with respect t o b.

(54) The absolute

Seebeck coefficient of a is given by Sa

=

Sab - Sb ,

(55)

where S^ is the absolute Seebeck coefficient o f b.

2.

Sample holder and sample mount i nq The thermal conductivity measurements were made w i t h the sample holder

shown i n Figure 5 mounted i n a l i q u i d helium cryostat of standard design. The sample holder was designed t o allow operation w i t h the sample holder immersed in the l i q u i d helium or nitrogen bath so that the thermal

re-

27

TRANSFER GAS TO VACUUM PUMPS STAINLESS —= STEEL TUBING COPPER

WOODS METAL VACUUM SEAL

STAINLESS STEEL TUBING

VARIABLE HEAT LEAK CHAMBER AMBIENT HEATER SAMPLE LEADS

COPPER — HEAT SINK

SAMPLE CLAMP COPPER COVER "

T- D

THERMOMETER CLAMPS GRADIENT HEATER

I

Figure 5.

Sample holder

1

28

s i stance between the sample and the bath would be small. transfer gas could be added or removed from the heat the thermal

resistance to the bath.

In addition,

leak chamber t o vary

The sample holder cover was soldered

to the sample holder with Wood's metal.

When i n operation,

the sample

holder was evacuated t o a pressure b e t t e r than 5 x 10"^ T o r r .

Heat con­

duction t o the sample was minimized by using small diameter (0.005 i n . o r less) wires and by thermally anchoring the wires to the copper heat sink. A 40 ohm heater wound on the heat s i n k was used t o o b t a i n ambient

temper­

atures above the bath temperature. The sample was connected t o the sample holder w i t h the sample heat sink clamp shown i n Figure 6A.

The thermocouples and the germanium re­

sistance thermometer were attached to the sample with the thermometer clamps shown i n Figure 6B, copper tabs.

The thermocouples were soldered t o the small

The nylon screws and phosphor bronze springs were used t o

compensate for the different thermal expansion coefficients of the sample and the clamp parts. clamps until

Good thermal contact was achieved by tightening the

the indium pad or knife edges deformed.

The copper tab on

the thermometer clamp nearest the gradient heater was e l e c t r i c a l l y insu­ lated by a layer of polyester f i l m tape so that a differential

thermo­

couple could be used t o measure the temperature difference. The temperature gradient i n the sample was established by a ten ohm heater wound on the free end o f the sample. The constantan wire of the thermocouple attached to the clamp on the "cold" end of the sample and a constantan wire attached t o 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 s i n k c l 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 c o e f f i c i e n t o f Cusack and Kendall

(16) and Borelius

(8)

Cu as t a b u l a t e d by

from t h e s e n s i t i v i t y o f Cu versus

constantan thermocouples as given in the tables published by Powell et a l . (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 In addition,

the ambient sample temperature.

the wide range of temperatures to be covered i n 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 t o the copper tab on the cold end of the sample.

The resistance o f the thermometer was measured by a 4-wire tech­

nique with the measuring current adjusted t o maintain a power dissipation below one microwatt

in the resistor.

The resistance thermometer was c a l i ­

brated by means o f a comparison o f i t s resistance w i t h the resistance o f 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 s i l v e r "normal" (Ag + 0.37 a t . % Au)

versus gold-

iron (Au + 0.03 a t . % Fe) versus s i l v e r "normal" d i f f e r e n t i a l thermocouple. One j u n c t i o n o f the thermocouple was soldered t o t h e copper tab on the " c o l d " end sample clamp and the other junction was soldered t o the e l e c t r i -

31

ca11 y insulated copper tab on the "hot" end clamp. been d e s c r i b e d by German ejL 2 1 -

(3)

This Lhermocouple has

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,

i t does not have the high thermal conductivity maximum a t low

temperatures that copper does.

The d i f f e r e n t i a l

measured with a Keithley model

I 48 nanovoltmeter.

thermocouple voltage was

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 o f the two thermo­ couple readings, and the temperature difference was determined by the difference between the two readings.

A calibration of silver "normal"

v e r s u s c o n s t a n t a n t h e r m o c o u p l e s h a s b e e n p u b l i s h e d b y P o w e l l e _ t a_l_. The e. m.

f . at liquid nitrogen temperature of the thermocouples used in

t h i s e x p e r i m e n t d i f f e r e d b y a b o u t 8 0 i^V f r o m t h e p u b l i s h e d v a l u e . e. m.

(45)

A new

f , , E, versus temperature, T, curve between 77 and 300° K was pre­

pared by f i t t i n g the equation E(T)

=

A(T - 273) + B(T - 273)^ + C(T - 273)^ ,

to the measured e. m. f . value at

(56)

liquid nitrogen temperature and the

d E / d T a t 2 7 3 ° K a n d t h e e . m . f . a t 3 0 0 ° K o f P o w e l l ^ _a]_.

(45)

A thermo­

couple table giving the e . m. f . and dE/dT a t 1° K i n t e r v a l s was calcu­ lated by evaluating Equation 56 and i t s f i r s t 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 i s shown i n Figure 7.

The

Leeds and N o r t h o u p 1 0),

they found K

.i4 1 / 3 5 \h/

7 T

(15)

SAMPLE SAMPLE SAMPLE SAMPLE

CO CO w

K-3 K-ll K-13 K-I5

THEORY UJ

100

120

140

160

180

200

220

240

260

200

500

TEMPERATURE ("K) Figure 10.

The thermal resistance o f Mg2Sn above 100^ K i s approximately l i n e a r i n T. (Tlie s o l i d I i n shows the thermal resistance calculated from the theory of Leibfricd and Sclil oc-.vr:)

45

M i s L l i r m e a n a L o i n i c i i u i b s , l-> t h e Deb yI ' t e m p e r a t u r e a n d

is

Line

c u b e r o o L o f L i i c - i l f j i n i c vo I u 'r , '• i s

i s the Grueneisen anharmon i c i l y p-i r-

7

Here P re|iresents the Lhree acoustic modes, we f i t Equation 15 t o the data w i t h find, with

7

=

1.4,

7

so we Find ' '

=

7'.. '

K.

11

as an adjustable paramrJcr, we

the solid line given in Figure 10.

between thr theory and L he experiment

jLer.

is satisfactory.

sistivity is linear in T for T > 120° K.

The agreement T h e Ih e r n i a I r e ­

In other words,

the T~'

depen­

dence of tlie thermal conductivity continues for a considerable distance belP.

3.

Holland (27) his pointed out a similar result

f o r Ge and S I .

Low temperatu re l a t t i c e therma1 conduct i v i t y I n tlie range 4 t o 80^ K, the theory developed by Callaway (10) w i l l

be used to interpret

the measured thermal conductivity.

k

fkTl3

re/T

^ h/ where x

=

Tico/kl and

ciprocal addition of

Callaway found

(e^ - 1)^

'

(68)

is a combined relaxation time found by the re­ the relaxation times for the resistive processes

plus the relaxation time for the phonon-phonon normal process.

The f i r s t

term o f Equation 33 i s given by Equation 68; the second term i s a cor­ rection to the theory that takes into account the effect of the normal processes (non-resistive processes). small and can be neglected.

Usually,

the correction term is

i t w i l l be evaluated to check i t s magnitude.

I n Equation 68, 0 i s the c h a r a c t e r i s t i c temperature o f one Debye mode which represents an average acoustic mode. Ti

0 is given by

^6 i t ^

® where a is the l a t t i c e constant.

(69) For Mg2Sn 0- =

154° K.

46

Holland (27) crystal

indicates that the average sound velocity, c,

in the

is found from c"'

which gives c

=

=

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

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|~'

=

Equation 35; Umklapp phonon-phonon processes, and normal

(70)

phonon-phonon processes,

— Î

=

Ao)^, w i t h A g i v e n by '

=

2 1

B j^CD T ^ .

Byexp (-0/aT) Tg is effective

from very low temperatures t o j u s t above the thermal conductivity maximum. T|

i s effective i n the region near the maximum.

cesses become important i n the region o f and for a l l higher temperatures.

The phonon-phonon pro­

the thermal

conductivity maximum

However, the phonon-phonon relaxation

times given above are correct only f o r low energy phonons so we can expect the theory t o break down a t high temperatures. Equation 68 w i l l be numerically evaluated with T^"'

=

c / L + Am^ + (Byexp ( - e / a T ) + B|^)

L is given by the Casimir (l4)

theory, L

the sample cross-sectional area. a are adjustable parameters.

=

IsC^ ^ ,2)

.

(y,)

where I ^ 1 ^ is

A is given by Equation 35.

By, B^ and

The value of a i s approximately 2.

11 shows the r e s u l t o f the c a l c u l a t i o n f o r the four samples. l i s t s t h e v a l u e s o f L , A , B y , Bj^ a n d a t h a t w e r e f o u n d .

Figure

Table 2

L and A are

values calculated from the theory. The correction term given in Equation 33 was evaluated for the case of sample K - l l .

The result i s given i n Figure 12.

The correction term

47

10.0

-

0 LU

5.0

h- d 1 5 O >-

> H o 3 û Z O

2.0 -


b S z

o o

< £C ^

0 K-13 L= 3.1 mm K-I3B L=Z.2mm

.5

WITH ELECTRONPHONON SCATTERING

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 t h e o r e t i c a l sizes were used i n the c a l c u l a t i o n . (The thermal conductivity calculated for these samples without electron phonon scattering i s also shown)

57

10.0

^ 5.0 d UJ Û I 2 O œ

I-

g 2.0 >-

H > O n Q Z

.0 —

o

o K-11 L=3.9 mm

o

Ù K-15 L = 4.8mm i 0-5

WITH ELECTRON PHONON SCATTERING

LU X

& 0.3

10

20

50

-A

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 o f the interaction was increased from i t s value f o r sample K-13 by the ratio of the uncompensated donors)

58

r e s u l t i s E^j

=

and f o r Ge, Ey

10.3 eV, which seems reasonable. =

For Si,

=

I I eV,

19 eV.

The results for r^,

A and E^ given above should not be interpreted

as the correct values needed t o describe the donor states i n Mg2Sn. are given only as an indication that

They

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 s c a t t e r i n g between the donor ground s t a t e

and the donor excited states.

The p o s s i b i l i t y e x i s t s t h a t some o f the

donor's excited levels are f i l l e d by thermal e x c i t a t i o n and we could have scattering between this level and other donor levels.

in addition, scat­

tering could take place between the f i l l e d donor level and the conduction band.

The relaxation times for these processes are not known.

and Carruthers (23) have argued t h a t i n Ge t h e i r e f f e c t the scattering between the ground state levels.

Griffin

is smaller than

Even i f the relaxation

times were known t h e i r addition t o 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 s t i l l be proportional t o the number o f uncompensa­ ted donors, n^^.

We have d e m o n s t r a t e d t h a t t h e a d d i t i o n a l

relaxation

time needed to account for the size dependence of these samples i s propor­ tional

t o Hgx-

59

C.

Seebeck Coefficient Results and Discussion

Tine Seebeck c o e f f i c i e n t s o f tine 4 n - t y p e samples K-1 1 , K - 1 3 , K-13B and K - ] 5 were measured a t the same time as

the thermal conductivity.

The impurity concentration i n a l l o f the samples was approximately the same.

I n order t o make a complete analysis o f the Seebeck c o e f f i c i e n t ,

a number of different

impurity concentrations should be measured, p-type

samples should be measured, and the measurements should be extended t o higher temperatures so that the samples become completely i n t r i n s i c . The analysis of the Seebeck c o e f f i c i e n t o f Mg2Sn i s complicated by the f a c t t h a t mixed conduction sets i n a t approximately the same temperature that the phonon drag contribution dies out. The absolute Seebeck coefficients o f the four samples i s shown i n Figure l 6 for the temperature range where the diffusion term i s dominant. From l i q u i d 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,

i t should have a higher diffusion Seebeck coef­

ficient. From Equation 42 we f i n d that i n the e x t r i n s i c region the d i f f u s i o n Seebeck coefficient, S^, can be expressed as k e

An - I n ( 7 h 3 / 2 e |R| (2amkT) 3 / 2 ) 1

where 7 |R|

and

ne

,

(74)

1000 o

800

S Q >

SAMPLE

K-ll

• SAMPLE

K-13

V A

K- 13 B K- 15

SAMPLE SAMPLE

600

3 O—o

200

0

100

200

300

TEMPERATURE (*K)

Figure 16.

The absolute Seebeck coefficient f o r the d i f f u s i o n range i s shown. (The curves represent the diffusion Seebeck coefficient calculated from Equation 74 and 75)

6l

Ae t

R is the Hall

=

iTig

coefficient and 7 i s a number which depends upon the scat­

tering mechanisms. 7

3

For acoustic mode scattering and small magnetic f i e l d s

%^/8. In the i n t r i n s i c range, S i s given by

Sy

=

^

I^P { A p - I n ( p h 3 / 2 ( 2 # m k T ) 3 / 2 ^ j

j

- nb | A n - I n (nh^/Z (2TtmkT)

,

(75)

where Aet'

"p'

3

~

mh

IT

and Aet

3

~ ^ Ï

mg

T •

n and p are respectively the electron and hole densities. In the extrinsic range,

the diffusion Seebeck coefficient w i l l be

calculated from Equation 74 with the value of calculated value f i t the experimental

adjusted t o make the

data at 150° K.

in the mixed con­

duction range the diffusion Seebeck coefficient can be calculated from Equation 75-

In this

region i t

coefficient.

The two equations n

is necessary to find n and p from the Hall

=

P + "ex ,

and

0 R

with b

=

=

7

p - b n

0 (p + bn)2

(76)

1.23 were solved t o find n and p as a function of temperature.

62

Since we are assuming that the mobility r a t i o , b,

is independent of tem­

perature we have 3

"p

mh

=

Api h a s b e e n t r e a t e d a s a n a d j u s t a b l e p a r a m e t e r .

The density of states

e f f e c t i v e masses r e p o r t e d by Lipson and Kahan (37) nig m^

=

1.3 m, were used t o calculate Ap from A^,

16 are the results of the calculation. and Ap that were found.

=

1.2 m and

The curves i n Figure

Table 3 gives the parameters Ap

The d i f f u s i o n Seebeck c o e f f i c i e n t was not calcu­

lated i n 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

K-13

-1.03

K-15

1.95

2.10

2.02

2.30

2.20

^For Equation 74. ^For Equation 75.

For acoustic mode scattering, for other types of scattering. range that

As ^./kT

=

Ae^/kT

With m^

=

=

2.0, and

Ae^/kT > 2.0

1.2 m we f i n d i n the e x t r i n s i c

1.54 and 1.87 for samples K - l l and K-15.

An ef­

f e c t i v e mass less than m i s required t o 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 t o check t h i s r e s u l t . Figure 17 shows the phonon drag Seebeck c o e f f i c i e n t 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. small

Above, 100° K,

t o be accurately determined.

a T~" temperature dependence. n

=

3.5.

Heller (24) n

=

the phonon drag contribution i s too The theory of Herring (26) predicts

For longitudinal acoustic mode s c a t t e r i n g

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

=

3.0 for Mg2Si.

Geballe and Hull

(20)

found

2 . 4 f o r n - t y p e Ge and 2.3 f o r n - t y p e S i . The maximum o f the phonon drag Seebeck c o e f f i c i e n t occurs near 18° K

while the thermal conductivity maximum occurs a t 13° K .

This result

demonstrates that different phonons are involved i n the two phenomena. The phonons which contribute to the phonon-drag have wavevectors of the same s i z e as the wavevectors o f thermal electrons.

Therefore, at 10° K

the phonons that contribute t o the phonon-drag have m

6 x lo"^ sec~'.

The phonons which contribute to the thermal conductivity have energies on the order of 3 kT so that their angular frequency i s near 4 x 10 sec"' at 10° K.

]9

Impurities, such as isotopes, scatter high frequency

phonons more effectively than low frequency phonons so impurity scattering does not make a s i g n i f i c a n t contribution t o 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 Seebeck coefficient of Mg2Si.

in the low temperature

The t o t a l Seebeck c o e f f i c i e n t a t

atures i s shown i n Figure 18 f o r the Mg2Sn samples.

low temper­

Sample K-13B i s sam-

64

10

o 1x1 O > 3 SAMPLE K-ll SAMPLE K-13 SAMPLE K-I3B SAMPLE K-15

10

AO

10

20

50

100

200

500

TEMPERATURE C K )

Figure 17.

The phonon drag Seebeck coefficient i s shown. 100° K the temperature dependence is T'^.S)

(From 30 t o

5x 10

O

:

y 2x10 Z3

O D V A

(f)

SAMPLE SAMPLE SAMPLE SAMPLE

K - II K-13 K-I3B K-15

5x10 10

20

50

TEMPERATURE (°l