The electron specific heat of a two-dimensional crystal

Physica 133B (1985) 181-195 North-Holland, Amsterdam

THE ELECTRON

SPECIFIC HEAT OF A TWO-DIMENSIONAL

CRYSTAL*

Fang-Xiao ZHAO* and J.O. LAWSON The University

of Texas at El Paso, El Paso, Texas 79968, USA

Received 18 April 1985 Revised 17 June 1985

The electronic specific heat of a two-dimensional, square lattice crystal is determined by considering only electron-ion core interactions represented by a sinusoidal-like potential. Two electrons per lattice site are assumed to be under the influence of the potential. The Green’s function equation of motion technique is employed in the second quantized formulation, with the approximation that the allowed linear momenta are restricted to two Brillouin zones. The result shows a specific heat variation with temperature, characteristic of an intrinsic semi-conductor or metal, dependent upon the well-depth of the potential.

1. Introduction

The electron specific heat of a one-dimensional crystal with a sinusoidal potential was recently obtained [l] by one of us (J.O.L.). Two electrons per lattice site were assumed to be under the influence of the potential; electron-electron interactions were ignored. The allowed linear momenta were restricted to the first two Brillouin zones. The specific heat indicated metal, semiconductor to metal or insulator behaviour as a function of temperature, dependent upon the well depth of the potential. In this paper, the electronic specific heat for a two-dimensional, square lattice crystal will be obtained. The Hamiltonian is modified from the previous paper [l] to reflect the one-dimension to two-dimension change. As before, the wave vectors associated with the electronic motions are restricted to within the first two Brillouin zones, excluding the edges of the second zone bordering the other higher zones. Two electrons from each atom at each lattice site are allowed to take part in the conduction so that only the first Brillouin zone is filled at the absolute zero of temperature. Numerical methods were used to calculate the specific heat in the temperature range from 10 to 50000 K and under various potential well-depths. (Temperatures above 5 000 K have little meaning, of course.) As will be seen, the time correlation functions like (B(t')A(t)) and their associated Green’s functions form the core of the calculation herein contained. The basic theory [l, 21 has been extensively reported previously so that it is now well known. The theory is now applied to the two-dimensional model.

2. The two-dimensional

Hamiltonian

The model of a Hamiltonian including a sinusoidal potential in the one-dimensional cases [l, 3,4] proved to be satisfactory to some extent. In this paper, a Hamiltonian needs to be introduced on the same basis with a sinusoidal-like potential. The proposed potential energy is the product of the space functions in * Much of the present work is from F.-X.Z.‘s master’s thesis of the same title as this paper El Paso (May 1984) in partial fulfillment of M.Sc. degree requirements.

0378-4363/85/$03.30 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

presented

to the University

of Texas at

182

F.-X. Zhao and J.O. Lawson

/ Electron specijk

heat of a 20 cryslal

the one-dimensional potentials [l, 3, 41 such that it reduces to the one-dimensional sinusoidal potential along any line containing the lattice points which are a distance, a, apart. The Hamiltonian to be used for one particle is, in the first quantized form,

(1) where U is the well-depth at any lattice site, and a is the nearest distance between any two lattice sites, The coordinate system is chosen in such a way that the origin sits at an arbitrary lattice site; the x- and y-axes are along two perpendicular lines of lattice mentioned before (see fig. 1). Equipotential surfaces (lines in this two-dimensional case) are shown in fig. 2. Near the lattice sites, one observes that the equipotentials are circular and approach a square halfway between lattice sites. As was previously pointed out, the potential in eq. (1) is a first order approximation to the real one. However, the resemblance is quite satisfactory in the sense that the potential in eq. (1) has its lowest value (-U) at every lattice site and has zero value in between these lattice sites, i.e., when either x or y is equal to (n + ~)Lz,where n is an arbitrary integer of either sign. Thus, for U sufficiently large, the proposed potential can approximate the strong Coulomb attraction from the lattice sites. The Hamiltonian (for all electrons) may now be expressed in the second quantized form. One assumes that the matter field is quantized in plane-wave form as [l]

WI =

&

C (aK,oeiK’W),

(2)

K r

where A is the sample area, K = XX + jrC, is the two-dimensional wave vector (or the linear momentum vector if h is taken to be unity), r = ix + jy is the position vector, u is the spin state and LZ~,~is the annihilation operator of an electron in the state (K, v). The many particle Hamiltonian for stationary lattice

.

.

.

.

.

.

.

.

*

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

,

.

.

.

.

.

.

.

L

.

.

.

.

7

.

.

.

-

.

Fig. 1. Two-dimensional

a

.

.

.. .

.

.

.

square lattice.

Fig. 2. Equipotentials.

F.-X. Zhao and J.O. Lawson / Electron specific heat of a 20 crystal

sites or stationary ion cores is obtained in the second quantization X =

formulation

183

[5] to be (in this example)

dA$*(r)h$(r)

I A

>(

(3)

where akm is the creation operator of an electron in the state (K, a), and the chemical potential /.L is introduced in accordance with a grand canonical ensemble. Substituting the expression

2Tx ’ a = i

cos

(e i2m/a+ emi21rxln)

into eq. (3), one finds X =

C a+,.,,.a,, K’.K,d.(r -- :

C

g

K2 - T-

/+(K’,

K)S(a’, a)

a:,,~,aK,~(S(K’,K+i~)+S(K’,K-i~)

K’.K,u’,(r +S(K’,K+j?)+S(K’,K-jT))6(d,u) -3 K’,K, d,

a~,,,.aK,u(S(K’,K+~(i+j))+6(K’,K+$(i-j)) CT

+6(K’,K-2f(i+j))+S(K’,K-~(i-j)))S(~’,~).

(4)

In deriving eq. (4), use has been made of the expression 1 z I e

i(K-K”rdA = S(K’, K), (u’)(a) = S(u), a)

;

A

here i?(K’, K) and a((~‘, a) are the Kronecker

delta functions [6].

By defining A, to be AK=;K2-+

one may write eq. (4) as

(5)

F.-X. Zhao and J.O. Lawson / Electron specific heat of u 20 crystal

184

+ ‘i,

uaK-Z?r/a(i-j),

D +

ak

caK+2n/a(i+j),

w+

ak

ma K+Zx/a(i-j),

Since A, is independent of 0; and spin-spin and spin-magnetic may drop the CTnotation and rewrite eq. (4) as

x

=

2

2

‘#kK

-

4

2 K

K

-f

C

(a~aK-i21,/a

(a~aK-2nL$-j)+

+

a:aK+i2r,n

akaK-2wh(i-j)S

+

n)

*

field interactions are not considered, one

a~aK-j2Th

a:aK+2?r/a(i+j)f

+

aiaK+j2wh)

aLaK+27da(i-j))

3

(6)

K

where u is to be understood.

3. The equations of motion and the Green’s functions solutions The Green’s function equation of motion technique

E((A 1W) = -;

is now used as before [l, 2,7,8] where one has

([A, Bl,) + (([A, HI_ I B))) .

(7)

The correlation functions, (a&aK1), and so the associated spectral intensities, ((a,, 1a&)), are now to be evaluated. In eq. (7), replacing A with aK, and B with a& leads to

where the Hamiltonian [a K,, [a

CT”

K,,u’,

(6) is to be utilized. One now has the need of the relations,

ai2,,l+= WI, K,FW’,~1, I+ =

aKz,o

laK,,

“KaK,]_

IaK,,

xl

=

=

A KlaK,

-g

(10)

a&,J+ = 0 y

[a&,,,

6(KlT

-

(9)

K)aKz

f

(aK,-2n,n(i+j)

(11)

?

(aK1-i2nja

+

+

aKl+i2do

a K1-2v,n(i-j)

+

’

where the same spin state is to be understood.

aK1-j2?rlo

a K,+Zdo(i+j)

+

aKl+j2da)

+

a K1+2nlnCi-d

’

Substitution of eqs. (9)-(12) into eq. (8) yields

(12)

F.-X. Zhao and J.O. Lawson / Electron speciJic heat of a 20 crystal

-

$

(((“K1-2n/n(i+j)

+((a

K1+*m/o(i+j)

I ‘LJ) + ((“Kl-*r/a(i-j) I ‘L_J)+ ((“K1+2r/a(i-j)

185

1‘LJ)

I ‘ii*))).

(13)

Because of its form, eq. (13) may be used to generate a system or chain of equations (by appropriate substitutions for K, and KJ which can be solved simultaneously, if one utilizes some approximation methods to reduce the resulting chain of equations to a finite set. The approximation to be used is that of a two-band model. The wave vectors are restricted to the first two Brillouin zones [l]. Implicit in this assumption is that energy band gaps occur at a Brillouin zone boundary. In the reciprocal space, one has the first two Brillouin zones as indicated in fig. 3 (see, for example Kittel [9] for Brillouin zone construction). One has for each zone the following: 1st Brillouin zone: -% U

K,,

Ky
2nd Brillouin zone:

Fig. 3. The first two Brillouin zones.

186

F.-X. Zhao and J.O. Lawson

27T --
-~-Kx
a

27r

--
a

2i?

“
a 277

E
a

/ Electron specijic heat of a 20 crystal

a

-g-Ky
a

-2+Kx
a

-2+Ky
a

Any Green’s functions with K outside of the first two Brillouin zones (including the edges of second zone bordering the other higher zones) will be neglected [l]. From fig. 3, one sees that problem will be more tractable if one makes a transformation to coordinates K: and Ki which rotated 45” counterclockwise with respect to the K, and KY axes. (The same thing is accomplished rotating the reciprocal space by 45” as seen in fig. 4.) One now has, i =

T$

(i’

_

j'),

j

=

$

In the primed coordinates, V\/271--
K;<-

(i + j’)

.

the first two Brillouin zones are conveniently

(14) expressed as

VC% a

the the are by

(15)

.

Fig. 4. The first two Brillouin zones in K’ space (via a 45” counterclockwise

rotation).

F.-X. Zhao and J.O. Lawson / Electron

By substituting Hamiltonian as

the new coordinates

and dropping

specificheat of a 20 crystal the primes afterwards,

187

one may rewrite the

X = 2 2 A@h;aK - y C (UhK-fir/,(i-j)+ UbK+tir,a(i-j) + ‘kK-~r,a(i+j)+UhK+ti?r/a(i+jJ K

K

-4

c

(“LuK-i2firla

+

u~“K+j2fir/o

’

uhK+i2X&ln

+

uhK-j2fis/a)

(16)

7

K

where the new coordinates are to be understood. The four terms under the last summation in eq. (16) (u&,_,,~,,,,, etc.) are seen to be all outside of the first two Brillouin zones. The terms under the second summation will survive in a certain quadrant of the first two Brillouin zones (e.g., ~~a,_,,,~~_~~ survives in the fourth quadrant: 0 < K, < d/Zn-/a, --d/27r/u < K,, < 0). Hence, in determining the average energy within the first two Brillouin zones only, no contributions will come from the last summation. Also, three out of four terms in the second summation would be eliminated in a certain quadrant, as

Cx) = 2 C

AK(~;‘K)

-

T

C

K

=

2

((“iuK-fir/n(i-j))

+

(‘~‘K+fir/aci-j))+

(“:uK-fiw/o(i+j))+

(‘fY’K+fiv/a(i+j)))

K

C

+zK)

(A,(

-

;

(a’

K

a

K

_

V%a~i+j$)

+

2

K++

2

(‘K(‘hK)

-

f(“hK+6r/o[i-j$)

K-+ u

+2

c

(‘K(

u:“K)

-

~(‘:‘K+6/n(i+j)

))

+

2 z_

(AK(u;uK)-

f

(“:a,fiv/a(i-j$)

3

(17)

K--

where the symbol CK++ means summing over the first quadrant of the first two Brillouin zones in the new coordinates (see fig. 4) and so on. With these arguments in mind, one then transforms the equation of motion (eq. (13)) to the new coordinates. One then has

(E - A ~,)((a~1

I a+,>>

=

-&

‘(KIY

K2)

+

((“Kl-V?/Z?,la(i+j)

-

i

(((“KI-fir/a(i-j)

I a;))

FE

(((“KI-iZ\/Tr/lo

+((a

Kl+i2fir/a

’

((‘K,+fir/a(i-j)

I ai>>

1uZ>)

((‘Kl+jN%r//a

((“Kl-j2fir/a

+

I a&)))

((“Kl+6r/a(i+j)

1‘LJ) +

I a&)) +

I a&>>

I ‘LJ))*

(18)

As will be shown, by constraining K, and K2 to be within the first two Brillouin zones, all the required Fourier components of Green’s functions can be obtained from eq. (18) due to symmetry considerations. Moreover, these Fourier components of Green’s functions may be paired off when solving the pertaining simultaneous equations, which makes the problem significantly more manageable. Because of eq. (17), one may treat the equation of motion (eq. (18)) within a suitable quadrant. In the process, one may omit all the Green’s functions which are outside the said first two Brillouin zones. In the first quadrant (K + +: 0 < K, < ~//ZTT/U, 0 =CKY < d2?r/u), for K, = K2 = K + +, one has tE

-

A K++)(@K++

I u:++))

=

-11277

-

;

((a,,,

_,&,aci+jj

1 a;++));

(19)

188

F.-X. Zhao and J.O. Lawson / Electron speciJic hear of a 20 crystal

for K, = K++ (E-A

- d?n/a(i

+ j), K2 = K+ f, one has

K++-V7?da(i+j))((aK++

I ‘L++)) = -f

-VTrr/a(i+j)

(20)

(ta,++ ) a;++>> .

One finds out immediately that these are a pair of solvable equations which yield the Fourier components of Green’s functions ((a,,, 1a;,,)) and ((cz~++_~~,~(~+~)1a;++)) in terms of E, AK++, A K++_fitinCi+jj and U The same situation exists for the other quadrants. In the second quadrant (K- +: -df!n-la < K, < 0, 0 < KY -C d\/2r/a), for K, = K2 = K- +, one has (E-

‘,-+)((a,-+

for K, = K-+ (E - ‘K-+

I a:-+>> = -‘I’m

+ ldr/a(i

- j), K2 = K-+,

+fi*/a(i-j))((“K-+

In the third quadrant

(K--:

+V%/a(i+j))((“K--

-fin/u


And in the fourth

+\lTw/a(i+j)

I


one has

a;-->> ;

(23)

one has I a;__>> = -~~(u~--

(K+-:

quadrant

(22)

(t”,_+ )uL_+)> .

~0, -l&n-la

-& - F ((a,--

+V%/o(i+j)

(21)

one has

+ Xhn-/u(i + j), K2 = K--,

(E - A,__

((a,-+ +an/o(i-j) I at-+>> ;

I u:-+))= -f

+fin/a(i-j)

(E- A K--)((uK--I a;--)) = for K, = K--

- f

0<

K, < ~/~IT/u,

(24)

1a:__>) .

-Xbrrl~

<

KY CO), for K, = K2 = K+

one

has

)a;+_))= -&- f ((a,+--fir/n(i-j) I a”,+-)) ;

(E - A K+-)((a,+-

for K, = K-t - - d\/2r/u(i

(E - ‘K+For brevity,

- j),

-\/Tnla(i-j))((“K+-

one defines

K2 = K+ -,

-VZdo(i-j)

(25)

one has

I u:+_>> = -;((a,+_ I u;+->,.

(26)

the following: 1a:):>>3

A, = A,,

A, = ((a,

A,=A K-V%/o(i-j)7

A2 =

A,=A K+V%a(i-j)7

A, = ((a,+aa(i-j)

I

A,=A

K-V%lo(i+j)

)

A, =

((aK-V%/a(i+j)

I

A,=A

K+V%/a(i+j)

7

A5

((“K+V%r/a(i+j)

=

((aK-V%h(i-j)

I (27)

F.-X. Zhao and J.O. Lawson / Electron specijc heat of a 20 crystal

189

In these notations, eqs. (19) through (26) can be expressed in one pair of equations:

(E-A,)A,=-&-;A,,

(28)

(E - AJA,

(29)

= -;

A,,

where n = 4, 3, 5 or 2 for the lst, 2nd, 3rd or 4th quadrant. Note that A, appears to vary from quadrant to quadrant in this formulation, but because of the symmetry and eq. (5) it must be dependent on the magnitude of K only. Now, substituting eq. (29) into eq. (28) one finds

1

U/8

(30)

A”=27T(E-A,)(E-~n)-(~j8)27 E-A,,

A&

(31)

2dE-A1)(E-A,)-(U/8)2’ With a little algebraic manipulation,

eqs. (30) and (31) can be written as (32)

(33) where Dn,, Dn2 are defined as on1 =

;(A, + A,)+

(;(A, - A$+

(~/8)2)1’2,

(34) (35)

The A,‘s are the required Fourier components

4. Evaluation

of the correlation

for the Green’s functions.

functions

The desired correlation functions in determining the average of energy (eq. (17) are to be found by the standard method [l, 2] to yield (from eqs. (27), (32)-(35)) Dn1-A” (&J

=

(a&,-V??r/a(i-

Dn2-4

eaDnl+ 1 - esDn2+ 1 ’ j)>=

-

u/8

(36)

(37)

190

F.-X. Zhao and J.O. Lawson / Electron specijic hear of a 20 crystal

(Gk

-

-\/Wa(i

+a

=

-D

““,

(u&

-&) ,

j-p&

41

42

+ d/2n-/u(i + j)) = -D

(40) 52

Attention must be given to which seems to assume different forms in different quadrants, i.e., n = 4, 3, 5 or 2 for the lst, 2nd, 3rd or 4th quadrant.

5. Determination

of the specific heat

One now substitutes eqs. (36)-(40) into eq. (17) to get the desired average of energy. One has

(x)=2;+(D f, (

41(041-

41

42

d4)+

v-@~,,)

(U’gY +

d,(D42-

1

31

‘,

(

32

+

A,@,,-

expW,,)+

(U/8)*

ev(PD,,)

-

A,P3,-A3)+WN2

+2K:+(D

d4)+

1

A3)+

1)

(UW2

evW3,)

-

1

+

1

))

A,(& - As)+(LrI8)*A,@,,- A,)+ U4’8)2

+2K&‘, ( A (Dz- 42) + ( U8)2 +2K&‘, ( 51

exp(PD,,)

52

+

-

1

ew(PD,,)

22

+

1

+

)I

1

A, (Dz2 - A,) + (US)’

r

21

exp(PD,,)

-

exp(PD,,)

+

11

1

(41)

’

From the definitions of the A’s and D’s (eqs. (5), (27), (34) and (35)), one may show that contributions from each quadrant in eq. (41) are equivalent, yielding the result for the average energy to be Al(D41-

cx,=8;+(D4,~D42(

A4)+

ev(PD,,)

(u/8)2 +

A,(D42-

1

-

A4)+

exp(PD,,)

(u/8J2 +

1

11

(42) ’

By assuming a large sample so the states are dense enough that the summations in eq. (42) may be replaced by integrations [5], one obtains v’m.

vmn

A (W=g8

A,(D42 - AJ + (u/8)2

dK, dKY A1(D42- 4) + (u/8)* I

I

0

0

041 -

(

042

ev(PD4J

+

1

-

evW42)

+

1

1

(43) ’

where A is the total area of the sample. The specific heat is v%lo fi?rlo dK, dKY 2&I I 041 - 042 c = (27r)*(K,T)* I x

(A dD41(

$

+

C ~f312P4,

cosh*($D,,)

(A AD42

-

-

44)

+

( W9*P42

cosh*( f j3D4*)

>’

(44

F.-X, Zhao and J.O. Lawson / Elecfmn specify hear

ofa 20

crystal

191

In order that numerical methods can be employed for integration of eq, (44), one needs to tailor the integrand. First of all, the chemical potential p plays a decisive role. For free electrons, it is merely the Fermi energy at the absolute zero temperature, i.e.,

h*K; P=&f=2m’

(T=O)

(45)

a circle in K space for the two-dimensional model. Here Kt is the associated Fermi momentum. For higher temperatures, the chemical potential is slightly less than ey and is a function of temperature. For this paper, one assumes that the chemical potential p is independent of temperature. Additionally, one notices from the definitions of D,,r and A,, and eqs. (36)-(40) for the pertinent correlation for low temperatures. This observation implies that for low functions that one has (uLuK) % (a&J,,, temperatures one has from eq. (6),

and that the electrons (from the definition AK) are “moving” in a well of depth U/4. Hence, one may assume h2K; ---

U

P=22m

4

to be the first order approximation to the actual chemical potential. At absolute zero, all states are filled to the Fermi level. Hence, the number of electrons one has is Kr

N’=yq

A

If Kr

dK,dK,=2=2r

A

I

A

KdK=gK:.

0

One now requires that the first Brillouin zone be filled at absolute zero. This imposition can be realized by assuming that the area of the Fermi circle is equivalent to the area of the first Brillouin zone (equivalent to equating the number of states in the Fermi circle to the number of states in the first Brillouin zone), i.e., 27 ?rK;=-.a or

One then has

2lr a

192

F.-X. Zhao and J.O. Lawson

By realizing A=

/ Electron specijic heat of a 20 crystal

that one also has

Na2,

where N is the total number of lattice sites, one immediately zone to be filled at absolute zero and the second zone empty, Ne=z

a2

finds that in order one must have

for the first Brillouin

=ZN,

or that the number of electrons taking part in conduction or transport or atoms, assuming one atom per lattice$te. By choosing a to be two angstroms (A), the Fermi energy,

is twice the number

h2K; ii2 4~

of lattice sites

(49)

Ff=-=-g-g’

2m is found to be 11.97 eV for electrons. Taking advantage of eqs. (46), (48), and (49), one may rewrite the integrand tractable form. Using the definitions for A,, A,, DA1, and Dd2; changing variables

and

and defining

new functions

to be

fr(x, Y) = & (A,(x, Y)+ A&T Y))

2

f

f2k

Y) = 2’; (A&,

Y)- A&,

Y)) 9

f3( U) = ul(~cf) 3 f&‘J

= c,l(KJ)

d&,

Y) = i ~&?x,

KY) 3

d&,

Y) = i D&G

KY) 9

DW,

Y) = $

>

f

(DJKx,

KY) - D&x

DNl(x,

y) = cosh(;PQJK,,

KY)),

~N2(x,

y) = cosh(;PD,,(K,,

KY)) ;

KY)) 7

in eq. (41) in a more to

F.-X. Zhao and J.O. Lawson / Electron speciJic heat of a 20 crystal

193

one may write the specific heat to be

(50) The integration is now executed on an electronic computer for various values of well-depth U and temperature T expressed as /3 = (K,T)-’ which are controlled by the functions f&U) and f&T). The Gaussian quadrature formula [lo] is employed.

6. Results and conclusion The results of the calculation are plotted in figs. 5-7 showing the specific heat vs. temperature for various values of well depth U. A typical free electron specific heat linear in temperature is shown in fig. 5 for a well-depth smaller than one electron-volt (about one tenth of the Fermi energy). Such a linearity is as expected, since under a very small well-depth, the present model should approximate the free electron model. The maximum specific heat shown in fig. 5(b) (and any subsequent figures) which occurs at about 36 000 K does not signify any phase change beyond what has been discussed in the literature [ll, 1, 121, for the simple reason that it occurs at too high a temperature to be practical. I

k

I

I

1.0

I

I

I

I

I

2.0

TEMPERATURE (lo3 K)

Fig. Sa. Temperature U = 1.197eV.

dependence

of the specific heat for

Fig. 5b. Temperature U = 1.197eV.

dependence

of the specific heat for

F.-X. Zhao and J.O. Lawson

194

I

1

I

/ Electron speciJic heat of a 20 crystal

I

I

I

I

I

‘Y

Y 5 k

2.cl-

ki1.0 l%

2.0

1.0

1.0

TEMPERATURE (lo3 K)

Fig. 6a. Temperature U = 11.97 eV.

dependence

of the

specific

heat

for

Fig. 6b. Temperature U = 11.97 eV.

I

2.0

1.0

1.0

dependence

of the

dependence

4.0

of the specific

I

I

2.0

3.0

heat

for

heat

for

I

4,o

TEMPERATURE (10' K)

TEMPERATURE (lo3 K)

Fig. 7a. Temperature U = 23.94 eV.

2.0 3.0 TERERATURE (lo4 K)

specific

heat

for

Fig. 7%. Temperature U = 23.94 eV.

dependence

of the specific

19.5

F.-X. Zhao and J.O. Lawson / Electron specific heat of a 2D crystal

In fig. 6, where one has the well-depth equal to the Fermi energy (which is about twelve electron-volts), possibly two or three free electron metal-like phases appear to be evident for a temperature below 10000 K, but these are not quite clear. Such could be effected by a change of effective mass in practice. Linear temperature dependence of the specific heat seems still to dominate. What best can be stated is that semiconductor behavior is becoming evident. Once again, the maximum specific heat occurring at about 36 000 K does not signify any phase change. Although phase transitions are not fully clear in fig. 6, semiconductor behavior is evident in fig. 7 for temperatures below 1000 K. A well-depth twice the Fermi energy (U = 24 eV) is presented in fig. 7. A smooth semiconductor-to-metal transition is then observed in the temperature range 600-!JOOK. There seems still to exist another phase transition at about 8000 K. After this point, the free electron metal feature of the specific heat again covers a significant temperature interval. Comparing what one now finds to the previous results obtained in the one-dimensional case, one sees that although the specific heats in the two cases assume a very similar formulation, as far as the integrands in the specific heat (see eq. (44)) are concerned, a quite large deviation is observed in the results. In the one-dimensional case, the phase transitions are obvious for relatively low values (-lo-* Fermi energy) for the well depth and continue for a considerable range of well depth values up to and including values of the order of the Fermi energy. In the two-dimensional case, phase transitions were not apparent until the well depth was of the order of the Fermi energy. Also, the phase transition manifests a unique pattern in the one-dimensional case, but several varieties appear in the two-dimensional case. These observations demonstrate that electron motions in two-dimensional crystals (which will have something to do with the surface properties, or thin film properties, of materials) is much less restricted than electron motions in one-dimensional crystals. (The probability of an electron being at or near a scattering center is considerably decreased.)

References [l] [2] [3] [4] [S] [6]

[7] [8] [9] [lo] [ll] [12]

J.O. Lawson, 11 Nuovo Cimento 1D (1982) 449. D.N. Zubarev, Soviet Physics Uspekhi 3 (1960) 320. A. Tungare, unpublished thesis, University of Texas at El Paso (1982). A. Tungare and J.O. Lawson, The paramagnetic susceptibility of a one-dimensional crystal, A.T.‘s thesis [3]. C. Kittel, Quantum Theory of Solids (John Wiley, New York, 1%3). L.I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968). J.O. Lawson, S.J. Brient and R.E. Bruce, II Nuovo Cimento B20 (1974) 225. R.C. Smith and J.O. Lawson, Physica 832A (1976) 505. C. Kittel, Introduction to Solid State Physics, 5th ed. (John Wiley, New York, 1976). Z. Kopal, Numerical Analysis, 2nd ed. (John Wiley, New York, l%l). R.A. Bari and T.A. Kaplan, Phys. Rev. B6 (1972) 4623. J.O. Lawson, Phys. Lett. 48A (1974) 361.

unpublished

paper

based

upon