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TRESIS

 

 

This is to certify that the

thesis entitled
THE EFFECT OF D—ELECTRON CORRELATION ON THE
MIXED-VALENCE PHASE 0F SAMARIUM SULFIDE

presented by

GLENN FLETCHER

has been accepted towards fulfillment
of the requirements for

PL} degreein PHYSICS

 

 

\D Malia/ICC.

Major professor

é; l2. 1981

 

0-7639

THE EFFECT OF D-ELECTRON CORRELATION ON THE

MIXED-VALENCE PHASE OF SAMARIUM SULFIDE

by

Glenn Fletcher

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics

1981

ABSTRACT

THE EFFECT OF D-ELECTRON CORRELATION ON THE
MIXED-VALENCE PHASE OF SAMARIUM SULFIDE

BY

Glenn Fletcher

We present the results of calculations made on the
two-band Hubbard model. Unlike previous investigators we
have included electron correlation in both bands. We have
used various approximation schemes to investigate the
ground state of the model, in which we calculated the
Green function by using the equation of motion decoupling
technique of Zubarev.

We have applied the two-band Hubbard model to the
problem of mixed-valent systems, in particular to SmS.
Schweitzer showed that the Falicov-Kimball model could
account for the first-order transition to a mixed-valence
phase observed in SmS. We find that including conduction-
electron correaltion as an intrasite interaction between
electrons of opposite Spin (Hubbard model) does not
qualitatively change the results of Schweitzer. The crit-
ical values of the parameters are changed in a way that
should give a better agreement of Schweitzer's results

with the experimental values.

DEDICATION

To Maurine, with love.

ii

 

ACKNOWLEDGEMENTS

I would like to thank Professor S.D. Mahanti for
the help and guidance he gave to me as my thesis advisor.
His encouragement was greatly appreciated and his skill
both as a researcher and as a teacher were an inspir-
ation to me.

I would also like to thank Professor Thomas A.
Kaplan for his constant help and particularly for his
role as my surrogate thesis advisor.

I am grateful to Professor Carl Foiles, Professor
Wayne Repko, and Professor Truman O. Woodruff for their
service on my guidance committee.

I would like to thank Professor John Robinson for
his help and encouragement during my stay at Indiana U.-
Purdue U. at Fort Wayne.

Finally, I would like to thank Maurine for her

patience and her constant support.

iii

LIST OF
LIST OF
Section

I.

II.

III.

IV.

TABLE OF CONTENTS

FIGURES

TABLES

INTRODUCTION

A. Physical Properties of Sms

B. Theoretical Models

C. D-electron Correlation

THE FALICOV-KIMBALL MODEL

A. The Ramirez, Falicov, and Kimball
Approximation

B. The Schweitzer Approximation

THE HUBBARD MODEL FOR ELECTRON CORRELATION

A. The Atomic Limit

B. Finite Bandwidth

THE TWO-BAND MODEL WITH MEAN-FIELD

SOLUTIONS

A. Mean-field Energy Approximation

B. Mean-field Green Function Approximation

C. Mean-field Results

iv

page
vii

viii

12
14

15
19
29
29

31

35
35
36

39

V. THE TWO-BAND MODEL IN THE ATOMIC LIMIT

A. The Green Function

B. Derivation of the f-d Correlation
Function Using the Partition Function

VI. HUBBARD APPROXIMATION OF THE TWO-BAND MODEL

A. The Green Function

B. The Total Energy

C. Results

VII. IMPROVED APPROXIMATION OF THE TWO-BAND MODEL

A. Derivation of the D—electron Green
Function
A.l. Derivation of Fdd
A.2. Derivation of Ffd
A.3. Derivation of Ffdd

B. Derivation of the f—d Correlation
Function

C. The Total Ground-state Energy
C.l. Derivation of the d-d Correlation

Function
D. Results
VIII. SUMMARY AND DISCUSSION
APPENDICES

A. Zubarev Green Functions

page

47

47

50
54
55
58
61

65

66
67
68

71

74

76

77

79

84

88

 

 

B. Total Energy
C. General Method for Obtaining The Ground
State

REFERENCES

vi

page
91

96
98

Figure 1.
Figure 2.
Figure 3.
Figure 4.
Figure 5.
Figure 6.
Figure 7.
Figure 8.
Figure 9.
Figure 10.
Figure 11.
Figure 12.
Figure 13.
Figure 14.
Figure 15.
Figure 16.
Figure 17.

Figure 18.

 

LIST OF FIGURES

SmS during insulator-metal transition

Density of states in Falicov-Kimball approx.

nc vs. band gap in Falicov-Kimball approx.
Phase diagram for Falicov-Kimball approx.
Density of states in Schweitzer approx.

nC vs. band gap in Schweitzer approx.
Phase diagram for Schweitzer approx.

.1 (mean-field).

Phase diagram with U/W

Phase diagram with U/W .3 (mean-field).

nC vs. band gap with U # O. (mean-field).

Phase diagram with U/W .l (mean—field).

Phase diagram with U/W

.3 (mean-field).
nc vs. band gap with U # O. (mean-field).
Density of states in Hubbard approx.

nC vs. band gap in Hubbard approx.

Phase diagram for improved approx.

nc vs. band gap for improved approx.

Density of states in improved approx.

vii

18
20
21
25
26
27
40
41
42
44
45
46
62
64
80
82

83

LI ST OF TABLES

Table 1. Possible states in atomic limit.

viii

52

 

 

I. INTRODUCTION

This thesis contains an analysis of the general prob-
lem of two interacting, correlated bands of electrons.
Specifically, this model is applied to the description of
the.mixed-valence compound Samarium Sulfide (SmS).

The problem of electron correlation in narrow bands
arises in many areas of solid state physics. For example,
the experimental data on the magnetic properties of tran-
sition metals seem to require at the same time a localized
electron model and an itinerant electron, or band, model
for a satisfactory explanation. The observation of non—
half—integral atomic moments is difficult to explain with a
simple atomic or localized electron picture, while spin
waves are usually explained in terms of localized spins.
It is believed that correlation effects in the narrow d
bands can account for the atomic behavior and therefore
for the simultaneous localized and itinerant pictures.
Thus, the addition of correlation effects may radically
change the expected behavior of a band model of a physical
system.

Mixed-valence materials are usually described in
terms of the presence, near the Fermi energy, of both f
electrons and d electrons, and their interaction. Usually,
the f electrons are treated as a highly correlated band
with zero bandwidth; i.e., in the atomic limit. The d

electrons, on the other hand, are assumed to be in a wide

 

 

2
band and correlation effects are ignored. However, the
estimated values of the correlation energies of the d
electrons are comparable to the interband interaction
energies in SmS. This indicates that the effect of the
d-electron correlation probably cannot be ignored.

We have started with a model which successfully pre-
dicts mixed valence and have added a two-particle inter-
action between the d electrons. Thus, we are dealing
with a two-band model which contains interband and intra-
band interactions. This model is solved using various
approximations and the resulting change in the ground-

state prOperties is investigated.

A. Physical Properties of SmS

 

Samarium Sulfide (SmS) is one of a large number of
.materials which exhibits the property of intermediate
valence.l At atmospheric pressure, SmS is a semiconduc-
tor with Sm2+ and 82' ions in a NaCl lattice. At a pres-
sure of 6.5 kbarz, a first order semiconductor-metal
transition takes place with a volume decrease of about
10-12% and an increase in conductivity. In addition,
experiments indicate that the valence of the Sm ions in
the collapsed phase is intermediate between Sm2+ and Sm3+.

The samarium atom has the electronic structure

Xe+4f65d0632 and sulfur has Ne+3523p4. When Sm forms a
compound with S, the two outer 6s electrons are given up

by the Sm atom. The Ed band broadens due to the crystal

3
field effects and hybridizes with the 65 band, forming the
Sd-GS conduction band. The 4f electrons are relatively
unaffected by their surroundings (except for a constant
energy shift) since they are highly localized.

At room temperature and pressure, SmS is a semi-
conductor with a gap of approximately .1 eV between the
4f levels and the conduction band. If we now increase the
pressure, the d-band (which is Split by the crystal field)
broadens and moves closer to the f-levels. Before the gap
closes, a first-order insulator-metal transition takes
place. During this transition, the volume decreases by
about 13%, the conductivity increases, the color changes
from black to gold, but the crystal structure remains the
same. The volume of SmS vs. pressure is plotted in Fig-
ure 1.

Because the volume decreases suddenly and especially
because the conductivity increases, the obvious explan—
ation would be that one f—electron per Sm ion has been
promoted to the conduction band, leaving behind Sm3+ ions.
Since Sm3+ has a smaller volume than Sm2+, this would
account for the volume decrease. In addition, the conduc—
tion band would then be occupied, thereby increasing the
conductivity. However, other experiments have yielded
results which cannot be explained by such a simple model.

The results of the volume measurements (see Figure 1)
show that in the collapsed phase the lattice parameter is

30% larger than it should be if all the sites were Sm3+.

 

 

 

 

 

~~~~~~
Q...
~~~~.~~~
Q...
~~~~
~~~~

‘s

|

' |

|

|

l

|

|

|

|

|

|

l

(19' ‘

\

\/ |

|

__ K
VO ‘s

§~~~~
§
,
0.8 -
I _L i 1
O 2 4 6
Figure l.

SmS during

8
P (kbar)

insulator—metal transition.

 

 

S
Chatterjee et al.3 have deduced from the lattice data that
for SmS the valence is 2.8, which is intermediate between
+2, for all sites Sm2+, and +3, for all sites Sm3+. This
is the origin of the term "intermediate valence". Another
experiment which gives the valence of SmS is that of Cam-
pagna et al.4 Their XPS (x-ray photoemission spectra) data
from SmS in the collapsed phase show two peaks which cor-
respond to emission of an f electron from Sm2+ and Sm3+

ions. This experiment gives a valence which agrees to

 

within 20% of the lattice constant determinations.

From the fact that the valence is not +3, we could
infer that not all of the Sm ions lose an f electron to
the conduction band. A possible description would be a
spatially inhomogeneous distribution of Sm2+ and Sm3+
ions. However, experiments using the Mossbauer effect to
determine the valence in the collapsed phase have also
been done5, with surprising results. Mossbauer experi-
ments provide a measure of the density of the 55 electrons
at the nucleus. This density will be different for the
Sm2+ and Sm3+ ions, since the 5s electrons in the latter
are screened from the nucleus to a lesser degree. In the

collapsed phase, the Mossbauer line is found to be inter—

2+ 3+

.mediate between the lines expected from the Sm and Sm
ions alone. If the two kinds of ions were simultaneously
present at fixed sites, we would expect to see two sep-

arate lines.

The physical picture that emerges which includes the

6

experimental results outlined above is as follows: each
Sm site is equivalent, with an f electron hopping back
and forth between the f level and the conduction band.
The lifetime of an f electron at a given site can be
determined to within a range of from 10'16 sec to 10'9
sec. The upper limit is determined from the Mossbauer
data. Since only one line is seen, the hopping time must
be less than the resolution of the experiment, which is
estimated to be 10'“9 sec. The lower limit is due to the
XPS experiments which are estimated to have a time reso-
lution on the order of 10'16 sec. This picture of an
f electron hopping between the f level and the conduc—
tion band is the source of the term "fluctuating valence".

Theoretical models must therefore explain at least
the following properties of SmS:

i) A pressure-induced first~order phase transition
from a semiconductor to a conductor.

ii) Intermediate or fluctuating valence in the
collapsed phase.

iii) A homogeneous distribution of Sm ions with

intermediate valence.

B. Theoretical Models

 

In discussing the various theoretical models which
have been suggested to explain the pressure—induced phase
transition in SmS, we follow Robinson1 and separate the

theories into two broad categories. The first category

 

7
consists of those theories which emphasize the Coulomb
interaction between the f electrons and the conduction

electrons as the principal cause of the transition.

 

These theories are usually based on the model proposed by
Falicov and Kimball.6 The second category of theoretical
models includes those in which the electron—lattice inter—
action is primarily responsible for the transition. We
shall compare the two categories briefly and then discuss
the electronic models in detail, since these are the
theories of interest in this presentation.

Hirst7 showed that a first—order pressure—induced
phase transition may occur if one considers only the
volume dependence of the energies of the lattice and the
electrons. Physically, his "compression—shift" model pro—
poses that a decrease in the volume of the lattice shifts
the energy of the f electrons upward relative to the con—
duction band and thereby causes them to delocalize. This
promotion of the f electrons further compresses the lat—
tice, since the ionic size decreases after losing an f
electron. Then the parameters can be adjusted to produce
a first-order phase transition. In order to prevent the
collapse from proceeding all the way to +3 valence (in
which one electron/site is promoted to the conduction
band), Hirst considered a volume dependent bulk modulus
which increases with compression.

Others have considered models which rely on the elec-

tron—lattice interaction. Varma and Heine8 proposed a

8

.model very similar to that of Hirst, in which the elastic
energy depends on the different ionic volumes. The sta-
bility of their model depends on a nonlinear lattice con-
traction with volume change. The nonlinearity is treated
as an adjustable parameter and they are able to achieve
agreement to within 20% of the experimental results.
Their model neglects all Coulomb correlations.

Coqblin and Blandin9

presented a model in which the
compression-shift mechanism is reSponsible for the first—
order phase transition. However, they use the virtual
bound states of Friedel-Anderson10 to broaden the f level
and allow for the possibility of non-integral valence.
Others have used the periodic Anderson10 model,using
electron-phonon interactions to bring about the phase
transition. For example, Entel, Leder, and Grewell use
the periodic Anderson model, which consists of an Anderson
local impurity at each site which can exchange electrons
with the conduction band. They claimed that within the
Hartree—Fock approximation it is not possible to have a
discontinuous transition with a periodic array of Ander-
son impurities alone. They included the interaction of 4f
electrons with the longitudinal optical phonons which can
shift the 4f electron energies and can induce 4f-5d inter-
band transitions, thereby renormalizing the explicit hy—
bridization and bringing about a discontinuous change in

the number of f electrons.

The models considered so far have assumed that the

 

 

f electrons are delocalized through promotion into the
conduction band. However, Hall coefficient measurements12
on Sm 75Y.255I which is an example of chemically collapsed
SmS by the introduction of Y atoms, suggest that the extra
electron doesn't go into the conduction band. In addition,
optical experiments13 done on SmS in the semiconducting
phase show rather narrow peaks in the d band. This exper-
imental evidence, along with the difficulty in explaining
the magnetic prOperties of SmS in the collapsed phase, led
Kaplan andMahantil4 to propose an excitonic or "essen—
tially localized" model in which all but a few (about .1
electron/ion) of the electrons are localized. So for SmS
the transition is predominantly f6 to fsdl, where d1
stands for a localized d electron. Since the one-electron
mixing interaction between 1:2 and 1:3 orbitals on the
same site vanishes by inversion symmetry, Kaplan and Mahan-
ti suggested that two Sm sites simultaneously mix f and d
electrons. This avoids the symmetry difficulty. In order
to explain the phase transition, Mahanti et a1.15 added
the elastic energy of the lattice and were able to predict
a discontinuous valence change by the compression-shift
.mechanism.

In the lattice model, the compression—shift mechanism
gives a sharp transition and the stiffness of the lattice
halts the valence change at a non-integral value. In the
electronic models, which we now consider, the volume

change is treated as an effect rather than as the cause of

 

 

10
the transition, with the f electron-conduction electron
Coulomb interaction being the cause.

When a Samarium ion gives up an f electron to the con-
duction band, there is a residual interaction between the
conduction electron and the "f hole" left behind. This in—
teraction is usually attractive, even when screening is
taken into account. This attractive force lowers the en-
ergy of the conduction band, so that as more f electrons
are promoted to the conduction band, it lowers further,
allowing more f electrons to enter it, and so on. This is
the mechanism for the insulator—metal transition suggested
by Ramirez, Falicov, and Kimball6. This model has been
studied by many authors16 and will be discussed in detail
in Section II.

The model proposed by Ramirez, Falicov, and Kimball
was solved by them in the mean-field approximation. They
found that for different values of G, the electron-hole
interaction strength, they could predict either a first—
order or a continuous phase transition. However, the dis-
continuous phase transition is always to a state in which
all ions are Sm3+; i.e., each Sm ion has given up an f
electron and intermediate valence is precluded.

In order to predict a first—order transition to an
intermediate valence state,.many authors17 have added a
hybridization term to the Falicov—Kimball Hamiltonian
which.mixes the localized and itinerant electrons. This

gives the f level a finite width by giving the f electrons

 

11
a finite lifetime. It is then possible to have the Fermi
energy pinned inside the f band, producing intermediate
valence. This also accounts for the large density of
states at the Fermi energy which is indicated by the large
electronic specific heat.18

Khomskii and Kocharjan19 attempted to obtain an inter—
.mediate valence state without an explicit hybridization
term in the Hamiltonian. They started with the Falicov-
Kimball model (for a single impurity) and made a mean-field
approximation in which they retained terms which represent
the quantum mechanical admixture of f electrons and conduc-
tion electrons. Therefore, in their solution, the f elec-
tron—conduction electron mixing arises from the Coulomb
interaction, G. With this approximation scheme they found
first-order phase transitions to intermediate valence
states. However, hewson and Riseborough20 showed that
this problem is exactly soluble and that the first—order
transition found by Khomskii and Kocharjan is probably due
to their approximation.

Although it is likely that mixing terms, either one-
or two-particle, are necessary to explain the ground—state
and low lying excitation properties (susceptibility,
neutron form factor, etc.) of mixed-valence systems, it was
shown by Schweitzer21 that these terms may not be neces-
sary to account for the phase transition to mixed valence.
He derived an approximate solution to the Falicov—Kimball

model in which a first—order transition to a state with

 

 

 

12
intermediate valence is possible. However, since he did
not include a mixing term, his solution must be an inhomo—
geneous distribution of Sm2+ and Sm3+ ions. Presumably,

a small mixing term would give a homogeneous mixed-valence

 

ground state without affecting his results. His approx-
imation included exciton—like correlations between f
electrons and conduction electrons. Schweitzer's work

will be discussed in Section II.

C. d Electron Correlation

 

The conduction electrons in the collapsed phase of SmS
move in a band made up from 5d and 65 bands. These elec—
trons will be referred to as d electrons. In all of the
previous models for the phase transition of SmS, the Coul—
omb interaction between d electrons has been neglected.
Schweitzer's approximation includes exciton—type correl—
ations which tend to localize d electrons around f holes.
However, localization can also take place due to the mu—

22 of the

tual repulsion of the d electrons. A calculation
Coulomb correlation energy in transition metals gives val—
ues of U/W between .3 and .8, where U is the Coulomb cor—

relation and W is the bandwidth. No such calculation ex-

ists at present for the 5d band in the rare earths. How-

ever, rough estimates can be made which predict values of

U/W on the order of .5 eV. Since this is the range of

values for the f—d interaction, G/W, it is not obvious

that the d—electron Coulomb repulsion can be excluded

13
without affecting the phase transition.

The model we consider here is essentially a two-band
Hubbard23 model with correlation in both bands. This mod-
e1 is interesting in its own right. In addition we be-
lieve that it has an important application in the area of
intermediate valence systems. We shall refer to this mod—
el as the "two-band model" in the remainder of this thesis.

In Section II we discuss the Falicov—Kimball model and
the approximate solutions due to Ramirez, Falicov, and Kim—
ball and due to Schweitzer. In Section III we discuss
electron correlation in narrow bands and the Hubbard model.
Section IV contains a description of the model being con—
sidered here, along with two approximate solutions, in
which the d—electron correlation is treated in a mean—field
approximation. The two-band model is exactly soluble in
the infinitely-narrow-bandwidth limit and the solution is
discussed in Section V. An intermediate solution is dis-
cussed in Section VI, in which the Hubbard decoupling
scheme is applied to the two-band model, using the exact
solution of section V. An improved approximate decoupling
scheme is derived in Section VII along with the results
of calculations made using this approximation. A summary

and conclusions are presented in Section VIII.

 

 

II. THE FALICOV-KIMBALL MODEL

In order to explain the metal-insulator transitions
observed in various materials, Falicov and Kimball6 pro—
posed a model in which both localized and itinerant qua—
siparticle states exist simultaneously. The interaction
responsible for the electronically driven phase transi-
tion is taken to be the intra-atomic Coulomb repulsion
between the two kinds of quasiparticles.

This model was applied to the d-d phase transition

 

in metallic Ce by Ramirez and Falicov. They assumed
that there are two different types of electron states:
(a) an extended, uncorrelated band, obtained from hybrid-
ized s— and d-like Bloch states and (b) a set of local—
ized, highly correlated f-like states. The.mechanism for
the phase transition is the short-range Coulomb interac—
tion matrix element between f—states and conduction-band
states.

The Hamiltonian first introduced by Falicov and Kim—

ball can be written

= i +
H E:(k)akoaic + E0 Ebio bio
+ +
- G igo' bioaio'aio'bio ' (l)

where a: (a+ ) creates (destroys) an electron in state
kc ko
k with spin 0 and energy e(k), bio creates a hole with

spin 0 at site i with energy E, and

14

15

-> ->

+ 1 iq-R. +
a. . = — e l a+ . (2)
1" AT 53 q°'

They assumed the interaction strength, G, is positive,

correSponding to an attractive electron-hole force.

A. The Ramirez, Falicov, and Kimball Approximation

The Falicov-Kimball model was solved for the case of

a finite bandwidth by Ramirez, Falicov, and Kimball6 in

the Hartree (or mean-field) approximation. The last term
in the Hamiltonian, which represents the interaction be—
tween itinerant and localized quasiparticles at the same

site, can be written as

- G ,a.

+ + + +
b. + — G g<b0bo>§0,a. io

. a. a.
100' io io' io' 10 10

co

= -Gnc f p(€)f(s)d€ , (3)

—00

where nC is the number of conduction electrons/site,

which equals the number of f—holes/site; f(e) is the
Fermi function and p(e) is the density of states for
electrons with both spins in the conduction band. Since

for a non—interacting band

(X)

; 5(E) <aloako> = I p(e)f(e)d€ , (4)
ko

—(X)

if we set E = O, we obtain for the total energy

 

 

16

E = I (e-nC G)p(e)f(e)de . (5)

For the ground state, T +0 and f(e) + 0(u-e), where
l x>O
6(X) = (6)
O x<O
and u is the chemical potential or Fermi energy. Then at
zero temperature

u
E = [ (e-nC G)p(€)de . (7)

—00

The chemical potential is determined from the constraint

equation u
nCN = J p(€)d€ . (8)

The density of states, p(e), is derived from the
Green function for the itinerant electrons. In the mean—
field approximation the Green function is easily found,
since the first equation of motion is exactly soluble.
(See Appendix A) If we define the Fourier transform of

the double-time Green function by

+
G°(k,w) = <<a§0; a§0>>w . (9)

it satisfies the equation of motion

3 >> . (10)

+ +
w<<a+ + >> = < a+ ,a+ > +<< a+ ,H ;a
w 0 k0

ko;ako ko kc k

Using the Hamiltonian (1) we have

 

17

aKc'H = e(k)ak>O - 2Gncak>O (ll)
. . + .
and, uSing the fact that a§0,afio = l, we find
w<<a+ ~ai >> = 1 +{e(K) + 2Gn }<<a+ -ai >> (12)
kc' kc C kc’ kc
or c + +
c (k,w) = {w-e(k) - 2nCG}—l . (13)
The density of states is given by
p(w) = l Im ; Go(k,w-i0+) (14)
n
kc
and we get p(w) = p0 (w — ZnCG) , (15)

where pO (w) is the unperturbed density of states.

So, the unperturbed density of states is rigidly
shifted by chG. Ramirez, Falicov, and Kimball used for
their unperturbed density of states an approximation to a
simple-cubic lattice. In order to compare their results
with the approximation due to Schweitzer (to be discussed
later), we shall use the semi-elliptic density of states

given by

 

00(0)) = 82 m2/4 _ E2 ' (16)
WW

where W is the bandwidth. This is plotted in Figure 2,
along with the effect of the interaction in the mean—
field approximation, Eq. (15).

The occupation of the conduction band at zero tem-
perature is obtained by finding the absolute minimum of

the total energy with respect to no, the number of con-

 

18

 

 

Unperturbed
F-K (MF T) ------------------

»'

 

 

I
I
l
l
l
l
l
l
l
l
l

 

 

Figure 2. Density of states in Falicov—Kimball approx.

 

19

duction electrons. This was done for various values of
G and A, the gap energy, which is the energy difference
between the localized levels and the bottom of the con—
duction band. The results are shown in Figure 3. Whe-
ther the transition is first order or sicond order de—
pends on the value of G.
In Figure 4 we plot a phase diagram in which the

parameters are G and . As was shown above, there are
first and second order transitions possible where n

C

goes from O to 1. However, there is no region of inter—

 

mediate valence that can be reached by a first order
transition.

As we discussed in the introduction, there have been
many attempts17 to modify the Ramirez, Falicov, and Kim—
ball model in order to explain intermediate valence.
However, we will describe in the next section an approx-
imate solution of the Falicov-Kimball model due to Schwei—
tzer which goes beyond the mean—field approximation and

is able to predict intermediate valence states.

B. The Schweitzer Approximation

 

Schweitzer21 assumed for the Falicov-Kimball Hamil—

tonian the idealized model

ic —2 kc

c f d
+fi .2 nic'nic ' (17)

20

 

 

 

 

 

 

 

 

 

 

G/w=.35 .3 .25 .2
1.0 -
n. "
.6-
.2 -
.i .65 0.0 - .6 5 -.i
A/w

Figure 3. nc vs. band gap in Falicov—Kimball approx.

21

 

 

  
  

 

 

 

 

 

 

.5
1st order
----- 2nd order
G/w he:
25* 1
, / ’ I
,." l
r’ nczo

O < n,<1 l
l
l
l
l

J 4 J J— J

- 2 -1 0.0 1 2

zQ/Vv

Figure 4. Phase diagram for Falicov—Kimball approx.

22

where nio is the number operator which counts the number
of localized electrons with spin -c and energy cf at site
i, ngo counts the number of itinerant electrons with spin

c and energy 6(k), and

e-i(E-E').§i C: C+
k'c kc °

d-
n.—

10 (18)

ZHA

r15:

Uff is the intra—atomic Coulomb correlation energy

of a localozed electron and G is the f-d interaction.

For the case of SmS, the localized states corres-
pond to the 4f levels of Sm. The state with two "f-elec—
trons" corresponds to the 4f6 configuration and the state
with one "f—electron" corresponds to the 4f5 configura—

tion. The state with zero "f—electrons" is projected
ff

out due to the large U ; i.e.,
f f _ —£ —f _
<(1 - nio)(l - nio)> — <ni0 ni0> — o , (19)

where 3:0 counts the number of holes at site i with Spin c.

It should be pointed out here that Since the f elec-
tron number operator, nfo , commutes with the Hamiltonian,
Eq. (17), the number of f electrons at Site i is a good
quantum number and therefore a homogeneous mixed-valence
ground state is ruled out. However, a small hybridiza-
tion or f—d mixing term should be included in order to
give a homogeneous mixed-valence state without affecting
the phase transition.

The total energy is completely determined as a func-

 

23

tion of nc by the conduction electron Green function. It

can be shown that, since the conduction band is non—inter-

acting,
e(nC)/N = (28f + Uff) - (cf + Uff)nC
+ 2 I €p0(€)f(e)d€ , (20)
where
1 .
pO(€) = - FE" Im E cg (k,e+10+) . (21)

We define the gap parameter A to be the energy re—
quired to excite an electron out of the localized state
into a state at the bottom of the band, ignoring exciton—

ic correlations. Then

_ ff
A — (8f + 2G + Ed — W/2) - (2Ef + U ) . (22)

Ignoring the constant first term in (20), we can

write the energy per particle as

C!)

+ W/2 - 2G)nC + 2 J Epo(€)f(€)d€ . (23)

(D

€(nc)/N = (A - Ed

In order to derive the Green function in Schweit-
zer's approximation, we again use the equation of motion
method. Since Schweitzer did not make the Hartree ap-
proximation, the decoupling of the equations of motion

was made by using the following approximation:

—f + —f +
X <<n, 'n, "c, ;c+ >> : nCX <<n. 'C, ;0+ >>
. 1c 30 1c kc , 1c 10 kc
cc 0
+ n 2 <<n. c. ; c+ >> — n2 <<c. ;ci >>. (24)
C 10' 3c kc C 1c kc

 

24

This approximation terminates the hierarchy of equa-
tions of motion by writing three-particle Green functions
in terns of two-particle Green functions. The third term
in (24) is necessary to avoid double-counting. Phisical—
1y, this approximation corresponds to two particles "prop—
agating" in the presence of the "mean field" of the third
particle. The two-particle Green function can be thought
of as representing a particle in the d band and a hole in
the f band prOpagating together and for this reason ,

Schweitzer refers to these terms as "exciton—like correl-

 

ations".
Using the above approximation and Eq. (10), Schweit-

zer obtained the following Green function:

 

 

60(E.w) = {w - 5(E) - 2(a)}"1 , (25)
where
nc(l-nc)G2F{w—(2-nC)G}
2(a)) = (2 — nC)G + (26)
1+(1-2nc)GF{w-(2-nC)G}
and
Fun) =§ 2:, 1 1 (27)
k w-€(k)

We can obtain the perturbed density of states from
the imaginary part of the Green function, as in Eq. (21).
(See Figure 5.) At zero temperature, the number 11C of
conduction electrons/site is determined from the absolute
minimum of the total energy. We have plotted the results
for various values of the parameters in Figure 6. In

addition, the phase diagram is plotted in Figure 7. It

25

 

Unperturbed ______ ..
Schweitzer

 

‘LSP

 

 

 

 

 

 

 

Figure 5. Density of states in Schweitzer approx.

26

 

 

 

  

G/w = .35 .3 .25
1.0- e
L.
HC
.6—
- .15
.21L

 

 

 

 

 

.1 .05 0.0 -.05 -.1
A/W

Figure 6. nc vs. band gap in Schweitzer approx.

27

 

 

    

 

 

 

 

.5
1st order / z
—————— 2nd order
G/w
n¢=1
/
/
/
/ /
'25" /" I 0<n,,<1

/ / I

/’/' I

/ ’ I

i I rHZZC)

O<ng1 :

I

I

I

:2 -J (10 .1 2
AM

Figure 7. Phase diagram for Schweitzer approx.

28

is clear from Figure 7 that there is a region in which
there is intermediate valence and which can be reached
by a first order transition.

Schweitzer's approximate solution of the Falicov-
Kimball model is able to produce a first order transition
to an intermediate valence state. However, the Falicov—
Kimball model neglects correlation in the conduction band.
In Section IV we present a model which extends the Fali—
cov-Kimball model to include d-electron correlation. Sec—
tion III contains a summary of the Hubbard23.model and
the Hubbard approximation, which forms the basis for our

treatment of the d-electron correlation.

 

III. THE HUBBARD MODEL FOR ELECTRON CORRELATION

In this section we introduce d-electron correlation
and discuss how it will be treated within the Green func-
tion technique. We discuss the Hubbard23 model of correl-
ation in narrow bands and the Hubbard 124 approximation.
Hubbard considered a single narrow band, built from Bloch
functions, in which electrons interact when two electrons
are in Wannier orbitals on the same atom. Thus, Hubbard
only considered intra-atomic interactions among electrons

in the band. The Hubbard Hamiltonian is given by

+

H = €..c. c. + E Z
1] 1c 30 2

ic

n. n.—4 (l)
ijO 10 .10

+ O I O 0
where 0. creates an electron at Site 1 With Spin c and
1c
0 O +
n. is the corresponding number operator; n.— = c —c.— .
1c 10 1c 1c

A. The Atomic Limit

 

Hubbard first considered the limiting case of zero
bandwidth, which corresponds to negligible overlap of the
d-electron wavefunctions on different atoms. This reduces
the problem to a system of isolated atoms and atomic the—
ory gives the exact solution. The method of Green func-
tions also gives the exact results, as we now Show.

Following Hubbard we use the mithod of double-time

Green functions (see Appendix A). We define the center of
the band by l +
T0 = N i €(k) . (2)

29

 

30

In the limit of zero bandwidth, 9(R) + To, so that

ii. (ii-R5,)
gij = fi 1% $326 -> Tosij. (3)

The Hamiltonian of Eq. (1) now becomes

U
2 n. + — 2 n. n.-.'. (4)
0 i0 2 ic 1c 10

The Green function is defined as

O +
t I = << . o 0
G13 (E) Cic'cjc>>E (5)

The equation of motion for this Green function is given by

. + _ + , +
E <<CiO,CjO>> — < CCiO'CjO]+> + ((ECiG'HJICjO>>‘
(6)
Using (4),
[Ci ,H] = TOCi + uniocio (7)
d sin e C C+ - 0 e et
EGO (E) - a + T G0 (E) + UFO (E) (8)
ij ‘ ij 0 ij ij
where
c _ . +
Fij(E) — <<niccic'cjc>> . (9)

. . . c
If we now write the equation of motion for Tij(E), we get

a term proportional to

2 4-
<< . . . . >>.
nicCic’Cjc

 

31

2

Since ni6'= niE-for fermions, the sequence of equations

of motion is terminated and we get
6-- < n.->

E - To - U

 

r8. (E) =
13

h is independent of i and G (paramagnetic limit) so

 

i6
that n13 = n/2. Finally we obtain
c l - n/2 n/2
G..(E) =5.. __ + . (ll)
13 13 _ _ _
E TO E TO U

The density of states is given by

0 _ 1 c . + c .-+]
E — ——— .. - - . +
p ( ) 2M ;[013 (E 10 ) Gij(E IO )

J

(l - n/2) 0(E - To) +'n/2 6(E - To - U).(l2)

B. Finite Bandwidth

 

Hubbard next considered the case of finite bandwidth,
which involved additional terms requiring approximations.
In order to terminate the hierarchy of equations of motion,

he used the following approximations:

+ +
<< . , >> 2 < ,_> << ; . >>
niECkc'Cjc nlc CkO C30
+ + +
<<c._c _c. ;c. >>z<c,_c -> <<C, ;c‘.L >> (13)
lo kc lo 30 1c kc ic Jc

+
_> <<C. ;C. >> .

+ + +
<<C _C _c. ;C.—>>:<C C.
Jc kc 1c 1c Jc

kc ic ic

 

32

The condition that Hubbard set is that the resulting Green
function reduce to the correct zero-bandwidth result.

Instead of reproducing Hubbard's derivation of the
finite-bandwidth problem, we present here an alternate
derivation introduced by Doniach.25 This method will be
used later on the two-band Hubbard model being considered
in this thesis.

Doniach observed that for the usual tight-binding model

of band theory, where

 

+ +
H = Z €..C.C, =2 Tn. +2 €.,C.C.
TB ijc 13 1c 3c ic 0 1c i#j 13 1c 3c’
0 (14)
the Green function defined by
G0 (E) - <<C ‘C+ >> (15)
ij ' ic’ jc
satisfies the equation of motion
(E - T )G0 (E) - 6 + 2 e G0 (E) (16)
0 ij - ij £¢i 12 2j °

If we Fourier transform the Green function we get

c + 1
G (k,E) — E—g(K)+io+ . (17)

 

Equation (16) can be written as a perturbation series

by iterating GE (E):

3
G9. = G + G s G
1] empty empty ij empty
+ G X‘s. G e G + -——- (18)
empty 2 11 empty Qj empty

33

where

1
+ (19)

G = ________.
em t
p y E-To+io

is the propagator for the empty atom.
Hubbard's decoupling procedure is equivalent to a

generalization of (18) in which G is replaced by the

 

 

empty
atomic-limit solution, equation (11). Thus,
G0 E “'G0 <3 +G° O (20)
ij( ) _ atomic ij atomic i E:iILGRjLE)’
Taking the Fourier transform yields
GOULE) = G q l ..
[Gatomic] — €(k) + To
1
= (21)
(E-T ) E‘Tn'U + To -e(E)

 

n
E-TO-U(l-7)

which is the same as that derived by Hubbard.

So, physically, Hubbard's decoupling scheme is equiv-
alent to starting with the exact isolated-atom limit and
then turning on the "hopping" term and letting the electrons

propagate to nearest neighbors with a probability given by

Since the Hubbard model is concerned with a band of

interacting electrons, the total ground-state energy is

given by (see Appendix B)

E = 1; [511211191 A°(E,w)dw, (22)
kc 2

—oo

 

34

where p is obtained from the constraint

M O +
n N = § I A (k,w)dw
c
kc

_m,

and

AO(E,w) = l [GO(E.w-i0+) — GO(K.w+iO+)]
2ni

 

is the single particle spectral weight function.

(23)

(24)

 

IV. THE TWO-BAND MODEL WITH
MEAN-FIELD SOLUTIONS
The model which we present in this section is that of
. . 6 . .
Falicov and Kimball plus the addition of the d-electron
correlation. The interaction between the electrons in the

conduction band is included as an intrasite interaction,

 

as in the Hubbard model (see Section III). The Hamiltonian
is
ff
f f d
H = 2 (efn + U2 . i— § €(E)n
10 10 10 0 k0 EC
f d d d
+G z n n, + n . (1)

In this section we present two approximate solutions
of the Hamiltonian in (1), both of which are mean—field
treatments of the d-electron correlation. The first
approximation consists of adding to the total energy in
Schweitzer's approximation (see Section II) a mean-field
energy term. The second solution consists of calculating
a new d-electron Green function with the d-electron corre—
lation treated in the mean-field approximation and using
this Green function to obtain the total energy.

A. Mean-Field Energy Approximation.26

 

The model Hamiltonian Eq. (1) can be written as

d nd
ic ic '

(2)

U
H = H + — Z n
PK 2 ic

35

36

where HFK is the Falicov-Kimball Hamiltonian, eqn (11.1).

We can calculate the total energy by using

<nc.i né—>
IO 10

E = <H> = <HFK> + g- E o (3)

ic
If we apply a mean-field approximation to the d-electron

correlation term, we get

2
d d n Un
_2 <n.n_>—>[_J. <nin.—>-gZ-—C-=N——E,(4)
ig 10 10 2 lo 10 10 2 1c 4 4
d
where n = Z <n. >.
c 0 1c

 

As a first approximation to the total energy we cal-
culate <HFK> using Schweitzer's approximation. Then the
total energy is just the sum of the energy calculated in

Schweitzer's approximation and the d-electron interaction

term;
Un2

_ c
ET(nC) — Esch(nc) + ‘Z—'° (5)

It should be emphasized that E (nc) contains no con-

sch
tribution from the d—electron correlation. The ground
state then corresponds to the value of the conduction band

occupation number, nc, which gives the absolute minimum

of the total energy.

B. Mean-Field Green Function Approximation.

 

The Hamiltonian Eq. (1) with the d-electron inter-

action treated in the mean-field approximation is given by

 

H = HFK + — Z <n >n,— + <n.—>n, - <n ><n —>
2 1c ic ic 1c ic 1c 1c
2
Unc d UHC
= H + _— Z n. - o
FK 2 io, 10 4

The last term is a constant and drops out when the Green

function is derived. HFK contains a term

—> d
S(k)n
c Kc

. d d
ans, Since 2 n, = § n1 ,
id 10 k0 k0

the effect of the d-electron correlation can be accounted

for by defining
EIR’) = so?) +-——-‘?- .

in which case the Hamiltonian reduces to the Falicov-
Kimball Hamiltonian with €(k) replaced by S(R).

Then, within the Schweitzer approximation,

 

GO(EILU) = ~ :1; I
w-€(k)-Z(w)

where 2(w) is given by Eq. (11.26).

Since we have an interacting system, we can no longer
calculate the total energy as was done in the Schweitzer
calculation. The derivation of the correct ground-state
energy equation is given in appendix B, Eq. (B.13), with

E(E) replacing e(fi):

 

38

~+ d f (1
§ e(k)<n+ > + G 2 <n. 'n. >
kc kc 100. 10 1c

(X)

§ wi°(fi,w)f(w)dw.

kc
Therefore,
~ + d
§ wAO(R,w)f(w)dw = § e(k)<n >
kc kc R
f d UT!
+ G 2 <n. ,n, > + __E z <n§ >
icc' 0 1c 2 kc k0

d
I 6(E)<n+ > + G X <n£ ,n§—>

 

kc k0 icc' 10 10
2
NUn
+ C ,
2
But,
_>
E = <H> = Z e(k)<nd > + G X <n£ ,né >
T kc EU icc' 10 10
+ g. E <nq_><né > .
i0. 10 10
So, 2
m ~ Un
E = ; EAO(E,w)f(w)dw - N—IE— .
T kc

In this equation, AO(R,m) is calculated from the Green
function which contains the d-electron correlation in the
mean-field approximation. The last term is subtracted from
the expression to compensate for double—counting the

interaction.

 

39

C. Mean-Field Results.

 

The approximations described in Sections A and B were
used to find the ground state of the two-band model. The
results are presented in this section.

In order to find the conduction band occupation in
the ground state, we used the procedure outlined in Appendix
C. This procedure yields the value of nc, the number of
conduction electrons, which minimizes the total energy for
different values of the parameters G/W and U/W, where W is
the conduction band width.

Phase diagrams for the mean-field energy approximation
(Section A) are plotted in Figures 8 and 9 for U/W = .1
and U/W = .3, respectively. These phase diagrams Show
possible first-order (solid lines) and second-order (dashed
lines) phase transitions as the gap parameter, A, changes
with pressure. A comparison with Figure 7, which is the
phase diagram for Schweitzer's approximation (U=O), shows
that the mixed—valence region in phase Space accessible by
a first-order phase transition has increased in area due to
the addition of d-electron correlation. The reason for this
can be found by looking at FigurelO. FigurelO is a plot of
the value of 11C on the mixed-valence side of the first-order
transition for G/W = .35. As U is turned on, three changes
become apparent. First, the value of the gap parameter, A,
at which the phase transition takes place shifts to a smal-

ler value. Second, the value of nc, after the first—order

 

4O

 

 

   

 

 

 

— 1st Order
2nd Order ————————
L0~
G/w
.6L
L
/ I’
2‘ ,I/ I
/ l _
// o<nc<1 : "c-O
/
y 1 l 1 r
'5 -.25 01) .25 .5

A/w

Figure 8. Phase diagram with U/W .l (mean—field).

41

 

 

l.
lst Order
2nd Order

to—

U/w =.3

 

 

h—

 

(10
A/w

Figure 9. Phase diagram with U/W

.3 (mean—field).

 

 

42

 

 

 

 

 

 

 

 

Figure 10. nC vs. band gap with U # O. (mean-field).

43
transition, is lowered. Third, the rate at which the con-
duction band fills up as the gap closes (i.e., the slope of
the line as A decreases) decreases. Therefore, when U#O,
a smaller gap is required to obtain the same nC as when
U=O so that the mixed-valence region increases in size.

Figures.1l and 12 are phase diagrams for the mean-
field Green function approximation (Section B). The effect
of d-electron correlation is seen to be qualitatively the
same as in the previous mean-field approximation. Figure1u3
shows more clearly that there are quantitative between the
two approximations. The value of A at which the discontin-
uous transition takes place does not shift as dramatically
as in Figure U). However, the occupancy of the conduction
band at the transition is lower in Figure2t3 and the slope
is smaller in this approximation.

Both approximations yield the same qualitative results
which can be summarized as follows. The Coulomb repulsion
between conduction—band electrons opposes the promotion
of f electrons into the conduction band, thereby increasing
the critical pressure (or decreasing the critical gap
parameter,A) and decreasing the value of nC at the transi—
tion for a particular value of the f-d interaction strength,

G.

 

44

 

 

 

 

 

 

Figure 11. Phase diagram with U/W = .l (mean—field).

45

 

 

t4~

r~ u/w =.3
tOr

G/w

.6F

_ n6
.2-

/ 0< nc<1
I I / 1

 

 

 

 

Figure 12. Phase diagram with U/W = .3 (mean-field).

 

46

 

 

     

 

 

 

 

 

 

1.0 ~
he ~—
.6 —
G/w =.35

.2 —

1 l l 1

.1 0.0 -.05 -.1

A/w

Figure 13. 11c vs. band gap with U # O. (mean—field).

V. THE TWO-BAND MODEL IN THE
ATOMIC LIMIT

A. The Green Function

 

In this section we present a derivation of the d-elec-
tron Green function for the two—band model in the atomic
limit. Although this Green function can be found exactly,
we are interested in the case where the f—electron corre-
lation energy, Uff, is so large as to preclude the simul-
taneous excitation of two f electrons into the conduction
band. Therefore, we project out the state with two f holes.
The required Green function is derived using the equation
of motion method and is found to be exact due to the termin-
ation of the hierarchy of equations of motion.

In the atomic limit, each atom is isolated and the
model reduces to N independent atoms. Therefore, the

Hamiltonian can be written as

f d fd dd
H=ZH.=Z(H,+H.+H.+H.)
i 1 i i i l i
f d d
= 2 (H? + ed 2 nn + G 2 n ,nd + 2'2 n—n ). (1)
i 1 c 0 cc' 0 O 2 c O

f f d .
We have left H. as it is since both I10 and n0 commute Wlth
i
it and it therefore doesn't enter into the equations of
motion for the Green function. In addition, we set ed=0

and define H: = l-nZ, so that the Hamiltonian can be re-

written as

47

 

48

. d
nand = G 2 fif,n , (2)
0' C 00.1 G O

.‘1‘.
H
m
+
N
C)
M
B
+
sud

Z
G d

where we have dropped the site index, i.

The Green function for the d-electrons is defined by

c +
G (w) = <<c ;c >> . (3)

c c
I O G O
The equation of motion for G (w) is
+ +

<< ° >> = + << - >>

w co,cO l [CO,H],CG ~(4)
Using (2), we get

d _f
= + _
[cc,H] 2Gc0 Ungco G Canon:O . (5)

Therefore,

+ d + _f +
(m-2G)<<c ;c >> = 1 + U<<n_c ;c >> - G 2 << n ,c :c >>.
O c o 0 O c c c

O!
(6)

We now write equations of motion for the two-particle
Green functions that were generated. For the d-d Green
function,

d + d d d +
(w-2G)<<n_c ;c >> = <n—> + u<<n_n_c ;c >>
O c c c c c c c

_f d +
- G X <<n "n_c ;c >>
c c c c

d . .
where we have used [nE,H] = 0. Since for Fermions,

d d _ h
n6n6-- n5-, we ave
d + d +
(w-U—ZG)<<n_c ;c >> = <n_> - G 2 <<n "n—c :c >>. (8)

 

49

Likewise, for the f-d Green function we obtain

_f + —f _ +
(w—G)<<no,c0;co>> = Z <n > + U 2 <<nfungc ;c >>.

T T 0'" O O O O
(9)
The equation of motion for the three-particle f—d-d

Green function is

_ + _f d
(w-U-2G)<<nf"ngc ;c >> = <n "n—>
0 c c c O c

_ _ +
- G 2 <<nfnfnngc ;c >>. (10)
T c c c c .

T
. —f —f —f . . . . 6
Since nc"nc" = no" and Since, folloWing Falicov-Kimball
we have chosen Uff and sf such that
—f—f
< :
n0n8> 0, (ll)
_ + _ +
z <<nfnfun§c ;c >> = <<nf ngc ;c >>. (12)
r T c c c c c" c c c
Therefore, (10) becomes
_ + _f d
(w-U-G)<<nf"n9c ;c >> = <n "n_> . (13)
0 GO 0 0' 0

Substituting (13) into (8) and (9), and substituting the

result into (6) gives

 

 

d -f
+ U<n_> G X, <n ,>
(w-ZG)<<c :c >> = 1 +.____Q_ - c 0
0 0 w-U-ZG w-G
-UG Z <Hf'ng> 1
c' 0 0 (w-U—G)(w-U-2G)

 

1 . (14)
(w-G)(w-U-G)]

I

 

 

 

 

If we define nc = Z <H§,> = Z <n%>, we can rewrite (14)
c' c
as
__ d 3n _
c n - Z. <nf'n_> 1 - .72.+ zi<nf.n§>
G (A) = C c c c + c c c
w-G w-ZG
2533,1151? nC/2 - 2.<H§,n§>
+ Om U G + 3 2G 0 ' (15)
_ _ w- _

The Green function just derived can be written in
the form

0
A.(w)
G00») =2 3 (16)
J

 

wwi

 

. . . c

where the wj are the exc1tation energies and the A,(w)
J
are the Spectral weights. A condition which must be satis-
fied by the spectral weights is
c

E A.(m) = l. (17)
j 3
It can be Shown that this sum rule is satisfied in the pres-

ent case.

B. Derivation of the f-d Correlation Function Using the
Partition Function.

 

 

The correlation function, Z,<H§'ng>, which appears in
the atomic-limit Green functionocould be derived using the
fd Green function and Eq. (A,8). However, since the ener-
gies are known for all possible states, the partition
function, and therefore the correlation function, is easy

to find.

51

The possible states of the system along with their
energies and degeneracies are listed in Table (l). The

correlation function is given by

68me (2 Hang) , (18)

Z <Hf ng> = lvTr
Z 0.!

0" O" 0

where u is the chemical potential, B=l/kT, N'is the total

number Operator, and
A = Tr e'B(H-UN). ‘(19)

In order to facilitate the following expressions, we

define the variables

_ Uff
s = eBu x = e 8
-B(€ -u) _
t = e f y = e BU , (20)
-B(G-u)
v = e

Then the partition function can be written

2 2 2 2 2 4 2
Z = 2t + 4tv = 2tv y + t x + 2t v sx + t v xys .
(21)
The correlation function is
—f d 1 2
Z <no,n3> = Z<2tv + 2tv y). (22)

0'
We can eliminate the partition function Z by using the fact

that

<H£> = nC/2 = l/Z(t + 2tv + tvzy). (23)
c

52

 

 

 

 

 

 

 

IQ

Table 1. Possible states in the atomic limit.
STATE E - UN Degeneracy
0 d
8f - [.l 2
8f +'f
—+—«
8f+G-2p 4
—I——«
“4—1—4
8f+26+U-3p 2
___¢;__—f
—————d
22f+u"-2u 1
—I=—I—«
———$———-d
I I Zef-I- Uff-I- 26-311 2
f
—I—I—«
zc,+u"+4c+u-4p 1

 

 

 

 

 

 

53

 

 

Then
2 <Hf.n.C—1-> = HG) , (24)
0.1 O O l - f(U+G) + f(G)

where f(x) = l is the Fermi function which, at

zero temperature, goes to 0(u-x), the step function.

To summarize, we have derived the atomic-limit Green
function for the two-band model. This Green function is
exact in the case being considered here; i.e., the intra-
atomic Coulomb interaction between two f electrons, Uff,
is so large that the possibility of exciting two f elec—

trons into the d level from the same Site is zero. The

Green function is given by

 

 

3
_ — + . _
Go(w) = nC CF + 1 inc CF + CF + nC/2 CF (25)
“(-3 w-ZG w-U-G WG- '
_ d
where C = Z <nf'n—> (26)
F O" G O
and is given by Eq. (24). This Green function will be

used in the next section in an "intermediate solution"

2 . .
calculation based on Hubbard's 3 approx1mation.

VI. HUBBARD APPROXIMATION OF THE
TWO-BAND MODEL

Recently, Mazzaferro and Ceva27 applied the Hubbard 123

approximation to the Falicov-Kimball model (no d-electron
correlation). They referred to their calculation as an
"intermediate solution", since Hubbard I is thought to be
an improvement over mean-field approximations but not as
good an approximation as CPA (Coherent Potential Approxi-
mation).28 They were concerned with the possibility of a
phase transition as a function of temperature and found no
first-order transition.

Schweitzer's21 approximation goes beyond mean-field to
include excitonic correlations. However, this approximation
is also an improvement over the Hubbard I decoupling scheme,
since (see Section III) the approximations are made on the
three-particle instead of the two-particle Green functions.
It is interesting to compare Schweitzer's approximation
with the CPA method. In a later paper29 Schweitzer pres-
ented the results of a CPA calculation on chemically induced
intermediate valence in SmS. He started with the Falicov—
Kimball Hamiltonian and generalized it for the case of the

ternary alloy Sm S, where R denotes any rare-earth ele—

l-xRx
ment that is trivalent in the monosulfides. His results
included the case where x=O, which corresponds to SmS.
Although we were unable to Show a mathematical equivalence
between the self-energy derived in ref. 21 and 29, the num-

erical results of the two calculations are nearly identical.

54

55

Therefore, Schweitzer's improved approximation of the
Falicov-Kimball model is very similar to CPA, and the
calCulation of Mezzaferro and Ceva is intermediate between
the mean-field approximation and Schweitzer's approximation.
In this section we present a calculation using the
Hubbard I approximation to the two-band model. When the
d-electron correlation is turned off (U=O), we recover the
results of Mezzaferro and Ceva. However, since we are
interested in the possibility of a pressure-induced transi-
tion, our calculation is done at a fixed temperature (T=O)
as a function of the gap parameter, A. We find that a
first-order transition to an intermediate valence phase is
possible even in this simpler approximation. The two-band
model is also solved with UfO in order to find the effect

of the d—electron correlation on the phase transition.

A. The Green Function

 

Mazzaferro and Ceva27 used Hubbard's23 decoupling
scheme (see Section III), which is equivalent to starting
with the atomic-limit Green function and allowing hopping
to nearest-neighbor sites. Therefore, for the Falicov-

Kimball model,

606;») = , (1)

 

1
[612(0)] ‘1 - e (E)

where G:(w) is the exact atomic-limit Green function for the

Falicov-Kimball Hamiltonian; we have chosen the atomic

56

level to be TO=O. We can obtain this Green function by
letting U=O in the atomic—limit solution of the two-band
model derived in Section V. We then have

n l - n

0’ C C
G (O) = + —————— . (2
A w - G w - 2G )

 

Using this equation in (l), we obtain for the Hubbard I

approximation to the Falicov-Kimball model

 

GGIE.w) = ‘w’G"w'ZG) - eIE) ’1 . (3)
w-(l+n )G
c
This is equivalent to the Green function derived in ref. 27.
The intermediate solution for the two-band model can
be constructed in a similar way. From Section V we have for

the exact atomic-limit Green function

 

3
nC-CF 1-2nc+CF + CF nC/Z—CF

Go(w) =
A w-G w-2G w-U-G w-U-ZG

I (4)

 

where CF is the f—d correlation function,
C = Z <n nd> . (5)
c
c
The finite-bandwidth solution using the Hubbard decoupling
scheme is then given by

c + _ 1
0’ -1 +
[GAm] - e(k)

 

We now derive the perturbed density of states

57

0+
2 A (k,w) (7)
i
along with

g e(K)A°(E,w), (8)

w

both of which are required for the total energy.
If we define the poles and the spectral weights of

the atomic-limit Green function as

 

l c F
m — 2G A = l - in + C
2 2 2 c F
(9)
= + =
m2 U G A3 CF
2 + 2 = —
w4 U G A4 nC/2 CF,
then
c 4 Ai
G ((0) = Z _ . (10)
A i=1 w ”i

G (krw) = (11)

 

 

The spectral weight function for this Green function is

AOIEE) = ler—I[GG(}:Iw—in) — GOII‘E.w+in) . (12)

58

 

or
AO(R,O)) = 5 S((0) - HE) , (13)
where
4 A. -1
8(0)) = z 1 . (14)
1=1 w-wi
Then
2:, AGO—$.01 = o [5(0)] (15)
k 0
and
; eIEIA°Ii,w) = sum [sun] . (16)
k 0

B. The Total Energy

 

The total energy is found by using an equation similar
to one derived in Appendix B. However, the following diffi—
culty arises. If we use Eq. (B.19), we have the following

equation for the total energy:

E = [§i§l:9] A0(k,w)dw + E; Z <nf.n_> (17)
%c[ cc' 0 0

—00

When G=O, the f—d correlation term drops out and the
expression reduces to the Galitskii-Migdal expression,

Eq. (B.8). When G#O, we must evaluate the f—d correlation
function. However, if we use the atomic-limit correlation
function, which was derived in Section V Eq. (v.24) , we

do not recover the correct energy expression when U=O.

59
Therefore, we will use Eq. (8.17),
p + d d
E = [ w+Z A0(k,w)dw - g§.2<n—n > , (18)

kc 2 c 0 0

—-oo

and we will evaluate the d-d correlation function in such
a way that (18) gives the Galitskii-Migdal expression when
G=O.

In order to derive the d-d correlation function which
gives the correct energy when G=O, we first derive a similar
expression for the Hubbard model (G=O) and show that the
total energy using this expression is equivalent to the en-
ergy obtained by using the exact expression, Eq. (B.8).
Then <n%n:>, the d-d correlation function for the present
model, is approximated in such a way as to give the Hubbard
value for the total energy when G=O.

In Hubbard's approximate solution to his Hamiltonian,

, dd . . . .
the d-d Green function, P , is never explic1tly derived.

However, we find from his derivation that

 

dd _ d +
IT — I <<niECic7cic>>
1 i
— nC/2 N + z (fi)<<c+ - + >>
w-U K 8 kc'cRc °

From Eq. (A.8) we find

°° +

a nd no 2 + A0(k,w)f(w)dw
_— e m-U

 

(20)

Then, using (18) we obtain for the total energy

60

' c + d
E = wf(w) § A (k,w)dw — H X <n n,—>
kc 2 i O

= [ m ; AG(E,w)f(w)dw
kc

co

4

Un m + +,
C l ; €(k) Aq‘k'w’f‘w’d“ (21)
k0 (1)-U

00

For the Hubbard model,

 

A°(E,w) = 5[ w(w’U) — 2(E) , (22)
w-U(1-nC/2)

and therefore

n 4
U C/

E = g [ f(w) w - m-U(l-nc/2) dw. (23)

—oo

 

If we use the Galitskii-Migdal expression Eq. (B.8)

00

E = [ [EiELEL] AO(R,w)f(w)dw (24)
J

I
kc

.+
and substitute Eq. (22) for A0(k,w) we obtain

Unc/4
w-U(l-n /2)
c

 

E = XI f(w) w - dw, (25)

0

--00

which is the same as Eq. (23) above.
We can use the preceding derivation as a guide for
the case we are considering in this section; i.e., the two-

band model. For this model,

61

 

_ - r —> 0' +
w_U_2G + w-U-G] [l+e(k)G own] (26)

and, using Eq. (A.8) we obtain

/d d n /2-C. ,WC'
2 <n n > [ [L—i-l- F I; G(E)AO(R,w)f(w)dw.

 

w-U-ZG w-U-G

 

i k
(27)
The total ground-state energy at T=O is then
u
0+
E = (A + .5 - 2G)n + g A (k,w)dw
C kc
U D nC/Z-CF CF + c +
...._ ________+ .
2 w-U-2G w-U-G £0 €(k)A (k,w)dw (28)

~00

The chemical energy,U , is found by using the condition

ll
2 A00? )d (29)
n = ,w m.
C Kc

-®

Then, the total ground-state energy can be written as

E = (A + .5 — 2G)n
c

 

 

[ U nc/Z-CF CF
w - -———-———-+

2 w-U-ZG m-U-G)S(w)] 00 3(4)) dw’ (30)

J
_m

where S(w) is defined in Eq. (14).

C. Results
The density of states is plotted in Figure14 . The

solid line represents the unperturbed density of states.

62

 

 

 

 

 

 

3 1
I I 5
__ __ =.
/w /w c
G u n
_d
e
_b
r
u
t
"Li
M____
nuWLW
u.
.
.
.
.
.
.
.
.
.
.
.
.
.
_ .
5 0.
4| 4|

Figure 14.

Density of states in Hubbard approx.

63
The dotted line is the perturbed density of states for
W=l,, where W is the bandwidth. The dashed line is the
perturbed density of states for W=.l (scaled by 1/10).
As the band narrows, the density of states approaches the
atomic limit which corresponds to four atomic levels at
G, G+U, 2G, 2G+U.

Figure 15 shows the value of nc, the number of conduc-
tion electrons, which minimizes the total energy. This was
found using the procedure outlined in Appendix C. The three
curves correspond to U=O., .l, and .3. When U=0. we recover
the results of Mazzaferro and Ceva at zero temperature.
However, as the gap parameter, A, varies with pressure,
we get a first-order transition to a value of nC < l;

i.e., a state with intermediate valence. The presence of
d-electron correlation affects the critical gap and the
value of nC just after the transition in a way similar to

the mean-field treatments in Section IV.

64

 

 

 

 

 

 

 

 

 

LA
2

_. L.
O
O
A
I
'0

Figure 15. nC vs. band gap in Hubbard approx.

VII. IMPROVED APPROXIMATION OF THE
TWO-BAND MODEL

In this section we present an improvement of the
previous approximations in Section IV, which treated the
d-electron interaction in the mean—field approximation.

In Section A the equation of motion method is used to
derive the d-electron Green function. The decoupling
scheme is guided by the following conditions:

i) The Green function should reduce to the results of
Schweitzer when U=O.

ii) The Green function Should reduce to the results of
Hubbard when G=O.

iii) In the atomic limit 8(R) + ed=0 , the Green func-
tion should reduce to the exact result.

The f—d correlation function which arises in the deri-
vation of the d-electron Green function is derived in Sec-
tion B. Section C contains the expressions for the ground—
state energy of the system and Section D contains the
results of calculations of the ground-state properties.

The Hamiltonian for the model being considered here
will be written in two representations: the localized and
the itinerant representations. Each one is more convenient
when dealing with the Hubbard and the Schweitzer approxi-
mations, respectively. In addition, we will use the hole
notation for the f-electrons:

_f f

65

66

The Hamiltonian in the localized representation is

L f
H = H + Z (6 + 2G5 )C C
. rsr rS rs Irr ST
+ g 2 nd nd_ - G X Hf ,nd . (1)
ST ST ST srr' ST $1

In the itinerant representation, we have

f d d
HI=H++Ze(q)+2Gc:Tca+EZnn—
qr q -T 2 ST ST ST
_f i -+' °§ +
- g 2 nst' +§ e (H q ) S C+T'C§'T" (2)
ST' qq'r q

A. Derivation of the d-Electron Green Function

 

We are interested in finding the d-electron Green
function for this model, since from this we can find the

total energy of the system. We define the Green function as
c + +
G (k,w) = <<CKO;CEG>> . (3)
This Green function obeys the equation of motion

+ I +
w<<C+O;C+O>> = l + <<[C H ];CEO>> , (4)

k k Ec’

. . . . I
where we have used the itinerant Hamiltonian H . The

commutator is found to be

+ +
ik'R
[C+ ,HI] = €(E) + 2G CR + _E.Z e S n "Cs .
kc c /fi 8 so c
ikR
- E E e S Hf .C . (5)
[N ST' ST 50

67

Therefore,
‘ +->-
+ + ' . d +
(w - e(k) — 2G)<<C+ ;c >> = 1 + 9.2 elle<<n,_c, ;c+ >>
kc kc [fi i 10 1c kc
"iii?
10
-'§ 2 e 1<<H€ C. ;C+ >>
fN ic' 10' 1c kc
s 1 + rdd + rfd (6)

. . dd
A.l. Derivation of I

The equation of motion for the two-particle Green

. + . . . .
function Tdd = <<niaciG;CkO>> can be Simplified if we use
the localized Hamiltonian HL and write CEO in terms of

+
CiO , so that
12?:
+ ’1 ' d +
<<anC. ;C+ >> = l 2 e £<<n.—C. :C >> (7)
1 1c kc /fi g 10 1c 2c

Then we have for the equation of motion for the d—d Green

function, Pad,
6. + d + + —C >
. ° = . > .
w<<nificic'clc>> ([nificlc'c ]> << n 1c'C£c
(8)
d L _ 26 + d
Now, [nificic'H ] — niECic : €isnic so
+ +
+ E, EismiFI'CSECic Csacifi'ciO)
d —f d

68

and

d c+]— c16
[niacic’ 1c _ mi? in '
and therefore

+
>>

d
(w-U-ZG)<<n,.C. :C
10 1c 2c

23
Using Hubbard's approximation (Eq.

d + + +
i 8is <<niECsc;C£c>> + <<CiECsECic‘CRc
+ +
- <<Csficificic;czc>>
d +
: <n_> Z 8- <<C 7C >>
c s¢i is sc 2c
Then, <<n.ECiO;C£O>>= — 2 e <<niECi :C
1 A? II

(10)

+
>>.

20 (ll)

(III.13) we have

>>

(12)

<. + —> +
= I'L—> l e-lk'Ri + i Z e-lk.RQ’
w-U-2G )[fi m 2
+
x z e <<C ,c >>
s is SO £0
G _f d +
’ 5:11:21? f “niTniaCio’CiZo” ° (13)
. dd . .
Finally F 18 given by
dd ikR.
= H z e l<<nq‘_C. ,C+ >>
/fi' 1 1c ic kc
U<n_..> +
= ___2_ 1 + e <<C —C >>
w-U-ZG k k0, k0
ikR.
_f
- __g§__. i z e l<<n. nq_C. ;C+ >>. (14)
w-U-ZG /fi 1T 1T ic 1c kc
fd

A.2. Derivation of P
In order to simplify the derivation of the two—particle
f—d Green function rfd, in which we will use Schweitzer's
approximation, we write the equation of motion using the
itinerant representation. Also, we first derive the Green

function

f». +

—f +lk ’Ri _ +

<<n 2 e <<n. C. ;C+ >>.
1' 10' 1c kc

. >>
10' k'c kc

Elle-

(15)

70

Using the itinerant Hamiltonian (2), we find

-* i—f
_ ' — o =
(w e(k ) 2G)<<nic'ck'o’ck >> <n0,>
32' i3
— O . — +
+ E 2 e l<<n€ ,n§_c. ;C >> (16)
/fi j 1c 30 3c kc
:1 E
-1 0 .
- E I e j<<nf n. u ; + >> .
/fi 30.0" 1c' 3c jc kc

If we now use Schweitzer's approximation (11.24) on

the three-particle f—f—d Green function, we obtain

 

 

'E "*
fd - G l .Ri <<—‘ + >>
I" ’ 7N: 133.9 niE'Cic’Ckc
-n G
_ c 2 2 i
— ——_—_—— + G F ' << ; >>
w'-€(K) nC (w ) Ckc Ckc
2 2 —iEO§o iE"(§j-EQ’)
— E—jia—-<<C+ -c+ >>.. I“; E e 3e
w'-e(k> ko' kc N3/2 j,§. w'-e(K')
0'

_f d + nG
x <<n'c'n2—C2c7ckc>> /[l + (l-2nC)GF(w') + ___2___],

l O w"€(K)
—f
where n = z <n '>, w' = w — (2—n )G, and
C 0.1 0’ C
l 1
F( ) = - z . (18)
m N g w-eIFI

If we combine (6), (l4), and (17) and rearrange, we
obtain the following equation for the d-electron Green

function:

71

 

 

 

 

 

I 2 2”, .H + ,
+ ‘Unc/Z + nc G..1-(wF-e(k))F(w')
w-e(k)-2G-______ €(k) + +
m-U-ZG w'-e(k) 1+(1-2nC)GF(w') +nCG
x <<C+ - 1 >>
kc’Ckc
2
= 1 + Unc/ _ nCG
w-U-ZG w'-e(E) l+(l-2nC)GF(w') +nCG
'E R <<_f d +
.0. .1. 2 .l i nwniaciorck.»
/N ic' w-U-ZG
+ (if-€612) 1
+ 3/2
w'-€(k) 1+(l-2nC)GF(w') +nCG N
-iE.§. iE'(§.—§,)
x z e 3e 3 x <<Hf nd—C -C: >> (19)
jgk‘ w'-e(k') jc' £c 20' kc
0|

In the present form this Green function gives the
correct Hubbard Green function when G=O. and the correct
Schweitzer Green function when U=O. This is easily seen
since the additional "cross terms", which are different
from those arising in the Hubbard and Schweitzer solutions,
are multiplied by the product UG and are therefore not
present in these two limits. Therefore, whatever approxi-
mation is made on the f—d-d Green function, we still keep

the correct limits of U=O and G=O.

A.3. Derivation of Pfdd

The approximation for the decoupling of the equation of

fdd

motion for P was guided by the fact that the atomic limit

72

is exact. Therefore, our approXimation will be such as to

—).
reduce to the atomic limit as e(k) + e

d = O.
The first approximation we make is
<< d c+ >> <<—'f d + >> 2 '
. 0 + = o —) =
njc'niaczc’ kc njc'njECjc’Ckc J
o 1753'
(20)

This is within the spirit of the other approximations in
which intrasite interactions are thought to be the most

important. Then we have for the last term in (19)

 

 

"ikR- ik' (Rn-R2,)
1 X e 3e 3 <<Hf nd—C °C+ >>
' l I
N372 j£k' w'-e(k') jc 2c Rc kc
0'
-> +

= 1 g (l 2 1+ )e J«Ht,- n, 7C+ >>I

Nl/2 30" N Kiwi-€(kl) 30" 30 30'

+
= F(w') 7: 2 e <<njc'njECjc7CRc>> , (21)
N jc'

where F(w') is defined in (18).

We now have one intrasite f—d—d Green function to

derive. The equation of motion is

d + < _f d + >
< ' o o . > = . . ' n+
w< njc‘nJECjc’CRc > njc'nJECJc’Ckc
_f d L +
< . - >> ,
+ <n. , njECjc'H 'Ckc

Using (1) we obtain

73

. ikR.'
(w-U-ZG) 11 X e 3<<fif n3 c. ;c+ >>
/fi jg. 30' 30 30 k0
._ d ikR' ' _ +
= Z <n:,n3> + 7% 2 e 3 5.1 <<n§0.n§5C£O;CEG>>
c' N jkc' J
+ <<Hf C+ —C C+ >> <<"f 4C+ C —C C+ >>
30' jUCfic jc' k0 njO' £3 jc jc' kc
_ .9 2 e j<<n§ ,n. n.—C. ;C >>. (23)
m“ jG'O" 30 30" 30 30 kg

The second term on the right hand side can be found using

the Hubbard-type approximation and is equal to

z <—f d> k <<C > (24)
no'na €( ) .

- >
c' kc'Ckc

The third term can be found by using the idempotency of the
f—electron number operator and the condition which projects

. . _f
out the state With two f-holes; i.e., <n0n0> = 0. Then

 

z <<E‘f _f nd_c C+ >> <<—f d_c c: >> (25)
o+ = . . .- ,
O" jc'njc" jc jc' kc nJO'njo 30' k0
We thus arrive at an expression for Pfdd:
”12?:
i : ~ d +
_l 2 e J<<fi'f, n,—C, ;c+ >>
yfi jc' jc' jc jc kc
_f d +
= 1 z (n 'n_> 1+€(§)<<c+ 'c+ >> . (26)
w—U-G O. c c kc kc

Upon substituting this equation into (19) and rearranging

terms we get for the d-electron Green function

74

 

 

 

 

 

 

Gc
+ ' ‘NUM
<<C—12070120» = 8-5—— , (-27)
DEN
where
0 UnC/2 ncG
GNUM = l + T""‘" '
w-U-ZG [w'-S(R)][1+(1-2nc)GF(w')]+nCG
- UG g, <E§,ng> l + [w'-€(K)] F(w')
w-U-G w-U-2G [aw-5(2)] [1+(l-2nc)GF(w' )]+ncG
(28)
and +
c + UnC/Z €(k)
G = w - €(k) - 2G -
DEN w-U-ZG
ng G2[l-[w'-€(E)]F(w')]
+
[w'-e(kx][1+(1—2nC)GE(w')]+nCG
06.2.- 63.11% . 1
+ e(k) ——————-
w-U-G w-U-ZG

w'-€(R) F(w')
+ + . (29)
[w'-€(k)][1+(l-2nc)GF(w')]+nCG

 

B. Derivation of the f-d Correlation Function

 

We will now derive the correlation function

fd _ —f d
C - g,<n0,n3-> (30)

which appears in the d—electron Green function. Since we

75

don't know the exact eigenstates of the Hamiltonian, the
partition function cannot be calculated and another method
must be used.

If we try to derive Cfd from the f—d Green function,
we find that, since Pfd depends on G0(k,w), the correlation
function we seek would have to calculate self-consistently.
That calculation would be sufficiently complicated that a
simpler approximation is necessary. Therefore, we will
adhere to the basic requirement that the Green function be
exact in the atomic limit and begin with the atomic-limit
f—d correlation function.

The f-d correlation function, Cfd, can be found by
extending the atomic-limit correlation function to a finite

band. From Eq. (v.24),

 

d
cfd = z <fif.n_> = f(G’ (31)
c' O O l-f(U+G)+f(G)
where
8(x-u) -1
f(x) =[e + 1] (32)

is the Fermi function.
Since we are dealing with finite bands and not atomic
d-levels, we will replace the Fermi function at a discrete

-> +
energy by a sum over k of the Fermi function at e(k):

1

H(X) fi%[e

00(E)dE
e8 (E+X- 11) +1 '

B[€(k)+x-u] ]-1
+1

(33)

 

76

where pO(E) is the unperturbed density of states for the
d band centered on ed=O. Then
_I d .., .
cfd = 2 <n n_> = H(G) . (34)

c' 0' O 1-H(U+G)+H(G)

This ansatz requires calculating Cfd by using the
unperturbed density of states, 00(E). Therefore, the
calculation is not self-consistent. However, the Green
function is exact in the atomic limit and gives the correct

Green functions when U or G is zero.

C. The Total Ground—State Energy

 

The ground-state energy of the model system described
by the Hamiltonian (2) can be calculated using the d—electron
Green function only. This is fortunate, since the f-elec-
tron Green function would be difficult to derive. The d-
electron Green function is easier to calculate since n:
commutes with the Hamiltonian (2), whereas n30 doesn't.

The total energy is given by
E = (A + .5 — 2G)n + E , (35)
c b

where A is the gap energy, defined in Section II, nc is the
number of conduction electrons in the d-band, and Eb is the
total energy of the band electrons. Since the electrons in
the conduction band interact not only with the f-electrons
but also with themselves, the energy of the band electrons

can be calculated by using the expression derived in

77

Appendix B. We can use either (B.17) or (8.18), depending
on which correlation function we use.

If we use (8.18), the f-d correlation function is
required. As stated before, the f—d correlation function is
difficult to calculate. We also find that if we use the
same approximation for the f-d correlation function in this
expression as we did in the approximate calculation of the
Green function, we would not obtain the same ground-state
energy when U=O as in the Schweitzer calculation. This is
because, whereas the term including the f-d correlation
function in the Green function drops out when U=O, it remains
and is necessary in the expression for the total energy.
Therefore, we cannot accept this approximation to the energy.

Equation (B.17), on the other hand, uses the d—d corre-
lation function. Therefore, we shall use (B.17) and we

derive the d-d correlation function in the next subsection.

C.l. Derivation of the d-d Correlation Function

If we attempt to use the atomic—limit d-d Green func-
tion and to use this in Eq. (B.17) for the energy of the
present model, we find that we do not recover the ground-
state energy of the Hubbard model when G=O. Therefore, we
must derive an expression for the finite-bandwidth d-d
correlation function which gives the correct Hubbard limit.

The equation for the total energy is given by (B.17)

u c + d d
E = +2 Guam (k,w)dw — 3115'. >3 <n—n > . (36)

—oo

78

We follow the example derived in Section VI.B. and the
d-d correlation function for the present model is approxi-
mated in such a way as to give the Hubbard value for the

total energy when G=O. The d-d Green function is given by

 

 

d + nC/2'-.C.fd Cfd
z <<n'6cic'cic>> = +
i 1 w-U-ZG w-U-G
+ +
- N + g €(k)<<ck0;cko>> . (37)

W

Then the d-d correlation function is given by

 

 

 

 

 

fd
u
d d r nc/2 ' C- cfd + c +
Z<n. n.—> = + g e(k)A (kIw)dw
i 10 1“ w-U-ZG w-U-G k
(38)
and the total ground-state energy is
u
0+
E = (A + .5 — 2G)n + { w 2 A (k,w)dw
T C Rc
u fd fd
U nC/Z-C C +0.,
- 3 + § e(k)A (k,w)dw. (39)
w-U-ZG w-U-G kc

—<X)

Therefore, we have the total energy given by Eq. (39)
which agrees with Schweitzer's results when U=O, since then
AO(E,w) is just the spectral weight function derived from
Schweitzer's Green function. In addition, when G=0 (39)
agrees with the Galitskii-Migdal expression for the total

energy of a band of interacting Fermions.

79
Equation (39) has the proper limits when G=O. and
U=O. When U and G are not zero,Cfd is required and is
given by Eq. (34). This expression is not self—consistent

but gives the correct atomic limit for the total energy.

D. Results

In this section we present the results of calcula-

tions using the improved approximation described above.
The ground state occupation of the conduction band was
found by the method described in Appendix C.

Figure 18 shows the perturbed density of states
(the unperturbed density of states is given by Eq. 11.16)
for G/W=.3, U/W=.3, and nc=.4, for which the chemical poten—
tial is —.221. The band has an increase in the density of
available states at the bottom, which is the contribution
of the attractive particle—hole potential of the Falicov-
Kimball model. In addition, the band is split, due to the
conduction-electron correlation. According to the Hubbard
model approximation, this splitting takes place for any
non-zero value of U.

Figure 16 is the available phase space calculated
within the improved approximation. The region of intermed—
iate valence accessible via a first-order transition is of
the same general shape as the same region in the mean—field

approximations of section IV.

80

 

 

 

 

 

1.0 ~
“I“, =.35
"(w = ,1
G/w
.6 - “6:1
/ 7‘ /
/
/
2-,/’
I o<nc<1
i I
-.2 -1

 

Figure 16. Phase diagram for improved approx.

81

In Figure 17 we have plotted the value of the conduc-
tion band occupation against the gap parameter. This is
compared with the case where U=O, which is Schweitzer's
result. Once again, the critical values of the gap param-
eter and the number of conduction-band electrons are shif—
ted by the addition of the correlation effects in such a
way as to oppose the first-order transition and the occu—

pation of the conduction band.

10

Figure 17. nC VS.

82

 

 

 

 

 

 

band gap for improved approx.

 

 

83

 

Unperturbed -— —. - _ .. _ ..
Improved Approximation

 

1.5 r'

D(E)

 

 

 

 

 

Figure 18. Density of states in improved approx.

VIII. SUMMARY AND DISCUSSION

We have presented the results of an investigation into
the ground state of a two band Hubbard model. This
model differs from other two-band models in that both
bands have electron correlation. Although this model

is interesting in general, our motivation was the appli—

 

cation of this model to the problem of the mixed-valence
systems.

We started with the Falicov—Kimball Hamiltonian and
added a d-electron interaction term modeled after
Hubbard's correlation term. This correlation term was
approximated using two mean-field approximations. In
the first approximation, the contribution of the correlation
term to the total ground-state energy was calculated using
the mean-field energy. This total energy was then minimized
to find the ground state. The result was that Schweitzer's
original prediction of a first order transition to
an intermediate valence phase was unchanged. However,
the critical parameters had values different from those
found by Schweitzer.

The second mean-field approximation consisted of
making the approximation in the Hamiltonian and then

deriving a new Green function. From the Green function,

84

85

the ground-state energy was calculated and the ground state
found by minimization. As with the first mean-field
approximation, Schweitzer's results were not changed
qualitatively but required different critical values of

the parameters.

As an intermediate calculation, we applied Hubbard's
decoupling scheme to the two-band model Hamiltonian. This
was done by Mazzaferro and Ceva for the Falicov-Kimball
Hamiltonian only (without d-electron correlation). Their
work Shows no first order transition as a function of
temperature. We found that a first-order transition as a
function of the gap parameter is possible and that an
intermediate valence phase can be stabilized for certain
values of the parameters. This implies that Schweitzer's
decoupling scheme is not necessary to explain the first
order transition to an intermediate valence regime but
that Hubbard's approximation is sufficient.

An improved approximation was also presented in which
the Green function which was derived has the following
limits:

i) agreement with Schweitzer when U=O

ii) agreement with Hubbard when G=O

iii) agreement with exact atomic limit when bandwidth
goes to zero.

This Green function was used to calculate the ground state

of the system. The results were essentially the same as

86

the mean—field Green function results. The density of
states derived using the improved appeoximation is sig—
nificantly different from the mean-field results and
therefore the excitation Spectra will be different for
the two approximations. However, the ground—state oc-
cupation of the conduction band has a similar dependence
on the parameters in both approximations. Therefore,
within this approximation, which agrees with the limiting
cased mintioned above, Schweitzer's results are not
changed significantly by the addition of conduction—elec—
tron correlation.

Schweitzer used his model to calculate the critical
pressure required to give an intermediate valence of 2.6.
He found a critical pressure of 5 kbar, which is lower
than the experimental value of 6.5 kbar. We have shown
that including d-electron predicts a lower occupation of
the conduction band at the forst order transition. This
implies that the two-band model should give a higher
critical pressure and a better agreement with the exper-
imental results.

Possible extensions of these calculations include:
i) improvement of the treatment of the d—electron cor—
relation beyond Hubbard I. Since part of Hubbard's
later approximation (Hubbard III) is equivalent to CPA
and since Schweitzer's approximation seems to be at least
numerically equivalent to CPA, it would be interesting

to begin with the two-band Hamiltonian and to do a CPA

87

calculation on it.
ii) finite temperature. All of the calculations present-
ed in this thesis are ground state calculations. It would
be useful to extend these calculations to investigate the
properties of the two-band.model at finite temperatures.
The ground state of the two-band Hubbard model within
the approximations described in this thesis can be either
an insulator, a mixed-valent conductor, or a monovalent
conductor, depending upon the values of the parameters.
This suggests that SmS, a mixed-valent material under
pressure, is a good candidate for a system which can be

described by a two-band Hubbard model.

APPENDICES

APPENDIX A
ZUBAREV GREEN FUNCTIONS

The Green functions used in the derivations presented
here were introduced by Zubarev. We present a brief
summary of the notation and the basic equations which are
used throughout the present work.

We consider the grand canonical ensemble such that the

average of a quantum mechanical operator A is given by

~

<A> = l/Z Tr Ae"JH , (A.1)

 

where H = H -uN and Z = Tr e . H is the Hamiltonian, N
is the total number operator, and u is the chemical poten-
tial. We write the time dependence of an operator in the

Heisenberg representation so that

A(t) = elHtAe'lHt (A.2)

For any operators A and B, the retarded and advanced

Green functions are defined by

Gr a(t't') <<A(t);B(t')>>r a

I I

-i0(t-t')
< A(t),B(t') >I (A.3)

i8(t-t')

where the upper (lower) term denotes the retarded (advanced)

Green function and where

88

89

l x>O
0(x) = (A.4)
0 otherwise
is the step function.
The equation of motion for the Green function can be

derived using the equation of motion of a Heisenberg

operator. The result is:

.d . I
ldE<<A(t)’B(t )>>

0(t-t')<[A,B]>
+ <<[A(t),H];B(t')>>. (A.5)

This equation applies to retarded or advanced Green
functions. The second term on the right hand side of (A.5)
usually involves a more complicated, higher-order double-
time Green function. The equation of motion for this new
Green function is then derived, which, in general, involves
a still more complicated Green function, and so on. This
chain of equations is terminated either automatically,
as in an exactly soluble case, or by some approximation
involving writing a Green function in terms of a less
complicated one.

It can be shown that Gr

(t,t') = G (t-t'), so that
a r a

I I

we can define the Fourier transforms

r a a iE
Gr'a(E) = <<A;B>>E' = r, e tdt.

NH
:

J <<A(t);B>>
"°° (A.6)

The corresponding equation of motion is

 

9O

1
E<<A;B>>E = 2..fi..<[zs(,B]> + <<[A,H];B>>E. (A.7)

Correlation functions can easily be derived from the

corresponding Green function by using the following equa-

 

tion:
<B(t')A(t)> = i [a [<<A;E>>E+iO+ -<<A;B>>E_io+]f(E)
e‘E(t’t')dE (A.8)
where f(E) = 1 is the Fermi function.

e8 (E'U) +1

 

APPENDIX B

TOTAL ENERGY

We now derive several expressions for the total
energy of a system of interacting Fermions, all of which
use the double-time Green function.

We begin with the equation of motion (A.5):

iécG(t) = i§_<<A(t);B(0)>>

dt dt 0(t)<[A,B]>

+ <<[A(t),H];B(O)>>.

 

(3.1)

+
Let A(t) = Ck(t) and B(O) = CTE and consider the general

Hamiltonian
+ + 1 + +
= ._ P .
H % €(k)CECk + 2 §ngpraarC§C§C§ucau (B 2)
I
Then

'd C t C+ 6 t + k < C t C+ >
a—<< - = < - +>
l t 1':( )I E>> ( ) €( ) E( )I k

+ +
+ I I ; >0
2 'VE§.§§ <<Ca(t)CE (t)C§,(t) C+>

ma k
(B.3)
+
If we now let t + o and use (A.8) we obtain
1 . + . +
—— mf(w)[G(w+10 ) - G(w-10 )]dw
2n
(k)<n > + 2 v <c+c c c+> (B 4)
= + +++~> ++++, .
E k 3'43. kp'qq' q p: q. k

91

92

Since ETot = <H>, then using the general Hamiltonian (B.2)

we have
+ 1
ET = <H> = § €(k)<n+> + — Z V++'++' o
k k 2 5-5. PP qq
(if?
- <c§c§c§.ca.>. (8.5)

Therefore, if we sum (3.4) over K we get

 

g mf(w)A(E,w)dm = § e(£)<n+> , (B.6)
k k k
where A(E,w) = 3— [G(E,w+io+) - G(E,w-io+)]. (3.7)
Zn

In order to recover the factor of 1/2 difference between

equations (B.5) and (B.6), we add
2 5(E)<n > (B 8)
i ‘1? °
to both sides and divide by 2. We then obtain

= + i + + + +
E }% €(k) <n.}? + Z V§§,q—qp,<C§CTCp,Cq'>

= Z w+€(E) f(w)A(k,w)dm, (B.9)
K J 2

---w

 

which is the Galitskii-Migdal expression for the total
energy of a system of interacting Fermions.
If we now consider a non-interacting band of electrons

interacting with an external potential, we can write a

93

Hamiltonian of the form

+ +
H = Z e(k)n+ + 2 V++,C+C+ . (B.10)
'1; R “5'5! Pp p p'

In this case

i§_<<c+(t);ci>> = 6(t) + e(§)<<C+(t);C:>>
dt k k k k

+
+ z V++<<C+(t);C+>>. (B.11)
E kp k p

Converting the Green functions to correlation functions

by using (A.8) gives

wf(w)A(E,w)dw = Z €(;)<n+
1% J K k

+
> + figVEE<CECE>. (B.12)

—(X)

The total energy is given by

ET = <H> = % e(i)<ni> + E+VEE<CEC+>° (B.13)
P
Therefore
_).
ET = Z [wwA(k,w)f(w)dw. (B.14)
1'2

Finally, we will have a need for an expression for the
total energy of a system in which both an external poten-
tial and a mutual interaction are present. In this case

the Hamiltonian can be represented by

 

94

H=;€(i)n++-]-'- 2V++ ++,C,,C1Ci+C-*
k k 2 ++. 99' qq' p q p' 9'
l
+
+ 2 V++ C+C+ (B.15)
gg' PP' P P'

As was done in the previous derivations, we find

= = + —> 1 o
E <H> %€(k)<nk>+'2_ .Vfifi'fifi'

Q'UiM
WU

+ + +
' <C C_).C_> C3.) + Z V§§'<C§C§'> (B016)

:5 q p' 55.

Therefore, (B.15) and (8.16) differ by a factor of 1/2

on the potential energy term. We now have two options,

depending on whether we can calculate independently the

correlation function <CiCiC+ C+ > or <CiC+ >. The two
P g P Q' P'

possible results are found in the same way as in the two

previous derivations. The results are

E = i J wf(w)A(E,w)dw

—oo

+
éE.V§P' 55' <CPCECP 'Cq '> (B°17)

qq'

N|I-'

and

 

E 2 w+ 5‘?) f( )A(E )d + 1 Z V <C+C >
= _ + —> .
T K 2 w ,3 w 2 55. PP' P P'

-” (B.18)

 

95
When the external potential is due to another band of

electrons with number Operator n, the total energy is

co

E = ; w+€(k) f(w)A(§,w)dw

 

T k 2
1 +
+ — Z V++'<n.C-+C+,>. (3.19)
2 155' PP 1 P P

 

APPENDIX C

GENERAL METHOD FOR OBTAINING THE
GROUND STATE.

The method used to find the occupancy, nc, of the
conduction band is described in this Appendix. This method
was used in Chapters IV, VI, and VII.

Since the ground state of any system corresponds to
the state of minimum energy, the usual procedure for find—
ing the ground state is to calculate the total energy as
a function of the variable of interest and then to minimize
the energy with respect to this variable. The most general
expression derived in this thesis for the ground-state

energy as a function of n can be written as
c

E(n ) = [ f(w,n ,u)dw. (C.l)
c C

—00

The chemical potential,u, is found using the condition that

CD

F(w,nc,u)dw. (C.2)

 

J

-CD

In all of the calculations done in this thesis, the
chemical potential, u, was found by calculating the root

of the equation

I(u) = nC - I F(w,nc,p)dw (C.3)

.00

for a given value of nc. A "root-finding" subroutine

was used which calculated the root within specified bounds

96

 

 

97

very quickly. The functions f and F contain the sum over
E of the single-particle spectral weight function,
0+

2 A (k,w), (C.4)

+

k
and this sum usually had to be performed numerically.
Therefore, each evaluation of 1(3) required that a double
integration be done numerically.

Once u is calculated for a given value of nc, the
total energy is given by Eq. (C.l). Thus, the total energy

as a function of no, E(nc), is calculated and plotted vs.

 

n Then the occupancy of the conduction band is given by

C.

the value of nC which has the lowest energy.

 

BIBLIOGRAPHY

10.

ll.

12.

l3.

l4.

 

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A brief list of excellent review articles on mixed va—
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{3, 1324(1973); K. Kanda, K. Machida,and T.

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D.I. Khomskii and A.N. Kocharjan, Solid State Commun.
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concerned here with reference 23, referred to as
Hubbard I.

S. Doniach, Advances in Phys. l§,819(l969); see also:
S. Doniach and E.H. Sondheimer, Green's Functions for
Solid State Physicists, (W.A. Benjamin, Inc., Mass.,
1974) Chapter 8.

 

 

 

The results of this subsection were reported earlier
in G. Fletcher and S.D. Mahanti, Bull. Am. Phys.
Soc. 23, 231(1978)

J. Mazzaferro, H. Ceva, Solid State Commun. 22,
137(1979)

 

28.

29.

30.

31.

100

In a later paper, Hubbard (see reference 24) improved
his approximation calculation of his model by

adding a scattering correction and a resonance—broaden-
ing correction. It was shown by Velicky, Kirkpatrick,
and Ehrenreich, Phys. Rev. 175, 747(1968)that, when
only the scattering correction is included, Hubbard's
approximation is formally equivalent to CPA.

J.W. Schweitzer, J. Appl. Phys. 12, 2090(1978)

D.N. Zubarev, Usp. Fiz. NaukLZl, 71 (1960) (Sov. Phys. Usp. 3,
320 (1960))

V.M. Galitskii and A.B. Migdal, Sov.Phys. JETP 33, 96 (1958)

 

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