l—"w r3
_ ' n. . .T—uui ‘ .-

ABSTRACT
ENERGY BANDS, SURFACE STATES, AND RESONANT TRANSMISSION
’ OF ELECTRONS IN FINITE ONE-DIMENSIONAL CRYSTALS
By
Willard M. Gersbacher, Jr.

The energies and wave functions of electrons in
finite one-dimensional potentials designed to simulate
various interesting physical situations have been calcu;
lated and are discussed. A general finite one-dimensional
periodic potential was considered, and the form of the“
wave function in all regions of energy was established.
without recourse to the BlochsFloquet theorem. ‘Various
types of potential terminations were considered, namely,
step-function terminations at a potential minimum or
maximum.or at an arbitrary point in the end cell, and
terminations by an arbitrary potential. The energy states
(both band and surface) were calculated in each case and
conditions were obtained which exhibit clearly how these
states depend on the properties of the potential.

The band structure above the vacuum level was in-
vestigated by calculating the reflection coefficient
for free electrons incident upon the periodic potential,
for several types of terminations, and interesting
relations between band-structures and tunneling were

obtained. The resonant emission of electrons from.metal

Willard M. Gersbacher, Jr.
and nonsmetal surfaces covered with.impurity layers was
investigated for energies above the vacuum level by
methods similar to those used in investigating the band
structure.

In each area of investigation specific calculations
were performed for Kronig-Penny type periodic potentials
with appropriately selected parameters. The results of

these calculations are summarized in a number of graphs.

 

ENERGY BANDS, SURFACE STATES, AND
RESONANT TRANSMISSION OF ELECTRONS
IN FINITE ONE-DIMENSIONAL CRYSTALS

. , rm
Willard MI’VGersbacher, Jr.

A THESIS

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

DOCTOR OF PHILOSOPHY
Department of Physics
1970

1 J

(1..

‘5
\_
.3

{.,

.3“.

'i

i, 3’8

o A."_!'

To my wife and my parents

11

ACKNOWLEDGMENTS

I would like to thank Professor Truman O. WOOdruff
for suggesting the subject area of electronic properties
of solids and for his help, encouragement, and patience
throughout the course of this work.

For the financial support, I would like to thank
the National Science Foundation and Michigan State
university.

I would also like to thank Mrs. Frances Strachan
for the help she has given in the typing of the final
draft of the thesis. -

Last, but not least, I would like to thank my wife,
Victoria, for the typing and for the constant encourage-
ment she has given.me throughout.

111

 

TABLE OF C ONTENTS

LIST OF TABLES. O o o e e o o o e o e o o e o e o 0
LIST OF FIGURES

O O O O O O O O O 0

INTRODUCTION 0 -

sectioneoeeeeeoeeeeeeeeeeeeee
I DETERMINATION OF TIE FORM OF TIE WAVE

FUNCTIONS . AND ENERGY BANDS FOR FINITE

PERIODICPOTENTIALS........ ..

II TERMINATION OF THE PERIODIC POTENTIAL -
DETERMINATION OF THE CONDITIONS FOR

BANDANDSURFACESTATES. . . . . . . .

III TERMINATION OF A KRONIG-PENNEY POTENTIAL

A e SHWHIEY’TYE TERMINATION o o e o 0

Be TAMM‘TYPE TERMINATION. e o e o e 0

IV ARBITRARY TERMINATION OF A FINITE PERIODIC

PWIAL O O O O O O O O O 0 O O O O O

V LOW ENERGY ELECTRON DIFFRACTION FROM FINITE

ONE-DIMENSIONAL PERIODIC POTENTIALS . .

VI RESONANCE TRANSMISSION IN ELECTRON EMISSION

FROM SURFACES WITH ADSORBED ATOMS. . .

Page

vi

223
43
45

'70

87

«.99

116

A. ADSORBED ATOMS ON A METAL SURFACE. . .118
B. ADSORBED ATOMS ON A NON-METAL SURFACE-126
summaylfi
BIBLIOGRAPHY......................138

iv

LIST OF TABLES

Table
2.1 Existence conditions for surface states
case 1 O O O O O O O O O O O O O 0 O
2.2 Existence conditions for surface states

Case 2 O O O 0 O O O O O O O O O O O

Page

39

#0

LIST OF FIGURES

Figure Page

1.1 Determination of the energy band
regions 0 e e o e e e o e e o e o e o 15

2.1 Effective wave number versus
energy - case 1 e e e e o e e e e e e 32

2.2 Effective wave number versus
energy ' C886 2 o e e e e e e o e e e 37

3.1 Shockley- and Tamm-type potential
taminations O O O O O O O O O O O O "'4

3.2 Band structure as a function of
well depth for a Kronig-Penney
pOtential With 2b: 00758.. e e e o e o 48

Band structure as a function of
well depth for a Kronig-Penney
p0tential With 21)" 0.508.. e e o e e o "'9

\N
W

3.# Band structure as a function of
well depth for a Kronig-Penney
p0tent1a1 With 2b: 0.258. e o o o o e 50

3.5 Determination of pass bands and
attenuation bands . . . . . . . . . . 54

3.6 Determination of the eigenvalues
for the Shockley type termination.
The crossing of the K curve with
a dotted curve determines an
eigenvalue. . . . . . . . . . . . . . 57

3.7 Energy versus (k and K). The x's denote
eigenvalues which were determined
in Figure 3.6 (Shockley). . . . . . . 59

3.8a Eigenfunctions for the first pass band
(ShOCkley) C O O O C O C C I O O O C O 61

3.8b Eigenfunctions for the second pass band
(ShOCkley) O O O O O O O O O O O O O O 62

vi

Figure Page

3.8c Eigenfunctions for the third pass
band and surface state eigen-
function which has emerged into
the third attenuation band from
the tOp of the third pass band
(Shookley) O O O O O O O O O O O O O O 63

3.8d Eigenfunctions for the fourth pass
band (Shockley). . . . . . . . . . . .64

3.8e Surface state eigenfunction which
has emerged into the fourth
attenuation band from the bottom
of the fifth pass band and eigen-
function of the fifth partial ‘
pass band (Shockley) . . . . . . . . .65

3.9a Surface state eigenfunctions in the
third attenuation band. The
eigenstates for the top of the
third and the bottom of the
fourth pass bands have moved into
the third attenuation band to form
two surface states (Shockley). . . . .67

3.9b Surface state eigenfunctions in the
fourth attenuation band. The
eigenstates for the top of the
fourth and bottom of the fifth
pass bands have moved into the
fourth attenuation band to form
two surface states (Shockley). . . . .68

3.10a Surface state eigenvalues for the
third attenuation band versus
the number of cells. . . . . . . . . .69

3.10b Surface state eigenvalues for the
fourth attenuation band versus
the number of cells. . . . . . . . . .69

3.11 Comparison of Tamm and Shockley
band structures. . . . . . . . . . . .73

3.12 Determination of the eigenvalues
for the Tamm type termination.
The crossing of the K curve with
a dotted curve determ nes an
eigenvalue 0 O O O O O O O O O O O O O 76

vii

Figure
3.13

3.14a

3.14b

3.14c

3.14d

3.1he

3.15

4.1

4.2

Energy versus (k and K). The
X's denote eigenvalues which
were determined in Figure
3012 (TaM) o o o o e e o e e o o o

Eigenfunctions for the first pass
band and surface state eigen-
function which has emerged into
the first attenuation band from
the top of the first pass band
(TM). 0 O 0 O O O O O O O O O O 0

Surface state eigenfunction which
has emerged into the first
attenuation band from the
bottom of the second pass band
and eigenfunctions of the second
pass band (Tamm). . . . . . . . . .

Eigenfunctions of the third pass
band (Tawn) O O O O 0 O 0 O O O O O

Eigenfunctions for the fourth pass
band (Tamm) . . . . . . . . . . . .

Surface state eigenfunction which has
emerged into the fourth attenuation
band from the bottom of the fifth.
pass band and eigenfunction of the
fifth partial pass band (Tamm). . .

Surface state eigenfunctions in the
fourth attenuation band. The
eigenstates for the top of the
fourth and the bottom of the
fifth pass bands have moved into
the fourth attenuation. . . . . . .

Termination of the periodic potential
at an arbitrary point in the end
0,911. O O O O O O O O O O O O O O 0

Energy eigenvalues as a function of
the cell termination parameter A .

viii

Page

78

8O

81

82

83

84

85

88

Figure
5.1a

501b

5.2a

5.2b

5.3a

5.3b

6.1a

6.1b

6.2

6.3

6.4

6.5

Reflection coefficient versus energy
for the Shockley-type termination
of four cells . . . . . . . . . .

Reflection coefficient versus energy
for the Tammrtype termination of
four cells 0 O O O O O O O O O O 0

Reflection coefficient versus energy
for the Shockley-type termination
of twenty cells . . . . . . . . .

Reflection coefficient versus energy
for the Tamm-type termination of
twenty cells. . . . . . . . . . .

Reflection of electrons from four
identical barriers with Shockley
terminations at both ends . . . .

Reflection of electrons from four
identical barriers with Tamm
terminations at both ends . . . .

metal'Impurity e o o o e o o e e e e
Non-Metal’lmpurity e e o e o e o o 0

Transmission coefficient for 1,2,3
adsorbed atoms on a metal -
case 1. O O O O O 0 O O O O O O 0

Transmission coefficient for 1,2,3
adsorbed atoms on a metal -
case 2 O O O O I O O O O O O O O 0

Transmission coefficient for 1,2,3
adsorbed atoms on a non-metal -
case 1 0 O O 0 O O O O 0 O O O O 0

Transmission coefficient for 1,2,3

adsorbed atoms on a non-metal -
case 20 O O O 0 O O O O O O O O 0

ix

Page

110

111

112

113

ll#

115
119
119

124

127

13a

135

INTRODUCTION

Progress in understanding surface phenomena has been
slow, because physical and chemical processes at the sur-
face are inherently more difficult to analyze than those
in the bulk. The forces acting on the atoms at the sur-
face are not symmetrical, as in the bulk, and consequently,
the atoms are usually diSplaced from their ideal lattice
positions. Moreover, Just the fact that the surface con-
stitutes an abrupt termination of the crystal lattice re-
sults in a deformation of the crystal potential -- its
periodic nature is lost at the surface. This has far-
reaching consequences for the electronic processes in the
underlying region of the crystal close to the surface.

At the same time, unsaturated forces from the surface
atoms make them highly reactive towards various atoms out-
side the crystal. Thus, except when produced and main-
tained in a high-vacuum, the surface is covered by one

or more layers of foreign matter, greatly increasing the
complexity of an already difficult problem.

Although theoretical interest in electronic surface
states has existed since the 1930's, little was accom-
plished because there was negligible technological moti-
vation or opportunity for experimental confirmation. The

great impetus for surface-state study came with the advent

2
of the transistor in the late 1940's. Since then further

motivation has rapidly developed in conjunction with a
variety of solid-state technologies.

Theoretical studies of the electrical properties of
surfaces have progressed along two different lines. Tamm's1
theoretical work, which treated a rather simplified model,
was extended by Shockley2 and by other workers to cover more
general situations. It was shown that in covalent crystals,
surface states may be associated with the unfilled orbitals
or dangling bonds of the surface atoms, which may trap an
electron at the surface. The historical development and a
summary of the various theoretical techniques used to calcu-
late surface states is given quite completely by Davison
and Levine}. Suffice it to say that most of the calcula-
tions since 1950 have used a LCAO (or MO) type approach
which was first introduced by Goodwin” in the late 1930's.

The second approach to this problem has been essen-
tially phenomenological. Its aim was to determine the
characteristics of the surface states by fitting a few
specified parameters of the theory to experiment. The
theory was completely analogous to that of bulk impurity
states. This approach has proved extremely fruitful in

characterizing the surface of several crystals, such as

germanium and silicon. However, the correspondence is

3
still small between the experhmentally observed surface
states and the theoretically proposed Tamm or Shockley
states.

It is the purpose of this paper to establish a
better understanding of the Tamm- and Shockley-type sur-
face states by showing specifically what parameters these
states depend on, and how these states change as the para-
meters which specify the periodic potentiai are varied.
Since most calculations which have been performed to date
deal only with semi-infinite crystals, many interesting
effects which are peculiar to finite crystals have been
neglected; consequently, investigation of some of these
effects seems appropriate. It was thought better to per—
form simplified exact calculations relating to several
different questions concerning finite crystals and using
several different types of boundary conditions in each
case rather than to carry out a lengthy approximate cal-
culation designed to illuminate only one aspect of real
crystals. For simplicity, the one-electron scheme is
used throughout, although it is probable that surface
polarons and many-body interactions between surface states
can occur. The Justification for using this approximate
scheme is that practically all the new concepts which have

been introduced into physics through solid-state theory,

a

such as energy bands, effective mass, and Brilluoin zones,
have been obtained via the one-electron approximation. HOw-
ever, what one achieves in simplicity of calculation by
using this scheme one pays for with certain ambiguities
which would not otherwise arise.

In the one-electron approximation, one attempts to
represent all the various forces acting on a single elec-
tron by a single static field acting independently on each
electron. This one field includes both the interactions
between electrons and those between ions. The one electron

Schrodinger equation, then, may be written as
. 1 _
’33-" v2 + “(kmfl‘Pnéb-r): End“) 44515.0.

The significance of the potential in this equation has been
the subject of much study. In the one-electron scheme one
assumes the existence of such an average potential acting
on each electron. From the fact that (in the Born-
Oppenheimer approximation)5 the interaction potential has)
the periodicity of the lattice, one infers that Vh (k,r)

has the same periodicity. What is usually done is to pick
a physically plausible potential for each state and solve

the one-electron problem. It is common practice to assume
the same potential for all electron states (i.e. Vh(k,r) =

V (r) ). The accuracy of this approximation is at present

5
not known. Also, when a V (r) is determined in an ag_hgg.
fashion it is clear that only those results which are
reasonably insensitive to the choice can be trusted. Thus,
one expects only qualitative agreement between calculation
and experiment.

The technique used in this paper is based on the
so-called cell-matching procedure. In this technique, the
assumed crystal periodic potential is divided into cells,

a cell being one period of the periodic potential, and
solutions of Schrodinger's equation are found in each cell.
By connecting the solutions in each cell continuously to
those in the next cell, a wave function is constructed which
is across the part of the crystal in which the potential is
perfectly periodic. This wave function is then matched to
the solution of Schrodinger's equation in the surface re-
gion to form a wave function for the crystal as a whole.

To solve any problem by this procedure in three di-
mensions is quite difficult, since the wave function must
be matched continuously from cell to cell at an infinite
number of points on the cell boundaries. The LCAO (or M0)
methods are better suited for this type of problem, al-
though they are not conceptually as clear. Since the
interest here will be in clarifying various qualitative
questions concerning finite crystals and not in quantita-

tive results, only one-dimensional situations will be

6

considered. Although this limitation may lead to neglect
of many interesting effects, it is believed that most of
the results obtained will have important analogs in three
dimensions.

In Section I, the wave function for the periodic
part of the crystal potential is constructed for any value
of energy without recourse to the Bloch-Floquet theorem.
Section II establishes the conditions under which surface
states may exist for the Tamm and Shockley type potential
terminations. Section III shows how to apply the princi-
ples discussed in Section II to the Kronig-Penney periodic
potential with Tamm- and Shockley-type terminations. In
Section IV, the effects of termination at an arbitrary
point in the end cell are considered. Section V deals with
the band structure above the vacuum level by investigating
the diffraction of normally incident free electrons from
the surface of a finite periodic potential. Section VI
deals with a related phenomenon: resonant electron emis-
sion from crystals covered with several layers of adsorbed

foreign atoms.

SECTION I

DETERMINATION OF THE FORM OF THE WAVE FUNCTIONS AND ENERGY
BANDS FOR A FINITE PERIODIC POTENTIAL

A suitable starting point for the calculation of the
energy bands and wave functions for a finite crystal, is
described in an article by James6. In this article, James
gives a particularly clear and elementary derivation of the
band structure of permitted energy levels for an infinite
crystal. Although he considers only infinite crystals, he
discusses the prOperties of all solutions of the Schrodinger
equation, including those which do not satisfy the infinite-
crystal boundary conditions, namely, the solution in the
forbidden bands. He also introduces a new parameter 0(E),
which, like the effective momentum p(E), depends upon and
partially characterizes the periodic potential. The essen-
tial reason for using James' results is that they may eas-
ily be applied to the case of finite crystals. This is
because, unlike most derivations of the form of the wave
functions for periodic potentials, James does not rely on
the BlocheFloquet theorem, which was derived for the case
of an infinitely extended periodic potential. The manner

in which the wave functions are constructed leaves no doubt

7

8
as to the form of the wave functions in all regions of
energy.

We shall, in summarizing and extending this work,
make the necessary modifications for the finite crystal.
Since we are modifying James' results, we shall give an
explanation of the various steps of James' derivation
which must be changed, and refer the reader to the article
when the analysis for the infinite crystal can be directly
carried over to the finite crystal. The results which
follow are valid for a general one-dimensional crystal, con-
taining a finite number of atoms each of which is repre-
sented by a potential well whose shape is symmetrical about
the center of the atom but is otherwise arbitrary. The
specification of a potential which is symmetrical about the
center of the atom is for mathematical convenience rather
than a necessity since the results which follow may be
altered to include potentials which can not be so defined.7

We consider the time independent Schrodinger equation
for a particle of mass m with energy E in a periodic poten-

tial V(x),

2 __ ..
- g $44100 + .VOQWOO " E4KX3) (1.1)

where V(x) is periodic with period a, and is defined only

for the range OSxSNa by

_. :: .., -1
V<X)=V(x+na>, ($330“ 1N , (1.2)

In the region where x>Na and x<O, we shall, for the moment,
leave the potential unspecified. The range of x will be
sub-divided into periods or cells of length a, such that in
the n-th cell

nasxflmlm , “n:- 0.1,” .,N-1, (1.3)

We choose the origin of our coordinate system so that the
potential in each cell is symmetrical about the center of
the cell.

We now fix our attention on the zeroth cell O\<x$ a,
and on a particular E. Since Schrodinger's equation is of
second order, it will have two linearly independent solu-
tions. For the potential as defined above, the solutions
considered are two real functions which we shall call g(E;x)
and u(E;x). The specification of g(E;x) and u(E;x) is com-
pleted by requiring them to be symmetric and antisymmetric,
respectively, about the center of the cell. That is,

10

50(6) 3(80) = gas a.) , (1.4a)

gyan?)

m

9’65») = —-9’(E;a). (1.4.)
u.(E) '5 21(50): -uCE;a), (1.40)

¢{(13) 55 117530) = 117530.). (Md)

For convenience, we shall normalize these functions so that
9(E;%)=1 . 9'CEsa/z>= 0.
U(s5%)=o . wag-6%): 1.

Since solutions to Schrodinger's equation have a constant

(1.5)

Wronskian, we then have, for any x,

w{9(E,-x),u(E;X)} 2: i , (1.6)
-: manila-D'- 9757>OW ;x).

It should be recalled that a necessary condition for the

linear independence of a set of solutions is that their

Wronskian be non-zero. This fact will be used later.
These functions need only be defined for the range

OSxSa. Since the potential has the same form in every

11
cell, g(E;x-na) and u(E;x-na) will be the independent solu—
tions in the nth cell. Within each cell, an arbitrary solu-
tion .4In. .of the wave equation can be expressed as an
appropriate linear combination of the corresponding cell

solutions:

111.10) = 0(1. 3(51’714Hfin1MEjl-M) , (1.7)

i 710.5 Z 4 (”+040

{01:7 0’1)2) ton)N’1.
These cell wave functions and their derivatives could now
be matched at the boundaries of each cell to form a con-
tinuous wave function for the entire crystal. H0wever, the
coefficients of our independent solutions depend on the cell
index and vary from cell to cell in a complicated manner,
making interpretation of the form of the wave function dif-
ficult. This difficulty of interpretation has been elimin-
ated by James, who defines a new set of linearly independent
cell solutions. Each of these new solutions is itself a
linear combination of g(E;x) and u(E;x), the combination in
each case being such that the new solutions are linearly
independent. He defines these so-called self-matching solu-
tions so that the dependence on the cell index is incorpor-
ated in each independent solution in such a way as to make

each independent solution in the n-th cell connect smoothly

on to the same independent solution in the next cell. With

12
the aid of the following definitions8

PE [0(5): (Qngo>/(ua’/llo); (1.8a)

 

 

a}. :.-= 01:03) =-- :6 (u.’/u.) pi. (1.81.)
i 1::
HELMET-1' 7 i;%>) (1.8c)
; >.... . J.
{1(BX) = C1Lfi)(—Q£€;£ + 71—57%!) 1): (1.8a)

James is able to write the independent solutions as

he é x {flu/)4,
41100 = E -(BX’M),{h ‘0’!) ..., ”.1. (1.9)

The significance of (If ) IE is made clear by the

relations

- £730) _ fi'{5;g.)

 

 

Oi: fi(£j0) - fi(£’,’0—) 1 (1.10)
.. f1(51‘_¢)
Y £10.70) (1'11)

.L
It should be noted that the quantity -( 111 / u.) ((17-
may be thought of as an effective wave number since it re-

duces in the free electron limit to the wave number k =

5171-177.

13
The general solution to Schrodinger's equation will be a
linear combination of these two linearly independent solu-

tions,
(«n = C+4‘+(x> + 0-41.00, (1.12)

where now 01+ and C... are constants independent of
the cell index. The nature of 4400 and I‘L[X)
will depend to a large degree on the value of f9(E). To
obtain an idea of how (DCE) will vary, it is necessary to
examine go(E), g'O(E), uo(E), u'o(E).

For very low values of E (E < Vfiin) the solutions

g(x), u(x) are damped and we must have g6, u' > 0, and

o
g'o, uO < 0. As E increases, g(x) and u(x) become oscill-
atory, and then oscillate with decreasing wavelength; it
follows that go, g'o, uo, u'o, oscillate between positive
and negative values, as E increases. According to James,
the order in which the quantities under consideration vanish
is
I, I I , I o a .

yc,(ja)uo)) (901%)) (yo)ao)) (yo)a0)j(1:13)
The order (within a parenthesis) of the zeroes of go and uo'
or of go' and uo is not fixed, but varies with the character
of the potential considered. These zeroes may in fact coin-

cide. The order in which these quantities go to zero has

in

important consequences for the existence of surface states,
as we shall see in the next section.

It follows from the above considerations that ’0
varies from - co to + w , depending on the values
of go, go', uo, uo'. A typical plot of f? versue E is
shown in Figure 1.1. Notice that the form which (3 takes
allows a natural separation of the graph into three regions,
depending on whether r< O, O <f< l, or F) 1. It should
be noted that this graph depends only upon the values of
go, go', no, uo'; that is, the graph depends only upon the
potential in one cell, and does not depend upon the number
of cells or the type of termination at either end of the
crystal. As we shall see below, this plot determines what
we shall call the band structure of the cell. (The term
band structure, as used here, refersto the structure of
this plot of '9 ‘versus E for a single cell solution. The
fact that several identical cells are Joined together means
that the band structure is constant across the identical
cells. Thus, instead of associating a band structure with
a periodic potential, we can in fact associate a band struc-
ture with one cell. The case of a periodic potential can
then be treated as a special case of combining the individ-
ual cell band structures.

We now investigate the dependence of the «PS on

in the various regions of interest. For each case, the

15

Hr

.msoflmmu pawn wanesm ecu mo coflpmsHEnopmo

hshym

M N

 

a

O a» N- Ma

 

 

.H.H musmflm

 

 

 

 

 

P .P h b 1|.I)IV|.I|1|'I'll¥lIleLilaiulll4L£l - AP! u - (1| VallefiL.-r.l Id
_
_ _ —
. i
1 I
. _
i .
. i
. .
_
u _
. .
_
_
. a
a
. 1 _
. . w
i .
.. _ u _
w . H u
N. nu .
m m w
.t. i
\0\ .r ..
\ p a . _
n . . _
a.
I i
)|\ w. k +
A i
. _
Il'l' III. I... pln -\v)]1l.ilci'|l!1 All-III.“ I .1 ATIIII.IOAIIIII|‘IIII LIOIIIEIIAIIII‘VIIII1170 ,1” by
_ I
n _ a.)
. u
u .
_ _ n

 

 

 

FIN/III;

 

 

 

 

 

 

 

 

_. .. ._.'.e._l._.._-_..

q-

01’

16
range of x and the value of n will be specified by
flaSXS(’IH/)6Z, h=0,1,.2., an, IV—!.
Case 1. P: ta),

In this case, r and r_ are equal to -1, 0+;o_=0

+
if uo'=,0, and 0+ = o_ = 1w ing= 0. Because of the nor-
malization, it is necessary to construct two new independent

solutions, which remain finite:

$.00: <¢+OQ£¢QQ> ) 4'10): (dfiaigr‘lj-ép , (1.11))

 

 

Letting uo' go to zero in these combinations, we obtain in

the limit

4’10) = {-01 “(12:14) 2 (1.15:1)

 

ta) = (’1)? 27m) fella-7111) 'Zlo yew]. (1.151.)
For go= O, we find,
4"“) = (‘21))! [(ZMImy(x.n¢).yo’u(x.M]’ (1-15a)

+100 = (’1)n gg'na) . (1.16b)

17
In both cases, one of the solutions repeats itself exactly
in each cell, except for a sign change, and the other solu-
tion increases linearly in amplitude as x increases.
Case 2. P<O .
In this case, (oi is imaginary and f1 , d"!

are complex. We may define
l. a 4k
’01 = i Z If] . (1.17)

Let us for the moment use the positive sign but keep in
mind that the negative sign is also acceptable. Then

I o J.
_ UHF!“ =(1‘llflz' (1.18)
n-(i-z'l'fli) ’ r’ 1+z'lfli)'

 

 

Since (Ix: r; ’rj‘: fl, and nr_=i ,we

may define a constant k by the relations,

= 61740. n = e—ZAR.‘ (1.19)

J

n
Then, characterizing our wave functions by k, we have

‘l’+k(X) = Cihd£(€5x-7za)j (1.2011)

18

4%“) = C [£1a£[£; X—M). (1.201))

We may define two new functions by

34 _
73mm = C I [x mag/6*”): (1'2“)

73100 = ci’é{”"“’£{5;x-m). (1.211)

It may easily be verified that

7311(x+a)= BM), 72,: (Ha):- 7.21/1). (1...)

Therefore, we may rewrite our independent solutions as

4’41“”: Ci'éxfll (X) J (1.23a)
4'40!) = 6.1%! file ()0) (1.23b)

where P;k(x), P;k(x) are periodic functions with period a.
Now we see that if the sign of 10* is chosen negative,
one need only change Ili+k to Ikk and qu

to llhk . These solutions are of the familiar Bloch
form and do not change in magnitude from cell to cell. They

may, however, change by a complex phase factor.

19
Case 3. P: 0'

In this case, r+ = r_ = 1, 0+
= 0. Constructing new solutions

=O_=01f8'o=0,

and c+=o_=;|;m ifuO
in the same way as for Case 1, we have for g'o = O:

|“()0 2 (X:7la) J (1.24a)

4’1(x)= 9.2101414) {ravage-M). (1.211.)

Similarly, for u0 = O, we obtain:

(m) = hiya—w-(zni/ga’uzx-m), (1.25.)

E (X-flQ . (1 .25b)

II’Z(X) = E

These solutions behave in the same way as do those for Case

1 except that these solutions do not have a sign change in

going from cell to cell.
29M. o<p<l.
Inthis case, P3.- )rt , fi- , (bi:
O< h<1 . i<n<+oo and

we may define a positive constant

are

all real. Since

r+ (fl-<1,

K by the relations

Ka ~ka
C (1.26)

20

Then,
_ kna
Ill-”(00 ‘- C £(EJX’7Z¢) J (1.27a)
~Km
W-ko‘) = C £05.; X-Zd)- (1.27b)
In the same way as for < O, we define periodic functions

P+K(x) and P_R(x) by

E]: (X) 2 C—k(X—M37€(PJX'7Z¢)J (1.28a)
fk(X—7la
P—kO‘) :2 C ).7€(EJX"M)2 (1.28b)

where BK ( X'I‘a) = Pix (X) as may be verified.

Then our two independent functions are

(IQKO‘): CKX HKO’) 1 (1.29:!)

[It/(0’) = C-kx RX (X) . (1.29b)

Qkfk_ increases exponentially in magnitude with increasing
x, while ’K decreases exponentially.
It Should be noted that the case = 1 never occurs
since this would imply that 90,170 = ual/[lo or
jo Ila, -- 30/ up = O, which would mean that g(x)

21
and u(x) are not linearly independent.
Case 5. P>1.
In this case-l < r_ < O, on < r <-l,' and $.13. 4’4,
Pi are all real. In view of the ranges of r+ and r_ .-

and the relation r r_ = 1, it is convenient to define a

+
constant K by
+I< -—Ka.
n="C a. ) r...=—-c . (1.30)
Then X I
7! 0L .,
4"” (X) = (’flnc £(tJX-M)) (1-30a)

4’4: (x) = we”? (5; x— m) . (1.301))
Defining new functions
G410) = 6-1)"c*("”");,€(5; X-Zd), (1.31:1)
G40!) = (‘1 )ncfka-MggOFJ/Y-M), (1 .3111)
it is easily seen that
Gik(x+a)= -G1sk(x) ., Guam): 6100) (1.32)

so that the 0+ K(x) are periodic functions with period 2a.

22

‘We then have as our independent solutions

KX
44K“): C aft“): (1.3311)

delX) =3 C.kx G—K (X) o (1.331,)

The magnitudes of 41K and 4"“ behave as in Case ’4.
For purposes of clarity in later sections, we shall

refer to energy regions in which the wave function is of

i161 -2
(ax) = C./( 6 721(0) + 6.; c £12m (1.32).)

as pass bands (P.B.) and energy regions in which the wave

function is of the form

4’00: C+kckxflkm + C-kgkxp'kooJ (1.31m)
4’00= Cuckaiklx) + 61,5”6140 (1.3%)

as attenuation bands (A.B.).

SECTION II

TERMINATION OF THE PERIODIC POTENTIAL - DETERMINATION OF THE
CONDITIONS FOR BAND AND SURFACE STATES

We shall now consider various boundary conditions
which may be imposed at either end of the periodic potential.
The type of boundary condition we wish to impose will, of
course, depend upon the specific physical phenomena we are
trying to represent. For example, near the surface the
atoms do not have symmetrical forces acting on them as in
the bulk. This results in a deviation from perfect period-
icity, which may take several different forms. One of these
deviations might be an increase in a basic lattice trans-
lation, the closer to the surface the atom is situated.
Another deviation might be total reconstruction of the sev-
eral layers of atoms near the surface into a different type
of structure from the bulk. Several other types of devia-
tions might occur, and in general we have to consider the
specific type of bonding and the lowest energy state of the
system to decide which type of deviation will occur. We
might also consider the dirty-surface case which arises
when the unsaturated forces or dangling bonds of the sur4

face atoms tend to attract foreign atoms which may be

23

an

floating around outside the crystal. This usually results
in h or 5 layers of foreign atoms being deposited on the
surface before the bonds are saturated and requires consid-
eration of interfaces between crystals with different per-
iodic potentials.

We shall start with the simplest boundary conditions
possible and try to understand most of the phenomena which
occur for this case. We shall also outline a procedure for
the computation of more complicated terminations. This
method is straightforward enough so that further calcula-
tions using it could be made without any major difficulties.

The simplest way in which to terminate the periodic
potential is by a step function at either end. (Incident-
ally, the relative smallness of the surface dipole contri-
bution to the work function bears witness that this type of
termination may not be too seriously in error in many
casesg.) For the present, consider the termination to be
made either at a potential maximum or at a potential mini-
mum (see Figure 3.1). The first type of termination is

10 whereas the second

type of termination is referred to as Tamm-typell. These

commonly referred to as Shockley-type

classifications are named for the authors who first showed
that these type of potential terminations could have sur-
face energy states associated with them. The Shockley-
type potential termination is generally applied to covalent

25

crystals where the surface perturbations are small whereas
the Tamm-type termination is thought to be valid in cases
where the surface perturbations are large. The'Shockley
termination is usually thought to have more physical valid-
ity; it corresponds to terminating the crystal between two
atoms. Both types of terminations allow us to divide the
periodic potential into cells in which the potential is
symmetric about the center of each cell. The Shockley-type
termination yields a cell with a potential minimum at the
center whereas the Tamm-type has a potential maximum there.

For either case, we go about solving for the energy
states by matching the wave functions in all regions at
their respective boundaries to form a smoothly varying func-
tion of position. This matching, of course, will only be
possible for particular values of the energy, and it is our
purpose to find those energies which do allow a continuous
wave function. These energies will be our eigenvalues. The
band structure (i.e. the regions of energy in which the wave
functions are either running waves of constant amplitude or
exponentially increasing and decreasing waves) is determined
completely by the periodic potential. The exact position of
the eigenstate in this band structure will be determined by
energy conditions which are derived below. A

For energies less than or equal to the lesser of V1

and‘Vé, we set up the matching conditions as follows: For

26

x 5; O, the solution of the Schrodinger equation yields

, X ——.
4'00 =AeK' , K, =V§e(w-£),- (2.1)

 

where we have used the condition that 1" go to zero at -m .
For 0 < x\< Na, we have five forms of the wave function
which the solution of Schrodinger's equation takes, depend-
ing upon the region of energy. In all cases, the general
solution will be a linear combination of the independent so-
lutions with constant coefficients which.must be determined
by the matching conditions. Let us for the present discuss
in detail only the case of the running waves and then list
the other cases, since the procedure is the same in all

cases. Then for Case 2, t91< O,

APO) == Cit/1 (III/26:) + C—k 44.1100, (2.2a)
q’ikm = ciz’ézPik/XL (2.2b)

For x 2 Na, we have

-k (x—Na) ‘ 'fi
4’(X)= BC 2 I Kz=fiii(lé-£).(2.3)

 

Continuity of the wave function and its derivative, or

alternately continuity of the logarithmic derivative at

27

0&0} + C~A 0.16

x = 0, requires

 

(:fié it (3 )é
_. C'flé— C-A .
" :04 C/év‘C-é I (21)
'where we have used
41% L17“; ’ 44‘ L: (7:, (2.5)
(kflfi]x: 1: , ‘IL‘HIX::;
(z; ==r-‘ GK: .

Similarly, at x = Na, we have

[Ma 46%
("A e —C—é€ )) (2.6)

“ “I "' C7- 16 a: “—Axyflh;
1 6M6” [4645
where again we have used the properties of the wave func-

(4),] == 016 60% (’1): 01-13%

flew. Xena.

£1410. __ 4.442
qkukijflé: C: )' ‘PLAh1 =jlf‘ca

Dividing Eq. (2.1) by Eq. (2.6), multiplying out the quan-

tion

(2.7)

titles and grouping terms, we obtain the quadratic equation

28
in C+k and C

Ciztfflf’szb)“2.1.(1276)5/‘Iz)éha%afi (2.8)

— Cid—(”26, 7%,) == 0 .

a:

-k:

Dividing through by (KIf ((2)6

terms of C-k’ we have

= ' "7) AV“ 1311111). )
CM [lérgkm (:évakghfi 9

and solving for C+k in

 

Now -—l€ %) girl and K1, K2 are always positive so

we may define a phase angle ’6 by the relation

' _ /K "kl ' (2.1 )
50¢ “ \éfldsmém O

 

Then, Eq. (2.4) becomes

=[+C0$¢+lsillé]€ “CLA (2.10s)

01?

{AI/4:411

CI]? = i C C“ . (2.10b)

29
We may now substitute back into the Eq. (2.4) or (2.6) to
obtain the conditions for matching. Substitution into (2.4)
yields the two conditions

a:=-z ’ut
k..=- ;aiith) M) (.11.)

If we had used Eq. (2.6), we would have obtained

K2,: ~iail‘an (M) > (2'12”
‘ Coll(‘é£/4‘:é) . (2.12b)

Both sets of equations give equivalent energies when the
equality is satisfied. We shall, for convenience, use the
first set of conditions. Both sets reduce to the same set
when V1 - Vé since the phase angle goes to zero.

For the other four regions, the matching conditions are
listed below.

(1... 1. ,9: too , g.u.’= a.

(El—2%KZ.) = ”6%) 91, = 0. (2.13)

2(S{#§3;.)‘=’ ;:—(E§?;) ) "

 

<32a£5e4}: ‘f7==:£9 ) ‘sz,€gh9-:: ‘7.

(M) =NGE') , now.

I
:2 (fik91F/Ki. :.J<%fi(c%%%:) ) J§h::=’<7‘

 

Casein 0<’o<1.

kl = '— a; Z‘anfi(flaz:§)>

 

 

‘—0;Caz%(M§fJ)>
K=—‘L[qy(1:___€o§)
= ”gifi

521% I: (:73215—3 )Sz'lzé kA/a .

(2.1#)

(2.15)

(2.16)

(2.17s)

(2.1711)

(2.183)
(2.18b)

(2.18c)

31
Case 5: ’9 :>.1. .

k. = wan/MW). (2.19.,
K, =: 'WCOZtA (W), (2.1911)

where

 

J.
Lac—I
k=_i[0~7(fF-£1l ) (2.20)

and a; , sinh I are defined as in Case ll.

These matching conditions are transcendental equations and
cannot be solved explicitly for the energy of matching.

The correct energy eigenvalues can be found by a graphical
method. The left-hand side of a particular matching condi-
tion is plotted as a function of energy on the same graph
as the right-hand side of the matching condition. The
points at which the curves of the left and right hand sides
cross yield satisfaction of the boundary conditions, and
consequently give the energy eigenvalues. The regions in
which the curves cross will determine the character of the
wave functions. The boundaries of these regions, as noted
in Section I, are determined by the zeros of g'o (E),

go (E), u'O (E), uo (E). It is important to note in what
order these functions go to zero as the energy increases be-

cause this order determines whether conditions are favorable

32
or not for surface states to occur. Consequently, it is in-
structive to follow the matching conditions as a function of
energy for a particular sequence in which g'o, go, u' , u

o 0
go to zero to obtain insight into what conditions determine

whether a surface state might occur.

For simplicity, we consider the case where V1 a V: so
thatKl-K2 and «fi- 717 :0.

Let us first consider the case of the zeroes occurring

in the following order with increasing energy.

3:4)?» 119’. unyofiyoflofi 11., ~-- (“1)

l
A qualitative graph of -(ZIGM) [Ioli is given for this
case in Figure 2.1. Notice that a discontinuity in the

VW'

slope occurs at g'o = 0.

+

i.
z.

 

 

l
|
|
I
1
I
I

— ’(flfil/uoflfl

 

 

l

H: “I ‘Q‘
‘4 ‘V (3

§—|3 A|g_3——V| MII?‘

0.Q’1°"I IOQ'Q'I I°l

 

9§=O 941:0 11¢=0 u,— -0 ga=o gza Ill-:0 14,: 0 go’eo

Figure 2.1. Effective wave number versus energy - Case 1.

33

The above characteristics are those for the reduced zone
scheme where we have limited k to the range 0 5 k S. 7r/a. .
If we had taken the sign on I'M-g; to be positive for the
odd-numbered bands and negative for the even-numbered bands,
we would have obtained completely positive values for the
graph above with discontinuities in the slope occurring at
u'o - 0 also. In such a case, an energy versus k plot would
be in the extended zone scheme.

The left-hand side of each of the matching conditions
given by Eqs. 2.11-2.20 is 111 =W(V,— E) ‘, and

when plotted as a function of energy is Just half of a

 

parabola with its axis along the E coordinate. It is posi-
tive for all values of energy. The right-hand side of the
matching conditions will have alternating regions in which
we must use the appropriate formulae.

The first region, corresponding to the energy at the
bottom of the potential well, is an attenuation band_
0 < < 1. In this region, -(YIo’/%)I Ii is positive, as

are tallA(Kfi/Q./Z) and Cot (KA/a/z), so that
" (’%:Fi> ltdlzé (%)< O) (2.229.)
-<—%rewz%<%<o. (2.2...

31+
As the energy increases to the zone boundary where go' a O,

, .L
K approaches zero, as does —(”0/[L0)IFI 1' . Therefore, the

I . I 1
product '~%Ifliéfl£{%) goes to zero while the product -116; FI".
5 c (8%) approaches the limit -(-%)/N< O. This last

limit is a result of the fact that -€4flto) I'm-Ii goes to
zero at the same rate as €051,099) goes to infinity
so that a finite limit is achieved at go' a 0. Both func-
tions are less than zero for the whole region so that no
crossing with the K1 curve is possible. At go' a O, we
have f- o, k = 0 so that -—7£-:/ [Zia/Mb?) is zero and
—(—2%If/£cot(-&g£’) has the value véuvl/fla/l’,’ provid-
ing a smooth connection into the next region of a pass band.
From the go' - O boundary to the go -= O boundary, (a varies
from O to - co, and consequently, It varies from 0 to “IT/a.
~(lloI/llaNf/t is positive in this region.-cof(éjg) is Just
tall(éla’-¥) so that both functions have the same form,

one being shifted by 1172. Both functions have periods of
71' within which they vary from - 00 to + 00. Now, as k
varies from O to 1r /a, tan(k//a/z) goes through N/2
periods so, that it passes through any positive number ”/2
times if N is even and (N+1)/2 times if N is odd. "Cot( 1‘49)
also goes through n/z periods and goes through any positive
number N/2 times if N is even and (N~1)/2 times if N is
(odd. Multiplication of talz(kA/a/v.) and —60f(,4/\/a/z)

by -(llo//llo)q0r£ changes the slope of these functions,

35

but is is true that the total number of intersections with K1

will be N, and consequently there will be N eigenvalues. As
I
the boundary go = O is approached, —%IIpI%/é£é goes to the
I
value ~(—%)}I< o if N is even and 1- 00 if N is odd. Like-
I f I
wise, ~(—%fo/£)Coz€éz4" goes to -_I- 00 if N is even and -é-%3)Af
< 0 if N is odd. At go = O, we connect on to an attenuation
’ 1
band where F > 1. 0n the boundary, -(‘%I Iz)z‘alli(%)
. I I 1
is equal to -(-”H‘-o’)l\/ and - (’filfll)&fi( ‘91s equal to
-_1_- 00. As the energy increases in going to the next bound-
1 .l
ary uo' - O, -(’%Iflz)w(gf/a)increases to zero and
’ .I. “a. 5
(.%lfl9&0d(¢)increases from - 00 to the value 5;; O.
The two functions are both negative for the whole region so
that no crossings are possible. At uo' .. 0, (0: - 00, k a
1 J.
"IT/a, and "(Ila/2b)?!" is zero, going negative in the next
region of a pass band. Since k varies from T/a to 0 with
increasing energy, the behavior ofW) and -Cdl(é¢,t)is the
reverse of that found in the first band. However, since
ygl .L ' 1
- I I” is negative for this region, the products {—filfl‘)‘
”" as ’ s ~
'tam( z, ) and -—[—%, If! )cdfl‘) yield the (same form for the

I .1
' a - 3 0
functions as in the first band. At no ’0, we have( %3Ifl)

'tdfl[fiAh/z) = 0 if N is even and \fi;; < 0 if N is odd
while —6%Ifl/£)Caf¢i¢a)is equal to fyfly, if N is even

and zero if N is odd. In going from u0' = O to 110 a O, we

again have a total of N crossings of the K1 curve. At 11 a

o, —Z‘a0’;lrlefbn(éé€4) goes to Ng< 0 and reg/#leégg)

36
goes to + 00. These functions connect on to the forbidden

band boundary conditions (for 0 < f < l) at 110 - O:
—‘€.3:,é)m(m)- ”ff and -(—%I/I“)ca2é(m)
- + 00. Continuing across this region to the boundary gO '

- 0, “($19, I‘M/h) increases to zero while -(- %0l [‘9‘
M(W) goes to - (- %/%)/N<0. Again both functions are
entirely negative in this region so that no crossings of K1

are possible. At go' - O, we are again back to the same
type of band as considered before. As may be seen by the
above arguements, for the sequence of zeros of the functions

30', 30’ uo' and 110 considered, no surface states are poss-

ible.

we now consider the case when the order in which go',

g , uo', uo go to zero deviates from the order given above.

For example, suppose tht order is given by

.90]; 9' J ”0’, 11.0) 90’) (WI)90))%J%S'°' (2-23)

In this case, the order of the functions in parenthesis,

110' and go , is reversed. A qualitative graph of -(a”/Uo)lf/"

:18 given below in Figure 2.2. Comparison of this graph
with that of the previously considered sequence shows that
'the main result of switching the order of go: no. is to
cause -‘("°’/llaa)l/’It to be zero on the upper band edge at u0'
= O, at which point it goes negative in the region f) l.

37

l w
|

' ll
: ll
! II
[I II
I I
l l
I 1

fl

 

1

 

 

 

‘31 I
\.
~§ i i-
\'J "" | “' I Isl
V O ._, v o 0 v
3"V A-IVIfi—VIAIV $4
I‘ O :I Q— Q...l 0 LI Q... Q..." 0"
grog .9o=0 115:0 uo=ogo= -o uo’eg 90:0 ans—0 0-0

Figure 2.2. Effective wave number versus energy - Case 2.

At g o - O, -(”9’¢l,)lf(* is equal to - 00 at which point it
increases to some negative value before going to negative
infinity again at the upper band edge “0 a O. The essential
feature to notice is that the form of -(“0%(o)lpli is
altered in the two band regions adjacent to the attenuation
region in which it goes negative. In the lower band, be--
tween go' a O and u0' = O, ‘(%élf#)bn(é%) will go
through N/2 periods if N is even or odd achieving the value
oantuo'=OifNis evenand W‘(gé)>01fNis odd.
Notice that the ratio ya/fi is greater than zero in con-
trast to the previous situation. For N odd, the magnitude
of the product #(fi) will determine whether the crossing

38

of K1 occurs in a pass-band or .in an attenuation band.
For N small and the potential weak, the value (61%) > K1,
and the crossing will occur in a pass band region. For

N large, the ratio 95/ , which depends only on the per-
iodic potential, remains the same but is reduced by a
large number N so that flgk K1, and no crossing occurs
in the passeband region. The function -(-%élf(£)6d2" ( £21?)
behaves in a similar way, attaining the value of #69791)
> 0 if N is even and 0 if N is odd. Thus the number of
band states which occur will be either N if 7619990)) K1
or N-l if fi(.7%)< x1. At uo' - o, -(—%Ifr£)tan/z(£44’)
- O and —('%llfllé)dobé(5g- # (fit/f.) . In going from
no, - O to go‘ 8 0, --[-‘-‘g/ [ably—gnomes“ to the
value ”(M/”0) > O and -- -%If/£)&fl('§éld/ goes to + 00.
Again, if N is small such that N (“.510)< K1, then no cross-
ings will occur, but if N is large such that A” (4/14)) K1,
we obtain crossing in the attenuation band. It should be
pointed out that an intermediate ease with #6907?” > K1
and ”(ual/lb) > K1 may occur, in which case there is only
one crossing in the forbidden band. For N large, such
that #(YOT/yo>< K1 and N(.%€)> x1, we obtain two cross-
ing in the forbidden band. By analogy with the band con-
sidered previously, the allowed band bordering this at-
tenuation band from above will have N allowed states if

iv(%:)<x1 and N-l states if AM“: > K

1 . We may

39
summarize the results of the above reasoning by the fol-

lowing table. (The function written above each column.is
.to indicate at what energy the conditions are to be eval-

uated.)

Table 2.1. Existence conditions for Surface States - Case 1.

.. k"0 (90‘30

 

I-VL %)>k' N(%)<kl No surface states

 

One surface state

 

fi(%)<l<l N (%;)</(, Coming from lower band
r I I On f t t .
# §§)>k, N (flip/(g cogizgurfgg; 3P3; band

5 I
t (3m, N(-Z’;:)>I<. assistants“... 1......

 

 

 

 

 

 

We see that for large N, one of the pass-band states from
the two adjacent pass-bands~moves into the attenuation
band to form two surface states.

The situation is similar when the zeroes of no: 30'

are interchanged; that is, for
I I ( l a
902.90) 5L0) .90) 0)) ' H (2.24)

In this case we may represent the conditions for surface

state occurrence as

#0

Table 2.2. Existence conditions for Surface States - Case 2.
fzg=<9 _ IZD‘=¢9
I I

fi%)>kl Ngfl<kl No surface states
’ One surface state

1. I

N (_‘%)<k’ N(%)(k’ from lower band

One surface state

.L (w) >k' ”6%) >/(' from upper band

IV
I ’ 2 r
‘5’!’( 10%) < k) N653) kl refuggirameiower ”and“

 

 

 

 

 

 

 

 

 

Thus, we see that a necessary condition for the
possible occurrence of surface states is that the band
edges cross. When this condition is fulfilled, the actual
occurrence depends upon the number of atoms considered
and on the strength of the potential.

The effect of unequal terminations (i.e. V1 K Vé)
at either end of the crystal is to spread the energy
levels. Hewever, this effect is not large since the
difference between‘v1 and‘Vé is manifested in K1, K2 which
vary only slowly with changes in'V1 and'v .

2
For a periodic potential terminated at an arbitrary

41
point in the end cell by a step function, the formalism
which we are using is not the most convenient. However,
the matching calculations may be done in the same manner
as above, matching the wave function and its derivative
at the termination point to obtain a contiruous wave
function. This procedure is carried out in Section IV

for a Kronig-Penney12

type periodic potential and the
results are examined. It is found that this type of
termination gives rise to many other surface states be-
sides the ones considered above.

For arbitrary terminations of the crystal potential,
it is necessary to use iterative techniques to integrate
the Schrodinger equation. For the region of the crystal
in which the crystal is perfectly periodic, we know the»
wave function and its derivative at all points. At the
point where the potential starts to deviate from perfect
periodicity, we may iterate through the arbitrarily vary-
ing potential to a point outside the crystal where the
potential is constant. In this region, the wave function

'KIXI , so that

must have the exponential decay form, 8
we may match this to form a continuous wave function in
all regions. This method may also be used to connect

two different types of periodic potentials by a region
of non-periodic potential. In this way, a wide variety

of problems may by dealt with, with few changes in the

42

calculational apparatus necessary to change from consider-
ation of one phendmunon to another..

Another application which can be made of this method
is to the calculation of the reflection and transmission
coefficient through several identical barriers. This
topic will be discussed in Sections V and VI.

SECTION III

TERMINATION OF A KRONIG-PENNEY POTENTIAL

In order to obtain a better understanding of the
principles discussed in Section II, we next consider
the Tamms and Shockley—type terminations of a Kronig-
Penney13 type potential, as shown in Figure 3.1. By
the Kronig-Penney type potential, we mean an array of
a finite number of rectangular well potentials. The
reason fer choosing this type of potential is that it
leads to a Schrodinger equation which can be solved
easily, and the energy band structure associated with
it is similar in many respects to that of a real crystal.
Two important features which serve to illustrate this
point are the existence of points of contact between
different allowed bands (a special type of band cross-
ing) and the behavior of the band structure as the energy
approaches infinity; namely, the attenuation band gaps
go to zero. Mane,“ and later Stat2,15 and Koutecky,16
have derived conditions for the crossing of bands. In
these papers it is shown that crossing occurs for poten-
tials which require large numbers of Fourier components

to represent them, with some of the components being

43

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negative. Since a rectangular well potential abounds in

harmonics, as does a real potential, we expect the Kronig-
Penny model to be a useful model of a real potential.
A . SHOCKLEY-TYPE TERMINATION
We begin by considering the Shockleyetype termination.
Dividing the periodic potential into cells yield a cell
with a pctential minimum at the center.

 

 

 

 

LT— -b 'w "—4
It). «~—r———~ I
o! . j 1
I | 7
I + l I
C) 1011 (L

In the region, a/2-b€ x€ a/2+b, the Schrodinger equation

has two linearly independent solutions,

 

 

300: scam-4i?» (3.1.) .-
. k W) k=\f%(FH/o/1>2
1(0):: 5772,51“ 21’ - (3.11:)
Note that

9(%)=1 , 97%):0, (3.2a)
U(%)=0 ) “7%)=1. (3.21:)

If we now connect these solutions individually to the

45
solutions of Schrodinger's equation in the region
osxSa/2-b, we obtain for E<VO/2, the values of

g(x), g'(x), u(x), u'(x) on the left cell boundary:

90 z 603 )1, as; [43-5)— £52744 Sabra—5), (3 .1.)

‘9; = -Kcoséési11/(Z-a)+lé Sz’méb/Mk/Qz-é), (3 .41»)
2L. = wt 605M SIM/((72.1 — isiraazas/W-éu .uc )

710’ = 605% CasMflj-afl' {Sagas/MB .ud )
K -= \féfl‘m/z - 55‘.

For E - vo/2, and E>Vo/2, we must connect in a similar

 

way to obtain expressions for go': go’ uo', uo. To
determine the pass bands and attenuation bands, it is
necessary to make a plot of F versus E, where F has

been defined in Section I as

F ::: (95/30/[af/flo). (35)

Such a plot was described in Section I. The exact posi-
tions of the band edges (i.e. energies at which one of

the quantities go', go, u ', uO equals zero) will depend

c
on a, b, V6. To obtain a qualitative idea of what effects
these parameters will have on the band struetures, we may

make energy versus potential depth plots for several values

47
of the well width and the lattice constant a. Such plots
are shown in.Figures 3.2, 3.3, and 3.4. Figure 3.2 has
a well width of 0.75a, while Figure 3.3 has a width of
0.5a, and Figure 3.4 has a width of 0.25a. These graphs
were obtained by making a f, versus E plot for a given
I’u’

o 0
go to zero (i.e. the band structure). The parameter‘Vb

a, b,‘v6 to obtain the points where go', go, u

was then changed incrementally and another (3 versus E
plot was made. In this way, the dependence of the band
structure on‘Vb was obtained for a given a, b. Allen17
has obtained similar but more detailed graphs, varying
the constant b.for 12 values, for the Kronig-Penney type
potential, and the reader should refer to his article for
more detail.

Note: In all graphs contained in this thesis,

—'hawe s 11 use units in which 22b” 1. To this

end we choose our unit of length to be -

strom and our unit of energy to be (1.97) '

3.88 electron volts. This energy unit will

be denoted by E.U. hereafter.
For the above mentioned figures, note that for large values
of the potential‘vb, the pass-band regions are very narrow,
corresponding to the tight binding of electrons to the
constituent wells or atoms. For large energies E, the
effect of the potential is not "felt" as much, allowing
the pass-band to widen. As we decrease the strength of
the potential, the pass-bands widen allowing some of the

band edges or boundaries to cross or Just make contact.

 

48

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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m °"°"‘"‘°='="===‘:E=E=E55;:§:;. ,,,,,,
o "'°-'°'-:-::::=:;. ....
go=0
g'o=0
-— 2 L-
Uo=0
-4 _ A = 5.4 U'o=0
B = 2 . 0 2 5
g'o= go=0
"' 6 J L 1 l 1 4
0 2 4 6 8 10 12 14
< v
0

Figure 3.2

Band structure as a function of well
depth for a Kronig-Penney potential
with 2b= 0.75a.

49

 

 

 

 

 

 

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fififififififiF”” _w~3g££££ °°°°°°°°°° U'0=0
T 4 """"
a 2 """" 9°=°
Hr Efififififii 111
Q) .;.:. ........
a ifififitws ......... ,9'0‘
0
Uo=0
-2 -
U'o=
A 5.40
-4 - B - 1.35
go=0
._ 6 4 1 1 l l 1 \
0 2 4 6 8 10 12
<E————Vo——————

Figure 3.3 Band structure as a function of well
depth for a Kronig—Penney potential
with 2b= 0.50a.

50

 

 

 

 

 

 

_2 __ 90§0
A = 5.4 9' o§
-4? B = .675
-6 1 i 1 1 1 1
0 2 4 6 8 10 12 14

Figure 3.4 Band structure as a function of well depth
for a Kronig-Penney potential with 2b= 0.25a.

51

This is best illustrated in.Figure 3.3 in which the upper
band edge of the third pass-band edge crosses with the
lower band edge of the fourth passeband, the upper band
edge of the fourth pass-band crosses with the lower band
edge of the fifth passéband, and the upper band edge of
the fifth passeband Just touches the lower band edge of
the sixth pass-band. For the tight binding situation,

the band edges are determined by the sequence of zeroes,

v.91)?“ Zlo’,l£o,90’) 901m/fllo, .. . . (3.6)

As we saw in Section II, this sequence of zeroes does not
allow the emergence of surface states. As the potential
is decreased, the bands which cross have an attenuation
region between them, which is favorable for the appearance
of states.. The points at which the bands cross can be
shown to occur only for E >‘Vb/2.18' By consideration of

the equations for go', go, u ', no, one can show that the

o
crossing points are given by

_. mr" W’-
$71155 ”+ -—-Z§2—’ V

°CVBSS> O. (3.7)
M'::ing- fit .2 .))

where n,‘m are integers. It should be noted that the

appearance of band crossing only for E > Vb/a is not

52

peculiar to our model potential but can be proved to hold
for any arbitrary periodic one-dimensional potential.19
(This does not hold for three dimensional crystals.)
Since the value of the vacuum level energy is of the order
of 20 eV’for most real solids,2O we see that to obtain
surface states V6 must be weak enough to allow band cross-
ings within this range, and consequently, surface states .
of the Shockleytype would be expected to appear for nearly
free electrons. we note also in passing that_the width
of the attenuation regions does not decrease monotonically
with increasing energy but varies in width_due to the
crossings. Hewever, on the average, the width will go to
zero as the energy goes to infinity. As Vb goes to zero,
the attenuation band-widths go to zero yielding a con-
tinuous pass-band region, or a free electron region.

To proceed, we consider specific values for a, b, v0.
As a-reasonable approximation for a real crystal, we let
V6 have the magnitude of the'firzt Fourier component of
the O.P.W. potential for silicon along the (111) direc-
tion and ta‘ke'V1 - V: - V600, the average potential for
silicon. Along this direction, Si has a lattice constant
of approximately 5.4 A0. The well width will be taken
to be equal to the barrier width. For the present, we
shall limit ourselves to four cells. Thus

53

V. =Va= raid/=- 4.5’.e’.u., (3.8.)
Va = I3.2 eV= 3.45%, (3.811)

a= £¥A° (3.8c)
A = /.35flo (3.3a)

N: 17‘ (3.8e)

A plot of P versus energy is given in Figure 3.5. Notice
that the lowest pass-band is very narrow, of the order

of .02 E.U. in width. The attenuation-band above this
pass-band is large, of the order of 1.75 E.U} or about
6.8 ev, which is much too large a gap, the experimentally
observed gap for 31 being about 1.2 ev. However, one

of the difficulties associated with the choice of a po-
tential in the one-electron approximation is that the
Fermi energy is not unambiguously known.22. By inspece
tion of the widths of the attenuation energy widths up

to the value of the vacuum, 6.5 E.U., we see that a
width which is of the same order of magnitude as 1.2 ev
would be that width occurring between 5 E.U. and 6 E.U..

54

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1. me an an m e s a a m m a a- m
. . . ...
u n O
u a u r—«u
.. _. . a.
..._ _. . . m.
a . .
M. u. a . .0
new. "M a
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u
u M v 7‘
. n
r
7 q:
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o o o o o o o
b __ = = = = = _. __ = = = ._ .s
0 0 0 0 0 0 0 0 Q 0
L A. _ M 1.

55
If the Fermi energy were assigned to this region, the

work function would be of the correct Order of magnitude
as would the attenuation band width. However, this width
is still rather wide. This discrepency may be accounted
for by the fact that, while we are using an O.P.W. po-
tential, we are not orthogonalizing our wave functions
to the core wave functions, so this potential is incor-
root for our wave functions. Antoncik23 has shown that
this orthogonalizationumay be taken into account by sub-
tracting an effective potential from the O.P.W. poten-
tial, the effective potential being calculated from the
core wave functions. The sum of the two potentials is
known as the pseudo-potential. The valence and conduo;
tion wave functions may then be calculated without an
orthogonalization procedure. For our purposes, we as-
sume that the O.P.W. potential is reduced in.magnitude
(i.e. VB is reduced). By inspection of Figure 3.3, we
see that if V6 is reduced to about 2.4 E.U., the atten-
uation width between 5 E.U. and 6 E.U} becomes about
1.2 eV'which is approximately the energy gap observed.
Since this calculation is for illustrative pur-
poses only,we shall use the value of V6 - 3.4 E.U} in
the calculations which follow, being satisfied that we
are in the correct range.

using the boundary conditions derived in Section

56
II, (Eqs. 2.11-2.20) we plot the left hand side of these
equations on the same graph as the right hand side to
obtain Figure 3.6. The vertical lines indicate band
edges. The solid curve is Just the left hand side of
the boundary conditions, K1 --?\/"7"(vl - E)‘. The dotted

 

curves are the two functions on the right hand side of
the boundary conditions. The crossing of the K1 curve
and the dotted curves determines the eigenvalues. This
procedure for finding eigenvalues is exactly analogous
to the method of finding eigenvalues for an electron
in a box with finite ends. Notice that in Figure 3.6,
the band edges are determined by the sequence of zeros

of go', go. uo', uo in the order,

$5.90 ) Zia/2741:2912 (”o/J90): (90/1210). (3-9)

From the considerations of Section II, we expect to

find surface states occurring between (uo' - 0, go a 0)
and between (go' - O, 110 - O). This is indeed the case.
A pass-band state from the third pass-band has moved in-
to the attenuation region. The number of cells is not
great enough or the potential is not strong enough to
move the upper pass-band state down into the attenuation~
band, so there is only one surface state. The same sit-

uation applies for the next higher attenuation region,

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58
except that now for the partial pass-band (which would

be pass-band 5 if‘V 'Vé were large enough to include

a
the whole pass-band) one of the pass-band states has
moved down into the attenuation region. Note also the
effect of varying Vi or V2. Since the dotted curves
do not depend on‘V1 or‘v2 (if‘vl =‘Vé), we can vary

K1 -V%gyl - Ejto see that increasing and decreasing V

 

l
raises and lowers the crossing points and consequently

the energy eigenvalues. Since K1 depends on the square
root of‘vl, we do not expect the energy crossing points
to change very much as Vi is varied except for energies
near the vacuum level.

An energy versus (k and K) plot in the reduced-
zone scheme is shown in Figure 3.7. The horizontal
lines indicate the band edges. The solid 99?V°3 are.
the E versus k plots in the pass bands whereas the dash:
ed curves are the E versus K plots in the attenuation
regions. Te be entirely correct, we should have plotr
ted E versus K along the imaginary axis perpendicular
to the real values of k. This would then yield a band
structure with pass bands connected by imaginary loops
across the attenuation bands. These imaginary loops

24 as "real lines", their exact

are referred to by Heine
meaning coming from the analytic continuation of k in

E(k) into the complex plane with only real values for

ENERGY

-2.00

59

 

n——..c———.————.——— _-—————-———-———-———————

V um Level

P.B.5

 

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Figure 3.7.

WAVE NUMBER x (a/1r)
Energy versus (k and K). The X's denote
eigenvalues which were determined in
Figure 3.6 (Shockley).

60

E considered. The X's represent the eigenvalues as deter-
mined from Figure 3.6. Notice that in attenuation band
3, the value of K is very small so that our wave function
in this region is not attenuated very strongly going in-
to the interior of the crystal. In attenuation band 4,
the value of K is larger but is still not large enough
to attenuate the wave function strongly.

Figures 3.8 (a-e) are plots of the wave functions.
The pass-band functions are labeled as (hf where at is
the pass-band index, and F is the index of k); in the
pass-band ('e.g. 44,. denotes pass-band l, the lowest k
value labeled k1). The surface wave functions are de-
noted by “’1.S(~ 4).“? where ‘i } indicates a sur-
face wave function, 3' indicates the attenuation-band,
the numbering starting above the first passeband, and
8 indicates the label on kg (S=i or 2.) . The
symbolic notation rv l'Ju, (a is to indicate
which pass band function has moved into the attenuation
gap. Notice that the wave functions are Just modulated
box wave functions (i.e. modulated solutions to the
problem of an electron in a box). The surface wave
functions are hardly attenuated at all, as was to be
expected from the small attenuation constant K. In fact,
since the value of x is proportional to the band width,25
we do not expect to obtain appreciable attenuation for

61

 

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66
less than 5 cells. To obtain a feeling for the way in
which the surface wave functions behave for larger crys-
tals, the surface states were calculated for attenuation
bands 3 and 4 with N = 20. In this case, N is large
enough to pull into the attenuation regions two pass-
band states from the upper and lower passébands. The
results are plotted in Figures 3.9 (a,b). Notice par-
ticularly the attenuation in Figure 3.9b where the K
constant is larger.

The location of the surface state energies in the
attenuation region is not the middle of the gap as was
found by Shockley,26 (a result quoted by several other
authors). In fact, their location in the gap depends
on the potential considered, the height of the vacuum
level, and the number of cells. For the parameters con-
sidered in this example, we may observe the dependence
of the surface state eigenvalues on the value of N for
the two attenuation bands in which surface states occur.
A plot of the surface state energy versus N is given in
Figures 3.10 (a,b). The points on these graphs are not
connected by a smooth line because the variation of the
energy eigenvalues for fractional values of N is quite
complicated, as will be shown in Section IV. It is
evident from these graphs, that the surface states do

not occur in the middle of the gaps, and that for N be—

67

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_ _ _ A ‘4 a 4 A i _ 4 _ _ A. A _ 4 fi _ i
mmamm.mnm

>

, i/ ,. , i J ,, ,, a i,
: i i : 2:... is; ,3

68

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69

 

 

 

Figure 3.10a

4r
v C
O . .
.3 _ . O . O Q C
C
O

b
2 L: 1 1 1 J l I L I
4 6 8 10 12 14 16 18 20

 

Surface state eigenvalues for the third
attenuation band versus the number of
cells.

 

 

 

Figure 3.10b

.7"
.6'
0
.5”
e
e
. e
O . O .
e 0 '
4P . o
e
e
l J l I I l l L]
.3“
4 6 8 10 12 14 16 18 20

 

Surface state eigenvalues for the fourth
attenuation band versus the number of
cells.

70
coming very large, the two states approach a common
asymtote. That is, they both approach the same energy.
The rate at which they approach each other is dependent
on the K constant; the more attenuated the wave function,
the faster the states will come tegether with increasing
N. It is clear from the above that this behavior may
be explained quite simply.

Since the surface cells are terminated symmetrically,
an e1ectron.which spends most of its time near the surface
(in a surface state) and does not interact with the other
surface state will have the same energy on either end of
the crystal, for very large crystals. In other words,
the electron on the right edge does not "feel" the electron
on the left edge because of the localization. As the
crystal is made smaller, the electrons beginto "see" one
another and their interaction splits the degeneracy in
the energy levels, in analogy to splitting of the atomic
energy levels into bands when the atoms are brought to-
gether. For the surfaces very close together, the in-
teraction is great enough to push one state completely
out of the attenuation-band and change it into a pass-
band state.

B. TAMMéTYTE TERMINATION
For the Tamm-type termination, for comparison, we

shall use the same parameters a, b, V6,‘V1.'V2. N, as

71
we used for the Shockley-type termination. The cell for

this case has a potential maximum at the center as shown

 

 

 

 

 

below.
i + ' '
| ‘vg/z «i'ib—p |
oi '— L J
I ' T I
45/. :
LL | J
o 4/2— 0-

Proceeding as before, we may write down the solutions
to Schrodinger's equation for a/2-b é x ( a/2+b, I: < VO/2,

 

 

9(1): 653A k(Z-%>) (3.10a)
, k =\[§"f( 14/2 22:“), A
71.00: Slflék(g-%) . (3.10b)

Connecting these solutions to the solutions in the region
OS x S a/2-b yields the expressions on the left edge of
the cell of

a: cam oases-A) +fefiesmA/5Jge-na)
a]: A 605M$i7l (IQ/24) ~KSih/kA%£(%—£))( 3 . 11b)

72

Mo = ‘76 5211K$0$£[ 2*b-f MIAWIZ‘QUJn)

710, = 605A kLCaSHZ’b' [é SIR/{$114924}; ( 3 .11d)
A = V$ZE+ n/LT.

For E - Vb/z, E > V6/2, we connect to the appropriate

 

functions. With the parameters chosen earlier, a/2 is
equal to 2b, so that the functions go', go, uo', no:
for the Tamm-type termination may be related to the same
parameters for the Shockley-type termination. Thus we

see by inspection of the two sets of equations that

 

SHOCKLEY’ TANK
a’ «- go’, (3.12)

‘90, 3 Ha

I&» - 4&0

710 ' us

This may be illustrated by reflecting Figure 3.} across
the‘Vb - 0 vertical axis as shown in.Figure 3.11. The
band edges are labeled to clearly indicate which function
determines the boundary. For small values of‘Vb, the

sequence of functions is given by

9o;(ll5)9o), lb) 90,190) M) 6,0,) ”I))(”;)§o). (3 '13)

73

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xmukuokm

 

 

74
By previous considerations, we expect to find surface

states between (u0' = 0, go a 0), (go' a 0, u - 0),

o
and (u0' = 0, go = 0). Note also that some crossing

and uncrossing of the band edges has occurred at V0 = 0
in going from a Shockley-type termination to a Tamm-type
termination. As Vb increases, some of the bands again
cross to yield a tight-binding situation. The main
difference between this Tamm-type termination and the
Shockley-type termination is that for the tight-binding
case we still have surface states in the Tamm case but

we have no surface states for the Shockley case. Actually,
as used by most authors (and in this paper so far), the
designation Tammrtype termination is a misnomer. This
will become more evident later. Suffice it to say that
Shockley states as well as Tamm states may occur for the
Tamm—type termination. we shall distinquish between the
two types of states by the fact that Shockleyetype states
occur between pass-bands which have crossed once whereas
Tamm-type states occur between pass-bands which have
crossed twice. The Shockley-type states occur only for
weak potentials and disappear as the potential becomes
stronger. The Tamm-type states occur only for strong
potentials. Another characteristic which can be used

to distinguish the two types of states is the degree to
which they penetrate into the crystal, or in other words,

75

their localization. The Shockley-type surface state has
a large attenuation length, being able to penetrate rather
far into the crystal, as surface states go. The Tammr
type surface state, on the other hand, is quite localized
and is appreciable only in the first cell near the sur-
face. Both types of states may occur at the same time
for weak potentials. This is because for low energies,
close to or below the maximum of the periodic potential,
the electron feels the effect of the potential quite
strongly and thus is essentially tightly bound.ngor
higher energies, the effect of the potential upon the
electron is.much less, allowing the electron to behave
like an.almost free electron. Thus when both types of
states occur, we expect to find the Tammetype states
occurring at very low energies whereas we expect the
Shockleyetype to occur near the vacuum.1eve1.>

Proceeding as we did with the Shockleyetype termin-
ation, we apply the boundary conditions for the Tammrtype
termination to determine the eigenvalues. Figure 3.12
illustrates the graphical procedure. This figure should
be compared with Figure 3.6 for the Shockley case. Al-
though it is not evident from the graph, the crossing of
the K1 curve in the attenuation band region between
(uo' - 0, go - O) is actually two crossings, one curve

superimposed on the other. These two eigenstates are

76

.esae>cemue cm mesuEHeueo e>uso peuuocme cuu3 e>uso HM ecu mo mcummono

 

 

 

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en‘s oo.a amnm oo.n on.: no.3 om.m no.m an m ma.~ om.s ee.a can. 0 com.. oe.s- on a-
P F k i‘ P I] b 1‘ tP uoi‘ 'P F b L » hr r if,
.0 0 .0 .0 e .0 .0 0 0 O
> , > > .< > > < > >
0 . .O .. 0 T. .. 0 0 _.L O O
> .. . >
T. e I
d” v n a d “v a M d w. d v
E. a . _ 8 ..8 8 W a ”a a 8
. .7 .V E 8.... . Z T. T. 0

 

i

 

 

 

 

 

 

 

 

 

h

”.5

77

of the Tamm-type since they occur for very low energies.
The surface state occurring between (go' - 0. u0 = 0) is
exactly analogous to the surface state found for the
Shockley-type termination. The same considerations which
determine whether there will be 0, l, or 2 states in this
gap may be applied to this situation.

An energy versus (k and K) graph is shown in Figure
3.13. This graph should be compared with Figure 3.7;
note the differences and the similarity. In attenuation
band 1, note that the value of K in this range is actually
off the scale of the graph and was not plotted outside
the limits shown. However, the X's (there are two K's,
one superimposed upon the other) are positioned at the
point where the K curve would go if it had been plotted
on a larger scale. The X's of course represent the
eigenvalues obtained from Figure 3.12. The pass-band
states in pass-band 4 and 5 and the surface state in
attenuation-band 4 are approximately the same for both
Tammr and Shockley-type terminations. The surface state
in attenuation band 4 has approximately the same magnitude
of attenuation (i.e. K) as that of the Shockley-type
termination state calculated earlier, and we expect the
same general behavior of the wave function going into the
crystal. However, the two states occurring in attenuation-

band 1 have a very large attenuation constant K and should

ENERGY

b.00

-2.00

-.SCC

“1.25

78

 

V um Level

 

 

 

 

 

 

 

 

 

 

 

 

>t

 

 

 

j

Y
3 .LGU .203

Figure 3.13.

1 ‘—‘r 1

w 1 1 T—
' "I ' ' f“. 1‘ INN "r‘ . " n".
.JQe .HUQ .J?a .bfli .74. .BUu ,qCu l.ue

WAVE NUMBER x (a/w)
Energy versus (k and K). The X's denote
eigenvalues which were determined in
Figure 3.12 (Tamm).

79
be highly localized near the surface.

Figures 3.1# (a-e) are plots of the wave functions
for the energy eigenvalues obtained above. These wave
functions should be compared with Figures 3.8 (a-e) for
the Shockley-type termination. Note particularly the
Tamm-type states which are plotted at the bottom of
Figure 3.14a and at the top of Figure 3.l#b. They are
quite localized at the surface and in fact extend further
outside the crystal than the other surface states con?
sidered. Note also the energies at which each occurs.
They indeed lie practically at the same energy due to
the large attenuation and consequent lack of interaction.
This result has been discussed earlier for the Shockley-
type surface states except that in that case a larger
crystal was required to establish it. Figure 3.15 is
a plot of the surface state wave functions in the 4-th
attenuation band for 20 cells. As may be seen by com-
parison with Figure 3.9b, the differences between the
states are small and hence the designation of Shockley-
type states for both sets.

Finally, we should comment that the procedure out-
lined for finding the energy states by the graphical
method of plotting various quantities was only for il-
lustrative purposes. The actual computations were per-

formed on a CDC 6500 computer, with the processes

80

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one pcec mmem cumuu ecu mo Eouuoc ecu Eouu ccec couuesceuue
cuusom ecu oucu wemueae mec couc3 cOHuocSMcemue eueum eoeuusm .eeu.m eusmum

 

\._/ D \mw > A > 292:?)
.< <_< < < < C <../.\

a

..> \/s > D D > 22...>_

 

/_\ < < < < /\ < < ./_\

H msecH ese

85

.LEEeBL meumum eoemuom o3u
o3u show ou ccec cowuesceuue cuH50u ecu oucu ©e>oE e>ec mccmc meme

geese ecu eo souuoc one one season one eo sou ecu sou neeouenoouo
ece .ocec scuumsceuue cuusou ecu cu chHuocsucemue eusum eommusm .mH.m eusmum

 

 

 

 

L _ . Hmhmmm.mflm L
LL LL<<<<<<; :L :52 LL
Hemeéfimevaw

WON L L L L mmfi L L L r JOH L L L mm L L L H, W

 

mmammm.mnm

LL<L<L>§<§<$§LLL

   

o~.eeeLH.eeL

86
outlined taking place internally in the computer. The

total time necessary for the calculation of the band
edges and 18 eigenstates to 6-figure accuracy was of
the order of 10-15 seconds.

SECTION IV
ARBITRARI’TERMINATION OF A FINITE PERIODIC POTENTIKL

The problem of termination of the periodic potential
at an arbitrary point in the end cell will now be con-
sidered. To be more specific, we shall terminate the
potential at one end by a step function at an arbitrary
point in the end cell and at the other end by a step
function at the potential minimum, (see Figure 4.1)
Uhtil now, with one exception, general conclusions.have
been drawn from symmetry arguements for crystals ter-
minated in either a symmetric or antisymmetric manner.
In the literature, the various approximation methods
which have been used most extensively include the near-
ly-free-electron method, and the tight-binding (LCAO
or NO) method. The one exception is a paper by Levine,27
in which the problem of the arbitrary termination of a
semi-infinite cosine potential is treated. we expect
to find somewhat different properties for a finite
crystal; consequently, we shall carry out the calcul-
ations for the above mentioned problem to determine ex-
actly what the differences are.

This problem is made considerably more difficult

by the arbitrary termination since now.the wave function
87

88

 

 

mg“ a H “a . o w a e
e e a

 

.3:

 

 

 

em

 

 

MN

 

 

.Haeo cce

— _ — — ~ _
.—

 

 

.H.v eusmwm

 

89
and its derivative at the point of termination will not
reduce to a simple form. From Figure 4.1, we see that
the point of termination in the end cell is measured by
the quantityA from the right edge. By continuouslyvary-
ing A through this end cell and calculating the eigen-
values for each value of A , we obtain a general idea
of what effects an arbitrary termination has on the
energy eigenstates.

Proceeding as we did in Sections II and III, we
attempt to form a continuous wave function by matching
the wave function and its derivative in each region to
the wave function and its derivative in a neighboring
region.

For x S O, the wave function is of the form

 

W30: A C“ , l<——- \Ffz‘m—ET. (4.1)

For O < x s Na-A , we have

41(1)::- C+d'+[x)+ C- (I)- (x). (the)

where the form of ah. , i)... will depend on the energy

range we are considering. For x ? Na- A ,

K:(z-— (Ala-AD. (4.3)

W1) = B c.

90
Matching the logarithmic derivative at x = 0 yields the

condition,

K, =42: 2%))

while matching at x - Na-A yields

_ . Ct ¢‘I’(Na-A)+C- W ( Ila-21)
-K. " (Cd/K a-A)+C’.4L(Na-d)) ' (“'5’

If we now set the right hand side of Eq. (4.4) equal to

 

(#.u)

 

the negative of the right hand side of Eq. (4.5) and
define the quantities,

14‘:- WMa'A) , 76'; Oil/MX/zL—A), (n.6e)
y,= (MO/r11) , yz=03¢i(Ah-A),(t.6b)

 

 

we obtain 0
07" C-\____ Cal/+6-2, * '
C} ‘I’ C- I — Gill—6.192.) (m7)

Notice that Of - -— 0: , and that x1 becomes equal to 1:2
and y1 becomes equal to y2 if A is equal to some in-
tegral multiple of the lattice constant. multiplying

Eq. (#.7) through by the denominators, and grouping

91
terms, we obtain a quadratic in c+ and C_,

(Jim/m) + 0+6. ((XI-XJQMV/‘Jfl?
+ am, 1%) = 0.

Factoring this equation, and using the Wronskian relation

(1L8)

W{¢+)¢-} : *ZQF, (4.9)

we find that 0+ in terms of C is given by

C+— [-[wzzhe-ggltgn etdahwtflc .w 10,

This equation may be simplified in form by considering

 

the different forms .of the wave function for' the three
regions P< O, O < ’0 < l, and P) 1. In each region,
we may write the wave functions out explicitly and form
the various sums and differences which are contained
in.Eq. (4.10). By defining new variables, we may sim-
plify the form of Eq. (4.10) considerably. For f3 < O,

we define
37,011) + u(ava

0/ £
mfg _ (12:26: e1. gave If!
:i (2174-30)?) Zap—A 96-490?) (1;. llb)

 

(4.11a)

 

92
= J'1,(ig——l(a:A)+ ill—L1”) 51%), (It no)

A, =-_. (W - MwM—D (Inna)

7‘1: W- flg-y:A)Szn(éé), (A. lle)

T== ”A?“ )sinéz/I-ihfp'kasé/M—flz (It 111')
t‘ = V7311 + Re“; (4.113)

 

3271,31: "‘ 71/)”. (n.11h)

With these definitions, the matching condition given by
Eq. (4.10) becomes

+C-iA61/2zM-‘fié/ 2+ 2 I
Cfle= F

CL," (4.12)

For 0 < f < 1, the definitions are

(a -A) ) .
Mg: (3 wuia-A)f “a £&fi;\lf'£) (4.13s)

2(a—A >/
_____ 1(a(_c_z__-A) + ‘m)mg(§§)n.m)

121 = 5; j%)+ u(a—A) ——)$zz£/@(n. 130)

 

93

9“: (”721:“) -— 3"")Acosk5’f) (#13d)

~11 ’A'A .
)1: (We ) "’ 3;, ))51"A{1’),(*-139)

=(-;—:)S’z'ZA k(ALz)1+(’")c’o$1K66’1fh(4 .13r)

 

 

 

r = V’Rfi—Rf, (4.1m
Sifié ’32 :5 "‘ 77/? . (4.131;)

Then Eq. (4.10) becomes

C~l~k = :1" C‘K(Il’i)d+¢a/z iflz -/(° (“.1”)

Similarly, for ‘a> 1
a— ) (a—A

CoMé’=(1jz€:): fig: )llol'; (4.15.1)

R: i: ( fizz-3+ 3354))51'1164E), (4.1519)

12.. = fl "5.4) 7‘ ifffiwlfistu

-= (132% *- "2(,_:’TA)) SZMP?) (4.15d)

 

 

 

94

=(_zl_71(1'4) —‘2\'§-§9 603$ (€61), (Lise)
7.: (~35) W men + {—216 3,724 we)“ u .15.)

 

r = Viki-12?) (use)
Sin; ‘63 = 77/? . ' (n.15n)

Eq. (4.10) then becomes

Cm _ is - [41/494 + ¢% 1193*

(4.16)

Substitution of Eqs. (n.12), (M.l#), or (4.16) into Eq.
(u.n) yields '

 

K. = -i0¥D/L(’éw-i)f¢lélfl), (n .17.)

g<o _L
K. = z'arcd(kw’h{¢’/fifl), (4 1 >

I
where 0; = ‘i(%go)lflt.

o<p<1
!

 

95

-4 _. .. ,_
Kn -(K{dflA(k/12)da.é/z I).(4-18t)

1
K: = "’ 0+. 60% (Kai-akiM-ffzzwdeb)

where 0.7. = -( ”5%) fi.

 

H

 

The same formulae hold for f!) 1, except that the de-
finitions of K, ¢z and F; are changed.

¢néu ¢3 and PM!” P; will, in general, exhibit quite
complicated behavior as E or A is varied; thus is best
discussed in connection with a specific potential. The
procedure for finding the energy eigenvalues is the same
as used before. We plot the two sides of Eq. (4.17)
and Eq. (4.18) as functions of energy, and the points'
at which the equality is satisfied give us the eigene
values. This is illustrated for the Kronig:Fenney type
potential in Figure 4.2, where we have plotted the energy
eigenvalues as a function of the parameter A (see Figure
4.1) for N - 5 and a, b,‘v6.'vl, and‘v2 the same as in
Section III. The lowest pass-band is not shown because
it was so narrow that no details could be distinguished
on the scale used to plot the graph. Figure 4.2 shows
the change in the eigenvalues of the system as the
crystal is continuously changed from N - 5~to N - 4 atoms.

96

 

 

 

 

 

“'1 l 1 1 1
a .8a .6a .4a .2a 0

N=4 <~ A N=5

 

 

Figure 4.2 Energy eigenvalues as a function of the cell
termination parameter A.

97
Upon inspecting the eigenvalue spectrum calculated for
the Tamm-type terminations in Section III, we observe
that there are 2 surface states of the Tamm-type in
attenuation band 4. These surface states are shown
in Figure 4.2 for A- a or N - 4. As A is contin-
uously decreased the Tamm surface states are not much
affected, although there is some variation. The
Shockley state, however, oscillates in the attenuation
band quite noticable. The value of the Shockley state
energy eigenvalue as a function of N was given in
Section III for N an integer (see Figure 3.10). As
may be seen from this graph, we might easily be inclined
to draw a smooth line through the points for N integer,
which would be incorrect, because in between the in-
teger values, the eigenstate oscillates. we note also,
in Figure 4.2, the emergence of another type of surface
state which is absent in symmetric or antisymmetric ‘

terminations. Levine27

has found similar type surface
states for the semi-infinite crystal mentioned earlier.
However, the states which he finds cross from one pass-
band into the next pass-band, which is not the case for
the finite crystal. Indeed, the surface states which
emerge lie very close at all times to the pass-band
from which they originated and periodically disappear

and reappear with decreasing A . The behavior of the

98
pass-band eigenstates is interesting but not surprising.
As the crystal length increases, the number of pass-band
states must increase. The new states emerge from the
upper passéband edge while some of the old states dis-
appear into the lower pass-band edge. But the net effect
of increasing the crystal by one atom is to increase the
number of pass-band states by one.

The case of arbitrary termination, as may be seen
from the above, leads to extra energy states. Since
for real crystals we expect the termination of the crys-
tal by the surface to be a more or less arbitrary teré
mination, we might expect to find more states than are
usually considered in the calculations which have been
carried out to date for symmetric or antisymmetric ter-
minations.t These new states do not seem to be related
to the bulk properties of the crystal (i.e. the number
of cells, etc.) but depend only on the nature of the

termination.

SECTION V
LOW ENERGY ELECTRON DIFFRACTION FROM FINITE ONE-DIMENSIONAL
PERIODIC POTENTIALS

In this section, we shall study the elastic low
energy electron diffraction (LEED) for electrons normally
incident upon a crystal surface by calculating the re-
flection coefficient for a plane wave incident upon a
one-dimensional periodic potential at energies greater
than the vacuum level.. The applicability of this model
to real metallic crystals is questionable since the
energies involved are high enough to ionize the core
electrons, and many-body effects should be large.29 How-
ever, if we assume that we are only interested in very
low energy electrons, and that we are dealing principally
with insulators, then it is probable that the main con-
tribution to the elastic scattering is from the static
potential due to the ion-cores inside the solid. This
static potential should, in fact, be energy dependent
as are most calculations on real crystals, but we shall
assume that it is relatively constant over the energy
range ~onsidered. ' .

There has been.much interest in recent years in the
use of LEEDas a tool for the study both of surfaces and

the interaction of electrons with solids. Several methods

99

100

have been employed to study this problem. The method
used in this thesis may be classified as a "dynamical"
model calculation of which there are two different
variants. The first variant is based on a generalization
of Lax's multiple x-ray diffraction theory. The second,
with which we shall be concerned is based on a band
structure treatment. For more detail aboutweach method,
the reader is referred to the literature.3°'36 The
method, we shall use is essentially a matching procedure
in which one matches the plane wave at the surface to
the Bloch wave inside the crystal.

we proceed as follows: we consider a oneedimen-
sional periodic potential of period a with N atoms and
having either a Tammrtype or Shockley-type potential
terminationat either end. Our zero of energy will be
at the average value of the periodic potential. The
vacuum level on the left is at a height‘Vi above zero
energy and the vacuum level on the right is at a height
'Va. For x $70, we have a plane_wave of unit amplitude
incident from the left upon the periodic potential and
a reflected plane wave of amplitude R going to the left.
The solution of Schrodinger's equation in this region
yields the wave function
(’01): euéfi 726—2 ’7, (5.1)

fir" WMX£~MT.

 

101
For Oéxé Na, there are three regions of energy in each
of which the Bloch wave function assumes a different

form. These are
ZA _.
M1): Cue ‘3‘?!) 4 C+céézfllmha<o, (5.2a)
(’0!) = CI]: CK" Path) + C.kEk'7Z/((x);o((<1)(5.2b)

”(2) = 6&6?va C-ké-Kfiux) 5 fJ>1 . (5.2c)

As the energy of the incident plane wave is increased,
we must match tht appropriate Bloch functionto the in-
cident and reflected wave, depending on the band struc-
ture in the crystal. For x? Na, we have only an out-

going plane wave of the form
‘ (Jr-Ah)
¢In=716b42 , kz=Jitfl(E-pz>.(5.3)

These'wave functions must be matched at x a 0 and x - Na
to yield a continuous wave function throughout all space.
If the energy of the incident wave is such that
F < 0 inside the crystal, then we have at x .. o, the

‘matching condition

- 141): cit-C-k
‘Muai 0" emu

 

 

). (5..)

At x - Na, we have

 

6M6 —6—-A€ —- A
- __ “a — I 2.. (5 5)
(Cue’kfii‘éfie
If we now solve Eq. (5.5) for C_k in terms of C+k,’ we

obtain

 

a; -- [’42, 21M¢
C”: = 01 + 2351. >6 CM' (5'6)

Substituting this equation into Eq. (5. 4), we obtain

1’14 \: “‘(g; ) 2,114)
1+R/ i/é/ 11+(ai+ ikz)6w

afiib
__ M( —££Z; flail/1A
— 177 a; +19, furl/Ya /’

 

 

or

R : (Aefihai HawkézflanA/e (5 .7)
(bf/$2964“ (of. fitfimfifla'

Since aria-r -z (dd/Ma) Ifl” , we obtain ,a reflection
coefficient for F< 0 of

I'Rlz— (k: "W (N112 —£[Az)ZIZM
(147%)" 10.11% [Gill-(kiszw

 

.(5. 8)

103
For 0 < 2(1, we simply replace 1: by 41K and

I

04.- 9’.Ef in qu (5.7). to obtain

m1 = (t—Ae‘tarlfiam hefmé‘m (5 9,
(bf&)z10fl3—I-(IWIZ’— hkzfladkh' ‘

For F > 1, instead of calculating K by the formula
.1
.L 43? (173%
= - - 5.10a
K a 1 + (at ’ ( ,)

we use the formula

.1
: ’é[°](4‘:i{.‘) (5.101;)

and the reflection coefficient is the same as for 0 <

f? < 1 otherwise. ~'

 

For the case of a semi-infinite crystal, we need
only let N‘M, to obtain

 

1.. a.
3:3) , (“0:
Hut: 1 , (>0. (5.12)

I: we let v1 a v2, |RI2 will reduce for both (R o
and f!) 0 to

2.: ( tail”; Mam .
IRI 4|m141+(la;!‘+lef)‘£m2kM ’ (a (0’ (5 '13“)

104

2.__ ( 1W11+kf)zM1/<Ah
IR. ‘ ills/411+ (IdilééfFW/Ia’ F’O"5°1"°’

Notice that in this case, a; always enters Nil 2 as a
squared quantity. we saw in Section II that surface
states are obtained only if —(ll¢u(,¥l{goes negative
in an attenuation region. Since in an attenuation
region a; equals wflfi and is squared in the formula
for [BI 2, we do not expect to see in the reflection
spectrum.evidence of what would correspond to surface
states with energies lying within attenuation bands.
we should, however, be able to distinguish the pass region
and the attenuation regions for large N.

In a pass region, It varies from 0 to T/a or from
T /a to 0, so that tan(kNa) will exhibit N or N-l
zeroes, depending on the sequence of the zeroes of the
functions go', go, uo', uo for increasing energy. In
analogy with the surface states problem we find that
if the band edges have crossed (i.e. if the zeroes occur
out of order from the sequence go', go. uo', no, go',
go: “0" uo’ . . . ), then the neighboring pass regions
will have (depending on how large N is or how strong the
periodic potential is,) either N-l, N or N+l zeroes.

To see this, consider the form of the reflection -

105
coefficient in a pass region. The zeroes of IRlz will

occur in a pass region for either

ltdltafih)= 0 or [Gt/Mg. (5.14)

Considering only the tangent term, this condition.implies
that k - n'IT/Na, where n is an integer. Since 05 kx<
'1T’/a, and we are restricting ourselves to only positive
values for k, the possible values of n are 0, l, 2, . . .,
N. For ‘0 - 0 there is some difficulty since I: - 0, and
(0;, equals zero or infinity depending on which function
determines the boundary. The same difficulty arises at

P =- 00. By consideration of the limiting forms for

the boundaries P = 0 and €= 00,: we obtain

”2'1" texts/M)“ 9"""05 (5'15“)
mf: (WA/gaff}? 2 90:05
R11: h" fag/3w)" age—o; (5.15:)
mi: (95233;- , (to: a. (5.156)

Thus we see that k - 0 and k - (717a) yield no zeroes

'(5.15b)

 

of IRIZ. Then n is restricted to n -'l, 2, 3, . . .,

 

106

N-l. The extra zeroes may or may not occur depending
on whether q becomes equal to 1ch1 as k varies from
0 to 1173.. Inspection of the form of “(Wmflfli
in Figures 2.2 and 2.3 reveals that if the functions
80', so. uo'. no. 30'. so. uo'. “o’ . . . have zeroes
in this sequence, there will be one value where ICEjz'
becomes equal to k12 in each pass region, bringing the
total number of zeroes for IRI2 to N. For the sequence
of zeroes deviating from the above order, as in.Figure
2.3, we will obtain either no values or two values of
[Hill which are equal to k12 in the pass region be-
tween go' - 0 and uo' - 0, and either two values or no
values for the next pass region between go - 0 and
uo - 0, depending on the magnitude of klz. Thus in the
neighboring pass regions between which the band edges
have crossed, there will be either N-l or N+l zeroes.

For N becoming large, we see from Eq. (5.15) that
[R12 approaches 1 on the band edges. For a semi-
infinite crystal, the structure in the pass-band dis-
appears because the reflection coefficient oscillates
an infinite number of times, and only the envelope of
the peaks remain. In the attenuation regions, the re-
flection is perfect.

It should be pointed out that the above analysis

may also be applied to tunneling phenomena, For example,

107

we could let Vi and‘Vé be equal to the minimum value of
the periodic potential without changing the mathematics.
In this case, the calculations reduce to the solution
of a particle tunneling through N identical barriers.

The above analysis also resolves a question which
arises in applying the Born-von Karman boundary condition 36
to an infinite crystal. The Born-von Karman boundary
condition, as usually applied, uses only one of the in~
dependent Bloch functions and requires this function to
satisfy

cfim=1.

(5.16)

In this case, k = 2nT/Na where ~IM2 éhéfi/Z . There
are N values of k in all, counting both positive and

negative values. A better manner in which to impose

periodic boundary conditions would be to require the

function

M1) =-- Casi/”Pam +04: é’hfiw) (547)

to satisfy

thawed W). e...)

108
In this way, one obtains twice as many values of k given

by

k=7l7l'//Vd . -M/ZSX<///2. (5.9)

New there is still some doubt as whether to include

N/2 or - N/2 in the values for k, and one usually picks
either one or the other. However, we see by our analysis
that the choice should be neither of the above, but
should be such that

Lm _. “alez: \rflfVe/fi) (5 20)

(4 (in)

where Vb/e is the amplitude of the periodic potential

 

evaluated at a lattice site. From the formula for the

reflection coefficient, we see that k - n1r/Na, n - l,

2, 3, . . ., N-l, and 1k - a; correspond to values of

the energy for which the periodic potential is com-

pletely transparent to an incident electron and thus the

electron is not localized to any one region of the crystal.
To illustrate the features of the above discussion,

calculations were carried out on a Kronig-Fenny potential

for both Shockley- and Tamm-type potential terminations

at either end. For comparison, we use the same values

of a, b, V6,as we used in calculating the energy states

 

109
in Section III. Figures 5.1 (a,b) show the reflection

coefficients for v - v - 6.5 EA}. and N - 4 for both

1 2
the Shockley and Tamm terminations. The vertical lines
indicate the band edges, and the arrows indicate which
band edges have crossed. Figures 5.2 (a,b) have the
same potential configuration except now N e 20. Figures
5.3 (a,b) show the case of tunneling when‘v1 i‘Vé a Vb/a
- -l.7 3.0. and N - 4. The lowest pass-band is not
shown because it is so narrow that the plotter missed

it.

 

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</A \ jrf.\< Ill: /1\ ) JK 1.
\
n .o n ,n b an 6 n n b n b 6 nt,bn n
o o o o o o o o o o o o n. 00 o
L .l l I. .I I l- el I
= = = = __ = = = __ = = __ = = == =
A, +

 

 

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O O O O O O O O 05 O O O O O: O 05 O
5 II ‘ .4 | ‘ A I
= = = = = = = = _ = = = = 0 = “H =
0 0 Q 0 0 0 0 0 0 0 0 J 0 0

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SECTION VI

RESONANCE TRANSMISSION IN ELECTRON EMISSION
FROM SURFACES WITH ADSORBED ATOMS

In recent years there has been renewed interest in the

role of resonance effects in the emission of electrons from

a metal surface with adsorbed atoms’a'ns. Both qualitative

and quantitative“0 predictions indicate that structure in

38

the total energy distribution of field emitted electrons
should be related to resonance effects reflecting the
atomicplike energy level.speetrum of the adsorbed atoms.
Gadzukn5 has shown qualitatively that similar resonances can
occur for transmission over the surface barrier in which an
adsorbed atom is present: However, as we shall see later,
his results are questionable in several reSpects. This be-'
havior could be relevant to electron proceSses such as
thermionic, Auger, or photoelectron emission in which a
partial monolayer of an alkali metal is deposited on the
surface to reduce the work function of the material. Con-
sequently, it is interesting to try to calculate this effect.
The present section deals with a calculation which is
similar in many respects to calculations performed‘by

Gadzukn7. But whereas Gadzuk considered only the case of

116

117
transmission of an electron into free Space from a free
electron metal with a single adsorbed atom, we shall con-
sider the more general case of transmission of an electron
from one periodic potential across another periodic poten-
tial into free space. This includes the case of only one
period of a periodic potential, which is Just a single
adsorbed atom. we may relate this type of calculation to
tunneling phenomena,.a1though there are no classically for-
bidden regions. However, there are quantum mechanical at-
tenuation regions through which the electron must pass and
the analogy between the classical case and the quantum-
mechanical case will be clear in what follows.

A great many of the features of tunneling phenomena
in solids are essentially of a one-dimensional nature. If
the tunneling barrier (or effective barrier in our case) ex-
tends in the x direction the momenta in the y and z direc-
tions can usually be taken to be constants of the motion and
hence are merely fixed parametersna. Therefore, we shall
confine ourselves to one-dimension for the calculations
which follow, even though such a model neglects a variety
of geometrical effects.

We shall first consider the special case of electron
emission from a free electron metal across several layers

of foreign atoms into free Space. we will not consider the

 

118
mechanisms which excite the electron into states from which
itcan be transmitted out of the crystal but will assume _
that the electron has already attained such an energy state.
We shall then replace the free electron metal by a periodic
potential with the same layers of foreign atoms on the sur-
face and perform similar calculations. The model potene
tials we shall use for the first and second case are shown
in Figures 6.1a and 6.1b.
A. ADSORBED ATOMS ON A METAL SURFACE

As a model for a metal, we assume a semi-infinite free
electron region for x < 0, and we represent the foreign atom
layers as square wells for 0 < x.< Na2- For x > Nae, the
potential is zero representing free space. The three regions
are labeled appropriately by I, II, and III. By using a
square well for our model impurity potential, we have com-
plicated matters considerably, since we have to deal with
band-edge crossing effects also. waever, real impurity
potentials may be expected to have the same type of struc-

ture. In region I, the solution of Schrodinger's equation
yields plane waves which we shall write (for E > 0) as

(6.1)

41(1): (31%]; <I-RCi ,3: IéI=V2 fi(€+%/).

 

 

 

 

119

 

 

 

x HHH

 

 

 

 

 

 

 

QNII

 

 

 

 

 

 

.msoH Iasuozlsoz .na.m ouamwh

 
 

 

 

 

_ If?! at- V
Ho> We
He\\\\\w\\ \\\\}“
_ we

 

 

 

 

_l20
We want to find the transmission coefficient of an electron
incident from the left, past the foreign layers into vacuum.

Consequently, we may calculate lRl 2, the reflection coef-

ficient and use

I'T'lz': 1 "‘ ”RI” (6.2)

to obtain the transmission coefficient. In region II,

solution of Schrodinger's equation yields the Bloch waves

”12,: C+l|4+ C_(l}. (6.3)

the exact form of which is given in Section I. ,In region

III, the wave function is given by the out-going plane wave

 

4)) : TC; 3(1-A/az) k=W%XE) (6 .4)

 

121

The requirement that the wave function be continuous across
the three regions determines the coefficients B, C+, C_, and
T. If we recognize that this problem is Just a special case
of the problem treated it Section'V with the zero of energy
shifted and unequal potential terminations, we'may save cal-
culational duplication. By making the energy zero adjust-
ment and relabeling the correSponding variables according
to Figure 6.1a, we may directly carry over the calculations
of Section V to this section. we wish to calculate the
transmission coefficient rather than the reflection coef-
ficient as was done in Section‘V; so we must use Eq. (6.2)
to obtain the pertinent result. '

At this point, some discussion of the choice of po-
tentials is in order.. The potential around the foreign
atoms adsorbed on the surface of the metal is essentially
that due to the Coulomb potential of the fractionally
charged ion core. Thus, one would expect to use a strongly
attractive model potential whose radius is roughly that of
the ionic radius. However, if we incorporate the condition
of orthogonalization to the occupied tightly bound electron
as a separate, non-local term in the potential, the sum of
the two potentials yields a more smoothly varying function
of position. The pseudo-wave function will then be anal-

ogous to the free-electron wave function. This approach has

 

122

been used to calculate electron field emission distri-

butionsng. Other workers50

dispute this approach for the
following reaSons. The principle of the pseudo-potential
is that the exact Schrodinger equation for a system with a
complicated wave function can be transformed into a pseudo-
wave function equation with the same energy eigenvalue and
a simple pseudo-wave function. The simple wave function has
nothing to do with the real wave function. In doing trans-
mission coefficient calculations, the form of the wave func-
tion is intimately related to the details of the potential,
and the interference effects which are present in the real
wave function are glossed over in the pseudo-wave function
approach. .

we shall adopt the latter viewpoint in calculating
the wave function, so consequently, we must use strong po-
tentials in a model of the true potential rather than weak
pseudo-potentials. we shall assume the adsorbed atom is
displaced to a larger lattice spacing from the end of the
crystal than it would have in bulk material. As suitable

parameters to use in our calculations, we take (see Figure
6.1a)

8.2 3 900 A0, (605‘)

be

v02 - 51.5 ev - 13.28 11.11., (6.50)

- 1.5 1°, : (6.5b)

123
g 12eu5 'eV ‘3 3020 EeUe, (6 05d)

P

T2 3 3e88 817 = 100 EeUee (6'5e)

vcl

These are approximately the same values of the para-
meters as used by Gadzuk in his calculation. However,
Gadzuk was mistaken in his interpretation of the units of
the various parameters defined above so that when he thought
he was using a well depth V02 = 51.5 ev, he was actually
using a well deptth02 = 200 ev. As a matter of fact, all
of his energy units are too small by a factor of 3.88. Con-
sequently, the transmission coefficient which he calculates
has considerably more structure (i.e.the wave function has
a larger number of nodes near the well) then it should. ‘To
clarify this situation, and to generalize the results for a
larger number of adsorbed atoms than one, the transmission
coefficient was calculated using the above parameters for
the case of one, two, and three adsorbed atom layers. The
results are illustrated in Figure 6.2. The solid curve
represents the case of one adsorbed atom layer (N = 1),
whereas the 1/16" dashed curve is for two adsorbed atom lay-
ers (N - 2), and the 1/8" dashed curve is for three adsorb-
ed atom layers (N s 3). The range of energies in.which we

are interested is between zero and twenty electron volts,

124

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125
since calculations of this type for higher values of energy
are made irrelevant by inelastic effects. The most striking
feature about the N - 1 case is that the transmission coef-
ficient has very little structure, and is a slowly varying
function of energy. This is in agreement with previous
authors' assumptions about the nature of the transmission
coefficient with regard to photoemiSsion calculations, etc..
The small variation which does occur is a result of the at-
tenuation bands of the adsorbed atom. The attenuation band
edges are denoted by the vertical lines. For two adsorbed
atom layers, the situation is quite different owing to the
resonating effects between the impurity atoms. If we rea-
lize that this situation is analogous to the situation in
Section‘v, we may apply the same reasoning to explain the
structure of the transmission coefficient." The main dif-
ference between the calculations of this section and those
‘ of Section V is that we are now dealing with unequal bound-
ary conditions at either end of the "periodic potential".
The main effect which this has on the form of the trans-
mission coefficient is that, in a pass band, instead of
achieving the value one (corresponding to the reflection
coefficient of Section V going to zero) at various values
of k, the transmission coefficient reaches a relative max-

imum which may be shifted slightly from the values of

 

126

k = mr/Na2, n = 1, 2, . . . N-l. Otherwise, the two situ-
ations are analogous. The number of relative maxima which
the transmission coefficient will have in a given pass-band
will depend on whether the band edges which bound the pass-
band have crossed. Consequently, there will be either N-l,
N, or Nfl relative maxima in each pass-band. The attenua-
tion bands, of course, tend to lower the transmission coef-
ficient. Thus, the reason for the structure for the two
cases, N=2 and Ne}, becomes clear. To obtain a better under-
standing of what effect the potential depth of the impurity
layers has on the transmission coefficient, we have per-
formed the same calculations as above using the same values
for all the parameters except‘Vbz which was reduced to half
of its previous value. The results are illustrated in
Figure 6.3. This should be compared with Figure 6.2. As
'may be seen, the transmission coefficient is enhanced in an
overall manner, and the structure changes slightly due to
the change in the position and widths of the pass-bands
and attenuation bands. The fact that an attenuation band
becomes wider as the potential becomes weaker is purely
a band-edge crossing effect, as may be seen in Section III.

B. ADSORBED ATOMS ON A NON-METAL SURFACE

The situation of a nondmetal with adsorbed atoms on

its surface represents a more complicated problem mathema-

 

 

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a F P p n P elf b Pl ? P v b L. P 0
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4.
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128

tically than does the previous case, since now we must take
into account the band structure of the non-metal as well as
the band structure of the adsorbed atoms. The approach to
solving this problem.for the transmission of electrons from
the nonémetal into free space is, however, analogous to the
previous calculation. As before, we form a continuous wave
function throughout all space by Joining the appropriate
solutions to Schrodinger's equation at the boundaries of
the regions in which they are valid. We shall denote the
nonametal as region I, the adsorbed surface layers as region
II, and the vacuum.as region III.

Since region I is characterized by a periodic poten-
tial, the solutions to Schrodinger's equation will yield a

wave function of the form

“I = C: [bf 4' CE (1'?) 1(0) (5-6),

where the exact form of 1p} ) w. will depend on

the value of energy; i.e. on whether PI< O, O < PI< 1,
PI) 1. Region I will have a band structure which depends
only on the periodic potential in region I. We want to

have a wave traveling to the right (i.e. incident wave) and
a wave traveling to the left (i.e. reflected wave), in order

to calculate the reflection coefficient. Since the indepen-

 

129

dent solutions for an attenuation band are both real func-
tions, they do not carry any current. To be able to obtain
a current in an attenuation band, it is necessary to take a
complex linear combination of the two real solutions. wa-
ever, we shall not attempt to do this since the reflection
coefficient would depend exponentially on the depth inside
the nonametal at which the unstable state was created.
Since the time required for an electron to "tunnel" out of
the crystal is much larger than the time during which'the
electron would stay in such an unstable state, we can say
that the reflection coefficient is essentially one, or that
the transmission coefficient is zero. The independent sol-

utions for the pass bands, in contrast, carry currents of

,; ._.. l(tf¢f’£ we: - ”far we
,iét(q’1w-IW’* q)1¥'4,1’)_% I (6.71))
where a; - -- 01'— and 0;: i(-—fl€é)lfl:

“rt
Ill

l .1
From Section II, we know that the quantity ’(aO/flo)lflz’
is positive for the odd numbered bands and negative for the
even numbered bands. Since

We 65%!

(6.8a)

 

130

f" = *fi‘, (6.8b)

We must identify the wave traveling to the right as 41-}:
for odd numbered pass bands and “’3: for even numbered
pass bands. The situation is reversed for the wave travel-
ing to the left. Thus, the reflection coefficient is

given by

$11 odd Pass bands.

IR‘ 2 \ng‘z even Pass (smog?)

In region II, we will again have a band-structured situa-

tion, so we may write the wave function in this region as

1p = 01‘ 4).” + 61‘ (1)3“, owe/Varese)

The band structure in this region will, in general, be

different from that of region I.

In region III, the wave function takes the form of an

131

out-going wave

('1 : TC&3'(1’A/f) £3 =V1m£ / i Z; (6.11)
,z 2 Ala, .

 

If we now require the logarithmic derivatives to be equal

at the boundaries x - 0 and x - Nae. we obtain the reflec-

tion coefficients of

pI< o . ,0 1< 0

l l

IR 1" _ as, he . (gf- sews/a ,
— Z z

(eh/ere + (35+ 656 (a) 2W”

[1' :r g;
F’tfl’: ,
$1: - (Z!) I I) (6.12)

_— 1 A bdfldS)
E — {:1 211611 (24:115.

. p1<o. 1,015 0

H341“ ijwm (6 13)
" (6W e>t¢+ {yf— 651/ ezzéa/M- '

 

 

132
p.15 0, OI mantra/”1
W '1: (5 .14)

The transmission coefficient is Just

ITIZ': 1 - )le (5.15)

Notice that {Se}, is always positive and could be re-
placed by' I ll . As the periodic potential in region I
goes to zero, If, I goes to 1:1, where k1 was defined
earlier in Eq. 6.1, and we recover the results of the ear-
lier subsection.

In order to be able to draw comparisons between this
calculation and the calculation done earlier, we let
a2, b2,‘Vo 2, and T
let a1, b

2 have the same values as used before, and

1. V61. and T1 be such that the average value of

the periodic potential is the same as the'V’o1 chosen earlier.

Thus, we let

a1 - 3.0 A0 (6.15a)
b1 - .75 11°, (6.151»)
v01 - 2.0 E.U., (6.15c)
'r - 1.2 11.0.. (6.15d)

1

 

133
USing the above parameters for region I, and the two sets
of parameters for region II, the transmission coefficient
was calculated for the range 0 < E < 20 ev, and is illus-
trated in.Figures 6.4 and 6.5. ‘As before, the vertical
lines represent the band edges for region II. The band
edges for region I can clearly be distinguished by the
fact that the transmission coefficient goes to zero. These
graphs should be compared with Figures 6.2 and 6.3 and the
differences noted. There is only one attenuation band oc-
curring in region I for the range of energies considered.
It occurs more or less in the middle of a pass band of re-
gion II.

The structure in the transmission coefficient is en-
hanced in the pass band regions, as is readily seen by com-
parison. The various relative peaks in the transmission co-
efficient may be explained by previous considerations.

To summarize, we have seen that for the case of one
layer of adsorbed atoms on the surface of a metal or non-
metal, the transmission coefficient is a slowly varying
function of energy, except near a band edge for a nondmetal.
‘With the introduction of more than one layer, resonance ef-
fects predominate, leading to considerable structure in the
transmission coefficient. Hence, we would expect to observe
this type of effect in the total energy distribution of elec-
trons emitted from metals or nonpmetals with more than one

layer of foreign atoms deposited on the surface.

 

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135

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SUMMARY

By constructing the total wave function fOr a
one-dimensional potential from individual cell solutions,
we have found that such wave functions may assume three
forms, depending on whether the parameter F (defined in
Section I) is less than zero, between zero and one, or
greater than one. All three forms are of the Bloch-function
type; there is no restriction as to the form or magnitude of
the periodic potential. Via the parameter P ‘ , a band
structure may be associated with a single period of the per-
iodic potential. By applyingTamm and Shockley type poten-
tial terminations at either end of the periOdic potential,
we hav: found that the emergence of surface states is depenp
dent on whether the parameter -— (Ital/(lo) ”of; (defined
in Section I ; it corresponds to an effective wave number)
goes negative-in an attenuation region. The number of sur-
face states 1n an attenuation region was found to depend on
the number of cells in the periodic potential considered and
the width of the attenuation band. UBing a Kronig-Penney'
type periodic potential as an illustration, we have seen
quite graphically the effects which the two different types

of potential terminations have on the energy states and the

136

137

dependence of these states on the parameters which specify
the potential. For potentials terminated at an arbitrary
point in the end cell, we have found the emergence of a new
type of surface state which is not found for terminations a
which are symmetric. The band structure above the vacuum
level was investigated by considering the reflection of
free electrons from a finite periodic potential with the
Tamm.and Shockley type terminations. It was found that the
presence of surface states for energies below the vacuum

level may be inferred from the structure of the curves giving

 

the reflection coefficient versus electron energy. For the -
emission of electrons from crystals with from one to a few

layers of adsorbed atoms on the surface, resonance trans-

mission effects were found to be significant only for the

case of more than one layer of adsorbed atoms.

BIBLI OGRA PHY

 

BIBLI OGRA PHY

l. I. Tamm, Phys. Z. SowJet. l, 733 (1932); Z. Phys. 16,
849 (1932)- ‘

2. w. Shockley, Phys. Rev. 56, 317 (1939). ‘

3. S. G. Davison and J. D. Levine, to appear in Solid State
{lysics (Academic Press, New York, Vol. 21, 1968). ed.
by F. Seitz, D. Turnbull and H. Ehrenreich.

4. E. T. Goodwin, Proc. Cam. Phil. Soc. 2, 205, 221, 232
(1939)-

5. M. Born and J. R. Oppenheimer, Ann. Physik g4, 457 (1927).

6. H. M. James, Phys. Rev. 16, 1602 (1949).

7. Ibid.

8. All definitions of James have been changed by a sign
(i.e. our (11’ corresponds to his 0’. , our r; corres-
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9. D. S. Boudreux and v. Heine, Surf. Sci. g, 1126 (1967).

10. w. Shockley, 92. git.

11. I. Tamm, 92. git.

12. R. L. Kronig and w. G. Penney, Proc. Roy. Soc. A132,
499 (1931)- '

13. 119.43-

14. A. W. Maue, Z. Physik 9;, 717 (1935).

15. H. Statz, Z. Naturforschg. 2a, 534 (1950).

16. J. Koutecky, J. Phys. Chem. Solids .131: 233 (1960).

17. G. Allen, Phys. Rev. 9_1_, 531 (1953).

18. 121.1, '

19. B. A. Lippmann, Ann. Phys. 2, 16 (1957).

20. F. Herman, Rev. Mod. Phys. 39, 102 (1958).

21. T. 0. Woodruff, Phys. Rev. _1_<_)_3_, 1159(1956).

138

 

22.
23.
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27.
28.

29.
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33.
34.
35.
36.
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40.
41.

139

S. G. Davison and J. D..Levine, _p_. git.

E. Antoncik, J. Phys. Chem. Solids 21, 137 (1961).

V. Heine, Surf. Sci. g, l (1964).

19.1.9;-

W. Shockley, 92. git.

J. D. Levine, Phys. Rev. 1'11, 701 (1968).

1.91.9.-

C. B. Duke and C. W. Tucker, Jr., Surf. Sci. 15, 231
(1969).

E. G. McRae, J. Chem. Phys. 55, 3258 (1969);Surr. Sci.
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K. Hirabayashi and Y. Takeishi, Surf. Sci. 4, 150
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F. Hofmann and H. P. Smith,Jr., Phys. Rev. Letters 12,
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P. M. Marcus and D. W. Jepsen, Phys. Rev. Letters _2_g,
925 (1967).

J. S. Plaskett, Proc. Roy. Soc. (London) 520—, 363
(1967). ’

R. M. Stern, H. Traub and A. Gervais, .7. Vacuum Sci.
Tech. 5, 182 (1968).

R. M. Stern, J. J.’Perry and D. S. Boudreux, Phys. Rev.
.11. 275 (1969)-

see, for example, J. M. Ziman, Principles of the Theory
of Solids (Cambridge University Press, 1965), p. 25.
C. B. Duke and M. E. Alferieff, J. Chem. Phys. 56, 923
(1957).

E. W. Plunmier, J. W. Gadzuk and R. D. Young, Solid
State Commun. 1, 487 (1969).

J. w. Gadzuk, Phys. Rev. 183, 2(16 (1969).

R. Gomer and L. W. Swanson, J. Chem. Phys. 28, 1613

(1963).

42.

43.

44.
45.
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47.
48.

490
50.

140

L. D. Schmidt and R. Comer, J. Chem. Phys. 45, 1605
(1966).

A. J. Bennett and L. M. Falicov, Phys. Rev. 151, 512
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J. W. Gadzuk, Surf. Sci. 6, 133 (1967).

H. D. Hagstrum, Phys. Rev. 129, 495 (1966).

J. W. Gadzuk, Surf. Sci. 1§, 193 (1969).

E. 0. Kane, in Tunneling Phenomena in Solids, ed. by
E. Burstein and S. Lundquist (Plenum Press, New York,
1969) pp. 1-12. -

C. B. Duke and M. E. Alferieff, 22, git,

see reference 46.