Tight-binding methods for the study of surface states on semiconductors
by William Arthur Schwalm
A thesis submitted in partial fulfillment of the requirements for the degree of DOCTOR OF
PHILOSOPHY in PHYSICS
Montana State University
© Copyright by William Arthur Schwalm (1978)
Abstract:
The unifying concept which relates essentially all of the methods which currently appear in the
literature for computing surface electronic structure from a given, tight-binding Hamiltonian is shown
to be the difference equations satisfied by the Green functions. An efficient, generating function
formalism is developed through which the analytic resolvent method of Levine and Freeman may be
extended in order to solve more realistic surface problems. Two model Hamiltonians are completely
solved in order to demonstrate this method: One in which there is one state per layer and an s-p
Hamiltonia with one s-like and one p-like state per layer.
With the aid of the solution of the s-p Hamiltonian, a simple, bulk, bond-orbital model is introduced
and it is seen that there is a limit in which this approximation is exact. This model is used to assess a
bulk, bond-orbital model approach to the surface problem.
A bulk, bond-orbital model proposed by Pantelides and Harrison is used to study eigenstates of a
twelve layer film of (110) GaAs. The results compare well with those of calculations which do not
make bond-orbital approximations. Results are discussed and compared with those of other
investigations. \
TIGHT-BINDING METHODS FOR THE STUDY
OF SURFACE STATES ON SEMICONDUCTORS
by
WILLIAM ARTHUR SCHWALM
A t h e s i s s u b m itte d in p a r t i a l f u l f i l l m e n t
o f th e re q u ire m e n ts f o r th e degree
of
DOCTOR OF PHILOSOPHY
in
PHYSICS
Approved:
A (Am .
ChfSirman, Graduate Commit t e e
'He1Cid, M ^ p f 1 Department
Graduate ‘Dean
MONTANA STATE UNIVERSITY
Bozeman, Montana
June, 1978
ACKNOWLEDGMENTS
The a u th o r g l a d l y ta k e s t h i s o p p o r t u n i t y to thank D r. John
Hermanson who suggested and c o o r d in a t e d th e re se a rch r e p o r te d here as
p a r t o f th e e f f o r t o f o u r group i n t h e o r e t i c a l
s u r fa c e s t u d i e s .
He
a ls o wishes t o th a n k Dr. Hermanson f o r u n l im it e d tim e s p e n t in d i s ­
c u s s in g a l a r g e number o f t o p i c s
p a t i e n t and s k i l l f u l
in modern p h y s ic s .
His a d v ic e and
.
d i r e c t i o n have always been d e e p ly a p p r e c ia t e d .
S in c e re thanks a ls o t o Mizuho K a w a j ir i f o r many i n t e r e s t i n g d i s ­
cussions. and a g r e a t deal o f h e lp w i t h p r a c t i c a l m a t t e r s , and t o Dr.
G erald Lapeyre f o r c o n t in u in g encouragement and s u p p o r t.
Thanks t o th e members o f th e p h o to e m is s io n s p e c tro s c o p y group f o r
many e n l i g h t e n i n g and s t i m u l a t i n g d is c u s s io n s , and s p e c ia l
Dr. James Anderson f o r a c a r e f u l
o f th is
th e s is .
thanks to
r e a d in g and c o n s t r u c t i v e c r i t i c i s m
TABLE OF CONTENTS
Page
V I T A .......................................................................................
ACKNOWLEDGMENTS
.....................................................................................................
LIST OF FIGURES
..............................
iniii
vi
ABSTRACT............................................................................................................................v i i i
I.
II.
INTRODUCTION
I
A.
M a tr ix Slab C a lc u la t io n s .............................................................
I
B.
Green F u n c tio n C a lc u la t io n s
3
.....................................................
ONE-BAND C A S E ..........................
6
A.
Bulk H a m i l t o n i a n ...............................................................................
6
B.
Layer O r b i t a l s
...............................................................................
7
C.
Green F u n c tio n s
...............................................................................
8
1.
2.
D.
III.
............................................................................................ '
B u lk
.....................................................................
S u r f a c e ...................................................................................
8
IT
R e la tio n Between Green F u n ctio n s and D e n s itie s
and D e n s it ie s o f S t a t e s ..............................................................
11
E.
R e la t io n Between J and G
13
F.
Methods o f S o l u t i o n .............................................................
15
1.
2.
3.
T r a n s f e r M a t r ix Method . ............................................ .... .
E f f e c t i v e P o t e n t ia l Method ................................................
A n a l y t i c R e solve nt Method . ............................................
15
17
19
TWO-BAND C A S E ......................................................... ■..................................
33
A.
s-p Band, Normal E m i s s i o n ..........................
33
I.
33
B ulk
.........................................................
.............................................................................................
V
Page
2.
B.
Simple B o n d -O rb ita l Model
1.
2.
IV.
53
B u l k ................................................................................................
S u r f a c e .....................
53
62
..........................................................
69
B u lk C h e m i s t r y ...................................................................................
69
1.
2.
. ............................................
. . .
.................................
69
75
............................................................................
78
S u rfa ce m a t r ix elements .........................................................
B ra v a is l a t t i c e ..........................................................................
L a y e r - o r b i t a l s ..........................................................................
78
80
83
MATRIX SLAB CALCULATION..........................................................................
85
A.
I m p l e m e n t a t i o n ...................................................................................
85
1.
2.
85
85
B.
B.
Form ation o f b o n d - o r b i t a l s
B o n d - o r b it a l m a t r ix elements
S u rfa c e D e f i n i t i o n
1.
2.
3.
V I.
52
...................
BOND-ORBITAL MODEL OF HARRISON
A.
V.
S u rfa ce . . . ................................................................................
B u l k ..............................
S u r f a c e ...........................................................................................
R e s u lts and D i s c u s s i o n ....................................... - ........................
SUMMARY AND CONCLUSION
. . . . .
.....................................................
86
101
A.
G enerating F u n c tio n Formalism ................................
101
B.
T hin F ilm C a l c u l a t i o n ........................................................................... 109
REFERENCES................................... .............................................................................. 113
APPENDIX A - B u lk M a t r ix E l e m e n t s ..................................................................117
APPENDIX B - Layer O r b i t a l M a tr ix Elements
APPENDIX C - C o upling o f S u rfa ce t o Bulk
............................................
120
123
vi
LIST OF FIGURES
F ig u re
Page
1.
Argand Diagram f o r One-Band Model ..................................
24
2.
Layer D e n s it ie s o f S ta te s f o r th e One-BandModel
28
3.
In c o h e re n t Sums f o r th e One-Band Model
4.
Bands f o r th e s-p M o d e l ......................................................................
40
5.
Roots o f th e Two-Band C h a r a c t e r i s t i c Polynomial
..................
45
6.
Two-Band B u lk D e n s ity o f S ta te s . .................................................
51
7.
Layer D e n s it ie s f o r th e s-p M o d e l ................................................
54
8.
Form ation o f a Chemical B o n d .........................................................
56
9.
B o n d -O rb ita l A p p ro x im a tio n f o r Gap S ta te Energy . . . . .
65
10.
H yb rid O r b i t a l s in a Z in c b le n d e Semiconductor
..................
69
11.
Tetrahedron o f B o n d - O r b ita ls
.........................................................
76
12.
Formation o f Z in c b le n d e S t r u c t u r e ................................................
77
13.
B o n d -O rb ita l M a t r ix Elements
79
14.
B u lk - to - S u r f a c e C o upling C o e f f i c i e n t s
15.
Basis f o r th e fe e L a t t i c e
. . . .
...................
.........................................................
32
.......................................
81
.............................................................
82
16.
R e s u lt o f T r u n c a tin g th e Bulk . .....................................................
87
17.
More R e a l i s t i c S u r f a c e ..................................
88
18.
S urface B r i l l o u i n Zone
89
19.
F i l l e d S urface S ta te s on GaAs ( H O )
20.
P r o b a b i l i t y D i s t r i b u t i o n f o r B1
. . ..............................................................
.......................................
91
................................................
92
v ii
F ig u re
Page
21.
P r o b a b i l i t y D i s t r i b u t i o n f o r B2...........................................................
94
22.
P r o b a b i l i t y D i s t r i b u t i o n f o r B3
96
23.
B u lk Layer D e n s ity o f S ta te s f o r GaAs ( H O ) ............................105
24.
s - l i k e P a rt o f th e Layer D e n s it ie s f o r the
s-p Model
......................................................
. ...................................................................... ..........................107
v iii
ABSTRACT
The u n i f y i n g conce pt which r e l a t e s e s s e n t i a l l y a l l o f the methods
which c u r r e n t l y appear in th e l i t e r a t u r e f o r computing s u r fa c e e l e c ­
t r o n i c s t r u c t u r e from a g iv e n , t i g h t - b i n d i n g H a m ilto n ia n i s shown to
be th e d i f f e r e n c e e q u a tio n s s a t i s f i e d by the Green f u n c t i o n s .
An
e f f i c i e n t , g e n e r a tin g f u n c t i o n fo rm a lis m i s developed th ro u g h which
th e a n a l y t i c r e s o lv e n t method o f Levin e and Freeman may be extended
in o r d e r t o s o lv e more r e a l i s t i c s u r fa c e problem s. Two model H am il­
t o n ia n s are c o m p le te ly so lv e d in o r d e r t o dem onstrate t h i s method:
One i n which th e r e i s one s t a t e per la y e r and an s-p H a m ilto n ia n w ith
one s - l i k e and one p - l i k e s t a t e per l a y e r .
W ith th e a id o f th e s o l u t i o n o f th e s-p H a m ilto n ia n , a s im p le ,
b u l k , b o n d - o r b it a l model i s in tr o d u c e d and i t i s seen t h a t th e r e i s a
l i m i t in which t h i s a p p ro x im a tio n i s e x a c t.
This model i s used to
assess a b u l k , b o n d - o r b it a l model approach to th e s u r fa c e problem.
A b u l k , b o n d - o r b it a l model proposed by P a n te lid e s and H a rris o n i s
used to stu d y e ig e n s t a t e s o f a tw e lv e l a y e r f i l m o f ( H O ) GaAs. The
r e s u l t s compare w e ll w i t h those o f c a l c u l a t i o n s which do n o t make
b o n d - o r b it a l a p p ro x im a tio n s .
R e s u lts a re discussed and compared w ith
those o f o t h e r i n v e s t i g a t i o n s .
I.
INTRODUCTION
R e c e n tly , many c a l c u l a t i o n s o f s u r fa c e d e n s it ie s o f s t a te s have
been perform ed w i t h i n th e t i g h t - b i n d i n g i n t e r p o l a t i o n scheme^
A c co rd in g t o t h i s method, th e s i n g l e - e l e c t r o n H a m ilto n ia n o p e r a to r i s
re p re s e n te d by a s e t o f m a t r ix elements between lo c a l i z e d s t a te s cen­
te r e d a t each ato m ic s i t e .
U l t i m a t e l y , these m a tr ix elements are
t r e a t e d as parameters and are s e le c te d in such a way as t o reproduce
c e r t a i n r e s u l t s o f more a m b itio u s c a l c u l a t i o n s o r o f e x p e rim e n ta l
measurements
15
.
For example, th e param eters are o f t e n chosen t o r e ­
produce th e b u lk band s t r u c t u r e .
From th e b u lk H a m ilto n ia n m a t r ix elem e nts, th e program f o r com­
p u tin g the change in the d e n s it y o f s t a t e s caused by th e presence o f
th e s u r fa c e u s u a lly proceeds along one o f two ways:
E i t h e r a m a t r ix
H a m ilto n ia n c o rre s p o n d in g t o a la r g e s la b o f m a te r ia l i s d ia g o n a liz e d ,
o r Green f u n c t io n s are computed f o r th e s e m i - i n f i n i t e c r y s t a l .
A.
M a t r ix Slab C a lc u la t io n s
The f i r s t path along which th e c a l c u l a t i o n m ig h t proceed in v o lv e s
-] O
th e d i a g o n a l i z a t i o n o f a la r g e m a t r ix
. T h is corre spond s to s o lv in g
e x a c t ly th e problem o f f i n d i n g th e quantum s ta te s o f a t h i n s la b con­
s i s t i n g o f se v e ra l atom ic la y e r s .
T r a n s l a t i o n symmetry i n the d i r e c ­
t i o n s p a r a l l e l to th e s u r fa c e i s p re s e rv e d , and thus th e m a t r ix d ia g ­
o n a l i z a t i o n problem reduces t o f i n i t e
b lo c k s c o rre s p o n d in g t o the
2
i r r e d u c i b l e r e p r e s e n t a t io n s o f th e s u r fa c e t r a n s l a t i o n g r o u p ; in o t h e r
w o rd s , th e s u rfa c e p e r t u r b a t i o n does n o t mix those e ig e n s ta te s o f the
b u lk which have d i f f e r e n t components k j | o f wave v e c t o r p a r a l l e l to
th e s u r fa c e .
The g r e a t advantage o f a s la b ty p e o f c a l c u l a t i o n i s t h a t i t
e a s i l y implemented.
For a given va lu e o f k | | ,
and n u m e r ic a lly d ia g o n a liz e d .
is
th e m a t r ix i s f i l l e d
The energy e ig e n va lu e s and corre spond­
in g e ig e n v e c to r s may thus be o b ta in e d .
There are t h r e e m ajor o b j e c t io n s t o th e la r g e m a t r ix method.
F i r s t , f o r a p a r t i c u l a r c h o ic e o f k j j , th e l a r g e m a t r ix would be o f
rank nN, where n i s th e number o f lo c a l i z e d o r b i t a l s p e r u n i t c e l l ,
and N i s th e number o f l a y e r s .
I f t h e r e i s a r e f l e c t i o n symmetry
th ro u g h th e c e n t e r o f th e s la b , th e rank may be reduced t o nN/2.
There w i l l be o n ly nN energy e ig e n v a lu e s f o r each c h o ic e o f k | | ;
thus i t
i s d i f f i c u l t to produce enough e ig e n v a lu e s t o c o n s t r u c t a
l a y e r - b y - l a y e r d e n s it y o f s t a t e s f o r a p a r t i c u l a r v a lu e o f kj | , as
one would wish to do i n o r d e r to compare w i t h th e r e s u l t s o f an a n g le r e s o lv e d p h o toe m ission s p e c tro sco p y e x p e rim e n t.
As an example o f t h i s
d i f f i c u l t y , c o n s id e r a r a t h e r la r g e s la b o f
15 la y e r s (N=15) and e i g h t s t a te s on each la y e r ( n = 8 ) .
w ill
Then th e m a t r ix
be 120 x 120 and t h e r e w i l l be 120 energy e ig e n v a lu e s in a l l .
This w i l l
c r e a te a r a t h e r sk e tc h y h is to g ra m when these energy values
are p a r t i t i o n e d i n t o , say, 10 boxes o v e r th e e n t i r e energy range o f
3
th e spectrum.
The second o b j e c t i o n i s t h a t th e rank o f th e m a t r ix i s la r g e
enough so t h a t th e d i a g o n a l i z a t i o n must be done n u m e r ic a lly .
l u t i o n s do n o t p r o v id e any a n a l y t i c a l
The so­
i n s i g h t i n t o th e n a tu re o f the
s u rfa c e s t a t e s .
One f i n a l o b j e c t i o n i s t h a t even as th e s la b gets l a r g e r , the
s u r fa c e f e a t u r e s o f th e spectrum become l o s t in th e b u lk p a r t which
grows w i t h N.
E x tra steps would be r e q u ir e d i n o r d e r t o p r o j e c t the
d e n s it y o f s t a te s onto th e s u r fa c e la y e r .
B.
Green F u n c tio n C a lc u la t io n s
The second l i n e along which t i g h t - b i n d i n g c a l c u l a t i o n s have
evolved i s th e r e s o lv e n t o r Green f u n c t i o n method^
Here aga in the
problem i s reduced by the t r a n s l a t i o n symmetry along th e s u r fa c e .
T h is t im e , however, th e s e m i - i n f i n i t e c r y s t a l
is tre a te d .
For th e t i g h t - b i n d i n g H a m ilto n ia n s , th e e q u a tio n s o f m otion f o r
th e Green f u n c t io n s become a s e t o f c o u p le d , l i n e a r d i f f e r e n c e equa­
t i o n s w i t h c o n s ta n t c o e f f i c i e n t s .
These e q u a tio n s may be solved in
one o f a number o f ways, and a l a y e r - b y - l a y e r d e n s it y o f s t a te s can be
c a l c u la t e d from th e im a g in a ry p a r ts o f th e v a r io u s Green f u n c t i o n s .
The d i f f e r e n t methods o f s o lv in g th e d i f f e r e n c e e q u a tio n s have been
given d i f f e r e n t names such as th e a n a l y t i c r e s o lv e n t m e t h o d ^ , the
t r a n s f e r m a t r ix m e t h o d ^ , th e ansatz m e th o d ^ o r th e e f f e c t i v e
4
p o t e n t i a l method
19
.
Each method g iv e s r i s e t o a n o th e r v o c a b u la r y , b u t
th e u n i f y i n g concept i s th e s e t o f d i f f e r e n c e e q u a tio n s f o r th e Green
fu n c tio n s .
The f i r s t c r i t i c i s m o f th e m a t r i x method i s a u t o m a t ic a l ly avoided
when th e r e s o lv e n t method i s used.
The Green f u n c t i o n method e a s i l y
produces th e continuum as w e ll as th e d i s c r e t e p a r t o f th e spectrum ,
s in c e the s e m i - i n f i n i t e c r y s t a l i s t r e a t e d .
The use o f Green f u n c t io n s may a ls o a vo id th e second c r i t i c i s m o f
th e m a t r ix method, p ro v id e d t h a t c lo s e d form s o l u t i o n s o f th e equations
o f m otion can be fo u n d .
The d i f f e r e n c e e q u a tio n s possess s o l u t i o n s in
clo se d form o n ly i f th e t i g h t - b i n d i n g H a m ilto n ia n i s s im p le enough.
The e x is te n c e o f s o l u t i o n s i n terms o f elem e ntary f u n c t io n s cannot de­
pend upon th e method o f s o l u t i o n .
W ith each model H a m ilto n ia n is
a s s o c ia te d a c h a r a c t e r i s t i c p o ly n o m ia l, th e degree o f w hich i s d e t e r ­
mined by th e c o m p le x ity o f th e model.
When t h i s c h a r a c t e r i s t i c p o l y ­
nomial f a l l s o u t s id e th e c la s s o f p o ly n o m ia ls which are s o lv a b le by
• r a d i c a ls , then a s o l u t i o n i n terms o f ele m e n ta ry f u n c t io n s does not
e x is t.
The p o lyn o m ia l
i s th e c h a r a c t e r i s t i c p olynom ia l o f th e system
o f d i f f e r e n c e e q u a tio n s which are the e q u a tio n s o f m o tio n .
Even when a s o l u t i o n i n clo sed form i n terms o f e le m e n ta ry fu n c ­
t i o n s does n o t e x i s t , th e Green f u n c t i o n method may le a d t o some ana­
ly tic a l
i n s i g h t i n th e f o l l o w i n g way.
The la y e r d e n s i t i e s o f s t a te s
f o l l o w from Green f u n c t io n s which are e x p r e s ib le as r a t i o n a l f u n c t io n s
5
o f th e r o o t s o f th e c h a r a c t e r i s t i c p o ly n o m ia ls .
Near a s i n g u l a r i t y ,
th e y may be expanded and t h e i r b e h a v io r r e l a t e d to the m otion o f the
r o o ts in th e complex p la n e , th e r e b y p r o v id in g a c o n n e c tio n between the
b u lk and s u r fa c e e l e c t r o n i c s t r u c t u r e s .
To be f a i r ,
i t must be a d m itte d t h a t even in th e cases when e xa ct
s o l u t i o n s a re p o s s i b l e , th e r e i s sometimes l i t t l e
m o t i v a t io n to w r i t e
them down because t h e i r form al c o m p le x ity may re n d e r them incompre­
h e n s ib le ^ ” ^ .
N o n e - th e - le s s , these s o l u t i o n s , w hether sy m b o lic o r
p a r t l y n u m e r ic a l, r e p r e s e n t i n a most c o n v e n ie n t form e s s e n t i a l l y
e x a c t s o l u t i o n s t o the g iv e n model H a m ilto n ia n f o r th e s e m i - i n f i n i t e
geometry.
P a rt o f th e convenience o f th e Green f u n c t io n s i s t h a t th e den­
s i t y o f s t a t e s i s n a t u r a l l y o b ta in e d i n a form which i s f a c t o r e d
a c c o rd in g n o t o n ly t o th e i n d i v i d u a l l a y e r s , b u t a ls o a c c o rd in g to
the i n d i v i d u a l o r b i t a l s on each l a y e r .
The f i n a l o b j e c t i o n t o the
m a t r ix method i s thus circu m ve n te d by th e Green f u n c t i o n method.
II.
A.
ONE-BAND CASE
Bulk H a m ilto n ia n
The purpose o f t h i s s e c t io n i s t o in tr o d u c e one o f th e s im p le s t
problems which may be s o lv e d by th e r e s o lv e n t method and thus to demon
s t r a t e th e u n i t y o f v a r io u s s o l u t i o n procedures f o r th e r e s u l t i n g
d i f f e r e n c e e q u a tio n s .
C o nsid er a c r y s t a l w i t h sim p le c u b ic s t r u c t u r e c o n s i s t i n g o f an
s - l i k e a to m ic o r b i t a l a t each s i t e .
The b a s is f u n c t io n s are r e p r e ­
sented by |& ,m ,n>, where th e p o s i t i o n o f th e s i t e i s r = a&x + amy +
anx.
The l a t t i c e
spacing i s a and x , y , z are C a rte s ia n u n i t v e c t o r s .
The in d ic e s &,m,n a re in t e g e r s and s p e c i f y th e p o s i t i o n o f th e s i t e .
In t h i s b a s i s , th e n , th e b u lk H a m ilto n ia n i s d e fin e d as f o l l o w s :
<Vm'n'|H( o , |mn> = Us^ 1
+ y6V & 5m'm ^ n ' n + i + 6n ' n - i ^
+ y6Ji1a ^5IiVm+1 + 5ml m -i ^ 5n 1n
(I)
+ ^ ( 5& '& + i + 5Jil Ji-I^ 5m'm 5n 'n
where U and y are i n t e g r a l s
U =
rS
( x ) H (° ) ( x ) 6 ( x ) d \ ,
(2)
*
y = f ^ s (x )
( x ) tj)s ( x - a x ) d 3x.
In f a c t , w e . t r e a t th e i n t e g r a l s LI and y as c o n s ta n ts which are
7
a d ju s te d t o produce th e b u lk band s t r u c t u r e which may be computed by
a n o th e r means.
I t i s assumed t h a t a l l o t h e r m a tr ix elements o f the
H a m ilto n ia n va n ish and a ls o t h a t
B.
Layer O r b i t a l s
I f an a n g le - r e s o lv e d p h o toe m ission expe rim e n t were t o be p e r ­
formed on a (100) s u r fa c e o f the h y p o t h e t ic a l c r y s t a l d e fin e d above,
then th e a n g le - r e s o lv e d d e n s it y o f i n i t i a l
s t a te s m ig h t be o f i n t e r e s t .
Advantage may be taken o f th e t r a n s l a t i o n symmetry in the plane
o f th e s u r fa c e i f
th e b u lk H a m ilto n ia n i s r e w r i t t e n i n terms o f a new
b a sis which has Bloch symmetry along th e y and z d i r e c t i o n s .
Suppose f o r th e moment t h a t each l a y e r i s f i n i t e
and c o n s is ts o f
N atoms in th e y d i r e c t i o n and N atoms i n th e z d i r e c t i o n .
one c o u ld ta k e th e l i m i t N -> =°,
N w ill
but th is w ill
n o t be n ece ssary, s in c e
cancel o u t o f th e e q u a tio n s .
D e fine a la y e r o r b i t a l
on l a y e r & by th e r e l a t i o n
|i .k .k > = Tf I e1kyara eikzan|<t,m,n>.
'
U ltim a te ly ,
Y
z
N mn
(4)
The t r a n s f o r m a t i o n i s u n i t a r y and thus th e la y e r o r b i t a l s are o r t h o ­
normal .
^ ', k 'y .k ^ iA . k y .y
% k y
* k ',k ,.
(5)
Because o f t r a n s l a t i o n symmetry, H does n o t mix l a y e r o r b i t a l s
8
o f d i f f e r e n t wave v e c t o r , s in c e the se o r b i t a l s would then belong to
d i f f e r e n t i r r e d u c i b l e r e p r e s e n t a t io n s o f th e tw o -d im e n s io n a l t r a n s l a ­
t i o n group.
in itio n
Thus, H i s d iagon al w i t h r e s p e c t t o k ^ , k z .
From the d e f ­
(4) o f the l a y e r o r b i t a l s and fro m e x p re s s io n ( I ) f o r th e b u lk
H a m ilto n ia n m a t r ix elem e nts.
< t ' , k ' y , k ' z |H( o h t , k y , k z>
6k'..k.. V . k .
Z Z
y y
w V kZ1 6Jt1Z + v t 6 Z1Z+! * 5Z1Z - I 11
where e(k ,k ) = U + 2 y cosk a + 2 y cosk a.
j
z
'
t6>
For a p a r t i c u l a r k ,k ,
y
z
we have a one -d im e n sio n a l problem a s s o c ia te d w i t h the H a m ilto n ia n
u(o)
V
ji
=aZ1Z + v t a Z1Z+! + 5Z1Z - ! 1'
(7)
T h is b u lk H a m ilto n ia n is connected t o th e s u rfa c e H a m ilto n ia n in
a r a t h e r sim ple way.
Suppose, w i t h o u t lo s s o f g e n e r a l i t y , t h a t th e
s u r fa c e l a y e r i s a = I .
Z l
Then th e H a m ilto n ia n becomes
+ y(6
1Z 1Z 1
+
when
Z 1 , Z > I ; H1 1 = E, H12 - Hg % - y .
(8)
Hj l l j l = 0 o t h e r w is e .
C.
I.
B u lk .
Green F u n ctio n s
The common methods o f s o l u t i o n in v o lv e th e use o f one o f
two ty p e s o f Green f u n c t i o n s .
9
The f i r s t ty p e i s a Green f u n c t i o n f o r the b u lk H a m ilto n ia n .
b * , b^ c r e a t e and d e s tr o y an e l e c t r o n i n la y e r o r b i t a l I
Let
f o r a given
+
va lu e o f k y , k z .
The tim e dependence o f b^ and b^ in th e Heisenberg
p i c t u r e i s governed by th e b u lk H a m ilto n ia n H ^ .
Then th e o n e -e le c ­
t r o n b u lk Green f u n c t io n between l a y e r a -and la y e r z 1 i s d e fin e d by
5I , '- I
(t)
=
<<bv ;
=
(9>
- i e ( t ) <0|{bt , ( t ) , b+ (t=0)}|0>.
In e q u a tio n ( 9 ) ,
0 (t)
|0> i s th e s t a t e w i t h no e le c t r o n s in th e c r y s t a l ,
i s a s te p f u n c t i o n equal to 0 i f
t <
0, % i f t = 0 , and I i f t > 0.
The b r a c k e t i s an a n t i commutator w i t h ' t h e usual d e f i n i t i o n :
( A , BI =
AB + BA
(10)
I t should be noted here t h a t th e b u lk Green f u n c t io n s G ^ , ( t ) are
o n ly f u n c t io n s o f th e la y e r d i f f e r e n c e z ' - z ,
r a t h e r than o f z 1 o r z
s e p a r a t e ly , because o f t r a n s l a t i o n symmetry.
The o p e r a to r b ^ ( t ) e volves a cc o rd in g t o th e Heisenberg equ a tio n
4
Thus, G^ ,
b jt) = [b jt),
(ID
( t ) s a tis fie s
^ ^ z ' - z
(t) =
6 ( t ) < 0 | { b v ( 0 ) , b + ( 0 ) } |0 > - « [ H ( o ) , b£ l ] ;
bA» ,
(12)
where th e f i r s t term on th e r i g h t is j u s t & ( t ) G . b e c a u s e
(13)
10
The second term on th e r i g h t o f e q u a tio n (11) i s a h ig h e r o r d e r Green
f u n c t io n
« [H ^ \
- ie ( t)
b£ l ] ;
b£>> =
< 0 | { [ H ( o ) , b£ , ( t ) ] ,
b+ ( t = 0 ) } | 0 > .
(14)
I n t r o d u c in g th e F o u r ie r t ra n s fo r m
<v-* (“>=C V - * (t) e"iBt dt'
(’5)
and n o t in g t h a t in t h i s case th e h ig h e r o r d e r Green f u n c t i o n i s a
l i n e a r c o m b in a tio n o f th e o r i g i n a l
one e l e c t r o n Green f u n c t i o n s ,
e q u a tio n (12) becomes
“V-*
+ EGv
V t +v6Vtt-I
M
‘
M
+ %GV _1_% ( » ) • '
M
06)
T h is d i f f e r e n c e e q u a tio n i s th e e q u a tio n o f m otion o f the b u lk Green
f u n c t io n s .
It
i s im p o r t a n t t o n o t ic e t h a t f o r o n e - e le c t r o n H a m ilto n ia n s l i k e
t h i s one, th e Green f u n c t io n s in- th e domain are m a t r ix elements o f the
r e s o lv e n t o p e r a to r d e fin e d by
G = (w - H( o ) ) _1
(17)
A t p o in t s in th e w -p la n e where G i s n o t w e l l - d e f i n e d ( th e spectrum o f
i t can be seen from (14) and (15) t h a t G should be co n sid e re d as
th e l i m i t as o->0+ o f (w + in - H ^ ) ~ V
\
Using ( 1 7 ) ,
11
v
<“ +
- H<0)> V V
Gr - i = 6W
<18>
T h is i s e x a c t ly e q u iv a l e n t to e q u a tio n ( 1 6 ) .
2.
S u rfa ce .
A n oth er typ e o f Green f u n c t io n which i s used c o n ta in s
th e e f f e c t o f th e s u r fa c e and i s d e fin e d as
lV s ,
= - i0
(t)
<0|{a^, ( t ) , a^ ( 0 ) } | 0 > .
The tim e e v o l u t i o n o f a^ and a ^ , i s due t o th e f u l l
which i s t r u n c a te d a t th e s u r fa c e .
Jv j l
■ (T9)
H a m ilto n ia n H
Thus
(m) = < A '|( w - H ) " 1 |£>.
H i s e q u iv a le n t t o
(20)
b u t t r u n c a te d a t th e s u rfa c e a c c o rd in g t o (8)
J ^ ljl i s n o t a f u n c t i o n o n ly o f
because o f th e la c k o f t r a n s l a ­
t i o n symmetry.
D..
R e la t io n Between Green F un ctio n s
and D e n s it ie s o f S ta te s
The d e n s it y o f s t a t e s f u n c t i o n f o r a system governed by a
H a m ilto n ia n H i s d e fin e d as
D(E) = Z S(E-Ev ) .
The in d e x v runs o v e r each i n d i v i d u a l e ig e n s t a t e o f H.
(21)
D(E) is so
d e fin e d as t o enable a sum o v e r v t o be co n v e rte d t o an i n t e g r a l over
E.
E f ( E v ) = Z f ( E ) D(E) dE.
The d e l t a f u n c t io n s
in (21) may be re p re s e n te d by
( 22)
12
6<E- Ev> -
1 i ™+
n->0
7
o
(E-Ev )
9
+ n •
From t h i s p o i n t on, th e I im w i l l
TT^-O+
D(E)
' Ti Im- — Im.
( 23)
I { - I m ' <E - H n - E )}■
v
it
^
n o t be w r i t t e n e x p l i c i t l y .
Im ' ^ E + in - E .
<VI E + in - H
v>)
( 24)
(Trace J ) .
Rather than th e t o t a l
d e n s it y o f s t a t e s D (E ), i t
i s o f t e n more
c o n v e n ie n t t o have th e p r o je c te d d e n s it y o f s t a te s f o r some s t a t e
|j> ,
where | j > i s q u i t e a r b i t r a r y and n o t an e ig e n s t a t e o f H.
Dj (E) = E | < v | j > | 2 S ( E - E v ) .
( 25)
T h is would p e r m it e v a lu a t io n o f p r o je c t e d sums o f th e form
E | < j | v > | 2 f ( E v ) = I Dj (E) f ( E ) dE.
( 26)
As an example, the lo c a l d e n s it y o f s t a te s
Dj l (E) = E
|< j i|v
> | 2 6(E-Ev )
(27)
i s o f i n t e r e s t i n s u r fa c e problems when & i s a la y e r o r b i t a l c lo s e to
th e s u r fa c e f o r some va lu e o f k ^ , k z By th e a n a ly s is above i t f o l l o w s t h a t
Di ( E ) - - I
Tm. Ou ( E )
(28)
.
13
For t h i s
reason , th e n , th e o b j e c t o f th e Green f u n c t i o n method i s to
compute th e d ia g o n a l p a r t o f th e s u r fa c e Green f u n c t io n J.
E.
R e la tio n Between J and G
The r e s o lv e n t J c o rre s p o n d in g t o H must be r e la t e d t o th e b u lk
r e s o lv e n t G c o rre s p o n d in g t o
th e e f f e c t o f th e s u r fa c e .
v ia a p e r t u r b a t i o n V r e p r e s e n tin g
E v id e n tly , V w i l l
n o t i n any way be a
sm all p e r t u r b a t i o n .
L e t th e la y e r s o f th e c r y s t a l be separated i n t o two c la s s e s :
w ill
be th e c o l l e c t i o n o f a l l
la y e r s which w i l l
a f t e r th e s u r fa c e i s form ed, and B w i l l
b e lo n g in g t o a b a r r i e r r e g io n .
A
belong t o th e c r y s t a l
be the s e t o f l a y e r o r b i t a l s
A s c a l a r p o t e n t i a l o f magnitude
w ill
be added t o each d iagon al element o f th e H a m ilto n ia n in th e re g io n B.
In th e l i m i t t h a t p -*0 , th e s o l u t i o n w i l l
be excluded from re g io n B
and th e boundary c o n d it io n s in re g io n A w i l l
have been changed from
c y c l i c t o stan d in g -w a ve boundary c o n d it io n s .
In th e sample model s tu d ie d h e re , B may be taken t o c o n s i s t o f
o n ly one la y e r o r b i t a l , s in c e t h i s
bonds a t th e s u r fa c e .
i s s u f f i c i e n t to break a l l o f the
Take, f o r example, &=0, f o r B and l e t A con­
s is t o f & = 1 ,2 ,3 ...
W r i t i n g G and J i n b lo c k form i n th e la y e r o r b i t a l
0
G =
b a s is ,
0
V =
(29)
0
P
14
( E - H ^ - V ) -1 = [ ( E - H ( ° ) ) . ( I - G V ) ] ' 1
(30)
( V G V ) - 1 G, o r G = (I-G V ) J.
(31)
Using ( 2 9 ) , i t
f o ll o w s
I
-1
P
6AB \
(I-G V ) =
(32)
0
.
S u b s titu tin g
'" P
(32) i n t o
6B B 1
(31) and u sin g th e b lo c k form
(33)
give s th e two e q u a tio n s
j AA '
(1 '
p 2AB JBA
(34)
gAA9
(35)
p GBB} 0BA = 6BA
In th e l i m i t p + O, s u b s t i t u t i o n o f (35) i n t o
(34) y i e l d s ^
(36)
j AA = 6AA " 6AB ( gBB^ 6BA
E q uation (36) i s a v e ry im p o r t a n t r e l a t i o n which e x i s t s q u i t e
g e n e r a ll y between th e b u lk and s u r fa c e Green f u n c t i o n s .
noted t h a t e q u a tio n (36) i s an e x a c t r e s u l t and
o f th e new, stan d in g -w a ve boundary c o n d it io n s .
I t should be
has now th e e f f e c t
15
F.
I.
Methods o f S o lu t io n
T r a n s f e r M a t r ix Method.
T his method was fo rm u la te d by L. F a lic o v
and F. Y n d u r a in ^ and has been extended by Mele and J o a n n o p o u lo s ^ .
The t r a n s f e r m a t r ix method i s c l o s e l y r e la t e d t o th e a n s a t z ^ method
excep t t h a t the l a t t e r name i s u s u a ll y a p p lie d t o a method o f d i r e c t l y
s o l v in g the H a m ilto n ia n e ig e n v a lu e problem
(E-H)
|^> = O
(37)
w h ile th e fo rm e r name r e f e r s t o the same method a p p lie d t o th e s o l u ­
t i o n o f the d i f f e r e n c e e q u a tio n s re p re s e n te d by th e e x p re s s io n
(E-H) J = I
(38)
In both (37) and (38) th e H a m ilto n ia n in c lu d e s th e s u r fa c e .
(38)
leads t o a s e t o f d i f f e r e n c e e q u a tio n s f o r th e m a t r ix elements o f J .
In th e case o f th e s im p le model H a m ilto n ia n o f ( 8 ) , these are
SL"SL
I 'I
(39)
(40)
(E-e) J11 - IiJ21 - I
- IiJ11 + (E-e) J21 - WJg1= O
- VJni + (E-e) Jn+n - IiJ n+21
(41)
(E-e) J 12 - yJ22 = O
- IiJ12 = (E-e) J22 = IiJg2
- WJn2 + (E-e) Jn+12 =
e tc .
J n+22
16
The method o f s o l u t i o n depends upon the f a c t t h a t (40) i s homogeneous
fo r n ^ I.
Thus, f o r n ^ I ,
powers o f two numbers
Jn l = A
X1
X1 ,
th e s o l u t i o n i s a l i n e a r co m b in a tio n o f
X2 -
+ B x2 , n ^ I .
T ry a s o l u t i o n J n l = x 11.
(42)
Then from (40) we have
ux2 - (E-e)x + p = 0.
( 43)
The f u n c t i o n x here i s what i s known by F a lic o v and Y ndurain
t r a n s f e r m a t r i x , a lth o u g h i n t h i s s im p le example, i t
I7
as the
is not r e a lly a
m a t r ix .
T h is i s th e f i r s t example o f a c h a r a c t e r i s t i c p o ly n o m ia l.
two r o o ts are
X1
X i, 2 °
The
and x2 .
[( % )
- 1I
I t can be seen from (43) t h a t X 1X2 = I .
(44)
T h is i s a gen eral f e a t u r e :
r o o t s o f th e c h a r a c t e r i s t i c e q u a tio n s o c cu r in r e c ip r o c a l p a i r s .
Suppose E > E + 2, then
Ix 1 I > I > |x 2 | .
(45)
Since J n l must be bounded as n+<=°, th e c o e f f i c i e n t A o f (42) must
be z e r o .
Whence:
Jnl = B Xg, n >_!.
B u t, lo o k in g a t th e f i r s t o f e q u a tio n s ( 4 0 ) , we see t h a t
( 46)
17
(E-e) B
X2
- yB x l = I ,
B = ---------- --------- 2
(E-E)X2 - y x 2
(47)
=“ •
(48)
y
A t which e x p re s s io n (46) becomes
ni
(49)
y
Now, th e s u r fa c e d e n s it y o f s t a te s f o ll o w s from (49) v ia
di
(2 8 ) :
(so)
( e ) = i
One c o u ld go back to e q u a tio n s (41) and s o lv e f o r J22 and so on and
thus f i n d s u rfa c e d e n s it ie s o f s t a t e s on each la y e r ,
2.
E f f e c t i v e P o t e n t i a l Method
19
.
A n o th e r method f o r d e te r m in in g the
l a y e r d e n s it y o f s t a te s takes advantage o f a k in d o f hidden symmetry
o f th e t o t a l
H a m ilto n ia n H.
Foo and e t a l _ . T h e
T h is c le v e r idea seems t o be due to E-Ni
c o n n e c tio n o f t h i s method to th e t r a n s f e r
m a t r ix method was r e a l i z e d by Joannopoulos and M e l e ^ .
Suppose we have s o lv e d th e problem o f f i n d i n g th e Green f u n c t io n s
l1JI1JI
^ o r tlie s im p le s u r fa c e model above.
la y e r o r b i t a l
Then, i f we adsorb a n o th e r
i n t o p o s i t i o n on JL=O w hich i s e x a c t ly l i k e th e o t h e r
la y e r s and s o lv e f o r th e new Green f u n c t io n s R^1
iV t 5 j V - H , , A l '
(I - JV) R = J,
By equation (31)
we shou ld f i n d
18
O
oP
R01
w ith R
(51)
, R10
R11
I
W
I
LU
0
0
(52)
11
Thus
- jj_
E- e
I
(I-J V )
(53)
I
-UJ11
Whereupon (51) becomes
R __ y_
00
E-E
- yJ
11 R00
R
kIO
+
(54)
E-e*
R10
(55)
°'
E l im in a t i n g R ^ between (54) and (55) and using R(
00
U2
FsT
2
(56)
0 T l + J 11
f o r e - 2y < E <
E
J 11 gives
+
2y.
The minus s ig n in (57) must h o ld , i n o r d e r t h a t th e d e n s it y o f
s t a te s be p o s i t i v e , which means t h a t
( 58)
19
Equation (58) agrees w i t h
(5 0 ).
An i n t e r e s t i n g note co n ce rn in g th e meaning o f th e e f f e c t i v e
p o t e n t i a l method i s th e f o l l o w i n g :
The d ia g o n a l Green f u n c t io n J
can be w r i t t e n in terms o f an
i r r e d u c i b l e s e l f energy Z ^ (E ):
(E) = ( E - e - Z j l ( E ) ) ' 1 .
(59)
2?
(E) i s th e sum o f a l l terms i n th e Neumann expansion in which
a does not occu r i n an in te r m e d ia te s t a t e .
For th e sim p le
case, the
b u lk Green f u n c t io n
G ^
(E) = ( E - e - Z ( E ) ) " 1
(60)
c o n ta in s a s e l f energy Z(E) which i s made up o f a sum c o n t a in in g
as in te r m e d ia te s t a t e s
o n ly la y e r s t o th e r i g h t o f a g ive n
and a n o th e r sum o ver o n ly la y e r s t o th e l e f t .
la y e r I
Each sum i s equal to
Y z (E) because the b u lk H a m ilto n ia n connects o n ly n e ig h b o rin g la y e r s .
Thus
Z£=1 (E) = ^ Z (E )
(6 1 )
.
holds a t th e s u r fa c e la y e r .
3.
A n a l y t i c R e so lve n t M e t h o d ^ .
T h is a p p l i c a t i o n o f s c a t t e r i n g
th e o r y was made by S. Davison and J . D. L e v in e .
t i o n f o rm a lis m , which g r e a t l y f a c i l i t a t e s
v id i n g conce ptu al s i m p l i c i t y ,
K a w a jiri.
The g e n e r a tin g fu n c ­
com putations as w e ll as p r o ­
i s due t o Schwalm, Hermanson and
C onsider th e b u lk d i f f e r e n c e e q u a tio n
20
- - Gt + 1 + (E -= ) G, - Ij = , - , = 6» '
(SE)
The a n a l y t i c r e s o lv e n t method c o n s is ts o f s o lv in g
and then a p p ly in g ( 3 6 ) .
(62) in clo sed form
A more r e a l i s t i c model w i l l
le a d t o a more
c o m p lic a te d system o f b u lk d i f f e r e n c e e q u a tio n s .
In o r d e r t o in tr o d u c e the g e n e r a tin g f u n c t i o n method, d e f in e a
g e n e r a tin g f u n c t i o n :
f ( x ) = Z x* G,
M u ltip ly in g
- J f +
(63)
(62) by x* and summing o v e r Z w h ile a p p ly in g (63) y i e l d s
(E -e) f - x y f = I
(64)
or
f ( x ) = ---------------------- — y ~ .
(65)
(E-e) - u(x + —)
The e x p re s s io n f o r
expa nsio n.
f(x )
w i l l f o l l o w from (65) v ia a L a u re n t s e r ie s
B u t,
x
"
I_______
X2 -
U
( 66 )
X + I
has th r e e L a u re n t expa nsio ns.
The r o o t s o f th e denom inator (which are
zeros o f th e Fredholm d e te rm in a n t and r o o t s o f the c h a r a c t e r i s t i c
\29
p olynom ia l o f th e d i f f e r e n c e e q u a tio n )
E- E
Xl 2 =
2u
+
I .
are
(67)
21
These are th e same r o o ts a pp earing in ( 4 4 ) .
The g e n e r a tin g f u n c t io n
has become
" y
_
(X-X1)
I
y
(X-X2 )
___ ] _
( X 1- X 2 )
_
(X-X1)
“
I
y
(X2 - X 1 )
I
( 68 )
(x -x 2 )'
The terms must be expanded in th e f o l l o w i n g way:
( x - x ) 1 = - ” — ( l + v - + A r + • • • ) w h ile
r
xi
xi
(69)
x
(X-X2 ) 1 = + |(I + X
X
+ —
+•••) •
Xx
(70)
These c h o ic e s , which amount t o a ch o ic e o f th e annulus o f co n ve r­
gence o f th e La u re n t expansion o r e q u i v a l e n t l y to th e c h o ic e o f con­
t o u r f o r th e in v e r s i o n i n t e g r a l
f(x )d x
ZttI
C
(71)
X&+1
f o l l o w from th e p h y s ic a l boundary c o n d it io n s on
.
Suppose by way o f
example t h a t th e a l t e r n a t i v e expansion had been made i n
X1
r a t h e r than
(69).
X
(X-X1)
1 = 1"(1
X
X2
+ 1T" + ™T + ' ” )•
X
(72)
x"
Equation (72) i n d ic a t e s t h a t G_^~x1 as
But from ( 4 5 ) ,
Ix 1 | > I .
T h is i s im p o s s ib le f o r a p h y s ic a l s o l u t i o n , s in c e G must be bounded
22
i n th e b u lk as £ ^ ±
conve rg e n t when
T h u s, th e L a u re n t expansion i s chosen t o be
I x 2 M x ^ I x 1 1.
may be chosen as |x | = I .
The c o n to u r o f i n t e g r a t i o n in (71)
T h is c h o ic e o f c o n to u r r e s u l t s from p h y s i­
c a l boundary c o n d it io n s in th e same way t h a t the c h o ic e o f th e c o n to u r
f o r in v e r s i o n o f th e F o u r ie r t r a n s fo r m o f a p h y s ic a l q u a n t i t y is o f t e n
s p e c i f i e d by boundary c o n d i t i o n s .
O the r choices f o r th e La u re n t expan­
s io n correspond t o o t h e r s o l u t i o n s t o th e second o r d e r d i f f e r e n c e
e q u a tio n which are n o t in t h i s case th e r i g h t ones.
The above a n a ly s is has taken p la ce under th e s u p p o s it io n t h a t
E > e + Z \i.
The surd w hich appears in e x p re s s io n s (67) has branch
p o in t s a t E = e ± 2y.
P h y s i c a l l y , these are van Hove s i n g u l a r i t i e s
c o rre s p o n d in g t o th e edges o f the b u lk energy band o f t h i s sim ple
m odel.
As E becomes lo w e r than
e
+ 2y, th e r a d ic a l must be c o n tin u e d
p a s t th e branch p o i n t i n such a way as t o make i t a smooth f u n c t io n o f
E.
T h is i s done by remembering th e c a u s a l i t y c o n d it io n on G ^ ( t )
d is c u s s io n f o l l o w i n g
( 1 7 ) ) and r e p la c in g E by E + i n .
In a c tu a l c a l ­
c u l a t i o n s f o r more r e a l i s t i c m odels, n i s a sm all b u t f i n i t e ,
energy param eter.
(see
p o s itiv e
The ch o ice o f th e branch c u t i s then taken so t h a t
/ e - £ - 2y / e - E + 2y .
(73)
N o tic e t h a t (73) i s a branch c u t c o n v e n tio n and n o t an i d e n t i t y .
The
c u ts f o r th e two surd e x p re s s io n s on the r i g h t are along the r e a l E
a x is toward - °°, i . e . , both phases are zero f o r l a r g e , p o s i t i v e re a l
23
energy.
By t h i s c h o ic e , a branch c u t has been made using language a
computer m ig h t u n d e r s ta n d , between th e two b u lk van Hove s i n g u l a r i t i e s
which d e l i m i t a continuum .
T h is general procedure has a ls o been
f o llo w e d in l a t e r com putation s on r e a l i s t i c models where the continuum
s t r u c t u r e i s f a r more c o m p lic a te d .
The t r a j e c t o r i e s o f th e r o o t s X1 and x2 in th e complex x - p la n e
are p resen ted i n f i g u r e I w i t h E = O
and n = . I y .
When n i s q u i t e
s m a ll, th e c h a r a c t e r i s t i c p o lyn o m ia l
(43) has a lm ost r e a l c o e f f i c i e n t s .
In t h a t case, whenever X1 and x2 are complex, th e y are a lm o st complex
c o n ju g a te s o f one a n o th e r. ' T h e r e f o r e , because x2 = V x 1, these r o o ts
would have th e f o l l o w i n g form :
Xi
2 = e 1 1ka e ± Ka,
(74)
where a i s th e c r y s t a l
l a t t i c e spacing and ka and ca are r e a l numbers
w ith 0 <
would have a damped, s in u s o id a l form as a
Ka « 1 .
fu n c tio n o f I .
G co e+:| N ka e ' I 4 I ica.
The damping n goes t o zero as n
E- 2y < E < E + 2y.
(75)
O+ .
Thus in th e energy range
corresponds t o a d is c o n tin u o u s co m b in a tio n o f
two B l o c h - l i k e s o l u t i o n s to th e homogeneous H a m ilto n ia n e ig e n v a lu e
problem :
(E-H)
!*> = 0.
(76)
I t i s th e d i s c o n t i n u i t y i n G£ a t 2=0 which causes G to a c t as the
X PLANE
F ig u re I .
Argand Diagram f o r One-Band Model.
The arrows i n d ic a t e in c r e a s in g energy.
25
r e s o lv e n t o f H.
In t h i s c o n t e x t i t may be seen t h a t
i s th e i n t e r ­
X2
la y e r phase o f th e Bloch s o l u t i o n o f energy E, and when E i s o u ts id e
the range o f th e a llo w e d B l o c h - l i k e s o l u t i o n s , then
X2
corresponds to
th e decrement per l a y e r o f the a n a l y t i c a l l y c o n tin u e d Bloch f u n c t io n s
which decrease e x p o n e n t ia lly
21
Gjl ~ e ^ l & | a ,
I E-e I < 2y,
Gj, - e ~ K | * | a ,
I E - e | >2p.
(77)
From ( 6 8 ) , the e x p re s s io n f o r G^ i s seen t o be
X2
Ga (E) =
W
---------
.
(78)
2M/Ir
) 2 ' 1
Thus, from ( 2 8 ) , th e d e n s it y o f s t a t e s on any b u lk la y e r i s
D(E) = - I
Im. .< A | G (E)
= - — Im.
TT
| A >
(79)
Gn ( E ) .
0
Exp ression (80) holds w it h th e usual meaning o f th e square r o o t f o r
e n e rg ie s
fin ite ,
o ff.
e
- 2 v < E < e + 2 v , i n th e l i m i t t h a t n ^ 0.
I f n is l e f t
then th e in v e r s e square r o o t s i n g u l a r i t i e s o f (80) are rounded
26
E q uation (36) a f f o r d s an e x p re s s io n f o r the s u r fa c e Green f u n c ­
tio n .
Region B c o n s i s t s , i n th e s im p le model o f th e p re s e n t example,
o f s.=0,
w h ile re g io n A c o n s is t s o f S-=I , 2 , 3 , • • • .
( 81 )
S-S-
P u t t in g (78) i n t o
J
(81) y i e l d s
(82)
S-S-
f o r S-=I , 2 , 3 —
and f o r E > e + 2 y.
The d e n s it y o f s t a t e s on the s / t h la y e r can be o b ta in e d from (82)
v ia ( 2 8 ) .
The squre r o o t which appears i n th e denom inator as w e ll as
i m p l i c i t l y in
X2
must be a n a l y t i c a l l y c o n tin u e d i n t o th e band
|E - e I < 2 y , as i t was i n e x p re s s io n (79) t o o b t a in fo rm u la (8 0 ).
A ls o , s in c e from (74)
X2
~ e
i kci
in t h i s r e g io n , th e form o f th e Green
f u n c t i o n becomes:
l-c o s2 s,ka -isin 2 s,ka
J
«
<
when |E I
is
E
(83)
>
e
| < 2
y.
Thus, from (28) the d e n s ity o f s ta te s on la y e r
27
Dj l (E) =
l - c o s Z Aka
2" v /
(84)
- f i r * 2
The c o s in e may be made more a g re e a b le in form by n o tin g t h a t
coska = M x 1 + x2 ) = f | ^ - )
(85)
D e fin e th e a 1t h Tchebychev p o lyn o m ia l as
T ^ ( x ) = cos [itcos 1 ( x ) ] .
(86)
T h is d e f i n i t i o n a llo w s th e la y e r d e n s it y o f s t a te s on th e a 1t h la y e r
t o be w r i t t e n
t y E) ' I 1- T 2* < f r ) } D(E)
(87)
.
where D(E) i s given by (8 0 ) .
Both th e b u lk d e n s it y o f s t a te s per l a y e r D and th e l a y e r d e n s it y
Djl f o r X-=I , 2 , 3 and 7 are prese n te d i n f i g u r e 2.
th e l a y e r d e n s it y approaches th e b u lk va lu e
23
.
Note th e way i n which
T h is i s r e m in is c e n t o f
th e way in which th e squares o f th e harmonic o s c i l l a t o r wave f u n c t io n s
approach th e c l a s s i c a l p r o b a b i l i t y d i s t r i b u t i o n as the o s c i l l a t o r
quantum number n goes t o i n f i n i t y .
The p h y s ic s here i s n o t c lo s e ly
r e l a t e d to the o s c i l l a t o r case however.
The nodes on th e l a y e r d e n s it y
o f s t a te s f u n c t i o n on th e V t h la y e r are caused by th e f a c t o r
which appears i n e q u a tio n ( 2 7 ) .
| <x,|v>|2
The e x a c t e i g e n s t a t e which i s r e p r e ­
sented by Iv> i s fo rc e d to have a node on la y e r X-=O, i . e . , <01v> =0.
|v> i s thus a s ta n d in g wave whose wave le n g th depends upon energy.
At
28
t
o m (d
F ig u re 2.
Layer D e n s it ie s o f S ta te s f o r th e One-Band M odel.
A b scissa i s energy in q u a r t e r b a n d -w id th u n i t s and th e area under
each c u rv e i s one.
d e n s it y o f s t a t e s .
The dashed cu rv e shown on l a y e r 7 i s th e b u lk
29
c e r t a i n e n e rg ie s
w ill
|v> w i l l
have a node a t la y e r i .
have a node a t t h a t energy.
Then <a|v> =0 and
In t h i s sense, th e node s t r u c t u r e
i s a s ta n d in g wave e f f e c t .
I t may n o t be p o s s ib le to d e t e c t th e w ig g le s p r e d ic te d by t h i s
model i n th e l a y e r d e n s it y o f s t a te s by p e rfo rm in g an ang le re s o lv e d
p h o toe m ission e x p e rim e n t.
s te p m o d e l ^ ’ ^
The re a s o n in g , which i s based on th e th re e
i s as f o l l o w s .
What one m ig h t e xp e ct t o see coming o u t o f the c r y s t a l a t a given
va lu e o f ky and kz ( th e s u r fa c e k - v e c t o r ) and a given energy E, would
be a sum o v e r la y e r s o f th e i n i t i a l
d e n s it ie s o f s t a te s on each la y e r
w e igh ted by th e p r o b a b i l i t y t h a t an e l e c t r o n on a p a r t i c u l a r l a y e r ,
once e x c it e d by a p h o to n , can escape.
The p h o to c u r r e n t would then be
p r o p o r t io n a l t o
' A (a ,E ) = Z D0( E ) e " “ £
(88)
T h is e x p r e s s io n , which v a s t l y o v e r s i m p l i f i e s th e pho toe m ission process
ig n o re s the j o i n t d e n s it y o f s t a te s and m a t r ix element e f f e c t s as w e ll
as th e s u r fa c e t r a n s m is s io n o f th e e l e c t r o n and o t h e r im p o r t a n t as­
pects o f th e e x c i t a t i o n and d e t e c t io n o f th e e m itte d e l e c t r o n .
Most
o f these r e fin e m e n ts cannot be computed w i t h i n the framework o f a
t i g h t - b i n d i n g model.
The a i n
i n u n i t s o f la y e r s e p a r a tio n .
an e l e c t r o n on l a y e r £.
(88) i s a r e c ip r o c a l s c a t t e r i n g le n g th
Thus e a£ i s th e escape p r o b a b i l i t y f o r
One o f th e m ajor a p p ro x im a tio n s in v o lv e d in
30
(88) i s th e assumption o f in co h e re n c e , t h a t i s the assumption t h a t
th e p r o b a b i l i t i e s r a t h e r than p r o b a b i l i t y a m p litu d e s add.
For t h i s
reason A (a ,E ) m ig h t be c a l l e d an in c o h e r e n t sum.
The sum A (a ,E ) o f (88) may be most e a s i l y e va lu a te d by using the
e x p re s s io n (82) f o r J
, m u l t i p l y i n g by e
end t a k in g th e im a g in a ry p a r t .
A (a ,E )
D(E)
’
I
and summing, then in th e
The r e s u l t i s
1-e a cos2ka________
. l- e 'a
(89)
l - 2 e a cos2ka + e
where
(Ir)=coskaThe sum A (a ,E ) in (89) i s seen to have no zeros between th e
s in g u l a r i t i e s o f D (E ).
The reason f o r t h i s
i s c l o s e l y r e la t e d to the
f o l l o w i n g power s e r ie s a p p ro x im a tio n .
y jp - - I + p + p2 + p 3 + • • •
(90)
The l e f t hand s id e o f (90) has no zeros in th e f i n i t e
plane w h ile th e
l e f t hand s id e , t r u n c a te d t o a p o lyn o m ia l o f degree n , has e x a c t ly n
zeros on th e u n i t c i r c l e which become more and more dense as n i n ­
creases.
Two im p o r t a n t l i m i t s must o b t a in from ( 8 9 ) .
th e escape depth becomes small and so a
I im . A (o ,E ) ~ D (E)e~a .
In th e l i m i t t h a t
=°, A (a ,E ) must vanish as
(91)
31
A ls o , in th e l i m i t o f long escape d e p th , a ^ 0, e “
I , the
in c o h e r e n t sum must d iv e rg e i n p r o p o r t io n t o the escape depth and
approach th e shape o f th e b u lk d e n s it y o f s t a t e s .
n m . A (a ,E ) ~ I
D (E).
(92)
A s y m p to tic p r o p e r t y (91) f o ll o w s from (89) v ia power s e r ie s expansion
in the v a r i a b l e e a , w h il e (92) f o l l o w s from expansion o f (89) in the
v a r i a b l e a.
F ig u re 3 p re se n ts th e in c o h e r e n t sum f o r s e v e ra l d i f f e r ­
e n t va lu e s o f a in th e case o f th e sim p le model problem o f t h i s sec­
tio n .
The a s y m p to tic p r o p e r t ie s
(91) and (92) are e v id e n t as w e ll as
i s th e absence o f th e nodal s t r u c t u r e found in f i g u r e 2.
F ig u r e 3.
In c o h e re n t Sums f o r th e One-Band Model.
H I.
A.
I.
B u lk .
TWO-BAND CASE
S-P Band, Normal E m is s i o n ^
In t h i s s e c t io n a sim p le c u b ic c r y s t a l i s c o n s id e re d in
which t h e r e are on each s i t e both s and p - l i k e atom ic o r b i t a l s .
F irs t,
la y e r o r b i t a l s are formed e x a c t ly as was done i n ( 4 ) .
| a , £ , k y , k z> =
K mne l a ^mky+nkz^ l cx’ Jl’ m’ n>
(93)
where a i s s , p ^ , p ^ o r pz In th e f o l l o w i n g c o m p u ta tio n , th e r e s t r i c t i o n k =0, kz=0 w i l l
made.
be
T h is r e s t r i c t i o n would be o f i n t e r e s t in the i n t e r p r e t a t i o n o f
an ang le r e s o lv e d p h o to e m issio n expe rim e n t where o n ly those e le c tr o n s
are c o l l e c t e d which are e m itte d normal to th e s u r fa c e .
Note t h a t in
t h i s s-p model the normal d i r e c t i o n i s th e x - d i r e c t i on, as i t was f o r
the s im p le one-band model.
T h is normal e m ission case i s o f i n t e r e s t
here f o r a cou p le o f reason s, both o f which in v o lv e th e r o t a t i o n
symmetry o f th e s u r fa c e .
F irs t,
because o f t h i s symmetry, p^ o r b i t a l s on a p a r t i c u l a r l a y ­
e r w i l l mix o n ly w i t h p^ o r b i t a l s on o t h e r l a y e r s ; never w i t h o r b i t a l s
o f o t h e r symmetry.
Both p
L ik e w is e , Pz o r b i t a l s mix o n ly w i t h pz o r b i t a l s .
and s o r b i t a l s
w i t h each o t h e r .
t r a n s fo r m th e same way and so th e y w i l l mix
In o t h e r words:
The group o f the r o t a t i o n symmetry
o f the s u r fa c e and hence o f th e t o t a l
H a m ilto n ia n i s C^v -
p
and s
34
t r a n s fo r m a c c o rd in g t o th e i d e n t i t y r e p r e s e n t a t io n ,
and pz t r a n s ­
form a c c o rd in g t o d i f f e r e n t rows o f a tw o -d im e n sio n a l r e p r e s e n t a t io n
o f C4 v -
Al I m a t r ix elements o f an o p e r a to r between f u n c t io n s b e lo n g ­
in g t o d i f f e r e n t i r r e d u c i b l e r e p r e s e n t a t io n s o r t o d i f f e r e n t rows o f
th e same i r r e d u c i b l e r e p r e s e n t a t io n o f i t s
is h
27
.
symmetry group must van-
Hence, th e H a m ilto n ia n , i n c l u d i n g th e s u r fa c e e f f e c t , does not
mix them.
S e condly, aga in because o f s u r fa c e r o t a t i o n symmetry, th e r e are
w e l l - d e f i n e d s e l e c t i o n r u le s f o r a g iv e n p o l a r i z a t i o n o f th e in c i d e n t
photon.
and f i n a l
The reason i s t h a t th e m a t r ix elements between th e i n i t i a l
s t a t e o f th e e le c tr o n - p h o to n c o u p lin g term o f th e H a m ilto n ia n
reduce to m a t r ix elements o f th e momentum o p e r a to r which i s a v e c t o r
and thus tra n s fo rm s i n a p a r t i c u l a r way under r o t a t i o n .
So by s e l e c t ­
in g a p a r t i c u l a r p o l a r i z a t i o n s t a t e , th e expe rim e nt can s e l e c t i v e l y
probe i n i t i a l
s t a t e s w i t h a p a r t i c u l a r symmetry.
t r u e because, as Hermanson has shown
28
T h is i s e s p e c i a l l y
, th e f i n a l s t a t e must be even.
The o b j e c t o f th e p re s e n t c o m p u ta tio n i s to o b t a in l a y e r d e n s i­
t i e s o f s t a t e s in t h i s sim p le two-band case.
ky and kz which are both z e r o , w i l l
From now on th e in d ic e s
n o t appear in th e n o t a t i o n .
The
c o l l e c t i o n o f p^ s t a te s and t h a t o f pz s t a t e s form se p a ra te problems
which lo o k e x a c t l y l i k e th e one band model o f th e p re v io u s s e c t io n ,
and we c o n s id e r these problems s o lv e d .
35
The s and p
s t a t e s a re mixed by th e b u lk H a m ilto n ia n and the
m a t r ix elements between la y e r o r b i t a l s o f these s t a te s w i l l
be d e fin e d
as f o l l o w s , assuming o n ly n e a re s t n e ig h b o r i n t e r a c t i o n s .
<S ,Si 1 |H<°> I S ,£>
es6£ l £
(94)
+ ps;(^£' £+l + 6£ ' £ - l ^
<s,V
| H(o)
|p,£>
9 ^5£' £+l ™ 6 £ i £ - 1
<P, V | H( ° ) I S, £>
" 9 ^ £ , £+l " <S£ , £ - 1
<P, V |H(o) |p,£>
= V& 'A
) ^ £ 1£+1 + 6 £ ' £ - l ^ '
• The s t a te s used in (94) are r e l a t e d t o those d e fin e d i n
(93) by
I s ,£> = | s , £ , k = 0 , k z=0>,
(95)
|p,j>> = |p x , £ , k y = 0 , k z=0>.
As i n th e one-band case, because o f t r a n s l a t i o n symmetry, both
and G are f u n c t io n s o n ly o f th e l a y e r d i f f e r e n c e and th u s o n ly r e q u ir e
one la y e r s u b s c r i p t .
D e f i ne
~ss
Gt
I
PP
V6T
( 96)
/
The e q u a tio n s o f m otion f o l l o w from (17)
36
(E - H( o ) )G = I
(97)
-
(98)
Thus
+ O E - H j0 l IGz - H ' " '
where
H( 0 ) = n (o )T
-i
n+i
(99)
-
( I means t r a n s p o s e ) ,
Exp ression (98) re p re s e n ts a s e t o f c o u p le d , second o r d e r , l i n e a r
d i f f e r e n c e e q u a tio n s w i t h c o n s ta n t c o e f f i c i e n t s .
They w i l l
be solved
by th e method o f g e n e r a tin g f u n c t i o n s .
D e fin e a 2x2 m a t r ix g e n e r a tin g f u n c t i o n
F(s,E) = Z s \ . ( E )
Using (102) in
( 102)
(98) g ive s
(103)
Thus
37
F " 1 = ( IE - H^0 ^ - s H } ^ - ^ H ^ )
(E-E ) - y f.(s + I )
g(s - 7 )
(104)
- 9( s -
(E-Ep) - yp(s +
j )
An im p o r t a n t i n g r e d i e n t r e q u ir e d f o r th e in v e r s io n o f F
in
o r d e r t o produce F i s th e d e te rm in a n t o f F- 1 .
A (S 5E) = d e t F
■1
(E-Gs)(E-Ep) - Cyg(E-Ep) + Kp(E-Es) ]( s + j )
+ y S1V s + s')
+ 92 ( s - j )
(105)
.
The a n a l y t i c s i n g u l a r i t y s t r u c t u r e s o f F and thus o f G , are d e t e r &
mined by th e zeros o f
a.
t o the b u lk H a m ilto n ia n
29
b u lk band s t a t e s , as w i l l
a
i s th e Fredholm d e te rm in a n t co rre s p o n d in g
; it
c o n ta in s th e d is p e r s io n fo rm u la s f o r the
be shown i n th e f o l l o w i n g c o n s i d e r a t io n .
E x a c tly as i n the one band case, th e zeros o f A r e p r e s e n t s o lu ­
t i o n s o f th e homogeneous e ig e n v a lu e problem (76) f o r th e b u lk
H a m ilto n ia n .
In th e case when E i s in s id e a continuum o f H ^ ,
s is
o f th e form
s = elk a
where k i s a r e a l wave number, as i s X2 i n
(106)
(74).
Using t h i s
in (1 0 5 ),
th e zeros o f A (e l k a ,E) are found t o o c c u r when E takes on one o f two
values
38
E (k ) - jS1Ce5 + Ep + 2 (p s+ y p )c o s k a ]
(107)
± / Ces - E p + 2 ( y s- v ip )c o s k a ]2 + 16g2s i n 2ka.
The n o t a t io n may be s i m p l i f i e d i f energy is measured w i t h r e s p e c t to
E5 and i n u n i t s o f
=s " °
|y |.
Then,
" 'I
Ep -> E
Up
(108)
y
The reason f o r choo sing y s t o be - I
r a t h e r than I i s t h a t t h i s causes
the s-band t o bend up r a t h e r than down.
The e x p l i c i t r e l a t i o n between
th e new symbols and th e o ld which i s im p lie d by (108) i s
E =
V
£s
(109)
P = Up /
|p s l-
The energy param eter i s now
E-e„
(n o )
The e x p re s s io n (107) f o r th e band e n e rg ie s has become i n these
parameters
w+ (k ) = h e + ( y - l ) c o s k a ± jSy / [ E + 2 ( u + l ) c o s k a ] 2 + 16g2s i n 2ka.
The band s t r u c t u r e f o r t h i s two band model i n the d i r e c t i o n
( Ill)
39
p e r p e n d ic u la r t o th e (100) s u r f a c e , as p r e s c r ib e d by e q u a tio n (1 1 1 ),
i s presen ted i n f i g u r e 4.
y = .2 , e = l , g = . l .
The ch o ice o f y , e and g in f i g u r e 4 is
For o t h e r choices o f these param eters, th e bands may
lo o k even q u a l i t a t i v e l y d i f f e r e n t .
The e x is te n c e o f a gap and hence
o f a gap s u r fa c e s t a t e depends upon th e r e l a t i v e values o f y, e and
g
18 23 30
’
’
.
The values o f energy f o r which th e slo pe o f th e w versus
ka curve i s zero r e p r e s e n t van Hove s i n g u l a r i t i e s
o f s t a te s f u n c t i o n .
i n th e b u lk d e n s it y
As we have seen i n th e one b,and case, th e van Hove
s i n g u l a r i t i e s r e p r e s e n t branch p o in t s in th e d is p e r s io n fo rm u la s and
are th e p o in t s which d e l i m i t th e c o n tin u e o f the spectrum o f th e b u lk
H a m ilto n ia n o p e r a to r .
The form o f th e d e te rm in a n t A i s now
A(s,co)
= S - 1J
s
p(s,co),
(112)
where p i s a monic p o lyn o m ia l
p ( s ) = S4 - Qs3 + (R+2)s2 - Qs + I
(113)
w i t h Q(lu) = l r i k t l
g2-u
(114)
and R(w) =
(115)
g -y
g -y
T h is polyn o m ia l has r e c ip r o c a l symmetry.
then I / S g i s a ls o a r o o t .
p(s)
=
( s 2 + as
+
Thus, p
I ) ( s2 +
Bs +
I f Sq i s a r o o t o f p,
may be fa c to r e d
I )
(116)
40
F ig u re 4.
Bands f o r th e s-p Model.
K is wave number in u n i t s o f -
.
41
M u l t i p l i c a t i o n shows t h a t
a + g = -Q
ag + 2 = R + 2
(117)
Thus a and g are th e zeros o f
y 2 + Qy + R
(118)
In o t h e r words,
a
- -%Q + Jg/ D
3 = -%Q - Jg/ D,
(1 1 9 )
where D i s th e d i s c r i m i n a n t
D = Q2 - 4R
(1 2 0 )
In o r d e r t o f o r e s t a l l th e e ve n tu a l co m p u ta tio n a l c a t a s t r o p h y ,
beyond which th e a l g e b r a i c c o m p le x ity o f th e s o l u t i o n may obscure the
p h y s ic a l meaning, i t
i s im p o r ta n t a t t h i s p o i n t to c a ta lo g u e a number
o f r e l a t i o n s among th e r o o t s o f th e reduced, q u a d r a tic e q u a tio n o f
018) .
a + B = Q
cxB = R
(T2l)
a —B —/ D =
JQ2-4R
A ls o , th e re i s a re d u c e d , homogeneous d i f f e r e n c e e q u a tio n s a t i s f i e d
by powers o f a and B-
a
n _
n n -i
- - Qa
D n- 2
- Ra
0 22)
42
Bn = - QBn" 1 -RBn" 1
N ext, th e s i n g u l a r i t i e s o f D must be examined.
tio n
(120), i t
From the d e f i n i ­
fo llo w s th a t
D U ) = (ii+ lV
(9 % )^
[cj2 - 2am + b ] ,
(123)
where
( u + l) - 2q2
(124)
(y + 1 ) 2 - 4g2
and
b _ E2 + 16q2 (q 2- u )
(125)
(y+1)2 - 4g2
The r o o ts i n m o f th e p olynom ia l D are th e p o s i t i o n s o f th e van
Hove s i n g u l a r i t i e s which d e l i m i t th e s-p h y b r i d i z a t i o n gap which i s
m a n ife s t in th e e x p re s s io n f o r th e bands (1 1 1 ) , and i s demonstrated
i n f i g u r e 4.
The e x is te n c e o f such a gap depends upon the r e a l i t y in
CD o f th e r o o t s o f D.
Thus we examine th e p o in t s where th e n a tu re o f
these r o o t s changes.
The d i s c r i m i n a n t o f th e r o o ts o f D (a d i s c r i m i ­
nant o f a d i s c r i m i n a n t ) i s
d = a2 - b
(126)
We c o n s id e r zeros o f
[ ( y + 1 ) 2 - 4g2 ] ( a 2- b ) = 0
-
(127)
under th e s u p p o s it io n t h a t th e f i r s t f a c t o r i s not i d e n t i c a l l y zero .
43
[(y + T )
- A g ] ( a 2-b ) - 4g2{ I 6g
+ [ e 2 - 4 ( y + l ) 2 - 1 6 p ]g 2
+ [ 4 y ( y + 1 ) 2 - e2y ] } .
(128)
Thus th e zeros o f D, which depend upon e x p re s s io n (128) are given by
= [ ( yt 1 ) Jl..2g2l L ±
( ig g )
( y + 1 ) = - 4g 2
The f u n c t i o n p in (129) i s g ive n as
p ( 9 2 ,E , y ) = 16g2
+ [ e 2 - 4 (y + 1 ) 2 - 16 ] g 2 + [ 4 y ( y + 1 ) 2 - e2y ] .
(130)
The c e n t e r o f th e h y b r i d i z a t i o n gap is
gap c e n te r = I ( y +T) ~ 2g2] e ^
(131)
[ ( y + 1 ) 2 - 4g2]
and th e
w id th
gap w id th = 4 g / p ( g 2 , e , y )
(132)
( y + 1 ) 2 - 4g2
I t i s reason able to assume t h a t 4g2 < ( y + 1 ) 2 .
Under t h i s assump-
t i o n , th e complex square r o o t o f D a cc o rd in g to the branch c o n ve n tio n
X y + l )'
- 4g2
2
/
CO-CO +
/
OJ-Ol
(133)
g - y
N ext, th e zeros o f p d e fin e d in (113) must be computed using
s2 + as + T = O
s2 + gs + I = 0,
(134)
44
or
s
= - %x ± %v'a2-4
(135)
S3jlf = - Jgg ±
Zg2-4
The f o l l o w i n g p r o p e r t ie s among th e r o o t s (135) are o f c o n s id e r a b le im­
p o rta n c e :
S1 + S2 = a , S3 + S4 = e
(136)
Si S2 = I ,
(137)
S3S4 = I
S1 - S2 = Z i 2-4
S3 - S4 = / g 2-4
(138)
Now th e branch p o i n t s i n g u l a r i t i e s o f th e r o o t e x p re s s io n must be
e x p lo re d and must u l t i m a t e l y r e l a t e t o van Hove s i n g u l a r i t i e s
in the
spectrum o f th e b u lk H a m ilto n ia n H ^ .
S1 and S2 show branch p o in t s when a = ± 2.
e i t h e r w = 2, o r u = e-2y, w h il e
w = e+2y.
a
=
-Z
a = +2 im p lie s t h a t
c o r r e la t e s w i t h w = -2 and
These are th e band s i n g u l a r i t i e s a t ka = ± n and a t ka = 0
r e s p e c t iv e ly . The
e x is te n c e and p o s i t i o n in w o f these s i n g u l a r i t i e s
are n o t a f f e c t e d by th e h y b r i d i z a t i o n s t r e n g t h param eter g, and are th e
o u t e r l i m i t s o f th e s and p c o n tin u a in th e case g = 0 (see f i g u r e 4 ) .
S3 and s 4 have s i m i l a r branch p o in t s a t g = ± 2.
These p o in t s are
th e same as those f o r which a = ± 2.
In o r d e r t o understand th e paths o f th e r o o ts
S1,
s2 ,
S3
and
S4
as
th e y move around th e Argand diagram o f f i g u r e 5 as a f u n c t i o n o f w, i t
S PLANE
F ig u re 5.
Root T r a j e c t o r i e s o f a Model Two-Band H a m ilto n ia n .
46
i s c o n v e n ie n t t o begin w i t h w >> 2 and assume 0 < g < p < e < 2 .
3 is
a ll
n e g a tiv e and a i s p o s i t i v e .
Then
Thus a t t h i s ene rg y, th e r o o ts are
r e a l and
s2
<
-I
< S1
< 0 < s4 < I < s 3. '
(139)
As M becomes le s s than 2 ( w it h a s m a ll , p o s i t i v e im a g in a ry p a r t ) , then
r o o t s S1 and S2 le a ve th e r e a l a x is and approach the u n i t c i r c l e .
remains in s id e and S2 o u t s id e .
a t i o n in terms o f f i g u r e 4.
S1
T h is m otion o f S1 and S2 has an e x p la n ­
For energy j u s t le s s than 2 in f i g u r e 4,
t h e r e i s a p r o p a g a tin g Bloch s t a t e o f e s s e n t i a l l y s - l i k e c h a r a c t e r .
The i n t e r l a y e r phase o f t h i s s t a t e i s S2 , and th e v a lu e o f ka c o r r e s ­
ponds to th e p o l a r ang le o f S2 on th e Argand p l o t .
S1 and S2 would be
th e Bloch s o l u t i o n s f o r th e s - l i k e p a r t o f th e H a m ilto n ia n
l i m i t t h a t g -> 0.
As w i s lowered u n t i l w
in the
< w < e+2y, th e poles S3
and s4 approach th e u n i t c i r c l e and r e p r e s e n t p - l i k e Bloch s t a t e s .
The n e x t f e a t u r e s o f th e r o o t t r a j e c t o r i e s t o be co n s id e re d are
th e k in k s .
The loops and bumps which appear in f i g u r e 5 have a ve ry
in te r e s tin g s ig n ific a n c e .
For w values co rre s p o n d in g t o th e gap (see
f i g u r e 4) th e r e are no p r o p a g a tin g Bloch s t a t e s .
The a n a l y t i c a l I y
c o n tin u e d Bloch s t a te s are damped, and thus in t h i s re g io n S1 and s4
s h r i n k , i n m agnitude, away from th e u n i t c i r c l e .
As w goes below w+ ,
S1 and s 2 l e a v e t h e u n i t c i r c l e a t a v a l u e o f p o l a r a n g l e c o r r e s p o n d ­
i n g i n t h e band d i a g r a m t o t h e p o s i t i o n a l o n g ka o f t h e gap c e n t e r .
S1 and S2 e x e c u t e l o o p s i n t h e a t t e n u a t i o n r e g i o n and when w r e a c h e s
47
w_, th e
r o o ts aga in r e t u r n to th e u n i t c i r c l e a t e x a c t ly th e same
v a lu e o f ka, thus c l o s in g th e lo o p s .
Regarded as a f u n c t i o n o f Ra,
th e n , f i g u r e 4 may be co n s id e re d as a s u p e r p o s it io n o f two monotonic
b u t d is c o n tin u o u s f u n c t i o n s .
T h e re fo re th e r o o ts
S1
and s4 , which are
r o o ts o f th e c h a r a c t e r i s t i c p o lyn o m ia l o f H^0 ) , can be c l a s s i f i e d
a c c o rd in g to atom ic typ e even when g / 0.
S1
i s an s - l i k e c o n t r i b u t i o n
t o G , w h il e S4 i s p - l i k e .
A t th e e n e rg ie s where th e k in k s appear in the r o o t l o c i , the con­
t in u e d Bloch f u n c t io n s a r e e x p o n e n t ia lI y damped and w i l l
p r o v id e a
b a s is s e t w i t h which lo c a l i z e d s u r fa c e s t a te s may be expanded.
F i n a l l y , as w goes below e - 2 y ,
then S1 and s
S3
and
S4
leave th e u n i t c i r c l e and
le ave a t w = - 2 .
In o r d e r t o make th e r o o t s co n tin u o u s f u n c t io n s o f w, th e f o l l o w ­
in g branch c o n v e n tio n has been used:
v/a ^-4
=
Va2-2
/a + 2 ,
(140)
/ B 2-4 - / - 3 - 2 / - 3 + 2 ,
to g e th e r w ith
(135) f o r /~D.
Thus th e r o o ts are co n tin u o u s f u n c t io n s
o f c o n tin u o u s f u n c t io n s o f w when w i s r e a l except f o r a s m a ll, p o s i­
t i v e im a g in a ry p a r t .
R e tu rn in g now t o th e problem o f expanding the g e n e r a tin g f u n c t io n
in o r d e r t o o b t a in th e Green f u n c t i o n s , we f i n d t h a t
48
A..(s,w)
Fi j
i and j
Jl
(S,w)
(141)
A(s,a ))
being th e s t a t e s s o r p and A
( s ,<d )
r e p r e s e n t in g a c o f a c t o r .
Using (112)
S 2 A ^ ( S 5w)
Fi j ’ (s,a))
(142)
( g 2- y ) p ( s 5w)
Thus Fi j
i s a r a t i o n a l f u n c t i o n o f s w i t h th e p r o p e r t y t h a t Fi j
zero a t zero and i n f i n i t y .
A ls o , because o f the small p o s i t i v e im a g i­
n ary p a r t o f cu, th e poles o f Fi j
F1 j (s,u,)
I
g2-^
are always s e p a ra te d , whence
% S k A jjf= K '")
k=l
goes t o
(143)
P 1( S k ) ( S - S k )
The v a r io u s p a r ts o f (143) are q u i t e s im p ly computed, u sin g the
t o o l s t h a t have been noted in ( 1 2 1 ) , ( 1 3 6 ) ,
(137) and (1 3 8 ).
From
(1 0 4 ) , th e c o f a c t o r s are
A55 = (co-e) - (s + s)
(144)
Asp = ~ g (s - ~ )
(145)
Aps = 9 (s - I )
(146)
App = w + (S + I ) .
(147)
The p r e p a r a t io n o f G^s (cu) f o r example, i s as f o l l o w s .
49
Si Ass(Sl5 Uj) = (E-e)Si - y ( S i + l ) s 1 = (E - e + Va ) s \
Similar results follow in the case of s2 , S35 and S4 .
(148)
The derivative
is computed by factoring p ( s ) .
P( s ) = (S-S 1 )(S-S 2 )(S -S 3 )(S-S4 )
(149)
P1(S1) = (S 1-S 2 ) ( S 1- S 3 ) ( S 1-S4) + 0.
(150)
Using properties of the roots,
P1(S1) =
but
( s ^ + Bsi + I ) ,
(151)
s2 = - QS1 - I , so
P1(S1) =
(B-Ct)S1.
(152)
Fi na lly from (121), (a-B) = D.
P1(S1) = - Vr D v4 2-4 S1.
(153)
Results s im il a r to (153) also hold f o r the other roots.
Applying the above results to (143) yields
I
O2 -M
I
E-e+yct
S2
/D
/ x 2- 4
S-S2
s -s .
"3
s-s,
E - e+ p B
/ g 2-4
_
sI
S-S1
(154)
The c h o ic e o f annulus f o r th e expansions o f th e terms o f (154) depends
upon th e
a s y m p to tic form o f
by an argument s i m i l a r t o t h a t which
50
follows (70).
(S-S1) 1 =
( 1 / s ) (I + Si Zs +
S1Zs2 +
( s -s 2 ) 1 =
-S 1(I + S1S + S1
s2 + . . .
(S-S3) " 1 =
-S4 O + S4S + S4
S2 + . . .
(S-S4) " 1 =
( 1 / s ) (I + S4Zs +
S4Z s 2 + . . .
...
.(155)
Thus G|s is given by
( a)-e+ya)
G! S = ------i — T=
(y-g ) / D
I 5-I + (oi-e+yg)
(156)
/*2 -4
Sl
/ b 2-4
f o ll o w s from ( 1 56) thro ugh th e replacem ents
GPP = ____ I____
A
(to+a)
Za2-4
( y - g 2 ) yfl
|&|
1£ I + ( cq+ b ) c N
54
e
= 0, y
I .
057)
/ B 2-4
A s i m i l a r com putation produces
and
pSD
_ g s ig n ( £ )
f
O I
G«
" T T V w S
Ls i
U|
(158)
'
s i
GPs =' - GsP
&
&
]
■
(159)
The b u lk d e n s i t i e s o f s t a te s which f o l l o w from (156) and (157) are
p resen ted i n f i g u r e 6.
th e
The t o t a l d e n s it y o f s ta te s on a b u lk la y e r is
sum o f two c o n t r i b u t i o n s .
D(w) = D5 (oi) + Dp (Cj)
(160)
F ig u r e 6.
s -p M odel.
T o ta l B u lk D e n s ity o f S ta te s per Layer f o r the
The energy a x is i s in q u a r t e r s-band u n i t s .
area under th e c u rve i s two.
The
52
I t i s a c h a r a c t e r i s t i c o f th e r e s o lv e n t method t h a t the d e n s i t i e s o f
s t a te s occu r n a t u r a l l y in component fo rm .
The d e n s it ie s Ds and Dp o f
f i g u r e 6 are computed from
D1(O)) = - I
Im G11(O))
IT
(161)
O
where i i s e i t h e r s o r p.
2.
S u rfa c e .
Ji i
The s u r fa c e problem may now be solved u s in g (2 6 ).
Ji
( 162)
3Jl=O
k=s,p
There i s a p o s s i b i l i t y t h a t t h i s f u n c t i o n may have a p o le when the
d e te rm in a n t o f
has a z e ro .
From (1 5 6 -1 5 9 ), th e form o f G^ i s :
I
0
(163)
( y - g 2) /D
VO
The f u n c t io n s u and v are
(co -E + u a )
/a ^ -4
v =
w+a
_
( q) - s + u B )
(164)
/(32-4
+ _.w+B
/ a 2-4
/ b 2-4
Zeros o f u and v d e f in e th e zeros o f th e d e te rm in a n t and hence d e t e r ­
mine th e poles o f
.
The zeros o f u o c c u r a t th e p o in t s o) = ±2, th e
s-band edges, and a t w =
which i s th e p o i n t a t which th e sepa ra te
s and p-bands would have crossed i f
th e m ix in g param eter g were s e t to
53
z e ro .
See (1 1 1 ).
On th e o t h e r hand, th e zeros o f v are a t w = e±2y,
th e p-band edges, and aga in a t th e c r o s s in g p o i n t to = -jjfy.
seen from (162) t h a t th e zeros a t th e band edges w i l l
because these w i l l
th e r e w i l l
I t may be
n o t cause poles
be c a n c e lle d by o t h e r zeros in th e n u m e ra to rs .
But
always e x i s t an i s o l a t e d s u r fa c e s t a t e in such a gap e xa ct
I y a t the energy where the bands would have crossed i f
th e r e were no
s -p m i x i n g ^ 9^O.
F ig u re 7 i l l u s t r a t e s
th e d e n s i t i e s o f s t a te s curves on la y e r s
near th e s u r fa c e f o r th e s-p model i n normal p h o to e m issio n .
Note­
w o rth y f e a t u r e s i n f i g u r e 7 in c lu d e th e s t a t e in th e s -p h y b r i d i z a t i o n
gap, th e appearance o f standing-w ave nodal s t r u c t u r e s s i m i l a r t o t h a t
observed in th e one-band model
la ritie s
23
, and th e s o f t e n in g o f th e b u lk s in g u ­
on th e la y e r s near th e s u r fa c e ^ .
The s i n g u l a r i t i e s i n the
b u lk d e n s i t i e s are o f th e r e c ip r o c a l square r o o t t y p e , w h il e those
near th e s u r fa c e are s o fte n e d t o square r o o t s h o u ld e rs .
These f e a ­
tu r e s are r a t h e r t y p i c a l o f th e kin d s o f e f f e c t s produced i n t i g h t b in d in g models w i t h more r e a l i s t i c H a m i lt o n i a n s ''
B.
I.
B u lk .
.
Simple B o n d -O rb ita l Model
A number o f s i m p l i f i c a t i o n s have been proposed f o r the
tre a tm e n t o f sem icon ductor s u r fa c e s t a t e s which would a l lo w a reason­
a b ly r e a l i s t i c d e s c r i p t i o n o f th e H a m ilto n ia n and y e t keep th e compu­
t a t i o n as s im p le as p o s s ib le .
H a r r is o n ^
has proposed a v e ry s im p le model H a m ilto n ia n which
54
F ig u re 7.
T o ta l Layer D e n s it ie s o f S ta te s f o r th e s-p
Model on Layers I ,
u n its .
2 and 3.
Energy i s in q u a r t e r s-band
55
reproduces many o f th e b u lk p r o p e r t ie s o f a la r g e c la s s o f semiconduc­
to rs .
R e c e n tly , se v e ra l a u th o rs 32-34 Iiave a p p lie d H a r r is o n 's bond-
o r b i t a l model t o s u r fa c e s t a t e co m p u ta tio n .
a ls o .
T h is w i l l be done here
In th e p re s e n t s e c t i o n , a b o n d - o r b it a l w i l l
be developed as a
s i m p l i f i c a t i o n t o th e two-band model o f th e p re v io u s d is c u s s io n .
The
b r i e f t re a tm e n t below i s o f f e r e d as ah i n t r o d u c t i o n t o th e more e la b ­
o r a t e b o n d - o r b it a l model o f th e n e x t s e c t io n .
F ig u re 8 i l l u s t r a t e s v a r io u s steps i n th e c a l c u l a t i o n o f chemical
bonding f o r a sim p le d ia to m ic m o le c u le .
In f i g u r e 8a shows s and p
s t a te s on two i s o l a t e d atoms.
|p l> be s and p s t a te s cen­
te r e d on s i t e I and l e t
Let |s l> ,
|s2> and |p2> be th e same s t a t e s ce n te re d on
s i t e 2.
As th e two atoms are b ro u g h t t o g e t h e r , the i n t e r a c t i o n between
them i s re p re s e n te d i n terms o f H a m ilto n ia n m a t r ix elements between
th e o r b i t a l s .
< s l [ H | s l > = <s2|H|s2> = Eg
(165)
< p l I H | p l > = <p2|H |p2> = Ep
< s l I HI s2> = < s 2 |H |s l> = y s
< p l I H I p2> = <p2I H I p i > = y p
< s l|H |p 2 > = < p 2 |H |s l> = g
< s 2 |H |p l> = < p l|H |s 2 > = -g
Assume a l l
o t h e r m a t r ix elements are z e ro .
Ig n o re e l e c t r o n - e l e c t r o n
r e p u l s io n and o t h e r e f f e c t s and a ls o assume t h a t th e o v e r la p between
cn
Ch
F ig u r e 8.
Form ation o f a Chemical Bond.
57
o r b i t a l s v a n ish e s.
F ig u re 8b re p re s e n ts a h y b r id b a s is s e t formed from l i n e a r combi­
n a tio n s o f th e s and p o r b i t a l s on th e two atoms.
I h^+ h > = Jg( | s l> + | p l > )
Ih^ ^1> = % (I s I > -
( 166)
|p l> )
| h ( + ) 2> = % (I s2> + I p2>)
Ih^ ^2> = %(|s2> -
|p2>)
.
As th e atoms come t o g e t h e r , as i l l u s t r a t e d
in 8c, th e h y b r id o r b i t a l s
i n t e r a c t s t r o n g l y and th e c o rre s p o n d in g energy
l e v e ls s p l i t as shown in f i g u r e 8d t o form a bonding and an a n tib o n d ­
in g l e v e l .
These energy l e v e l s belong to a b o n d - o r b it a l which i s the
even co m b in a tio n o f th e two h y b r id s , and an a n t i b o n d - o r b i t a l , which is
th e odd c o m b in a tio n .
I f th e o r i g i n a l o r b i t a l s on th e two s i t e s were o n ly h a l f f i l l e d
w i t h e l e c t r o n s , then those e le c t r o n s would now occupy th e b o n d - o r b it a l
and p a r t o f the o t h e r h y b r id o r b i t a l s p o i n t i n g the o t h e r way (non­
bonding) , b u t th e a n t i b o n d - o r b i t a l would be empty.
to ta l
In t h i s way, the
energy i s lowered and a s t a b le bond i s fo rm e d .
Suppose a c r y s t a l were formed from an ensemble o f m olecules o f
t h i s sim p le ty p e .
Al I o f th e h y b r id o r b i t a l s would p a r t i c i p a t e in
bonding and so th e r e would be no non-bonding l e v e l s .
In the case
58
t h a t th e
broadening o f th e f i l l e d
b o n d - o r b it a ls which now form the
valence band does n o t mix th e bonding and a n tib o n d in g s t a te s ve ry
s t r o n g l y compared t o th e m ix in g o f th e b o n d - o r b it a ls among them selves,
then one m ig h t hope to r e p r e s e n t th e valence band as some l i n e a r com­
b i n a t i o n o f b o n d - o r b it a ls o n ly and n e g le c t the a n t i b o n d - o r b i t a l s a l t o ­
g e th e r .
T h is i s th e p h ilo s o p h y adopted when a b o n d - o r b it a l H a m ilto n ia n
i s used t o r e p r e s e n t th e valence bands o f se m i-c o n d u c to rs .
The s-p H a m ilto n ia n used in th e two-band c a l c u l a t i o n may be t r a n s ­
formed i n t o bond and a n t i b o n d - o r b i t a l s as f o l l o w s .
|b,&> = h { \ ^ + \ i >
+ |h /~ \ji+ l> )
= H.{ I S , £> + I p , £> + I S , 5.+1 > = bond o r b i t a l
|a,£> =
(166a)
| p , £,+!>)
near la y e r £.
| h ^ ,£> -
| h ( ~ ) , £ + l> )
= % (|s ,£ > + I P,£> -
I S,£+1> + I p ,£+1>)
= a n t i b o n d - o r b i t a l near £.
Using the d e f i n i t i o n s
(94) th e m a t r ix elements become
< b £ '|H |b £ > =
p ftu S V
+
(167)
- 1S
[tiv s + V p )
V
11+2
+ % ( E 5 - E p ) ] 6 Jl, !l4. 1
+ [>s(£s+ep ) + lI( V s -Up) +
+ M v /V p )
+ %(vs -Ep)]S%,%_,
59
+ [%(y +y ) 5
P
I ' 1-2
< b V |H|a&> =
ft(V
Ps ) +
^
+ f t t s P -6Sn 6 V 1H-I
+ t!s(Es - Ep ) ] 5t ' t - i
+ f t f t S-Vp) - ^
4V
j -Z
<a£‘ |H|a£> =
[>s(up-Ps ) + jSg]^, t + 2
+
f t f t s+ Vp)
+ ^ V p -s ’^ v v i
+
P s U s^ p ) + « p p- p s )
-
g ]\,,
+ f t f t St V p ) + y 6 p - 6 s ) ] 6 t ' t - i
+ W mp-Ps ) + > a ] \ , t . 2
Two t h in g s should be noted about th e H a m ilto n ia n d e fin e d in
(1 6 7 ).
F irs t,
it
r e p r e s e n ts th e same p h y s ic a l s i t u a t i o n as does (94)
and so both th e band s t r u c t u r e and th e t o t a l
t i e s o f s t a t e s would be th e same.
b u lk and s u r fa c e d e n s i­
Second, th e range o f th e i n t e r ­
a c t io n has been extended.
A H a m ilto n ia n which i s n e a re s t- n e ig h b o r in
atom ic l a y e r o r b i t a l s w i l l
be se co n d-neig hbor in b o n d - o r b it a ls .
It
i s e q u a ll y t r u e , o f co u rs e , t h a t a H a m ilto n ia n which i s
o n ly
60
n e a r e s t- n e ig h b o r in th e b o n d - o r b it a l r e p r e s e n t a t io n may c o n ta in second
n e ig h b o r atom ic p a r t s .
The a p p ro x im a tio n comes about when a l l m a t r ix elements are i g ­
nored e xcep t th e < bV |H|b£>.
In o r d e r to examine th e f e a s i b i l i t y o f
doing t h i s , c o n s id e r th e sums
E .( k a ) = I
z <b£, |H|b£> el k a ^ ' £ ^
b
N £ '£
11
= %(2g + e - I - u) +
h(2u
- 2 -e)co ska
- h ( v + I + 2g)cos2ka.
(168)
E ( ka) = I z
< a £ 1 1H I a£> e 1 k a ^ ' £ ^
a
N £15,
= Jg(e + I + y - 2g) + %(2y - 2 +e)coska
+ H W + I + 2g)cos2ka.
(169)
V(ka) = I
Z
< b £ '|H |a £ > e“ i l <a ( £ , _ £ )
N £ 1£
= ^ e i s in k a +
+ y + 2 g ) is i n 2 k a .
(170)
The b u lk band s t r u c t u r e would f o l l o w from th e d i a g o n a l i z a t i o n o f
Eb (ka) V(ka) '
H(ka)
(171)
*
W (ka) Ea ( k a ) /
.
The b o n d in g - a n ti bonding m ix in g i s p ro v id e d in (171) by V as d e fin e d in
( 1 7 0 ).
T h is w i l l
be e x a c t ly zero i f
e x i s t s an e x a c t b o n d - o r b it a l l i m i t .
e = 0, g = - i y l .
In t h i s l i m i t ,
Thus, th e re
th e eig e n va lu e s o f
61
H(ka) become:
a)+ (ka) = Eb = - (1+u) + ( p - l) c o s k a
w (k a ) = E
(172)
= + (1+u) + ( y - l ) c o s k a ,
a
which are e x a c t ly th e e xp re s s io n s o b ta in e d from (111) f o r th e valence
and c o n d u c tio n band e n e rg ie s in th e same l i m i t .
th e
The e x a c t v a l i d i t y o f
b o n d - o r b it a l model f o r th e valence bands in t h i s l i m i t w i l l be
im p o r ta n t when assessing th e f u r t h e r a p p l i c a t i o n o f t h i s H a m ilto n ia n
t o th e com putation o f s u r fa c e s t a t e s .
Computation o f th e b u lk Green f u n c t i o n f o r th e b o n d - o r b it a l
la y e r s i s e x a c t ly as in th e one-band case above.
x m
21 1
(173)
( , - i) /( S iiiiii) ^ T T '
V
U-I
= /lo+l+M _ /w± l + U ) 2 _ I
2
' p-1
'
/
.
(174)
P -I
T h is Green f u n c t i o n may be c a l l e d th e bond p ro p a g a to r.
(175)
e = - ( I+ p ) , m = ^ ( p - 1 )
Then
(176)
z
-
1
Exp ression (176) now lo o k s l i k e
(78).
62
2.
S u rfa ce .
In f a c t ,
(176) lo o ks so much l i k e
(78) t h a t i t m ig h t be
expected t h a t th e s o l u t i o n o f th e s u r fa c e problem c o rre s p o n d in g to
(176) would lead t o e x a c t ly th e l a y e r d e n s i t i e s o f s t a t e s expressed by
( 8 7 ) , w i t h th e replacem ents e -> e and u
th is
m.
I t must be s tre s s e d t h a t
i s d e f i n i t e l y n o t th e case.
When a c r y s t a l
i s c le a v e d , bonds are broken.
T h is r e s u l t s in a
la y e r o f d a n g lin g h y b r id o r b i t a l s which do n o t a r is e n a t u r a l l y in a
b o n d - o r b it a l model when th e c r y s t a l
o r b ita l. la y e r.
i s te rm in a te d a f t e r th e f i r s t bond-
A c t u a l l y , t h e r e would be a h y b r id o r b i t a l p o i n t in g o u t
i n t o space beyond l a y e r one.
Bond o r b i t a l
| h ( + ) , l > and |h ( ~ ) , 2 > b u t none o f | h ( ~ ) , l >
| b , l > c o n ta in s p a r t s o f
, and so i t
i s t h i s dang­
l i n g h y b r id o r b i t a l which must be appended t o th e s u r fa c e which a llo w s
th e s u r fa c e s t a t e s t o be computed w i t h i n th e o th e rw is e b o n d - o r b it a l
m odel.
Imagine a b a s is s e t c o n s i s t i n g o f bond o r b i t a l s 1 , 2 , 3 , • • • t o ­
g e th e r w i t h th e d a n g lin g h y b r id o r b i t a l .
The b u lk m a t r ix elements o f
th e H a m ilto n ia n become
<ba' |H|b£> = e5V j l + m(6V | ,+] + Sj l l t J ,
(177)
where e and m are r e l a t e d t o th e u n d e r ly in g two-band H a m ilto n ia n in
th e
b o n d - o r b it a l l i m i t by (1 7 5 ).
However, in p r a c t i c e , th e parameters
o f th e b o n d - o r b it a l H a m ilto n ia n such as e and m in th e s im p le example
under c o n s id e r a t io n are determ ined by b u lk p r o p e r t i e s , and th e param­
e t e r s o f th e unapproxim ated H a m ilto n ia n remain unknown and are o n ly
63
i m p l i c i t l y assumed t o e x i s t .
The d iagon al m a t r ix element which i s th e h y b r id energy
T = < h * " \ l | H |h ( " \ l >
may be
(178)
known from atom ic measurements.
O ther m a t r ix elements from
b o n d - o r b i t a l s o f th e b u lk t o the s u r fa c e h y b r id must be e v a lu a te d using
th e deco m p o sitio n o f h ^ , l
in terms o f bonding and a n t i bonding p a r t s .
< b ,£ |H |h ^ ,0 > =
I
= < b ,& |H |b ,0 > -
/2
i
= < b, £ I HI a , 0 > .
(179)
/2
However, th e
b o n d in g - a n ti bonding p a r t s o f these m a t r ix elements w i l l
be unknown in a b o n d - o r b it a l model.
I t i s c o n s i s t e n t t o suppose t h a t
th e y a re s m a ll, s in c e t h i s would have been th e assumption in th e b u lk
c a lc u la tio n .
In f a c t , t h e r e i s no o t h e r c h o ic e .
I f th e y are n o t s e t
t o z e r o , more parameters are in tr o d u c e d a t t h i s l a t e stage o f the c a l ­
c u la tio n .
may have
T h is s e r i o u s l y erodes any l e g it im a c y which t h i s approach
had up u n t i l
th is p o in t.
In th e p re s e n t example c a l c u l a t i o n ,
t h e r e i s no d i f f i c u l t y w i t h t h i s assumption s in c e th e b o n d - o r b it a l
l i m i t which has been taken fo r c e s a l l b o n d in g -a n tib o n d in g m a t r ix e l e ­
ments t o be e x a c t l y z e ro .
< b ,l|H |h (- ),l>
Thus
= I m ,
(180)
✓2
w h il e a l l
o th e r s are z e ro .
The com putation now proceeds as f o l l o w s .
I f th e h y b r id o r b i t a l
' 64
l a y e r were n o t connected t o the b u lk by th e m a tr ix element o f (1 8 0 ),
then th e p ro p a g a to r f o r th e stage on th e d isconne cted h y b r id would be
s im p ly
J qq - (w -x) ^
(181)
w h ile th e Green f u n c t i o n on th e s u r fa c e o f th e b o n d - o r b it a l s u b s tr a te
would be g ive n from th e r e s u l t s o f th e one-band problem by
(S f) -
J
w
l ) 2
'
1
I-
(182)
Using th e e x p re s s io n (51) which r e l a t e s th e f i n a l p ro p a g a to r to these
uncoupled Green f u n c t i o n s , th e r e s o lv e n t on th e d a n g lin g h y b r id becomes
R00 = U
- T - J p 2J i i ) " 1
(183)
The poles in w o f t h i s f u n c t i o n s ig n a ls th e o ccurrence o f a bound
s u r fa c e s t a t e .
co = J5( U
m)
The approxim ate gap s t a t e energy thus becomes
- V
(U
m) 2
- Js( U y ) 2
.
' (184)
The t r a j e c t o r y o f t h i s s t a t e i s p l o t t e d in f i g u r e 9 as a f u n c t io n
o f Mj t o g e t h e r w i t h th e p o s i t i o n s o f th e band edges.
s t a t e energy in th e two-band model was
o r b ita l
lim it,
In f i g u r e
e
/(1 + m )
The a c tu a l gap
, b u t i n t h i s bond-
£ = 0, so th e gap s t a t e should appear a t zero energy.
9 , th e a c tu a l gap s t a t e i s la b e le d A and th e b o n d - o r b it a l
a p p ro x im a tio n t o i t
i s la b e le d B.
The co n d u c tio n band minimum i s l i n ­
ear as a f u n c t i o n o f u and i s re p re s e n te d by a dashed l i n e .
The
65
F ig u re 9.
B o n d -O rb ita l A p p ro xim a tio n t o th e Gap S ta te Energy in the
Two Band Model.
The param eter determ ines th e band gap.
The valence
band extends between 2 and th e s la n t e d , s o l i d l i n e w h ile th e co nd uction
band extends between th e s la n t e d , dashed l i n e and +2.
q u a r t e r s-band u n i t s .
Energy i s in
A i s th e energy o f th e a c tu a l gap s t a t e , w h ile
th e b o n d - o r b it a l a p p ro x im a tio n g iv e s B.
66
valence band maximum i s a ls o l i n e a r and i s re p re se n te d by a s o l i d
l i n e w i t h equal b u t o p p o s ite s lo p e .
For any given v a lu e o f y the
valence band extends from -2 up t o - 2 y , w h ile th e c o n d u c tio n band is
between 2y and 2.
As th e bands become narrow compared t o th e gap, th e b o n d - o r b it a l
a p p ro x im a tio n is t o remove th e c o n d u c tio n bands.
C o upling t o the
c o n d u c tio n bands would re p e l th e s u r fa c e s t a t e downward by an amount
equal and o p p o s ite t o t h a t o f th e vale n ce bands.
In o t h e r words, a l ­
though th e vale n ce and c o n d u c tio n bands do n o t i n t e r a c t in th e bondo rb ita l
l i m i t , each i n t e r a c t s s e p a r a te ly w i t h th e s u r fa c e h y b r id .
The
a p p ro x im a tio n which i s being made i s t o n e g le c t th e c o u p lin g between
th e h y b r id and th e c o n d u c tio n band.
Thus th e upward push from the
vale n ce band i s n o t balanced by th e downward push which would have
been e x e rte d by th e c o n d u c tio n bands.
There are two reasons why th e b o n d - o r b it a l approach may y e t be
ju s tifie d
in th e s tu d y o f sem iconductor s u rfa c e s t a t e s .
F i r s t , the
graph o f f i g u r e 9 shows t h a t i f th e gap i s n o t e x tre m e ly narrow com­
pared t o th e band w i d t h , th e a p p ro x im a tio n may n o t be so bad.
Second,
in more r e a l i s t i c b o n d - o r b it a l models used f o r sem iconductor valence
bands, th e energy o f th e f i l l e d
h y b r id gap s t a t e i s much c lo s e r to
th e valence band maximum, and so th e in f l u e n c e o f th e c o n d u c tio n bands
m ig h t re a s o n a b ly be expected t o be much s m a lle r .
The purpose o f t h i s s e c t io n has been t o p o i n t o u t th e
67
a p p ro x im a tio n s in v o lv e d in a b o n d - o r b it a l model and t o warn th e reade r
o f th e p o s s ib le shortcom ings o f these a p p ro x im a tio n s when the model is
a p p lie d t o th e s o l u t i o n o f th e s u r fa c e problem.
IV.
BOND-ORBITAL MODEL OF HARRISON
A.
I.
B ulk C hem istry
Form ation o f b o n d - o r b i t a l s .
In t h i s s e c t io n a b o n d - o r b it a l model
w i l l be presented which was developed by H a r r is o n , P a n t e lides and
C
i r a
c i ^ and has been used by them t o i n v e s t i g a t e b u lk and s u r -
fa c e 3 2 -3 4 ,3 7 p r o p e r t ie s o f z in c b le n d e - ty p e s e m ic o n d u c to rs .
S t a r t w i t h an s and t h r e e p atom ic s t a te s on each c a t io n s i t e and
th e same on each anion s i t e .
For th e moment, i t
c o n s id e r th e d e t a i l s o f th e geometry.
i s n o t im p o r ta n t to
C o nsid er two s p 3 h y b r id s , one
on an anion p o i n t in g toward a n e ig h b o rin g c a t io n and one th e c a t io n
p o i n t in g back toward th e f i r s t as i l l u s t r a t e d in f i g u r e 10.
|ha> is
th e o r b i t a l on th e anion and |h c> i s th e o r b i t a l on th e c a t i o n .
The
o v e r la p
S = <hQI hc>
(189)
between two such h y b r id s which extend d i r e c t l y toward one a n o th e r has
been r e p o r te d t o be I a r g e ^ and cannot be n e g le c te d .
D e fin e two
parameters i n terms o f which th e bond f o r m a t io n may be d iscu sse d :
io n ic ity
M3 = %(ec-ea)
(190)
covalency
M2 = -< h a |H |h c>.
(191)
S e le c t in g as th e zero o f energy th e mean energy o f th e h y b rid s
69
A
A
F ig u re 10.
sp3 H yb rid O r b i t a l s on a C a tio n C P o in t in g Toward
N e ig h b o rin g Anions A.
70
energy zero =
(192)
th e bond f o r m a t io n becomes, as i t d id in th e p re v io u s s e c t io n , a 2
tim es 2 e ig e n v a lu e problem.
th e o r b i t a l s
T h is tim e , however, n o n - o r t h o g o n a l it y o f
must be c a r e f u l l y handled.
-M3
The 2 tim es 2 H a m ilto n ia n i s
-M2 \
(193)
-M2
w h ile th e o v e r la p m a t r ix
I
S
E
(194)
S
I
The f i r s t row (column) corresponds to th e anion in each case.
One
must s o lv e
(H - Ex)lip> = 0
.(195)
in o r d e r t o g e t th e m ix in g c o e f f i c i e n t s which determ ine th e bondo rb ita ls .
|^> =
N ext, tr a n s fo r m th e components o f |^> .
<t».
(196)
Then
(E '^ H E "^ - E) I <f» = 0.
(197)
Note t h a t s in c e H and E are H e r m i t i an, so i s E-jsHE- j ^.
Rather than
tr a n s fo r m th e components o f th e o r b i t a l , as in ( 1 9 6 ) , i t
t o t r a n s fo r m th e b a s is v e c to r s
|ha> and |hc > .
i s e q u iv a le n t
To do t h i s ,
E
2 is
needed.
The e ig e n v e c to r s o f E c o rre s p o n d in g t o i t s
e ig e n v a lu e s (1+s) and
71
(1 -s ) are
CHiLi1
I /I
(198)
Thus
(1+s,%C H 0-spsL )
(199)
1
And so
•f— + yT"
•/— — i/-F
( 201)
2 /l-S2
\ -J- - /+"
■/- + -fir I ,
where
/+ = / I + s , and /T" =
/I -s .
Thus th e new, o r th o g o n a li zed b a sis s e t becomes:
= = [ ( / ^ + / + ) ] h > + ( / - " - / + I I hc > ] ,
( 202)
2/ ^ s 2
I h *>
c
The b a sis
The
I
7------ [ ( / 2 /1 - s 2
- /n ih
> + ( / - + A ) 1h > ] .
a
c
o f (202) may be e a s i l y seen t o be orthonorm al using (189).
H m a t r ix o f (193) may now be expressed in terms o f |h*> and
Ih > as an H
m a t r ix .
I- V 3 ♦ SV2
V2
'
( 203)
-V,
V3 + SVg
72
where
V
2
(204)
V
3
(205)
The energy e ig e n v a lu e s o f t h i s m a t r ix are th e extreme values o f
th e bond e n e r g y . . The e ig e n v e c to r c o rre s p o n d in g t o th e minimum bond
energy g ive s th e m ix t u r e c o e f f i c i e n t s f o r th e b o n d - o r b it a l w h ile the
e ig e n v e c to r o f th e l a r g e s t e ig e n v a lu e g iv e s th e a n tib o n d m ix t u r e c o e f ­
fic ie n ts .
The bond and a n tib o n d e n e rg ie s are r e s p e c t i v e l y
SV2 - Z f T v f
(206)
sv2 + / V l + V*
(207)
where th e energy zero (192) must be added i n o r d e r t o f i n d th e abso­
l u t e e n e rg ie s .
In t h e i r b u lk c a l c u l a t i o n , P a n te lid e s and H a rris o n
parameters w i l l
them selves
35
, whose
be used in th e com putation below , have n o t concerned
w i t h th e o v e r la p S.
These o r t h o g o n a l i t y c o r r e c t i o n s e n t e r
e x p l i c i t l y a t th e s u r fa c e l a y e r where n o n -o rth o g o n a li zed h y b rid s form
th e
d a n g lin g h y b r id s u r fa c e s t a t e s .
Thus th e t r a n s f o r m a t io n between
th e b o n d - o r b i t a l s o f th e b u lk b a s is s e t and th e o r i g i n a l h y b r id o r ­
b i t a l s must be worked o u t in d e t a i l .
Strange as i t may seem, the
73
expansion c o e f f i c i e n t s f o r th e a n t i b o n d - o r b it a ls w i l l
a ls o be needed
so t h a t , as was done in th e s im p le r example above, th e h y b r id s may be
expanded in terms o f bonding and a n t i bonding p a r t s .
L e t |b> be a bond
o r b i t a l and |a> be an a n t i b o n d - o r b i t a l .
I b> = u Ih > + u Ih >,
a d
CC
(208)
where
U
3
-
—
- - --
-
[ ( y+ + /-)/l
+a
2 / 2 ( I -S 2 )
- ( / + - / - " ) / I -a ]
P
(209)
P
U = ---------------- [ ( / + + / - " ) / I -a - ( / + - / - ) / I +a ] ,
c
2 /2 ( l- S 2 )
P
P
(210)
and l i k e w is e
|a> = va |h a> + v c lh c>
(211)
w ith
V
= — — I-
3
v
— [ ( /+ + /^ ) /I -a
2/2(l-S2)
+
(/+ -/-") / l +a
P
],
P
= —-■
— [ ( / + - / - " ) /T-cT- + ( t/ + + / ^ ) / I +a ] .
c
2 /2 ( l- S 2 )
P
P
The symbol ap used in
(2 1 2 )
(213)
(209-213) i s d e fin e d by
Op = VgfVz + V s ) - \
(214)
I t i s r e l a t e d t o th e charge t r a n s f e r per e l e c t r o n in th e b o n d - o r b it a l
by
IUaI2 - IUcI = ap/Zl^S2
(215)
74
For t h i s reason ,
is c a l le d ^
th e bond p o l a r i t y , s in c e i t reduces
t o t h a t as S goes t o z e ro .
B e fore le a v in g th e s u b je c t o f bond fo r m a tio n and th e b a sis s e t
and moving on t o th e d e f i n i t i o n o f b u lk m a t r ix elem e nts, one l a s t
o p e r a tio n must be perform ed.
The t r a n s f o r m a t io n e q u a tio n s (208-213)
must be in v e r t e d in o r d e r t o o b t a in
I b> and |a>.
|h^> and |h^> as f u n c t io n s o f
T h is i s th e form in which th e c o n n e ctio n w i l l
be needed
when th e s u r fa c e p e r t u r b a t i o n i s d e f in e d .
I ha> = C1 1b> + C3 1a>
(216)
I hc> = C2 I b> + C4 1a>
(217)
C1 = vc/ A , C3 = - u c/ A ,
(218)
w it h
C2 = V ^ / A, C4 — U ^ /A,
where
a
= u . v . - U^v3 i s th e d e te rm in a n t o f th e t r a n s f o r m a t i o n .
Q u ite
s im p ly ,
A = ,- l//N S 2
.
(219)
Thus th e c o e f f i c i e n t s C1 and C2 , which w i l l
C1 = —— [ ( / + - / T ) / l -a
2/2
p
+ ( / + + / T ) / I +a
be needed l a t e r are
]
P
C2 - —— [ (/+ + /- ") / l -a + ( / + - / T j / I +a ^
2 /2
P
p
(220)
'
(221)
75
2.
B o n d - o r b it a l m a t r ix e le m e n ts .
Each io n in a s e m i-c o n d u c to r o f the
z in c b le n d e s t r u c t u r e i s t e t r a h e d r a l I y c o o r d in a te d .
The f o u r bond-
o r b i t a l s s u rro u n d in g a c a t i o n , f o r example, are each formed from a
l i n e a r co m b in a tio n o f h y b r id o r b i t a l s , one on th e c a t io n and one on a
n e ig h b o rin g a n io n .
F ig u re 10 i l l u s t r a t e s a t e t r a h e d r a l arrangement o f
h y b r id o r b i t a l s w h ic h , i n f i g u r e 11, have been used t o form bondo r b i t a l s in th e manner d e s c rib e d above.
Because th e ( H O ) s u r fa c e o f z in c b le n d e sem iconductors w i l l
t r e a t e d below, i t
be
<
39
i s e x p e d ie n t t o d e f in e a c o o r d in a te s e t as f o ll o w s
The z - d i r e c t i o n w i l l
be p e r p e n d ic u la r t o th e (110) s u r fa c e p o i n t in g
o u t o f th e c r y s t a l ; th e x - d i r e c t i o n w i l l
be p e r p e n d ic u la r t o th e (100)
p la n e , and th e y - d i r e c t i o n forms a r ig h t- h a n d e d c o o r d in a t e system.
The b a s is s e t f o r th e b u lk o f th e c r y s t a l c o n s is ts o f f o u r , i n ­
e q u iv a le n t b o n d - o r b i t a l s .
F ig u re 11 shows th e b asis s e t arranged in a
t e tr a h e d r o n about th e c a t io n p o s i t i o n .
W ith re s p e c t t o th e n e w ly-d e ­
f in e d c o o r d in a t e system, th e d i r e c t i o n s o f th e b u lk b a s is s t a te s are
|b l> + ( 1 / 3 , 0, 2 / 2 / 3 )
(222)
I b2> + ( 1 / 3 , 0, - 2 / 2 / 3 )
I b3> -> ( - 1 / 3 , 2 / 2 / 3 , 0)
I b4>
h.
( - 1 /3 , - 2 /2 /3 , 0).
The o r b i t a l s d e fin e d in (222) are arranged on an fe e l a t t i c e and
form th e z in c b le n d e s t r u c t u r e , as i l l u s t r a t e d
in f i g u r e 12.
N o tic e
t h a t th e use o f b o n d - o r b it a ls focuses a t t e n t i o n on th e bonds r a t h e r
i
76
F ig u re 11.
T etra h e d ro n o f B o n d - O r b ita ls .
The y - a x i s i s p o i n t in g o u t o f th e paper.
F ig u re 12.
The Z in c b le n d e S t r u c t u r e i s Formed by S ta c k in g th e Cation
Jacks o f B o n d - O r b it a ls as Shown in f i g u r e 11 i n t o an fe e L a t t i c e .
N o tic e how w e ll t h i s b a s is s e t corresponds w it h th e i n t u i t i v e , b a l l a n d - s t i c k n o t io n o f th e c r y s t a l
bonding.
78
than th e atoms them se lve s; thus th e vale n ce bands become a s s o c ia te d
w i t h a framework model o f th e c r y s t a l .
The c r y s t a l model i s ,
in f a c t ,
a g r a p h ic a l r e p r e s e n t a t io n o f th e b o n d - o r b it a l H a m ilto n ia n .
P a n t e lid es and H a rr is o n
35
have shown t h a t except in th e case o f
compounds c o n t a in in g th e carbon row elem e nts, th e b u lk p r o p e r t ie s o f
z in c b le n d e - ty p e sem iconductors a re w e ll reproduced u sin g o n ly th re e
r
m a t r ix element param eters.
These are presen ted in f i g u r e 13.
B1 and
A
B1 are minus th e n e a r e s t- n e ig h b o r m a t r ix elements between bonds which
have a common v e r te x a t e i t h e r a c a t io n s i t e o r an anion s i t e re sp e c­
tiv e ly .
Bit i s minus th e m a t r ix element between secon d -n e ig h b o r bonds
which are p a r a l l e l .
A n oth er m a t r ix element would be needed in o r d e r t o
t r e a t those compounds which c o n t a in a carbon row elem ent.
pounds w i l l
n o t be t r e a t e d in th e p re s e n t work.
Such com­
For convenience, th e
parameters
B1 = h { Z i + B1)
(223)
B1 = k t t i
- B1)
are in tr o d u c e d .
In a l l
t h e n , th r e e parameters s u f f i c e to determ ine th e b u lk band
s t r u c t u r e , namely B1 , B1 , and B4 .
B.
I.
S u rface m a t r ix e le m e n ts .
S u rface D e f i n i t i o n
Four c o u p lin g c o e f f i c i e n t s are r e q u ire d
i n o r d e r t o d e s c r ib e th e way i n which th e d a n g lin g h y b r id s connect to
79
F ig u re 13.
H a r r is o n .
The B o n d -O rb ita l M a t r ix Elements o f Pantel ides and
Open c i r c l e s r e p r e s e n t ions on a g ive n ( H O ) p la n e ,
w h ile th e c ro s s -h a tc h e d c i r c l e s r e p r e s e n t ions in a lo w e r plane.
80
th e b u lk .
These are presen ted in f i g u r e 14.
For example, c o n s id e r Q1.
g
I
=
<b4|H|h >.
c
(224)
Using (2 1 7 ),
g i = C2<b41HI b l > + C4< b 4 |H |a l>
(225)
As in th e two-band m odel, th e b o n d in g - a n ti bonding p a r t must be
dropped.
9i =
C2 ( - B f )
(226)
= — — [ ( / + + / T " ) / ] _ a + ( / T - / T " ) /T + c T T ] B f .
2 /2
P
P
L ik e w is e ,
02 =
C2 ( - B 4 )
(227)
= — — [ ( /++ /-") / l - a
2 /2
P
+ ( / + - / - " ) / I +a 3 B4 ,
P
9s = C i( - B ^ )
(228)
= ——[ (/+ -/-") / l
2/2
-a
P
+ (/+ +/-")/ l +a ] B^,
P
g4 = C i ( - B 4 )
-
= — — [ (/+ -/-") /l -a
2/2
2.
B ra v a is l a t t i c e .
illu s tr a te d
fe e .
+
P
(/+ + /-") /l +a
(229)
] B4 .
P
A s e t o f t r a n s l a t i o n v e c to r s a, S and c is
in f i g u r e 15.
These g en erate th e B ra va is l a t t i c e , which is
A t e t r a g o n a l u n i t c e l l which i s u s e fu l in c o m p u ta tio n s , a lth o u g h
81
F ig u re 14.
M a t r ix Elements Which Couple th e Dangling
H yb rid s Back i n t o th e B u lk.
82
F ig u r e 15.
N o n - P r i m it iv e , T e tra g o n a l U n it C e ll Used
in O rder to Demonstrate th e Ba sis V e cto rs a, b and c
c extends from th e c o r n e r to th e c e n t e r o f th e c e l l ,
a form s one edge o f th e square base and b forms th e
long edge.
83
it
is not p r im it iv e ,
i s a ls o i l l u s t r a t e d .
The components o f the
t r a n s l a t i o n v e c to r s in terms o f th e c o o r d in a te d i r e c t i o n s d e fin e d
above are
/I
0 ,
01
/01
/2 1
S= —a I , c =
a I
2/2 I
/2 Oi
The symbol a in (230) stands f o r th e le n g t h o f th e edge o f th e con­
v e n t io n a l
( c u b ic ) u n i t c e l l .
The p o s i t i o n s o f th e c a t io n s i t e s are
then d e fin e d in terms o f these by
= 4
+
+
where &, m and n are i n t e g e r s .
1231>,
The b o n d - o r b i t a l s d i s t r i b u t e d about a
c a t io n a t s i t e &, m, n are denoted by
|y,£.,m,n>
^
(232)
y d e n o tin g which b o n d - o r b i t a l , as in (222) and £, m and n p r e s c r ib i n g
th e p o s i t i o n o f th e c a t io n v e r t e x .
B u lk m a t r ix elements o f the
H a m ilto n ia n between s t a t e s so d e fin e d are presented in Appendix A.
3.
L a y e r-o rb ita ls .
N o tic e in (232) t h a t a change in th e in d ic e s z
and m i s a s s o c ia te d w i t h a change from one o r b i t a l
o rb ita l
t o an e q u iv a le n t
in th e same (110) l a y e r , w h ile a change in n im p lie s a change
in la y e r .
As in th e s im p le r examples, suppose f o r th e moment t h a t N1 by N2
s t a t e s o f a g ive n ty p e are on each la y e r .
d itio n s
With p e r io d ic boundary con­
in th e plane o f th e l a y e r , l a y e r - o r b i t a l s may be d e fin e d by
84
ik,I-r
I v , k i , l<2, n>
I e
II
Jlmn
(233)
y N1N2 Jim
The components k 2 , k2 o f
are th e c a r t e s ia n components p a r a l l e l to
th e s u r fa c e
I = k xx + k2y .
(234)
Thus th e exponent in (233) i s
I I •r
'll
mk a
k a
£ k ia + — — + 2"( k^a +
Jimn
/2
(235)
/2
A g a in , in terms o f these l a y e r - o r b i t a l s , m a t r ix elements o f th e b u lk
H a m ilto n ia n may be computed.
Because o f momentum c o n s e r v a tio n , o n ly
m a t r ix elements between s t a te s o f th e same
was, as w i l l
be n o n -z e ro .
T h is
be r e c a l l e d , th e m o t iv a t io n f o r th e t r a n s f o r m a t io n to
la y e r o r b i t a l s .
HjjV
| w ill
The d iagon al
( S m ) = ( S 1S2 ) ^ " ' )
11
(in t | | )
elements o f H are
Z s 1" 2£s2' 2m<M£mnl |H|v00n>,
Jim
(236)
where th e s t r u c t u r e f a c t o r s
ik i a /2
S1 = e
i k 2a/2/2
,
s2 = e
These m a t r ix elements are computed u sin g Appendix A and t a b u la te d in
Appendix B.
V.
MATRIX SLAB CALCULATION
A.
1.
B u lk .
Im plem entation
The procedure o f f i n d i n g th e e ig e n va lu e s and e ig e n v e c to rs
o f a th in f ilm
o f GaAs w i t h (110) s u rfa c e s i s e s s e n t i a l l y one o f
d i a g o n a l i z a t i o n o f a la r g e m a t r ix which has one row ( o r column) f o r
each o r b i t a l a t a g ive n 1 < ||.
two p a r t s :
z a t io n .
T h is problem i s l o g i c a l l y d i v i s i b l e i n t o
d e f i n i n g th e m a t r ix elements and p e rfo rm in g th e d ia g o n a li -
The d i a g o n a l i z a t i o n i s done n u m e r i c a l l y ^ and th e b u lk p a r t o f
the m a t r ix has elements as d e fin e d in Appendix B.
A r ra n g in g th e l a y e r - o r b i t a l s f o r a p a r t i c u l a r v a lu e o f
j in to
num erical o r d e r may be done a c c o rd in g t o
|m> = | y , k l 5 k2,n>
(238)
where
m = 4 ( n - l) + y
Thus, i f th e r e were 8 la y e r s in th e s la b , then th e b a s is s e t would
c o n t a in 32 s t a t e s .
The n o t a t io n o f (238) w i l l
o f th e p re s e n t c h a p te r .
Appendix B would now p r o v id e th e m a t r ix e l e ­
ments o f th e l a y e r - o r b i t a l
2.
S u rfa ce .
be used u n t i l th e end
b u lk H a m ilto n ia n .
Sim ply t e r m in a t i n g th e b u lk H a m ilto n ia n a t la y e r n=0
and la y e r n=N c re a te s a s u r fa c e o f th e ty p e i n v e s t i g a t e d by Shockley '
on
in 1939
.
As has been discussed in th e two-band case, t h i s Shockley
s u r fa c e i s u n r e a l i s t i c f o r a b o n d - o r b it a l b u lk H a m ilto n ia n .
86
C onsider a Shockley s u r fa c e formed in such a way as t o remove a l l
b o n d - o r b i t a l s beyond a given l a y e r , say f o r d e f in it e n e s s la y e r I , as
shown in f i g u r e 16.
Then th e b o n d - o r b i t a l s o f s t a t e
11> would s t i c k
o u t o f th e s u r fa c e a t th e c a t io n s i t e s w h ile no s t a te s a t a l l would
p r o tr u d e from th e anion s t a t e s .
Two im p o r ta n t changes must be made in o r d e r t o c o r r e c t l y r e p r e ­
se n t a r e a l i s t i c s u r fa c e as presen ted in f i g u r e 17.
b o n d - o r b it a l on each s u r fa c e c a t io n s i t e ,
both a t la y e r I and a t la y e r
N, must be re p la c e d by a c a t io n h y b r id o r b i t a l .
m a t r ix element in v o l v i n g e i t h e r
s la b H a m ilto n ia n .
T h is means t h a t each
| l > o r |4N-2> must be re p la c e d in th e
Second, s t a te s must be added t o th e b a sis s e t which
re p r e s e n t th e anion d a n g lin g h y b r id s .
M a tr ix elements i n v o l v i n g the
added s t a t e s must be added t o th e H a m ilto n ia n .
have been c a l l e d
F i r s t th e d a n g lin g
|0> and |4N+1>.
These added s ta te s
S u rfa ce c o u p lin g m a t r ix elements
needed t o e f f e c t these changes are presen ted in Appendix C.
E xp ressions (230) f o r th e l a t t i c e t r a n s l a t i o n v e c to r s induce a
s u r fa c e B r i l l o u i n zone as d e p ic te d in f i g u r e 18.
zone are a t k% = ± % and k2 = I - ^ .
Symmetry p o in t s and l i n e s o f the
s u r fa c e zone are la b e le d in th e n o t a t io n o f Jones
B.
The edges o f the
41
.
R e su lts and D iscu ssio n
We have used th e b o n d - o r b it a l model o f P a n te lid es and H a rris o n t o
compute e ig e n v a lu e s and e ig e n s ta te s o f a t w e l v e - l a y e r s la b acco rd in g
t o th e method d e s c rib e d in d e t a i l above.
D is p e rs io n p l o t s o f the
87
F ig u re 16.
R e s u lt o f T ru n c a tin g th e B ulk H a m ilto n ia n .
F ig u re 17
More R e a l i s t i c S urface w i t h Dangling H yb rid O r b i t a l s .
F ig u re 18.
S urface B r i l l o u i n Zone.
A 1 i s along x - a x i s w h ile A i s along y .
90
s u r fa c e s t a t e s appear in f i g u r e 19 t o g e t h e r w i t h th e p r o je c te d b u lk
bands.
The la b e l s B1 , B2 and B3 used t o denote the s u r fa c e s t a te s are
3
those in tr o d u c e d by Joannopoulos and Cohen
who have computed e ig e n ­
s t a te s f o r a t w e l v e - l a y e r s la b using an e i g h t s t a t e b a s is s e t o f hy­
b rid -o rb ita ls .
C o n s id e rin g th e s i m p l i c i t y o f th e p re s e n t m odel, the
r e s u l t s agree r a t h e r w e ll w i t h those o f Joannopoulos and Cohen.
are two s t a te s in th e o p t i c a l gap:
There
BJ (which i s n o t shown) i s the
c a t io n d a n g lin g bond s t a t e and B1 i s due to the d a n g lin g anion h y b r id s .
There i s a ls o a s t a t e B3 in th e h e t e r o p o la r gap near th e lo w e r valence
bands, and a s t a t e B2 which l i e s near th e bottom o f th e gap t h a t opens
in th e upper valence bands away from th e c e n t e r o f th e s u rfa c e
B r i l l o u i n zone.
F ig u re 20 shows th e p r o b a b i l i t y d i s t r i b u t i o n f o r an e l e c t r o n in
th e B1 s u r fa c e s t a t e a t th e r p o i n t , which corresponds t o normal p h o to ­
e m is s io n .
An e l e c t r o n in t h i s s t a t e spends 62% o f i t s
tim e on the
anion d a n g lin g bond, 10% on th e c a t io n back bond and a t o t a l o f 10% on
th e two p a r a l l e l
bonds which run along th e s u r fa c e .
z a t io n on th e s u rfa c e l a y e r i s about 82%.
The t o t a l l o c a l i ­
I f th e same s t a t e i s exam­
ined a t th e M p o i n t , th e e l e c t r o n d e n s it y i s 82% on th e anion d a n g lin g
bond and 9% in th e p a r a l l e l
bond, making a t o t a l
bonds.
A n other 5% i s in th e anion back
l o c a l i z a t i o n o f 96%.
T h is in cre a se d l o c a l i z a t i o n
i s due t o th e f a c t t h a t th e B1 s t a t e i s separated f a r from th e b u lk
91
IX
11_
O
Sf
CO
I
I
Aa
F ig u re 19.
o f GaAs.
CJ
I
ABjdUd
S u rface S ta te s on th e Unrelaxed ( H O ) S u rface
Zero o f energy i s th e valance band maximum.
92
Figure 20.
S ta te .
P r o b a b i li t y D is t r i b u t i o n f o r an E le ctro n in the B1 Surface
Note the la rg e d e n s ity on the anion dangling h y b rid .
93
continuum a t 'NI, as can be seen in f i g u r e 19.
E x p e rim e n ta ll y , n e i t h e r th e
gap^-^ .
nor B1 are found in th e o p t ic a l
Recent s la b c a l c u l a t i o n s by Randy, Freeouf and Eastman^
and o th e r s by C h a d i ^ , a l l o f which in c lu d e some form o f s u r fa c e r e ­
l a x a t i o n which i s n o t in c lu d e d in th e p re s e n t model, have shown t h a t
it
i s s u r fa c e r e l a x a t i o n which removes these s t a te s from th e gap.
Knapp^’ ^
has observed t h a t th e a c tu a l d is p e r s io n o f th e B1 s t a t e
seems t o be on th e o r d e r o f IeV , w h ile th e d is p e r s io n found in both
th e p re s e n t c a l c u l a t i o n as w e ll as in t h a t o f Joannopoulos and Cohen
i s about A e V .
The reason f o r t h i s d is c re p a n c y i s n o t a l t o g e t h e r
9 10
c l e a r , b u t t h e r e i s some evidence ’
t h a t i t may a ls o be due t o s u r ­
fa c e r e l a x a t i o n .
The p r o b a b i l i t y d i s t r i b u t i o n f o r s t a t e B2 a t th e M p o i n t i s shown
in f i g u r e 21.
A t M t h e r e i s 70% p r o b a b i l i t y in th e p a r a l l e l
14% i n th e c a t io n back bond and 11% in th e p a r a l l e l
second la y e r .
bonds,
bonds on th e
Thus, t h i s s t a t e i s lo c a l i z e d deeper in th e c r y s t a l
than i s th e B1 s t a t e .
T h is a n a ly s is o f th e co m p o s itio n o f th e B2
s t a t e agrees w i t h t h a t o f Joannopoulos and Cohen, who r e f e r t o i t as
being composed o f p a r a l l e l bonds along th e s u r fa c e .
In th e p re s e n t c a l c u l a t i o n , th e B2 s u rfa c e s t a t e i s seen to e x i s t
in th e gap a l l th e way from X1 t o X.
T h is i s n o t in agreement w ith
Joannopoulos and Cohen who see B2 d i e o u t b e fo re i t reaches X 1 o r X,
o r w i t h Chadi o r Pandi e t a l . who f i n d th e s t a t e a t X 1 b u t n o t a t X.
94
Figure 21.
P r o b a b i li t y D i s t r i b u t i o n a t M f o r an E le ctro n in the
B2 Surface S ta te.
95
The b in d in g e n e rg ie s o f B2 a t X 1 and a t X are t i n y , being .05 eV and
.001 r e s p e c t i v e l y .
There i s no a p p re c ia b le l o c a l i z a t i o n a t these ex­
treme p o i n t s .
In c o n t r a s t t o th e B2 s u r fa c e s t a t e , th e B3 s t a t e e x i s t s th ro u g h ­
o u t th e e n t i r e s u r fa c e B r i l l o u i n zone.
Joannopoulos and Cohen have
c a l l e d t h i s s t a t e th e anion s - l i k e s t a t e .
o r b i t a l s as a b a s is s e t ,
m odel.
it
Because o f th e use o f bond-
i s d i f f i c u l t t o see t h i s in th e p re se n t
A t th e zone c e n t e r , as re p re s e n te d by f i g u r e 22, th e r e i s a
41% p r o b a b i l i t y o f f i n d i n g an e l e c t r o n on a p a r a l l e l bond ( t h a t i s 20%
e a c h ), 15% on th e anion back bond and 9% on th e anion d a n g lin g h y b r id .
T h is i s a t l e a s t c o n s i s t e n t w i t h th e n o t io n o f an anion s - l i k e s t a t e .
T o ta l f i r s t - l a y e r l o c a l i z a t i o n
second l a y e r i s about 25%.
i s about 65% and l o c a l i z a t i o n on the
T h is s t a t e has n o t been observed in pho to­
e m is s io n , which co u ld be due t o i t s
s - l i k e c h a ra cte r.
Besides th e t h r e e s u r fa c e s t a te s analyzed above, C h a d i ^ has
n o tic e d a n o th e r s t a t e in h is u n re la xe d s la b c a l c u l a t i o n which occurs
in th e upper p a r t o f th e zone edge gap near th e X 1 p o i n t .
r e f e r s t o t h i s s u r fa c e f e a t u r e as S2 , f i n d s t h a t i t s
C h a d i, who
b in d in g energy
and l o c a l i z a t i o n are g r e a t l y enhanced by r e l a x a t i o n in th e several
models co n s id e re d by h i m ^ ’ ^ .
T h is s t a t e a ls o appears in th e re la x e d
• 9
models in v e s t i g a t e d by Pandey e t al_. .
A lth o u g h th e S2 s u r fa c e s t a t e appears t o be bound in th e p r e s e n t,
u n re la x e d c a l c u l a t i o n ,
its
b in d in g energy i s e x tre m e ly sm all and th e r e
96
Figure 22.
S ta te.
P r o b a b i li t y D i s t r i b u t i o n f o r an E lectro n in the B3 Surface
97
i s no r e c o g n iz a b le l o c a l i z a t i o n .
B in d in g energy o f S2 a t X ' i s
. 004eV.
Moving o n e - f i f t h o f th e d is ta n c e t o M from X 1 shows t h a t S2 has jo in e d
th e upper continuum and th e r e i s no lo n g e r any t r a c e o f S2 .
One
reason why o t h e r a u th o rs in c l u d i n g Joannopoulos and Cohen may not have
n o tic e d S2 i s because o f i t s
lo c a liz a tio n .
small b in d in g energy and poor s u rfa c e
On th e o t h e r hand, th e S2 s t a t e may n o t appear in t h e i r
m odel.
There are many d i f f e r e n t t i g h t - b i n d i n g H a m ilto n ia n s f o r th e z in c b le n d e - ty p e sem iconductors^ 6 ,9 -1 4 ,3 2 -3 7 ^
in which th e y d i f f e r from one a n o th e r.
Jjie r e are many kin ds o f ways
The e s s e n t ia l reason f o r the
p r o l i f e r a t i o n o f p a ra m e te rize d H a m ilto n ia n s i s t h a t th e se ts o f l o c a l ­
iz e d ato m ic f u n c t io n s are immensely o v e r-c o m p le te .
For example, any
f u n c t i o n in s i d e th e c r y s t a l co u ld be expanded in a s e t o f S l a t e r - t y p e
o r b i t a l s d e fin e d w i t h r e s p e c t t o a s i n g l e s i t e .
Three im p o r ta n t con­
s t r a in t s m itig a te t h is s it u a t io n :
I ) Only a f i n i t e
number o f basis
s t a te s are r e t a in e d on each s i t e .
2) M a t r ix elements between s ta te s
ce n te re d a t s i t e s which are sepa ra ted by more than a c e r t a i n d is ta n c e
are assumed t o v a n is h .
3) The la c k o f o r t h o g o n a l i t y between s ta te s on
d i f f e r e n t s i t e s must a f f e c t , e i t h e r e x p l i c i t l y o r i m p l i c i t l y , the p r o ­
cess o f bond f o r m a t io n and th e subsequent fo r m a tio n o f energy bands.
Because o f d i f f e r e n t ways o f t r e a t i n g th e above c o n s t r a i n t s , to g e th e r
w i t h d i f f e r e n c e s in choices o f b u lk band s t r u c t u r e , m a t r ix elements
which are n o m in a lly th e same in d i f f e r e n t param e te rize d H a m ilto n ia n s
98
may have v e ry d i f f e r e n t num erical v a lu e s .
In f a c t , t h e i r p h y s ic a l
meanings may be e n t i r e l y d i f f e r e n t .
Mele and J o a n n o p o u lo s ^ have r e c e n t l y in v e s t i g a t e d some o f the
d if f e r e n c e s between t i g h t - b i n d i n g H a m ilto n ia n s .
By s tu d y in g a c o l l e c ­
t i o n o f n e a r e s t- n e ig h b o r t i g h t - b i n d i n g H a m ilto n ia n s w i t h f o u r s ta te s
per s i t e and one w i t h an e x t r a anion 5s s t a t e , the above a u th o rs have
determ ined t h a t in o r d e r t o be u s e fu l as a t o o l f o r i n v e s t i g a t i n g s u r ­
fa ce s t a te s t h i s typ e o f H a m ilto n ia n must have th e f o l l o w i n g
p r o p e rtie s :
I ) The d ia g o n a l m a t r ix elements o f th e b a s is s t a te s on
each s i t e should be c lo s e t o th e atom ic H a rtre e -F o c k v a lu e f o r t h a t
s ta te .
2) The o t h e r m a t r ix elements must be chosen in such a way as
t o g iv e a good o v e r - a l l d e s c r i p t i o n o f th e c o n d u c tio n bands as w e ll as
to f i t
th e vale n ce band s t r u c t u r e . - C r i t e r i o n 2 was a ls o noted by
C h a d i^ .
A lth o u g h Mele and Joannopoulos have noted t h a t c r i t e r i o n I
i s i n t i m a t e l y r e la t e d t o th e change o f th e o v e r la p between s t a te s on
n e ig h b o rin g s i t e s f o r s i t e s a t th e s u r fa c e as opposed t o s i t e s in the
b u l k , none o f th e models which th e y have examined takes t h i s su rfa c e
o rth o g o n a lity c o rre c tio n e x p l i c i t l y
i n t o a ccou nt.
The p re s e n t c a l c u l a t i o n may be compared w i t h o th e rs appearing in
th e l i t e r a t u r e i n th e f o l l o w i n g ways.
F irs t of a l l,
H a m ilto n ia n s d iscussed by Mele and J o a n n o p o u Io s ^ 8
n e ig h b o r H a m ilto n ia n s .
each o f the
^
are n e a re s t-
Both th e Joannopoulos and Cohen c a l c u l a t i o n and
th e p re s e n t c a l c u l a t i o n c o n ta in more d i s t a n t i n t e r a c t i o n s .
The
99
u n re c o n s tru c te d model o f Pandy e t al_. a ls o c o n ta in s some second n e ig h ­
bor i n t e r a c t i o n s .
Only t w o ^ ’ ^
o f th e models t r e a t e d by Mele and
Joannopoulos produce th e c o r r e c t atom ic l i m i t .
C h a d i1s model does.
The H a m ilto n ia n o f Joannopoulos and Cohen does n o t.
C ira c i0
H a rr is o n and
have shown t h a t th e atom ic energy parameters i m p l i c i t in the
b o n d - o r b it a l model used in our c a l c u l a t i o n compare e x tre m e ly w e ll w i t h
the H a rtre e -F o c k e n e rg ie s in t e r p o l a t e d from th e t a b le s o f Herman and
S k illm a n ^ .
F i n a l l y , among a l l o f th e models discussed above o n ly the
b o n d - o r b it a l model takes th e change o f o v e r la p near th e s u r fa c e e x p l i c ­
itly
i n t o accou nt.
A t t h i s p o i n t i t must be mentioned t h a t G regory, S p ic e r , C ir a c i
and H a rris o n
have a p p lie d th e b o n d - o r b it a l model t o th e ( H O ) s u rfa c e
o f GaAs i n o r d e r t o e s tim a te th e p o s i t i o n s in energy o f th e d a n g lin g
h y b r id o r b i t a l
s u r fa c e s t a t e s .
I t i s p ro b a b le t h a t these a u th o rs have
n o t perform ed a s la b c a l c u l a t i o n , as t h e i r c o n c lu s io n s are based on th e
energy p o s i t i o n s o f th e uncoupled h y b r id o r b i t a l s r e l a t i v e t o the b u lk
band s t r u c t u r e .
They supposed t h a t th e occupied s u r fa c e s t a t e would
remain below th e vale n ce band maximum, which i s n o t th e case.
A n o th e r a p p l i c a t i o n o f the b o n d - o r b it a l model has been made by
C i r a c i , B a tra and T i l l e r ^ who s tu d ie d th e (111) fa ce o f z in c b le n d e ty p e sem iconductors u sin g a m a t r ix s la b method.
th e methods o f Haydock and K e lly
e x i s t on th e (1 1 1 ),
48
They have a ls o used
t o compare the s u r fa c e s t a te s which
( H O ) , and (100) s u rfa c e s o f Si w i t h i n the
.100
b o n d - o r b it a l model
34
U n f o r t u n a t e l y , th e method by which C i r a c i , Batra and T i l l e r have
appended th e d a n g lin g h y b r id o r b i t a l s t o th e s u rfa c e i s in e r r o r .
For
t h i s reason , o n ly those s u r fa c e s t a te s on the (111) s u r fa c e which do
n o t have s tro n g a m p litu d e on th e d a n g lin g bonds, may be expected t o .be
re p re s e n te d c o r r e c t l y by t h e i r r e s u l t s .
V I.
A.
SUMMARY AND CONCLUSION
G enerating F u n c tio n Formalism
The p r i n c i p a l t h r u s t o f th e re se a rch presented h e r e in has been to
develop and ana lyze v a r io u s methods f o r use in t h e o r e t i c a l s u rfa c e
p h y s ic s .
The method o f g e n e r a tin g f u n c t i o n s , as i t
above, i s an a d a p ta tio n which f a c i l i t a t e s
has been presented
th e use o f th e a n a l y t i c
r e s o lv e n t method t o s o lv e r a t h e r r e a l i s t i c t i g h t - b i n d i n g models o f
s u r fa c e s .
In c h a p te rs I I and I I I , th e method was used t o s o lv e models
w i t h one o r two s t a te s per l a y e r , and th e s o l u t i o n o f these models was
discussed in some d e t a i l .
G e n e r a liz a t io n o f t h i s method t o t i g h t -
b in d in g models o f a r b i t r a r y c o m p le x ity i s immediate and w i l l
be d i s ­
cussed b r i e f l y here.
I t may seem reason able t h a t any t i g h t - b i n d i n g H a m ilto n ia n may be
w r i t t e n in terms o f l a y e r o r b i t a l s p a r a l l e l t o a given c r y s t a l p lane.
Such a m a t r ix would be o f th e form
( s i , s2 ) ,
(239)
sia
where th e s u b s c r ip t s a and a ' denote la y e r s and i and j which la y e r o r b i t a l on l a y e r a and l a y e r &' r e s p e c t i v e l y .
S1 and S2 are s t r u c t u r a l
f a c t o r s r e la t e d t o t r a n s l a t i o n in th e plane o f the la y e r s in complete
analogy w i t h
i,j
elem ent.
(2 3 7 ).
Denote by H ^ 1 th e m a t r ix which has (239) as i t s
Note t h a t f o r a t i g h t - b i n d i n g H a m ilto n ia n , Hj^ ,
is o f
102
fin ite
rank n and i s zero when | ji- ji ' | > r f o r some range r .
b u lk Green f u n c t io n s are a ls o w r i t t e n
When the
in m a t r ix fo rm , then th e equa­
t i o n s o f m otion become
O ESnol - H00JG
(240)
Because o f t r a n s l a t i o n symmetry, H
)v"rp 5 Jv
wise G
P
may be w r i t t e n as G .
AfTp j
pL r
may be w r i t t e n H , and l i k e -
P
( , E P0 ' H - P ) G * + p = 16IO-
(241)
The g e n e r a tin g f u n c t i o n m a t r ix i s d e fin e d as
F(x) = Z GLx
( 242)
Then, when (241) i s m u l t i p l i e d by x * and th e r e s u l t i s summed over a l l
la y e r s
[ I E - H ( x ) ] F (x) = I ,
(243)
where
H(x) =
r
E
H xp
p = -r p
(244)
i s a f i n i t e m a t r ix o f rank n.
The d e te rm in a n t o f (E-H) i s o f th e form
a(x
) = d e t ( lE - H (x)) =
aN(xN +
where
+ 9N - I ^ n" 1 + ” N - V + * • * + aO5
(245)
N^nr and th e numbers a^ are f u n c t io n s o n ly o f E, S1 and S2 , as
103
w e ll as th e o r i g i n a l
t i g h t - b i n d i n g param eters.
M x ) = aN - N p ( x )
(246)
where p (x ) i s a monic polynom ia l o f degree ZN.
The r o o t s o f p may be found e i t h e r in clo se d form o r n u m e r ic a lly .
7 0
The l a t t e r i s by f a r th e more pow erfu l method o f p r a c t i c a l use
,
w h il e th e fo rm e r d iv id e d t i g h t - b i n d i n g models which can in p r i n c i p l e
be solved in terms o f e lem e ntary f u n c t io n s from those which cannot.
T h is i s because th e o t h e r steps r e q u ir e d t o produce s u r fa c e Green
f u n c t io n s r e q u i r e o n ly a f i n i t e
number o f a r i t h m e t i c o p e r a tio n s ,
though t h i s number may be q u i t e la r g e indeed.
There e x i s t s a s o l u t i o n o f a g ive n t i g h t - b i n d i n g model s u rfa c e
problem in c lo s e d form in terms e le m e n ta ry f u n c t io n s i f and o n ly i f
th e H a m ilto n ia n i s such t h a t p i s s o lv a b le .
dependence upon complex
Si ,
S2
Due t o th e p a ra m e tric
and E o f th e c o e f f i c i e n t s o f p, cases
o f s o lv a b le p w i t h degree more than f o u r must be v e ry e x c e p t i o n a l .
Denoting th e r o o t s o f p by x ^ , th e g e n e ra tin g f u n c t i o n m a tr ix may
now be w r i t t e n in th e form
F (x)
_ U(x)
" PW
U (x k )
There i s a ls o th e p o s s i b i l i t y t h a t F(«) f
(247)
0, but t h i s
i s v e ry u n p h y s i­
cal and may be ig n o re d in t h i s summary.
A f t e r L a u re n t s e r ie s expanding F, th e Green f u n c t io n s may be read
104
as th e c o e f f i c i e n t s o f th e expansion and t h e i r general form is
= Z t^J
(248)
where th e t ^ " are r e s id u e s from (2 4 7 ) , and k o f th e summation runs
o n ly o v e r those r o o t s o f p such t h a t |x ^ |
< I.
U n f o r t u n a t e ly t ^ J may
depend upon th e s ig n o f a .
Al I o f t h i s has been i l l u s t r a t e d
in c h a p te rs I I and I I I .
As
a n o th e r i l l u s t r a t i o n o f th e kin d s o f r e s u l t s which may be obta in e d
from th e b u lk Green f u n c t i o n , f i g u r e 23 p re se n ts th e t o t a l
b u lk la y e r
d e n s it y o f s t a t e s produced in t h i s way from th e same H a m ilto n ia n used
in c h a p te r VI f o r th e t h i n f i l m o f GaAs.
The v a r io u s curves o f f i g u r e
23 i n d i c a t e d e n s i t i e s a t d i f f e r e n t va lu e s o f th e x component o f £ | | ,
corresponds t o d i f f e r e n t va lu e s o f S1 .
Once th e b u lk Green f u n c t io n s are o b ta in e d from th e g e n e ra tin g
f u n c t i o n m a t r i x , th e s u r fa c e problem may be solved from th e r e l a t i o n
J =
O - G V f 1G.
(249)
In most cases, th e Shockley s u r fa c e i s o f i n t e r e s t and then
,-I
j AA
gAA ' 6AB( GBB)GBA’
(250)
where A i s in s id e th e c r y s t a l and B i s a b a r r i e r r e g io n , as e x p la in e d
in c h a p te r I I .
The method o f g e n e r a tin g f u n c t io n s as o u t li n e d in t h i s summary is
a fo rm a l re fin e m e n t o f th e a n a l y t i c r e s o lv e n t method used by Davison
F ig u r e 23.
B ulk Layer D e n s ity o f S ta te s f o r GaAs ( H O ) as a F u n c tio n o f Energy in
eV and k in U n it s o f th e f - X 1 D is ta n c e .
106
and L e v i n e ^ and developed by Freeman*^.
fo rm a lis m i s i n t r i n s i c a l l y
a ris in g
Since th e g e n e r a tin g f u n c t io n
i n t e g r a t i o n f r e e , many o f th e co n fu s io n s
in th e F o u r ie r tr a n s fo r m f o r m u la t io n s are c ircu m ve n te d .
I t may be noted t h a t th e g e n e r a tin g f u n c t io n f o r th e Green f u n c ­
t i o n in th e presence o f th e s u rfa c e c o u ld be found d i r e c t l y , thus by­
passing th e b u lk Green f u n c t i o n .
However, because th e b u l k problem i s
i t s e l f i n t e r e s t i n g and because o f th e ease w i t h w h ic h , th ro u g h equa­
tio n
( 2 4 9 ) , one may change th e s u r fa c e p o t e n t i a l ,
it
i s u s u a ll y b e t t e r
t o c a l c u l a t e th e b u lk Green f u n c t i o n f i r s t .
The ease w i t h which th e la y e r d e n s i t i e s o f s t a te s may be f a c to r e d
a cc o rd in g t o la y e r o r b i t a l , as p o in te d o u t in c h a p te r I I ,
is i l l u s ­
t r a t e d again in f i g u r e 24 where th e s - l i k e p a r t o f th e s u r fa c e la y e r
d e n s i t i e s o f th e two-band problem t r e a t e d in t h a t c h a p te r are presented.
These curves should be compared w i t h those o f f i g u r e 7 where th e t o t a l
d e n s i t i e s on th e f i r s t t h r e e la y e r s were p resen ted .
Using th e g e n e r a tin g f u n c t i o n method, several model c a l c u l a t i o n s
have been perfo rm e d , i n c lu d in g th e one and two-band models o f c h a p te r
I I and I I I .
A number o f im p o r ta n t q u a l i t a t i v e r e s u l t s have been d i s ­
covered in t h i s way.
The stan d in g -w a ve f e a t u r e s in th e la y e r d e n s it ie s o f s t a te s which
were p o in te d o u t in c h a p te r I I and I I I
f i g u r e 24 have been observed b e fo re
23
.
and which are a ls o obvious in
B u t, i t
i s now seen t h a t these
are always p re s e n t and t h a t th e y a r i s e because o f th e replacem ent o f '
F ig u re 24.
s - l i k e P a rt o f th e Layer D e n s itie s f o r th e Model o f Chapter I I I .
108
th e p e r i o d i c boundary c o n d it io n s o f th e b u lk by standing-w ave boundary
c o n d it io n s .
Shockley had d is c o v e re d t h a t t h e r e are s u r fa c e s t a t e s which a r is e
due t o th e f o r m a t io n o f h y b r i d i z a t i o n gaps
30
.
These s t a t e s occur in
th e s -p model a t e x a c t ly th e p o i n t in energy where th e u n h y b rid iz e d
bands would have cro sse d .
The e x is te n c e o f such a s t a t e does n o t de­
pend, in th e s-p m odel, upon th e s t r e n g t h o f th e h y b r i d i z a t i o n .
As
th e gap becomes narrow , th e gap s t a t e i s o n ly weakly l o c a l i z e d .
Using
th e g e n e r a tin g f u n c t i o n method, K a w a jir i
s i m i l a r gap s t a t e in -the s-d m odel.
49
has found t h a t t h e r e i s no
The o n ly d i f f e r e n c e between the
s-p and th e s-d models i s t h a t th e s -p i n t e r a c t i o n m a t r ix element i s
odd w i t h r e s p e c t t o s i t e in te rc h a n g e w h ile th e c o rre s p o n d in g s-d ma­
trix
element i s even.
< s £ I HI p V > = - <p&|H |s&'>
(251)
<s&|H |d&'> = + < d £ |H |s V > .
I t i s n o t known a t t h i s p o i n t w hether t h i s p a r i t y d i f f e r e n c e i s nec­
essary in o r d e r t o produce a gap s t a t e , so t h a t t h e r e can be no s t a t e
in an s-d gap u n t i l
some odd p a r i t y s t a t e i s mixed i n , o r whether the
a d d i t i o n o f second n e ig h b o r i n t e r a c t i o n s m ig h t a ls o produce a s t a t e in
a s t r i c t l y even p a r i t y gap.
T h is should be in v e s t i g a t e d .
In f a c t , th e s i m p l i c i t y o f th e g e n e r a tin g f u n c t i o n method makes
it
im p o r ta n t as a f o r m a l , m athem atical t o o l .
We have used i t ,
fo r
example, t o dete rm in e a c r i t e r i o n f o r s o l v a b i l i t y f o r t i g h t - b i n d i n g
109
H a m ilto n ia n s .
A t th e h e a r t o f th e s o l u t i o n f o r any s e m i - f i n i t e t i g h t -
b in d in g model i s a c h a r a c t e r i s t i c p o ly n o m ia l.
T h is p o ly n o m ia l, which
has been discu sse d above and a ls o in th e I n t r o d u c t i o n , i s th e ch a ra c­
te ris tic
p olynom ia l f o r th e d i f f e r e n c e e q u a tio n s ; th e homogeneous s o l u ­
t i o n s t o which are th e e ig e n v e c to r s o f th e H a m ilt o n ia n , and th e im pulse
s o l u t i o n s t o which are th e Green f u n c t i o n s .
o f s o l u t i o n must a l l
The v a r io u s e x a c t methods
a r r i v e a t t h i s same p o ly n o m ia l, and th e s o l u t i o n
must be expressed in terms o f i t s
ro o ts .
T h is c lo s e r e l a t i o n s h i p be­
tween th e v a r io u s methods has been demonstrated in c h a p te r I I f o r sev­
e r a l methods which in c lu d e most o f th e p o p u la r methods t o be found in
th e l i t e r a t u r e .
The Green f u n c t i o n method was used in c h a p te r I I I
t o produce th e
e x a c t s o l u t i o n t o a two-band model and t h i s s o l u t i o n in t u r n was used
t o i n v e s t i g a t e th e v a l i d i t y o f a s im p le b o n d - o r b it a l m odel.
Bond-
o r b i t a l models are e x tre m e ly u s e fu l in sem iconductor s t u d ie s because
th e number o f b a sis f u n c t io n s i s c u t in h a l f and th e number o f t i g h t b in d in g b a sis f u n c t io n s i s c o n s id e r a b ly reduced.
Perhaps more im­
p o r t a n t l y , th e b o n d - o r b i t a l s seem t o be a p h y s i c a l l y m eaningful b asis
f o r th e r e p r e s e n t a t io n o f th e f i l l e d
B.
vale n ce bands o f sem icon ductors.
T hin F ilm C a lc u l a t io n
The l a r g e m a t r ix method which has o f t e n been a p p lie d t o th e p ro b ­
lem o f f i n d i n g th e e ig e n s ta te s o f a t h i n f i l m
i s a ls o a method o f
no
s o lv in g th e system o f d i f f e r e n c e e q u a tio n s f o r th e model.
method, however, th e boundary c o n d it io n s are d i f f e r e n t .
In t h i s
There must be
s ta n d in g wave boundary c o n d it io n s a t each s id e o f a f i n i t e
t h i s makes th e spectrum d i s c r e t e .
r e g io n , and
I t a ls o makes th e s o l u t i o n more
d i f f i c u l t f o r m a l l y , b u t q u i t e s im p le n u m e r ic a lly .
When th e b o n d - o r b it a l model o f H a rris o n was used f o r a tw e lv e
la y e r s la b o f g a lliu m a r s e n id e , th e num erical s o l u t i o n became r a t h e r
n e a t.
The r e s u l t s are in e x c e l l e n t agreement w i t h those o f
Joannopoulos and Cohen who d id a s i m i l a r com putation f o r th e unrelaxed
( H O ) s u r fa c e o f GaAs, b u t w i t h o u t making th e b o n d - o r b it a l approxim a­
tio n .
T h is s u c c e s s fu l r e s u l t i s an im p o r ta n t c o n f ir m a t io n o f the
v a l i d i t y o f th e b o n d - o r b it a l model as a t o o l f o r th e i n v e s t i g a t i o n o f
s u r fa c e s .
The b o n d - o r b it a l b a sis i s founded on b u lk c h e m is tr y and i t
was n o t known w hether t h i s e x tre m e ly sim p le model w i t h o n ly th re e
parameters c o u ld be used a t th e s u r f a c e , even t a k in g th e o v e r la p c o r ­
r e c t i o n i n t o a cco u n t.
As we have seen in th e s im p le r m o d e l, th e s u r ­
fa ce p o t e n t i a l mixes and bonding and a n t i- b o n d in g s t a t e s .
B u t, i t
seems t h a t t h i s m ix in g must be q u i t e n e g l i g i b l e in th e case o f the
( H O ) s u rfa c e s o f z in c b le n d e - ty p e sem icon ductors.
A lth o u g h a b o n d - o r b it a l c a l c u l a t i o n f o r th e (111) s u rfa c e s o f
GaAs and o f Si has been c a r r i e d o u t by C i r a c i , B a tra and T i l l e r
4
,
these a u th o rs have made an e r r o r which i n v a l i d a t e s th e r e s u l t s t o some
e x te n t.
T h e ir r e s u l t s lo o k q u i t e reason able in them se lve s, b u t r a t h e r
Ill
than ig n o r in g th e b o n d in g - a n ti bonding m a t r ix e lem e nts, these auth o rs
have done what f o r m a l l y amounts t o n e g le c t in g some second neighb or
h y b r i d - o r b i t a l m a t r ix elements near th e s u r fa c e .
l i b e r t y t o do t h i s ,
m odel.
They are n o t a t
however, w i t h i n th e c o n te x t o f th e b o n d - o r b it a l
Thus, no f a i r t e s t s have appeared up t o now which p e r m it an
assessment o f th e use o f a b u lk b o n d - o r b it a l model in t h i s way.
The
r e s u l t s p resen ted above, however, do p r o v id e f o r such judgm ent.
H a rr is o n has used th e b o n d - o r b it a l concept in r a t h e r a d i f f e r e n t
way t o in v e s t i g a t e s u r fa c e r e c o n s t r u c t i o n
o r b ita l
33
.
By u sin g a f u l l
h y b r id -
b a s is s e t near th e s u r fa c e and then r e h y b r i d i z i n g t h i s s e t to
form s u r fa c e b o n d - o r b i t a l s , H a rris o n i s a b le to compute a b in d in g en­
e rgy d i f f e r e n c e as a f u n c t i o n o f s u r fa c e atom p o s i t i o n .
T h is use o f
th e b o n d - o r b it a l concept a c t u a l l y tra n sce n d s th e sim p le b u lk model
w i t h few param eters.
Whether o r n o t th e b u lk model may be used as i t
stands t o p r e d i c t
s u r fa c e r e l a x a t i o n e f f e c t s on th e e l e c t r o n i c s t r u c t u r e by s im p ly neg­
l e c t i n g th e b o n d in g - a n ti bonding p a r t o f th e p e r t u r b a t i o n i s n o t c l e a r .
It
i s a ls o n o t known a t t h i s tim e w hether c h e m is o rp tio n m ig h t be
t r e a t e d u sin g o n ly a b o n d - o r b it a l
im p o r ta n t a b so rb a te energy le v e l
b a s is s e t i f ,
f o r example, th e one,
l i e s deep w i t h i n th e s u b s t r a t e valence
bands.
The b o n d - o r b it a l p i c t u r e o f th e GaAs (110) s u r fa c e s t a t e s i s eas­
ily
r e l a t e d t o a b a l l - a n d - s t i c k model o f th e c r y s t a l s t r u c t u r e .
T h is
112
p i c t u r e i s p o s s ib le because o f th e r o l e p layed by th e f i l l e d
bands in th e c r y s t a l bonding.
valence
T h is i n t i m a t e co n n e c tio n o f th e bond-
o r b i t a l s w i t h th e vale n ce bands i s in t u r n r e s p o n s ib le f o r th e f a i l u r e
o f th e model t o c o r r e l a t e th e p r o p e r t ie s o f th e empty s t a t e s .
T ig h t-
b in d in g models in general do n o t r e p r e s e n t th e c o n d u c tio n bands ve ry
w e l l , and a f t e r th e b o n d in g -a n tib o n d in g i n t e r a c t i o n i s n e g le c te d ,
th e re is very l i t t l e
chance o f r e p r e s e n t in g th e c o n d u c tio n bands in
terms o f a n tib o n d in g o r b i t a l s o n l y ^ - ^ .
The r e s u l t s o b ta in e d u sin g th e model o f H a rris o n f o r th e z in c blende sem iconductors are e n c o u ra g in g , and i t
should be i n t e r e s t i n g to
c r e a te a s i m i l a r b o n d - o r b it a l f o r o t h e r types o f sem iconductors and to
use t h i s model t o i n v e s t i g a t e th e s u r fa c e e l e c t r o n i c p r o p e r t i e s .
REFERENCES
114
1.
K a n ji H ir a b a y a s h i, J. Phys. Soc. Japan,
27, 1475 (1969 ).
2.
K. C. Pandey and J.
3.
J. D. Joannopoulos and M arvin L. Cohen, Phys. Rev. BIO, 5075
(1975 ).
4.
S. C i r a c i , I . P. B a t r a , and W. A. T i l l e r ,
(1975 ).
5.
J.
D. L e vine and S.
6.
L.
M. F a lic o v and F. Y n d u ra in , J . Phys.
7.
J . Hermanson, M. K a w a jir i and W. Schwalm, S o l . S t . Comm. 21_, 327
C. P h i l l i p s , Phys. Rev. L e t t . 32^, 1433 (1974).
Freeman, Phys. Rev.
Phys. Rev. B12, 5811
B2_, 3255 (1970 ).
CS, 1571 (1975 ).
(1977) .
8.
M. K a w a j i r i , J. Hermanson and W. Schwalm, S o l. S t. Comm. 2^5, 303
(1978) .
9.
K. C. Pandey, J. L. F r e e o u f, and D. E. Eastman, J . Vac. S c i .
T e c h n o l., 14, 904 (1977 ).
10.
D. J . C h a d i, Phys. R e v ., in p re ss.
11.
J. Mele and J. D. Joannopoulos, S u r f . S c i . 66, 38 (1977 ).
12.
C. Calandra and G. S a n tr o , J . Phys. CS, L86 (1975 ).
13.
D. J. Chadi and M. L. Cohen, Phys. S t a t . S o l.
14.
J. D. Joannopoulos and E. J. Mele, u n p u b lis h e d .
15.
J. C. S l a t e r and G. F. K o s te r , Phys. Rev. 94^ 1498 (1954 ).
16.
S. Davison and J. D. L e v in e , S o l. S t. Phys. 25, 7 (1970 ).
17.
L. M. F a lic o v and F. Y n d u ra in , J. Phys. CS, 147 (1975 ).
18.
J a r o s la v K o ute cky, Phys. Rev. 10 8 , 13 (1957).
19.
E-Ni Foo, M. F. Thorpe and D. Weai r e . S u r f . S c i . 57, 323 (1976).
20.
D. N. Zubarev, S o v ie t Phys. Usp. 3^, 320 (1960).
(b ) 68, 405 (1 975).
115
21.
Smith Freeman, Phys. Rev. B2, 3272 (1970).
22.
P h i l i p I . T a y l o r , A Quantum Approach to th e S o lid S ta te ( P r e n t ic e
H a l l , New J e r s e y , 1970) Chap. 8.
23.
E-Ni Foo and Lambros G. Johnson, S u r f . S c i . 55^, 189 (1976 ).
24.
W. E. S p ic e r , Phys. Rev. 112, 114 (1958 ).
25.
W. E. S p ic e r , and F. Wooten, P ro c . IEEE 5]_, 1119 (1963 ).
26.
A two band model q u i t e s i m i l a r t o t h i s has been so lve d by Foo and
Johnson ( r e f e r e n c e 2 3 ).
They have made a s i m p l i f y i n g assumption
which has n o t been made here.
27.
Michael Tinkham, Group Theory and Quantum Mechanics ( M c G ra w - H ill,
C a l i f o r n i a , 1964) Chap. 4.
28.
J . Hermanson, S o l. S t. Comm. 22^, 9 (1977 ).
29.
M. B aker, Ann. Phys. 4_, 271 (1958 ).
30.
W. S h ockle y,
Phys. Rev. 56, 317 (1939 ).
31.
W. H a r r is o n ,
Phys. Rev. B8, 4487 (1973 ).
32.
S. C i r a c i , I . B a tra and W. T i l l e r ,
33.
W. H a r r is o n ,
34.
I . O r te n b u r g e r , S. C i r a c i and I . B a t r a , J. Phys. C. 9^ 4185 (1 976).
35.
S. Pantel id es and W. H a r r is o n , Phys. Rev. BI 1 , 3006 (1975).
36.
W. H a rr is o n and S. C i r a c i , Phys. Rev. B I 0 , 1516 (1 9 7 4 ).
37.
P. G reg ory, W. S p ic e r , S. C i r a c i , W. H a r r is o n , A p p l. Phys. L e t t .
25, 511 (1974).
38.
S. C i r a c i , Ph.D. T he sis (S ta n fo rd U n i v e r s i t y , 1974) u n p u b lis h e d .
39.
T h is c h o ic e o f c o o r d in a te s agrees w i t h t h a t o f Joannopoulos and
Cohen ( r e f e r e n c e 3 ) .
In r e fe re n c e s 9 -1 1 , th e x and y -a x e s have
been in te rc h a n g e d .
Phys. Rev. Bl_2j, 5811 (1975).
S u r f . S c i . 55^, I (1 9 7 6 ).
TT
TT
116
40.
Numerical d i a g o n a l i z a t i o n has been done w i t h th e FORTRAN sub­
r o u t i n e CEIGEN w r i t t e n by W. Bayl i s .
CEIGEN uses th e Jacobi
method extended t o complex, H e rm itia n m a tr ic e s .
I t is a v a ila b le
from Quantum C h em istry Program Exchange.
41.
R. Jones, Phys. Rev. L e t t .
42.
J. van Laar and J. J. Scheer, S u r f . S c i . 8, 342 (1 9 6 7 ).
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A. H u i j e r and J . van L a a r , S u r f . S c i . 52^, 202 (1975 ).
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(1976 ).
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J . A. Knapp, Ph.D. T he sis (Montana S ta te U n i v e r s i t y , 1976)
u n p u b lis h e d .
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F. Herman and S. S k i11 man. Atomic S t r u c tu r e s C a lc u la t io n s
( P r e n t ic e H a l l , New J e r s e y , 1963).
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49.
M. K a w a j i r i , u n p u b lis h e d .
20,
992 (1968 ).
APPENDIX A
118
B u lk m a t r ix elements o f th e b o n d - o r b it a l model o f H a rris o n f o r
z in c b le n d e sem iconductors are presen ted here.
1.
< l,V rn V |H |l,m n > = E ^ ,^ m 'm *n 'n "
A
‘ m( 6n'n+i + * n ' n - i )
+ 5j i , Ji,^6mlm - i 5n l n+i + 6ml m - i 6n l n+i ^
+ 5Jil £,-i5mlm -i6n , n+2 + 5jl, Ji+i5mlm+i6n l n-2^‘
2.
<1,& m n |H|2,&mn> - ~ B 1 6 5- ' 5-61T1' m5 n ' n " B i S£ , s , + i 6m , m + i 5 n , n - 2 -
3.
< l,A 'm 'n '|H |3 , A m n > =
4.
< l,% 'm 'n '|H |4 ,& m n > =
5.
< 2 ,& 'm 'n '|H |2 ,& m n > =
" B i^ '/r n 'm ^ 'n - i'
C ’
A
" Bi * A 'A * m 'm + i * n 'n - i
l5S,'A^rn'm5n 'n ~ B4^-5Ji, 5 ,-i5m 'm - iSn , n+i
+ 5Jil s,+i6ml m+i5n , n - i + lSml m^65,, £ - i 5n , n+i + 6Ji, 2,+i6n , n - i ^
+ 6$ ,'J i- i6ml m - i 6n l n+2 + 6S,1s,+i6ml m +i5n 1n - 2 ^ *
6.
<2,2 m n |H|3,2mn>
Bi 6£,, 5,6m'm'Sn , n " Bi (S£ , £ - i 6m,m - i 6n 'n + i '
7.
< 2 , V m 'n '| H | 4 , t m n > =
8.
< 3 .* 'm 'n '|* )|3 >ann> = V * ' A ' m 6n'n " ^ f V A ' m ^ n ' n + i + V n ' - i >
+ ^ 2 ' 2 + i^ m 'm + i ^ n 'n - i + 5 2 .l 2 - i 6m,m - i 6n l n+i
+ ^ 2 '2 ^ n 'n ( ^ m 'm + i + 6ml m - i ^ "
•
119
9.
< 3 ,V m V |H |4 ,™ > =
1°.
< 4 ,V m 'n '|H |4 ,m n > =
'r, " 6X ' A ' m +16n'n-
+ 6n 'n - , )
+ 5Ji, ^6ml m -i^ 6n l n + ^n'n+i^ +. 6ml m^6j i l Ji+ilSn l n -i + 6Jil Ji-I6I i1H+!
T rT
APPENDIX B
121
M a t r ix elements are presented f o r th e H a m ilto n ia n o f Appendix A
in terms o f l a y e r o r b i t a l s b u i l t on (110) planes.
1.
= SbSm - B4[<sm+2 + 6m _2)
+ (s2 + V s 2X s m rw S1 + Sm n tlZs1) ] .
2.
< M H | 2 , n > = -B^Smn - B \ w
.
3.
<1,m|H|3,n> = -B^Smn - Bjsm w S1S2.
4.
<1 ,m|H|4,n> = -B js mn - Bjsm w S1Zs2 .
5.
<2,m|H|2,n> = ^ s mn - Bt [(S m w + Sm J
+ (s2 + V s 2X s 1Smntl + Smn^ Z s 1) ] .
6.
<2,m|H|3,n> - -B js mn - Bjsm t l S1S2.
7.
<2,m|H|4.n> = -B js m - Bjsm t l S1Zs2 .
8.
<3,mjH|3,n> = EbSmn - B4U sJ + V s J ) Sm
+ ( S 1S2 -M Z s 1S2X s m t l + Sm w ) ] .
9.
<3,n|H |4,n> = -B js m - Bjsm ZsJ.
TT
122
10 .
<4,n|H |4,n> = eb6mn -
+ 1/ s 2)6mn
+ (s i/s g + S2Z s i) (6mn+1 + 5m n -i^ '
APPENDIX C
124
The m a t r ix elements r e q u ir e d in th e t h i n f i l m
c a l c u l a t i o n are
g ive n here in terms o f th e b u l k - t o - s u r f a c e c o u p lin g c o e f f i c i e n t s l i s t e d
in th e t e x t and i l l u s t r a t e d
in f i g u r e 14.
1.
<01H|,2> = g4s i ( s 2 + l / s 2 )
2.
<0|H|3> = g3SiS2
3.
<0|h|4> = ggSi/sg
4.
<01H15> = g3
5.
<01HI 6> = g4
6.
<1 IH12> = gj
7.
< 1 1H13> = g i
8.
<1|H|4> = Qi
9.
<1|H|5> = g2(s 2 + l / s 2) / s i
10 .
<1|H|9> = g2
11 .
<4NIH|4N+1> = g3S2/s1
12.
<4N-1|H|4N+1> = g3/ ( s 1s2)
13.
<4N-3|H|4N+1> = g4(s2 + l / s 2) / s 1
14.
<4N-6|H|4N+1> = g3
15.
<4N-71H14N+1> = g4
16.
<4n-21H14N> = gx
125
17.
<4N-21H14N-1> = 9 l
18.
<4N-31H14N-2> = Q1
19.
<4N -10|H14N-2> = g2
20 .
<4N-6|H|4N-2> = g2 ( s 2
I Zs2 Jsi
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