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Surface states in a simple crystal model
Physica 43 (1969) 1-16 © North-Holland Publishing Co., Amsterdam
S U R F A C E S T A T E S IN A S I M P L E C R Y S T A L M O D E L M. SCHERER and P. PHARISEAU Laboratorium voor Kristallogra[ie en Studie van de Vaste Sto/, Rijksuniversiteit Gent, Belgie" Received 9 October 1968
Synopsis We calculate the surface energy bands in a semi-infinite simple cubic lattice in which the potential is assumed to be constant within non-overlapping spheres surrounding the atoms, and zero in the interspace. Considering the case that the crystal is limited by a (100) plane, numerical calculations showed that the existence of surface states is strongly dependent on the choice of the crystal surface. An analogous conclusion could be made by investigating the case that the crystal boundary is a (110) plane. 1, Introduction. A Green function m e t h o d was used b y E n g e l m a n n 1) to calculate the b a n d s t r u c t u r e in a three-dimensional lattice in which the p o t e n t i a l is supposed to h a v e a c o n s t a n t non-vanishing value in a spherical region a r o u n d each atom, and vanishes outside these non-overlapping spheres. T h e t r e a t m e n t of such simple crystal models m a y be of great i m p o r t a n c e in a t t e m p t i n g to f o r m u l a t e a general m a t h e m a t i c a l method. In this p a p e r we i n v e s t i g a t e the surface states in the semi-infinite E n g e l m a n n model for a cubic crystal. W e find t h a t there exists a two-dimensional b a n d s t r u c t u r e , localized at the surface, and which is superimposed on the three-dimensional b a n d s t r u c t u r e of the bulk. N u m e r i c a l applications yield results which are in complete agreement with those of Guillemin2), i.e. if the p e r t u r b a t i o n due to the existence of a b o u n d a r y increases the distance between the surface level and the e n e r g y b a n d also increases and the degree of localization is better. 2. G e n e r a l / o r m u l a t i o n o / t h e problem. We consider a semi-infinite crystal h a v i n g only one a t o m in a unit cell. F u r t h e r m o r e we restrict our discussion to a crystal p o t e n t i a l of the " m u f f i n - t i n " form, i.e. spherically s y m m e t r i c a b o u t each a t o m within a sphere of radius a, and c o n s t a n t in the space b e t w e e n t h e m . Shifting the zero of the e n e r g y scale in an a p p r o p r i a t e m a n n e r we m a y choose this c o n s t a n t zero. Calling b the distance b e t w e e n the surface plane a n d the plane f o r m e d b y the centers of the atoms of the t o p layer we suppose t h a t b > a. Outside the crystal we assume a c o n s t a n t potential.
2
M. SCHERER AND P. PHARISEAU
L/
Fig. 1. Illustration of the muffin-tin potential used for calculating surface states in a semi-infinite three-dimensional crystal. The origin of a rectangular coordinate system is an arbitrary lattice point. F u r t h e r m o r e we call q the distance from the origin to the b o u n d a r y plane, and u the unit vector along the x axis, normal to the surface and pointed inward the crystal. W o r k i n g in atomic units, the one-electron Schr6dinger equation for the whole system is given b y
[A + E -- U(r)l ~b(r) = 0
(1)
where the potential energy U(r) is defined b y
U(r) =
V0 = constant
for
x + q < 0,
V(r) = • v(]r -- rsl)
for
x+q>O,
(2)
rs
with v(p) = 0
for
p ) a.
(3)
As is well-known from the t h e o r y of inhomogeneous differential equations, eq. (1) is equivalent with the integral equation
~b(r) = Vo ~ g(rlr') ¢(r') dr' + x'+q
~ g(r[r') V(r') ~b(r') dr',
x'+q>O
(4)
SURFACE
STATES
IN A SIMPLE
CRYSTAL
MODEL
3
where g(r It') is the outgoing-wave Green function for the Helmholtz operator A+E; g(rl r') = -- (27r)-~ lim S d K [ K 2 -- (E + ie)] -1 exp[iK- (r -- r')] = s--~+ 0
= --(4n) -1 Jr -- r'1-1 exp iKlr -- r']
(5)
with K=
x/E
for positive energy values E,
ix/-- E
for negative energy values E.
(6)
The wave function ¢(r) in the region outside the crystal can be written x+q<0; ¢(r) = ~ A~j exp i(K2~y + KsF) exp xptj,
(7)
i,i
where K2t, K~I and A ij are still arbitrary, the first two being real and p,j is given by p,j = (Vo + K~i + K~j -- E)L
(8)
Since we are only interested in states which represent an electron bound in the crystal (E < Vo), Plj is real and positive so that $(r) is damped exponentially from the surface away. Substitution of eqs. (5) and (7) in the first integral of the right hand side of eq. (4) yields two integral equations x+q>0; ~(r) = -- Vo ~ Aij[2B,j(p,j + B~j)] -1 exp i(K2,y + K3F) exp(--xB,l)" i,]
•exp[--q(p,~ + B~j)] +
~ g(rlr') V(r') ¢(r')dr',
(9)
x'+q>O
and x+q<0; 0 = - - V o Y, A,j[2Btj(ptj -- Btj)] -1 exp i(K~ly + K3jz) exp xBtt" if
•expE--q(ptj -- B,~)] +
S g(rlr') V(r') •(r') dr',
(lo)
x'+q>O
where we used the shorthand notation
] (K~i + K]i -- E)~ B,j = [ --i(E -- K~, -- K]i)~
if
K~i + K~j -- E > 0,
if
K~, + K]j -- E < 0.
(I I)
As a result of the translation s y m m e t r y of the lattice, the wave function
4
M. SCHERER AND P. PHARISEAU
inside the crystal has the form x+q>0;
¢(r) = u,(r) exp i k . r = ~ Og(k) exp i(k + g).r,
(12)
g
where uk(r ) is periodic with the same periodicity as the lattice, and where the summation ~g runs over all reciprocal lattice vectors g. If k is real we are dealing with the well-known Bloch waves, but when k is complex, the expression (12) defines a Tamm evanescent wave or surface wave. Taking into account the relation (2) of the muffin-tin potential, and the structure of the wave function (12) inside the crystal it is easy to show that
g(rrr') V(r') ~(r') dr' ---- ~ G(r[r') v(r') ~(r') dr',
(13)
r'<_a
x'+q>0
where the structural Green function G(r] r') is defined b y
G(rlr' ) = ~ g(rlr' + rs) exp ik.rs.
(14)
fs
In the right hand side of relation (14) the summation X,. runs over all the translation vectors rsrof the semi-infinite lattice.
3. Application to a simple model. In order to simplify the mathematics we assume from now on that we are dealing with a very special muffin-tin potential, viz. constant within the atomic spheres. Hence v(p) = V = constant
(p _< a).
(15)
On the other hand, if we suppose that the wave function inside the spheres satisfies all the conditions required to apply the mean-value theorem for integrals and taking into account the equality (13), we have
S g(r Jr') V(r') ~(r') dr' = VQG(r I~)
(2 < a),
(16)
x'+q>O
where we used the notation Q = S $(r') dr' # 0.
(17)
r'<_a
According to the definition (14) of the Greenian G(r]r') and the explicit form (5) of g(r, r'), it is easy to show that in a first approximation the following equality holds if [r -- rs] > a :
G(rl~) ~-, G(rlO).
(18)
As shown in appendix I the structural Green function G(r[0) can be expressed as a summation over all the reciprocal lattice vectors g. Using the approximations (16) and (18) and taking into account the results
SURFACE STATES IN A SIMPLE CRYSTAL MODEL
5
obtained in appendix I, eq. (9) and (10) reduce to x+q>0;
~b(r) : --Vo Y, A , j [ 2 B o ~ j + B~j)] -1" i,i
• exp i(K2,y + Kajz) e x p ( - - x B o ) exp[--q(p,i + Bij)] --
- - V Q r-1 X [(k + g)2 _ E]-lexp[i(k + g).r] - g
--iVQr -1 Y, {2C(g.)[(k + g ) . u -- iC(g.)]} -1. g
•expEi(k + g ) . . r -- (x + q) C(g.) -- iq(le + g ) . u ] ,
(19)
and x+q<0;
0 = --Vo Y, A,j[2Btj(p~j -- Bij)] -1" i,i
•exp i(K2,y + Katz) exp xBli exp[--q(p~l -- B,t) ] -- i V Q r - 1 Z {2C(g.)[(k + g ) . u + iC(g±)]} - L g
•exp[i(k + g ) . . r + (x + q) C(g.) --iq(k + g).u].
(20)
As eq. (20) must hold in each point (x, y, z) of the semi-infinite space outside the crystal, this gives the equality (K2,, Ksj) ---- (k + g ) .
(21)
and eq. (20) reduces to
VoA (g.)Ep(g.) -- C(g-)] -1 exp[--qp(g.)] + + iVQr-~ y~ E(k + g ) ' u + iC(g.)] -z e x p E - i q ( k + g).uJ = O, (22) g.u
where according to eq. (21) we changed the notation A tJ --)" A ( g . )
p,j -+ p(g.) = [Vo + (k + g)~, -- E]~.
(23)
On the other hand, if we require that the wave function ~b(r) defined b y eq. (19) presents the structure (12) we get
VoA (g.) EP(g.) + C(g.)1-1 expE--qp(g.)~ + + iVQr-1 y, [(k + g ) . u -- iC(g.)] expt--iq(k + g).u] = 0,
(24)
g.tt
and x+q>0;
4,(r) = - - V Q r -1 Y, [(k + g)2 _ E]-I exp i(k + g).r, g
where we used the notations (23).
(2s)
6
M. S C H E R E R
AND P. PHARISEAU
The condition t h a t the d e t e r m i n a n t of the coefficients of the two linear homogeneous eqs. (22) and (24) in the unknowns A ( g . ) and Q vanishes, gives the equation Y~ [(k 4- g ) . u 4- ip(g.)~[(k 4- g)2 _ E ] - I expE--iqg.u~ = 0.
(26)
g-u
According to the approximation (18) the wave function in the crystal, defined b y eq. (25) is only valid in the region between the atomic spheres. We assume now t h a t inside and just outside these spheres the wave function is spherically symmetric about the lattice point. Hence eq. (25) reduces to ~bout(r) = - - V Q r-1 E [(k 4- g)2 _ E~-I Ilk 4- girl -1 sin [k 4- glr g
(27)
and inside the sphere the wave function is given by
(28)
~bin(r) = D(~r) -1 sin ,~r where h2
=
E --
(29)
V.
The constants D and Q can be evaluated if we express explicitly t h a t the wave function is continuous and has a continuous derivative on the surface of the atomic sphere r = a. This gives a system of two linear homogeneous equations in D and Q. The necessary and sufficient condition t h a t this system has a nonzero solution is given b y h -1 s i n h a E [(k + g)2 __ E ] - I cos [k + g] a = g
= cos Jta ~ Ik + g]-I [(k + g)2 _ E ] - I sin [k + g[ a.
(30)
g
Eq (30) defines the three-dimensional bandstructure. For real values of the wave vector k it is completely equivalent with the result of E n g e l m a n n 1) ~" (Krs)-1 e x p ( - - i k . r s ) exp i~rs = = -- [K sin ,ta cos Ka -- ,t cos ha sin ~a] -1 × × [i~ sin ha -- i cos 2a~ exp iKa, where the s u m m a t i o n E'r. runs over all the translation vectors crystal except the vector rs = 0.
(31) rs
of the infinite
4. Sur/ace states in a simple cubic crystal with a (! 00) sur/ace. Taking into account the choice of the origin of the rectangular coordinate system we have q = b + cm, where m is an integer and c is the lattice constant. Eq.
SURFACE STATES IN A SIMPLE CRYSTAL MODEL
7
(26) reduces now to +.o X [(kl Av 2rOll/C) 2 "3r- (k2 Av 2~12/c) 2 Av (k3 -J- 2~I/3c) 2 - - E l - 1 × 11~ --oo
X {kl + 2nll/C + iVV0 + [k2 + 2~12/c) 2 + (k3 + 2nl3/c) 2 -- EJ~}"
• expt--ibll(2=/c)] = 0,
(32)
where 12 and 13 are two integers. As shown in appendix 2 the sum on the left hand side of eq. (32) can be calculated explicitly• Using this result eq. (32) is equivalent with exp(--iklc) : L(b -- c)/L(b)
(33)
where we introduced the shorthand notation L(a) = [Vo + (k~. + 2~12/c) 2 +
(k3 + 2~13/c) 2 - - El½"
•sinh{aE(k2 + 2nl2/c) 2 + (ks + 2nl3/c) ~ -- E]~} + + E(k2 + 2nl2/c) 2 + (k3 + 2nl3/c) 2 -- E]~"
•cosh{c~[(k2 + 2r:/2/c) 2 -}- (k3 + 2rd3/c) 2 -- El½}.
(34)
As the right hand side of eq. (33) is always real, eq. (33) can generally not be satisfied for real kl values. Separation of real and imaginary parts in eq. (33) gives k l : rd/c -b ifl,
l being an integer;
(35)
(-- 1)1 exp tic : L (b -- c)/L (b).
(36)
On the other hand we can show that eq. (31) still holds for complex k values if 0 < / 5 < x/--E.
(37)
Confining ourself to this case, and taking into account the result (35), eq. (31) becomes +,oo
E(--l)Zexp flc]n,[izcx/n ~ -¢- n~ + n~] -1-
2~ ~1, n2, n3~
--oo
• expE-i (k :
+ k3 8)]
+
+
- -
E# sin ha cosh/,a -- ~ cos ha sinh/~a] -1-
• [4 cos ha + / z sin hal exp(--/za),
(38)
x/--E.
(39)
where ,u :
Using Cauchy's theorem the absolute convergence of these multiple series is straightforward because of eq. (37). In order that eqs. (36) and (38) have
8
M. SCHERER AND P. PHARISEAU
at least one common solution fl the necessary and sufficient condition is t h a t all parameters do satisfy the resultant. The elimination of fl between eqs. (36) and (38) is very easy as it can be performed b y a simple substitution. This gives an equation which can be written formally Fl(k2 + 2~12/c, k3 + 2~13/c, E, V0, V, a, b, c) = 0.
(40)
Eq. (40) defines a two-dimensional b a n d s t r u c t u r e with wave vector (k2, ka) and b a n d i n d e x (12, 13), and which is localized at the surface. The surface states depend on the crystal potential (a and V), the lattice parameter c, the height of the surface barrier V0 and the choice of the crystal surface b. In order to illustrate the influence of this last parameter we calculated numerically the surface states corresponding to a total wave vector k normal to the crystal boundary. As the terms of the left h a n d side of eq. (38) decrease exponentially we only used these terms corresponding with nearest and nextnearest neighbours of the a t o m at the origin. Eq. (38) becomes B = 4Ix -1 exp(--x) + (xx/2)-i
exp(--xx/2)] +
+ (A + A-1)[x -1 exp(--x) + 4(xx/2)-i exp(--x~/2)],
(41)
where we introduced the s h o r t h a n d notations A = (--1) t exp ~c = [x/V0c 2 + x 2 sinh ex + x cosh ex] - 1 . • [x/V0c 2 + x 2 sinh(e -- 1) x + x cosh(e -- 1) x], B = Ex sin ~ , / m - - x2 c o s h x~ - - , / m - - x2 cos ~ , / m - - x2 s i n h x ~ - I × E , / m - - x~ cos ~ , / m - - x2 + x sin ~ 4 m
- - x~l e x p ( - - x ~ )
(42) × (43)
with x = #c;
c -= b/c;
b = a/c;
m = - - V c 2.
(44)
The transcendental eq. (41) is solved numerically for the parameter values Voc 2 = 0.5;
m~ 2 = 81; = 0.10; 0.15; 0.20; 0.25; 0.30; 0.35; 0.40; e
= 8; ~ + 0.05; ~ + 0.10; ... ; 0.90;
10 < x < 100. Not all solutions x of eq. (41) correspond to surface states. Only those which yield a value A, satisfying the condition [A] > 1 are physically acceptable. These are summarized in table I. The bandedges are given for the values ]A[ = 1. F r o m table I it is obvious t h a t large forbidden energy gaps are advan-
SURFACE STATES IN A SIMPLE CRYSTAL MODEL TABLE I Physically acceptable solutions x of e q . ( 4 1 ) f o r various parameter values e and 6. The bandedges for the different cases are also given. I ~
0. I0
0.15
0.20
0.25
0.30
0.10
35.69636828 70.37475558 85.45934298
0.15
35.69638577 70.37475680 85.45934316
23.79755010 46.91650131 56.97289502
0.20
35.69638626 70.37475680 85.45934316
23.79758703 46.91650403 56.97289541
17.84813708 35.18737329 42.72967086
0.25
35.69638628 70.37475680 85.45934316
23.79759057 46.91650405 56.97289542
17.84819593 35.18737803 42.72967155
28.14989494 34.18373651
0.30
35.69638628 70.37475680 85.45934316
23.79759076 46.91650405 56.97289541
-35.18737817 42.72967157
-28.14990210 34.18373758
35.69638628 70.37475680 85.45934316
. 46.91650405 56.97289541
.
0.35
70.37475680 85.45934316
. 56.97289541
. --
--
--
A=+I
35.69638628 70.37475680 85.45934316
23.79759081 46.91650405 56.97289542
17.84821804 35.18737817 43.72967157
14.27943507 28.14990264 34.18373761
II.90831832 23.45827641 28.48644839
A=--I
35.69638628 70.37475680 85.45934316
23.79759077 46.91650405 56.97289542
17.84820153 35.18737817 42.72967158
14.27884588 28.14990249 34.18373761
II.90191771 23.45926014 28.48644795
0.40
. 35.18737817 42.72967156
.
-23.45825582
. -34.18373761
--
.
t a g e o u s for t h e a p p e a r e n c e of s u r f a c e s t a t e s . I n d e e d b y i n c r e a s i n g S - v a l u e s the f o r b i d d e n gaps n a r r o w a n d the surface states disappear. A n o t h e r i n t e r e s t i n g f e a t u r e is t h a t b y i n c r e a s i n g e - v a l u e s t h e s u r f a c e l e v e l s do c o n v e r g e n c e t o t h e b a n d e d g e s . T h i s is i l l u s t r a t e d i n fig. 2, w h e r e we p l o t t e d /~c as a f u n c t i o n of ~. If/~ b e c o m e s zero t h e s u r f a c e s t a t e d i s a p p e a r s a n d does n o t e x i s t a n y m o r e for g r e a t e r e - v a l u e s . T h e s e n u m e r i c a l r e s u l t s p e r m i t u s
10
M. S C H E R E R
AND P. PHARISEAU
60
50
40
30 "\ \
"\. \.
% 'N. \. 20
\
"\ "\
\. \.
"\ "X
"~'x
\.
• •
\
"%.
.~.
\
"x..
~ "~°
,,\ • "\'\. "% X.
1C
%.
0.10
0.20
0,30
0.40
E Fig. 2. T h e d e g r e e o f l o c a l i z a t i o n o f t h e s u r f a c e s t a t e s a s a f u n c t i o n of t h e r e l a t i v e d i s t a n c e of a (100) s u r f a c e p l a n e t o t h e f i r s t a t o m i c l a y e r .
to conclude that the smaller the distance between the surface plane and the plane, formed by the atoms of the top layer (i.e. the greater the deformation of the potential caused by the crystal surface) the larger the distance of the surface level to the bandedge and the degree of localization. These conclusions are in agreement with those obtained by Guillemin2), who used the L.C.A.O. method.
SURFACE STATES IN A SIMPLE CRYSTAL MODEL
11
5. Sur]ace states in a simple cubic crystal with a (110) sur]ace. In order to simplify the calculation we choose the rectangular coordinate system such that g . u ---- 2~(ll + 12)/cx/2, g . = [27z(12 -- ll)/Ca/2 ~ v + (2~13/c) w
(ll, 12, 13 integers),
(46)
where v and w are two m u t u a l l y normal unit vectors parallel with the surface. As in the left-hand side of eq. (26) g± is a constant, and q = mc/~/2 + b (with m an integer), we g e t . +00
X {[(k.u + ~l/cl) + 2~ll/Cll 2 + ll=
--oo
~- ( k . t ) AV 7r.l/Cl) 2 AV (k,~V AV 2,:13/c) 2 -- E} -1 ×
× {(k.tt -- xl/Cl) + 2roll/el + iVV0 +
(k,v
+ rcl/Cl)2 q-
+ (k. w + 2rd3/c) 2 -- Eli} exp(--i2rcbll/cl)
(47)
with Cl = c/~/2, l = 12 -- ll.
Eq. (47) is formally equivalent with eq. (32). Hence we get k . u = 7~m/cl + ifl
(m integer),
(--1) m+1 expflcl = M(b -- Cl)/M(b)
(48)
with M(a) = [Vo + ( k . v + rd/Cl) 2 -J- ( k . w ~- 27zl3/c) 2 --EJ~. • sinh{a[(k.v + ~l/cl) 2 + ( k . w + 2~13/c) 2 -- E l i } + "AV [ ( k , V "~- 7~l/Cl) 2 + (~.iV "AV 27:la/c)2 -- EJt.
• cosh{a[(k.v + =llCl) 2 + ( k . w + 2rd31e)2 -- Eli}.
(49)
Confining ourselves to the case t h a t the inequality (37) is satisfied we m a y use eq. (31) instead of (30): +00
X hi,
n~, n3 =
[ ( - - 1) ra e x p ~c1~ nx+n` [~c4n21 -AV n~ -q- n3~]-1" --¢o
• exp{--i(cl(n2 -- nl) k . v + cksn3]}
exp[-~c~/n~ + n~ + n~]=
= ~/z sin 2a cosh/za -- 2 cos 2a sinh/za] -1. • [2 cos 2a + # sin ~a I exp(--#a).
(50)
The absolute convergence of these multiple series is straightforward b y Cauchy's theorem. After elimination of fl between eqs. (48) and (50) the resultant has the form F 2 ( k . v + rd/Cl; k . w + 2rd3/c; E, Vo, V, a, b, c) = O.
(51)
12
M. S C H E R E R
A N D P. P H A R I S E A U
30
J
20
\ 10
\ %. \.
\ •\ \ \
\
\
"\
%. N. OL 0.10
0.15
0.20
0.25
0.30
Fig. 3. T h e degree of l o c a l i z a t i o n of t h e surface s t a t e s as a f u n c t i o n of t h e r e l a t i v e d i s t a n c e of a (110) surface p l a n e t o t h e first a t o m i c layer.
Eq. (51) defines the two-dimensional surface bands with wave vector (k.v, k.w) and bandindex (l, la). The surface states depend on the same parameters as in the case that the crystal is limited b y a (100) plane. As in the former case we made some numerical calculations under the assumption that the total wave vector k is normal to the crystal surface. To calculate the lefthand side of eq. (50) we confined ourselves to the terms corresponding with nearest and next-nearest neighbours of the atoms at the origin. Eq. (51) gives B = 2[x -1 exp(--x) + (xx/2)-* exp(--xx/2)] + + 2(C + C-1)[x -1 exp(--x) + 2(x~/2) -1 exp(--xx/2)] + + (xx/2) -1 (C 2 + C -2) exp(--xx/2 )
{52)
SURFACE STATES IN A S I M P L E CRYSTAL MODEL
13
TABLE I I Physically acceptable solution x of eq. (52) for various parameter values e and 6. The bandedges for the different cases are also given. 0.10
0.15
0.10
35.69638627 70.37475680 85.45934316
0.15
35.69638628 70.37475680 85.45934316
23.79759069 46.91650405 56.97289542
0.20
35.69638628 70.37475680 85.45934316
46.91650405 56.97289541
-
-
m
0.20
C:-]-I
C=--
1
0.20
-
-
35.18737818 42.72967156 m
70.37475680 85.45934316
56.97289541
35.69638628 70.37475680 85.45934316
23.79759081 46.91650405 56.97289542
17.84821804 35.18737817 42.72967157
35.69638628 70.37475680 85.45934316
23.79759073 46.91650405 56.97289542
17.84818503 35.18737817 42.72967157
-
-
m
w h e r e B is d e f i n e d b y eq. (43) a n d C = ( - - 1 ) m exp(flCl) = [(V0c 2 + x2) ½sinh ex + x cosh ex] -1 × X [(V0c 2 + x2) ½sinh(e - - l / x / 2 ) x + x cosh(e - - l / x / 2 ) x]
(53)
a n d w h e r e we u s e d t h e n o t a t i o n (44). T h e t r a n s c e n d e n t a l e q u a t i o n (52) is s o l v e d n u m e r i c a l l y for t h e p a r a m e t e r v a l u e s (45) (with e < 0.70). T h e p h y s i c a l a c c e p t a b l e s o l u t i o n s are s u m m a r i z e d in t a b l e I I w h e r e t h e b a n d edges are n o w g i v e n for ]CI = 1. F r o m t h e n u m e r i c a l c a l c u l a t i o n we find a n a l o g o u s conclusions as for t h e s i m p l e c u b i c c r y s t a l l i m i t e d b y a (I00) plane, as i l l u s t r a t e d in fig. 3.
14
3/L S C H E R E I ~
AND P. PHARISEAU
APPENDIX
1
As m e n t i o n e d before we can express G(rlO ) as a s u m m a t i o n over all the reciprocal lattice vectors g. For this purpose we use the s t a n d a r d results +00
~(~) = (2n) -1 1 exp(iat) dt, +00
(A 1)
c~+~
/(a) = I / ( t ) 8(t -- o~) dt = I / ( t ) 8(t -- a) at, --co
(A 2)
c6--~
y 0 8(p -- rs) = r -1 ~ exp(ig.p), ts
(A 3)
g j
where the summations y,,0. and ~ , respectively run over all the translation vectors of the infinite crystal lattice and reciprocal lattice• F r o m the definition (14) follows G(r[0) ----=--(27t) -a lim IdKEK2 -- (E + i~)] -1. e--~+0
• ( ~ exp i(k -- K).rs) exp i K . r .
(A 4)
Using successively eqs. (A 2), (A 3) and (A 1), one can easily show t h a t expi(k--K).rs=T r,
-ly~ g
I
dpexpi(k+g--K).p----
p•u+q~O +co
~'-1(27:) 2 ~ BE(k + g ) - -- g ± ] I dp1 exp ipl(k + g -- K ) . u , g
--q
(i
5)
where we introduced the s h o r t h a n d notation (k+g).--K2_=k+g--K--
[u.(k+g--k)]u.
(AS)
If we are dealing with a T a m m wave we know t h a t k. u is complex and t h a t as a consequence of the choice of the rectangular coordinate system I m k . u > 0. Hence Y, exp i(k -- K ) . r s ---- i~"-1 (27~)2 Y~ 8[(k + g ) . -- K j_] × ts
g
× [(k + g -- K ) . u ] -1 exp[--iq(k + g -- K ) . u ] .
(A 7)
If on the contrary k is real (which happens if we are dealing with a Bloch wave) we m a y transform eq. (A 5) b y using the s t a n d a r d result +00
I dt exp(--ita) = exp(--iqa) lim (ia) -1. --q
t--~+oo
• [1 -- exp(--ita)] = exp(-iqa)E~8(a) - i~(1/a)],
(A 8)
where the sign ~ indicates t h a t we need to take the principal value of the integral. The equality (A 5) reduces t h e n to Y~exp i(k-- K ) . rs = ~--1 (2u) 2 y~ (~E(k+ g) j.-- K . exp] iq ( K - - g - - k). u × rs
g
× {~8[(k -t- g - - K ) . u ] - - i ~ [ 1 / ( K - - g - - k ) . u 3 } .
(A 9)
SURFACE STATES IN A SIMPLE CRYSTAL MODEL
15
After substitution of eqs. (A 7) or (A 9) into (A 4) the remaining integration can be performed by the methods of residus and we finally obtain for k real or complex: x+q>O;
G(rlO) = - - r -1 Y~ [(k + g)2 _ E]-I exp i(k + g).t
-
-
g
- - i t -1 Y~ ½C(g.) -1 [(k + g). u -- iC(gx)] -1. g
•exp i(k + g)x.r exp[--(x + q) C(g.)] exp[--iq(k + g).u], (A 10)
x+q
•exp i(k + g)±.r exp[(x + q) C(g±)] exp[--iq(k + g).u],
(A I1)
where we used the notation
C(g±) = / [(k + g)~ -- El½ t --i[E -- (k + g2]½
if if
APPENDIX
(k + g)~ -- E > 0, (k + g)~ -- E < O.
(A 12)
2
The left-hand side of eq. (32) can formely be written +o0
S = ~ [(kl + 2rd[c) 2 + B2]-I [kl + 2nl/c + iA) exp(--ibl2~/c), (A 13) l=--oo
where b, c, A and B2 are real (the first three are even positive), and Im kl is not negative. To evaluate the summation (A 13) we m a y use the property (A 2) and Poisson's summation formula +00
+00
~(t -- 2nl/c) = (c/2n) N g = --OO
~=
exp itnc.
(A 14)
--OO
Hence, +00
+00
S = (c/2~) • n. . . . .
I dt[(t -~- kl) 2 -[- B2] -1 It -~- kl -~- iA] exp[--it(b -- nc)]. ( a 15)
In order to calculate this integral by the method of residus we assume that the distance b between the crystal surface and the plane formed by the centres of the atoms of the top layer is not greater than the distance between two parallel planes going through the centres of two neighbouring layers parallel to the surface. For a (100) and (1 I0) surface this is respectively expressed by the conditions b < c and b < ct.
16
SURFACE STATES IN A SIMPLE CRYSTAL MODEL
If B 2 is positive we have to consider the following two cases:
a) If 0 < I m kl < B, the expression (A 15) reduces after calculation of the residus, to co
S = (ic/2B) exp ibkl{(A + B) exp bB Y~ exp[--nc(B + ikl)] + n=l
0
+ (A -- B) e x p ( - - b e )
Y~ exp[nc(B -- ikl)]}. n~
(A 16)
--oo
As the infinite geometric series in eq. (A 16) converge absolutely we finally obtain S = (ic/2B)(cosh cB -- cos ktc) -1" • exp ibkl{[A sinh bB + B cosh bB] exp(--iklC) -k q- A sinh (c -- b) B -- B cosh(c -- b) B}.
b)
(A 17)
If I m kl > B > 0 the relation (A 15) is equivalent to 0
S = --(ic/2B) exp ibkl{(A + B) exp bB Y, exp[--nc(B + ikl)] -n =
-oo
o
-- (A -- B) exp(--bB)
E
exp[nc(B -- ikl)~.
(A 18)
Taking into account the sum of the two convergent geometric series it is easy to show t h a t eq. (A 18) is equivalent with eq. (A 17). If B 2 is negative, the calculation of S needs some care. P u t t i n g S=
lim S ( s ) = e--~+0
+00
= lim (c/2=) • s-->+ 0
n=
S dt[(t + kl) 2 + --00
+ (B 2 -- ie)] -1 It + kl + iA] exp[it(nc -- b)],
(A 19)
we m a y calculate S(s) as in the former case. We only have to make now a difference between the cases t h a t kl is complex or real. The final result can in b o t h cases be transformed to the form (A 17). Acknowledgement. We wish to t h a n k Professor Dr. W. Dekeyser for his continuous interest and discussions.
REFERENCES 1) E n g e l m a n n , F., Z. Phys. 145 (1956) 430. 2) Guillemin, D., Thesis, U n i v e r s i t y of Lyon, 1968.
S U R F A C E S T A T E S IN A S I M P L E C R Y S T A L M O D E L M. SCHERER and P. PHARISEAU Laboratorium voor Kristallogra[ie en Studie van de Vaste Sto/, Rijksuniversiteit Gent, Belgie" Received 9 October 1968
Synopsis We calculate the surface energy bands in a semi-infinite simple cubic lattice in which the potential is assumed to be constant within non-overlapping spheres surrounding the atoms, and zero in the interspace. Considering the case that the crystal is limited by a (100) plane, numerical calculations showed that the existence of surface states is strongly dependent on the choice of the crystal surface. An analogous conclusion could be made by investigating the case that the crystal boundary is a (110) plane. 1, Introduction. A Green function m e t h o d was used b y E n g e l m a n n 1) to calculate the b a n d s t r u c t u r e in a three-dimensional lattice in which the p o t e n t i a l is supposed to h a v e a c o n s t a n t non-vanishing value in a spherical region a r o u n d each atom, and vanishes outside these non-overlapping spheres. T h e t r e a t m e n t of such simple crystal models m a y be of great i m p o r t a n c e in a t t e m p t i n g to f o r m u l a t e a general m a t h e m a t i c a l method. In this p a p e r we i n v e s t i g a t e the surface states in the semi-infinite E n g e l m a n n model for a cubic crystal. W e find t h a t there exists a two-dimensional b a n d s t r u c t u r e , localized at the surface, and which is superimposed on the three-dimensional b a n d s t r u c t u r e of the bulk. N u m e r i c a l applications yield results which are in complete agreement with those of Guillemin2), i.e. if the p e r t u r b a t i o n due to the existence of a b o u n d a r y increases the distance between the surface level and the e n e r g y b a n d also increases and the degree of localization is better. 2. G e n e r a l / o r m u l a t i o n o / t h e problem. We consider a semi-infinite crystal h a v i n g only one a t o m in a unit cell. F u r t h e r m o r e we restrict our discussion to a crystal p o t e n t i a l of the " m u f f i n - t i n " form, i.e. spherically s y m m e t r i c a b o u t each a t o m within a sphere of radius a, and c o n s t a n t in the space b e t w e e n t h e m . Shifting the zero of the e n e r g y scale in an a p p r o p r i a t e m a n n e r we m a y choose this c o n s t a n t zero. Calling b the distance b e t w e e n the surface plane a n d the plane f o r m e d b y the centers of the atoms of the t o p layer we suppose t h a t b > a. Outside the crystal we assume a c o n s t a n t potential.
2
M. SCHERER AND P. PHARISEAU
L/
Fig. 1. Illustration of the muffin-tin potential used for calculating surface states in a semi-infinite three-dimensional crystal. The origin of a rectangular coordinate system is an arbitrary lattice point. F u r t h e r m o r e we call q the distance from the origin to the b o u n d a r y plane, and u the unit vector along the x axis, normal to the surface and pointed inward the crystal. W o r k i n g in atomic units, the one-electron Schr6dinger equation for the whole system is given b y
[A + E -- U(r)l ~b(r) = 0
(1)
where the potential energy U(r) is defined b y
U(r) =
V0 = constant
for
x + q < 0,
V(r) = • v(]r -- rsl)
for
x+q>O,
(2)
rs
with v(p) = 0
for
p ) a.
(3)
As is well-known from the t h e o r y of inhomogeneous differential equations, eq. (1) is equivalent with the integral equation
~b(r) = Vo ~ g(rlr') ¢(r') dr' + x'+q
~ g(r[r') V(r') ~b(r') dr',
x'+q>O
(4)
SURFACE
STATES
IN A SIMPLE
CRYSTAL
MODEL
3
where g(r It') is the outgoing-wave Green function for the Helmholtz operator A+E; g(rl r') = -- (27r)-~ lim S d K [ K 2 -- (E + ie)] -1 exp[iK- (r -- r')] = s--~+ 0
= --(4n) -1 Jr -- r'1-1 exp iKlr -- r']
(5)
with K=
x/E
for positive energy values E,
ix/-- E
for negative energy values E.
(6)
The wave function ¢(r) in the region outside the crystal can be written x+q<0; ¢(r) = ~ A~j exp i(K2~y + KsF) exp xptj,
(7)
i,i
where K2t, K~I and A ij are still arbitrary, the first two being real and p,j is given by p,j = (Vo + K~i + K~j -- E)L
(8)
Since we are only interested in states which represent an electron bound in the crystal (E < Vo), Plj is real and positive so that $(r) is damped exponentially from the surface away. Substitution of eqs. (5) and (7) in the first integral of the right hand side of eq. (4) yields two integral equations x+q>0; ~(r) = -- Vo ~ Aij[2B,j(p,j + B~j)] -1 exp i(K2,y + K3F) exp(--xB,l)" i,]
•exp[--q(p,~ + B~j)] +
~ g(rlr') V(r') ¢(r')dr',
(9)
x'+q>O
and x+q<0; 0 = - - V o Y, A,j[2Btj(ptj -- Btj)] -1 exp i(K~ly + K3jz) exp xBtt" if
•expE--q(ptj -- B,~)] +
S g(rlr') V(r') •(r') dr',
(lo)
x'+q>O
where we used the shorthand notation
] (K~i + K]i -- E)~ B,j = [ --i(E -- K~, -- K]i)~
if
K~i + K~j -- E > 0,
if
K~, + K]j -- E < 0.
(I I)
As a result of the translation s y m m e t r y of the lattice, the wave function
4
M. SCHERER AND P. PHARISEAU
inside the crystal has the form x+q>0;
¢(r) = u,(r) exp i k . r = ~ Og(k) exp i(k + g).r,
(12)
g
where uk(r ) is periodic with the same periodicity as the lattice, and where the summation ~g runs over all reciprocal lattice vectors g. If k is real we are dealing with the well-known Bloch waves, but when k is complex, the expression (12) defines a Tamm evanescent wave or surface wave. Taking into account the relation (2) of the muffin-tin potential, and the structure of the wave function (12) inside the crystal it is easy to show that
g(rrr') V(r') ~(r') dr' ---- ~ G(r[r') v(r') ~(r') dr',
(13)
r'<_a
x'+q>0
where the structural Green function G(r] r') is defined b y
G(rlr' ) = ~ g(rlr' + rs) exp ik.rs.
(14)
fs
In the right hand side of relation (14) the summation X,. runs over all the translation vectors rsrof the semi-infinite lattice.
3. Application to a simple model. In order to simplify the mathematics we assume from now on that we are dealing with a very special muffin-tin potential, viz. constant within the atomic spheres. Hence v(p) = V = constant
(p _< a).
(15)
On the other hand, if we suppose that the wave function inside the spheres satisfies all the conditions required to apply the mean-value theorem for integrals and taking into account the equality (13), we have
S g(r Jr') V(r') ~(r') dr' = VQG(r I~)
(2 < a),
(16)
x'+q>O
where we used the notation Q = S $(r') dr' # 0.
(17)
r'<_a
According to the definition (14) of the Greenian G(r]r') and the explicit form (5) of g(r, r'), it is easy to show that in a first approximation the following equality holds if [r -- rs] > a :
G(rl~) ~-, G(rlO).
(18)
As shown in appendix I the structural Green function G(r[0) can be expressed as a summation over all the reciprocal lattice vectors g. Using the approximations (16) and (18) and taking into account the results
SURFACE STATES IN A SIMPLE CRYSTAL MODEL
5
obtained in appendix I, eq. (9) and (10) reduce to x+q>0;
~b(r) : --Vo Y, A , j [ 2 B o ~ j + B~j)] -1" i,i
• exp i(K2,y + Kajz) e x p ( - - x B o ) exp[--q(p,i + Bij)] --
- - V Q r-1 X [(k + g)2 _ E]-lexp[i(k + g).r] - g
--iVQr -1 Y, {2C(g.)[(k + g ) . u -- iC(g.)]} -1. g
•expEi(k + g ) . . r -- (x + q) C(g.) -- iq(le + g ) . u ] ,
(19)
and x+q<0;
0 = --Vo Y, A,j[2Btj(p~j -- Bij)] -1" i,i
•exp i(K2,y + Katz) exp xBli exp[--q(p~l -- B,t) ] -- i V Q r - 1 Z {2C(g.)[(k + g ) . u + iC(g±)]} - L g
•exp[i(k + g ) . . r + (x + q) C(g.) --iq(k + g).u].
(20)
As eq. (20) must hold in each point (x, y, z) of the semi-infinite space outside the crystal, this gives the equality (K2,, Ksj) ---- (k + g ) .
(21)
and eq. (20) reduces to
VoA (g.)Ep(g.) -- C(g-)] -1 exp[--qp(g.)] + + iVQr-~ y~ E(k + g ) ' u + iC(g.)] -z e x p E - i q ( k + g).uJ = O, (22) g.u
where according to eq. (21) we changed the notation A tJ --)" A ( g . )
p,j -+ p(g.) = [Vo + (k + g)~, -- E]~.
(23)
On the other hand, if we require that the wave function ~b(r) defined b y eq. (19) presents the structure (12) we get
VoA (g.) EP(g.) + C(g.)1-1 expE--qp(g.)~ + + iVQr-1 y, [(k + g ) . u -- iC(g.)] expt--iq(k + g).u] = 0,
(24)
g.tt
and x+q>0;
4,(r) = - - V Q r -1 Y, [(k + g)2 _ E]-I exp i(k + g).r, g
where we used the notations (23).
(2s)
6
M. S C H E R E R
AND P. PHARISEAU
The condition t h a t the d e t e r m i n a n t of the coefficients of the two linear homogeneous eqs. (22) and (24) in the unknowns A ( g . ) and Q vanishes, gives the equation Y~ [(k 4- g ) . u 4- ip(g.)~[(k 4- g)2 _ E ] - I expE--iqg.u~ = 0.
(26)
g-u
According to the approximation (18) the wave function in the crystal, defined b y eq. (25) is only valid in the region between the atomic spheres. We assume now t h a t inside and just outside these spheres the wave function is spherically symmetric about the lattice point. Hence eq. (25) reduces to ~bout(r) = - - V Q r-1 E [(k 4- g)2 _ E~-I Ilk 4- girl -1 sin [k 4- glr g
(27)
and inside the sphere the wave function is given by
(28)
~bin(r) = D(~r) -1 sin ,~r where h2
=
E --
(29)
V.
The constants D and Q can be evaluated if we express explicitly t h a t the wave function is continuous and has a continuous derivative on the surface of the atomic sphere r = a. This gives a system of two linear homogeneous equations in D and Q. The necessary and sufficient condition t h a t this system has a nonzero solution is given b y h -1 s i n h a E [(k + g)2 __ E ] - I cos [k + g] a = g
= cos Jta ~ Ik + g]-I [(k + g)2 _ E ] - I sin [k + g[ a.
(30)
g
Eq (30) defines the three-dimensional bandstructure. For real values of the wave vector k it is completely equivalent with the result of E n g e l m a n n 1) ~" (Krs)-1 e x p ( - - i k . r s ) exp i~rs = = -- [K sin ,ta cos Ka -- ,t cos ha sin ~a] -1 × × [i~ sin ha -- i cos 2a~ exp iKa, where the s u m m a t i o n E'r. runs over all the translation vectors crystal except the vector rs = 0.
(31) rs
of the infinite
4. Sur/ace states in a simple cubic crystal with a (! 00) sur/ace. Taking into account the choice of the origin of the rectangular coordinate system we have q = b + cm, where m is an integer and c is the lattice constant. Eq.
SURFACE STATES IN A SIMPLE CRYSTAL MODEL
7
(26) reduces now to +.o X [(kl Av 2rOll/C) 2 "3r- (k2 Av 2~12/c) 2 Av (k3 -J- 2~I/3c) 2 - - E l - 1 × 11~ --oo
X {kl + 2nll/C + iVV0 + [k2 + 2~12/c) 2 + (k3 + 2nl3/c) 2 -- EJ~}"
• expt--ibll(2=/c)] = 0,
(32)
where 12 and 13 are two integers. As shown in appendix 2 the sum on the left hand side of eq. (32) can be calculated explicitly• Using this result eq. (32) is equivalent with exp(--iklc) : L(b -- c)/L(b)
(33)
where we introduced the shorthand notation L(a) = [Vo + (k~. + 2~12/c) 2 +
(k3 + 2~13/c) 2 - - El½"
•sinh{aE(k2 + 2nl2/c) 2 + (ks + 2nl3/c) ~ -- E]~} + + E(k2 + 2nl2/c) 2 + (k3 + 2nl3/c) 2 -- E]~"
•cosh{c~[(k2 + 2r:/2/c) 2 -}- (k3 + 2rd3/c) 2 -- El½}.
(34)
As the right hand side of eq. (33) is always real, eq. (33) can generally not be satisfied for real kl values. Separation of real and imaginary parts in eq. (33) gives k l : rd/c -b ifl,
l being an integer;
(35)
(-- 1)1 exp tic : L (b -- c)/L (b).
(36)
On the other hand we can show that eq. (31) still holds for complex k values if 0 < / 5 < x/--E.
(37)
Confining ourself to this case, and taking into account the result (35), eq. (31) becomes +,oo
E(--l)Zexp flc]n,[izcx/n ~ -¢- n~ + n~] -1-
2~ ~1, n2, n3~
--oo
• expE-i (k :
+ k3 8)]
+
+
- -
E# sin ha cosh/,a -- ~ cos ha sinh/~a] -1-
• [4 cos ha + / z sin hal exp(--/za),
(38)
x/--E.
(39)
where ,u :
Using Cauchy's theorem the absolute convergence of these multiple series is straightforward because of eq. (37). In order that eqs. (36) and (38) have
8
M. SCHERER AND P. PHARISEAU
at least one common solution fl the necessary and sufficient condition is t h a t all parameters do satisfy the resultant. The elimination of fl between eqs. (36) and (38) is very easy as it can be performed b y a simple substitution. This gives an equation which can be written formally Fl(k2 + 2~12/c, k3 + 2~13/c, E, V0, V, a, b, c) = 0.
(40)
Eq. (40) defines a two-dimensional b a n d s t r u c t u r e with wave vector (k2, ka) and b a n d i n d e x (12, 13), and which is localized at the surface. The surface states depend on the crystal potential (a and V), the lattice parameter c, the height of the surface barrier V0 and the choice of the crystal surface b. In order to illustrate the influence of this last parameter we calculated numerically the surface states corresponding to a total wave vector k normal to the crystal boundary. As the terms of the left h a n d side of eq. (38) decrease exponentially we only used these terms corresponding with nearest and nextnearest neighbours of the a t o m at the origin. Eq. (38) becomes B = 4Ix -1 exp(--x) + (xx/2)-i
exp(--xx/2)] +
+ (A + A-1)[x -1 exp(--x) + 4(xx/2)-i exp(--x~/2)],
(41)
where we introduced the s h o r t h a n d notations A = (--1) t exp ~c = [x/V0c 2 + x 2 sinh ex + x cosh ex] - 1 . • [x/V0c 2 + x 2 sinh(e -- 1) x + x cosh(e -- 1) x], B = Ex sin ~ , / m - - x2 c o s h x~ - - , / m - - x2 cos ~ , / m - - x2 s i n h x ~ - I × E , / m - - x~ cos ~ , / m - - x2 + x sin ~ 4 m
- - x~l e x p ( - - x ~ )
(42) × (43)
with x = #c;
c -= b/c;
b = a/c;
m = - - V c 2.
(44)
The transcendental eq. (41) is solved numerically for the parameter values Voc 2 = 0.5;
m~ 2 = 81; = 0.10; 0.15; 0.20; 0.25; 0.30; 0.35; 0.40; e
= 8; ~ + 0.05; ~ + 0.10; ... ; 0.90;
10 < x < 100. Not all solutions x of eq. (41) correspond to surface states. Only those which yield a value A, satisfying the condition [A] > 1 are physically acceptable. These are summarized in table I. The bandedges are given for the values ]A[ = 1. F r o m table I it is obvious t h a t large forbidden energy gaps are advan-
SURFACE STATES IN A SIMPLE CRYSTAL MODEL TABLE I Physically acceptable solutions x of e q . ( 4 1 ) f o r various parameter values e and 6. The bandedges for the different cases are also given. I ~
0. I0
0.15
0.20
0.25
0.30
0.10
35.69636828 70.37475558 85.45934298
0.15
35.69638577 70.37475680 85.45934316
23.79755010 46.91650131 56.97289502
0.20
35.69638626 70.37475680 85.45934316
23.79758703 46.91650403 56.97289541
17.84813708 35.18737329 42.72967086
0.25
35.69638628 70.37475680 85.45934316
23.79759057 46.91650405 56.97289542
17.84819593 35.18737803 42.72967155
28.14989494 34.18373651
0.30
35.69638628 70.37475680 85.45934316
23.79759076 46.91650405 56.97289541
-35.18737817 42.72967157
-28.14990210 34.18373758
35.69638628 70.37475680 85.45934316
. 46.91650405 56.97289541
.
0.35
70.37475680 85.45934316
. 56.97289541
. --
--
--
A=+I
35.69638628 70.37475680 85.45934316
23.79759081 46.91650405 56.97289542
17.84821804 35.18737817 43.72967157
14.27943507 28.14990264 34.18373761
II.90831832 23.45827641 28.48644839
A=--I
35.69638628 70.37475680 85.45934316
23.79759077 46.91650405 56.97289542
17.84820153 35.18737817 42.72967158
14.27884588 28.14990249 34.18373761
II.90191771 23.45926014 28.48644795
0.40
. 35.18737817 42.72967156
.
-23.45825582
. -34.18373761
--
.
t a g e o u s for t h e a p p e a r e n c e of s u r f a c e s t a t e s . I n d e e d b y i n c r e a s i n g S - v a l u e s the f o r b i d d e n gaps n a r r o w a n d the surface states disappear. A n o t h e r i n t e r e s t i n g f e a t u r e is t h a t b y i n c r e a s i n g e - v a l u e s t h e s u r f a c e l e v e l s do c o n v e r g e n c e t o t h e b a n d e d g e s . T h i s is i l l u s t r a t e d i n fig. 2, w h e r e we p l o t t e d /~c as a f u n c t i o n of ~. If/~ b e c o m e s zero t h e s u r f a c e s t a t e d i s a p p e a r s a n d does n o t e x i s t a n y m o r e for g r e a t e r e - v a l u e s . T h e s e n u m e r i c a l r e s u l t s p e r m i t u s
10
M. S C H E R E R
AND P. PHARISEAU
60
50
40
30 "\ \
"\. \.
% 'N. \. 20
\
"\ "\
\. \.
"\ "X
"~'x
\.
• •
\
"%.
.~.
\
"x..
~ "~°
,,\ • "\'\. "% X.
1C
%.
0.10
0.20
0,30
0.40
E Fig. 2. T h e d e g r e e o f l o c a l i z a t i o n o f t h e s u r f a c e s t a t e s a s a f u n c t i o n of t h e r e l a t i v e d i s t a n c e of a (100) s u r f a c e p l a n e t o t h e f i r s t a t o m i c l a y e r .
to conclude that the smaller the distance between the surface plane and the plane, formed by the atoms of the top layer (i.e. the greater the deformation of the potential caused by the crystal surface) the larger the distance of the surface level to the bandedge and the degree of localization. These conclusions are in agreement with those obtained by Guillemin2), who used the L.C.A.O. method.
SURFACE STATES IN A SIMPLE CRYSTAL MODEL
11
5. Sur]ace states in a simple cubic crystal with a (110) sur]ace. In order to simplify the calculation we choose the rectangular coordinate system such that g . u ---- 2~(ll + 12)/cx/2, g . = [27z(12 -- ll)/Ca/2 ~ v + (2~13/c) w
(ll, 12, 13 integers),
(46)
where v and w are two m u t u a l l y normal unit vectors parallel with the surface. As in the left-hand side of eq. (26) g± is a constant, and q = mc/~/2 + b (with m an integer), we g e t . +00
X {[(k.u + ~l/cl) + 2~ll/Cll 2 + ll=
--oo
~- ( k . t ) AV 7r.l/Cl) 2 AV (k,~V AV 2,:13/c) 2 -- E} -1 ×
× {(k.tt -- xl/Cl) + 2roll/el + iVV0 +
(k,v
+ rcl/Cl)2 q-
+ (k. w + 2rd3/c) 2 -- Eli} exp(--i2rcbll/cl)
(47)
with Cl = c/~/2, l = 12 -- ll.
Eq. (47) is formally equivalent with eq. (32). Hence we get k . u = 7~m/cl + ifl
(m integer),
(--1) m+1 expflcl = M(b -- Cl)/M(b)
(48)
with M(a) = [Vo + ( k . v + rd/Cl) 2 -J- ( k . w ~- 27zl3/c) 2 --EJ~. • sinh{a[(k.v + ~l/cl) 2 + ( k . w + 2~13/c) 2 -- E l i } + "AV [ ( k , V "~- 7~l/Cl) 2 + (~.iV "AV 27:la/c)2 -- EJt.
• cosh{a[(k.v + =llCl) 2 + ( k . w + 2rd31e)2 -- Eli}.
(49)
Confining ourselves to the case t h a t the inequality (37) is satisfied we m a y use eq. (31) instead of (30): +00
X hi,
n~, n3 =
[ ( - - 1) ra e x p ~c1~ nx+n` [~c4n21 -AV n~ -q- n3~]-1" --¢o
• exp{--i(cl(n2 -- nl) k . v + cksn3]}
exp[-~c~/n~ + n~ + n~]=
= ~/z sin 2a cosh/za -- 2 cos 2a sinh/za] -1. • [2 cos 2a + # sin ~a I exp(--#a).
(50)
The absolute convergence of these multiple series is straightforward b y Cauchy's theorem. After elimination of fl between eqs. (48) and (50) the resultant has the form F 2 ( k . v + rd/Cl; k . w + 2rd3/c; E, Vo, V, a, b, c) = O.
(51)
12
M. S C H E R E R
A N D P. P H A R I S E A U
30
J
20
\ 10
\ %. \.
\ •\ \ \
\
\
"\
%. N. OL 0.10
0.15
0.20
0.25
0.30
Fig. 3. T h e degree of l o c a l i z a t i o n of t h e surface s t a t e s as a f u n c t i o n of t h e r e l a t i v e d i s t a n c e of a (110) surface p l a n e t o t h e first a t o m i c layer.
Eq. (51) defines the two-dimensional surface bands with wave vector (k.v, k.w) and bandindex (l, la). The surface states depend on the same parameters as in the case that the crystal is limited b y a (100) plane. As in the former case we made some numerical calculations under the assumption that the total wave vector k is normal to the crystal surface. To calculate the lefthand side of eq. (50) we confined ourselves to the terms corresponding with nearest and next-nearest neighbours of the atoms at the origin. Eq. (51) gives B = 2[x -1 exp(--x) + (xx/2)-* exp(--xx/2)] + + 2(C + C-1)[x -1 exp(--x) + 2(x~/2) -1 exp(--xx/2)] + + (xx/2) -1 (C 2 + C -2) exp(--xx/2 )
{52)
SURFACE STATES IN A S I M P L E CRYSTAL MODEL
13
TABLE I I Physically acceptable solution x of eq. (52) for various parameter values e and 6. The bandedges for the different cases are also given. 0.10
0.15
0.10
35.69638627 70.37475680 85.45934316
0.15
35.69638628 70.37475680 85.45934316
23.79759069 46.91650405 56.97289542
0.20
35.69638628 70.37475680 85.45934316
46.91650405 56.97289541
-
-
m
0.20
C:-]-I
C=--
1
0.20
-
-
35.18737818 42.72967156 m
70.37475680 85.45934316
56.97289541
35.69638628 70.37475680 85.45934316
23.79759081 46.91650405 56.97289542
17.84821804 35.18737817 42.72967157
35.69638628 70.37475680 85.45934316
23.79759073 46.91650405 56.97289542
17.84818503 35.18737817 42.72967157
-
-
m
w h e r e B is d e f i n e d b y eq. (43) a n d C = ( - - 1 ) m exp(flCl) = [(V0c 2 + x2) ½sinh ex + x cosh ex] -1 × X [(V0c 2 + x2) ½sinh(e - - l / x / 2 ) x + x cosh(e - - l / x / 2 ) x]
(53)
a n d w h e r e we u s e d t h e n o t a t i o n (44). T h e t r a n s c e n d e n t a l e q u a t i o n (52) is s o l v e d n u m e r i c a l l y for t h e p a r a m e t e r v a l u e s (45) (with e < 0.70). T h e p h y s i c a l a c c e p t a b l e s o l u t i o n s are s u m m a r i z e d in t a b l e I I w h e r e t h e b a n d edges are n o w g i v e n for ]CI = 1. F r o m t h e n u m e r i c a l c a l c u l a t i o n we find a n a l o g o u s conclusions as for t h e s i m p l e c u b i c c r y s t a l l i m i t e d b y a (I00) plane, as i l l u s t r a t e d in fig. 3.
14
3/L S C H E R E I ~
AND P. PHARISEAU
APPENDIX
1
As m e n t i o n e d before we can express G(rlO ) as a s u m m a t i o n over all the reciprocal lattice vectors g. For this purpose we use the s t a n d a r d results +00
~(~) = (2n) -1 1 exp(iat) dt, +00
(A 1)
c~+~
/(a) = I / ( t ) 8(t -- o~) dt = I / ( t ) 8(t -- a) at, --co
(A 2)
c6--~
y 0 8(p -- rs) = r -1 ~ exp(ig.p), ts
(A 3)
g j
where the summations y,,0. and ~ , respectively run over all the translation vectors of the infinite crystal lattice and reciprocal lattice• F r o m the definition (14) follows G(r[0) ----=--(27t) -a lim IdKEK2 -- (E + i~)] -1. e--~+0
• ( ~ exp i(k -- K).rs) exp i K . r .
(A 4)
Using successively eqs. (A 2), (A 3) and (A 1), one can easily show t h a t expi(k--K).rs=T r,
-ly~ g
I
dpexpi(k+g--K).p----
p•u+q~O +co
~'-1(27:) 2 ~ BE(k + g ) - -- g ± ] I dp1 exp ipl(k + g -- K ) . u , g
--q
(i
5)
where we introduced the s h o r t h a n d notation (k+g).--K2_=k+g--K--
[u.(k+g--k)]u.
(AS)
If we are dealing with a T a m m wave we know t h a t k. u is complex and t h a t as a consequence of the choice of the rectangular coordinate system I m k . u > 0. Hence Y, exp i(k -- K ) . r s ---- i~"-1 (27~)2 Y~ 8[(k + g ) . -- K j_] × ts
g
× [(k + g -- K ) . u ] -1 exp[--iq(k + g -- K ) . u ] .
(A 7)
If on the contrary k is real (which happens if we are dealing with a Bloch wave) we m a y transform eq. (A 5) b y using the s t a n d a r d result +00
I dt exp(--ita) = exp(--iqa) lim (ia) -1. --q
t--~+oo
• [1 -- exp(--ita)] = exp(-iqa)E~8(a) - i~(1/a)],
(A 8)
where the sign ~ indicates t h a t we need to take the principal value of the integral. The equality (A 5) reduces t h e n to Y~exp i(k-- K ) . rs = ~--1 (2u) 2 y~ (~E(k+ g) j.-- K . exp] iq ( K - - g - - k). u × rs
g
× {~8[(k -t- g - - K ) . u ] - - i ~ [ 1 / ( K - - g - - k ) . u 3 } .
(A 9)
SURFACE STATES IN A SIMPLE CRYSTAL MODEL
15
After substitution of eqs. (A 7) or (A 9) into (A 4) the remaining integration can be performed by the methods of residus and we finally obtain for k real or complex: x+q>O;
G(rlO) = - - r -1 Y~ [(k + g)2 _ E]-I exp i(k + g).t
-
-
g
- - i t -1 Y~ ½C(g.) -1 [(k + g). u -- iC(gx)] -1. g
•exp i(k + g)x.r exp[--(x + q) C(g.)] exp[--iq(k + g).u], (A 10)
x+q
•exp i(k + g)±.r exp[(x + q) C(g±)] exp[--iq(k + g).u],
(A I1)
where we used the notation
C(g±) = / [(k + g)~ -- El½ t --i[E -- (k + g2]½
if if
APPENDIX
(k + g)~ -- E > 0, (k + g)~ -- E < O.
(A 12)
2
The left-hand side of eq. (32) can formely be written +o0
S = ~ [(kl + 2rd[c) 2 + B2]-I [kl + 2nl/c + iA) exp(--ibl2~/c), (A 13) l=--oo
where b, c, A and B2 are real (the first three are even positive), and Im kl is not negative. To evaluate the summation (A 13) we m a y use the property (A 2) and Poisson's summation formula +00
+00
~(t -- 2nl/c) = (c/2n) N g = --OO
~=
exp itnc.
(A 14)
--OO
Hence, +00
+00
S = (c/2~) • n. . . . .
I dt[(t -~- kl) 2 -[- B2] -1 It -~- kl -~- iA] exp[--it(b -- nc)]. ( a 15)
In order to calculate this integral by the method of residus we assume that the distance b between the crystal surface and the plane formed by the centres of the atoms of the top layer is not greater than the distance between two parallel planes going through the centres of two neighbouring layers parallel to the surface. For a (100) and (1 I0) surface this is respectively expressed by the conditions b < c and b < ct.
16
SURFACE STATES IN A SIMPLE CRYSTAL MODEL
If B 2 is positive we have to consider the following two cases:
a) If 0 < I m kl < B, the expression (A 15) reduces after calculation of the residus, to co
S = (ic/2B) exp ibkl{(A + B) exp bB Y~ exp[--nc(B + ikl)] + n=l
0
+ (A -- B) e x p ( - - b e )
Y~ exp[nc(B -- ikl)]}. n~
(A 16)
--oo
As the infinite geometric series in eq. (A 16) converge absolutely we finally obtain S = (ic/2B)(cosh cB -- cos ktc) -1" • exp ibkl{[A sinh bB + B cosh bB] exp(--iklC) -k q- A sinh (c -- b) B -- B cosh(c -- b) B}.
b)
(A 17)
If I m kl > B > 0 the relation (A 15) is equivalent to 0
S = --(ic/2B) exp ibkl{(A + B) exp bB Y, exp[--nc(B + ikl)] -n =
-oo
o
-- (A -- B) exp(--bB)
E
exp[nc(B -- ikl)~.
(A 18)
Taking into account the sum of the two convergent geometric series it is easy to show t h a t eq. (A 18) is equivalent with eq. (A 17). If B 2 is negative, the calculation of S needs some care. P u t t i n g S=
lim S ( s ) = e--~+0
+00
= lim (c/2=) • s-->+ 0
n=
S dt[(t + kl) 2 + --00
+ (B 2 -- ie)] -1 It + kl + iA] exp[it(nc -- b)],
(A 19)
we m a y calculate S(s) as in the former case. We only have to make now a difference between the cases t h a t kl is complex or real. The final result can in b o t h cases be transformed to the form (A 17). Acknowledgement. We wish to t h a n k Professor Dr. W. Dekeyser for his continuous interest and discussions.
REFERENCES 1) E n g e l m a n n , F., Z. Phys. 145 (1956) 430. 2) Guillemin, D., Thesis, U n i v e r s i t y of Lyon, 1968.