Electron surface states and surface entropy for a simple cubic lattice

SURFACE

SCIENCE

ELECTRON

15 (1969) 101-108 0 North-Holland

SURFACE

STATES

FOR A SIMPLE

AND

CUBIC

Publishing Co., Amsterdam

SURFACE

ENTROPY

LATTICE

G. ALLAN and P. LENGLART Institut

Suptrieur d’Electronique du Nord, Physique 3, Rue Francois Baes, Lille, France

Received 17 December

des Solides,

1968

We study the surface states for a simple cubic lattice in tight-binding approximation. The Green function method, already successfully employed for the study of localized defects and surface lattice vibrations, is used to determine the variation of the density of states due to a [OOl] surface. Then, the variation of the electronic specific heat and surface entropy are calculated for a half-filled band. Among many methods used to study the effect of surfaces on electron and phonon states, the Green function method, seeing how it is profitable for the study of localized defects, seems one of the most interesting. We use it with the tight-binding approximation to calculate the electron density of states for a finite simple cubic lattice in section 1. On the other hand, we show in section 2 that the electron surface entropy can be not negligible as this was already assumedr).

1. Variation of the density of states for a [OOl] surface We shall use the formalism of Brown2) who applies the Green function method to the extended defects. In the tight-binding approximation for a non-degenerate band (typically an s band) of an infinite crystal, Bloch waves $k are written as a sum over all the lattice sites R, of the atomic functions 4(v - R,) ijk = $

eik.Red(r

- R,).

(1)

c R. They are eigenvalues of the one electron hamiltonian of a simple cubic crystal for the energy E(k) which is, if we take only the nearest neighbourss) E(k)=6+2/?cosk,a,

(2)

where d=E,+cr-2y(cosk,a+cosk,a) 101

and

p=-2~.

(3)

102

0. ALLAN

AND

P. LENGLART

As Brown2), we shall define now the Wannier functions ]R,, Y, k,) localized in a plane [OOl] parallel to the surface; R, is the distance from the plane to the surface and k, is the component of the wave vector k parallel to the surface 1 IR,, r, k,) = N+ $k exp(ik;R,). (4) c kZ In this basis, the matrix

elements

of the Green

G(L R,, K) = 4, r, k,

function

Gz are

lE_$ -+<1 R:, r, k, > 0

1 =N

c k:

and those of the surface potential V(k,,

exp Cikz (R, E-E(k)+is

WI

’

(5)

V(r)

R,, R:) = CR,, r, k,l T’(r) IR,, y, k,) 1 =

v exp

N2 -

[ik,

(&

-

WI

t

R*, Re' x V(r)c$(r-

R:--

Ri)dr.

r

J

4 (r - 4 - &.I (6)

of the surface atoms; we also neglect the We neglect the displacements variations of the integrals involving atomic wave functions on the same atom (called a in eq. (2)).

ll%E R,=O~R,=a

Fig. 1.

Model of the surface.

To create a surface in an infinite crystal between the planes R,=O and R, = a (fig. l), we cut the connections between both parts of the crystal and we take equal to y the resonance integrals of the surface potential when both atomic functions in V(k,, R,, RL) are centred on nearest neighbours

103

ELEcTRoN~URFACESTAT~~~FORASI~PLECUBICLATTICE

and on each side of the surface. So (7)

I/ (k,, Rz, R:) = Ya~*,R.'hzz,,o 6Rz, a Y the matrix of V(v) is only a 2 x 2 matrix I/=:

(o 0’y>

(8)

Y

and the corresponding

elements of Gz are 1

exp (ik,a)..-^. -

G c E-EE(k)+is k, 1 1 & c E-E(k) k ‘ii k, To simplify the calculations, ,s>

-_

(9)

we shall use the following basis-functions

Ip,y>W + la, r, k,>,

,AS)

=

I&r, W - la, r, W -.--~,

42

J2

WV

so that

01) and

(12) where Gs =

1

1 i- cos k,a

_. -... N c E-E(k)fic=-j+j k.

1

l-cosk,a ______.--G, = N c E-E(k)+is

1

i J(

1 i =: .- f jj’ ,8

(13)

E-6-/3 -(E--6-t+)

kz

_) ’

if E is between the energies S+p and 6-B and Gs=_;+;!&c!j,

G*+J~.??~$j,

(14)

if E is outside the previous interval of energy. The partial phase shift q(k,, E) is given by the sum of the arguments of Gs and GA, : rs(k,, E) and qAs(k*, E) which differ $r, so the denisty of electrons are the same on both sides of the surface (fig. 2). Note that we do not find bound surface states: this result seems to be due to the approximations.

104

G. ALLAN

AND

P. LENGLARl

*

E

Fig. 2.

Partial

phaseshift.

Then rl(k,,E)=arcsin(E-6/2y) q (k,, E) = 0

for

elsewhere

S+j3
.

Put AN(E) equal to the total charge displaced for the energy states below E and An(E) the variation of the density of states: E

AN(E)

An (E’) dE’ ,

=

(16)

s -6~

+nia +z/a

AN(E)

= ,2n! ss

rl (L

E) dk

dk,

(17)

We find (fig. 3) that the derivative of AN(E), i.e. An(E), is proportional IE+6yl’ at the edges of the band. It also has logarithmic divergences

Fig. 3.

Variation

of the displaced

charge

for the energy

states

below

E.

to for

ELECTRON

SURFACE

STATES

FOR A SIMPLE

CUBIC

LATI’ICE

105

E= -&2y. With the same approximations as in refs. 4 and 5 for the surface tension ys by surface atom versus the filling up of the band, that is to say: EF

N3 Ys = - 2$

s

El=

N

AN(E) dE

AN(E) dE = - z

-6Y

t3)

s -6Y

we obtain the result of fig. 4.

Fig. 4.

Surface tension versus the filling up of the band.

Let us remark that we get exactly the same result with this method as other authors through the moment method4es). This is not surprising because we choose the same approximations of tight-binding and of surface potential. 2. Variation of the electronic specific beat and surface entropy From the variation of the density of states that we got in the first section, we can derive the electronic specific heat of a crystal with a [OOI] surface and the electron contribution to the surface entropy. We shall consider a crystal with a half-filled band. In this case, the Fermi level does not change with the temperature in both cases of an infinite crystal and a crystal with a [OOl] surface. The variation of the electronic specific heat between these two cases is equal to : 4-m

wheref(E,

T) is the Fermi function with I&=0.

106

G. ALLAN

AND

P. LENGLART

Variation of the electronic specific heat versus temperature.

Fig, 5.

After numerical integration, we get the curve of fig. 5. The surface entropy by surface atom S, is given from the next formula:

(20) 0

0 We can compare

this value (fig. 6) with the phonon

part. For a simple cubic

4 Seiik 0.02.

0.01

000

3.0

Fig. 6.

31

3:

Q?

(? .$I

35

kT

.

y

Electronic part of the surface entropy.

lattice, the contribution of the phononsr) S,, to the surface entropy constant at high temperatures (where S,,, is generally measured)

is almost

ELECTRON SURFACESTATESFORA SIMPLE CUBIC

107

LATTICE

We apply this formula where TD is the Debye temperature and our results to the cases of silver and molybdenum (these metals are not simple cubic lattices but we shall look only at the relative values of S, and S,,). In both cases (table 1) the variation of the surface entropy versus temperature is weak and a great part comes from the electron surface entropy. TABLE1

Contributions of the electrons and phonons to the surface entropy T= 1000°K

Approximative value of y (ev) Silver Molybdenum

0.7 0.5

Se/k

215* 380*

0.305 0.296

0.003 0.017

0.309 0.307

0.007 0.044

* Ref. 6.

The electronic contribution is weak for silver (which has a broad s band with only one s electron) even at 1000°K; for molybdenum (which has a narrower d band with 5d electrons) the electronic contribution is also small at 350°K but becomes not negligibIe at 1000°K (about fifteen per cent of the phonon surface entropy). The Green function method applied to a [OOI] surface in a simple cubic lattice gives the same results as the moment method. We do not get surface bound states with our approximations; but it can be shown that the electron surface entropy cannot always be neglected with respect to the phonon term. Moreover, the variation of total entropy versus temperature comes almost totally from the electron contribution. This simple calculation may be extended to a degenerate d band. It would be interesting to see whether, with the same approximations, we get surface bound states. Let us remark that the simplicity of the model makes possible the study of surface defects for a simple cubic lattice.

We wish to thank Professor J. Friedel for his helpful encouragement, Dr G. Leman and Dr L. Dobrzynski for stimulating discussions. References I) L. Dobrzynski, These de 3&me Cycle (Orsay, 1968) (not published). 2) R. A. Brown, Phys. Rev. 156 (1967) 889.

108

G. ALLAN

AND

P. LENGLART

3) N. F. Mott and H. Jones, The Theory of Properties

of Metals

and Alloys

New York, 1936) p. 68. 4) F. Cyrot-Lackmann, to be published. 5) F. Cyrot-Lackmann, These d’Etat (Orsay, 1968) (not published). 6) Ref. 3, p. 14.

(Dover Pub].,