PHYSICAL REVIEW
8
VOLUME 31, NUMBER 2
Model-potential
15 JANUARY 19SS
approach to metal surfaces
P. Raghavendra Rao and G. Mukhopadhyay
Indian Institute of Technology, Powai, Bombay
Physics Department,
400076, India
(Received 29 June 1984)
A simple and realistic Inodel. potential of %'oods-Saxon type is used to determine the one-electron
states for a semi-infinite metal (jellium). The various potential parameters are determined by the use
of certain physical constraints which ensure the self-consistency approximately. To judge the accuracy and the effectiveness of this potential model, we compare the work-function values obtained in
the present scheme with those of the fully self-consistent calculations of Lang and Kohn. We also
extend the present analysis incorporating the lattice effects via local-pseudopotential
theory to obtain the work-function values for a few simple real metals, and the results are compared with the
polycrystalline experimental values.
I. INTRODUCTION
A detailed study of metal surfaces is needed in view of
the latest developments in experimental techniques (both
elastic and inelastic low-energy electron diffraction experiments, molecular beam scattering, etc.). Many of the
various metal surface properties and their related phenomena (e.g. , photoemission, response to external field,
etc. ) can be understood via the ground-state one-electron
wave functions (OEWF). Normally these OEWF are obtained through the self-consistent local-field approximation of Lang and Kohn' (LK) Pocal-density approximation (LDA)] using the Kohn-Sham approach of densityfunctional theory2 (DFT). In this approach a set of
Schrodinger-type equations for particles moving in an effective potential are solved numerically. The effective potential is written as a sum of electrostatic potential and a
local exchange and correlation potential which 'are functionals of electronic charge density that must then be
determined self-consistently from OEWF. This procedure
is rather involved numerically, so that even for a simple
metal with planar surface represented by a semi-infinite
jellium, heavy computation is essential making its utilization (e.g., photoemission ) completely numerically oriented at the outset. To circumvent this problem, attempts
have been made to use simple models for the one-electron
effective potential for a semi-infinite jellium. Such model
potentials do give fairly accurate results for ground-state
properties compared with other LDA calculations. '
&1th th1s ln mind~ wc plcscI1t herc a paramctrIzed oncelectron potential, V(x) of Woods-Saxon type, to obtain
OEWF in analytic form. The parameters will be determined iteratively with the imposition of certain physical
conditions. These physical constraints will ensure, though
approximately, the self-consistency of the potential. The
constraints used here follow from the intrinsic requireinent of total charge neutrality, the self-consistency in the
surface dipole barrier, and the application of a theorem
due to Budd and Vannimenus
(BVT). The last requirement has been used before by Sahni et al. for the finite-
linear-potential-model
calculations, producing the workfunction and surface-energy values of reasonable accuracy.
We use a semi-infinite jellium model to represent the
metal wherein the positive background is extended over
the region x &0 with yz plane parallel to the planar surface. The model potential considered here is of the form
(see Fig. 1)
V(x) =
1
+exp[(x
—xo)/aJ
'
where the parameters Vo, xo, and a are to be found. This
model potential has the advantages over the earlier proposed finite-Barrier" (FB) potential and finite-linear (FL)
potential models in that it has continuous derivatives with
respect to x, and does not require finding the solutions in
different regions of x and then matching them and their
first derivatives at the interface of these regions. This
also has a feature that as a~O, this potential model
reduces to a finite-step-barrier
model. Below we obtain
for this potential the OEWF in terms of hypergeometric
series, which may be used for further studies, viz. , linear
response functions, etc. These solutions are mathematically and computationally simple and many of the spatial
integrals related to parameter-determining
constraints can
be performed analytically. We present here the calculations of work-function (4 ) values for jellium model and
Yac.
2
A
Y()
=—
CF
g=0 xokF
0
FIG. 1. Schematic representation of the jellium model and
the one-electron effective potential v,ff of %'oods-Saxon type.
1985
The American Physical Society
P. RAGHAVENDRA RAO AND G. MUKHOPABHYAY
868
compare them with the fully self-consistent LDA results,
in order to judge the effectiveness and the accuracy of the
present model. A preliminary account of this has already
been communicated.
We also present the work-function
(4) values for a few simple metals incorporating the ionlattice effects through the local-pseudopotential
theory,
following Sahni and Gruenebaum.
They treat the contribution of the difference 5U(r) between the pseudopotentials of the semi-infinite lattice of ions and the electrostatic potential of the jellium background as an additional
term in the total energy functional, thereafter treating the
total energy variationally. This is different from the approach of LK who have introduced these ion-lattice effects via first-order perturbation theory. In Sec. II we
present a detailed analysis to obtain the expressions for
density profile, dipole barrier, electrostatic potential, and
metal surface position parameter.
We also specify the
physical constraints and obtain expressions for them in
terms of the three potential parameters, namely, Vo, xo,
and a. In Sec. III we present the results and conclusions.
The OEWF for the semi-infinite system are written as
ik
y+ik
(in atomic units, where
+ V(x) Eg(x) = —k2—
g(x),
(3)
—
—
where
=
Vo
V(r) )
eF
—
A,
1+exp[(ri
e~ and
The electron
charge
is obtained
density
as
p(ri) =n (r))p(r)~ oo ) where n (ri) is the electron density profile, and in terms of OEWF is defined as
n
(i))=3
f
dt(1
t
)
—
g, (r)) ~'.
(6)
~
The solutions for g, (ri) in Eq. (5) can be written as
g, (r) ) =y "u (y),
where
y
= —exp[(r) —bio)/a] .
t )'~, Eq. (5) can
setting p =a(A. —
Now,
hypergeometric
be recast into a
equation of the form
y(1 —
y)u
"+(1—y)(l+2p)u' —1, a u =0,
2+1(p+tpo p
(2)
2
with V(x) given by Eq. (1). Here k =k, +k„and E is
the energy of the state. For convenience, we introduce dimensionless
variables
as q =xkz, qp —
xpkz, and E
'
—,k =t eg, where kp is the Fermi wave vector and eF
is the Fermi energy of the metal. In terms of these new
variables, the potential has the form
U(ri)
(5)
(8)
z
where g(x) satisfies the equation
A'= l=e =m, )
I d
dx
(3)] now takes the form
where u' and u" are the first and the second derivatives
of u with respect to g, respectively. Here the boundary
condition is such that u(y=0)=constant. From Eq. (8)
we obtain u as
II. THEORY
V(x, y, z) =g(x)
31
(4)
i)o)/a:]
a=akF.
at.
po —
1+2p y»
to unity for y=0,
tpo
where
u goes
as required by
the boundary condition, and I' converges absolutely
throughout the entire unit circle in the complex y plane.
Thus the bound-state solutions which enter in all quantities of our interest here are obtained as
A(ri)
=
(q —go)p/a
2c
Xz+i(p+tpo p
tpo
e—
(q — /a
1+2p — go)
.
(10)
where c is the normalization constant.
One can make use of various transformation formulas
for hypergeometric functions'
and then by writing in
series form,
(r)) can be written both for r) &r)o and
'g ('gp as
f,
The wave equation
[Eq.
I
2c
P, (ri)= —Im e
1+
(k/a)(g
bt, e
g
k=1
—i f 5(t)+(q —qo)~]
1+
—qo)
g ai e
k=1
for 'g&'go
—(k/a)(g —go)
for r) &r)o,
(12)
where
2
t2
1
2C
(
bk ——
1)~
k!
oo
2t. 2
1+ 4 k',
—2
1+
cx A,
(13a)
k(k+2p/tr)
I'(k+p+ipo)I (k+p ipo)I (1+2p—
)
I (0 +1+2p)I (p+ipo)I (pi po),
—
(13b)
MODEL-POTENTIAL APPROACH TO METAL SURFACES
ak
1}k
——
r(k+P, +il p)I.(k —p+t'pp)I (1 —2p)
«+1 —2V)1 (I +t Vp)1'( —@+tap)
Here 5(t) is the phase shift defined as
(13c)
[making use
pp
5(t) = tan-
of Eq. (16)]
J(a, &) = —
p
Po
2po
—tan
p+n
n=1
P, (g)~ sin[t(g
n
—gp)+5(t)]
as
q~oo
.
can be seen that for a = 0, 5( t) reduces to
t ) '~ }, which is the phase-shift factor obtan '[t(A, —
tained in the FB model. An expression for gp, the metal
position parameter, is obtained from the overall chargeneutrality requirement, i.e.,
It
q=0,
(g) —e(g)
nT q
where nT(g)=n
function
~0
(g)dg+
n
(16)
e(g} is
and
f
go
the usual step
[n (g) —1]dg .
(17)
u
=—
8
F
nT(g) .
(18)
condition, together with
Applying the charge-neutrality
the boundary conditions u„( oo ) =0 and v'„( —oo ) = 0, the
solution for u„ is obtained as
u
8
(g)=- 3mkF
nT(n')(q'
q)dg',
to the work
and the surface dipole barrier contribution
function hP is given by the expression
bP = —3m.
kF
f
nT(g')(g'
rtp)dg'
—
.
—
(20)
We need the jump J(a, A, ) in the potential as g varies
u ( oo ) ] which is reconfrom —oo to + oo [i.e., u ( —oo ) —
structed within LDA using the density profile in Eq. (6).
The expression for the effective potential has the form
u
(g) = —
8
3mkF
2
2
f
oo
nz. (rl')(g'
p„,(gazoo)+
2
2
rl)d
g'—
p„,(n (g)),
(21)
where p, „, is the exchange and correlation part of the
chemical potential. The jump now can be obtained as
nT(g')(g'
oo
g'
gp)d—
}
2
, p.,(g ~),
(22)
where b, P is given by the expression in Eq. (20). Now, as
a self-consistent requirement this should be the same as
A,
(
= Vp/ez), i.e.,
J(a, )=A,
(23)
A,
Lastly, we impose another physical constraint which relates the difference in electrostatic potential between that
at the surface and deep inside the metal to the bulk properties. According to the theorem due to Budd and Vannimenus, this difference is related to the energy per electron for the uniform electron gas:
Au„=u„(g =0) u„(g~ oo—
)
The electrostatic potential u„(g} is obtained in the
present choice of origin of the axes as a solution of
Poisson's equation:
d
co
2
g Pxc('tI~
kF
(14)
such that as gazoo, g, (g) tends to the form [to be consistent with the definition of n (g) in Eq. (6)]
f
8
3mkF
2 tan-
q, =
869
0.4 —0.0829rs-
0.0796rs
(rg+7. 8)
(24)
where r, is the electron density parameter, r, =3/4mn,
and n is defined by 3n n =kz. In the present case, this
difference is equal to
b,
8
v„= —3mkF
f
oo
[n (g) —
e(q)]qdg .
(25)
The right-hand side of Eq. (24) depends only on r, which
is a bulk parameter. Thus, by this simple theorem, the
difference in the electrostatic potential from its surface to
inside value is related to r, .
In all these expressions [Eqs. (17), (22), and (25)] it is
possible to evaluate the spatial integrals analytically using
the expression in Eq. (5) for the charge density. Finally,
the expressions we obtain for computation are in the form
of infinite series with a few integrals (integral over
t =k!kF) to be evaluated numerically.
These equations
are solved iteratively for A, and a using the NewtonRaphson numerical solution method, with the Gaussian
method for evaluating the integrals.
quadrature
The
software developed for this program are quite general for
the use of the Newton-Raphson method for two variables
and a separate subroutine for Gaussian quadrature is used.
This is well optimized and usually takes about a minute
on a Digital PDP-10 system.
The various expressions for gp, hP, and b, u„ in terms
of A, and a are as follows:
(a) Expression for gp.
P. RAGHAVENDRA RAO AND G. MUKHOPADHYAY
—
go ——
3m.
8
+3
sin[25( t) ]
I—
2)+
t —
) a(Ii
dt (1
o
4t
~here
1
1
2c
2p
"
+2
k
i
.
'
bk
4bk
+
k+2p
kk.
k+k +2p
i
and
gk e
I, =Re
—2i5
—2i5
2
go
2
'4'=
8
8
k+k'+2ipo
f
t')a' S—
, +S,
k
+2tPo
(b) Expression for AP (in units
—3mkF
' kk~,
5'(t
(k+k')
of Ep):
=0) —3
dh(l
0
+,
——,'
(2p)2
f
0
dh
cos{25)+{1—t')
5'(t)
(27)
where
Si-
CO
(2c)
(k+2p),+Re
(k+2ipo)
k
and
S2 —
00
bkbk
g
(2c)
(c) Expression for 4v
(k+k'+2p)
(in units
2
+Re
~r«k'e
2(k
&k~k'
+ k'+ 2ipo)
2(k
+k')
of eF):
[5'(t =0) —q, ]+3a'
f
—2i5
1
dt I cos[2(got
f
dt(l
—t')ReS,
—5)] —(1 —t )sin[2(got —5)][5'(t)—go] I,
(28)
where
k=i
ake
(k/a)qo
e
—2i (5+got)
{k+»po)
2
k
+T
e
In all these expressions, it is easy to see that as a~O,
these tend to the expressions obtained in the FB model.
The work function for a semi-infinite jellium
is simply obtained as (in units of ez)
4
@ (ep)=A,
—1.
t
(k +k')/a]qo
k
via the local pseudopotentials,
the expression
N(ez) =)i.'
(5v),„
—1 —(5v ),„,
These in turn are reflected in the change
(5v),a„= 3
2r,c
d'
for the choice of Ashcroft pseudopotential"
(31)
co(r)
of the
orm
(30)
is the average value of 5v(r) over the
of the semi-infinite crystal. Here A, ' (different
from A, in value) is determined iteratively from the jump
in the potential with the inclusion of lattice effects. As
the density profile changes slightly on introduction of
ion-lattice effects, the results for surface dipole barrier
where
volume
QkQk~
(k'+ k)
+k'+2ipo)'
show a change.
For the real metal case, wherein the ion-lattice effects are
introduced
~k~k
of values in A, and @. For the densely packed crystal faces
(5v ),„ is obtained as
(29)
for the work function reduces to
—2i (5+got)
to(r)=
——[1 e(r, —r)] .
T
—
(32)
Here z is the ionic charge and r, is a cutoff radius. The
choice of r, for each meta1 is such that it gives a good
description of the bulk properties. d in Eq. (31) represents
the interplanar spacing. .
MODEL-POTENTIAL APPROACH TO METAL SURFACES
871
4
TABLE I. Results of work-function
(e~) values and the potential parameter, viz. , the barrier
height parameter A, , the metal position parameter qo, and the exponential parameter G. , for the oneelectron potential model of Woods-Saxon type are shown in the jellium-model approximation.
(e~)
values of (a} fully self-consistent calculations of Lang and Kohn (LK), (b) finite-barrier model-potential
calculations (FB), and (c) finite-linear potentia1-model calculations (FL) are quoted. r, is the WignerSeitz radius.
4
e'(eF)
Present
Qo
2.0
2.5
3.0
3.5
4.0
0.514
0.431
0.343
0.242
0.035
0.462
0.521
0.532
0.501
0.407
1.096
1.173
1.245
1.313
1.379
0.201
0.376
0.550
0.724
0.902
0.31
0.464
0.628
0.797
0.978
FBb
FL'
0.053
0.342
0.639
0.807
0.891
0.334
0.463
0.600
0.743
0.891
'See Ref. 1
Ref. 4.
'See Ref. 5.
bSee
4
III. RESULTS
In Table I the values for A, , a, and go, obtained from
our approximate self-consistent scheme as discussed in
Sec. II, are presented. In the same table, the values for
for the jellium model obtained in the
work function
present scheme as well as those of Lang and Kohn obtained via fully self-consistent calculations and other
LDA calculations ' are also presented.
In Table II the work-function values for the principal
faces of a few simple metals (Al, Zn, Li, Ca, Sr, Ba, and
Na) obtained by using the expression in Eq. (30) for @(e~)
and Eq. (29) for @ (ez) are presented. A detailed comparison of these results with experimental values is not possible since the experimental values are available only for
polycrystalline samples. Nevertheless, we compare our results with the available polycrystalline
experimental
values only, ' since the anisotropies among different faces
are typically of the order of 10% of the mean workfunction values. ' For comparison sake we also present in
4
the same table the results for both 4(eF) and
(e~), obtained by Sahni and Gruenebaum
for the finite-linear potential model. They have obtained 4(eF) for these simple
metals, with wave-vector correction to the local exchange
and correlation energy functional.
As Table I indicates, our results for semi-infinite jellium compare well with those of Lang and Kohn for the
range r, =2 to 4. As r, increases, the comparison improves and it is interesting to note that for about r, =4, all
the three model potentials, namely, FB, FL, and the
present model potentials, give almost the same results.
This is true since as we have mentioned earlier, as n~O
our model reduces to the FB model and it can be seen that
from Table I for r, =4, the value of a (=0.035) is very
small. This is also true in the FL potential-model case
since it reduces to the FB model as the slope parameter
tends to zero. From the FL model as well as from the
present model analysis it can be drawn that for r, & 4, it is
safe enough to use the finite-barrier potential model.
However, for smaller r, (-2), the FB model fails com-
TABLE II. Work-function values of real metals 4(eF) with local pseudopotential approximation for ion-lattice effects, as well as
for the jellium model N (eF ). The results of Sahni and Gruenebaum for both +(ez) as we11 as 4 (ez), and the polycrystalline experimental values are quoted from Ref. 9. Faces shown are the most densely packed crystal faces of the metals, respectively, and
),„
is the average of the discrete-lattice perturbation over the volume of the semi-infinite crystal. r, and r, are the Wigner-Seitz and
pseudopotential core radii, respectively.
(»
+ (ep)
Metal
Face
Al(fcc)
Zn(hcp)
Li(bcc)
(0001)
(110)
2.07
2.30
3.28
Ca(fcc)
Sr(fcc)
Ba(bcc)
Na(bcc)
(111)
(111)
(110)
(110)
3.28
3.57
3.69
3.99
'See Ref. 9.
(111)
Sahn-
SahniGruenebaum'
(» ).,(~r)
Present
1.12
1.27
1.06
—0. 145
—0.073
—0.180
0.226
0.304
0.649
0.287
0.371
0.749
0.255
0.314
0.661
0.297
0.385
0.831
0.358
0.457
0.665
1.73
1.93
2.01
1.67
—0.026
0.649
0.748
0.788
0.749
0.875
0.927
0.790
0.908
0.976
0.888
1.054
0.649
0.747
0.786
0.889
(0.498)
0.616
0.659
0.734
0.857
ls
0.013
0.071
0.001
iGrueneba'
Present
1.057
Experiment'
P. RAGHAVENDRA RAO AND G. MUKHOPADHYAY
pletely whereas the other two models give better results.
We have an advantage over the FL model in that ours
does not require much mathematical labor and is computationally a simple one. In the FL potential-model calculations, the OEWF are obtained in terms of the Airy functions and their first derivatives (see Ref. 5) whereas in the
present case, these are obtained in simple infinite series
form that converges sufficiently fast. This makes the use
of OEWF obtained in the present scheme for the further
study, viz. , in linear response theory, simpler compared to
the FL potential-model calculations.
From Table II it can be seen that our results for simple
metals are tallying well with the polycrystalline experimental values. For the simple metals we considered here,
the difference between C&(ez) and
(eF) is very small
4
N. D. Lang and W. Kohn, Phys. Rev. B 1, 4555 (1970); 3, 1215
(1971).
W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965); N. D.
Lang, in Solid State Physics, Advances in Research and Applications, edited by H. Ehrenreich, F. Seitz, and D. Turnbull
(Academic, New York, 1973), Vol. 28, p. 243.
P. J. Feibelman, Phys. Rev. B 12, 1319 (1975); 22, 3654 (1980).
~V. Sahni, J. B. Krieger, and J. Gruenebaum, Phys. Rev. B 12,
3505 (1975); P. Raghavendra Rao and G. Mukhopadhyay,
Nucl. Phys. Solid State Phys. (India) 24C, 1 (1981).
5V. Sahni, J. B. Krieger, and J. Gruenebaum, Phys. Rev. B 15,
1941 (1977); V. Sahni, C. Q. Ma, and J. S. Flatnholz, ibid 18,
31
~10% of
each other) owing to the fact that
finite and cancels the effect of change in the relaxation dipole barrier on inclusion of the ionic pseudopotential. Our N results for the jellium model show an imfor
provement over the results of Sahni and Gruenebaum
larger r, values in comparison with polycrystalline experimental values. The work-function values
for the simple metals Li, Ca, Sr, Ba, and Na are within 10% of the
polycrystalline experimental values. However, for Al and
Zn, they are about 20%%uo different from their respective experimental values.
In conclusion we note that the present analysis with a
Woods-Saxon type of potential model does lead to consistent results, and the OEWF thus obtained are simple
enough to enable us to use them for further studies.
(within
(5v
),„ is
4
3931 (1978).
P. Raghavendra Rao and G. Mukhopadhyay,
Nucl. Phys. Solid
State Phys. (India) (to be published).
7H. F. Budd and J. Vannimenus, Phys. Rev. Lett. 31, 1218
(1973); 31, 1430(E) (1973); J. Vannimenus and H. F. Budd,
Solid State Commun. 15, 1739 (1974).
SP. Raghavendra Rao and G. Mukhopadhay, Solid State Commun. (to be published).
V. Sahni and J. Gruenebaum, Phys. Rev. B 19, 1840 (1979).
'OI. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series
and Products (Academic, New York, 1965), p. 1043.
. W. Ashcroft, Phys. Lett. 23, 48 (1966).
N.