- Email: [email protected]
A note on the electron density profile at the surface of metals
Surface Science 105 (1981) L245-L248 North-Holland Publishing Company
SURFACE SCIENCE LETTERS A NOTE ON THE ELECTRON DENSITY PROFILE AT THE SURFACE OF METALS *
B.V. PARANJAPE and G.C. AERS ** Institute of Theoretical Physics, University of Alberta, Edmonton,
Alberta, Canada T6G 2J1
Received 12 November 1980; accepted for publication 14 January 1981
A two step model of electron density profile at the surface of a metal is in use for the study of surface plasmas. The same model is used here to calculate surface energy and work function. The parameters in the model are determined by minimizing surface energy. It is shown that the model is a reasonable approximation to the true profile only in low density metals.
The shape of the electron density profile at the surface of a metal plays a very important role in the dispersion of the surface plasma modes. Calculation of the dispersion equation becomes very simple if the profile is a single step function [I]. Although the surface plasma frequency thus calculated gives correct answer for zero wave number, it gives an incorrect wave number dependence. In a paper from this institute it was proposed [2] that the actual surface profile may be best approximated by taking a double step profile at the surface. Thus the electron density was assumed as n_(z)=n,,
z<-a,
n-(z) = yo ,
-a
n-(z) = 0 ,
z>b.
(1)
Positive charge density is assumed as
n+=no
for
z
n+ = 0
for
z>O.
(2)
*Work supported in part by the Natural Sciences and Engineering Research Council of Canada. ** Now at Daresbury Laboratory of the Science Research Council, Warrington WA4 4AD, England. 0039-6028/g
l/0000-0000/$02.50
0 North-Holland Publishing Company
B. V. Paranjape, G.C. Aers /Electron
L246
4.
(2) is the assumption
density profile at metal surfaces
of the jellium model. Charge neutrality
condition
1-Y
b=---a
Y
gives (3)
’
Plasma dispersion can be calculated with the profile given in eq. (1). The parameters y and a can be chosen to fit the experimental curves showing the wave number dependence of the surface plasma frequency. This profile has been adopted by several authors [3]. Dispersion relation can be written down analytically in a simple form if the above profile is assumed. With a real profile one has to make calculations numerically. The electron density profile given by eq. (1) has not been used to calculate other physical properties of the surface, such as surface energy u and work function W. Of course elegant calculations for the evaluation of these properties are available in the literature [4]. Any estimates made with a profile such as a double step will give extremely poor results. We still feel it is useful to calculate u and W from the profile of. eq. (l), because it then provides another check on the quality of the double step as a profile. We have found that a double step model is quite reasonable for metals with low electron densities. In metals with a high density of electrons the double step model gives poor results for surface energy and work function. In the present paper we follow Smith [S] who has used the density functional approach developed by Hohenberg and Kohn [6]. In this method one assumes a reasonable form of the electron density profile in terms of a few parameters. Positive charge density is assumed constant throughout the metal which extends from z = -00 to z = 0. The ground state energy E&r) of the electron gas is then written as a functional of the electron number density n. For an inhomogeneous electron gas H-K have given an expansion of the energy E&r) in terms of gradients of the electron density. Surface energy u is the energy necessary to cleave a metal per unit area of new surface formed. Thus u is the total energy of the separate pieces after splitting minus the total energy of the unsplit block. Thus 0 = &/(n_) - E&I+) . Here E.,(n+) does not include the gradient terms. Contributions ous terms are (see Smith [5]) ukinetic
=
&(3n)2’3
uexhange =i(3/n)“3 ucorrelation
= 0.056n~‘3a(l
3
- Y*‘~)
(’ - ‘*) 7
’
1 1 - 37 +47*
n, = 72
-
;
1-r*
1) ,
(4)
- Y“~) ,
nz’3a(l
o,.o,,r,,& = 2r& ’
ugradient
ni’3a(y2’3
to u from the vari-
.
0.079 (0.079 + y”3n:‘3)(0.079
+ r&/3)
’
B. V. Paranjape, G.C. Aers /Electron density profile at metal surfaces
Calculation mately, for
of eqs. (4) to (7) is straightforward.
L247
osradient is evaluated approxi-
.
(9)
In the double step function model that we have chosen Vn = 00 at two points z = -a and z = b where the density changes in steps. Estimates of ogadient can however be made by replacing finite differences through the gradient and by substituting the average value of n in each interval. Thus we estimate (3wadient assuming dn(z)/dz = +(I
- 7)/k
n = ne(7 + I)/2
-2a
(10)
O
(11)
I
dn(z)/dz = -7no/2b n = 7ne/2
i
Minimizing the total surface energy with respect to 7 and a one can determine these parameters. Physical quantities u and W (eq. (2.4) in ref. [5]) can now be determined [5]. It can be seen from table 1 that the model gives satisfactory results at
Table 1 Parameters a and 7, obtained by minimizing a; values for no and Wexp are from ref. [4] Metal
cs Rb K Na Li Ag Au CU Ca Mg Cd Zn La Tl In Ga Al
“0 (au X 10w3)
Y
1.33 1.67 1.95 3.77 6.92 8.73 8.80 12.6 6.90 12.8 13.8 19.5 12.0 15.4 17;o 22.0 26.9
0.44 0.44 0.43 0.42 0.41 0.40 0.40 0.39 0.41 0.39 0.39 0.38 0.39 0.39 0.38 0.37 0.37
a (au)
ku X 104)
0.70 0.68 0.67 0.62 0.57 0.56 0.56 0.53 0.57 0.53 0.52 0.50 0.53 0.52 0.51 0.49 0.48
0.25 0.31 0.36 0.64 1 .oo 1.14 1.14 1.27 1 .oo 1.28 1.28 1.05 1.27 1.25 1.20 0.80 0.22
Wtheo (eV) 2.5 2.6 2.6 2.5 2.1 1.9 1.0 1.4 2.1 1.3 1.2 0.4 1.4 1.0 0.75 0.04 -0.57
1.8 2.2 2.2 2.4 2.4 4.3 4.3 4.4 2.8 3.6 4.1 4.2 3.3 3.7 3.8 4.0 4.3
L248
B. V. Paranjape, G.C. Aers /Electron
density profile at metal surfaces
low densities. Use of a double step model is doubtful in Al since the work function for Al thus calculated gives negative values. It is well known that surface energy calculated from a jellium model gives reasonable results for low density metals and for high density metals (I becomes negative. Even the selfconsistent calculation of Lang and Kohn [4] gives negative values for u in the high density region. L-K have introduced in their jellium model pseudopotentials to improve their results. u calculated from the density profile in eq. (1) gives results similar to those obtained by Lang and Kohn [4] (see fig. 4) without the pseudopotentials. Values of work function calculated by Smith using exactly the same procedure as used in this paper are in very good agreement with the experimental values. Smith has used a much more reasonable profile. The profile that we have used gives poor results for the work function in high electron density metals. In the context of the present calculation which does not include the effects of lattice structure the double step model used in (1) and (2) is reasonable only in low electron density metals. The authors wish to thank Dr. Summerside
for help with the numerical
work.
References [l] R.H. Ritchie, Progr. Theoret. Phys. 29 (1963) 607. [2] A.D. Boardman, B.V. Paranjape and R. Teshima, Surface Sci. 49 (1975) 275. [3] A. Eguiluz and J.J. Quinn,Phys. Letters A53 (1975) 151; Forstmann and H. Stenshke, Phys. Rev. B17 (1978) 1487. [4] N.D. Lang and W. Kohn, Phys. Rev. B12 (1970) 4555. N.L. Lang, Solid State Phys. 28 (1973) 225. 0. Gunnarsson, M. Jonson and B.I. Lundqvist, Solid State Commun. 24 (1977) 765. [5] J.R. Smith, Phys. Rev. 181 (1969) 522. [6] P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864.
SURFACE SCIENCE LETTERS A NOTE ON THE ELECTRON DENSITY PROFILE AT THE SURFACE OF METALS *
B.V. PARANJAPE and G.C. AERS ** Institute of Theoretical Physics, University of Alberta, Edmonton,
Alberta, Canada T6G 2J1
Received 12 November 1980; accepted for publication 14 January 1981
A two step model of electron density profile at the surface of a metal is in use for the study of surface plasmas. The same model is used here to calculate surface energy and work function. The parameters in the model are determined by minimizing surface energy. It is shown that the model is a reasonable approximation to the true profile only in low density metals.
The shape of the electron density profile at the surface of a metal plays a very important role in the dispersion of the surface plasma modes. Calculation of the dispersion equation becomes very simple if the profile is a single step function [I]. Although the surface plasma frequency thus calculated gives correct answer for zero wave number, it gives an incorrect wave number dependence. In a paper from this institute it was proposed [2] that the actual surface profile may be best approximated by taking a double step profile at the surface. Thus the electron density was assumed as n_(z)=n,,
z<-a,
n-(z) = yo ,
-a
n-(z) = 0 ,
z>b.
(1)
Positive charge density is assumed as
n+=no
for
z
n+ = 0
for
z>O.
(2)
*Work supported in part by the Natural Sciences and Engineering Research Council of Canada. ** Now at Daresbury Laboratory of the Science Research Council, Warrington WA4 4AD, England. 0039-6028/g
l/0000-0000/$02.50
0 North-Holland Publishing Company
B. V. Paranjape, G.C. Aers /Electron
L246
4.
(2) is the assumption
density profile at metal surfaces
of the jellium model. Charge neutrality
condition
1-Y
b=---a
Y
gives (3)
’
Plasma dispersion can be calculated with the profile given in eq. (1). The parameters y and a can be chosen to fit the experimental curves showing the wave number dependence of the surface plasma frequency. This profile has been adopted by several authors [3]. Dispersion relation can be written down analytically in a simple form if the above profile is assumed. With a real profile one has to make calculations numerically. The electron density profile given by eq. (1) has not been used to calculate other physical properties of the surface, such as surface energy u and work function W. Of course elegant calculations for the evaluation of these properties are available in the literature [4]. Any estimates made with a profile such as a double step will give extremely poor results. We still feel it is useful to calculate u and W from the profile of. eq. (l), because it then provides another check on the quality of the double step as a profile. We have found that a double step model is quite reasonable for metals with low electron densities. In metals with a high density of electrons the double step model gives poor results for surface energy and work function. In the present paper we follow Smith [S] who has used the density functional approach developed by Hohenberg and Kohn [6]. In this method one assumes a reasonable form of the electron density profile in terms of a few parameters. Positive charge density is assumed constant throughout the metal which extends from z = -00 to z = 0. The ground state energy E&r) of the electron gas is then written as a functional of the electron number density n. For an inhomogeneous electron gas H-K have given an expansion of the energy E&r) in terms of gradients of the electron density. Surface energy u is the energy necessary to cleave a metal per unit area of new surface formed. Thus u is the total energy of the separate pieces after splitting minus the total energy of the unsplit block. Thus 0 = &/(n_) - E&I+) . Here E.,(n+) does not include the gradient terms. Contributions ous terms are (see Smith [5]) ukinetic
=
&(3n)2’3
uexhange =i(3/n)“3 ucorrelation
= 0.056n~‘3a(l
3
- Y*‘~)
(’ - ‘*) 7
’
1 1 - 37 +47*
n, = 72
-
;
1-r*
1) ,
(4)
- Y“~) ,
nz’3a(l
o,.o,,r,,& = 2r& ’
ugradient
ni’3a(y2’3
to u from the vari-
.
0.079 (0.079 + y”3n:‘3)(0.079
+ r&/3)
’
B. V. Paranjape, G.C. Aers /Electron density profile at metal surfaces
Calculation mately, for
of eqs. (4) to (7) is straightforward.
L247
osradient is evaluated approxi-
.
(9)
In the double step function model that we have chosen Vn = 00 at two points z = -a and z = b where the density changes in steps. Estimates of ogadient can however be made by replacing finite differences through the gradient and by substituting the average value of n in each interval. Thus we estimate (3wadient assuming dn(z)/dz = +(I
- 7)/k
n = ne(7 + I)/2
-2a
(10)
O
(11)
I
dn(z)/dz = -7no/2b n = 7ne/2
i
Minimizing the total surface energy with respect to 7 and a one can determine these parameters. Physical quantities u and W (eq. (2.4) in ref. [5]) can now be determined [5]. It can be seen from table 1 that the model gives satisfactory results at
Table 1 Parameters a and 7, obtained by minimizing a; values for no and Wexp are from ref. [4] Metal
cs Rb K Na Li Ag Au CU Ca Mg Cd Zn La Tl In Ga Al
“0 (au X 10w3)
Y
1.33 1.67 1.95 3.77 6.92 8.73 8.80 12.6 6.90 12.8 13.8 19.5 12.0 15.4 17;o 22.0 26.9
0.44 0.44 0.43 0.42 0.41 0.40 0.40 0.39 0.41 0.39 0.39 0.38 0.39 0.39 0.38 0.37 0.37
a (au)
ku X 104)
0.70 0.68 0.67 0.62 0.57 0.56 0.56 0.53 0.57 0.53 0.52 0.50 0.53 0.52 0.51 0.49 0.48
0.25 0.31 0.36 0.64 1 .oo 1.14 1.14 1.27 1 .oo 1.28 1.28 1.05 1.27 1.25 1.20 0.80 0.22
Wtheo (eV) 2.5 2.6 2.6 2.5 2.1 1.9 1.0 1.4 2.1 1.3 1.2 0.4 1.4 1.0 0.75 0.04 -0.57
1.8 2.2 2.2 2.4 2.4 4.3 4.3 4.4 2.8 3.6 4.1 4.2 3.3 3.7 3.8 4.0 4.3
L248
B. V. Paranjape, G.C. Aers /Electron
density profile at metal surfaces
low densities. Use of a double step model is doubtful in Al since the work function for Al thus calculated gives negative values. It is well known that surface energy calculated from a jellium model gives reasonable results for low density metals and for high density metals (I becomes negative. Even the selfconsistent calculation of Lang and Kohn [4] gives negative values for u in the high density region. L-K have introduced in their jellium model pseudopotentials to improve their results. u calculated from the density profile in eq. (1) gives results similar to those obtained by Lang and Kohn [4] (see fig. 4) without the pseudopotentials. Values of work function calculated by Smith using exactly the same procedure as used in this paper are in very good agreement with the experimental values. Smith has used a much more reasonable profile. The profile that we have used gives poor results for the work function in high electron density metals. In the context of the present calculation which does not include the effects of lattice structure the double step model used in (1) and (2) is reasonable only in low electron density metals. The authors wish to thank Dr. Summerside
for help with the numerical
work.
References [l] R.H. Ritchie, Progr. Theoret. Phys. 29 (1963) 607. [2] A.D. Boardman, B.V. Paranjape and R. Teshima, Surface Sci. 49 (1975) 275. [3] A. Eguiluz and J.J. Quinn,Phys. Letters A53 (1975) 151; Forstmann and H. Stenshke, Phys. Rev. B17 (1978) 1487. [4] N.D. Lang and W. Kohn, Phys. Rev. B12 (1970) 4555. N.L. Lang, Solid State Phys. 28 (1973) 225. 0. Gunnarsson, M. Jonson and B.I. Lundqvist, Solid State Commun. 24 (1977) 765. [5] J.R. Smith, Phys. Rev. 181 (1969) 522. [6] P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864.