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Variational calculation of metal surface energies
Solid State Communications, Vol. 21, pp. 463-465, 1977.
Pergamon Press.
Printed in Great Britain
VARIATIONAL CALCULATION OF METAL SURFACE ENERGIES V. Sahni* The New School of Liberal Arts, Brooklyn College of the City University of New York, Brooklyn, NY 11210, U.S.A. and L Gruenebaum Department of Physics, Brooklyn College of the City University of New York, Brooklyn, NY 11210, U.S.A.
(Received 6 October 1976 by G. Bums) A one-parameter model potential variational calculation of jelliummetal surface energies in the local density approximation is performed, the results very closely approximating those of the self-consistent calculations of Lang and Kohn. The use of this model potential for the determination of superior upper bounds for the surface energy and the study of various density gradient expansions is indicated. IN RECENT WORKx - a we have examined the question as to whether by employing simple model potentials to represent the effective potential at a metal surface together with certain theoretical constraints it is possible to obtain meaningful results for metal surface properties. Together with the intrinsic requirement of charge neutrality, 4 typical constraints applicable to such calculations are those of the self-consistency of the surface dipole barrier, x the application of a sum rule due to Budd and Vannimenus 5 and that of the variational principle 6 for the energy. 2 The use of the BuddVannirnenus theorem (BVT) allows the exact determination of the contribution to the surface dipole barrier from charge inside the metal. This contribution, particularly for high density metals, can be as large as 40% of the total dipole moment. Thus the application of this theorem leads to accurate results for the surface dipole barrier and work function. 1'3 On the other hand, application of the variational principle for the energy leads not only to a determination of the energy correct to second order but also to an upper bound for the result. In this communication we present the results of a oneparameter variational calculation of jeUium metal surface energies as def'med within the local density approximation 7 (LDA). We assume that the total energy is the sum of the bulk and surface energy contributions and that it is only the latter which is a function of any parameters used to define the effective potential. 2,s Variational minimization of the energy thus leads to an upper bound for the surface energy. We assume here, as in reference 3, the effective * Supported in part by a grant from the City University of New York Faculty Research Program. 463
potential V~f(x) at a metal surface to be given as
Ve~(x) = Fx O(x)
(1)
where F is the field strength defined in terms of the slope parameter x e as F = k } / 2 x e, k~/2 is the Fermi energy9 and O(x) is the step function. For this model potential, the electronic wavefunction is --
sm[kx+~(k)]
Ck(x) - [ CkAi(~)
forx~ 0 (2)
where Ai(~) is the Airy function, ~ = (x -- E/F)(2F) 1/3, and E is the energy. The phase shift 5(k) and the normalization factor Ck are
1 A{(-- ~o) 8(k, x v ) - X~o " Ai(-- ~o) Ck = --
2
sin 8(k, xF)[Ai(-- ~'o)]-1
(3) (4)
where ~'o = (k2/k~)(kFXF) 2/a and where Ai'(~) is the derivative of the Airy function. The electronic density per unit surface area of metal is defined as kF
L Oe(x) = 2re---I / (k~--k2)l~kl 2 dk
(5)
0
and the uniform positive background of density p+ = k~/3rc 2 is assumed to end abruptly at the metal surface position at x = a. For a given field strength, this position is determined by application of the phase shift rule of Sugiyamaa'x° according to which
464
VARIATIONAL CALCULATION OF METAL SURFACE ENERGIES
Vol. 21, No. 5
Table 1. Results for the surface dipole barrier ~¢ in eV, the surface energy Es and its kinetic E~ electrostatic E e e and exchange-correlation Ex~ components in ergs/cm.2 The columns YF and Ya represent the values of the slope parameter and metal surface position respectively rs
YF = kFXF
Ya = kFa
Surface energy components
Surface dipole barrier Present work Lang-Kohn a
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 a
3.446 2.733 2.179 1.745 1.410 1.149 0.949 0.791 0.668
1.364 1.080 0.854 0.666 0.509 0.376 0.264 0.168 0.085
6.21 3.68 2.37 1.62 1.17 0.88 . 0.69 0.55 0.46
6.80 3.83 2.32 1.43 0.91 0.56 0.35 0.16 0.04
Ek --5449 -- 1822 -- 725 -- 325 --158 --82 -- 45 -- 25 -- 15
Surface energies E s
Exc
Ee,
Present work
3184 1438 753 438 275 184 129 95
1298 429 175 83 45 27 17 12
71
8
--967 45 203 196 162 129 101 82 64
l_ang-Kohn a -- 1008 36 199 194 158 124 98 77 60
See reference 11.
3rr 8kF
a =
kF
3 / k3F k6(k) dk.
(6)
The surface energy E s in the LDA is the sum of the kinetic Ek, electrostatic Ees and exchange-correlation Exc contributions where 3,7 k~- [1
80 /3
2 k~0
--
kF
dk]]
0
]_1
f (Vetf[Pe;X] -- Vef~[Pe;--°°])Pe(X) dx
(7)
-oo
=
PT(X)V
,(X) dx
Exc = f (exc [pc(x)] -- exc [Pe])p~(x) dx.
(8) (9)
In the above expressions PT is the total charge density, Ves(x) the electrostatic potential, exe the sum of the average exchange and correlation energies per particle for a uniform electron gas and De the mean interior electronic density. For the correlation energy we have employed the Wigner interpolation formula. With a change of variables y = kFX, and k/kF = q such that the metal surface position is at Ya = kFa, the quantities Ya, Pe/k~, A¢/kF (A¢ is the surface dipole barrier), Ves/k~, Ek/k~ and Ee~/k~ are all universal functions of the slope parameter YF = kFXF. With the exception of the exchange-correlation energy, the spatial integrals for all other properties can be performed analytically so that the determination of these properties reduces to a simple numerical calculation of k-space integrals ranging
from 0 to 1. The explicit expressions employed in the present calculations together with the universal plots of the various properties are given in reference 3. For a specific metal, defined by its Fermi momentum kF = 1/otrs; a -1 = (91r[411/3, we have varied the slope parameter YF until a minimum of the surface energy is obtained. The results of this variational minimization for the surface energy and its components together with the appropriate slope parameter YF and metal surface position Ya are given in Table 1. We have also included in the table the results for the surface dipole barrier and for purposes of comparison the results of Lang and Kohn 11 (LK) for both these properties. A study of the results for Es indicates that with the exception of the case of r8 = 2.5, the results for the total surface energy differ by less than 6.6% from the LK values over the entire metallic range, differing by as little as 1% for r 8 = 3.5. However, for r, = 2.5, the electrostatic component is precisely the same to three significant figures, the percentage differences in the individual kinetic and exchange-correlation components being only 1.5% and 1.3% respectively. It is for high and medium densities that the results for the individual components very closely approximate those of LK. For example for r~ = 3.0, the results for Eh and Eke are precisely those obtained by LK whereas Ee, differs by only 2% from their value. For r, > 4.5, the electrostatic energy is again the same as that of LK but the kinetic and exchangecorrelation energies are different. We also observe that for r s = 2-4.5 there is little difference in the results for the electronic density when compared with the LK densities. For example, the density normalized with respect to the bulk value for r, = 2 differs from LK by less than a percent inside the metal in the range from
Vol. 21, No. 5
VARIATIONAL CALCULATION OF METAL SURFACE ENERGIES
1.2 to -- 0.2 Fermi wavelengths, and is within 5% up to a quarter Fermi wavelength outside the metal where the density has dropped to a fifth of its value at the surface. However, for low density metals, these differences are larger, although the Friedel oscillations occur at approximately the same positions in space. In comparison with the results for the surface energy as obtained 8 by application of the BVT to this model potential, we note that the two sets of values are very similar although the values of the slope parameter YF are quite different and that furthermore, the present energies more closely approximate the LK results. This indicates that the surface energy near its minimum is fairy insensitive to variations in the parameter YF, and thus any reasonable choice for the parameter should lead to good results for Es. The results for the surface dipole barrier, however, differ more from the LK values than do the results of the application of the BVT. The latter set of results differed by at most 0.32 eV from LK over the metallic range. This greater difference in the present results for A¢ are a consequence of the fact that although we have employed wavefunctions obtained by energy minimization, properties other than those represented by operators appearing in the assumed -
-
465
Hamiltonian can be determined only to the same order as that of the wavefunction employed and are thus not as accurate as the energy. In conclusion we note that this model potential leads to very accurate results for the surface energy on application of the variational principle for the energy. As pointed out in reference 3, the fact that the effective potential does not become constant but increases indefinitely is unimportant as it is in error only in a region far from the metal surface where the electronic density is a small fraction of its value at the surface. Superior and more physically meaningful bounds for the surface energy, particularly for high density metals, could be obtained by adding to this approximate Hamiltonian the gradient term contributions to the exchangecorrelation energy, ~ the classical cleavage energy of a neutralized lattice la and the ion pseudo-potential contribution, xx and then performing the variational calculation. Finally, we note that since various criteria, such as the BVT exist for the determination of the field strength for arbitrary rs, the model proves ideal for the study of higher density electron gases and various density gradient expansions. ~'la
REFERENCES 1.
SAHNI V., KRIEGER J.B. & GRUENEBAUM J., Phys. Rev. BI2, 3503 (1975).
2.
SAHNI V. & GRUENEBAUM J., Phys. Rev. B (to be published).
3.
SAHNI V., KRIEGER J.B. & GRUENEBAUM J., Phys. Rev. B (to be published).
4.
BARDEEN J., Phys. Rev. 49,653 (1936).
5.
BUDD H.F. & VANNIMENUS J., Phys. Rev. Lett. 31, 1218 (1973); 31, 1430(E) (1973).
6.
MOISEWITSCH B.L., Variational Principles, p. 153. Interscience, New York (1966).
7.
LANG N.D., Solid State Physics, Advances in Research and Applications (Edited by EHRENREICH H., SEITS F. & TURNBULL D.), Vol. 28, p. 243. Academic Press, New York (1973).
8.
MAHAN G.D.,Phys. Rev. B12, 5585 (1975).
9.
Atomic units are used: lel = h = m = 1. The unit of energy is 27.21 eV.
10.
SUGIYAMA A.,J. Phys. Soc. Japan 15,965 (1960).
11.
LANG N.D. & KOHN W.,Phys. Rev. B1, 4555 (1970);Phys. Rev. B3,1215(1971).Theself-consisteneyprocedure has been refined by Lang and the improved results for the surface energy and dipole moments quoted in the text.
12.
GELDART D.J.W. & RASOLT M.,Phys. Rev. B13, 1477 (1976).
13.
JONESW.&YOUNGW.H.,J. Phys. C4,1322(1971);HODGESC.H.,Can. J. Phys. 51,1428(1973); SHY-YIH WANG J. & RASOLT M.,Phys. Rev. BI3, 5330 (1976).
Pergamon Press.
Printed in Great Britain
VARIATIONAL CALCULATION OF METAL SURFACE ENERGIES V. Sahni* The New School of Liberal Arts, Brooklyn College of the City University of New York, Brooklyn, NY 11210, U.S.A. and L Gruenebaum Department of Physics, Brooklyn College of the City University of New York, Brooklyn, NY 11210, U.S.A.
(Received 6 October 1976 by G. Bums) A one-parameter model potential variational calculation of jelliummetal surface energies in the local density approximation is performed, the results very closely approximating those of the self-consistent calculations of Lang and Kohn. The use of this model potential for the determination of superior upper bounds for the surface energy and the study of various density gradient expansions is indicated. IN RECENT WORKx - a we have examined the question as to whether by employing simple model potentials to represent the effective potential at a metal surface together with certain theoretical constraints it is possible to obtain meaningful results for metal surface properties. Together with the intrinsic requirement of charge neutrality, 4 typical constraints applicable to such calculations are those of the self-consistency of the surface dipole barrier, x the application of a sum rule due to Budd and Vannimenus 5 and that of the variational principle 6 for the energy. 2 The use of the BuddVannirnenus theorem (BVT) allows the exact determination of the contribution to the surface dipole barrier from charge inside the metal. This contribution, particularly for high density metals, can be as large as 40% of the total dipole moment. Thus the application of this theorem leads to accurate results for the surface dipole barrier and work function. 1'3 On the other hand, application of the variational principle for the energy leads not only to a determination of the energy correct to second order but also to an upper bound for the result. In this communication we present the results of a oneparameter variational calculation of jeUium metal surface energies as def'med within the local density approximation 7 (LDA). We assume that the total energy is the sum of the bulk and surface energy contributions and that it is only the latter which is a function of any parameters used to define the effective potential. 2,s Variational minimization of the energy thus leads to an upper bound for the surface energy. We assume here, as in reference 3, the effective * Supported in part by a grant from the City University of New York Faculty Research Program. 463
potential V~f(x) at a metal surface to be given as
Ve~(x) = Fx O(x)
(1)
where F is the field strength defined in terms of the slope parameter x e as F = k } / 2 x e, k~/2 is the Fermi energy9 and O(x) is the step function. For this model potential, the electronic wavefunction is --
sm[kx+~(k)]
Ck(x) - [ CkAi(~)
forx~
where Ai(~) is the Airy function, ~ = (x -- E/F)(2F) 1/3, and E is the energy. The phase shift 5(k) and the normalization factor Ck are
1 A{(-- ~o) 8(k, x v ) - X~o " Ai(-- ~o) Ck = --
2
sin 8(k, xF)[Ai(-- ~'o)]-1
(3) (4)
where ~'o = (k2/k~)(kFXF) 2/a and where Ai'(~) is the derivative of the Airy function. The electronic density per unit surface area of metal is defined as kF
L Oe(x) = 2re---I / (k~--k2)l~kl 2 dk
(5)
0
and the uniform positive background of density p+ = k~/3rc 2 is assumed to end abruptly at the metal surface position at x = a. For a given field strength, this position is determined by application of the phase shift rule of Sugiyamaa'x° according to which
464
VARIATIONAL CALCULATION OF METAL SURFACE ENERGIES
Vol. 21, No. 5
Table 1. Results for the surface dipole barrier ~¢ in eV, the surface energy Es and its kinetic E~ electrostatic E e e and exchange-correlation Ex~ components in ergs/cm.2 The columns YF and Ya represent the values of the slope parameter and metal surface position respectively rs
YF = kFXF
Ya = kFa
Surface energy components
Surface dipole barrier Present work Lang-Kohn a
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 a
3.446 2.733 2.179 1.745 1.410 1.149 0.949 0.791 0.668
1.364 1.080 0.854 0.666 0.509 0.376 0.264 0.168 0.085
6.21 3.68 2.37 1.62 1.17 0.88 . 0.69 0.55 0.46
6.80 3.83 2.32 1.43 0.91 0.56 0.35 0.16 0.04
Ek --5449 -- 1822 -- 725 -- 325 --158 --82 -- 45 -- 25 -- 15
Surface energies E s
Exc
Ee,
Present work
3184 1438 753 438 275 184 129 95
1298 429 175 83 45 27 17 12
71
8
--967 45 203 196 162 129 101 82 64
l_ang-Kohn a -- 1008 36 199 194 158 124 98 77 60
See reference 11.
3rr 8kF
a =
kF
3 / k3F k6(k) dk.
(6)
The surface energy E s in the LDA is the sum of the kinetic Ek, electrostatic Ees and exchange-correlation Exc contributions where 3,7 k~- [1
80 /3
2 k~0
--
kF
dk]]
0
]_1
f (Vetf[Pe;X] -- Vef~[Pe;--°°])Pe(X) dx
(7)
-oo
=
PT(X)V
,(X) dx
Exc = f (exc [pc(x)] -- exc [Pe])p~(x) dx.
(8) (9)
In the above expressions PT is the total charge density, Ves(x) the electrostatic potential, exe the sum of the average exchange and correlation energies per particle for a uniform electron gas and De the mean interior electronic density. For the correlation energy we have employed the Wigner interpolation formula. With a change of variables y = kFX, and k/kF = q such that the metal surface position is at Ya = kFa, the quantities Ya, Pe/k~, A¢/kF (A¢ is the surface dipole barrier), Ves/k~, Ek/k~ and Ee~/k~ are all universal functions of the slope parameter YF = kFXF. With the exception of the exchange-correlation energy, the spatial integrals for all other properties can be performed analytically so that the determination of these properties reduces to a simple numerical calculation of k-space integrals ranging
from 0 to 1. The explicit expressions employed in the present calculations together with the universal plots of the various properties are given in reference 3. For a specific metal, defined by its Fermi momentum kF = 1/otrs; a -1 = (91r[411/3, we have varied the slope parameter YF until a minimum of the surface energy is obtained. The results of this variational minimization for the surface energy and its components together with the appropriate slope parameter YF and metal surface position Ya are given in Table 1. We have also included in the table the results for the surface dipole barrier and for purposes of comparison the results of Lang and Kohn 11 (LK) for both these properties. A study of the results for Es indicates that with the exception of the case of r8 = 2.5, the results for the total surface energy differ by less than 6.6% from the LK values over the entire metallic range, differing by as little as 1% for r 8 = 3.5. However, for r, = 2.5, the electrostatic component is precisely the same to three significant figures, the percentage differences in the individual kinetic and exchange-correlation components being only 1.5% and 1.3% respectively. It is for high and medium densities that the results for the individual components very closely approximate those of LK. For example for r~ = 3.0, the results for Eh and Eke are precisely those obtained by LK whereas Ee, differs by only 2% from their value. For r, > 4.5, the electrostatic energy is again the same as that of LK but the kinetic and exchangecorrelation energies are different. We also observe that for r s = 2-4.5 there is little difference in the results for the electronic density when compared with the LK densities. For example, the density normalized with respect to the bulk value for r, = 2 differs from LK by less than a percent inside the metal in the range from
Vol. 21, No. 5
VARIATIONAL CALCULATION OF METAL SURFACE ENERGIES
1.2 to -- 0.2 Fermi wavelengths, and is within 5% up to a quarter Fermi wavelength outside the metal where the density has dropped to a fifth of its value at the surface. However, for low density metals, these differences are larger, although the Friedel oscillations occur at approximately the same positions in space. In comparison with the results for the surface energy as obtained 8 by application of the BVT to this model potential, we note that the two sets of values are very similar although the values of the slope parameter YF are quite different and that furthermore, the present energies more closely approximate the LK results. This indicates that the surface energy near its minimum is fairy insensitive to variations in the parameter YF, and thus any reasonable choice for the parameter should lead to good results for Es. The results for the surface dipole barrier, however, differ more from the LK values than do the results of the application of the BVT. The latter set of results differed by at most 0.32 eV from LK over the metallic range. This greater difference in the present results for A¢ are a consequence of the fact that although we have employed wavefunctions obtained by energy minimization, properties other than those represented by operators appearing in the assumed -
-
465
Hamiltonian can be determined only to the same order as that of the wavefunction employed and are thus not as accurate as the energy. In conclusion we note that this model potential leads to very accurate results for the surface energy on application of the variational principle for the energy. As pointed out in reference 3, the fact that the effective potential does not become constant but increases indefinitely is unimportant as it is in error only in a region far from the metal surface where the electronic density is a small fraction of its value at the surface. Superior and more physically meaningful bounds for the surface energy, particularly for high density metals, could be obtained by adding to this approximate Hamiltonian the gradient term contributions to the exchangecorrelation energy, ~ the classical cleavage energy of a neutralized lattice la and the ion pseudo-potential contribution, xx and then performing the variational calculation. Finally, we note that since various criteria, such as the BVT exist for the determination of the field strength for arbitrary rs, the model proves ideal for the study of higher density electron gases and various density gradient expansions. ~'la
REFERENCES 1.
SAHNI V., KRIEGER J.B. & GRUENEBAUM J., Phys. Rev. BI2, 3503 (1975).
2.
SAHNI V. & GRUENEBAUM J., Phys. Rev. B (to be published).
3.
SAHNI V., KRIEGER J.B. & GRUENEBAUM J., Phys. Rev. B (to be published).
4.
BARDEEN J., Phys. Rev. 49,653 (1936).
5.
BUDD H.F. & VANNIMENUS J., Phys. Rev. Lett. 31, 1218 (1973); 31, 1430(E) (1973).
6.
MOISEWITSCH B.L., Variational Principles, p. 153. Interscience, New York (1966).
7.
LANG N.D., Solid State Physics, Advances in Research and Applications (Edited by EHRENREICH H., SEITS F. & TURNBULL D.), Vol. 28, p. 243. Academic Press, New York (1973).
8.
MAHAN G.D.,Phys. Rev. B12, 5585 (1975).
9.
Atomic units are used: lel = h = m = 1. The unit of energy is 27.21 eV.
10.
SUGIYAMA A.,J. Phys. Soc. Japan 15,965 (1960).
11.
LANG N.D. & KOHN W.,Phys. Rev. B1, 4555 (1970);Phys. Rev. B3,1215(1971).Theself-consisteneyprocedure has been refined by Lang and the improved results for the surface energy and dipole moments quoted in the text.
12.
GELDART D.J.W. & RASOLT M.,Phys. Rev. B13, 1477 (1976).
13.
JONESW.&YOUNGW.H.,J. Phys. C4,1322(1971);HODGESC.H.,Can. J. Phys. 51,1428(1973); SHY-YIH WANG J. & RASOLT M.,Phys. Rev. BI3, 5330 (1976).