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Russian Physics Journal, Vol. 41, No. 12, 1998

C A L C U L A T I O N OF THE E L E C T R O N W O R K FUNCTION AT M E T A L SURFACES BY A DENSITY F U N C T I O N A L M E T H O D M. V. Mamonova and V. V. Prudnikov

UDC 539.612.001

We develop a method for calculating the electron work function on the surface of both simple and transition metals, using electronic density functional theory. We include corrections due to inhomogeneities in the electron density, the effects of surface relaxation in the ionic plane, and the electron-ion interaction, for which we use a generalized Heiney-Abarenkov pseudopotential description. The results lie in good agreement with experiment.

One of the important surface characteristics is the electron work function at the surface of a metal. The complexity of the theoretical description and the comparison of calculation results for the work function with the experimental values, in comparison with the similar problem for the surface energy of metals, stems from the small relative changes in the experimentally observed work function for a large number of materials which exhibit metallic properties. Thus, molybdenum exhibits the largest surface energy among metals, which exceeds by nearly twenty times the value of the surface energy for potassium, while the values of the work functions for them differs by no more than a factor of two. Until now, no satisfactory agreement between the calculated and experimental work function values has been obtained [ 1-3 ]. In the present work, we perform a calculation of the work function for a series of metals, using a density functional approach. We study the influence of various approximations, taking into account the discreteness of the crystal structure and the inhomogeneity of the electron gas in the surface layer of metals. We take into account the effects of lattice relaxation at the surface. We obtain additional contribution to the work function by taking account of the influence of the electron-ion interaction, using a generalized Heiney-Abarenkov pseudopotential. The work function is defined as the minimum energy required to remove an electron from the bulk of a solid material. Its nature is related to the existence of a potential barrier close to the surface of a metal. The value of the work function is defined as the difference between the height of that potential barrier and the chemical potential: W = eq0 = D - g .

(1)

The potential of a dipole electric layer, VD, acting on the electron near the surface, may be obtained in terms of a "jelly" model by a solution of the Poisson equation AVD(z)

= -4rC0(z)

(2)

where 9(z)

no0(-z)] =

= e[n(z)-

(3)

enD(z).

The result is () VD z

= 4r~e z

dz

z'

-

", , z no z

.

(4)

Then the height of the dipole potential barrier Do in the jelly model is given by the expression Omsk State University, Omsk. Translated from lzvestiya Vysshikh Uchebnykh Zavedenii, Fizika, Vol. 41, No. 12, pp. 7-12, December, 1998. Original article submitted January 14, 1998. 1174

1064-8887/98/4112-1174 $20.00

9 1999 Kluwer Academic/Plenum Publishers

CALCULATION OF THE ELECTRON WORK FUNCTION AT METAL SURFACES BY A DENSITY FUNCTIONAL METHOD

I 175

~o

Do = VD(~176 -VD(-~176= 4rte I dz'z'nD(z').

(5)

The distribution of electron density, n (z), is found as the function which provides the minimum functional for the total energy of the inhomogeneous system. In practical calculations the method of test functions is used, when the distribution of electron density n (z) is chosen as a solution of the linearized Thomas-Fermi equation:

no[(1-1e-~Z)O(-z)+le-f~ZO(z)],

n(z)=

(6)

where no is the bulk electron density, and 13 is a variational parameter determined by minimizing the function of the total energy of the inhomogeneous system. Then, alter the integration in (5), we find that 4/tn0 Do = "132

(7)

In order to go beyond the "jelly" model, the expression given for the dipole potential barrier must be supplemented by corrections in the electron-ion interaction by taking account of the discrete distribution of the ion charge at the crystal lattice sites. The influence of the electron interaction on the work function is related to the difference in the electrostatic interaction of the ions with the electron density in the ground state and in the state with one electron removed. Following [4], we shall write the additional contribution to the potential barrier in the form

-

Del :

I2

~ g(z) ne (z)

dz ,

(8)

I2ne(z) dz where d is the interplane distance, and 8 V(z) has the meaning of the sum of the ion potentials, averaged over the planes, after subtraction of the potential for a semi-infinite homogeneous positive background. Since the expression (8) is homogeneous relative to ne, we may consider ne equal to the surface charge density induced by a weak electric field with strength Ez in the semi-infinite model of a homogeneous background:

no

he(Z)

=

L 8rt +

-Fk 8rt

~

exp(13z), z < 0,

] exp(-13z),

(9) z > 0.

To calculate the correction to the electron-ion interaction we used the pseudopotential of Heiney and Abarenkov, widely employed in metal physics:

Vel(r)

=

Ir--r,lnm,

(10)

where Z is the ion valence. Under the condition V0 = O, the Heiney-Abarenkov pseudopotential becomes the expression for another well-known pseudopotential of Ashcroft. For 8 V(z) for - d < z < 0 we find the following expression:

1176

M.V. MAMONOVAANDV. V. PRUDNIKOV

~V(z) = 2gno{(d(R m - z+d ) - ~-~[R 2 - (z+d)2])O(Rm - z + d ) - [z + dO(-z-d)]2}. (11) Performing the summation over the ion planes from z = - ( i + d / 2 ) , i = 1, 2 .... , and using the periodicity of the potential 8 V(z - d) = 8 V(z), from (8) we obtain

De1 =

4~n 0 exp(-~d/2) { ~d cosh([3Rm) + V0d[sinh([~Rm)- ~Rm cosh(~Rm)] - 2 sinh(-[]e/']~. \ 2JJ ~2 (2-- exp(-~d/2))

(12)

Taking into account the effects of displacement of the surface ion planes relative to their bulk position by an amount 8 leads to an additional term in the magnitude of the dipole barrier: Da =

4gn 0~_y_(2:e_~p(_~.d_7~exp(~(8-d/2)){2(l_el~8)sinh~d _2

~28d} + D e l ( e ? ' 8 - 1 ) - 4 r t n o d S ,

(13)

where d is the interplane distance. The resulting value of the dipole potential barrier is given by the sum of the terms: D = Do + Dei + DS. The chemical potential I.t for the electron gas is defined by

bw an '

g-

where w is the bulk energy density for the crystal. In a local density approximation, the bulk energy of a metal attributable to one atom and expressed in terms of the density parameter rs (4rtrs3/3 = n -l) is given by ( 1.105 Evol =

w(n)Zn = Z[

rs2

0.056 + lSRm2 l+0.127rs r3

0.458 rs

0 "9 Z 2/3 rs

Vo R 3 ]

(14)

r3s ).

As a result, the chemical potential la may be written, taking account of the ion-correlation and pseudopotential corrections, as 0.056n~/3 + 0.0059n~/3

B = 0.5(3~2n0)2/3-(~-~) 1/3

(0.079 + nlo/3)2 + 0.5Z 2/3

(_~)~/3

[

_ 4tenor" +

8rtZoR3,,,no 3Z

(15)

In the present work we have defined the parameters of the Heiney-Abarenkov pseudopotential by minimizing the bulk energy of the metal. Minimization of (14) with respect to rs leads to expressions which relate V0 and Rm:

Vo :

rs2 ( 2 . 2 1 _ 0 . 9 Z 2 / 3

+

4.5R_~__0.458 rs

0"00711rs2 ) (1 + 0.127rs) 2

(16)

The problem thus arises of how to determine the second parameter in the potential (10). Typically it is determined by comparison of calculations performed using a given pseudopotential with some empirical characteristics. In the present work, we have used the value of the surface energy as this experimental characteristic. The values of the parameters [3 and 8, in terms of which, according to the expressions presented, we may obtain the value of the work function, will also be determined by minimizing the surface energy.

CALCULATION OF THE ELECTRON WORK FUNCTION AT METAL SURFACES BY A DENSITY FUNCTIONAL METHOD

I 177

According to methods based on density functions [5-7], the surface energy of a metal may be written as the following sum:

(17)

(Y = (~0 + ~ i i + (Yei,

where G0 is the contribution from the electron system in the "jelly" model, Gii arises from the electrostatic interaction of the ions among themselves, and (Yei is related to the difference in the electrostatic interaction of the electrons with the discrete ions and with the homogeneous "jelly" background. Expressions for these components of the surface energy, taking account the effects of displacement of the surface ion planes, according to our work [7], have the form

(--4n(a-5))

4~z 2

oii(13, 5) -

c3

exp[

~c

'

(18)

where c is the nearest neighbor distance in a plane parallel to the surface, and (19)

Cei(13, 5) = ~ei(f~, O) + aGei(~, 5),

s,-

(Yei(~, O) -

~3 [

_

}

_

AOei(13,5)- 2nn~ 132 d(1-exp[3d) exp(--~)cosh(l~Rm) + 2rtn~d52 when the Ashcroft potential is used, which makes it possible to calculate the value of the surface energy of metals without introduction of an additional parameter V0. The surface energy, in terms of the "jelly" model is the difference between the total energy values when the electrons are distributed according to the function n (z) and in a step-function, that is, as n(z) = n o O ( - z ) :

(20)

Go = i {win(z)] - w[noO(-z)]}dz.

Here the electron gas energy density win(z)] includes the electrostatic, kinetic, exchange, and correlation energies, as well as the gradient correction to second order i~the electron gas inhomogeneity for the kinetic energy in the Weitzeker-Kirzhnitsa approximation and the exchange-correlation energy in the Vashishta-Singwi (VS) approximation [8]. In order to improve the quantitative agreement of the surface energy values for transition and noble metals, in this work we have taken into account the effect on the surface characteristics due to fourth order terms in the gradient n (z) which appear in the kinetic energy density:

W4, kin =

1.336n

s4o(3=

)

I(9)

2

9(V2n),Vn[

-

VG-)

l lVn] 4-

2

I.I + 3 Inl -

-

(21)

-

and in the exchange-correlation energy:

W4,xc

=

2.94 - 10 .5 exp (-0.2986n -~

/9) 2

(22)

1178

M . V . MAMONOVA AND V. V. PRUDNIKOV

Table 1. Calculations of the surface energy q (erg/cm 2) and the work function W (eV) for a number of metals using an Asherofl potential with fourth order gradient corrections in the VS approximation for the most densely packed facet (the underlined values lie in best agreement with experiment)

Metal

Quantity

Na



(bcc)

W

K

o

(bcc)

W

AI

o

(fcc)

W

Cu

o

(fcc)

W

Fe

~

(bcc)

W

Cr

o

(bcc)

W

2 nd order gradient correction 267 2.63 151 2.07 941 3.25 872 3.98 635 3.44 641 3.5

4 th order gradient correction to the kinetic energy 300 3.1 166 2.38 1165 4.2 1104 4.95 1043 4.56 1034 4.57

4 th order gradient correction to the exchangecorrelation energy 374 3.8 196 2.87 1860 6.3 1798 7.07 2394 7.31 2334 7.27

The calculated values of the surface energy for a number of metals, using the Ashcroft potential, are shown in Table 1. Clearly, for simple metals it is best to use only the second order corrections in the gradient. For A1 it is necessary to use gradient corrections to fourth order in the kinetic energy, which is due to the large inhomogeneity in the electron gas when there is s-p hybridization of the electron quantum states. For noble and transition metals the best agreement with experimental values of the surface energy was obtained by use of the fourth order gradient corrections to the surface and exchange-correlation energies, due to the effects of s-p-d hybridization. Copper, in its compounds and electronic properties, may appear as either monovalent or a divalent metal. We performed the surface energy calculation for both cases. However, for monovalent copper all of the approximations we used degraded the results, while for the divalent copper we obtained better agreement between the calculated and experimental values of the surface energy when we used the fourth order gradient corrections to the kinetic and exchange-correlation energies. We used these conclusions as to the applicability of various types of gradient correction for the description of surface properties of simple and transition metals when we replaced the Ashcroft potential with the Heiney-Abarenkov one. The values of the variational parameters ~ and 5, necessary for the calculation of the potential barrier D, were determined in this work by minimization of the surface energy using the Heiney-Abarenkov potential, and are shown in Table 2. Application of the Heiney-Abarenkov potential [9] leads to the appearance of an additional term in the expression (19) for the electron-ion component of the surface energy:

XA(~, 5) -- 2nn02 A(Y ei ~3 d (exp[38 - 1)dV0 exp(__~){[3R m cosh([3Rm)

sinh(13Rm)},

(23)

The introduction of the additional parameter V0 in the pseudopotential, determined by comparison of the calculated surface energy with experimental values, makes it possible to vary the calculated work function values, obtaining simultaneous agreement of the values of both metal surface characteristics.

CALCULATION OF THE ELECTRON WORK FUNCTION AT METAL SURFACES BY A DENSITY FUNCTIONAL METHOD

1179

Table 2. Calculations of the Heiney-Abarenkov pseudopotential parameters R m and V~ and also the corresponding values of the work function, taking account surface relaxation (Ws, in eV) and without those corrections (W, in eV). The experimental work function values are given as Wexp

[ Metal

Z I n~ atomic units

Na (bcc)

1

K (bcc)

I

d, atomic units

c, atomic units

Rm,

We•

eV

atomic units

V0, atomic units

W, eV

W~, eV

0.0038

5.71

6.99

2.35

2.7

0.326

2.59

1.99

1

0.00196

6.99

8.55

2.137

2.5

0.087

2.13

1.66

AI (fcc)

3

0.0269

4.92

5.25

4.24

1.2

0.19

4.25

2.85

Cu (fcc)

2

0.0252

3.92

4.80

4.98

1.4

0.603

7.16

4.93

Fe (bcc)

4

0.0504

4.84

4.70

4.31

1.2

0.475

7.24

4.43

Cr (bcc)

4

0.0492

3.85

4.72

4.58

1.3

0.53

7.18

4.44

From Table 2, where we show calculations of the electron work function for the most densely packed surface plane in a number of metals, it is clear that without taking account of the effects of the surface ion planes we obtain better agreement with the experimental values for simple metals and significantly elevated values for the transition metals. Including the effects o f surface relaxation leads to values for the transition metals which agree with experiment, but lowers them for the simple ones. The far stronger dependence of the work function values on the ion plane displacements (20 to 50 %) found in this work, in comparison to the effects on the surface energy [7], makes it necessary to determine more precisely the relaxation parameters. To do so will apparently require taking into account the displacement of several ion near-surface layers, and also the influence o f temperature on the parameter ~. REFERENCES 1. B.V. Vasil'ev, M. I. Kaganov, and V. L. Lyuboshits, Usp. Fiz. Nauk, 164, No. 4, 375-378. 2. G. Paash and M. Khitshol'd, in: P. Pishe and G. Leman (editors), Toward an Electron Theory of Metals [Russian translation], Mir, Moscow (1987), Vol. 2, pp. 466-540. 3. MarchLundquist (editor), Theory of an Inhomogeneous Electron Gas [Russian translation], Mir, Moscow (1987). 4. N.D. Lang and W. Kohn, Phys. Rev., No. 3, 1215 (1971). 5. R.M. Kobeleva,B. R. Gel'chinskii, and V. F. Uldaov, Fiz. Met. Metalloved., 48, No. 1, 25-32 (1978). 6. V.F. Ukhov, R. M. Kobeleva, G. V. Dedkov, and A. L Temrokov, Electron Statistical Theory of Metals and Ionic Crystals [in Russian], Nauka, Moscow (1982). 7. A.N. Vakilov, V. V. Prudnikov, and M. V. Prudnikova, Fiz. Met. Metalloved., 76, No. 6, 38-48 (1993). 8. P. Vashishta and K. S. Singwi, Phys. Rev. B, 6, No. 3, 875-887 (1972). 9. M.V. Mamonovaand R. V. Poterin, and V. V. Prudnikov, Vestn. Omsk Univ., No. 1, 41-43 (1996).

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