M.V. Mamonova (mamonova_mv@mail.ru ), M.A. Bartysheva

Omsk State University

Accounting for the effect of Friedel oscillations and lattice relaxation on surface characteristics of metals

Abstract: A form of the variational trial functions, taking into account the oscillations of the electron density distribution near the surface is considered. Within the approach the density functional method, the calculation of the surface energy and work function of some metals is carried out. Relaxation of two ionic layers is taken into account in complex with Friedel oscillations.

Keywords: Surface energy, work function, Friedel oscillations, distribution of electron density

It was shown decades ago within the jellium model that the redistribution of the itinerant electrons at a simple metal surface results in damped electron density oscillations propagating into the bulk known as Friedel oscillations. These quantum oscillations of electron density are features which makes a significant effect on electronic density distribution. Main causes of their existence are any defects of three-dimensional symmetry of the crystal lattice. A self-consistent study of Friedel oscillations at metal surfaces was done by Lang and Kohn [1] within the jellium model of an electron gas. In terms of quantum mechanics approach using ab initio calculations the following form of electronic density distribution function near surface was obtained [1]:

(1)

Density functional theory (DFT) – is one of the most widely used theoretical approaches to studying of surface features of different substances. Trial functions method applied within DFT [2] despite it’s relative simplicity provides results which are in very good correspondence with experimental values. However such features as Friedel oscillations could not be taken in account in case solution to linearized Thomas-Fermi equation is used as the trial function for electronic density distribution.

Consider semi-infinite metal which has flat surface. Average electronic density of the metal is equal n0. Presuming that metal is uniform in XY plane we shall further consider that all magnitudes depend only on Z variable.

Within DFT energy of the ground state could be defined as gradient decomposition:

(2)

where - energy density including kinetic, electrostatic, exchange and correlation terms. All terms are provided in [2]. According to present study, for noble and transition metals with relatively high density values best results are obtained by using fourth-order gradient corrections to kinetic energy density term: (3)

and to exchange-correlation term:

‑ . (4)

For less dense alkali metals it is enough to use second-order gradient corrections.

Another essential surface characteristic is work function. According to [3] it could be defined as difference between chemical potential of the system and it’s dipole barrier value.

(5)

Dipole barrier is stated as sum of multiple components. In case we consider influence of surface relaxation up to second ionic layer, dipole barrier will consist of:

(6)

Formulae for dipole barrier components are taken from article [2].

Within jellium model:

(7)

Component describing influence of discreet distribution of ions in the crystal lattice:

(8)

Where a cut-off radius and d is is interplane distance.

Consideration of first ion layer shift on value from it’s ground position is performed by adding following term to dipole barrier value:

(9)

For component describing shifting of second near-surface ion layer we shall use the formula presented in article [4]:

(10)

Chemical potential of the system is described as:

(11)

In present work we present new form for electronic density trial function which allows considering electronic density oscillations near the surface. Area within the metal in which oscillations could be described by formula (1) needs to be sewed with near-surface area in which electronic density experiences exponential falling. Near-surface layer in which abrupt collapsing is represented is defined as separate area with width is determined as . Width value is calculated self-consistently. By this means trial function could be represented as:

(12)

where:

(13)

Values for the set of variational parameters , and are obtained on the basis of surface energy condition minimum on all three parameters:

(14)

Expressions for surface energy and work function with terms describing discreet distribution of ions in lattice and gradient terms are presented in [2].

In table 1 results of calculations are shown. Obtained results are in a good correspondence with experimental data.

Me	kmin	βmin	γmin	σmin	W
	a.u.	a.u.	a.u.	mJ/m^2	eV
K	-0,867	1,34	0,641	162	1,33
Na	-0,866	1,35	0,602	251	1,84
Al	-1,031	1,22	0,185	1448	2,93
Cr	0,529	1,39	-0,325	2369	4,17
Cu	-0,474	1,25	0,276	1624	4,13

Table 1. Values for variational parameters, surface energy and work function for set of metals with Friedel oscillations taken into account.

According to [5] quantum density oscillations are closely related to another feature of surface existence – effect of lattice relaxation which also represents oscillatory behavior. The Friedel oscillations in electron density contribute to drive the ions to relax in an oscillatory fashion.

Methodology described in [2] had been extended and we have obtained surface energy terms describing relaxation shifting effects of first and second ion layers. Forms of work function terms were taken from [2]. Experimental data were used as fixed values for relaxation parameters and . Results of calculation with account of density oscillations and relaxation effects are represented in table 2 in comparison with experimentally obtained values and values presented in [2].

In the formula (10) fixed values and were used instead of variation parameters, where and are experimental values taken from different sources and d is distance between layers in bulk metal. Actual values for and used for calculations and values of surface energy and work function are shown in the table 2.

Me	consideration δ1	consideration δ1, δ2	σ relax mJ/m^2	W relax eV
Li(100)	δ1/d=-0,068	δ2 /d=0	385	3,05
	δ1/d=-0,068	δ2 /d=0,006	372	3.28
Al(110)	δ1/d=-0,085	δ2 /d=0	2273	3,91
	δ1/d=-0,085	δ2 /d=0,05	1121	4.22
Table 2. Values for surface energy and work function obtained with relaxation parameters equal to δ1 exp/d and δ2 exp/d Cu(111)	δ1/d=-0,031	δ2 /d=0	1771	5,03
	δ1/d=-0,031	δ2 /d=0,019	1167	4.47

In the table 3 results of our calculations are compared to experimental data and values obtained with trial function method without taking in account complex influence of Friedel oscillation and surface relaxation [2].

Me	σ[2] mJ/m^2	σ exp mJ/m^2	σ relax mJ/m^2
			consideration δ1	consideration δ1 , δ2
Li	368	380	385	372
Al	728	1140	2273	1121
Cu	1894	1750	1771	1167
Me	W[2] эВ	Wexp эВ	W relax eV
			consideration δ1	consideration δ1 , δ2
Li	2,33	2,99	3,05	3.28
Al	6,98	4,06	3,91	4.22
Cu	7,27	4,58	5,03	4.47

Table 3. Comparison of obtained values for surface energy and work function with experimental data and values presented in [2].

It is obvious that simultaneous consideration of quantum electronic density oscillations and lattice relaxation effects increase accuracy of surface energy end work function of metals, especially in case of metals with relatively high electron density.

Figure 1. Electronic density distribution with different surface effects consideration for different values of density parameter r_s.

Behavior of electron density distribution with provision for Friedel oscillations is displayed on figure 1.

It is clear that electron density oscillation amplitude depends on magnitude of density parameter as it was shown in [1]. Relative influence of quantum oscillations is for alkali metals is greater (12%) in comparison with transition metals (5%). However, despite it is common for Friedel oscillations to have multiple peaks; obtained electron density has only one local maximum. It is necessary to point that in case relaxation effects are taken in account amplitude of the electron density oscillation is greater by contrast to consideration of quantum oscillations effect alone.

References:

[1.] Lang N.D., Kohn W. Theory of metal surfaces: Charge density and surface energy // Phys.Rev. 1970. V.B1. №12. P.4555-4568

[2.] Мамонова М.В., Прудников В.В., Прудникова И.А. Физика поверхности. Теоретические модели и экспериментальные методы. Москва: ФИЗМАТЛИТ, 2011. - 400 с.

[3.] Lang N.D., Kohn W. Theory of metal surfaces: Work function // Phys.Rev. 1971. V.3. №4. P.1213-1223

[4.] А.В. Матвеев, Влияние решеточной релаксации металлических поверхностей на работу выхода электрона. Вестн. Ом. ун-та. 2008. № 1. С. 14–18

[5.] J.-H. Cho., Ismail, Z. Zhang, E. W. Plummer. Oscillatory lattice relaxation at metal surfaces// Phys.Rev. 1999. V.В59. №3. P.1677-1680