Friction parameter of an ion near a metal surface

Solid State Communications, Vol. 32, pp. 419—422. Pergamon Press Ltd. 1979. Printed in Great Britain. FRICTION PARAMETER OF AN ION NEAR A METAL SURFACE* T.L. Ferrell, P.M. Echeniquet and R.H. Ritchiel Health and Safety Research Division, Oak Ridge National Laboratory, Oak Ridge, TN 37830, U.S.A. (Received 20 November 1978; in revised form 4 June 1979 by R.H. Silsbee) The force on an ion located near a condensed-matter surface determines its Brownian motion along the surface. We have studied the dissipative component of this force using a model in which the ion interacts with a metal while moving parallel with its surface. We recover the known asymptotic form of the force for large separations when only electron— hole excitations are accounted for but find a different form when damping of collective states is included. A BROWNIAN MOTION MODEL has recently been developed to treat the kinetics of chemical species near metal surfaces [1, 2]. Applications of the formalism seem to have been restricted so far to a semi-infinite electron gas embedded in a positive ion jellium and bounded by an infinite repulsive barrier, In view of the importance of this problem and the several approximations necessary to attain numerical results for the friction parameter [2], we have used a rather different approach. The specular reflection condition is assumed for electrons incident on the surface from deep in the metal, and an expression for the retarding force on an ion is derived in terms of the dielectric response function of the metallic system. Our result is applicable at arbitrary ion velocity v but in the special case considered here, where v ~ VF and where damping of collective states’is neglected, agrees with the results of Schaich [2] when Lindhard dielectric function is employed. If damping of plasmon states in the metal is accounted for, however, we find additional frictional losses that may be comparable with those due to singleparticle excitations in some situations, Schaich’s [2] friction parameter ‘qyy(z0) is defined for a particle ofmassM moving with constant velocity v parallel with the surface. It is related to dW/dR, the energy loss per unit length of travel by the particle, by

=

1 dW MvdR’

(1)

Thus i~, is the fractional momentum loss of the particle per unit time. The quantity dW/dR is identically equal to the retarding force exerted on the moving particle by the medium due to polarization charges induced by the particle. In this paper we will work in terms of dW/dR from which ~ may be found immediately by using equation (1). The retarding force on an ion moving parallel to a bounded electron gas may be found from the specular reflection model. This model was first used to study the dispersive properties of the surface plasmon some time ago by one of us [3] and has been shown to describe some aspects of the response of a bounded electron gas with surprisingly good accuracy [4]. Many other workers have used this model subsequently to study various aspects of the image potential of a charge near a metal surface [5]. We apply this model to the case where an ion of charge Ze moves with speed v parallel to the plane boundary of an electron gas when it is located a distance z0 from the surface in the region outside of the metal. The energy loss per unit length of travel is given by 2 ca.’ dk~J—Im dW (Ze) 1_—_-1exP(_2Kzo)dw~ K 1+1) 0 (2) where ,c = ~./(~,/V)2 + k~.The integral I may be written

~ —~—J

_____________

*

~

li—il

___________

Research sponsored by the Division of Biomedical and Environmental Research, U.S. Department of Energy, under contract W-7405-eng-26 with the Union Carbide Corporation.

5

~

dk~ k~~k, w (3) 2 = K2 + k~,and has been termed the surface where k function by Newns [5]. We employ an dielectric =

it

t Present Fisica Estado Solido, Faculdad de Ciencias,address: University of del Barcelona, Barcelona 28 Spain.

-

approximate form for 6k,w, the dielectric function of the metal, which is appropriate when energy transfers, h~,are small compared with the Fermi energy

1 Also Department of Physics and Astronomy,

University of Tennessee, Knoxville, TN 37916, U.S.A. 419

420

FRICTION PARAMETER OF AN ION NEAR A METAL SURFACE 2

I

10

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111111

I

r4

jiiii~

(

Vol. 32, No. 5

I

2

~e

5—

—

2

—

2—

—

=

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4

.—-.--.—

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-

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-

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-

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10_i

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100

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101

z0/a0

Fig. 1. Contributions to the friction parameter of a charged particle moving parallel with the surface of a semiinfinite electron gas and at the distance z0 from the surface. The dashed curves show the dimensionless function Mfj~(z0),originating in the creation of electron—hole pairs and defined by equation (5), as a function of z0. The values of r~, 2/hyao)M~ the one-electron radius, used in the calculation are shown for each curve. The solid curves show the quantity (e 2(zo),originating in the creation of surface plasmons, as it depends on z0, for various values of r~. EF

=

(m/2)v~of the medium. We take =

2 1+ ~/{s —

2

k 2 [1

—

nrwO(2kF

w(w + i7)}

—

collective states here and may be taken from experiment k)/2kvF] (4)

where w~is the plasma frequency and kF = mvFlh. The step Equation function 0(x) [1 + (x/IxI). (4) is=a ~simple generalization of the longitudinal dielectric function of an electron gas as derived from the hydrodynamical model [6] in which the propagation velocity s = VF/%./~.The w-proportional term multiplying s2k2 in the denominator describes damping due to electron—hole excitations. It is chosen so that when equation (4) is expanded in a power series in c~it agrees with the small-w expansion of ~ the Lindhard dielectric function [6], to first order. The presence factor O(2kF when k) accounts forThe the term fact that of Im the {l/e~~~,)vanishes k> 2kF. containing 7 in equation (4) describes damping of —

for a given metal. Electrons which are participating in collective motion may scatter on thermal fluctuations, impurities or defects in the lattice: such scattering is represented by the factor y. To be consistent with the approximation that hw <
J

—

—

[~

~i)

—

~

Vol. 32, No.5

FRICTION PARAMETER OF AN ION NEAR A METAL SURFACE 2

I

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zo/a 0 Fig. 2. Contributions to the dimensionless friction parameter of a charged particle moving paralled with and at the distance z0 from the surface of a semi-infinite electron gas characterized by the one-electron radius r8 = 2.5 M~,(zo) arises from the electron—hole pair creation, while M172(z0) is due to surface plasmon generation. The latter is shown for various values of fry, where y is the damping rate of surface plasmons. 2 j~ x exp and

[—

2kFzouJ du

(5)

1 (d4’\ V ~\dR/ 2

ao

=

2

x

f

h (dW~ 2 my kdR )2

(Ze)+ u2’l

2\

‘

I

‘

—

d

0

-~

s

=

ha0 7k~rF m

— —~~-

exp

—

2s

1 2kTFzou du. (6)

I

-~

The abbreviations s = (1 + d = u + s, t = (u~ u2)”2, a±= UrnS ±t have been used. Here Urn = 2kF/kTF=’ ~(9ir/4)2~’3r~’2for the electron gas in which the one-electron radius is r~,k~F= 3~/v~,kF = mvFlh and a 2/me2 is the first Bohr radius. Energy 0 = h by (dW/dR) losses represented 1 originate from electron— hole excitations and those represented by (dW/dR)2 arise from surface plasmon excitation, The asymptotic forms ~2)i~2

—

1 IdWt

—I—I

I.-1DI

\

/i

m(Ze) 24ln(kTFzo/l.4475) 2

=

(7)

—

t~. ‘~‘0

I__)1

~

~~~TFZ0)

\4

,

(8)

(Ze)

—

w,~ 4z~

—

may be derived easily from equations (5) and (6). It is worth remarking that, for large enough distance from the surface, equation (8) must dominate equation (7). That is, damping of the surface plasmon state must determine frictional losses at large enough distances from the surface. An analogous contribution to the energy loss of a slow electron via bulk plasmon creation in an electron gas has been evaluated by Ashley and Ritchie [8]. Undamped plasmon states cannot be excited by swift electrons having energy less than the plasmon energy. When damping is included, it is found that an electron with arbitrarily small energy above the Fermi level may deliver its energy to the plasmon field. Figure 1 shows M~,(z0)plotted as a function of z0/a0 as dashed curves for various values of r8, the oneelectron radius of the metal (w = 3e /ma0r ). Also .

2

P

3

3

8

422

FRICTION PARAMETER OF AN ION NEAR A METAL SURFACE

shown is (e2/hao~(Mf72(z0) vs z/ao for various values of r8. These are drawn as solid curves. Figure 2 displays Mf~~(zo) vs zo/ao as a dashed curve, while the functions M172(zo) vs zo/ao, for various values of fry, are graphed as solid curves. All values shown in this figure have been computed for r8 = 2.5, a representative value for metals of interest here. Experimental values of fry h/r, where r is the Drude damping time against collisions in metals, are ~ I eV for most metals. However, one sees from Fig. 2 that the frictional effect may amount to as much as a 10% correction when y is large. In a subsequent publication, these results will be described more fully; application to the case where the particle moves inside, and perpendicular to, the surface will be considered also. REFERENCES 1.

E.G. D’Agliano, W. Schaich, P. Kumar & H. Suhl, Nobel Symposium XXIV, Aspendasgarden, Academic Press, New York (1974); W. Schaich, .1. C7~em.Phys. 60, 1087 (1974); E. G. D’Agliano, P. Kumar, W. Schaich & H. Suhi, Phys. Rev. B! 1, 2122 (1975).

2. 3.

4. 5.

6. 7.

8.

Vol. 32, No. 5

W. L. Schaich, Solid State Commun. 15, 357 (l975);W.L. Schaich, Surf Sci~49, 221 (1975). R.H. Ritchie & A.L. Marusak, Surf Sci. 4, 234 (1966). This model was also considered independently by D. Wagner, Z. Naturf 21a, 634 (1966). V. Celli, Surf Phys. p. 393. IAEA, Vienna (1974). D.M. Newns,Phys. Rev. Bi, 3304 (1970); J. Harris & R.O. Jones, .1. Phys. C6,1346 3585(1973);J.I. (1973); J. Heinrichs,Phy~ Rev. B8, Gersten & N. Tzoar, Phys. Rev. B8, 5671 (1973); D. Chan & P. Richmond, J. Phys. C9, 163 (1976). J. Lindhard, Kgl. Danske Vid. Sels. Mat-Fys. Medd. 28, No. 8 (1954). The corresponding limit of Lindhard’s function in his (3.9). The asymptotic form,is given equation (7)equation is given by Schaich [2]. Note also that Shier [Am. J. Phys. 36, 245 (1968)] has derived an expression equivalent to the asymptotic form of l/v(dW/dR)2, equation (8), using a pleasingly clear argument from from classical electromagnetic theory. J.C. Ashley & R.H. Ritchie, Phys. Status Solidi (b) 83, K159 (1977).