Binding energies for positrons at metal surfaces

Surface Science 120 (1982) L491-L497 North-Holland Publishing C o m p a n y

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S U R F A C E S C I E N C E LETTERS B I N D I N G ENERGIES FOR P O S I T R O N S AT METAL S U R F A C E S

D.B. L A M B R I C K and G. SIOPSIS School of Mathematical and Physical Sciences, University of Sussex, Brighton BN] 9QH, UK Received 6 May 1982

The binding energies of surface states of positrons in the adiabatic approximation are estimated. The positron is allowed to penetrate the metal which is viewed as a dispersive plasma. Some other effects are also taken into account by introducing the positron work function as a phenomenological parameter. However, our results do not agree with experiment, which suggests that there are still effects unaccounted for by the theory.

There is a strong interest, at present, in the binding of positrons to metal surfaces [1-7]. The possibility of positrons being trapped by a deep potential well just outside a metal surface was first mentioned by Hodges and Stott [5]. Theoretical work [8-10] on the trapping of positrons at metal surfaces has yielded a form for the potential effectively experienced by the positron if the metal is described as a spatially dispersive metallic plasma without a cutoff and the adiabatic limit is taken. Estimates of the binding energy have thus been made [1,2,7,15] and also experimental values have been inferred from thermal activation of positronium emission [6]. In particular, we refer to the recent paper of Babiker and Tilley [2] who consider the case of a non-penetrating positron. By taking account of the form of the potential inside the metal and making some allowance for surface-dipole and ion-core repulsion effects, we aim to improve the calculated values of the binding energies. We present, then, the results of a variational calculation of the binding energies of certain metals, which tend to be in line with those of Barberan and Echenique [1] whilst being a half to a third of those inferred from experiment. An account of the experimental results is given by Mills [6]. We describe the metal by the hydrodynamic-jellium model and suppose that it occupies the half-space z < 0, z > 0 being a vacuum. A positron brought to the surface will induce an image potential which can serve to trap the positron just outside the metal. We name this potential VN, N standing for Newns who was the first to calculate it for z > 0 [8]. Subsequently Eguiluz gave its form for z < 0 [10]. Eguiluz gives the metal-interior form of the potential as well as the exterior. It is this form that we are using as our starting point. The material parameters 0039-6028/82/0000-0000/$02.75

© 1982 North-Holland

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D.B. Lambrick, G. Siopsis ,/ BE for positrons at metal surfaces

/_T entering the potential are then the velocity parameter fi = l,? v v and the bulk plasma frequency cop, where v v is the Fermi velocity. The positron is assumed to be localized normal to the surface and completely free parallel to it. We have then a one-dimensional single-particle problem. The image-induced potential is then

Vn(z)--

~2 o¢~ 2f0 dqL' e-2q'~: cop

COp 2q~+flT -q"

..2 q_ coP q' fit

'

(la)

for z ~> 0, and % f12 / " % VN(Z) = - 2--fi + ~2O~p - T / "I0 {aq,,

2 )1,2 fop

q~+fi-; (lb)

for z <~ 0, where q~h is a wavenumber parallel to the surface. The parameters cop and fl can both be written in terms of r,, the electron concentration parameter (defined so that volume per electron = ~r& 2) as

cop~ ( 3 / 4 ) W2, j S ~ - ( 3 / 5 ) l / 2 ( q r r / 4 ) 1 3 r ~

I~- 1.4866& i

We also note that VN is an increasing function of z and V N ( - oC) = -- (cop/2~),

(2a)

Vn {0) = -- ~ (cop/2fi),

(2b)

VN(OC ) = 0.

(2c)

We write VN in a more convenient form by introducing dimensionless variables. Thus for z 1> 0 we put y = ( f l / % ) q l l to obtain 1 cop

oe

) 1/2

where # = 2copZ/~, and for z ~< 0 the substitution •~2 2-- 2] 1/2 y = (1 + / J qlq/COp)

yields VN(Z) = ~'- ~ -

-- 1 -1- fl

)2

_

]2.

(3b)

We attempt, in our final potential, to take into account two other effects.

D.B. Larnbrick, G. Siopsis / BE for positrons at metal surfaces

L493

Firstly, the surface-dipole layer due to electron leakage creates a repulsive (for a positron) potential inside the metal and secondly the are excluded from the ion-cores by C o u l o m b i c repulsion which represent here as a constant repulsive potential. Hence we add potential step at z = 0 of height V0. N o w by setting VN(-~

) + V0 = - q , + ,

constant positrons we also to VN a (4)

where ¢,+ is the positron work function, we c o m p e n s a t e for any difference between V y ( - o e ) and the true correlation energy. This is a method that was first suggested by Barton [15]. Thus in our model the work function is an input p a r a m e t e r and provided that we have accurate values for it we should have a physically reasonable potential. Our single-particle H a m i l t o n i a n is therefore

H= ~p~ + VN(z ) + O(--z)[-q,+ -- V N ( - ~ ) ] ,

(5)

where 0 is the step function: 0(z)=

1, 0,

for for

z >0 z<0,

and atomic units (h = e 2 = m = 1) have been adopted. For a b o u n d state to exist we must have ( H ) < 0 as well as ( H ) < - ~ + , for some q,. We e m p l o y the variational method to estimate the ground state energy and a d o p t the trial function

~Nae :/', qJ(z)=LN(a+z)e-"/h,

for for

z~0,

(6)

as used by Mills and Evans [ 11] and N i e m i n e n and Hodges [7]. The variational p a r a m e t e r s are b and c while N 2 = 4/b2(b + 2c) is determined by the normalization condition ( g , ] } ) = 1 and a = bc/(b + c) by requiring that + ' be continuous. For given r~ and ~+ we now require the values of b and c that minimize (q, IHIq~), for which we must find an expression. N o w H = T + V, where T = ½p~ = -- 1 d 2 / d z 2 and since Pz ]~b) = - i[~k'), we have (q, l T [ + ) =

½(q/l~b') =

~UZb.

(7)

T h e potential term divides into interior (Vi,) and exterior (Vou,) parts. We find that for the exterior of the metal, ,N 2 = 1),/2 _ y ] 2 (~P[ V°u'[q~) = - a f0 d y [ ( y 2 + x

(y+ v )-,

,

(8)

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D.B. Lambrick, G. Siopsis / BE for positrons at metal surfaces

where we have defined y

=/~/¢opb. For

the metal interior,

(~]Vin[+)=-½N2a2cq~+ +¼N2a2f°~dy[y-(y2-1)l'2]2(y+rl)

',

(9)

where the first integral has been done and 7 = fl/COpC. Although tedious, these integrals are straightforward to evaluate and we find that (4'1Voml~b ) = -

(Wp/2fl)N2( a2I + 2aJ + K ),

(10a)

where the integrals I, J and K are given by I = - (fl/4C0p) [1 + 43' + (4y 2 + 2) log(2y) + 8 y ( y 2 + 1)R],

(10b)

J = (fl2/w2)[ylog(2y)+ 1+ 1/4y+ (2y2+ I)R],

(10c)

K--

B

2~0pa l ° g ( 2 y ) +

+

43`3 + Y2 +2 +23`1)

]

+3'(23`-+1~2 3) R ,

(10d)

and R=(3`2+1)

,/210g[(¢y2+l

_l+y)(¢3`2+l_3`)(2y),/2].

(10e)

In the interior of the metal we must distinguish between the two cases 7 ~> 1 and ~ < 1. We obtain then ( ¢ ] V~n[¢) =

-½N2a2cq~+ +¼N2a2127-½+(1-2,2)log(2+27)+F(7)],

(lla)

where

(1 + 7-¢:2-l )(7+ ¢e-1 ) 2 ¢ -,log (1+7+¢¢ 1)(7

for

7 ~> 1

--4T/¢1 --72 tan -1

for

7 < 1.

F(~) = 1 +r/

tan

1 --7

1

7 V1 _72

'

(lib) We obtain then a cumbersome, if not complicated, function of b and c with r~ and qs+ as input parameters,

H(b, c) = ~U2b +

(~b[Voutl~b) + (~[V,n[~),

(12)

the last two terms being given by (10) and (11) and where (7) has been used. We have minimized expression (12) numerically for certain metals and the results are reproduced in table 1. We took our values of ~+ from four sources,

D.B. Lambrick, G. Siopsis / BE for positrons at metal surfaces

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Table 1 Variational optima of H(b, c) (eq. (12)) for certain metals; b and c are the variational parameters in eq. (6); q,+ is the positron workfunction from source indicated in the column "Refs"; B is the optimum value of - H(b, c), i.e. an estimate of the binding energy of a positron; where no number appears, no bound state for the positron exists; atomic units throughout Metal

Crystal face

rs

e/,+

Refs. a)

bopt

Copt

B

Zn

2.3

5.60 2.35

0.0479 0.0347

2.59 2.07

HS NO HS HS M M M HS HS HS HS HS NO NO NH M M NH NH NO M M M

3.05 3.70

Cd AI

0.035 --0.044 0.065 0.025 0.0015 -- 0.007 --0.0018 0.075 0.095 0.13 0.1 0.14 0.15 0.13 0.029 -0.015 -0.0048 0.088 0.040 0.059 --0.096 --0.093 -0.040

3.05 3.30 3.35 3.30

4.10 3.30 3.05 3.15

0.0462 0.0414 0.0401 0.0409

3.20 3.60 3.50

5.15 2.90 3.00

0.0442 0.0360 0.0373

3.10

7.35

0.0466

3.60 3.65 3.45

1.85 1.90 2.40

0.0360 0.0352 0.0379

(111) (100) (110) Ga In TI Sn Pb

2.19 2.41 2.48 2.21 2.3 OlD, (100) (110)

Cu

2.67 (111) (110)

Ag Au Mg Cr W Ni

(0001) (100) (111) (100)

3.01 3.01 2.65 1.48 1.62 1.81

~,1 HS: Hodges and Stott [12]; NO: Nieminen and Oliva [13]; NH: Nieminen and Hodges [7] (all being theoretical); and M: Mills [14] (experimental).

t h r e e b e i n g c a l c u l a t e d v a l u e s , t h o s e o f H o d g e s a n d S t o t t [12], N i e m i n e n a n d O l i v a [13] ( p e r h a p s t h e m o s t a c c u r a t e v a l u e s ) a n d N i e m i n e n a n d H o d g e s [7]. T h e e x p e r i m e n t a l v a l u e s a r e g i v e n ( w i t h r e f e r e n c e s ) b y M i l l s [14]. T h e a b s e n c e o f a b o u n d s t a t e is i n d i c a t e d i n t h i s m o d e l b y c ~ ~ ( t h e p o s i t i o n e s c a p i n g i n t o the metal) and ( H ) m i . ~ - q , + . In our model we always have a bound state for q,+ < 0 . It m i g h t h a v e b e e n f e a r e d t h a t t h e v a l u e s o f t h e b i n d i n g e n e r g y w e o b t a i n e d u s i n g t h e v a r i a t i o n a l m e t h o d a r e n o t i n f a c t g o o d e s t i m a t e s o f it. T h a t t h i s is n o t t h e c a s e c a n b e i n f e r r e d f r o m ref. [3] w h e r e l i t t l e is lost b y n o t a c t u a l l y solving the Schr6dinger equation. The values we have found are roughly 1 eV i n a g r e e m e n t ( s e e t a b l e 2 ) w i t h t h o s e o f B a r b e r a n a n d E c h e n i q u e [1]. N o t i c e

D.B. Larnbrick, G. Siopsis / BE for positrons at metal surfaces

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Table 2 Binding energies for positrons at certain metals; atomic units throughout Metal

Our results a)

BE b)

NH c)

Zn

0.0479 0.0347

0.0408

0.074

0.0441 0.0415

0.077 0.077

Cd AI (111) (100) (110)

0.0462 0.0414 0.0401 0.0409

Ga In

M d)

B e)

0.031

0.0511

T1 Sn Pb

0.106 0.111 0.107

0.037 0.036 0.037

0.103 0.109

0.032 0.033

0.085 0.088 0.099 0.092 0.103

111),(100) (11o) Cu (111) (ll0) Ag Au Mg Cr W Ni

(0001) (100) (111) (100)

0.0442 0.0360 0.0373 0.0466

0.044

0.0360 0.0352 0.0379

0.033 0.032 0.034

~) Our estimates. b) Estimates by Barberan and Echenique [1]. c) Estimates by Nieminen and Hodges [7]. a) Experimental results quoted by Mills [6]. e) Estimates by Barton [15].

however the remarkable disagreement with the results of Babiker and Tilley [2], which shows the size of the effect of penetration. The usefulness of the adiabatic approximation is demonstrated by our close agreement with the results of Barton [15]. Thus useful results may be obtained by simple means. Our values are roughly one-half to one-third of the experimental values quoted by Mills [6]. This suggests that there are still some effects unaccounted for by the theory. We wish to thank Dr. G. Barton for guidance and for introducing us to this field. For financial support G.S. thanks the Schilizzi Foundation and D.B.L. thanks the Cheshire LEA.

D.B. Lambrick, G. Siopsis / BE for positrons at metal surfaces

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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

N. Barberan and P.M. Echenique, Phys. Rev, BI9 (1979) 5431. M. Babiker and D.R. Tilley, Solid State Commun. 39 (1981) 691. G. Barton and M. Babiker, J. Phys. C14 (1981) 4951. G. Barton, J. Phys. C14 (1981) 3975. C.H. Hodges and M.J. Stott, Solid State Commun. 12 (1973) 1153. A.P. Mills, Jr., Solid State Commun. 31 (1979) 623. R.M. Nieminen and C.H. Hodges, Phys. Rev. BI8 (1978) 2568. D.M. Newns, J. Chem. Phys. 50 (1969) 4572. G. Barton, Rept. Progr. Phys. 42 (1979) 963. A.G. Eguiluz, Phys. Rev. B23 (1981) 1542. E. Evans and D.L. Mills, Phys. Rev. B8 (1973) 4004. C.H. Hodges and M.J. Stott, Phys. Rev. B7 (1973) 73. R.M. Nieminen and J. Oliva, Phys. Rev. B22 (1980) 2226. A.P. Mills, Jr., International School of Physics "Enrico Fermi", 83rd Session at Varenna, 1981. [15] G. Barton, University of Sussex Report (1982).