Structures of liquid metal surfaces and the specular reflection of fast electrons

662

Journal of Non-Crystalline Solids 117/118 (1990) 662-665 North-Holland

STRUCTURES OF LIQUID METAL SURFACES AND THE SPECULAR REFLECTION OF FAST ELECTRONS* Masayuki HASEGAWA+ and Toshihiro ICHIKAWA§ +School of Mathematics, Faculty of Engineering, lwate University, Morioka 020, Japan §Institue for Materials Research, Tohoku University, Sendai 980, Japan# In this contribution we seek f o r the p o s s i b i l i t y of deducing the atomic density p r o f i l e s of l i q u i d metal surfaces from the r e f l e c t i o n high energy electron d i f f r a c t i o n (RHEED) experiments. For this purpose, we have calculated the glancing angle dependence of the specularly reflected i n t e n s i t y of f a s t electrons f o r assumed atomic density p r o f i l e s . Comparisons of these results with the RHEED experiments suggest that monotone density p r o f i l e s are plausible f o r both In and Sn in cont r a s t to the previous results for other metals obtained by the Monte Carlo simulations and the X-ray r e f l e c t i o n experiments. I t is also found that the surface width of Sn is much larger than that of In. I . INTRODUCTION

tions are i n e v i t a b l y involved in the calcula-

Over the l a s t two decades increasing attention has been paid to the microscopic theory of

tions of the e f f e c t i v e Hamiltonian which determines the energetics of the ions.

We also note

l i q u i d metal surfaces and various type of theo-

that a s t r a t i f i e d structure, i f any, of the

ries have been developed I ,

density p r o f i l e is reflected as a small struc-

These theories are

generally successful in predicting the surface

ture in the angle dependence of the X-ray re-

tension of the l i q u i d simple metals such as the

flectance and making analysis of such data

a l k a l i metals.

without ambiguity is not easy.

However, these theories are too

much complicated or too crude to be used for calculating surface density p r o f i l e s .

Recently, one of the authors (T. I . ) has

Rice and

performed the r e f l e c t i o n high energy electron

co-workers have p r i m a r i l y been concerned with

d i f f r a c t i o n (RHEED) experiments in an attempt

t h i s problem and performed Monte Carlo simula-

to determine the surface structures of l i q u i d

tions for l i q u i d Na, Cs and Hg2.

In and Sn.

The atomic

Electron beams couple much more

density p r o f i l e s obtained by these studies show

strongly with the atoms than the X-rays and, in

o s c i l l a t o r y or s t r a t i f i e d structures near the

a sense, are more adequate for studying surface

surface.

structures.

The X-ray r e f l e c t i o n experiments on

l i q u i d Cs and Hg also support the existence of such structures in the density p r o f i l e s 3. Unfortunately, the results of these studies for the atomic density p r o f i l e s are not very conclusive because of various d i f f i c u l t i e s .

density p r o f i l e s of l i q u i d metal surfaces from the RHEED experiments.

We

f i r s t note that the numerical simulations f o r l i q u i d metal surfaces are f a r from exact in the sense that crude and u n j u s t i f i a b l e approxima-

The purpose of this contribution is to seek for the p o s s i b i l i t y of deducing the atomic For this purpose, we

calculate I ( 0 ) , the glancing angle dependence of the specularly reflected i n t e n s i t y of f a s t electrons, for assumed atomic density p r o f i l e s . Then, we look f o r the plausible density p r o f i l e

*This work is supported in part by the research grant of I n s t i t u t e for Materials Research, Tohoku U n i v e r s i t y , through No. 880168. #Present address: Department of Applied Physics, Faculty of Engineering, Meiji University, Kawasaki 214, Japan 0022-3093/90/$03.50 (~) Elsevier Science Publishers B.V. (North-Holland)

M.Hasegawa,T. Ichikawa/ Struetures of liquid metal surfaces

663

by comparing these results with experiments.

e ik-r

2. SPECULAR REFLECTION OF FAST ELECTRONS I f the dynamical e f f e c t of the atomic d i s t r i -

ei(k,q).r

9

VAPOUR ,

~

bution ( i . e . the energy of phonon-like excita-

\/"

tion) can be ignored, we may consider that the external electrons in l i q u i d metals are scattered by the s t a t i c superposition of the atomic potentials.

LIQUID~

Then, the potential f e l t by an exter-

eik°.r

nal f a s t electron may be given by V(r) = ~ Vatom(l~ - r i l )

(I)

1

where Vatom(r ) is the atomic p o t e n t i a l , i . e . the

FIGURE 1 Specular r e f l e c t i o n of f a s t electrons from the l i q u i d metal surface.

e l e c t r o s t a t i c potential due to the nucleus and atomic electrons.

In this quasi-elastic approx-

where E =~2k2/2m is the energy of the incident

imation, the e f f e c t of the thermal diffuse scat-

electrons and the wave function ~(r) is subject

tering (phonon-like e x c i t a t i o n ) which dominates

to the asymptotic conditions in Fig. I .

the e l a s t i c scattering is i m p l i c i t l y taken into

convenient to rewrite eq. (2) as

I t is

account through the s t a t i c d i s t r i b u t i o n of the [-(~2/2m)~2 + Vs(z) + AV(~)]~(r) = E~(r) (3)

atoms. In metals, i t is desirable to t r e a t the va-

where Vs(Z) is the potential averaged over the

lence electrons as the extended conduction elec-

d~rections p a r a l l e l to the surface and AV(r) =

trons.

V(r) - Vs(Z ).

In heavy metals, however, the electro-

The specular r e f l e c t i o n of a

s t a t i c potential due to the core electrons domi-

s u f f i c i e n t l y extended plane wave is e s s e n t i a l l y

nates the electronic potential and any d i f f e r -

determined by Vs(Z), whereas AV(r) may be

ence arising from d i f f e r e n t treatments of the

treated as giving r i s e to the damping of the

valence electrons is i n s i g n i f i c a n t .

incident, reflected and refracted waves.

We consider the specular r e f l e c t i o n of fast electrons in the above approximation.

We take

Fur-

thermore, we may expect that the electrons incident with small 0 are reflected before any

the z axis to be perpendicular to the l i q u i d

damping e f f e c t becomes important.

surface and assume that the incident electrons

in the f i r s t approximation, the problem may be

can be treated as a plane wave.

reduced to the one-dimensional scattering prob-

Let e be the

glancing angle of the incident electrons with wave-vector k = (k x, O, kz), where kx : kcos8 and kz= -ksine (see Fig. I ) . Then, the asymp-

Therefore,

lem represented by the Sch~dinger equation -(~2/2m)d2w(z)/dz2 + Vs(Z)W(Z) : (~2k~/2m)w(z)

t o t i c form of the specularly reflected wave may

(4)

be given by the plane wave with wave-vector k +

The wave function w(z) in eq. (4) s a t i s f i e s the

q, where q : (0, O, q) and q = 2ksine.

asymptotic condition

A practical method of calculating l ( e ) may be to solve the scattering problem represented by the Sch~6dinger equation [-(~2/2m)V 2 + V(~)]~(r) : E~(~)

(2)

w(z)

f

aeikz z + be-ikz z

I eikr z

(z + ~)

(z ÷ -~)

where kr = kz[l - Vs(_=)/Esin2e]I/2

(s)

664

M. Hasegawa, T. Ichikawa/ Structures of liquid metal surfaces

Then, the r e f l e c t i v i t y

is given by

1.6

l ( e ) = Ib/al 2

(6)

We note that the approximate treatment in the

~(z)

/

/'\D \

1.2 i"

~\

it/ /t

above is v a l i d only f o r small 0 and the damping e f f e c t may become important as 8 increases.

0.8

3. APPLICATIONS TO LIQUID In AND Sn

0.4

i,

"~

---,I

\

STEP

I,

The RHEED experiments have been made f o r In and Sn near above t h e i r melting temperatures.

0

These metals have been chosen f o r practical reasons, i . e .

-3

I

I

-2

-1

0

z/R s

1

low melting temperatures, low vapour

pressures and s t a b i l i t y

against oxidations.

The

energy of the electron beams adopted in these experiments is 20keV.

The experimental d e t a i l s

FIGURE 2 Atomic density p r o f i l e s assumed f o r c a l c u l a t i n g the surface potential Vs(Z). p(z) = p(z)/p b.

w i l l appear elsewhere. In order to deduce the atomic density prof i l e s of In and Sn from these experiments, we have applied the formalism in §2 to these metals.

0

Vs(z)

The surface potential Vs(Z ) may be most -0.5

conveniently taken to be the average of V(r) over the atomic d i s t r i b u t i o n , i . e .

'~...__. , . I I ~

Vs(Z ) : < ~ Vatom(Ir - r i l ) ) a v I

":S ' TEP

-1.0

= fdr'vatom(Ir - r'I)p(z')

,,__iD

(7)

where p(z) is the atomic number density.

1.5

To calculate Vs(Z ) we assume parametrized

I -2

I -1

I

z/R s

0

I 1

density p r o f i l e s of the form p(z)

:

{

Pb[l - cexp(81x)cos6x]

(x < O)

Pb(l - c)exp(-B2x)

(x > O)

(8)

where Pb is the atomic number density of the

FIGURE 3 Surface p o t e n t i a l s of In calculated by using p(z) in Fig. 2. The results of Sn are almost i n d i s t i n g u i s h a b l e from those of In in t h i s scale. Vs(Z ) = Vs(Z)/IVs(-~) I.

bulk l i q u i d metal and x = z/Rs, Rs being the atomic-sphere radius defined by (4~/3)R~ = pb -1

steep potential b a r r i e r ( f o r the atoms) near

Two of the four parameters c, BI, 62 and 6 can

the surface.

Such a behaviour may be s i m i l a r

be eliminated by requiring that p(z) and dp/dz

to that of the radial d i s t r i b u t i o n function of

are continuos at z = O.

the bulk l i q u i d metal.

Figure 2 shows p(z) used in our calculations. The parameters of each curve in Fig. 2 are A; 61 : 6.5, ~ = 0

C; 61 : 2.0, ~ = 3.8

The value of 6 = 3.8

(curves C and D in Fig. 2) has been chosen f o r t h i s reason.

Other p o s s i b i l i t i e s , of course,

cannot be excluded but d i f f i c u l t

B; B1 = 2.0, ~ = 0 D; BI : 0.7, 6 : 3.8 O s c i l l a t o r y behaviours of p(z), i f any, may be

on physical basis.

considered as a r i s i n g from the presence of a

lated by using p(z) in Fig. 2.

to i n t e r p r e t e

Figure 3 shows the surface potentials calcu-

M. Hasegawa, T. Iehikawa / Structures of liquid metal surfaces The results of calculations f o r l ( e ) are

665

1

compared with experiments in Fig. 4. The experimental results are normalized so that the

ItS}

smooth extrapolations of the reflected i n t e n s i t i e s ( f o r small e) to e = 0 is unity.

This

i 0 -~

normalization is almost unique for In, but

"'"'"-,. STEP

there is some ambiguity for Sn since the reflected i n t e n s i t i e s are not smooth probably be-

10-2

\\

cause of the experimental errors.

""

We see from Fig. 4 that monotone density p r o f i l e s are plausible for both In and Sn and that the surface with of Sn is about three times larger than that of In.

10 -3 _

We find that the behav-

iour of l ( e ) in the range of small e is essentially

determined by the steepness of Vs(Z) near

the surface.

Once we allow for the physically

10 -& 0

5

10

15

20

GLANCINGANGLE9 Lmrod)

acceptable o s c i l l a t i o n s in p(z), the steepness of Vs(Z ) near the surface is almost uniquely determined. Such a Vs(Z) is too soft ( f o r In) or too steep ( f o r Sn) to reproduce experimental

l(e).

FIGURE 4 Glancing angle dependence of the specular ref l e c t i v i t y of fast electrons (E = 20keV). F i l l ed and open c i r c l e s are observed results for In and Sn respectively and curves are results calculated by using p(z) in Fig. 2.

4. DISCUSSIONS We f i r s t note that the atomic density prof i l e s p(z) of l i q u i d In and Sn which reproduce

ondly, we have considered only the l i m i t e d type of p(z), but other p o s s i b i l i t i e s cannot be ex-

RHEED experiments are quite d i f f e r e n t from the

cluded.

previous results for other l i q u i d metals: p(z)

analysis of the X-ray reflectance data.

of l i q u i d Na, Cs and Hg obtained by Monte Carlo

This statement also applies to the

In conclusion, the present analyses of the

simulations or X-ray r e f l e c t i o n experiments show o s c i l l a t o r y behaviours near the surface.2'3 I t

RHEED experiments for l i q u i d In and Sn cast a

is not clear at this stage i f this difference is

density p r o f i l e s of l i q u i d metal surfaces.

doubt on the previous results for the atomic

due to the inadequateness of the present or previous studies or to the d i f f e r e n t natures of the individual metals. There remain some problems in the present anslyses.

F i r s t l y , we have ignored the damping

e f f e c t which would become important f o r large e. I t is e a s i l y understood that, i f we take into account this e f f e c t , 1(e) would decrease more rapidly than otherwise. In f a c t , the observed l ( e ) decrease more r a p i d l y at large e than the calculated l ( e ) and t h i s difference may be understood in terms of the damping e f f e c t .

Sec-

REFERENCES I . M. Hasegawa, J. Phys. F 18 (1988) 1449 and references therein. 2. M.P. D'Evelyn and S.A. Rice, J. Chem. Phys. 78 (1983) 5225; J. Gryko and S.A. Rice, J. Chem. Phys. 80 (1984) 6318; J. Harris, J. Gryko and S.A. Rice, J. Chem. Phys. 86 (1987) 1067. 3. B.C. Lu and S.A. Rice, J. Chem. Phys. 68 (1978) 5558; D. Sluis and S.A. Rice, J. Chem. Phys. 79 (1983) 5658; L. Bosio and M. Qumezine, J. Chem. Phys. 80 (1984) 959.