Jellium sphere model in overlayer-substrate systems

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Jellium sphere model in overlayer-substrate systems

Chen Lu-Jun, Wang Ning, Luo Enze

To cite this version:

Chen Lu-Jun, Wang Ning, Luo Enze. Jellium sphere model in overlayer-substrate systems. Journal

de Physique I, EDP Sciences, 1993, 3 (4), pp.1053-1058. �10.1051/jp1:1993176�. �jpa-00246769�

Classification

Physics

Abstracts

73.20Hb 73.30Y 79.60Gs

Jellium sphere model in overlayer-substrate systems

Chen Lu-Jun

(I), Wang Ning (2)

and Luo Enze

(3)

(1)

Department

of

Physics,

Box 271, Xidian

University,

Xi'an, 710071, China (2)

Beijing

Vacuum Electronic Devices Research Institute,

Beijing,

100016, China (3)

Department

of

Physics,

Box 271, Xidian University, Xi'an, 710071, China

(Received 22 May 1992, revised 30 September 1992, accepted 16 December 1992)

Abstract. It _was found in calculations that the

jellium/slab

model reduces the

ability

of the adsorbed atoms to attract the valence electrons. As _a result, the calculated minimum of the work function is _too low

compared

witli the experiment. In this paper, as an

improvement,

_we

proposed

a

jellium sphere/slab

model, in which the

positive charge background

is described by discrete

jellium

sphere. Test calculations show that this _treatment lifts the work function minimum by 0.4eV and 0.8eV for Cs/~V(001) and Cs/Ir(001) system,

respectively, resulting

in better agreement ^with experiments.

1. Introduction.

Using

the

jellium/slab

model

[II

and the Film-LAPW method

[2-5],

the coverage

dependence

of work

functions,

_energy bands and surface states for alkali

metal/transition

metal

systems

can be well described

[6-8]. However,

the effect of discrete

charge

distribution of adsorbed atoms in those systems on the work function is very remarkable. As _a matter of

fact, Lang [9]

used the

pseudopotential

in his

jellium

model to simulate the discreteness for _a bare metal surface. The first order

pseudopotential

corrections (&W

)

of work functions _are different for various

crystal

faces. For

example,

the corrections for bare surfaces of

Cs(110), Cs(100)

and

Vs(I it)

_are

0.23,

0.61 and 0.67 eV

respectively.

The

negative correction,

in the bare

surface _case, shows the decrease of the

dipole

moment which is formed

by

the

positive

background

and the electrons

spreading

out to the _vacuum

region.

This shows also that the

attracting ability

of ion-lattice

represented by

the

pseudopotential

is

stronger

than that of uniform

background. Moreover,

_we consider that _as the discreteness in both substrate and

overlayer

_are

neglected simultaneously

in

Lang's jellium/jellium

model

[10],

the result in this

case is consistent with the actual _case in the _sense that

positive

discrete

charges

in both

regions

have _an

opposite

and

nearly

counteractive force _on electrons.

Therefore,

the better calculated

results,

_were obtained in that time.

However,

this counteraction does not yet ^{exist the}

ieilium/slab model,

which leads to a remarkable influence _on the calculated

quantities,

such _as

1054 JOURNAL DE

PHYSIQUE

I N° 4

the work function etc, because of

unilaterally ignoring

the discreteness of

positive charge

^{in the}

adlayer.

Hence, the

jellium sphere/slab

model is

proposed

^{in this} paper to simulate the discreteness of

positive charge

ⁱⁿ

overlayer.

The

corresponding

mathematic treatment for this

model is

given

^below. Calculation results show that this model has well

improved

the

calculated work

function,

which is the _most sensitive

quantity

_among all calculated

quantities.

2. Jellium

sphere

model and it's mathenlatic treatment.

In order to consider the discreteness of

adatoms,

but without

expanding

the calculation unit because of the

computational capability,

the total

charge

in the

jellium

of _a cell is concentrated in _a

jellium sphere

which is laid _{on a} 2-dimensional cell _as shown in

figure la,

and _{we assume}

that the

charge

^is

homogeneously

distributed in the

sphere.

The main purpose ^{of this} paper is _to get a clear

understanding

of the effect of discrete

charge

_on the work function and _to know

what _an

important

role it

plays

ⁱⁿ the

adsorption

_process.

The parameters are calculated

according

to

figure16. Dj

is

given

to be the thickness of

jellium

and R _to be the radius of

jellium sphere

which is

positioned

at z = zo. We have zo = zi ⁺

Dj/2,

where _z

= zi denotes the interface between the interstitial

(TNT) region

and the

jellium-vacuum region. r(z)

ⁱⁿ

figure16

is

given by

r(z)= ~/R~- (z-zo)~ zo-R«zwzo+R.

z

z

~ ~

ⁱ

y

³

j

j i

ii

^~

R

t~iJflilt~

ai bj

Fig.

I. al The schematic drawing of

jellium

sphere/slab model b) The schematic drawing for parameters calculation.

Suppose

_Qvai is the number of valence electrons and is filled in the

jellium sphere.

Then at the coverage @, the number of electrons in the

sphere

is Q~~i.

They

distribute

homogeneously

in the

jellium sphere

and thus

Pi =

~~~ (in sphere)

p =

wR3

3

P2 "

°

(out

of

sphere)

which _can be

expanded

further into the 2-dimensional _star functions

P

(rj

₌

jj pG(zj

_q~G(rjj

j

GE jG,j

with

N~

PG(Z )

"

$

^{P (~}

⁾ Pi

_{(~(( d~((}

s

V~G(ri =

~~

~j~

e'~~

~< <R>

In _our

problem, _t~

=

0 and G

=

(R t~)

is the discrete part ^{of the} 2-dimensional _{space group,}

Nop

is the order of the group. The substitution of

q7~(r)

into

p~(z) gives

PG(Z)

_"

(~

i jj~

^{PI ~ '~~ '~ ~~(}

ReG

~~~~RG)~,

G~

=

(RGIy

^~~ld

^~~~~~

^~~~

~~~~°~~~~~~

~~~~~~~~

~~~z j

=

ii ~j

~j~

i

_Sin

_iGx /

_~~~

~~~

'

_~'~

~~~

~

_~~~

~~~~

when

G~

₌ ⁰ or

G~

=

0,

the result of

p~(z)

is

just

the _{same as} the limitation of the above

equation

at

G~»0

_or

G~»0,

^I-e- when G

=

0, p~(z)

is

equal

to

~°~~ ~~~~~°~~ ~j~ "~

~~

~i~~~

~

=

~~~~

"

[R~

(z zo)~]

4

~~3

^A

3

This distribution is _a

parabolical

_curve, I-e- the first term in

expansion

of _p

(r corresponds

to an

inhomogeneous jellium

flake with _a

parabolical

distribution.

3. Results and

analysis.

In order to

investigate

how remarkable the influence of the discreteness of

positive charge

in the

adlayer

_on the calculated

quantities,

such as the work function etc.,

is,

and what _an

important

role the discreteness

plays,

^the

Cs/Mf(001)

system ^and

Cs/Ir(001)

system are

calculated under _some coverages

respectively.

The

crystal

lattice constants for W and Ir _are chosen to be 5.9813 _a-u- and 5.1212 _a-u-

respectively.

The valence electrons for both of them

are taken to be

5d~6s~

and

5p~5d~6s~,

and _core electrons _to be

[(Xel4fl~]

and

[(Xe-sp~l4f~~]

respectively.

The

muffin-tin(MT)

radii _are 2.55 _a-u- and 2.50 _a-u-, and the radii of

jellium spheres

_are 2.5 _a-u- and 2.4 _a-u-

respectively

for both systems. ^The calculation results _are

given

in

figure

2 and table I. It _can be _seen that the discreteness has resulted in about 0.4 eV,

arising

from the work function minimum for the

Cs/Vf(001)

system

(at

m

0.55,

or

NaflQsm0.21),

and about 0.8eV

arising

for the

Cs/Ir(001)

system

(@=0.6,

or

Na/Ns

»

0.17).

In

practice,

the calculated work functions for

Cs0V(00 Ii

and

Cs/Ir(001) using

the

jellium/slab

model _are 1.34 eV and 0.2 eV _at

= 0.5 and

= 0.6

respectively.

When

1056 JOURNAL ^DE PHYSIQUE I N° 4

o

(cv)

2

0.1 0.2 0.3 0.4

Na/Ns

Fig. 2. The

comparison

of calculated results I) Calculation _curve of CsliV (001) system by flake model in reference [8] and (O) ^is a calculated value at e

=

0.5 by

sphere

model _;2)

Experimental

_curve of Cs/~V (001) system

reported

in reference [I Ii 3) Calculation _curve of Csflr (001) _system

by

flake model in reference [7] 4) Calculation points

(symboled

by (o)) of Csflr (001) _system, the upper branch

by

sphere ^model and the lower

by

flake model, (li) denotes the

experiment

value (Ref. [6]).

Table I. The

comparison ofcalculated

results

by

the

jellium/slab

and the

jellium sphere/slab model,

where Na/Ns ₌ 0.283

for Cs/Ir(00i)

and

Na/Ns

= 0.387

for Cs0V(00i).

~ Jellium

sphere

Jellium flake

d System ~~

^fi

~~ ~fi

(eV) Q~~

^W

(eV) Q~~

^~VUC

0. ii 0.4 Ir-Cs I.4i 0.460 0.98 0.44i 4.3 §b

0.17 0.6 Ir-Cs 1.02 0.515 0.20 0.470 9.7 §b

0.19 0.5 W-Cs 1.74 0.740 1.34 0.685 2.7 §b

using

the

jellium sphere/slab model, they

become 1.74eV and 1.02 eV. The

experimental

work function minimum is 1.3 eV for

Cs/Ir(001) [6].

This indicates that the contribution of

discrete

charge

distribution in

adlayer

is very

important

for work function determinations.

The calculated values

by

the

jellium sphere/slab

model must have _a

sensitivity

to the

change

of

jellium sphere

radius

R, since,

for

example,

the work function minimum of

Csflr(001)

should be

changed

to 0.2 eV from 1.02 eV when the

jellium

is transferred _to the flake from the

sphere.

However,

the work function is insensitive to the

change

of the radius in the

large

^value

region,

at least, say, in the

region

from R

= 1.5 _{a.u, to} R

= 2.4 _a.u. When the radius R is reduced to

1.5 _a,u., the calculated work function and the total

charge

in _{vacuum at}

=

0,4 _are 1.42 eV and 0.453 _e

respectively

which _are much closer _to the values

II.41eV

and 0.460

e)

obtained

1/2

by

R

= 2.4 _a,u. Since

self-consistency

_was

assumed,

when AV ^~ dz _~ 0.03

R~d,

ⁱⁿ

our

calculations,

an

uncertainty

of about _± 0.2 eV for the work function may exist

according

to

our

experience.

That is _to say, a variation of the

jellium sphere

radius within _a range from 1.5 _a-u- to 2,4 _a,u, does not

obviously

influence _our results.

Some

change

of valence electron

charge

in the

jellium-vacuum region

has arisen from the

discreteness,

and the

charge

redistribution has resulted

(see Fig.

3 and

Fig. 4).

The

charge

redistribution _can be considered _as the result of the fact that the

attracting ability

of the

jellium sphere

is stronger ^{than that of the uniform flake. The}

negative

centre of electrons in the

region

between

adlayer

and substrate surface

layer is,

in the

sphere model,

closer _to the _centre of

positive charge

in

adlayer

^than that in the uniform model.

According

to

figure

³ and

figure 4,

it

can be considered that there exist three

dipoles (as schematically

shown in

Fig. la)

in the surface

layers,

which influence calculation work functions. The first

(pi )

and the second

(p~ point respectively

to the slab and the

jellium sphere

from the

negative

centre of electrons in the

region

between the

outlayer

slab and the

jellium sphere.

The third

(p~) points

to the

jellium sphere

^{from the} centre of electrons in the _outer half-side of the

jellium layer

and those

spreading

into the _vacuum

region.

The contributions of the three

dipoles

to the work function

are

symbolized

as ~fi

i, W~ ^and W~

respectively, then,

the total contribution _can be

regarded

as

AW

= W

i + W~ ~fi~. When the

negative

centre is

getting

closer _to the

jellium sphere (I.e.

far away from the

slab), Wi

^will be

increasing

and W~

decreasing.

While W~ ^is not

remarkably changed according

to the

charge

redistribution in the _outer half-side

jellium layer

and in the

vacuum

region

in

figure

3. Hence, the total contribution leads _{to a net} increase of the work

function. In other

words,

the attraction

discrepancy

between the

jellium sphere

and flake to the electrons has taken on an

important

role in the work function. Less than lo §b of the

change

of electron

charge

has made the work function extend to _a

large region (see

Tab.

Ii.

Indeed _a

quite precise description

of the

positive charge

distribution in the

jellium-vacuum region

is very necessary. Of _course, there _are other factors which could influence the calculated work function

[8].

The main

significance

of this paper is make _us

know,

that the discrete

charge distribution,

beside the other factors is _an

important

factor in the work function. Once the other factors have been taken into account, there is _a very

important

reason for

considering

the effect of

positive charge

distribution

by

the

jellium sphere/slab

model. Here _we stress

again

the

important

effect of discreteness and the

corresponding

treatment.

In

Summary,

as the

jellium/slab

model reduces the

ability

^{of the} adsorbed atoms to attract the valence

electrons,

the

jellium sphere/slab

model is

proposed

to simulate the

positive

discrete

charge

distribution of the _true adatoms in the

adlayer.

The calculation shows that the

lip(Z)

x 10~

a,u.~~

l,

p(0.5)-p(0.0)

flake model

,

2

1',

2,

p(0.5)-p(0.0)sphereinodel

D,

o 50 loo x o.15 _a-u- Z

Fig.

3. The

discrepance

_curves of valence

charge

distribution in the

jellium-vacuum region

(I.e. in the

adlayer) by sphere

and flake models for the Cs/~V

(001)

_system at

= 0.5.

JOURNAL DE PHYS>QUE -T 3, N'4, APR>L >993

1058 JOURNAL DE

PHYSIQUE

I N° 4

6p(Z)

_x 10~

a.u.~~

lo

o

Fig.

4. The contour line of

charge density discrepany

Ap(

= p~~jj,~~~~~ p~~j~~~w~) on the

profile

W (I lo) in _a had calculated cell for CsliV

(001)

system at coverage e

= 0.5.

influence of discreteness upon the work function is very

important

and

gives

a contribution of

about 0.4eV~0.8eV to the work function minimum for different

overlayer-substrate

systems.

^{This shows} us that

charge

discreteness in

adlayer

is _an

important

factor in

influencing

the determinations of work function.

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