The sticking of a reactive particle on a metal: electron-hole pair and phonon energy loss

Journal of Electron Spectroscopy and Related Phenomena, 45 (1987) 391-402 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

THE STICKING

OF A REACTIVE

AIID PIIONON ENERGY

G.P.

BRIVIOl,

Donnan Liverpool

ON A METAL:

dl

ELECTRON-BOLE

PAIR

LOSS

T.B. CRIIILEY2 and A. DEVESCOVI

‘Dipartimcnto I-20133 Milano, 2

PARTICLE

391

Fisica Italy

Laboratories, 1~69 3BX,

dell’Universlt2

di

University Kingdom

United

of

1

Milano,

via

Liverpool,

Celoria

P-0.

16,

Box

147,

S WMARY The sticking of hydrogen and deuterium atoms into a Morse chemisorption potential on a metal, with energy loss to either electron-hole pairs or phonons is considered. In the electronhole pair case, computations are reported which indicate that the inelasticity may be large with sticking coefficients for both atoms essentially unitv for energies UP to 80 meV. For the whonon problem some details of the app;oach using adiabatic phonons are presented. In this latter case, within the approximations involved in our model, the sticking coefficient of hydrogen atoms is essentially one for energies up to 30 meV, then it drops to zero.

INTRODUCTION The

sticking

conversion

of

elementary stuck

of

to the

reactive

gas

siderable

practical,

models loss

bond for

is

although problems

atom

when

quantum

of

where

instantaneous

atom's

chemisorption

stuck

particle

The

process because

Fully

been

quantum

is

step,

pairs,

a

acting

to

the

and

of

of

a

con-

a chemi-

are computational

actual

in the

classically, excitations if

case one

the

gas

distribution of

energy

because

on the classical

0368-2048/87/$03.500 1987Elsevier Science Pub1ishersB.V.

itself

the energy

handle

difficult

of the

a semi-classical

is treated

easier

to

"(refs.2-5)"

for the elementary

So obtaining

force

In

the

mechanical

when

computed

there

to

atom

sticking

dynamical

motion

only

is an essential electron-hole

and/or

gas phonons)

"(ref.3)".

is used

is given.

have

is due

importance

is strong,

overcome

the gas

the

of this process,

pairs,

inelasticity

with

The

in the process.

strong

trajectory

pairs

theoretical

mechanics

trajectories

exchange

be

of

excitations.

inelasticity

to

surface

energy

a review).

the dynamics

the

still

a solid

a fundamental

and

electron-hole

"(refs.l,6)" atom's

is

is formed

investigation and

for

elementary

studying

to

to

(electron-hole

(see ref.1

contributes

atom

translational

excitations

system

sorption

a gas

the

the

particle

392

denends

on

ron-hole

the

spectral

pair

complication

is absent

chemisorption

semi-classical

for

low

enough

the

task

of theory

lf?VC?l,

To ease

and

Brivio

using

of sticking (u) atoms

and Grimley by

a

investigated

square

hias dominated the solid

1have failed.

using

a Morse-potential

potential,

but

unaltered.

with

The

potential

were

obtained

by

atomic

the square improved alism ness

well

of

the

phonons,

and We

lmplnglng

onto

one-phonon

present

effects

Rule

we can

out

that

gas

chemisorption

theoretical of

results

approximation

of using

ones

those

these

on

the

the

for

(Cu) same

to

use

of this

Morse

the

as

the

adiabatic theoretical

coefficient atoms,

the

exactly

potential)

sticking

copper

In the

for computing

to phonons

an outline

form-

the useful-

systems.

model

with

need to be

further

is

were

obtained

comment

here

Morse

approximation

not the T-matrix

Morse

for

chain

model

the

calculations

(the

we give

obtained

formula,

for

(O-80 motion

approach

results

the unbound

from

whole

atom's

wavefunctions while

the

atoms

classical

the

method

paper

excitations,

gas

some

of

mechanic-

chemisorption

to the actual

loss

Our

fail it is

of hydrogen

pair

in the

report

approach

case.

a linear

scattering

that a semiwe

energy

a to

fundamental

quantum

the actual

different

potential

in this

fully

incoming

the theoretical

with

pair

approach.

the

before

chemisorption

it will

in any case,

at the most

but the present

semi-classical

electron-hole

that

but

phonon

is not

objections

Wentzel-Warners-Brillouin

potential,

coefficient

course

turned

the

analytically,

we are developing

sticking

it

of

state

(they use the Golden

meantime

of

features

are quite

"(refs.3,4)")

same

other

the

( WXBA) . The results

are

elect-

by

trajectory

electron-hole

paper

bound

computed using

the

approximation

all

unaffected

is

of

atom,

Then,

quantum

In this

the static

In their

This meant

surface.

would

that

replaced

energies

by

be assumed

and inelastic

well.

of

particular

mechanics.

with

"(refs.2-5)"

range

this

phenomena

quantum work

instantaneous

importantly

gas

to elucidate

deutcrium

potential

most

of the

the computational

al investigations

near

There

approach,

the

phonons,

surface

calculation

energies

of to

it can

energy

the

and this means

meV),

loss

pro!,lcm "(refs.7,8)".

the

(iI)

For

because

potential Evcnso

excitation. trivial

distribution

excitations.

of

H

computed

in

potential

as

before. The

limitations

and phonon

energy

of our calculations loss phenomena

and the electron-hole

are discussed.

pair

393

LOSS

TO ELECTRON-YOLE The basics

cold

solid

and

D

of

have

have

made

These

i)

used

ii)

employed

for

approach

"(refs.3,5)"

a

to

sticking

on

and calculations

square-well

chemisorption

a

for H

potential

calculations

adiabatic

atom plus

gettering-theory

been published

been

"(refs.4,5)".

PAIRS

the

electron

states

of the coupled

system,

gas

solid; a T-matrix

iii) calculated

formalism;

the T-matrix

in an approximation

intermediate

Interactions

iv)

allowed

a single

v)

included

only

adiabatic

electron

with

energy

electron-hole

which

neglects

nonconserving

states;

pair excitation

in the

states;

multi-step

transitions

between

initial

and final

states. A

convenient

alism

was

cascading The not

square-well

through

the

element adiabatic

the

well

will

the

lead

incoming

We consider

metal

with

atoms

stuck

atom.

Ro=1.3

in

the

bound

gas

states

coupling energy

a more

near

expected Since

top

of

expectation

the

matrix

between

two

initial

denominator

(see

description

the

coefficient

but

near the

between

the

realistic

the

motion

resonances,

difference in

pair

potential

atom's

operator

motion

sticking

This

the

form-

states.

chemisorption

potential.

that

states

and unbound

transmission

actual

nuclear

at low energies

is borne

of

chemisorption of

out by computat-

Model

head-on

with

one

dimensional with

orbital

potential

model

where

the gas atom A

the end atom B of a straight per

atom.

The

ground-state

Il(ref.9)" energy

curve

is

chain elecfitted

potential

So(R)=-B(Ro)(exp[-2(R-Ro)a] with

the

a T-matrix electron-hole

for a llorse potential.

adiabatic

the Morse

the

a pseudo-one

M collides

tronically

of

below

The Theoretical

of

of

the

to a higher

gas

ions reported

of mass

density of

contains

of

to

effects

one may expect

below),

spectrum

in both bound

non-adiabatic

states

the

shell

that

adiabatic

Ramsauer-Townsend

high

states

final

is to say

single

quantum

limit

of

(l)-(v)

approximation

pronounced

lacks

Eq.(2)

of

included

introduces

dissociation

and

which

on the energy

only

also

summary

used

ao,a--0.88aoe1,

- 2expE(R-Ro)a]) a0 is the Bohr radius,

(1) and E(Ro)=-D.lH

394

(Bartree). define

An

infinitely

a discrete

The matrix

high

spectrum

element

state,

assumed

QAB(R)

and

and

to

to

a pair

index refers

the

gas

atom

of adiabatic

to the electron-

state

on

Eo(R)

is

(3)

B;

atomic

orbitals with

distribution

at

on

in sticking

the

of an electron

its value

cascading

changes

are

th e spectral

only

only

<"II>

interaction

iS

nAB(e;R)

if

coupling

the Latin one

ao

=

the Coulomb

and

Greek

at R=b=40

to be "(ref.2,4)"

<:IA>

Here

the

1s placed

gas atom states.

of the operator

states 1 pB> and 1 qy> where ic

barrier

of unbound

are small

shell

B, HEI

is

the ion core of A, and

of the bond

EF, the Fermi

the energy

on A and

level

order

is needed

is allowed,

between

in eq.(2)

because

on the scale of electronic

A

energy

energies.

and n are obtained from Hartree-Fock embedded cluster comQAB AB putations for hydrogen chemisorption "(ref.9)" in the manner described

before

"(ref.2)".

energy

us

on

Eo(R),

and

therefore

Eo(R)

negative

imaginary

the

of

sum

=ra

r,(~~) In

the

the

part

transitions

under

la >

state

in an inital

W to other

acquires

rgy (Em) determining

state

states

a self-energy

the lifetime

Istlck(sa) and inelastic

JCL> with

scattering

on

whose

of Ia > is r inel(Ea ) a

"(refs.2,'3)";

stick(Ea )

first

only bound

makes

sticking

contributions

The gas atom

+

rinel(Ea

approximation

states

)

(4)

when

are considered

all

cascading

as possible

is

final

neglected

and

ones,

rcr inel(Ea) =0 and

ra

(Ed

In

)=

this

sticking

s=R

where

rltiCk(ca case

the

)=m

v coll

~-bound(s~

rate

coefficient

stick'(Rstick

1 _

of

-a~) I tBI

sticking

s is calculated

w 14

2

is Rstick=2ra

(5) stick/pl and

from

(6)

+ "~011)

is the collision

the

frequency,

vcoll=(sa/2Mb

2 l/2

)

,

395

Results

and discussion

The

:?otential

Because near

of

the

(1) has

22 bound

denominator

the top of the potential

for

:I in

results

Fiq.1

were

well

computed

for D in Fig.2

used

states

(eV - sB)

the

II, and

31 for

D.

bound

states.

The

states.

BQ Elm&)

90

Pig.1. Sticking coefficient for hydrogen incoming atOi.1 energy, owing to electron-hole

PO

for

(21, bound state 16 > stick T . So the results

topa

top 5 bound

O.SJ

so

CCi.

dominate

with

the

in

the

ElmoV)

68

40

as a function of pair excitations.

Fig.2. Sticking coefficient for deuterium as a function of the pair excitations. incoming gas atom energy, owing to electron-hole

As pointed

out in the Introduction,

were

function

calculated has

potential, WKBA.

between lated

are

the top bound

most

within

it, by finding states

and with

important

the WKI3A. Since

oscillations

working

tested

analytically

The

by using

several

we

We also

bound

state

were computed analytically, while the unbound ones,

wavefunctions Ia>,

the atomic

inside

the

limits

overall

of the Korse

the gas atom wavethe of

validity

a very

potential

chemisorption

good

of

the

agreement

in eq.1 calcu-

the IdKBA.

feature

of

our

results

(Figs.l,Z)

is

that

396

the sticking

coefficient

in the energy

range,

in the box, of energy We expect a short

(s) of both II and D are essentially

spanning

a fraction

the zero energy range

mechanical

These square and

results

well

Morse

requires

are quite

chemisorption

5 of ref.4).

the box

for for

beyond

energy

b=40

a fully limrt

s

CCL>>80 meV. from

those

potential

(the

broken

to find

known

and by using

The high

state

80 meV.

"(ref.10,

length

different

To illuminate

and 4, the probabilities

0, well

se+

potential,

let Ed go to zero,

unity

atomic

, to about

say Fa

gas atom wavefunction.

, apparently

'0, as aa-

unbound

s+ 0, as

the

by increasing

ao, so that to effectively quantum

the first

of meV,

limit, like

potential

to be reached

example)'

from

this

difference

the gas atoms

obtained lines we

show

with in

the

Figs.3

in Figs.3

II and D in the

-7

Fig.3. energy.

P(Q

) for

hydrogen

as

a function

of

the

incoming

atom

the

incoming

atom

L

eo

Fig.4. P(Q) energy. Morse

for

chemisorption

s,.This probability

deuterium

well

40

as

eo

a function

for the initial

is calculated

E(m.V)

as

of

state

la> as

functions

of

397

P(E~)

= ,," dRI
where

a is the classical

(7)

and d is the R-value

turning

for which

depth,

a

for Ii nor D is there

Xeither

0.

transmission

resonances

ned the energy slower

wavefunction behaviour

(with

no

Grimley-Pisani

the

largest

Eo(R),

is one tenth

of

d = 4.6747

for the square

range

well

(;-, -80 meV)

involved)

potential

P increases

slighly

for both H and D.

a fully nearly

gover-

withsu

quantum

mechanical

reproduces

the

same

parametrizations

model

for nAB(EF,R)

"(ref.9)",

non-adiabatic

sticking

we used rates

here

in eq.2,

that

from

which

for a square

well

gave

poten-

"(refs.2,4)".

Owing eq.(2) than

to

the

is much

properties smaller

for the square

the

overlap,

initial the width

depth, are

much

than' the cpllision coefficients The

inadequate. states,

by

the

such

sticking those

sticking us

reported

on

the

intermediate

like

(5) and moreover

approximation

yet in a position

than

that used

to undertake

inside

the

our

same in

. here

model

states

have

and

sticking

through

are

used

multi-excitation

will

the

calculated

of sa

shell

hitherto

about

computed

the energy

this

between

why

theoretical

to compute

states

in refs.2,4

explains

inelasticity,

and off

in

discussed;

extended

of

rates,

coefficients

that

bound

operator

well

is

final single

to be considerstick(,e ) to compute Ta

ed. In other words, it will be necessary inel and T (Ed ) from the gettering T-matrix formulte

top

as already

is more

a square

This

strong

and multi-excitation

potential ones,

coupling

than

denominator

"toll' are so flat as functions

tells

and cascading

energy

wavefunctions

inside

larger

our

unity For

Morse

number,

that

essentially

a) the

non-adiabatic

in Figs.l,2

fact

the

state than

the

this, paper,

for

determined

potential

and

that:

well corresponding

and the bound

Morse

LOSS

Eo(R)

on

in Figs.3,4.

the possible

the

er

atom

any sign of the Ramsauer-Townsend

of P with

WKBA

gas

by 10Eo(d)=E(Ro),

of s. Instead

calculation

as that

Among

b)

which

than v'Fb,in the energy

A preliminary

tial

d is defined

dependence

of the

the potential

its maximum

i.e.,

point

not

from

approximated

this T-matrix

in a high-

"(refs.3,4)".

We are not

task.

TO PIIONONS

In this energy

loss

section to

the

we will phonon

consider system.

some aspects We will

use

of sticking adiabatic

with

phonons

398

(phonons

in

case

more

than

term

"(ref.3)"

no

coupling operator

the

for

colliding

presence

generalization intend

Let

the

the

atom,

We

fixed

of

til:

where phonons number

atoms

first

adopt

R from

Cd q

in which by

details

of

case

of

a gas

substrate

the atom

atoms.

is straightforward, model

the

The

but

we

left

to

first.

m be

numbered

atom of mass

adiabatic

from

ii?approach

phonons

the equilibrium

that

position

from

the

gas

substrate

for the substrate

chain

is

bq bq

b ' and b are creation and destruction operators for q q of frequency wc. Imposing the boundary condition that atom 1 0 is fixed, the atomic displacements are

we have

the q-values

tan(qa/2)

q

(9)

+b +) q

a, the interatomic

wavevector.

The

distance,

eigenvalue

so that q is

condition

determining

is

= cot(Nqa)

N real

roots

For finite

(10)

in the range

R, we assume

on the chain

to the right)

from

is an extra

=

of

R),

caused

+

9

introduced

II’

some

simple

chain

at be

.

, the Hamiltonian

l-dimensional

there

fixed can

exhibit

the

mass

the

for R*

where

k(R-xN)

of

require

l/2 sin(nqa)(b

with

for

dimensions

y_,= Cq(ti/Nmmq)

a

will

and let the gas

at distance

For R+m

=

we

straight

a

atom

model

chain

right.

gas

processes

the l-d mensional

O.l,..,n,.,N,

atom N, would

I30

with

to compute

right,

and

to three

The theoretical

the

e ents

inelastic

head-on

of

two-phonon

O
that the gas atom exerts

atom N where

its equilibrium term

J"Ndx k(R-x) =

a force

xN is its displacement position

for R+-

(positive

. Consequently

in the Iiamiltonian

xNk(R)

(11)

0

for

small

phonon

displacements

operators

using

xN. eq.(9),

Next

we

express

xN

in

terms

of

the

399

XN

1,

=

h pq

bq+)

+

(12)

where

A

(I

=

l/2

(WNmwq)

From eqs.(8),

sin(Nqa)

(ll)-(13)

H=til

qw4

which,

as is well

bq+bq

we have

the Hamiltonian

(14)

+ k(R) lquq(bq+bq+) "(see ref.11

known

by a transformation

to displaced

Bq = bq + k(R)xq/tiwq,

in terms

(13)

~q+

for example)"

is diagonalized

phonons:

= bq+ + k(R) 'q/Hwq

(15)

of which

(16) Thus

the

phonons

gas

have

A moving phonon E,(R) gas

atom

at R polarizes

the same frequencies gas

atom

atom

these

Vmn(R)

moving

transitions

and

From

see

we

at most

Then we go over acting

adiabatic

on

phonon

E,(R)

can

to

with phonon

this

interaction exchange

E,(R).

The

adiabatic

state,

and

let

potential.

energy operator

with

A the

causing

(17)

that

Vmn (R) can

to second

the

create

or

destroy

two

adia-

"(ref.))".

phonon

quantization state

Im;R>

and express

the operator

in

in

eq.(17),

terms

of

operators:

d/dR=(tiNm) -1'2(dk(R)/dR)lq Similarly

energy

adiabatic

adiabatic

phonons.

= (@2/M)(d/dR

n;R>)

phonons

on

transfer

chain;

substrate

is "(ref.12)"

=

eq.(17)

an

substrate

atom-substrate

initially

phonons

+ 1/2
d/dR,

exchange be

be the corresponding

adiabatic

batic

can

1 m;R>

Let

system.

the

as bare

we can work

~q-~'~

sin Nqa(Bq-8q+)

out the operator

d2/dR2

in eq.(17).

(18)

400 The eigenvalue eq.(18) N+-

condition

by cos(qa/2)

(10) enables

and this allows

us to replace

us to pass

sin(Nqa)

in

to the limit

when

without tial

difficulty.

(eq.(l)

equation,

the

elements and

gas

atom

wavefunctions

third

chemisorption

through to

be

of the

derivatives

the

used

to

sticking

of this

poten-

Schroedrnger form

matrix

coefficient,

potential

determine

V itself.

Finally imaginary

that the static

determines

in a calculation

and

second,

the operator

we

write

part

of

Golden-Rule

down the

the

gas

chain

formula

atom's If

approximation.

the substrate

so

example)

of Vmn(R)

the

w =a

We remark

for

is written

for

rcr(Ed ), the

self-energy

the

phonon

In

the

dispersion

negative one-phonon

relation

in the form

sin(qa/2)

that

cold

Q

for

(19)

is

the

substrate,

highest

and only

(Debye)

allowing

phonon

bound

frequency,

states

then

as final

for

ones

a

as in

eq.(5):

r,( so ) = r stickcca)

= (n2 - U& )%/'w& I<6 I (dk/dR)d/dRla>1'

= (2ti2/M2ma2)~B

where Ha,6 final

=sa-s6

gas atom

Results

onto

we display

a linear due

calculated bound

the sticking of

between

of

the

same

energy one

range

and this

the initial

and

(Fa -30

potential

shows

pair

multi-excitation

region,

meV)

that

sticking

states

for

used

in

the

as before.

inelasticity

stressed

final

as

by eq.(6)

the

a large as

of the incoming = 2r stickiti is a stick states and the two top

R

the unbound

Morse

(s) of H impinging

function

s is determined

In

problem,

as

excitations.

excitations.

this

coefficient

atoms,

in eq.(20)

pair case.

essentially

Cu

one-phonon

by using

states

the

chain

to

electron-hole In

difference

dk/dR=-d2Eo/dR2.

and discussion

In Fig.5

energy

is the energy

states;

(20)

the

coefficient

is

due to phonon electron-hole

and cascade

401

00’

I PO

40

Em

E(m.Vl

Fig.5. Sticking coefficient for hydrogen as a function incoming atom energy, owing to phonon excitations. processes,

presently

In the energy to zero,

owing

ignored,

interval

to a much

ions and to the Debyc difference

(30-40

between

lower

frequency

this

will

have

meV)

the sticking

to be taken

probability cutoff.

behaviour

of

the

account.

coefficient

of one-phonon

In order the

into

of

drops

excitat-

to understand

sticking

the

coefficient

in

Fig.5

and that due to electron-hole pair excitations in Fig.1, stick compare T in eqs.(2),(5) and (20) as function of (E~-EB). 0 stick Finally we remark that, while in both Ta for electron-hole pairs inverse

(eq.(5)

al models), tation pairs

and phonons

squared such

values and

properties

mass

(eq.20)

dependence

a dependence

of

phonons.

the

of the bound

is

is the same

is completely

coupling

This

there

(not predicted

operator

a quantum

and unbound

obscured for

effect

atomic

incoming

atom

by the semiclassic-

both

by the expecelectron-hole

determined

by

states.

REFERENCES

1

2

3 4 5 6

7 8

K. Schoenhammer and 0. Gunnarsson in "Many Body Phenomena at Surfaces" eds. D. Langreth and 13. Suhl, Academic Press, Orlando, 1984, p.421. G.P. Brivio and-T.B. Grimley, Surface Sci. 89 (1979) 226; G.P. Brivio and T.B. Grimlev, Surface Sci. 131 (1983) 475; G.P. Brivio and T.B. Grimle;, Surface Sci. 161 (1985) L537. G.P. Brivio and T.B. Grimley, Phys. Rev. B 35 (1987) 5959. G.P. Brivio, Phys. Rev. B 35 (1987) 5975. G.P. Brivio and T.B. Grimley, J. Vat. Sci. Technol. (1987) in press. K. Schoenhammer and 0. Gunnarsson in "Dynamical Processes and Ordering on Solid Surfaces", eds. A. Yoshimori and M. Tsukada (Springer Verlag, Berlin, Heidelberg) 1985, p. 57. D. Kumamoto and R. Silberg, J. Chem. Phys. 75 (1981) 5164. S. Shindo, J. Phys. Sot. Jpn. 52 (1983) 562; S. Shindo in

the

402 "Dynamical Processes and Ordering on Solid Surfaces" eds. A. Yoshimori and Eel.Tsukada (Springer Verlag, Berlin Heidelberg) 1985, p-20. 9 T.B. Grimley and C. Pisani, J. Phys. C: Solid State Phys. 7 (1974) 2331. 10 W. Breing, Z. Physik B; 36 (1980) 227. 11 G.D. Mahan, "?+lany-Particle Physics" (Plenum Press, New York) 1981, p.269. 12 T. B. Grimley, Phllos. Trans. R. Sot. London, Ser.A, 318 (1986) 135.