Physics of Solid Surfaces

284 Pages·1982·15.901 MB·iii-iv, 1-282\284

by  M. Láznička (Eds.)

Studies in Surface Science and Catalysis 9
Proceedings of the Symposium on
PHYSICS OF SOLID SURFACES
Edited by
M.L8znitka, Institute ofPhysics, Czech. Acad. Sci., Prague
ELSEVIERSCIENTIAC PUBLISHING COMPANY
Amsterdam - Oxford - Nnw York 1982
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Prinh!d 10 CZedl')slov.I~la
QUANTUM THEORY OF PHONON MEDIATED ADSORPTION AND DESORPTION
CeroId Doyen
Institut fur Physikalische Che.ie, Sophienstr. II, 08000 Munchen Z,
Federal Republic of Germany
For a theoretical understanding of surface
reactions it is necessary to investigate the
quantum mechanical movement of the reactants
on given potential energy surfaces. The solid
surface provides inelastic channels, which
consist of electronic or vibrational excitations.
Metastable helium atoms show an abnormally large
de-excitation probability at surfaces as compared
to gas phase scattering. The explanation of this
is a dimensionality effect, which reduces the
three dimensional elastic scattering fTom small
particles to a quasi one dImensional one in the
case of a flat surface. This dimensionality
effect is ~lso operating for phonon inelasticity
and leads to unit sticking probability for zero
kinetic energy of the incoming particle and zero
substrate temperature. If the coupling to the
phonon field is sufficiently strong, many phonon
transitions are dominating. In this case the gas
atom is temporarily trapped in a transient resonance
state, in which it exchanges energy with the phonons
before decaying into the possible final channels.
Time reversal yields a close relationship between
sticking and desorption. Assuming perfect
accommodation of the parallel motion of the adsorbate
on the flat metal surface, a relation between the
mean energy, the normalized speed ratio and the
angular distribution of desorbing particles is
derivcd, which agrecs well with the available
experimental data. The developed formalism is
also applied to inelastic reflection from surfaces.
1. Introduction
The application of the adiabatic approximation to the electronic
motion leads to the concept of potential energy surfaces. It i~ the
task of chemisorption theory to calculate them for the gas - surface
interaction. Surface chcmistry investigates the motion of the particles
on these potential energy surfaces and the rates of transitions
between different states on the same or on different potential energy
surfaces. Elas\ic scattering is a (physical) part of this
problem and the theory of clastic scattering from surfaces appears
to be in rather good share (I) • AithouJh the problem of cal-
culating or esti_ating the potential energy surface is largely un-
solved, it oight be very helpful for surface chemistry to investi-
gate, how the particles move under the influence of assumed forces.
1
A detailed understanding of this scattering process could even lead
to a successful reversal of the theoreticai attack, i.e., deducing
important properties of the potential energy surface from the
scattering data provided by atomic and molecular beam experiments.
A wealth of experience and results is available from the theory
of reactive gas phase collisions. An important question is, if
qualitatively different effects can be expected for inelastic surface
scattering. One such effect has recently been observed experimentally
{l} • Although it does not Involve phonons, it is the starting point
of this paper, because it demonstrates a principle which is also
of importance for phonon mediated inelasticity, but is harder there to
observe directly in experiments.
II. Reflection coefficients for metastable Ite-atoms
The scattering of electronicaily excited helium atoms from gaseous
atoms and molecules has been used for a long time to obtain detailed
information about the interaction potentials of the colliding particles
(l}.The cross sections can often be calculated from the experimental
data. In the case of lte(ZlS) and He(lJS) scattering from gaseous CO
the clastic cross section at a relative kinetic energy of 25meV has
been found to be more than hundred times larger than the inelastic
cross sectIon for the dc-excitation process{4}. The Munich group
H. Conrad, C. [rtl, J. KUppers and W. Scssclmann studied the scattering
of metastable Ite-atoms from a dense layer of CO-molecules adsorbed
on a Pd(lll)-surface in extremely carefully performed experJments{l}.
4•
They found a reflection coefficient of the order of 10- A radiative
decay of the metastable lie-atoms due to a long range interaction with
the surfdce was excluded by time-of-fllght measurements. One therefore
has to conclude that the large inelasticitr is due to Auger dc-
excitation processes, whIch are very similar to those occurin~ in
the gas phase, because carbon monoxide is known to adsorb oon-
dlssoclatively on Pd with the gross features of Its electronic
structure being unchanged.
Oeing confronted with this discrepancy between y~s phase and
surface scattering behaviour, it Is useful to recdll the different
nature of clastic scattering in hoth cases. lor an absolutelr flat
surface clastic scattering means Sp~cuidr reflection and this is a
uniquely determined final state. On thc other hand qUdntum mechanlcai
elastic scattering from a potentlaJ JocdliLcd in all three dimensions
is possible in anr direction. Using ho~ normallLdtlon for counting,
14
one could say that there arc 10 eutuoLnn st.ates avatLabLe for ,1.1S
2
phase elastIc scatterIng but only one state for surface elastIc
scatterIng. Hearly all the eidstic channels opcn in gas phdse
scatterIng are closed in the surface case and a drastic change of
the clastic reflectIon coefficIent should not be surprIsIng.
For a theoretical investigation of this effect the HamlJtonIan
is split into five parts. This splitting turns out to be very
convenient for dny kind of Ineiastic scattering from surfaces (~)
and .ill be used below to study phonon scattering.
(1) H =Hin + Vin_loc + Hloe + Vloc-out + Hout
H describes the contInuous set of incoming scatterIng states
in
decaying exponentially at the surface. In the notation of second
quantizatIon it can be written in thd for~l
(2) Hin = Ik t k0nk0
t is the kinetic' energy, k denoles the ~omentum of the incomIng
ko
particle, the zero refers to inncr '1uantum states,i.e., the electronic
confiljuration of the he11U1.1 olnd lhe solId before the collIsion In the
case of Auger de-e~cltation dnd th~ vibratIonal configuration in the
case of phonon scat terJuu, lero kInetIc energy and the i,dt LaI
cOllfl'JlJr.tllon ,h'fin('s the cllergy zero. n Is the eccepetLon nur-,Iler
ko
ouerator for the :;LIl,· /1.;0>. " compriscs the states, .herr the
loc
1','rti"lc I" loealLzcu 'W.lr the solid surface. IIcrc.1 simplr model
ls .I:;SIJ""·,,, where for ('''"'ry internal confilJuration there 15 only
one locdlllrd stolle IA> for the partlclcl
The suuserlpt '\0 reLers lo till! localIzed lJartlclc with the total
sysl"m beIu. In U,c luIt loI conf Iyuratlon "0". Tht· subscrIpt Am
drnntes that tIu- 511 II 1\J('.11 lzed particle has Induccd a trdnsJtion
to tlo.· cIJnfilfur.ltilln ",I". i.e., has untlerlJon" a ,\uIJer de-cltcJtatioll
In llo(' 1'.151' 111101,,1' di scuss Len, Thl' finotl confIlJur.ltiun Is cloaractrrlzed
"II"" I~ t d
hy lilt' total « ,,,' .)1)<1 by the transltJon amplJtude Wm• aAPo,alld Am
oIrc the creat lou .11". oI;slrlJellnn ouerators for the eorrc~pondJn!l
stetes, Lrausi t lons h"llu','" dill .1IIt! IIl0c' I.e., penetrat Lon or the
Inc..,,,in,! p.lrlil'I,· iuIo 1I", Loc.sI lied state IA> .llItI eLrst fe rrrJ~ctJon
·,rl' 'h'5criL"d II)' ll.<: lerL' 'ill-Inc' In :if'contl 'IualltlLdtion this r...dsl
Magnitude and .o.entum dependence of the matrix elements V in
Ak
competition with the de-excitation matrix elements W. determine the
reflection coefficient R. H describes the continuum of scattering
out
states of the particle after the inelastic transition. The hopping
between these states and the localized states is given by the term
Vloc-out. The latter two ter.s, Hout and Vloc-out' need not be
specified for the moment. for the present purpose they can be absorbed
into the Creen function C describing the effective inelasticity in
A
the localized state.
It should be noted that eqs.(l) to (4) refer to many body states
in contrast to the .ore co••on use of one particle states in connection
with the notation of second quantization. Hence the described model
Ha.iltonian characterizes interactions between configurations. The
before_.en~loned special nature of surface elastic scattering is now
contained in the density of states of the involved .any body states.
The density of e!~enstates of H is one di.ensional, whereas H
in loc
and H have three dimensional density of states, because energy
out
and .o.entu. can be transferred into all three dimensions.
The scattering .atrix ele.ent for elastic reflection is given byl
(5) <ko-Iko+> = 1 + ilCA 1m qA
C is the exact Creen function for the state lAo> and the embedding
A
function qA describes its interaction with the continuum of H
in•
It is convenient to assume that C and qA are regular functions of
A
the energy. This means, however, that the limit of infinite nor.ali-
zation volu.e has been taken, in which case the singularities of C
A
and qA transfor. fro. isolated poles to branch cuts. Now the imaginary
parts do no longer tell, whether they arise from one or three
di.ensional densities of states and this involves a danger in cal-
culating the elastic scattering probability. It can be shown that this
.atters only for the states degenerate with the incoming scattering
.ave function and they can be taken into account properly by
introducing an additional resonance at the incoming energy (6).
The final result is thenl
The superscript -r- indicates that C and qA are regular functions
A
in the above mentioned sense. 6N is the percentage of the wave function
k
Iko> contained in the neighbourhood of the energy level (ko. If 6W
k
is unity then the scattering matrix element eq.(6) will be zero and
the reflection coefficient given by
as well. The physics depends now critically on the penetration
potential Vin-loc. If the penetration strength is smail, i.e., if
V is small, the state Iko> has a narrow width in the local conti-
Ak
nuum and nearly all the spectral density Is concentrated in an infinitely
small energy interval around (ko. There will be nearly no coupling to
other scattering states of H but s~rong mixing with the local
in,
continuum, because the local density of states is much higher than
the density o( scattering states.
The interaction of a metastable He-atom with a CO-molecule is
Z._
very weak. The binding energy has been measured to be meV {_}.
6H will be practically unity, because the weakness of the potentiai
k
Vin-loc allows only coupling to deqenerate states. In gas phase
scattering there are equally many states available for elastic and
inelastic transitions at every energy, i.e., there arc no restrictions
by a dimensionality effect. Eq. (5) can therefore be applied with
C and qA taken as rc~ular functions. The iMagInary part of the
A
embedding functIon is given by!
ThIs becomes very small if V gets small and the scatterinll matri.
Ak
element eq.(5) tends then to unity. This e.plains the large clastic
reflection coefficient found in the gas phase experiments.
How assume that the CO-molecules form a surface with nothing
else being changed. TI,en clastIc processes w1ll be reduced to specular
reflection and the dimensionality effect comes Into play. There will
be infinitely m~ny states of Il degenerate _ith /ko> but only one
loc
state of Hi"' Therefore all the intensity has to go into thc inelastJc
channels. lhis is e~pressed by the additional factor in eq.(6). lhus
the proposed model gives a simple ~nd natural explanatIon of the
initially surprising c~perimental findings.
It should be noted that the dimensIonality effect operates only
for weak cuupling. 'or strong coupling the width uf the state Iko>
wIll be large and 6H small. In thIs lImIt eq.(6) becomes identIcal
k
wIth eq, (S). lIence for sufficiently strong interactlon non vanishIng
elastIc reflectIon coeffIcIents from surfaces can be eKpected and
have been found {7}.
III. StIcking coefficIents
In the foilowiny the vibrationai degrees of freedom of the
solid surface are InvestIgated In their Influence on a scatterIng
partIcle. Eqs.(l) to (8) remaIn valId wIth the necessary change in
interpretatIon. H has essentIally the same meaning, eKcept that
in
the subscrIpt "0" now refers to the Initial vIbrational configuratIon
of the solid. ~hen appiyIng eqs.(l) to (&) this InItIal phonon
confIguration Is always assumed to be eKactiy specIfIed. Uefore
comparIng the calculated results,e.g. the elastIc reflectIon coeffI-
cIent eq.(7), wIth eKperiMent one has, of course, to perform the
thermal average over all InItIal states. ThIs (non trIvIal) problea
Is brIefly treated In the neKt two sections. The eIgenfunctIons of
"loc dre gIven by the set{IA>lnd>} , where Ind> Is a product state of
phonons modifIed by the presence of the partIcle In the locailled
state IA>. This Ileans that H descrIbes the Lnteraet Lnq particle-
loc
phonon sytell when the forller Is localized near the surface.
"out Ircludes the backscattered particle when It has eKcIted or de-
eKclted one or several phonons. The eigenstates of H are the
out
final states for InelastIc reflection, .hIch .111 be cKamined In
scction V.
Surface chcmlstry cKperiments do ver~ often not distInquish
between elastIc and Inelastlc reflectIon. in thIs case the relevant
quantity to eKamine is the st LekLnu coeffIcIent, "hlch is defIne•• b)'1
where Hel Is gIven by eq,(7) dnd iilnel Is the pereenta"e of the fluK
scattered Into "out. Assume thdt thc substrate temperaturc Is zero,
I.e., before the SCdtterin~ event there 15 onl) the Lero-polnt motIon
of the Lat t tce vibratIons. There Is a fal.lOUS !.roblt',. of hoI' the
stlckln, coefficient wIll behave In this cdse, If the ~Inetic enery)
of the lncorolnq l'i1rticle tends to zero It'}. riO(, eoufuslon erosc,
because the lonl' estabLlshed ulstcrtcu-wavc dorn d:.;.ro"",.ltlon (V,,,,\)!')}
predicts d vdnishinrJ st Lckln.] eoef f LcIr-ut ill LIds Llntt , which is
,,,It'
clearly a'Jdlnst c"i,er!u;elltal trl,ncls llltuitilln.
6
For the here discussed model the sticking rate in the DWUA Is
given by:
(10)
with Prho the phonon density of states. This formula
has an intuitive interpretation: The particle hops into the localized
state IA> with the transition amplitude YAk and then excites a phonon
with the transition amplitude W • For a wave function IA> decaying
m 2
Iv
exponentially away from the surface , has the following behaviour
Ak
ncar zero kinetic energy:
(II)
V is a number characterLs tLc of tile shape and depth of the adsorption
well, H is the mass of the sticking particle and l is the normalization
length for the one dimensional continuum of "in. Inserting into eq.(IO)
and dividin~ by the incident flux k/(Ml) shows that the OlGA -
sticking coefficient tends to zero proportional to k.
Instead of applyln'j this convcntional procedure, one could use
eqs.(~) and (6), since at zero kInetic energy and zero substrate
temperature H is zero, hecause lncLest ie ref Lect don would require
inel
eneryy transfer from the metal to the ~as atom and there arc no
phonons available to supply it. Therefore the st LckLnq coefficient
Is Just S = I - Hel• The .ipproxLmat Lons enter now as approxin,atiolls
to the Cre"ll funet,Ion C1\. D;Ili!\ corresponds to nejIeet in<) the tel'us
1\ru,)";'\.0aAIn in III.ofor caleuLect Jnq the scat,terLnu weve funetIon.,kc.'o. nJln'J
to "'I.(n) the 11"d~Jlnd'Y ,)dl't of the embedding function 'I~, 15 ZCI'O at zero
kinetic eneryy dnd hence the scatterinu matrix clement eq.(5) would
be unity. It 15 obvious that e'l.(5) has to he used in the D\\I;,\.
because neqLectLnq the \'1"1 - terms nieens that the Green function does
not contain the eouplinq to the three dlmcnsLonaI phonon continuum.
Usinq thc full JoeaI :r.tuilt.Ollldll cq.(J) to calculate the Green
functIon requires, however , the U~H' of eq, (il) for evaluat Lnq the
forw,trd scatt.er iuu "latri, cLencnt , AccordInn to eq, (11), Vin-loc
tends to zero r'Jr k t.endln.j to zcro .md this ImplIes that cll tends
k
to un Ity, Till, UIl..t stlckfn., prnbabfLl ty at zero kinetIc energy is
now cxplalne:f as d conscquence of the dlmensLonalIty effect, Ifltleh
comes into opcrat.Lon, because thr- behavLuur of Vin-Ioc ncar threshold
It''dch is ofl."n referred t.o .15 "trililsmiss.ion l'robll,nl" {.IO} ) makes
7
the effective p.rtlcle surface Inter.ction weak.
For this deduction of unit sticking coefficient it has been
tacitly .ssuMed that the lower edge of the local continuum lies below
the threshold given by zero kinetic energy. This Might .ppe.r obvious,
If the w.ve function IA> is thought of .s a ordinary bound vibr.tion.l
.t.te. The restriction to Ju.t one such state In the HaMiltonian Is
not obviously a good appro_1Metlon. It can, however, be Justified, If
the st.te fA> Is given a special nature. In order to clarify this
point, the followtng for. of the locai Ha.l1toAlan Is lAvestlgated
lA SOMe dchUI
rw r Att,(q)C~Ct,(bqt
(ll) Hloc • tf£tnt + q qbqtbq + t5' + bq)
C~
and c are the creation and destruction operators for the gas atoM
t
In the one p.rtlcle b••ls functions, which are localized In the
t
.dsorptlon well. b and b are the phoftoft operators. The atOM - .olld
• q q
InteractloA Is lincar in the phonoft operators, which Mcans that only
a••ll dlsplace..nta of the Metal atOMS are considered. The coupling
const.nts define a local phofton ~de vi••
tr,'
(1) m. ( l I l
tt,(q»-l/l tt,(q)bq
tl'
The spectral resolution po(w) of the local phonon .ode will u.ually
pe.k at .... phOfton frequency w • For siMplicity aSSUMe that It has
o
a 6 • function ahape.
If the other ph."on .odes arc now orthog_.lhed tilt thc _de 0, the)'
_Ill decouple and lhe local H••lltonlan can be handled by deflnln~
a.
a dlaplaced local phonon ~ode
r
(I') Ao• trAtt.(q)<ctCt ,>
All .pprod....te self-consistent version of .lloc Is Lhen "hen b)' {Ll] I
8