Photoemission Prom Oriented Orbitals

PHOTOEMISSION FROM ORIENTED ORBITALS I. Bartos institute of Physics, Czech. Acad. Sci., Prague Differential and total photoemission cross-sectlon of oriented orbitals are evaluated for a short-range atomic potential and their dependence on the energy and the polarisation of the incident radiation Is discussed.

1. Introduction Atomic and molecular electron orbitals, which are oriented arbitrarily in space when in the gas phase, get uniquely oriented when atoms and molecules are regularly adsorbed on crystal surfaces. The valence electrons form directed bonds to their substrate neighbours and also core levels get polarised when their magnetic quantum number degeneracy is lifted in the anisotropic field at the surface. A

less Integrated information can be obtained In photoemission from

such oriented centers than It is for gas atoms where many effects. especially those connected with the angular distribution of photoelectrons, are averaged out. This can contribute to a better understanding of adsorbate crystallography and bonding. Photoemlsslon cross-sections depend on parameters of the incident radiation, e.g. on its energy and polarisation. The strong energy dependence, which is of great importance for quantitative determinations of Individual atomic species at the surface of a

sol-

id, Influences also strongly angular distributions of photoemitted electrons. Thus it Is important to evaluate correctly the cross-sections; in doing so, also substantial differences between the final electron state in a free atom and in a

solid should be taken into account, while a created hole has a

long-range effect on the outgoing electron in the gas phase, the screening effects in a

solid transform this Interaction Into a short-range one. In

addition, electron scatterings from other atoms take place in a

solid.

As the reflectivity of low energy electrons from surfaces is small (of the order of one percent), we neglect here any further scatterings of the outgoing electrons with other atoms. We arc then left with a one-center problem as proposed In

(11,

for which we shall use the short-range

potential of the muffin-tin type. For such a

spherically symmetrical problem,

symmetry conditions only allow to draw conclusions about the angUlar distribution of photoelectrons, excited by a linearly polarised radiation of any polarisation. Still further simplification, consisting In neglecting completely any Interaction between the Ion and the outgoing electron, the plane wave approximation for the final electron state, Is discussed In the following

164

paragraph. 2. Plane wave approximation for a single center In the one-electron scheme, the theoretical description of photoemlssion reduces to the evaluation of relevant optical matrix elements. Similarly to the analysis of optical phenomena, initial states (occupied), are bound electron states; in contrast to it final states in photoemlssion are unbound scattering electron states in the continuous part of the energy spectrum( above the vacuum energy level). If there were zero coupling between the photoemltted electron and the ionized centre left behind, the final state would be simply a plane wave and the matrix element could be easily evaluated. This simplification has appeared not to be generally valid [2J, however, as will be also demonstrated next.

1c

or 10 ~

Fig. 1

~-------

........

---

Photoemlssion cross-section 6 of the s-level In the square well 2 = 0.25 eVnm 2 (a-radius. V-depth of the well): in

with the strength va

the plane wave approximation (dashed) and exact (full line). In Fig. 1, the energy dependence of the photoemlssion cross-section 6 IE) for the s-level in a square well potential is given

(3). The result,

obtained In the plane wave approximation, deviates appreciably at low energies where a sharp structure connected with a p-type resonance is present. The angular distribution of photoemltted electrons, which is governed by the differential cross-section, can also be easily evaluated in the plane

wave approximation. There, the optical matrix element gets related to the Fourier component of th e initial orbital, which has the same angular part as the orbital itself

[4]. Thus, spatial distribution of photoemltted electrons

should be closely related to the shape of the Initial state orbital. This approximation has been used e.g. for plotting angular distributions of electrons photoemltted from an s-orbital, coupled with underlying d-orbitals In

[5].

Again, more elaborate calculations as well as experimentally measured anIsotropies appear to be more complicated, e.g. qualitative changes of the profiles are observed with changes of energy of the exciting radiation. These facts stress the Importance of an adequate description of the final electron state in photoemission and Instead of its plane wave appro165

xlmatlon we shall be uslog a true eigenstate of the slogle center Hamiltonian. 3.

Outllne of the theoretlcal descriptlon The usual approach. leadlog to the golden-rule form with an incom-

Ing wave state for the final electron state. can be replaced by a more straightforward procedure ba~
[6J.

From the Lippmann-Schwinger equation for a system of interacting electrons and photons, follows an lotegral equatlon for the wave function of emitted electrons:

'+1(1) -

(1)

If;.

where

€

J G'(r,r'iEj) r..e

'fI;.(r.) d 1 r '

is the wave functlon of the initlal bound state wIth energy E • i denotes polarisatlon of locldent photons and the Green function of an

= E + taW. f i For a spherically symmetrical potentlal of a centre. to be studied

electron is taken at energy E

here, It is natural to use the angular momentum repl'esentation, Then (for initial (lLmi.l-state):

and G

• the l-th radlai component of the Green functlon. can be expressed I in terms of solutlons of the radial Schr8dlnger equatlon. satysfying one-sIded boundary conditions. as:

G I ( r,r ') • Bh (rl R1ULl WI

2m,.,:

for

~

r> r'

Flnally. the Wronskian of these two solutlons for the short-range potential Is simply determIned by Its phase shifts

di :

WI.... e'\ 6'"1 (E) From the asymptoties of (1). the electron cu,"rent density and subsequentty the relevant. differential cross-section of photoemlsslon from the loittal state i is derlved:

166

Here, the angular Integral of the product of three spherical harmonies (

Gaunt coefficients)

11 -

Ii

!

1, m

l defined as Rltl

... m

=

After Integration of

(3)

l

contains optical selection rules'

' m

f

i

!

1 and the; Rlil

, radial matrix elements are

d r r 3 Rli (r) R1< (r) d 5/dJl over.o. , the total

photoemlsslon

5

Is obtained:

O~ (~t~nw5T2IM;i(nV1 2 2 I M;p'\) 1 ~:I R1;) Y, m-ml. (n€)· JdJl yrm Y1m-mi Y!imt 12

It Is Interesting to compare formulas (2)

with (3)

for differential and

total cross-sections' while In the angular resolved case (2) contributing (I,m) -

the Individual

channels contribute with the amplitUdes, which can give

rise to Interference effects, In Its integrated counterpart (3)

only the Inten-

sities of these channels are to be added.. Thus, for example, the phase shifts

do not appear In the expression for (5

•

4. Application For photoemlsslon of electrons from atoms In solids we shall use a short-range potential mstead of a long-range Coulomb potential which has to be used in treating the photoemlsslon from gas atoms. We shall use a muffin-tin type potential and we avoid the plane wave approximation for the final state: that means that both, the initial and the final electron states will be described by eigenfunctions corresponding to this potential. Numerical results will be given for oriented inner levels of aluminium (2p). First, the sensitivity of the cross-sections to small changes In the shape of the potential has been tested

[6]. Two atomic potentials for Al

crystal have been used: one obtained by means of Mattheiss construction and another one obtained from the selfconsistent procedure for the crystal.

It appears that the dominant d-channel contributions are practically identical in the whole energy region studied: the s-channel contribution is somewhat smaller for the self-consistent potential, however. Therefore, at least as far as tightly bound inner electron levels are concerned, there Is only a small sensitivity of the results to th~ details of potential. Also, our comparisons with calculations for long-range free atom potentials

[7]

displayed

only minor dlfferencles for energies above the vacuum level. Substantial differences have been found at energies just above the muffin-tin zero, but this energy region Is only of academic interest for external photoemlsslon. a/ Energy dependence of cross-sections In fig. 2a, the energy dependence of the cross-sections for the 2p level of Al is shown decon.posed into two contributing channels sand d. The contribution of the s-channel decreases on Inc roe slng the energy. That of the d-channel Is small at lowest cnel"gies because of the repUlsivo effect of the term in the effective potential, but g(>ls rapidly enhanced

167

at higher energies and represents there a

substantial contribution to the

total photoemlssion cross-section. The two regimes, where s- or d-channel dominate, manifest themselves distinctly in the angular distribution of the photoemltted electrons. These

effects

illustrate for the 2pz orbital in fig. 2b, have been dis-

cussed in detail in

[6]. Qualitative changes of the angular distribution

with energy as well as strong interference effects between the two channels cannot be obtained in the plane wave approximation; there only the Intensity of

dfJldfl but not the shape can vary with energy.

30

•

E,

.0

0:11/2

E ttY]

Fig. 2 Photoemission cross-sections

5

6'2~~~Of

the 2p z level of AI,excited

by the z -polarised radiation:

al

s- and d-contributlons to the total cross-sectlon

f5

(energy E

is refer-

red to the muffin-tin zero)

bl

polar plots of the differential cross-section

d6Pz/d.C1.;

the fourfold symmetry is used to place plots for four energies into one picture (

bl

E

-

1,5,9.5 and 30 ev)

Polarisation dependence of the cross-sections

If the energy of the incident radiatlon is kept fixed and only the polarisation direction is varied, then instead of energy dependent phase shifts and radial matrix elements only their values at this energy are needed. The situation gets further simplified for excitations from the 2p-level if the energy region is treated at which the role of the d-channel is dominating. Then,contribution of the s-channel can be neglected and we are left with interferences between terms with different m-values within one I-channel (1 -

2). The differential cross-section of the Pz orbital is then:

It is to be noted that tho phase shift got completely eliminated from this

168

expression for do/do and the radial matrix element R enters only as a multiplicative factor, scaling the intensity of photoemitted electron flux. Polar plots os do/do for a linearly at several angles

pz-orbital, excited by a

Ele:

radiation, polarised by

with respect to this orbital, are shown in

fir. 3.

I

1f/8

1f/4

Fig. 3 Polar plots at 'f - 0 at various polarisations

g-O

1f/2

31f/8

of differential cross-sections for a

p z -orbital

6€ of the incident linearly polarised radiation ( s-

-channel contribution neglected). 'The inversion symmetry in El

is used to

reduce the plots to angles from -':iT/2 to ~Ji"!2 ; other polarisations are given by the symmetry: (El,€Ie:)-+ (-El,-Ele:l. In this context, a support for the assumption that the role of indirect processes with electron scatterings from the substrate atoms is small in comparison with that of direct processes should be mentioned. 'The results shown in fig. 3 are closely similar to the relevant results of angular distributions for 0

(2p-level)

[8], where

on NI have been evaluated including

the substrate scattering. So, even in angular studies the anisotropy of the emitter may be a

dominating factor determining the directional distributions

of photoelectrons emitted from the adsorbate atoms. Finally, a special case of a simultaneous change of both El such a way that vector)

El

= Ele:

and Ele: in

(emission in the direction of the polarisation

is of importance. Such profiles can be obtained at a

special rigid

configuration of the spectrometer by rotating the sample only. Then, for

e

= El

e

, we get from (4):

M", cos 8 (3cos 2 8-1) + Sin2. which is a (cos

El

e. 3 cos EJ •

2· cos

EJ ,

result which follows from the ple.ne wave approximation too

is the angular part of the Pz -orbital). This result, which has been

shown here for a

special case. can be obtained quite generally from (2)

and it has been used [9]

to explain the fact that some of the earlier

experimental XPS data were successfUlly interpreted in the plane wave approximation. c/ Simulation of oriented molecular orbitals In contrast to core electron levels, which can only get oriented by 169

lifting their m-degeneracy in the anisotropic field from surrounding atoms (the effect is rather small [~oJ

).

the valence electrons can form oriented

bonds between the neighbouring atoms by hybridisation, Le. by mixing states with different I ' s. Once the initial orbital is known, the photoemission process can be described by evaluating corresponding optical matrix elements. Multiple scattering processes take place in the final state and should be taken into account. Here, we use the same approximation as in case of core levels: i.e. going beyond the plane wave approximation by considering the effect of atomic potential on the final state, but neglecting any further scattering effects. For a description of photoemission from a

6

molecule, oriented along the z-axis, we adopt a ding to

[~J.

Then::

M (11,n l;)~ cos Ell; [-3 (cos2 8 - 1 )+~ cos El 1+ sin

-bond of the diatomic

simple s-p model, accorE)€

[-l sin 20+~ si n 0J

and corresponding polar plots of d5/d.a , which in contrast to those from fig.3 do not possess the inversion symmetry, are shown in fig.4.

9:0

Fig. 4 Polar plots of d6/dn

for a z-oriented 6 -bond at several polarisa-

tions of the incident radiation E>l;' Plots for other polarisations can be obtained from the symmetry: (E>,0l;)-(-O,-O£) i (-El,9T-O e). A

rather strong characteristic polarisation dependence of the polar plots is

obtained. Surprisingly, the angular distributions shown for 8£ are very similar to those obtained in though in

[~~J

[~~J

for the 5 6

0

and 'jj/2

orbital of CO,

a full multiple scattering in the final state has been con-

sidered.

~70

5. Conclusion Pronounced changes in the spatial distribution of photoelectrons emitted from oriented orbitals are predicted when energy and polarisation of the incident radiation are varied. As the orientation and the shape of orbitals for atoms and molecules adsorbed on crystalline surfaces is governed by the geometry of the adsorption site. the angular resolved photoemission both fron- the core and valence states should

give an

additional or comple-

mentary information about the surface crystallography and U·.e type of bonding of adsorbates. 'I'hough the interpretation of profiles in most cases cannot rely upon the plane wave approximation for the final state, the short-range atom representation used here promises to give adequately the dominant part of the anisotropy of pnotoemteston, being still a sim.ple procedure to work with. In addition, the results for oriented orbitals may serve as an anisotropic input into schemes where subsequent electron scattering effects have to be taken into account. References

[I]

N. J.Shevchik, J. Phys. C: Solid State Physic g( 1978) ,3521

[2]

A. Liebsch, Phys. Rev. Letters

E

(1974) ,1203, Phys.Rev. B13 (1976),

544

[3] 1. Bartos, F.Maca, Proc.6th Conf.Czech.Physicists 13-07,Ostrava( 1979) (in Czech.) [4]

J.W.Gadzuk, Solid. State. Commun.

[5)

J.W.Gadzuk, Phys. Rev. B10 (1974) .5030

[6]

I.Bartos, F.Mlka. Phys.Stat. Sol.( b)

[7]

15 (1974). 1011 99 (1980) .755

S.'I'.Manson. in Photoemission in Solin., I, Eds. M.Cardona end L.Ley. Springer Verlag. Berlin (Heidelberg) New York 1978

[8]

M.Scheffler, K.Kambe. F.Forstmann. Solid State Common. ~'5 (1978) .93

[9]

S.M.Goldberg, C.S.Fadley. S.Kono, Solid State Commun. 28

(1978) ,459

(10)W.Eberhardt, G.Kalkoffen. C.Kunz. Solid State Commun. 32 (1979) ,901 [11] J.W.Davenport, J.Vac.Sci.'I'echno1. 15 (1978),433

171