Angle-Resolved (Inverse) Photoelectron Spectroscopy

Introduction

The roots of angle-resolved photoelectron spectroscopy can be traced back to at least 1887 when, for the first time, Hertz [1] discovered the photoelectric effect. Its interpretation as electron emission induced by ultra-violet radiation was given by Thomson [2] after he announced the discovery of the electron and justified independently by Lenard [3]. In 1905, Einstein published an article [4] including the quantum theory of the photoelectric effect for which he received the Nobel Prize in 1921.

The advent of commercial ultrahigh-vacuum (UHV) systems capable of routinely delivering pressures of 10-10-10-11 mbar, the construction of high-resolution electron energy analyzers based on electrostatic deflection and the development of synchrotron radiation as a tunable, intense, polarized source of ultra-violet (UV) photons made angle-resolved photoemission spectroscopy (ARPES) one of the most advanced and productive methods for the investigation of the electronic structure of atoms, molecules and solids.

Angle-resolved inverse photoelectron spectroscopy (ARIPES) in the ultra-violet energy range is a technique complementary to the now widely used photoelectron spectroscopy, since it can probe electronic states in an energy regime not accessible to ordinary photoemission. It is based on the effect that electrons impinging on a solid surface may emit radiation which has now been known for almost hundred years in the case of keV electrons as X-ray emission [5].

Experimental Arrangement

Figure 1: Schematic illustration of the arrangement of the (inverse) photoemission experiment. J and j are polar angle and azimuth of the outcoming (incident) electrons, respectively (picture from S. Woedtke, Ph.D. thesis).

A typical experimental arrangement is shown in fig. 1. Photons of energy hn are incident at the surface of a single crystalline material. Photoemitted electrons are detected at a polar angle J and azimuth j and their kinetic energy distribution is measured. With incident electrons and outcoming light, you have the set-up of an inverse photoemission experiment.

It has been shown by Pendry [6] that a close relationship between the photon current in inverse photoemission and the electron current in the corresponding photoemission experiment exists. Therefore, the photoemission theory, outlined in theoretical aspects holds with minor differences also for inverse photoemission. Its main difference comes from the phase space and normalization factors when electrons and photons become interchanged. Since the wavelength of ultra-violet photons lph is much longer than that of electrons lel with comparable energy (e.g. E 10 eV yields lph 1200 Å and lel 4 Å) there are much less photon states per energy interval than electron states. Taking all normalization and phase space factors into account, the ratio of the cross sections becomes:
 

(1)

In the typical vacuum ultra-violet (VUV) energy regime this ratio is of the order of 10-5 which demonstrates the experimental difficulties inherent in an inverse photoemission process.

Theoretical Aspects

Following the Green function approach, the photocurrent within the sudden approximation 1 is simply given by the spectral representation of the one-particle Green function:

(2)

where the summation runs over all initial states Yi and final states Yf with energy ei and ef, respectively and denotes the energy of the incident photons. The perturbation to the system caused by the incident radiation is given by

(3)

 

where and Φ are the vector and scalar potentials of the incident light field. Several approximations for M reduce the problem to a managable size. By neglecting multi-photon processes the term in (3) vanishes. If local-field corrections considering the inhomogeneity of the electron system are neglected, one has ∇ = 0 (taking the commutation relation of operators into account) and in the Coulomb gauge Φ vanishes. Since in the UV spectral range wavelengths are large compared to typical atomic distances, can be regarded as independent of (dipole approximation). Within this approximations M becomes

(4)

 

In the non-interacting limit, the photocurrent becomes simply

(5)

 

with one-particle matrix elements

Photoemission data are usually analyzed on the basis of the more simple ''three-step-model''[8,9] consisting of bulk optical excitation followed by subsequent electron transport to, and escape through the surface. This model dates back to at least 1945 [10]. It was shown [11] that in the limit of sufficiently weak electron damping, the transition matrix elements for photoemission become the usual crystal-momentum conserving matrix elements. Thus, besides the energy conservation in (5) the crystal momentum is conserved within a reciprocal lattice vector :

 

(6)

 

The momentum of the photon can be neglected for energies in the VUV and thus, only vertical transitions are allowed. If the final state energy is large enough, and in the absence of inelastic scattering, the electron can be emitted into the vacuum. By passing through the surface, the components parallel to the surface of the final state electron momenta inside and outside the solid are related by

 

(7)

 

where  is a reciprocal lattice vector of the surface and

 

(6)

 

with Ekin, kinetic energy of the outcoming electron and emission polar angle J and azimuth j. Therefore, a Bloch wave of crystal momentum can be transmitted into the vacuum in any of a number of beams. The value of k thereby remains undetermined because of the broken translational symmetry perpendicular to the surface. However, suitable assumptions on that part of the band structure complementary to that under investigation (e.g. free-electron like final states) allow, in principle, the determination of the vertical component of wave vector k within the direct transition model. Nevertheless,improved methods of exploitation such as electron-transition plots (ET-plots [12]) where initial state energies of spectral features are displayed as a function of the corresponding final state energies avoid the problem of determining k and experimental results can be directly compared with theory.

References

[1] H. Hertz, Ann. Physik (Leipzig) 31, 983 (1887).

[2] J.J. Thomson, Phil. Mag. 44, 293 (1897).

[3] P. Lenard, Wien. Ber. 108, 1649 (1899); Ann. Physik 2, 359 (1900).

[4] A. Einstein, Ann. Physik 17, 132 (1905).

[5] W.C. Röntgen, Sitz. Ber. Med. Phys. Ges. Würzburg, 1895, 137 (1895).

[6] J.B. Pendry, Phys. Rev. Lett. 45, 1356 (1980); J. Phys. C14, 1381 (1981).

[7] L. Hedin and S. Lundqvist, Solid State Phys. 23, 1 (1969).

[8] W.E. Spicer, Phys. Rev. 112, 114 (1958).

[9] C.N. Berglund and W.E. Spicer, Phys. Rev. 136, A1030 (1964).

[10] H.Y. Fan, Phys. Rev. 68, 43 (1945).

[11] P.J. Feibelman and D.E. Eastman, Phys. Rev. B 10, 4932 (1974).

[12] L. Kipp, R. Manzke M. Skibowski, Proc. SPIE -- Int. Soc. Opt. Eng. (USA) 1361, 794 (1991).

Footnotes:


1 In the sudden approximation for the final state negigible interaction between the photoelectron and the remaining N-1 electron system is assumed [7].