Department of Physics, The Flinders University of South Australia, Adelaide 5001, ... Engineering, Australian National University, Canberra, ACT 0200, Australia.
PHYSICAL REVIEW B
VOLUME 57, NUMBER 20
15 MAY 1998-II
Electron-momentum spectroscopy of crystal silicon Z. Fang, R. S. Matthews, and S. Utteridge Department of Physics, The Flinders University of South Australia, Adelaide 5001, Australia
M. Vos Research School of Physical Sciences and Engineering, Australian National University, Canberra, ACT 0200, Australia
S. A. Canney, X. Guo, and I. E. McCarthy Department of Physics, The Flinders University of South Australia, Adelaide 5001, Australia
E. Weigold Research School of Physical Sciences and Engineering, Australian National University, Canberra, ACT 0200, Australia ~Received 5 January 1998! Electron-momentum spectroscopy based on the (e,2e) reaction has been used to observe the energy-momentum density of valence electrons in the @110# direction for an ultrathin, free-standing film of crystalline silicon. An asymmetric scattering geometry is used in which the incident, scattered and ejected electron energies are 20.8, 19.6, and 1.2 keV, respectively. The measurement is complicated by the possibility of diffraction of the free electrons. The theory of the reaction including diffraction is summarized and applied to experiments with different target orientations. The orientation is determined from an independent electron diffraction experiment. Very good agreement between theory and experiment is observed. @S0163-1829~98!01220-X# PACS number~s!: 71.20.2b, 72.10.2d
I. INTRODUCTION
The kinematically complete observation of the ionization of a target by a beam of electrons forms the basis of electronmomentum spectroscopy.1 If all external electrons have sufficiently high energy, the difference between the total initial and final momenta for one event is the momentum of the bound electron at the collision instant. The energy difference is the separation ~or binding! energy of the bound electron. In an (e,2e) experiment the energy-momentum density of target electrons is measured. The experimental criterion for sufficiently high energy is that the apparent energymomentum density does not change when the total energy is substantially increased. In the past few years, energy-momentum densities have been successfully observed for a range of very thin (;10 nm) amorphous or polycrystalline solid targets. Some references are given in a brief review by Vos and McCarthy.2 The energies of the external electrons in this series of experiments are about 20.8, 19.6, and 1.2 keV, which very easily satisfy the high-energy criterion for atomic and molecular targets. The overall energy and momentum resolutions are 0.9 eV and 0.15 a.u. ~1 a.u. of momentum corresponds to 1.89 Å 21 !, respectively. The energy-momentum density is interpreted in terms of the independent-particle model for a large crystal, spherically averaged for random orientation. Experiment and theory have closely corresponded for the previous (e,2e) results on solids. Generally, at a certain binding energy only one limited set of plane waves, all with the crystal momentum, contribute to the wave function. The measured intensity of events at a given energy-real momentum combination corresponds to the band-occupation density at that point. It is given by the 0163-1829/98/57~20!/12882~8!/$15.00
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absolute square of the unit-cell orbital for the observed direction in momentum space. In practice the band-intensity distribution is broadened in energy by experimental resolution and by lifetime effects. Further departures from the interpretation of the intensities as occupation densities are of three kinds for amorphous and polycrystalline targets. The first occurs also for atoms and molecules and is due to electron correlations, which split the independent-particle-model state into a manifold of satellites whose momentum-density structure is characteristic of the manifold. The remaining departures are caused by multiple collisions of the free electrons in the solid, before and after the ionizing event. Collisions with energy loss greater than the resolution are mainly due to plasmon excitation. Phonon excitations ~thermal-diffuse scattering! cause imperceptible energy loss but broaden the momentum observation in a way that is modeled by an elastic collision with a target atom, which recoils to excite the vibration. These mechanisms have been included in a Monte Carlo simulation of the experiment by Vos and Bottema.3 The model has had sufficient success in describing the reaction on aluminum4 to justify the claim that the reaction is understood. For a single-crystal target, elastic scattering from target atoms can be coherent resulting in diffraction. This changes the result of the experiment from the expectation of the single-collision interpretation. The simple interpretation of diffraction is in terms of umklapp processes, in which the momentum is changed by the addition of a reciprocal-lattice vector. A formal description in terms of dynamic diffraction theory5 has been given by Allen et al.6 The first crystalline target in the present series of experiments was graphite.7 The energy-momentum densities for different orientations were understood very well on the basis of the independent-particle 12 882
© 1998 The American Physical Society
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ELECTRON-MOMENTUM SPECTROSCOPY OF CRYSTAL SILICON
model, and no diffraction effects were observed. The formal theory of diffraction in the (e,2e) reaction6 has been implemented by Matthews8 to describe the experiments. Strong diffraction effects are predicted for certain orientations of crystalline silicon. The present work describes the experimental observation and the calculation of these effects, as well as the measurement of the energy-momentum density for crystal orientations where these effects are negligible. II. THEORY
The amplitude for the high-energy (e,2e) reaction in the single-collision interpretation is
^ k f ks C N21 u T u C N k0 & 5 ^ 21 ~ k f 2ks ! u t u 21 ~ k0 2q! & 3^ k f ks C N21 u C N k0 & ,
~1!
where q5k f 1ks 2k0 , N21
«5E f 1E s 2E 0 .
~3!
The experimental external-electron momenta are chosen so that they are high enough to describe the electrons by plane waves and the electron-electron amplitude factor of Eq. ~1! is essentially constant. The differential cross section1 is then given by d s 5Kz^ qu i & z2 , dV f dV s dE f 5
~4!
where the kinematic factor K is essentially constant. For a crystal target the orbital quantum-number set i is divided into the crystal momentum k and the band quantumnumber set a. The structure factor of Eq. ~1! is
^ qu i & [F a k~ q! 5 ~ 2 p ! 23/2
E
d 3 r exp~ 2iq•r! C a k~ r! , ~5!
where the orbital in coordinate space is the Bloch function C a k~ r! 5N
N21 21/2
( c a~ r2Rn ! exp~ ik•Rn ! . n50
~6!
The lattice points are denoted Rn and the number of unit cells is N. The coordinate-space orbital of the unit cell is c a (r). The (e,2e) amplitude factor ~5! reduces to F a k~ q! 5N 1/2f a ~ q! d k2q ,
where f a (q) is the momentum-space orbital of the unit cell
f a ~ q! 5 ~ 2 p ! 23/2
~7!
E
d 3 r exp~ 2iq•r! c a ~ r! ,
~8!
and the factor N21
d k2q5N 21 ( exp@ i ~ k2q! •Rn # , n50
~9!
for a large crystal, equates the crystal momentum k to the observed momentum q. In band theory the crystal momentum is arbitrary with respect to the addition of a reciprocallattice vector, but in this reaction its value is restricted to the observed momentum q. The theory has provided excellent descriptions of the energy-momentum densities observed in the previous measurements. Observed events occur on the band-dispersion curves
~2!
are the wave functions for the initial N and C and C electron and final N21 electron states. Here, the momenta of the incident, fast, and slow emitted electrons are k0 , k f , and ks , respectively. The electron-electron collision operator is t. The set of quantum numbers of the bound-electron orbital is denoted by i. The one-electron overlap function ^ C N21 u C N & is the quasiparticle orbital u i & , which in the independent-particle model is the independent particle or Hartree-Fock orbital. The momentum of the bound electron is q. Its energy eigenvalue is «, given in terms of the external-electron energies by N
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«5« a ~ q! ,
~10!
with intensities given by Eqs. ~4!, ~5!, and ~7!. The intensity distributions are broadened by resolution and thermal diffuse scattering and are reproduced at energies lowered by plasmon excitation, whose energy usually is larger than the band spread so that plasmon excitation does not interfere with the valence-band observations. At certain orientations of the crystal target one or more of the external-electron beams experiences significant diffraction. Rather than the plane waves in Eq. ~1!, the motion of an external electron must now be described by a distorted wave that takes the form of a sum of Bloch waves6 inside the crystal,
x ~ 6 ! ~ k,r! 5 ~ 2 p ! 23/2
(l a l (g C lg ~ k!
3exp@ i ~ kl 1g! •r# ,
0
