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
57
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
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© 1998 The American Physical Society
57
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
12 883
«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<z<d.
~11!
The superscripts 1 and 2 refer to the boundary conditions
for ingoing and outgoing electrons, for which the distorted
wave ~11! is matched to a plane wave at the entrance and exit
surfaces, respectively. The crystal target is assumed to be an
infinite slab in the x and y directions, with finite thickness d
in the z direction. The quantities kl and C lg are eigenvalues
and eigenvector components of the dynamic-diffraction eigenvalue problem determined by k and the crystal potential.
The Bloch-wave coefficients a l are determined by the
boundary conditions.
The eigenvalue problem is obtained by substituting the
distorted wave ~11! in the Schrödinger equation for an electron in a crystal,
1
2
¹ 2 x ~ 6 ! ~ k,r! 1 @ E2V ~ r!# x ~ 6 ! ~ k,r! 50,
~12!
where V(r) is the static-crystal potential if the kinetic energy
is so high that polarization effects are negligible. The present
calculation includes a constant imaginary potential that gives
an exponential attenuation factor representing perceptiblyinelastic ~mainly plasmon! excitations. The mean free path is
known empirically. The eigenvalue problem is
12 884
57
Z. FANG et al.
@ 2 ~ E2V 0 ! 2 u kl 1gu 2 # C lg 1
(
hÞg
2V g2hC lh 50,
~13!
where V g is the Fourier component of the crystal potential
for the reciprocal-lattice vector g, which indexes a diffracted
beam, and l indexes a branch of the dispersion surface5 for
k.
The plane-wave amplitude of Eq. ~1! is replaced by the
distorted-wave amplitude
^ qu i & 5 ^ x ~ 2 !~ k f ! x ~ 2 !~ ks ! u C a kx ~ 1 !~ k0 ! & ,
~14!
which is a sixfold linear combination of plane-wave terms
~7! with coefficients defined by Eq. ~11!. The differential
cross section ~4! is integrated over the reaction space, which
is infinite in the x and y directions and has constant thickness
d in the z direction. Details are given in Ref. 6.
III. IMPLEMENTATION OF THE THEORY
The unit-cell orbitals for silicon are calculated by the empirical tight-binding9 method, implemented by Matthews.8
The parameter set used in the construction of an empirical
Hamiltonian is limited to the inclusion of first- and secondnearest-neighbor interactions. The energy-band structure
~10! is then obtained from solving the Schrödinger equation.
For each valence band the Bloch wave function C a k(r) is
constructed from the linear combination of symmetrically
orthonormalized atomic orbitals. Since the wave-function
overlap with neighboring core states can be safely neglected,
only the valence orbitals were included in the evaluation of
the orthonormalized orbitals. The orthonormalized orbitals
are used as the basis states in forming the Bloch wave functions and the coefficients arrive independently through the
solution of a secular equation introduced by Löwdin.10 The
momentum-space representation of the wave function is calculated from the Fourier transform of the orthonormalized
atomic orbitals, from which the energy-momentum density is
obtained for direct comparison with experiment.
The formulation of the electron-momentum density of
crystalline silicon is so far independent of (e,2e) kinematics.
In order to compare with experiment, a particular geometry
must be specified. We use (X,Y ,Z) to label the experiment
coordinate axes and (x,y,z) the target-crystal axes. The experimental coordinate system is shown in Fig. 1, where the
incident electron is in the Z direction and the electron momentum is measured along the Y direction. The polar angles
of the detectors, measured from the incident direction, are
chosen so that q is very close to zero for a coincidence event
in which the directions of k0 , k f , and ks are coplanar. Figure
1~a! shows the range of azimuthal angles over which ionization events are detected. A continuous range of target momenta is detected over narrow lobes in the q X 2q Y plane,
whose direction is essentially q Y . A range of directions of k f
and ks contributes to the same value of q. Details about the
spectrometer have been given by Storer et al.11
The conventional cubic-crystal directions @100#, @010#,
and @001# are denoted by x, y, and z, respectively. This
coordinate system is shown in Fig. 1~b! in which the z direction is normal to the target surface. The crystal slab is
aligned so that the direction of momentum to be measured is
close to the @110# direction, departing by a small angle of
FIG. 1. Schematic of the experimental arrangement. ~a! The
scattering geometry and the experimental coordinate system. The
range of azimuthal angles covered by the analyzers give momenta q
along the Y axis. ~b! The beam energies of the incident and outgoing electrons and the representation of the incident electron direction in the crystal coordinate system.
rotation about the Z axis. As indicated later, this small angle
can be accurately determined from an independent electrondiffraction measurement. Due to mechanical constraints the
experiment is not capable of controlling small tilt angles ~of
the order of 1°! about the X axis. The slab can be rotated in
an accurate way about the Y axis to reach the experimental
positions.
In simulating the experiment, the direction of the incident
beam ~Z axis! is fixed at a given set u x , u y , u z of incident
angles relative to the crystal axes. A method often used for
studying diffraction phenomena over a range of directions is
the stereographic projection. For a plane with Miller indices
(hkl) shown in Fig. 2~a!, the direction normal to this plane is
denoted by n5 @ hkl # . The projected coordinate (x p ,y p ) of n
is obtained from connecting the intersection point of n on the
unit sphere to the projection pole, and is written in terms of
h, k, and l as
x p 5h/ @ l1 ~ h 2 1k 2 1l 2 ! 1/2# ,
y p 5k/ @ l1 ~ h 2 1k 2 1l 2 ! 1/2# .
~15!
The standard ~001! stereographic projection of a cubic crystal, Fig. 2~b!, is constructed using Eq. ~15!. A number of
zone axes ~generated by the projection of the intersection
points between the unit sphere and a plane perpendicular to
the direction indicated by the zone axis label! are also plotted
in Fig. 2~b!. The zone axes indicate where the incident beam
will be strongly diffracted to the direction given by the zone
axis label.
The intersection of two or more zone axes is called a pole.
The @112# pole is of particular interest as it is close to the
incident beam direction. Using stereographic projection, the
incident electron direction can be described by either the
angles u x , u y , u z or the projected coordinate (x p ,y p ). The
ELECTRON-MOMENTUM SPECTROSCOPY OF CRYSTAL SILICON
57
12 885
FIG. 2. ~a! The stereographic projection of the surface normal direction n of crystal plane (hkl). ~b! The standard ~001! stereographic
projection of a cubic crystal. Zone axes in the graph indicate the directions with strong diffraction.
@112# direction, for example, has x p 5y p 50.225, or u x 5 u y
565.9° and u z 535.3°. Diffraction caused by the ~2̄20!,
~1̄1̄1!, ~3̄11!, and ~13̄1! planes will be strong for incident
electron directions close to this pole. However, if the incident direction happens to be exactly on a zone axis, the diffraction is not as strong as expected. The @2̄20# zone axis, for
example, is associated with either the ~2̄20! or the ~22̄0!
reciprocal-lattice vector ~g(2¯ 20) and g(22¯ 0) !. To cause a strong
diffraction k0 needs to be tilted by the Bragg angle towards
one of the vectors to form an exact Laue condition. In this
case the Bragg angle is 1.28° for 20-keV incident electrons.
The @2̄20# zone axis is therefore split into two tracks ~2.56°
apart! corresponding to the ~2̄20! and ~22̄0! reciprocal-lattice
vectors. The width of each track is quite small ~typically less
than 1°! so it is crucial to determine the incident electron
direction accurately.
For a given measured q the contribution of diffracted incident electrons must be averaged over the detection lobes.
The contributions from different outgoing electron directions
are integrated over the range of k f and ks that give q. This
range covers several degrees and therefore a wide range of
diffraction conditions. In practice this integration averages
out the diffraction effects of the outgoing electrons. The free
electrons travel a distance of only 1 or 2 nm in the crystal
due to the small inelastic mean free path of the slow ~1.2
keV! electrons. This makes it plausible that in describing the
experiment one needs to take diffraction into account only
for the incident beam, which traverses a distance of the order
of 10 nm in the crystal.
Using the distorted wave form ~11! for the continuum
electron within the crystal, the beam intensity diffracted to g
is given by
I g~ r! 5
U(
l
U
2
~ a l ! C lg exp@ i ~ kl 1g! •r# ,
~16!
and the transmitted beam intensity is I 0 (r). The average intensity obtained by integrating I 0 (r) over the slab thickness
d is shown in Fig. 3. It has covered a range of incident
directions ~0<x p , y p <0.6! for u k0 u 539.5 a.u. and d
512 nm. Dark regions in the figure indicate strong diffraction, and important zone axes are labeled as in Fig. 2~b!. It
can be seen that for u z ,30° the strongest diffraction happens near the @2̄20# zone axis.
The input data for each calculation, apart from the constant kinematic conditions of the measurement, are the inci-
FIG. 3. A map of the average intensity of the transmitted electron beam as a function of its projected coordinates. The intensity is
shown in linear gray scale with dark areas corresponding to low
transmission. In this figure the minimum transmission is 23% near
the ~101! and ~011! poles. Black dots refer to incident directions
with u z in the range 30°–35°.
12 886
Z. FANG et al.
57
dent angles u x , u y , u z and the target thickness d, which is not
strictly uniform in practice. Within the experimentally possible range, the results do not depend critically on d. To
determine the incident angles, we first align the coordinate
systems (X,Y ,Z) and (x,y,z) with each other. It takes three
consecutive rotational operations to reach the measurement
positions. The first is a rotation around the Z axis by an angle
of u Z ~positive in the counterclockwise direction!, and for
u Z 545° the Y axis is aligned exactly with the @110# direction. The second rotation is around the X axis with a small
tilt angle u X ;1°. The last rotation is around the Y axis by an
angle of u Y , leading to the final position. The incident angles
are obtained from the direction cosines of the Z axis in the
(x,y,z) system. The three rotational operations can be denoted by 333 matrices R 1 , R 2 , and R 3 whose elements are
basically sines and cosines of u Z , u X , and u Y , respectively.
The direction cosines of the Z axis ~cos ux , cos uy , cos uz!
are the last row of the matrix R5R 3 R 2 R 1 . The projected
coordinate of the Z axis is therefore written in terms of u X ,
u Y , and u Z as
x p 5 ~ sin u X cosu Y sinu Z
2sinu Y cosu Z ! / ~ 11cos u X cos u Y ! ,
y p 5 ~ sinu X cosu Y cosu Z
1sin u Y sinu Z ! / ~ 11cos u X cos u Y ! .
~17!
For u Z 5245°, u X 51.2°, and u Y 5230°, the projected coordinate is ~0.183, 0.196!, which is indicated in Fig. 3 as one
of the black dots. This dot and other dots for increasing u Y ,
are all on the @2̄20# zone axis with strong diffraction to
g(2¯ 20) . The theory is now ready to compare with the experiment.
IV. RESULTS AND DISCUSSION
The procedures for preparing ultrathin free-standing
single crystal silicon membranes have been described in detail elsewhere.12 In short, we use a commercial SIMOX wafer @200 nm Si ~100!/400 nm SiO2 on a Si substrate# as the
starting material. A protective layer (400 nm SiO2) is deposited on top of the wafer to ensure the Si substrate is removed
in a wet-chemical etching process, stopping at the SiO2 layers. A sandwich structure (400 nm SiO2/200 nm Si/
400 nm SiO2) with a diameter of 0.5–1 mm is obtained in
the middle of the sample. The SiO2 layers are then removed
in a HF dip. The remaining 200-nm Si membrane is further
thinned by reactive plasma etching using a 80:20 mixture of
CF4 and O2. The film thickness is constantly monitored by a
laser beam and the etching is stopped when the film is about
10 nm thick. The target then undergoes a 30-min Ar sputter
cleaning ~400 eV, ;2 m A/mm2! to remove all the contaminants from the surface. Subsequent annealing removes the
surface damage introduced during the thinning and cleaning
processes.
A transmission electron diffractometer is mounted directly above the (e,2e) scattering chamber. The target orientation in the (e,2e) experiment is accurately determined
from the diffraction pattern. The target manipulator has three
translational (X,Y ,Z) and two rotational ( u Y , u Z ) move-
FIG. 4. The transmission electron diffraction pattern measured
with a 20 keV incident electron beam. The target is oriented with
~a! its surface ~001! plane perpendicular to the beam and ~c! the
~1̄ 1̄2! plane perpendicular to the beam ~after 35.3° rotation around
the Y axis!. The crystal orientation is determined from the analysis
of the diffraction pattern in ~b! and ~d! as described in the text.
ments. The target is first positioned with its surface normal
direction within the Y -Z plane. A diffraction pattern, as
shown in Fig. 4~a!, is then recorded for 20-keV incident electrons. Using the method for analyzing diffraction
patterns,13,14 all the spots in Fig. 4~a! are indexed and the
target orientation is determined to be u Z 5242.2°, u X 5
20.8°, and u Y 50.8°.
The above conclusion is obtained from the analysis illustrated in Fig. 4~b!. The Y axis should be aligned with @1̄10# if
u Z 5245° ~momentum density measured in this direction is
the same as in @110#!. The small deviation angle, which cannot be corrected due to limited rotational range in u Z , is
measured from the horizontal mark ~a dark line in the image!, which leads to u Z 5242.2°. The intensity of ~220! and
~2̄ 2̄0! diffraction spots should be the same if u Y 50. However, in Fig. 4~a! we intentionally adjusted u Y 50.8° to
match the intensities of ~220! and ~2̄20!. This allows the tilt
angle around X axis to be determined as u X 520.8°. Further
confirmation comes from the ring pattern formed on the firstorder Laue zone. The intersection of the Ewald sphere with
the first-order Laue zone, by assuming u X 520.8° and u Y
50.8°, is also plotted in Fig. 4~b! as a circle that fits the ring
pattern nicely.
The incident electron beam should be perpendicular to the
~1̄ 1̄2! plane if the target is rotated around the Y axis for u Y
535.3°. The diffraction pattern in this case is shown in Fig.
4~c!, and similar intensities in the ~111! and ~1̄ 1̄ 1̄! diffraction spots indicate the zero position of u Y is well defined. At
a first glance one may be confused by the stronger intensity
in ~22̄0! compared to ~2̄20!, suggesting a positive u X instead
of u X 520.8°. This is actually caused by the combination of
57
ELECTRON-MOMENTUM SPECTROSCOPY OF CRYSTAL SILICON
FIG. 5. Illustration of the trajectory of incident directions as the
target is rotated around the Y axis. The initial position ~point A! is
close to the ~2̄20! strong line due to a small negative tilt angle u X .
The small deviation angle between the Y axis and the @1̄10# direction is responsible for the change of intensity from ~2̄20! to ~22̄0! at
the final position ~point B!, as observed in Fig. 4.
u Y and u Z as illustrated in Fig. 5. The projected coordinate
for ( u X , u Y , u Z )5(20.8°,0°,242.2°), according to Eq.
~17!, is ~4.6931023 , 25.1731023 !. This is indicated in
Fig. 5 as point A. The trajectory of varying u Y from 0° to
35.5° ~point B! indicates a change of intensity from ~2̄20! to
~22̄0!. The crystal is indeed tilted around @111# by a small
positive angle even for u X 520.8°. This agrees well with
the analysis of the ring pattern and Ewald sphere in the firstorder Laue zone as illustrated in Fig. 4~d!.
The (e,2e) measurements are carried out at six positions
of u Y , varying from 30.1° to 35.1° in increments of 1°.
These positions are indicated by the black dots shown in Fig.
3. The (e,2e) results are shown in Figs. 6 and 7 in the form
of gray scale intensities as a function of « and q Y , and the
FIG. 6. The measured and calculated electron-momentum density for ~a! u Y 530.1°, ~b! u Y 531.1°, and ~c! u Y 532.1°. The intensities of diffracted beams are determined from the target orientation and thickness d. Calculations have been broadened by
experimental resolutions in energy and momentum. The major diffracted beam intensities are indicated at the top of the plot. The
calculated momentum intensities caused by the diffracted beams
agree well with the experiment.
12 887
FIG. 7. The measured and calculated electron-momentum density for ~a! u Y 533.1°, ~b! u Y 534.1°, and ~c! u Y 535.1°. Diffracted beam intensities are indicated as in Fig. 6. The existence of
strong ~1̄ 1̄ 1̄! diffraction is responsible for the poor quality of the
data, even after the data collection time is considerably increased to
obtain better statistics as in ~b!.
theoretical calculations are shown next to the measurements.
Compared with the theory the measured intensity is smeared
in momentum due to elastic scattering. The intensities below
20 eV are caused by plasmon excitation, which is absent in
the theory. The effect of inelastic scattering on (e,2e) intensity is treated as an imaginary part of the crystal potential
@ V(r) in Eq. ~12!#, and the representation of inelastic scattering itself has been neglected. The measured electronmomentum density in Fig. 6 has an extra parabola shifted by
1.7 a.u. in the 2q Y direction, which is very well described
by the theory. This is caused by the diffraction to ~22̄0! as
indicated in Fig. 3 since the vector g(22¯ 0) gained by the incident electron is interpreted by the (e,2e) spectrometer as
q Y , and its length is 1.73 a.u. for silicon.
From Figs. 5 and 3 one can see that when u Y is increased
above 33°, diffraction to ~1̄ 1̄ 1̄! becomes significant. What
happens if g(1¯ ¯1¯1 ) is gained by the incident electron? Since g
is now in the 2q X direction and the spectrometer sits on the
Bethe ridge,11 an (e,2e) event will be detected only if the
target electron has the momentum components @2g(1¯ ¯1¯1 ) ,
q Y , 0#. Unfortunately, the occupation number for electrons
with u qu > u g(1¯ ¯1¯1 ) u 51.06 a.u. is low so the diffracted beam
hardly makes any contribution to the coincidence (e,2e)
counts.
In terms of signal-to-background ratio, the situation
would be improved if diffraction to g(1¯ ¯1¯1 ) simply reduced the
overall count rate. The undesirable effect is that the singles
count rate is not reduced so that the random coincidence
count rate is significantly increased relative to the signal.
This is shown in Fig. 8, in which the (e,2e) coincidence
signal to background ratio is plotted and the maximum coincidence count rate ~at u Y 531.1°! is about 90 counts per
minute. The signal-to-background ratio becomes less than 1
for u Y .33°. The quality of the (e,2e) data is poor even with
good statistics at u Y 534.1° in Fig. 7~b!. Such a strong dependence on u Y is not observed in amorphous or polycrystalline targets. Details about random and coincidence count
rates and their effects on (e,2e) data reduction can be found
12 888
Z. FANG et al.
57
FIG. 9. The electron-momentum density of silicon as measured
along the @1̄10# direction in a target orientation with no significant
diffraction effects. This is achieved by introducing a slightly larger
tilt angle of u X 522.8°. Discussion on the differences between
theory and experiment is given in the text.
FIG. 8. The coincidence (e,2e) signal to background ratio
shows a strong dependence on the rotation angle u Y , which is not
observed in amorphous and polycrystalline targets.
in Ref. 15. The effects of random coincidences are not considered in the theory so the calculations shown in Fig. 7 give
a nice picture of what would be expected if only genuine
(e,2e) events are recorded.
It is, however, not difficult to avoid diffraction in the
(e,2e) experiment. Since in our spectrometer only the diffraction of the incident electron beam is important, by choosing k0 in a light area of Fig. 3 the diffraction effects should
be small. This is achieved by mounting the target with a
slightly larger tilt angle u X , which is determined from the
diffraction pattern similar to Fig. 4. In this case, the crystal
@1̄10# direction is aligned exactly with the Y axis. In Fig. 9
the (e,2e) result with no observable diffraction effects is
shown and the target orientation is measured to be
( u X , u Y , u Z )5(22.8°,31.5°,245°). The theory also predicts minimal diffraction intensities ~about 5% of the incident beam is diffracted! for this orientation.
A detailed comparison between the theory and experiment
reveals more interesting information. In Fig. 6, we see that
the momentum intensity caused by the ~22̄0! diffraction has
increased noticeably with increasing u Y in both the theory
and experiment. This is explained by the trajectory of the
incident beam, shown in Fig. 5, getting closer to the strongest ~22̄0! diffraction line. The ~22̄0! intensity would become
even stronger if other diffracted beams, especially ~1̄ 1̄ 1̄!,
~13̄1̄! and ~3̄11̄!, are absent. In reality, however, more than
50% of the incident beam is diffracted to g(1¯ ¯1¯1 ) for u Y
534.5°. This is the situation demonstrated in Fig. 7, which
one would like to avoid.
Instead of finding directions to minimize diffraction phenomena, the possibility of using diffraction to an advantage
in (e,2e) experiments has been suggested.6 If the direction of
a fast electron is aligned with a low index crystal direction,
reduced phonon and single-particle excitations will occur
due to channeling. One therefore expects a substantial increase in coincidence count rate if the incident and outgoing
electrons are all channeled. This is the case of extreme departure from plane-wave conditions, and the Bloch-wave
electron density is localized at a specific position within the
unit cell. The observed momentum distribution then reflects
the localization function instead of the momentum distribution of bound electrons. Measurement for this purpose was
first suggested by Baker, McCarthy, and Porter.16 To complete such an experiment is quite a challenge.
As indicated before the (e,2e) intensity caused by inelastic scattering is not included in the calculations shown in
Figs. 6 and 7. Current implementation of the theory treats
inelastic scattering as merely an attenuation of the (e,2e)
electrons. Also the theory does not include an explicit implementation of thermal diffuse scattering into the dynamical
theory, so the broadening effect in momentum due to elastic
scattering from atomic cores, which recoil to excite phonons,
is not predicted. Elastic and inelastic scattering in amorphous
or polycrystalline targets has been described using Monte
Carlo simulation;3 however for crystal targets this approach
has yet to be developed to include the diffraction phenomenon. In the absence of significant diffraction as in Fig. 9 the
Monte Carlo approach may be able to provide a detailed
analysis of the electron-momentum density, which will be
presented in a forthcoming paper.
V. CONCLUSIONS
The application of the (e,2e) reaction to crystals enables
one to observe the occupation probability of electronic bands
as a function of energy and momentum along a given crystal
direction. This is achieved in the usual situation when electron diffraction does not cause ambiguity in momentum determination. Difficulties can be introduced when the incident
beam is significantly diffracted as the principal-beam intensity is reduced and the random coincidences increase. The
current theoretical description of the (e,2e) reaction in crystals is based on the dynamical diffraction theory using a
Bloch-wave model. The theory has been successful in describing how the diffracted beams change the observed intensity. A complete description of the reaction and its possible advantageous use in (e,2e) experiments remain for
future investigations.
ACKNOWLEDGMENTS
We would like to thank Dr. O. Reinhold of the Australian
Defence Science and Technology Organisation for his assistance in the Si sample preparation. The study of the electronic structure of solids using (e,2e) spectroscopy was supported by the Australian Research Council.
57
ELECTRON-MOMENTUM SPECTROSCOPY OF CRYSTAL SILICON
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