ARTICLES
Spectral momentum density of electrons in copper
X. Guo,* Z. Fang, A. S. Kheifets,
†
S. A. Canney, M. Vos,
†
and I. E. McCarthy
Electronic Structure of Materials Centre, The Flinders University of South Australia, Adelaide, South Australia 5001, Australia
E. Weigold
Research School of Physical Sciences and Engineering, Institute of Advanced Studies, The Australian National University, Canberra,
ACT 0200, Australia
Received 28 July 1997
The spectral-momentum density of electrons in a copper thin film has been directly measured using electron
momentum spectroscopy. The measured spectral-momentum density shows two distinct features. The first is a
free-electron-like parabola with dispersion spanning 10 eV in energy and 0.65 a.u. in momentum. The other is
a weak and extended band located in a narrow range of energies from about 2 to 5 eV below the Fermi level.
A spherically averaged linear muffin-tin orbital LMTO calculation of copper reproduces these features in
both the dispersion pattern and the intensity. After taking into account the elastic and inelastic multiple
scattering through a Monte Carlo simulation, the agreement between the calculation and the measurement is
good. The measurement and the LMTO calculation are also compared with an available linear-augmented-
plane-wave calculation for the energy-integrated electron momentum distribution of the valence band and the
agreement is also good. S0163-18299800812-1
I. INTRODUCTION
The electron spectral-momentum density is the energy-
resolved momentum density distribution of electrons in sol-
ids, which provides detailed information on the electronic
structure of solids. For example, the three-dimensional
energy-momentum dispersion pattern of electrons in a solid
is obviously contained in the measured spectral momentum
density. Over the last ten years electron momentum spectros-
copy EMS or ( e ,2e ) spectroscopy has developed into a
powerful technique to measure directly the electron spectral-
momentum density of crystalline solids and, in particular,
structurally disordered solids.
1,2
Electron momentum spectroscopy is based on the ( e ,2e )
reaction.
3
In an ( e ,2e ) experiment all kinematical parameters
are accurately measured. These parameters are incident elec-
tron kinetic energy E
0
and momentum p
0
, scattered electron
fast electron kinetic energy E
f
and momentum p
f
, and
ejected electron slow electron energy E
s
and momentum
p
s
. At high incoming and outgoing electron energies and
large momentum transfer K=p
0
-p
f
, the ( e ,2e ) cross sec-
tion is dominated by binary collisions of the incident electron
and the bound electron in the target. The binding energy
and momentum q of the bound electron before the collision
are then determined via energy and momentum conservation,
neglecting the recoil energy of the ion:
E
0
- =E
f
+E
s
, 1
and
p
0
+q=p
f
+p
s
, 2
where q is the real momentum of the bound electron.
Within the independent particle approximation the ( e ,2e )
cross section is proportional to the modulus square of the
bound electron momentum space wave function | ( , q) |
2
,
i.e., the electron spectral momentum density ,q.
4,5
For
different kinematic conditions that correspond to different
momenta q and different binding energies via Eq. 1 and
Eq. 2, the measurement of ( e ,2e ) cross sections is thus a
direct measurement of the spectral momentum density ,q
of the bound electrons in the target. Thus the method is com-
monly referred to as electron momentum spectroscopy. Since
the measurement does not rely on the crystal momentum,
EMS applies equally to ordered crystalline or disordered
samples.
5,6
Conventionally, density of states DOS is used to de-
scribe the electronic structure of solids. The DOS is propor-
tional to the number of one-electron states integrated over
momentum space in the energy range from to +d .
Calculated densities are often compared to x-ray photoemis-
sion spectroscopy XPS spectra. The spectral intensity of a
band state as measured by photoemission depends signifi-
cantly on the dynamics of the process since the cross-section
is related to the atomic orbital origins of the band state and
the incident energy of the light source.
7
We take the photo-
emission measurement of CuO as an example.
8
The cross-
section ratios, weighted by the number of electrons per atom,
are ( O 2p )/ (Cu 3d ) 2.16, 1.05, and 0.03 for He I 21.2
eV, He II 40.8 eV, and XPS AlK , 1486.6 eV light
sources, respectively. That is, in XPS one has primarily
emission of d electrons, whereas in He I and He II spectra
one sees more or less equally both d and p electron emis-
sions. Therefore, detailed information on the DOS and on the
PHYSICAL REVIEW B 15 MARCH 1998-I VOLUME 57, NUMBER 11
57 0163-1829/98/5711/63339/$15.00 6333 © 1998 The American Physical Society