Dynamic screening and electron–electron scattering - Donostia ...

V.M. Silkin et al. / Surface Science 601 (2007) 4546–4552 4547
[17]. In this study it was proven that e–e and the e–ph
mechanisms are equally important for the hole decay for
1 ML coverage. Very recently a joint experimental and theoretical
investigation of electron dynamics for Cs and Na
adlayers on Cu(111) was performed as well [18]. Here,
we investigate the energy dependence of the e–e contribution
for several positions of the QWS band relative to the
Fermi level E F , which, in turn, depends on the Na coverage.
For the sake of comparison with the experiment, a reasonable
assumption could be that the e–ph contribution
should not change significantly for such small variation
of the QWS energy.
As a second case study, we analyze dynamic screening
properties in metal clusters. In clusters and nanoparticles,
chemical, optical, and electronic properties can be tuned
by varying the size and/or the shape of the system. Besides
its fundamental interest, tailoring the electronic properties
can be of vast importance for many processes of technological
interest, such as photochemical reactivity. Over the last
years, the advent of femtosecond lasers has fueled the
experimental study of electron relaxation processes. Particular
attention has been paid to the dependence on size of
the electron–electron interaction processes [19–21]. We will
show that the interplay between screening effects and space
localization of the initial excitation makes the lifetime of
the excitation vary with respect to the bulk equivalent.
2. Theoretical methods
In very recent years, improvements in experimental
methods, in particular in the field of ultrafast laser spectroscopies,
are making it possible to study electronic excitations
at time scales below the femtosecond [22,23]. In this
time scale, the screening of excited electrons in condensed
matter can be incomplete and the description of the electronic
excitations as quasiparticles is questionable [24].
However, most standard experimental techniques provide
information on time scales for which the screening of excited
electrons can be approximated as instantaneous in
practice. In this case, the description of the many-body
electronic excitation can be simplified by means of the quasiparticle
picture. In the following, we focus into this
situation.
The scattering rate of an excited electron or hole due to
inelastic e–e scattering can be evaluated by means of
many-body calculations based on the electronic self-energy
[25,26]. In this approach the lifetime broadening of a quasiparticle
in a quantum state characterized by an energy E 0
and wave function w 0 is obtained as the projection of the
imaginary part of the self-energy R(r,r 0 ,E 0 ) onto the state
itself (atomic units are used throughout, i.e., h = e 2 =
m e = 1, unless otherwise is stated)
Z Z
C e–e ¼ s 1
e–e ¼ 2 drdr 0 w 0 ðrÞImRðr; r0 ; E 0 Þw 0 ðr 0 Þ: ð1Þ
Frequently the self-energy R is represented by the so-called
GW approximation [25], which is the first term in a series
expansion of R in terms of the screened Coulomb interaction
W. Usually, the non-interacting Green function G 0 is
used to replace the full one-electron Green function G. In
the following we provide a brief description of the numerical
methods used in the calculation of scattering rates in
the two examples considered in this work, namely, quantum
well states at metal surfaces and metal clusters. A more
detailed account can be found elsewhere [26,27].
2.1. Quantum well states
In the case of Na adlayers on the Cu(111) surface, we
assume that the charge density and the one-electron potential
only vary along the z-direction perpendicular to the
surface and are constant in the (x,y) plane parallel to the
surface. This assumption is valid for Na induced QWS’s
at coverage rates close to 1 ML, since the wave functions
lie mainly in the Na layer and in the vacuum side, i.e., in
a region with little potential variation in the direction parallel
to the surface [28]. This leads to significant simplification
of Eq. (1), which now takes a form
C e–e ¼ s 1
e–e ¼ 2 X 0
f
Z
Z Z dqk
ð2pÞ 2
dzdz 0 / 0 ðzÞ/ f ðz0 Þ
Im½ W ðz; z 0 ; q k ; E 0 E f ÞŠ/ 0 ðz 0 Þ/ f ðzÞ; ð2Þ
where the summation is performed over all available final
states f, W(z,z 0 ,q k ,x) is the two-dimensional Fourier transform
of the screened Coulomb interaction, and / n (z) is the
z-dependent component of the wave function w n (r):
w n ðrÞ ¼p 1 ffiffiffi e iq kr k
/ n ðzÞ
ð3Þ
S
with S being a normalization area. To describe the Cu(111)
surface covered by Na adlayers, we employ a slab containing
31 atomic layers of Cu together with a region corresponding
to Na and a vacuum region corresponding to 20
Cu interlayer spacings. For the description of the surface
electronic structure, the model potential of Ref. [17], based
on the model potential of the bare Cu(111) surface [29], has
been employed. This potential reproduces the QWS energy
position of E QWS ¼ 0:127 eV at the surface Brillouin zone
C
center for 1 ML Na coverage [30]. To investigate the linewidth
dependence on the QWS energy position relative to
the Fermi level and demonstrate the dramatic change of
the screening upon the QWS band shrink below E F , we evaluate
C e–e for the cases with QWS energies E QWS ¼ 0eV 1 and
E QWS
C
¼ 0:042 eV. We assume that the small variation of
the QWS binding energy at the scale of 0.1 eV does not
significantly affect the QWS wave function, and therefore
use the same set of wave functions obtained for the
E QWS
C
¼ 0:127 eV case, just shifting the QWS band to the
corresponding energy positions.We also assume here that
1 As the QWS energy position at the C point is gradually moved down
with increasing Na coverage [16] this energy position should roughly
correspond to h 0.9.
C