Surface Science 601 (2007) 4553–4556
www.elsevier.com/locate/susc
Electron–phonon coupling in a sodium monolayer on Cu(1 1 1)
S.V. Eremeev a, I.Yu. Sklyadneva a,b, P.M. Echenique b,c, S.D. Borisova a,
G. Benedek d, G.G. Rusina a, E.V. Chulkov b,c,*
a
c
Institute of Strength Physics and Materials Science, pr. Academicheskii 2/1, 634021, Tomsk, Russia
b
Donostia International Physics Center (DIPC), 20018 San Sebastian, Basque Country, Spain
Departamento de Fı́sica de Materiales and Centro Mixto CSIC–UPV/EHU, Facultad de Ciencias Quı́micas, UPV/EHU,
Apdo. 1072, 20080 San Sebastián, Basque Country, Spain
d
Dipartimento di Scienza dei Materiali, Università di Milano-Bicocca, Via Cozzi 53, 20125 Milano, Italy
Available online 22 April 2007
Abstract
We present calculation results for electron–phonon (e–ph) coupling in one monolayer (ML) of Na on the Cu(1 1 1) surface. We show
that the e–p coupling parameter k decreases compared to that for clean Cu(1 1 1) due to the significant decrease of the Na vertical vibrational mode contribution to the Eliashberg function in the 1 ML Na/Cu(1 1 1) system. The corresponding phonon induced lifetime broadening Ce–ph of a quantum-well state at low temperature decreases by 30% compared to that on clean Cu(1 1 1).
2007 Elsevier B.V. All rights reserved.
Keywords: Electron–phonon interaction; Metal surfaces; Quantum-well states
1. Introduction
During the last decade, electron–phonon (e–ph) interaction in surface structures has been a subject of intensive
theoretical and experimental investigations [1–22].Most of
these investigations have been performed for surface states
(SS) formed on clean metal surfaces [1–17] while only few
have been devoted to the study of an e–ph coupling parameter k in quantum-well states (QWS) formed in ultrathin
metal films on metal substrates [17–22]. In this paper we
present the calculation results on e–ph interaction in
QWS for a monolayer (ML) of Na on Cu(1 1 1) which
has been studied both theoretically and experimentally.
We discuss here the Eliashberg spectral function, a2F(x),
*
Corresponding author. Address: Departamento de Fı́sica de Materiales and Centro Mixto CSIC–UPV/EHU, Facultad de Ciencias Quı́micas,
UPV/EHU, Apdo. 1072, 20080 San Sebastián, Basque Country, Spain.
Tel.: +34 943 01 8220; fax: +34 943 01 5600.
E-mail address: [email protected] (E.V. Chulkov).
0039-6028/$ - see front matter 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.susc.2007.04.140
the e–ph coupling parameter, k, and an e–ph contribution,
Ce–ph, to the lifetime broadening (linewidth) of the Na
quantum-well state. As was shown experimentally by using
photoemission spectroscopy [18], two-photon photoemission technique [23], and scanning tunneling spectroscopy
[24], the C QWS in this system is located just below the Fermi level, EF, (see Fig. 1). Such a position of QWS has been
confirmed theoretically [25–27]. E–ph interaction in the
1 ML Na/Cu(1 1 1) has been calculated by Hellsing et al.
[18,19,16] simulating the entire phonon spectrum of the
system by a single frequency (Einstein model) that corresponds to vertical vibrations of the rigid Na monolayer.
This can lead to overestimation of k and Ce–ph despite the
use of accurate one-electron potential and wave function
of QWS. Simple estimations of k and Ce–ph within 2- and
3-dimensional Debye models significantly lowered these
quantities [26]. However, the role of the whole phonon
spectrum of 1 ML Na/Cu(1 1 1) in the e–ph interaction in
QWS of the 1 ML Na/Cu(1 1 1) system has not been studied yet. In this work we investigate this effect and conclude
that the role of vertical Na vibrations in k and Ce–ph is
rather small.
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S.V. Eremeev et al. / Surface Science 601 (2007) 4553–4556
The kk,i can directly be obtained from measurements of
temperature dependence of the linewidth of surface and
quantum-well states by using high temperature asymptotic
behavior [1–8,12,22]
Ceph ðk;i Þ ¼ 2pkk;i k B T
Fig. 1. Schematic electronic structure of 1 ML Na/Cu(1 1 1). The shaded
area shows the bulk Cu electron states continuum, solid line parabola is
the Na quantum-well state, and dotted line indicates the Fermi level.
2. Theory
The phonon-induced lifetime broadening Ce–ph(k,i) of
an electron state with momentum (k, i) and energy k,i is related to the Eliashberg a2Fk,i(x) spectral function through
the integral [28]
Z xm
Ceph ðk;i Þ ¼ 2p
a2 F k;i ðxÞ½1 f ðk;i xÞ
0
þ f ðk;i þ xÞ þ 2nðxÞdx
ð1Þ
Here, f and n are the Fermi and Bose distribution functions, respectively, and xm is the maximum phonon frequency. The spectral function is given by
X
dðx xq;m Þ
a2 F k;i ðxÞ ¼
q;m;f
2
jgðki ; kf ; q; mÞj dðk;i k;f xq;m Þ
ð2Þ
where g(ki, kf, q, m) is the e–ph matrix element which reflects
the probability of electron scattering from the initial state
wki and energy k,i to the final state wkf with energy k,f
by the phonon xq,m. The sum in Eq. (2) is carried out over
all final electron states wkf and all possible phonon modes.
The e–ph matrix element is
1=2
1
gðki ; kf ; q; mÞ ¼
hwki j^eq;m rR V SC jwkf i ð3Þ
2Mxq;m
where M is the atom mass, ^eq;m are the phonon polarization
vectors, and $RVSC is the gradient of the screened oneelectron potential with respect to atom displacements from
their equilibrium positions R.
The strength of the e–ph coupling described by kk,i is defined as the first reciprocal moment of the Eliashberg function [28]
Z xm 2
a F k;i ðxÞ
dx
ð4Þ
kk;i ¼ 2
x
0
ð5Þ
In the calculations, we simulate the 1 ML Na/Cu(1 1 1)
semi-infinite system by using a slab model with 31 atomic
layers of Cu(1 1 1) and Na atoms located on both sides of
the Cu slab. A unit cell of our model system contains nine
atoms in each Cu layer and four Na atoms on each side of
the Cu slab [25,29]: a total number of atoms per unit cell is
287. With this number of atoms ab-initio computations of a
phonon spectrum and a gradient of one-electron potential
related to the vibrations are not feasible currently. Therefore, in the present calculations we use a model recently
proposed for studying e–ph interaction in surface states
[6,9,17]. This model combines three independent approximations to evaluate the e–ph coupling matrix elements
(Eq. (3)): (1) one-electron wave functions and energies
are calculated with one-dimensional potential [30]; (2)
phonon frequencies and polarizations are obtained from
a one-parameter force-constant model [31]; (3) a gradient
of one-electron potential is represented by the Ashcroft
pseudopotential [32] screened within Thomas–Fermi approximation. We apply this model to 1 ML Na/Cu(1 1 1)
with two important changes: (a) one-dimensional model
potential of Ref. [26] designed for 1 ML Na/Cu(1 1 1) is
used instead of the Ref. [30] potential which was designed
for clean metal surfaces; (b) an embedded atom model is
employed to calculate vibrational spectrum for 1 ML Na/
Cu(1 1 1) [29] since the one-parameter force-constant model
[31] does not work for this system. The embedded atom
interatomic interaction potentials produce the 1 ML Na/
Cu(1 1 1) surface crystal structure in excellent agreement
[29] with ab-initio calculations [25]. These potentials also
produce phonon spectra of the Na/Cu(1 1 1) system for different coverages of Na in good agreement with available
experimental data [29].
3. Calculation results and discussion
In Fig. 2 we show the calculated layer density of phonon
states (LDOS) for 1 ML of Na which forms the (3/2 · 3/2)
crystal structure on Cu(1 1 1). We compare it to LDOS
computed for Cu(1 1 1). One can see that LDOS of the
Na layer is completely different from that of the clean
Cu(1 1 1) surface layer: LDOS of vertical (z-polarized)
vibrations of Na is much more broad than the corresponding LDOS for the Cu surface layer (as well as for systems
with lower coverages of Na, see Ref. [29]). Another important feature of the Na adlayer DOS is the very clear two
peak structure of horizontal vibrations of Na. In the top
Cu substrate layer, LDOS is slightly modified with respect
to the clean Cu(1 1 1) surface layer.
In Fig. 3 the evaluated Eliashberg function for QWS is
shown. Two broad peaks are seen in the figure at energies
S.V. Eremeev et al. / Surface Science 601 (2007) 4553–4556
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Fig. 4. Temperature dependence of the lifetime broadening Cep of the C
surface state in Cu(1 1 1) (dashed line) and of QWS in 1 ML Na/Cu(1 1 1)
(solid line).
Fig. 2. Calculated phonon layer DOS for the Cu(1 1 1) surface (left) and
for 1 ML Na/Cu(1 1 1) system (right). S, S-1, and S-2 indicate the Cu
surface, the first and second sublayers of Cu, respectively. The topmost
right panel shows the Na-adlayer DOS.
tions of atoms of the top Cu substrate layer. The second
peak is related to both vertical and horizontal vibrations
of Na atoms as well as to vertical vibrations of the top layer
Cu atoms. This picture is very different from the clean
Cu(1 1 1) surface where Cu vertical vibrations (Rayleigh
mode) provide 30–35% of the full phonon-induced contribution to the surface state decay rate [6,9]. It is also very
distinct from the results obtained by using the Rayleigh like
mode (Einstein mode) to simulate the entire phonon spectrum of 1 ML Na/Cu(1 1 1) [18,19]. The calculation of k
with the Eliashberg function shown in Fig. 3 gives
k = 0.14. This value is close to k = 0.16 obtained theoretically [6,9] and k = 0.14 ± 0.02 deduced from photoemission measurements [1,12] of the surface state on clean
Cu(1 1 1). Despite this coincidence it is worthy to note that
k in 1 ML Na/Cu(1 1 1) and k in Cu(1 1 1) are formed by
different phonon modes, therefore, the coincidence should
be considered to a large extent as accidental.
In Fig. 4 we show the temperature dependence of Ce–p
for Cu(1 1 1) and 1 ML Na/Cu(1 1 1). The slope of Ce–p,
which determines k, is practically identical for both these
curves. Nevertheless, at T = 0 K, Ce–p = 5.4 meV obtained
for 1 ML Na/Cu(1 1 1) is significantly smaller than Ce–p =
7.8 meV calculated for Cu(1 1 1).
4. Conclusions
Fig. 3. Eliashberg function, a2F(x), for QWS at the C point of 1 ML Na/
Cu(1 1 1).
around 10 and 15 meV. The first peak (at 10 meV) results
mostly from the interaction of the quantum-well state with
horizontal vibrations of Na atoms and with vertical vibra-
We have studied e–ph interaction of an excited hole in
the C quantum-well state with all vibrations of 1 ML Na/
Cu(1 1 1). We have found that vertical vibrations of Na
atoms are not very important for e–ph interaction of
QWS in 1 ML Na/Cu(1 1 1) as assumed before for this system. The e–ph coupling parameter k was found to be 0.14,
i.e., close to that in the surface state on clean Cu(1 1 1). The
phonon-induced contribution to the hole decay rate has
been obtained to be equal to 5.4 meV.
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S.V. Eremeev et al. / Surface Science 601 (2007) 4553–4556
Acknowledgements
Partial support by the Basque Departamento de Educación, Universidades e Investigación, the University of
the Basque Country UPV/EHU (Grant No. 9/UPV
00206.215-13639/2001), the Spanish MEC (Grant No.
FIS2004-06490-C03-00), the Ministry of Science of Russia
(Grant No. 40.012.1.1.1153) and NATO science programme (Grant PST. CLG. 980395) is acknowledged.
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