JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 23 (2011) 484006 (8pp)
High-resolution phonon study of the
K L Kostov1,2, S Polzin1and W Widdra1,3
1Institute of Physics, Martin-Luther-Universit¨ at Halle-Wittenberg, 06099 Halle, Germany
2Institute of General and Inorganic Chemistry, Bulgarian Academy of Sciences, 1113 Sofia,
Received 4 April 2011
Published 16 November 2011
Online at stacks.iop.org/JPhysCM/23/484006
Using high-resolution electron energy loss spectroscopy the phonon dispersion of Ag(100) has
been studied at two different sample temperatures of 86 and 300 K. The dominant feature in the
spectra corresponds to the Rayleigh wave. Its full dispersion is determined along the¯Γ¯X high
symmetry direction in the first and second Brillouin zones. The Rayleigh phonon maximum at
the¯X point shows a redshift with increasing temperature. This is explained based on a surface
anharmonicity with an anharmonicity constant of 0.014, comparable to the value reported for
Cu(100). In the vicinity of the¯Γ point two additional phonon features have been discovered at
about 110 and 160 cm−1, which are tentatively assigned to high density of states features from
the bulk phonon bands. However, the observed steep dispersion is in contrast to theoretical
calculations. Along¯Γ¯X two surface resonance branches have been observed with maximum
frequencies in the range of 90–110 cm−1near to the zone boundary. These branches agree with
helium atom scattering data where available, but are not predicted by theory.
(Some figures in this article are in colour only in the electronic version)
The study of the surface phonon dispersion along the high
symmetry directions of the surface Brillouin zone (SBZ)
provides deep insights into the dynamical properties within the
surface region of a single crystal, and helps us to understand
such processes as surface relaxation and surface reconstruction
occurring at the point where the translational invariance of
the bulk crystal is lost in the direction perpendicular to the
surface .The first experimentally determined surface
phonon dispersions using high-resolution electron energy loss
spectroscopy (HREELS) were reported for clean and oxygen-
covered Ni(100) surfaces [2, 3].
revealed several modes in the¯Γ–¯X direction: (i) S6phonon
running through the gap of bulk phonons, (ii) S2resonance,
(iii) shear horizontal S1surface mode and (iv) S4, Rayleigh
phonon. The S6and S4modes were experimentally identified
based on an analysis of the HREELS selection rules and of the
energy dependence of the inelastic cross section [2, 4].
Compared with Ni(100) and Cu(100) similar vibrational
features are expected based on theoretical considerations for
The theoretical modeling
3Author to whom any correspondence should be addressed.
the dynamics at the Ag(100) surface . However, in the
first phonon dispersion study on clean Ag(100) by HREELS,
Rocca et al detected only the Rayleigh mode along the¯Γ–¯X
direction . The experimentally determined dispersion curve
was well reproduced by calculations assuming force constants
for the surface region similar to those of the bulk [5, 7].
The results for Ag(100) could be explained with a minor or
no change of the first interlayer spacing at 300 K .
a subsequent study the phonon dispersion along the ¯Γ–¯ M
direction of Ag(100) has been studied with special focus on
the zone boundary since there the Rayleigh phonon is more
sensitive to possible surface relaxations .
deviationsfrom bulk-deriveddynamicalproperties were found.
This is in contrast to the findings on Cu(100) and Ni(100)
where the force constants between the first and second layers
were increased with respect to the bulk values in order to
reproduce the experimentally measured RW dispersions [3, 9].
The modified dynamical properties at the surface are related
to structural changes, here a contraction between the first and
second surface layers [10, 11].
constant model with adjustable parameters was shown to be
misleading based on the predicted relaxation of the surface
However, later, the force
© 2011 IOP Publishing LtdPrinted in the UK & the USA
J. Phys.: Condens. Matter 23 (2011) 484006 K L Kostov et al
Besides the Rayleigh phonon,
dependent variation of the phonon frequency was observed by
HREELS for Ag(100) in the¯Γ–¯ M direction and interpreted as
a contribution of the S2mode . In a following paper this
conclusion was revisited and the new feature was attributed
not to S2 but most likely to the new longitudinal L1 mode
based on the inelastic cross-section calculations showing that
The similarities between phonon features of Ag(100) and
Cu(100) surfaces were confirmed by combined first-principles
phonon and multiple-scattering EELS calculations  and
molecular-dynamics simulations .
Ag(100) the S1, S2, S4(Rayleigh) and S6phonon modes have
been predicted, but only the S4phonon has been detected in
HREELS due to its significant cross section . However,
it was mentioned that S2and S6could also be favorable for
observation at some scattering conditions.
Using an out-of-plane scattering geometry in HREELS
where the axes of the analyzer and monochromator were not
in the same plane, Erskine et al  succeeded in detecting
the (for in-plane-scattering geometries) symmetry forbidden
S1shear horizontal mode and found it at a surprisingly low
frequency (3.3 meV or 27 cm−1at the¯X point). To explain
this low frequency mode B¨ uscher et al  had to change
significantly the surface force parameters in contrast to the
theoretical studies considered above. However, in this case
the other experimental data could not be reproduced well.
The lowest frequency of the S1mode was found at 5.8 meV
(47 cm−1), below the Rayleigh mode, by the theoretical
considerations of Chen et al [12, 13].
The phonon dispersions of the Rayleigh and longitudinal
modes in the two high symmetry¯Γ–¯X and¯Γ–¯ M directions
were also measured using helium atom scattering (HAS) .
These experimental curves were better reproduced with
pseudocharge model calculations as compared to the previous
theoretical simulations. The frequency of the shear horizontal
S1 mode at the ¯X point has been calculated to 7.4 meV
(60 cm−1), significantly higher than the measured frequency
of 3.3 meV by Erskine et al . Therefore the assignment of
the latter feature remains still unclear.
In the present paper, we study the complete phonon
dispersion along the¯Γ–¯X direction of Ag(100) by HREELS
with significantly improved characteristics.
to measure and discuss new phonon features in comparison
with the earlier studies, for example in the frequency regions
of the S2, S6 and resonance modes and of the bulk phonon
edges. We study the dispersions at two sample temperatures,
86 and 300 K, to give more insight into the temperature
dependent surface dynamics. A small redshift of the Rayleigh
phonon frequency is measured with increasing temperature
and discussed based on the anharmonicity of the atom–atom
interaction potential at the surface.
an electron energy
At the ¯X point of
This allows us
2. Experimental details
The experiments were performed in a two-level ultra-high
vacuum (UHV) apparatus with a base vacuum in the low
10−9Pa range. The sample cleaning was carried out in the first
level (preparation) chamber equipped with XPS, LEED and
temperature programmed desorption facilities. The cleaning
procedure of the Ag(100) crystal included repeated cycles of
Ar+sputtering (3 μA at 1 keV ion energy) and subsequent
annealing at 700 K. The lower chamber houses an HREEL
spectrometer (Delta 05, Specs GmbH, Berlin) witha resolution
of 1 meV (∼8 cm−1), corresponding to a count rate up to
106s−1for the 4 eV electrons elastically scattered from well
ordered adsorbate layers [18–20]. Here for most cases the
of the low phonon signals. Only for some 4 eV measurements
near the¯Γ point did we use the high instrumental resolution of
∼1 meV in order to resolve the low frequency phonon at about
45 cm−1which is otherwise hidden in the tail of the elastic
peak. The electron scattering geometry includes an incidence
angle of 60◦with respect to the surface normal and a wide
range of off-specular angles up to 42◦. Due to the appropriate
azimuthal orientation of the sample the electron scattering
plane was spanned by the  high symmetry direction and
the surface normal. This allows the phonon dispersion in the
¯Γ–¯X direction of the surface Brillouin zone to be studied.
3.1. Rayleigh phonon
The phonon dispersion of the clean Ag(100) surface has been
investigated in the ¯Γ–¯X direction of the SBZ by HREELS
with primary electron energies in the range of 4–250 eV.
The high electron energies and large off-specular angles have
enabled us to measure the phonon dispersion also in the second
Brillouin zone. This is demonstrated in figure 1 where the
HREEL spectra for an electron energy of 250 eV are shown for
increasing off-specular angles along the¯Γ¯X¯Γ?direction (
direction). The spectra correspond to increasing momentum
transfer ?k?from top (?k?= 0) to bottom (?k?= 2.95˚ A−1).
The Rayleigh wave (RW) induced energy losses are the most
intense vibrational features in figure 1 in the loss as well as in
the gain region (negative energy losses). The approximately
sinusoidal dispersion of the RW is clearly observed from
28 cm−1(at 0.29 ˚ A−1) up to 70 cm−1at the¯X point. Note
that the 16th spectrum in figure 1, which corresponds to a
momentum transfer of ?k?= 2.18˚ A−1, is nearly identical to
the first one under specular scattering conditions. The elastic
peak has almost the same intensity and FWHM (24 cm−1) as
the specular peak. The momentum transfer of ?k?= 2.18˚ A−1
in the  direction corresponds to the center of the second
surface Brillouin zone. The similarity of the spectra measured
at the¯Γ and the¯Γ?points, which correspond to the LEED (00)
and (10) spots, respectively, and the absence of a zero loss peak
otherwise indicate the high structural quality of the Ag(100)
The data as presented in figure 1 were recorded
with different electron energies up to 250 eV for sample
temperatures of 86 and 300 K. The RW dispersion from these
determination of the RW dispersion through the first and into
thesecond surface Brillouinzone alongthe¯Γ–¯X direction. The
J. Phys.: Condens. Matter 23 (2011) 484006K L Kostov et al
Figure 1. Off-specular HREEL spectra at different parallel
momentum transfer values along the¯Γ¯X¯Γ?direction (
direction) using an electron energy of 250 eV and a total scattering
angle of 120◦. T = 300 K.
lowest RW frequency of 14 cm−1was measured at 0.13 ˚ A−1.
The RW dispersion ω(?k?) at both temperatures can be well
fitted with the periodic function
¯ hω = A1sin(?k?/(G/π)) + A3sin(3?k?/(G/π))
which is shown as solid line in figure 2. The fitting results are
summarized in table 1. The fitting yields 2.171˚ A−1for G, in
good agreement with the reciprocal lattice vector of 2.18˚ A−1.
The RW dispersion is dominated by a sinusoidal dependence
with maximum frequencies of 70.3 and 74.1 cm−1at the¯X
point at 300 and 86 K, respectively. For the deviation from a
sinusoidal dispersion which is described by the coefficients A3
and A5an upper limit of 2% as compared with the sinusoidal
coefficient A1can be given from the experimental data. Note
that the data beyond the¯Γ?point also follow this dispersion.
Despite the small frequency differences, a temperature
dependence of the dispersion curves can be clearly seen.
The beginning of the RW dispersion shows a characteristic
linear dependence on the wavevector at low momentum
transfer. However, at higher ?k?the two curves are clearly
Figure 2. Dispersion of the Rayleigh phonons at 300 K (full squares,
blue curve) and 86 K (down triangles, red curve). The points and
solid lines correspond to the experimental data and the fitting curves,
respectively. Inset: dependence of the RW frequencies measured at
86 and 300 K (solid circles, present work) and at 150 K (open
triangle, ) on temperature; the fitting curve is according to
Table 1. Rayleigh wave dispersion description as a series of odd
sine functions with amplitudes A1, A3and A5(in units of cm−1) with
periodicities of G, G/3 and G/5 (in units of˚ A
Fit parameter300 K86 K
distinguishable. At the¯X point the fitting procedure gives a
maximalRWfrequency of 70.3cm−1for 300K whichmatches
with the observation of Rocca et al . At 86 K we obtain
a higher value of 74.1 cm−1which can be compared with
the result of 73.4 cm−1measured by HAS  for a surface
temperature of 150 K. The¯X point RW frequency is depicted
in the inset of figure 4 as function of temperature.
3.2. Phonon dispersion along the¯Γ–¯X direction
Besides the intense RW peaks other phonon induced losses are
observed in off-specular HREEL spectra. In general they have
considerably lower intensities than the RW peaks but they can
be resolved due to the high energy resolution and the large
signal-to-noise ratio of the experimental setup.
to an experimental error in the peak-position determination
of the order of 0.5 to a few cm−1depending on the peak
intensity. To improve the separation of close lying phonons,
J. Phys.: Condens. Matter 23 (2011) 484006K L Kostov et al
Figure 3. Off-specular HREEL spectra at different electron energies
(indicated in the figure) and momentum transfer values of
obtained from the fitting procedure are also shown with blue curves;
the red ones denote their sum.
−1. The contributions of the different phonon modes
we analyzed simultaneously loss spectra obtained at different
scattering energies for a given momentum transfer ?k?. Due
to the significant energy dependence of the individual phonon
excitation cross sections  their relative contributions to the
spectra vary strongly. Simultaneous fitting of several spectra
fora givenmomentumtransfer therefore improvestheanalysis.
The fitting procedure also takes into account that each energy
loss peak is accompanied by a corresponding energy gain peak
of identical shape but with reduced intensity. The intensity
ratio between the loss and gain peaks is given by a phonon
energy dependent Boltzmann factor. An example is shown
in figure 3 for ?k?of 0.64–0.67 ˚ A−1. The four spectra are
deconvoluted according to four phonons at 57, 82.3, 97.6 and
137 cm−1which contribute quite differently for the different
electron energies. The high energy phonon would be badly
fitted in the 16 eV spectrum; however, its energy can be well
determined at 64 and 100 eV. For the phonon at about 97 cm−1
the spectra at 16 and 49 eV allow for a peak-position fitting,
whereas its cross section at 64 and 100 eV is too small.
The results derived from a similar analysis of all HREEL
spectra measured at 300 K are summarized in figure 4(a)
and compared to theoretical calculations along the ¯Γ–¯X
direction . The data of previous experimental studies using
HREELS (open squares, [6, 8]) and helium atom scattering
(HAS) (open circles, ) are also displayed.
Figure 4. Comparison between theoretical  and experimental data
for the surface phonon dispersion of Ag(100) along the¯Γ¯X¯Γ?¯X?
direction at a sample temperature of (a) 300 K and (b) 86 K. The
open squares and circles denote previous HREELS  and HAS 
results, respectively. The solid circles mark the present HREELS
data. The solid circles of a given color are obtained at the same
The phonon dispersion results for Ag(001) at a sample
temperature of 86 K are displayed in figure 4(b).
glance, the dispersions at 86 and 300 K are similar, but there
are somesignificant differences which willbe discussedbelow.
3.2.1. Phonons in the vicinity of the ? point.
frequency phonon features can be well separated from the RW
losses near to the¯Γ point using low electron energies, due to
the improved instrumental resolution. In addition to the results
of previous careful HREELS studies on Ag(100) [4, 12, 13],
we observe low-intensity peaks in the 100–170 cm−1range,
as shown in figure 5 for small parallel momentum transfers in
the¯Γ–¯X direction. Clearly there are two energy loss and two
energy gain peaks for ?k? = 0.1 and 0.13 ˚ A−1in figure 5.
The peaks are fitted with identical frequencies on the loss and
the gain sides and with a given loss to gain intensity ratio
as based on the Boltzmann factor. This results in phonons
at 111 and 165 cm−1, as indicated by solid lines.
on this observation the spectra at ?k? = 0.05, 0.03 and
0.0 ˚ A−1have been fitted also with two phonon features of
J. Phys.: Condens. Matter 23 (2011) 484006K L Kostov et al
Figure 5. Deconvolution of the loss and gain phonon features in
off-specular HREEL spectra at different parallel momentum transfers
?k?(indicated in the figure). The sample temperature is 300 K, and
the electron energy is 4 eV.
similar line shape as emphasized in figure 5. Note that at
?k? = 0 the deconvolution of the spectral feature into two
peaks as indicated in figure 5 gives only an upper limit for
the peak splitting; any smaller peak splitting would also be
compatible with the data at ?k?= 0. These and similar data
are summarized as solid black circles in figures 4 and 6, which
shows a close-up of their dispersion up to 0.4 ˚ A−1. The data
measured at 86 K in the same spectral region are marked as
open triangles in figure 6 and exhibit a similar dispersion as at
300 K. The data for ?k?> 0.07˚ A−1follow the dispersion of
the band edge of the projected density of phonon states as can
be seen by comparison with figure 4(a) when we allow for a
rigid shift of about 8 cm−1in comparison to theory. However,
for ?k? < 0.07 ˚ A−1there is a significant deviation with
decreasing momentum as both phonon frequencies approach
one another. This leads at the¯Γ point to a single broad spectral
feature at about 140 cm−1as shown in figure 5, which gives
an upper limit to the splitting of 15 cm−1. Theory predicts
a rather broad and structureless density of the projected bulk
phonons in the range with a nearly flat dispersion.
experimentally observed strong dispersion in the close vicinity
of the¯Γ point points to a strong long-range interaction within
this new phonon mode.
scattering , we detect below 100 cm−1between¯Γ and¯X
two additional phonon modes besides the RW. At 300 K one
phonon disperses from 50 cm−1at ?k?= 0.3˚ A−1to 99 cm−1
at the¯X point, as shown in figure 4(a). The other from 70
to 108 cm−1between ?k?= 0.25 and 1.1 ˚ A−1, respectively.
Similarly to helium atom
Figure 6. Comparison between the phonon dispersion curves
measured at (a) 300 K (black circles) and (b) 86 K (open triangles)
with an electron energy of 4 eV. The red curves denote the
theory-predicted contributions from the high density of states at the
band edges of bulk phonons .
Where available the data for both phonon branches agree
well with the HAS results .
disperse within the projected density of bulk phonons and
are therefore surface resonances. However, such resonances
are absent in ab initio calculations . This phenomenon of
additional experimental modes which are commonly referred
to as longitudinal resonances has been observed by HAS
also for other fcc metal surfaces.
controversially since its first discovery for the Ag(111) surface
by Doak et al . A detailed discussion of this issue can be
found in the work of Heid and Bohnen . There the authors
come to the conclusion that either (i) ‘theory wrongly predicts
important interatomic couplings’ or (ii) that the corresponding
features in the HAS spectra ‘are caused by special properties
of the He-surface interaction and are not related to any surface
mode or resonance’ .However, as seen in figures 4(a)
and (b), we also observe these phonon features by HREELS
in good agreement with the HAS data . Earlier HREELS
measurements have also detected similar resonances, for
example for clean Ru(0001) , again in good agreement
with HAS data , and for Cu(111) .
Both phonon branches
It has been discussed
S2 mode of Ag(100), which should show a weak parabolic
upwarddispersionaroundthe¯Xpointat 80cm−1(see figure3),
possesses a dominant vertical polarization in the second
surface layer with a small component in the first layer .
At 300 K we were not able to identify this mode although it is
allowed based on selection rules for HREELS. Its observation
may be hampered because of the proximity to the intense
Rayleigh peak (see figure 4). However, at 86 K we find a peak
in the predicted energy range, but only for the specific electron
energy of 49 eV. This is shown in the region of 0.85–1.04˚ A−1
in figure 7. As seen in figures 7(b) and (c) an intense peak
at and above 80 cm−1is clearly observed.
neither to the RW nor to the longitudinal resonance which is
J. Phys.: Condens. Matter 23 (2011) 484006 K L Kostov et al
Figure 7. Off-specular HREEL spectra measured with an electron
energy of 49 eV at 86 K and parallel momentum transfer values of
0.85˚ A−1(a), 0.95˚ A−1(b) and 1.04˚ A−1(c). The contributions of the
different phonon modes obtained from the fitting procedure are also
shown with blue curves; the red ones denote their sum.
located 10 cm−1higher at the¯X point. In the deconvolution of
the spectrum in figure 7(a) a component corresponding to the
Rayleigh phonon with a frequency taken from its dispersion is
included, but still there is a strong peak at 81.7 cm−1necessary
to describe the spectrum. One explanation for the detection of
the S2surface phonon only at low temperatures can be found
in the lower RW intensity (compared to the room temperature
measurements) which cannot overlap the S2signal. Another
reason can be found in molecular-dynamics simulations 
whichpredict a broad and weak spectral densityof theS2mode
at 300 K.
Similar considerations are valid for the S6mode which is
predicted as surface phonon in the vicinity of the¯X point in
the bulk phonon gap at about 135 cm−1. As indicated in
figure 8 we observe spectral features only at slightly higher
energies at 150–160 cm−1. Under most conditions the RW
dominates the spectra; only at 36 eV and a temperature of
86 K is the RW peak strongly suppressed (figure 8(b), lowest
spectrum) and is a well separated phonon at about 150 cm−1
visible. However, since its frequency is higher and since
it covers a wider momentum range as expected for the S6
mode, we tentatively assign it to high density feature of the
projected densityof states at the bulk band edges (see figure 4).
In contrast to our study on Ag(100), the S6 mode has been
4.1. Identification of the phonon modes
From the theoretical description four distinct surface phonon
modes are predicted for Ag(100) at the¯X point as labeled in
figure 4 . The S1phonon is polarized dominantly shear
horizontally in the first layer. The S4mode (Rayleigh phonon)
has a dominantly vertical character in the first layer. Whereas
the S2phonon exhibits a longitudinal displacement in the first
layer, it is vertical in the second layer. The S6mode has a
dominant first layer longitudinal polarization. Additionally the
molecular-dynamicsstudyof Yang, Rahmanand Daw reports a
resonance at 11meV(88.7 cm−1)associated withvertical atom
displacement . From the HREELS data in the vicinity of
the¯X point we derive four phonon branches with energies of
70.3(74.1) cm−1, ∼81 cm−1, 90–110 cm−1, ∼155 cm−1at the
X point. The first one is clearly assigned to the S4mode and
the modes between 90 and 110 cm−1to surface resonances as
discussed above. The second branch at ∼81 cm−1is clearly
surface phonon. From the remaining surface phonons the odd
S1mode is symmetry forbidden in the experimental geometry
and is accordingly not observed. Surprisingly, the theoretically
predicted S6mode could not be identified experimentally in
the corresponding spectral range. Furthermore, the band edges
which are discernible based on the increased density of states
there are found at approximately 10–20 cm−1lower energies
as compared to the ab initio calculations . In summary
we find discrepancies between the experimental dispersion
and the theoretical description. Additionally, the rather steep
dispersion observed close to the¯Γ point for the energy losses
at 110 and 160 cm−1is not compatible with any theoretical
study and is not yet understood.
4.2. Temperature-dependent dispersion of Rayleigh mode
The pronounced temperature dependence of surface phonons
is correlated to an enhanced anharmonicity of the interaction
potential between surface atoms which in turn is coupled
to processes like surface relaxation and surface reconstruc-
tion. Whereas atlowtemperatures theinteractionpotential
can beconsidered asharmonicandthe mean-square vibrational
amplitudes increase linearly with temperature as described
by a Debye–Waller law, at higher temperatures the mean-
square vibrational amplitudes increase more rapidly due to
the anharmonicity of the interaction potential. This has been
observed not only for the more open fcc(110) [26–29], which
can easily undergo a reconstruction, but also for the (100)
J. Phys.: Condens. Matter 23 (2011) 484006K L Kostov et al
Figure 8. Off-specular HREEL spectra in the vicinity of the¯X point as measured with different electron energies (a) at 300 K and (b) at 86 K.
The contributions of the different phonon modes obtained from the fitting procedure are also shown with blue curves; the red ones denote their
surfaces of Cu and Ni [30–32]. At higher temperatures the sur-
face anharmonicity leads to a line broadening and a frequency
redshift of surface phonons. Strong anharmonic effects were
also experimentally observed for the perpendicular vibrational
modes of atomic adsorbates on Ni(100) .
The consequences from anharmonicity have been cal-
culated by molecular-dynamics modeling for Cu(100) .
For the Cu(100) Rayleigh mode at the X point at about
12.4 meV (100 cm−1) a redshift of 0.37 meV (3 cm−1) has
been calculated between 150 and 300 K in good agreement
with temperature-dependent HAS measurements [14, 34].
Assuming a similar behavior for Ag(100) based on the
structural similarities and the similar Debye temperatures of
Cu(100) and Ag(100), a redshift of 0.27 meV (2.2 cm−1)
between 150 and 300 K is expected for the Ag(100) Rayleigh
wave at the X point(for thissurface the authors have calculated
a Rayleigh frequency of 8.7 meV (70 cm−1) but only at
300 K ). This is in good agreement with the measured
RW frequency of 73.4 cm−1at 150 K using HAS ,
giving a frequency shift of 3.1 cm−1with respect to the
value of 70.3 cm−1at 300 K as measured by us. For the
lower temperature of 86 K we observe a shift of 3.8 cm−1
compared to 300 K (see figure 2), in agreement with the above
considerations based on previous studies [14, 17].
These observations might lead to the conclusion that the
surface anharmonicity plays the dominant role for the RW
frequency shift as here observed. Similarly the temperature-
dependentdecrease ofthe phononfrequencies for Cu(110)
and for Cu(100)  has been interpreted and well reproduced
using a simple anharmonic Morse potential model. For the
anharmonic Morse potential the phonon frequency ¯ hω at a
given temperature is lowered with respect to the harmonic
frequency ¯ hωo (at vanishing mean-square displacement) as
given by 
¯ hω = ¯ hωo− χa¯ hωo(2no+ 1)
where χais the anharmonicity constant and nois the phonon
occupation number with no = [exp(¯ hωo/kT) − 1]−1. Using
this relation the experimental HREELS data for the Ag(100)
Rayleigh phonon frequency at the X point at 86 and 300 K
and also the HAS data of Bunjes et al for 150 K  have
been fitted. The result is shown as the solid line in the inset
of figure 2. The model describes the three data points well
with an anharmonicity constant of 0.014. This value agrees
within 20% with the anharmonicity for the Rayleigh mode for
Cu(100) which has been determined to be 0.0175 . This
behavior fits to the similar Debye temperatures and similar
inward relaxation of the topmost layer on Ag(100) and on
Cu(100) .On the other hand the anharmonicity for
Ag(100) is substantially smaller (a factor of 4) as compared
to the surface phonon resonance at 13 meV (∼105 cm−1) on
J. Phys.: Condens. Matter 23 (2011) 484006K L Kostov et al Download full-text
the more open Ag(110) for which an anharmonicity of 0.052
has been determined between 160 and 550 K .
As mentioned above, the enhanced anharmonicity might
induce a vibrational line broadening at higher temperatures.
The molecular-dynamics study by Yang et al for Cu(100) 
showed that this broadening is strongly mode dependent:
whereas the line broadening of the S4surface phonon between
150 and 300 K is negligible, a significant (almost three times
increase in width) broadening for the high-frequency S6mode
is found.This is consistent with our results for the S4
(Rayleigh) and the S6 spectral regions (figures 3, 7, 8) on
Ag(100). In the S6region we observe a broad peak at both
temperatures; however, the additional high density of bulk
phonons in this spectral region can also contribute to the
feature at 86 and 300 K. For the S2mode of Ag(100) a weak
and broad spectral feature has been predicted at 300 K .
This can explain why the S2phonon is clearly observable only
at low temperatures (86 K in figure 7).
Here we present a phonon dispersion study of Ag(100) using
high-resolution electron energy loss spectroscopy. Due to the
improved energy resolution and signal-to-noise ratio we detect
several new phonon features.
around 110 and 160 cm−1have been determined in regions
with an expected high density of bulk phonon states.
contrast to theoretical predictions they show a significant
dispersion in the vicinity of the ¯Γ point. (ii) Two surface
resonances are found along¯Γ¯X in agreement with earlier HAS
study .Here we were able to follow their dispersion
up to the ¯X point.
bulk phonon band are found to be 10–20 cm−1lower than
the theoretical predicted band edges. (iv) For the first time
an indication for the S2 surface phonon mode is observed
close to the¯X point; however, only under low temperature
conditions(86 K). The absence of a detectable feature at 300 K
in this region is explained by a large FWHM of the phonon
at higher temperature (300 K), as predicted by molecular-
The most intense features in the off-specular HREEL
spectra are the Rayleigh wave induced losses.
dispersion has been determined within the first and second
surface Brillouin zones. It is described by a dominantly sine-
like Rayleigh phonon dispersion with a ∼3 cm−1redshift at
the¯X point with increasing temperature from 86 to 300 K. This
temperature-dependent decrease of the phononfrequencies can
be well reproduced using a simple anharmonic Morse potential
model. It gives an anharmonicity constant of 0.014 for the
Rayleigh phonon which compares well with the anharmonicity
(i) Two spectral features at
(iii) The upper edges of the projected
Financial support by the Deutsche Forschungsgemeinschaft
DFG through the Sonderforschungsbereich SFB 762 ‘Func-
tional Oxidic Interfaces’ is gratefully acknowledged.
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