# High-resolution phonon study of the Ag(100) surface.

**ABSTRACT** 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 ΓX 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 ΓX two surface resonance branches have been observed with maximum frequencies in the range of 90-110 cm(-1) near to the zone boundary. These branches agree with helium atom scattering data where available, but are not predicted by theory.

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**ABSTRACT:**Surface phonon dispersion curves have been measured for the clean and hydrogen-covered Ru(0001) surfaces along the [110] and [100] high symmetry directions of the surface Brillouin zone using high resolution He atom scattering. Interatomic force constants and elastic constants were obtained by a lattice-dynamical analysis of the experimental data. Significant changes of the dynamical properties of the surface and the interaction strength between impinging He atoms and surface phonons are observed upon reaction of the surface with hydrogen.Surface Science 02/1997; 372:132-144. · 1.87 Impact Factor - Surface Science 09/1995; 339(1-2):1-7. · 1.87 Impact Factor
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**ABSTRACT:**Electron-energy-loss-spectroscopy measurements of Rayleigh phonon dispersion are reported for the Gamma¯-M¯ direction of the Ag(100) surface. The data are compared to lattice-dynamical calculations performed within a nearest-neighbor model. The comparison shows that a slight increase of the force constant between the first and second layer with respect to the bulk value is needed in order to fit the data. The force-constant change is, however, less than the accuracy with which one can reproduce the bulk phonon spectrum with this model and is therefore not physically significant. More sophisticated models are, on the other hand, shown to give wrong predictions for the surface spectrum. We conclude therefore that the Ag(100) surface exhibits smaller effects of surface relaxation, if indeed there are any effects, than the same surfaces of Cu and Ni.Physical review. B, Condensed matter 07/1990; 41(18):12905-12907. · 3.77 Impact Factor

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IOP PUBLISHING

JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 23 (2011) 484006 (8pp)

High-resolution phonon study of the

Ag(100) surface

doi:10.1088/0953-8984/23/48/484006

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,

Bulgaria

E-mail: wolf.widdra@physik.uni-halle.de

Received 4 April 2011

Published 16 November 2011

Online at stacks.iop.org/JPhysCM/23/484006

Abstract

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)

1. Introduction

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 [1]. 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 [5]. 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 [6]. 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 [6].

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 [8].

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

layers [12].

In

However, no

However, later, the force

0953-8984/11/484006+08$33.00

© 2011 IOP Publishing Ltd Printed in the UK & the USA

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J. Phys.: Condens. Matter 23 (2011) 484006K 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 [8]. 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

thecontributionoftheS2modeshouldbeextremelysmall[12].

The similarities between phonon features of Ag(100) and

Cu(100) surfaces were confirmed by combined first-principles

phonon and multiple-scattering EELS calculations [13] and

molecular-dynamics simulations [14].

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 [13]. 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 [15] 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 [16] 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) [17].

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 [15]. 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

resolutionwaschosentobeoftheorderof16–20cm−1because

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 [110] 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. Results

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 ([110]

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 [110] 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)

surface.

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

measurementsissummarizedinfigure2. Thedataallowforthe

determination of the RW dispersion through the first and into

thesecond surface Brillouinzone alongthe¯Γ–¯X direction. The

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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 ([110]

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))

+ A5sin(5?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, [17]) on temperature; the fitting curve is according to

relation (1).

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

−1).

T

Fit parameter300 K86 K

A1

A3

A5

G

68.95

−0.94

0.40

2.17

73.48

−1.29

−0.71

2.17

distinguishable. At the¯X point the fitting procedure gives a

maximalRWfrequency of 70.3cm−1for 300K whichmatches

with the observation of Rocca et al [6]. 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 [17] 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,

This leads

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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

0.64–0.67˚ A

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 [4] 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 [1]. The data of previous experimental studies using

HREELS (open squares, [6, 8]) and helium atom scattering

(HAS) (open circles, [17]) are also displayed.

Figure 4. Comparison between theoretical [1] 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 [8] and HAS [17]

results, respectively. The solid circles mark the present HREELS

data. The solid circles of a given color are obtained at the same

electron energy.

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.

At first

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

The high-

Based

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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.

The

3.2.2.

scattering [17], 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.

Surface resonances.

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 [1].

Where available the data for both phonon branches agree

well with the HAS results [17].

disperse within the projected density of bulk phonons and

are therefore surface resonances. However, such resonances

are absent in ab initio calculations [1]. 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 [21]. A detailed discussion of this issue can be

found in the work of Heid and Bohnen [1]. 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’ [1]. However, as seen in figures 4(a)

and (b), we also observe these phonon features by HREELS

in good agreement with the HAS data [17]. Earlier HREELS

measurements have also detected similar resonances, for

example for clean Ru(0001) [22], again in good agreement

with HAS data [23], and for Cu(111) [24].

Both phonon branches

It has been discussed

3.2.3. S2andS6surfacephonons.

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 [13].

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

Thetheoreticallypredicted

It corresponds

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J. Phys.: Condens. Matter 23 (2011) 484006K 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 [14]

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[13]. 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

experimentallydetectedinNi(100)[4]andalsoinCu(100)[13]

using HREELS.

4. Discussion

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 [13]. 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 [14]. 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

observableonlyatlowtemperaturesanditisattributedtotheS2

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 [13]. 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[25]. 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)

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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

sum.

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) [33].

The consequences from anharmonicity have been cal-

culated by molecular-dynamics modeling for Cu(100) [14].

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 [14]). This is in good agreement with the measured

RW frequency of 73.4 cm−1at 150 K using HAS [17],

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)[35]

and for Cu(100) [34] 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 [34]

¯ hω = ¯ hωo− χa¯ hωo(2no+ 1)

(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 [17] 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 [34]. This

behavior fits to the similar Debye temperatures and similar

inward relaxation of the topmost layer on Ag(100) and on

Cu(100) [37]. 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

7

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J. Phys.: Condens. Matter 23 (2011) 484006K L Kostov et al

the more open Ag(110) for which an anharmonicity of 0.052

has been determined between 160 and 550 K [36].

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) [14]

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

observedbroadlineshapeat86K.Thismightexplainthebroad

feature at 86 and 300 K. For the S2mode of Ag(100) a weak

and broad spectral feature has been predicted at 300 K [14].

This can explain why the S2phonon is clearly observable only

at low temperatures (86 K in figure 7).

5. Conclusions

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 [17].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-

dynamics calculations.

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

of Cu(100).

(i) Two spectral features at

In

(iii) The upper edges of the projected

The RW

Acknowledgment

Financial support by the Deutsche Forschungsgemeinschaft

DFG through the Sonderforschungsbereich SFB 762 ‘Func-

tional Oxidic Interfaces’ is gratefully acknowledged.

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