lattice dynamics simulation of ionic crystal surfaces: ii...

Lattice Dynamics Simulation of Ionic Crystal Surfaces:

II. Vibrational Modes Localised at Surface Steps and Corners

Andreas Markmann,1 Jacob L. Gavartin,2 and Alexander L. Shluger2

1Theoretische Chemie, Technische Universitat Munchen,

Lichtenbergstraße 4, 85 747 Garching, Germany

2Condensed Matter and Materials Physics Group,

Dept. of Physics and Astronomy, University College London,

Gower Street, London WC1E 6BT, Great Britain

Abstract

Attenuation parameter functions aline and apoint are used to quantify the degree of localisation of

normal modes at steps in the KBr(001), MgO(001) and CaF2(111) surfaces and at corners in a finite

cube of MgO as examples for one- and zero-dimensional defects in a surface. By the frequencies of

the localised modes thus obtained it can then be decided whether an adsorbed molecule is likely

to couple to the surface, since vibrational interaction between an adsorbed molecule and a surface

is maximal if a molecular vibration is resonant with a surface vibration localised at the adsorption

site. Crystal surfaces with steps are modelled within the periodic slab approach. Normal modes

are identified as localised if their attenuation parameter values exceed a certain threshold value.

Ambiguities exist due to the finite size of the supercell model but physically relevant criterion

cutoffs can be extracted from histogrammatic considerations. Out of the materials studied, step-

localised modes are predicted only in MgO and KBr. The maximum degree of localisation is weaker

in MgO, as it has no pronounced phonon band gap. Step modes with motion along and around

the step are predicted. Based on these findings, mechanisms for vibrational energy transport along

step funnelled by step localised modes are proposed. Normal modes localised at the corners of

an MgO cube are identified with an analogous criterion. They correspond to similar vibrational

modes at kinks in surface steps, whose properties are extrapolated from the displacements of the

corner modes.

1

I. INTRODUCTION

Molecules adsorbed on a crystal surface can exchange energy with it by vibrational inter-

action. The natural coordinates for describing vibrations in crystals are vibrational normal

modes, the modes of collective motion of atoms that do not interact with each other in the

harmonic approximation. These vibrational normal modes can be classified by their fre-

quency (or energy), their wave vector and by the degree to which certain atoms participate

in the mode.

A well-known example for a surface mode is the Rayleigh1 wave. First microscopic cal-

culations of surface modes of atomic lattices, using Green’s functions, were carried out by

Lifshitz and Rosenzweig in 19482. The first calculations of surface phonon dispersion curves

using an atomistic model were done by de Wette and co-workers in 19713–5. Experimental

measurements of surface phonon dispersion curves have been developed by Ilbach and Mills

using Electron Energy Loss Spectroscopy (EELS)6 and by Toennies and co-workers using

Inelastic Helium Atom Scattering (HAS)7. Furthermore, all-optical techniques using time-

resolved second-harmonic generation have been developed by Tom and co-workers, which

allow to probe modes at buried interfaces8.

Dispersion spectra of thin slabs modelling the MgO(001) surface presented in ref. 9 predict

normal modes up to a frequency of 21.5 THz, in good agreement with experiment. Dispersion

spectra from rigid ion calculations in the same ref., however, show the typical overestimate of

optical frequencies. This supports our view that we need to take polarisability into account

to gain useful results. A comparison of shell model slab calculation and HAS results for

the KBr(001) surface in ref. 9 shows a good quantitative agreement between theory10 and

experiment11, especially for the modes that are split from the bulk bands (see our own results

below). The maximum frequency for KBr is 4.9 THz. The CaF2(111) surface phonon Density

Of States (DOS) based on a shell model has been calculated and compared to the bulk DOS

by Allan and Mackrodt12. Theoretical CaF2 surface dispersion curves have been compared

with experimental results by Jockisch et al.13. They predict vibrational frequencies ranging

up to about 13.8THz.

In a recent paper14, we have performed periodic slab calculations of the MgO(001),

KBr(001), and CaF2(111) surfaces using shell model parameters that are specially geared to-

wards reproducing elastic properties of the crystals and identified surface modes based on the

2

“surface attenuation parameter”. Surface modes seen experimentally but not theoretically

reproduced previously were shown to have intermediate surface attenuation parameters.

In this article, we consider steps at the surfaces of the three ionic surfaces and corners of a

finite cube of MgO which play a role as a simple model for kinks at surface steps. These sites

are of particular interest as adsorption sites, as particularly the anions are more polarisable

at steps and corners/kinks due to their lower coordination and therefore more readily bond

with admolecules.

Step-localised and corner-localised normal modes identified with the criteria for normal

mode localisation introduced in ref. 14 are presented in section IV. Some of the modes found

are such that intriguing mechanisms for energy transport at the surface can be proposed.

Educated guesses at such processes based on our results are discussed in section V.

II. SETUP OF CALCULATIONS

A. Steps in the (001) Surfaces of MgO and KBr

The MgO and KBr bulk structures can, in analogy to NaCl, be built up as a face-centred

cubic Bravais lattice with a primitive unit cell consisting of the cation at (0, 0, 0) and the

anion at (1, 1, 1). Building on the slab shell model for surfaces used in ref. 14, periodic slab

models for steps in the (001) Surfaces of MgO and KBr were constructed by cutting away

half of the top and bottom surface monolayers. Supercells of the type shown in fig. 1 result.

They are two atoms deep and 2y atoms wide, were y is the width of the terraces between

the steps. The slab has two different thicknesses, b in the thinner and b + 2 in the thicker

half, modelling atomic steps at the top and the bottom of the slab. The surface lattice vectors

are collinear with the lines connecting nearest neighbour ions. In the largest calculations

used, y = 16 and b = 13. Note that each unit cell has four steps, A, B, C, D. All these

places of localisation are to be observed when deciding on whether a given normal mode is

a step mode.

With the exception of Γ-point and bulk calculations, all calculations in this chapter

were performed using eleven sampling k-points per segment connecting two corners of the

irreducible part of the Brillouin zone.

3

B. The CaF2(111) Surface

The CaF2 crystal can be built up using a face-centred cubic Bravais lattice with three

ions in the primitive unit cell, Ca2+ at (0, 0, 0), and one F− each at(

14, 1

4, 1

4

)

and(

34, 3

4, 3

4

)

.

Again building on the slab model14, a periodic slab model for steps at the CaF2(111) surface

was constructed by cutting away half of the surface monolayers. Unlike for the MgO and

KBr (001) surfaces, where a single layer of ions can be removed, the only stoichiometrically

correct way to remove ions from the CaF2 surface is to remove a triple ionic layer (which

makes up one monolayer), as shown in the lower panel (side view) of fig. 2.

The dashed grey circles specify the positions where ions have been cut away, leaving a

raised terrace in the middle of the supercell, analogous to the raised terrace in fig. 1 for MgO

and KBr.

It is not clear a priori which of the positions 1 or 2 the F ion at the left step will

take. According to ab initio Hartree-Fock calculations done by Huisinga, Reichling and co-

workers15, the structure with the fluoride ion at position 1 is more stable, so we have used

this configuration in the calculations presented here.

As before, b and b + 2 denote the number of CaF2 monolayers in the slab, respectively

(i.e. 3b to 3b + 6 ionic layers in CaF2), while y denotes the width of the terraces. The step

localisation analysis below has been performed on a supercell with y = 12 and b = 6.

Monolayers at the CaF2(111) surface consist of three ionic layers. For the purpose of

defining the aplane attenuation parameter, the ionic layer furthest from the middle of the

slab is postulated as location of the surface. The weighting function wR (r) assigns different

weights to different ionic layers within one monolayer. wR (r) is, however, flat enough near

the surface that this will not affect the physical relevance of the result. This argument holds

similarly for the step in the surface.

III. DETECTION OF SURFACE MODES

In order to allow for adequate distances between periodic images of the steps, compara-

tively large supercells were used, resulting in a large number of normal modes. Analysis of

these modes was enabled by the attenuation parameters aplane, aline and apoint, as introduced

in ref. 14.

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Briefly, the attenuation parameters are weighted averages of the normal mode compo-

nents squared p2n(jβ), where n enumerates the normal mode, j the ions and α are Cartesian

directions. The weighting function decays from near one to near zero with increasing dis-

tance from the location of interest, at which localisation is to be analysed. As the normal

mode vectors are normalised, localised modes yield an attenuation parameter near unity,

while delocalised modes yield one near zero.

We have used a weighting function wR (r) of the form

wR (r) =1

2−

1

πarctan

(

12(

r − R3

)

R

)

, (1)

where r measures the distance from the location of interest (surface plane, line along step,

point of defect/corner) and R is a parameter which is usually set to the maximum distance

L from that location encountered in the supercell, R = L. When several such locations exist

(such as there are four steps in the supercells used here), the areas up to the distance L

from different locations must not overlap. Otherwise, the normalisation of the normal mode

vectors cannot be exploited for a comparison of the modes.

The generalised attenuation parameter is then given by

agen (n) =∑

`

∑

j

wR (rj`)Patom (n, j) θ (R − rj`) ,

where “gen” can be replaced by “plane”, “line” or “point”, depending on the locations of

interest to study, which are enumerated by `, r`j is the distance of atom j from location `,

Patom (n, j) =3∑

β=1

p2n(jβ) is the participation in mode n of atom j and θ is the Heaviside step

function.

Mode n is considered a surface mode if the function aplane exceeds a certain critical

value acrit < 1 (also called criterion cutoff or threshold), i.e. mode n is called localised

if agen (n) > acrit. Note that the weighting function is not perfectly flat anywhere. If it

were nearly flat near the surface, surface and sub-surface modes could not be distinguished.

Due to the larger derivative (modulus) of the weighting function in the intermediate area,

however, surface and subsurface mode aplane values are still sufficiently separated from bulk

mode values.

The resulting criterion for normal mode localisation is numerically stable and scaleable,

gives comparable results for different supercells and is applicable to a wide range of local-

5

isation problems. A different choice of weighting function should yield comparable results.

The cutoff value acrit is arbitrary since the notion of localisation in a finite system is, strictly

speaking, ill-defined. However, it can be chosen such that a minimum number of normal

modes is left ambiguous. This can easily be done based on a histogram of the values ax (n)

over all n14.

IV. RESULTS

A. Phonon Modes Localised at a Step in the Surface, Detected with the aline

Criterion

The number of ions in the surface unit cell of a fifteen layer slab of MgO and KBr is

N = 30, i.e. the number of non-trivial normal modes in these calculations is 3N −6 = 84 for

each k-point. The number of ions in the step unit cell is 448, so the number of non-trivial

normal modes is 1338.

Due to this vastly increased number of normal modes, in the step calculations, a visual

inspection of all normal modes becomes prohibitive and an automated algorithm for detection

of localised modes is a necessary ingredient to the detection of normal modes localised at

steps (called step modes in the following).

1. Step Modes of the MgO(001) Surface

Fig. 3 shows histograms of (a) the surface attenuation parameter aplane and (b) the step

attenuation parameter aline calculated with an MgO supercell with a step (for details see

section IIA). Both densities are centred at lower values than for a flat surface (ref. 14). For

the measurement of surface localisation, this is due mostly to the fact that in comparison to

a flat surface, half of the ions at the surface (which would have received large weights) have

been removed. For the step criterion, this is due to the fact that step localisation is a more

special demand at the character of a mode than surface localisation.

Fig. 4 shows the dispersion spectrum of an MgO slab with a step (the wave vector is

parallel to the step). It can be seen that aline values above 0.4 occur resonantly around

16 THz and between k = (0, 0, 0) and k = (0, 0.2, 0) and below the main band around

6

5 THz near k = (0, 0.5, 0).

Below the main band, more step localised modes are identified at smaller k vectors if

the threshold value is relaxed to 0.33. Additional resonant step modes are then identified

around 10 THz at low k values and above 16 THz at high k values.

The normal mode at the Γ-point at 10.6 THz consists mainly of a longitudinal optical

motion of the step edge ions. Modes of this kind at k 6= 0 (including the modes marked

adjacent in fig. 4) may serve to transport an excitation along a step edge. Note that this

normal mode, however, is not localised enough to satisfy the higher cutoff value of 0.4,

i.e. energy transport is likely to be subject to considerable dissipation into the bulk of the

crystal. Following the spots marking this type of step mode through the Brillouin zone, it

can be seen that localisation is diminished towards the right in fig. 4. This means that bulk

participation and hence dissipation is likely to be greater for larger k-vectors.

The second class of step mode that can be found at the Γ-point at the higher frequency of

16.4 THz is illustrated in fig. 6b. Out of the two planes in the supercell, only the sublattice

in the paper plane is involved in the motion. There are several of these modes involving

different sublattices and step nearest neighbour ion movement in different directions relative

to the step ions. This normal mode involves motion perpendicular to the step edge and

around the step by adjacent ions. As it involves motion of step neighbour ions around the

step, such a mode may play a role in funnelling vibrational energy to the step. This type of

mode retains its character throughout the Brillouin zone and is detected at the right end of

the dispersion spectrum, fig. 4, when the 0.33 cutoff is used. In between, it is less localised.

At the lower right end of fig. 4, step modes split from the bulk spectrum can be seen.

These have the character of acoustic surface modes but are dominated by the motion of the

ions at the step edge due to their lower coordination. These normal modes are, however, quite

extended and with the small distance between two periodic images of the step (about 17A),

the corresponding degree of localisation for a real step cannot be predicted unambiguously.

2. Step Modes of the KBr(001) Surface


attenuation parameter aline. The step localisation function has a somewhat longer tail than

in MgO, indicating that step modes exist whose localisation is stronger than that seen in

7

MgO.

Fig. 7 shows the dispersion spectrum of a calculation of a KBr slab with a step. The band

gap around 2.8 THz in the vibrational spectrum that was seen in ref. 14 remains intact in the

system with a step; a lagoon between 2.13 THz and 2.3 THz can be seen from k = (0, 0.4, 0)

to k = (0, 0.5, 0).

At the upper end of both gaps, step localised normal modes are detected at a cutoff value

of 0.5, some of which are even above 0.6, i.e. more strongly localised than the step localised

modes in MgO, where values just above 0.4 were reached. Accordingly, many more step

modes are identified in KBr if the cutoff value of 0.4 is used (fig. 7).

A strongly localised (aline>0.6) resonant step mode exists at the Γ-point at 3.4 THz. It

consists mainly of a longitudinal optical motion of the step edge ions. This is analogous to

the mode identified in MgO that is a candidate for energy transport along the step edge.

The lowest frequency step mode seen at k = (0, 0.5, 0) is of the same character as the one

shown if Fig. 6b.

The second class of step mode that can be found at the Γ-point at the higher frequency

of 3.6 THz is illustrated in fig. 8a. This mode is an example for the stronger localisation in

KBr compared to MgO. Other than in MgO, where the lower coordination of the ions at the

step leads to localised modes involving most prominently these ions, the step mode shown in

fig. 8a demonstrates predominant motion of ions just below the step. An example for a step

mode with comparable degree of localisation to that found in MgO (∼4THz, in the centre

of the dispersion spectrum) is shown in fig. 8b. It also involves the sub-step ion, as well as

the higher terrace neighbour of the ion at the step.

3. Step Modes of the CaF2(111) Surface


attenuation parameter aline. The measure of localisation, fig. 9b, has a single peak between

0.075 and 0.10. The upper tail of the histogram is around 0.25. This means that step

localisation is greatly reduced compared to KBr, and even MgO.

Fig. 10 shows the dispersion spectrum of a calculation of an CaF2 slab with a step. In

the previous sections, a normal mode was considered localised at the step only if its aline

value exceeded 0.4. The maximum value of 0.265 reached in CaF2 is far below this. An

8

inspection of the displacements of the normal modes with this maximum aline value revealed

that the displacements of the ions at the step are not significantly larger than in the rest of

the supercell. This means that no step modes are predicted with the present model in CaF2.

B. Phonon Modes Localised at a Corner in MgO

1. Model

A simple continuation of the periodic approach used so far leads to a unit cell with peri-

odically repeated kinks. As this requires an equal number of ions in both lateral directions

along the surface, this dramatically increases the number of atoms in the unit cell, com-

pared to step calculations. Conversely, in order to keep the calculation computationally

viable, the lateral dimensions of the unit cell would have to be reduced compared to the step

calculations. A surface unit cell such as that shown in fig. 11 results.

With steps only three lattice spacings long, which is also the distance between kinks, this

clearly is not a good model for isolated kinks and thus would not produce useful results.

Kinks at a surface step therefore have to be simulated by the corners of a non-periodic

8 × 8 × 8 MgO cube.

This means that, unlike at a surface, the corners of the cube are not neighbours to a

lower-lying surface terrace as they would be at a kink. The ions below a surface kink are

more constrained than they are in the cube model. We will try to take this into account when

interpreting the cube results and extrapolate some properties of normal modes localised at

a surface kink.

2. Results

Fig. 12a shows a histogram of the one-dimensional point attenuation parameter apoint for

localisation around the eight corners of a 8×8×8 MgO cube. The distance is measured from

the corners of the cube. A large part inside the cube is then not covered by these spheres

which explains why the histogram is centred around 0.075 rather than the mean value of

the point attenuation parameter of 0.35. It can be seen that the histogram has a gap at 0.3

which makes for a canonical choice for a cutoff value of the criterion.

9

As the cube consists of 512 ions, 1530 non-trivial normal modes exist in this system.

Since the system is aperiodic and the defect considered is one-dimensional, the system was

simulated only at the Γ-point and the frequencies of the normal modes localised at the corners

(“corner modes”) are presented in the framework of the phonon density of states (DOS) rather

than a dispersion diagram as before. Fig. 12b shows the DOS with the frequencies of normal

modes exceeding an apoint value of 0.3 marked by dashed lines.

The lower-frequency normal modes at f = 2.97 THz and f = 3.28 THz are triply and

doubly degenerate, respectively. All are transverse acoustic modes. The higher frequency

corner modes have two-dimensional totally symmetric components and involve trapezium-

shaped distortions in the other directions. The lower frequency corner modes distort the

cube into a trapezium-shape in all directions. These corner modes play the role of the

generalised Rayleigh modes at a corner.

More normal modes of this type are detected at lower criterion cutoff values but they

are already quite extended so that they should not be regarded as corner modes for our

purposes. Fig. 13a shows a displacement diagram of one of the triply degenerate corner

modes exemplifying such a generalised Rayleigh corner mode.

The isolated corner mode at f = 7.11 THz, whose displacement diagram is shown in

fig. 13b, involves collective rotational motion of the three magnesium ions surrounding each

corner oxygen around the axis of C3 symmetry. The oxygen ions at the corners themselves

are not involved in the motion. This is an example for what could be termed a “sub-corner

mode”. The next diagonal layers of magnesium ions are also involved but only to a small

degree; their rotation is in the opposite direction to the corner neighbour magnesium ions

(not shown in the displacement diagram).

An analogue to this mode at magnesium corners is not detected at cutoff values down

to 0.24. This is understandable, as the oxygen ions are not involved in the motion of this

mode at all and hence the properties of the mode would change dramatically if the roles

of magnesium and oxygen were exchanged. This is due to the considerable polarisability

of the oxygen ions in MgO which affects (effectively damps) the motion of oxygen against

the magnesium sublattice. This appears to decrease the localisation of such a mode, if it

exists17. Finally, two high-frequency corner modes exist that involve optical motion of corner

ions against their neighbours. This type of mode is shown by the displacement diagram in

fig. 13c. Two forms of this mode exist, one each at oxygen and magnesium corners of the

10

cube. The corner mode at the magnesium corners is of lower frequency but at apoint = 0.45

significantly more localised than the other corner modes whose apoint values do not exceed

0.32. At present, however, it cannot be ascertained whether this mode corresponds to a kink

localised mode or is only due to the tetrahedral symmetry of the simulated cube.

3. Kinks in Surface Steps

At kinks in surface steps, the ions below the higher terrace are constrained and their

displacements diminished with respect to the MgO cube model. The corner modes presented

in the previous section then transform into kink modes whose ionic displacements are limited

to the atoms in the higher terrace. We will attempt in this section to extrapolate the results

from the previous section to kinks in surface steps of MgO.

Fig. 14a illustrates what the displacement diagram of a generalised Rayleigh-type kink

mode may look like. The corner modes presented in figs. 13b and (c) would also transform

into corresponding kink modes, with diminished motion of the subsurface ions.

Fig. 14b shows a double kink which is a typical defect in steps at ionic surfaces16. Gen-

eralised Rayleigh type modes can then only extend in one dimension along the step, with

strongest displacement at the kink. An optical double kink mode corresponding to the

optical corner modes presented in the previous section is illustrated in fig. 14b.

As indicated above, some of the modes predicted in this section may not exist, as the

modes they are based on are predicted only due to the symmetry of the cube. A larger,

non-symmetric calculation would have to be performed to answer this question, which is

outside the scope of this work. The extent to which the ions immediately below the kink are

displaced in a kink mode cannot be predicted reliably in analogy to the corner calculation.

This can become important for kink modes, as exemplified by the step mode presented in

fig. 8a, where motion of the ions below the step dominates.

V. DISCUSSION

Lattice dynamics provide information about the frequencies and spatial extent (localisa-

tion) of normal modes. However, as these considerations are based on the harmonic approx-

imation, no specific statements about the coupling between normal modes, i.e. vibrational

11

energy dissipation, can be made. Speculations about dissipation from molecules adsorbed

on the surface can be obtained only from the spectral properties and the localisation of the

normal modes.

If a molecule adsorbed at a step has molecular normal modes of vibration at frequen-

cies where the crystal possesses step-localised modes, vibrational coupling is likely to be

enhanced. Similarly, if the step-localisation of a lattice mode is particularly high, the am-

plitude of ionic motion at the surface at a given temperature will be particularly high.

The special character of the ionic displacements of the step modes in MgO and KBr that

involve motion along the step and around it (presented in sections IVA1 and IVA2) invites

speculation about the possibility of funnelling and steering of the vibrational energy of an

incoming vibrational wave at a step.

Such a fictitious process is illustrated in fig. 15. An incoming vibrational wave packet

consisting of an infinite number of normal modes (on the top left) that may originate from

the vibrational photoexcitation of a defect site outside the image reaches the step and leaves

three classes of waves due to overlap or collisions with the respective normal modes:

1. A transmitted wave packet,

2. a reflected wave packet and

3. vibrational motion along and around the step.

The last point is of main interest in this discussion. It means that the energy of the initially

free wave packet is funnelled to the step. This will have immediate repercussions for the

amount of energy reaching a molecule adsorbed on a step.

A particular case of funnelled energy transport along the step is illustrated in fig. 16. The

direct vibrational or electronic excitation of a photosensitive molecule causes it to vibrate

(1). This vibration will sooner or later involve a vibrational normal mode mode along the

step, either immediately during the excitation due to an overlap with the step mode or later

due to dissipation (2). The vibrational excitation can then reach another molecule adsorbed

on the step (3), which was not excited in step (1). The vibrational energy transmitted could

then be measured by IR spectroscopy or dissociation/desorption of the second molecule.

This particular process presents a viable candidate for simulation using molecular dy-

namics within the shell model. However, it requires a very large unit cell and is therefore

12

computationally quite demanding. It was also not the task of our project and is therefore

left as an idea for the future.

VI. CONCLUSION

We have used shell model lattice dynamics to predict the normal mode structure in the

(001) surfaces of MgO and KBr and the (111) surface of CaF2 with steps in the surface.

One-dimensional surface defects were modelled by the corners of a large finite cube of MgO.

The localisation of modes can be quantified by the aline and apoint attenuation parameters.

If these measures for localisation lie beyond a certain threshold value, the mode is said to be

localised at that location. The threshold values can be based on a histogrammatic evaluation

of the attenuation parameters over all normal modes. Using this criterion, vibrational modes

localised at steps of MgO, KBr and CaF2 are identified among the modes resulting from shell

model lattice dynamics calculations.

The aline criterion was used to detect step localised modes in the three ionic crystals.

Normal modes of vibration localised at steps in surfaces of MgO, KBr and CaF2 were pre-

dicted from shell model calculations for the first time. The step modes in KBr were most

strongly localised, with aline values exceeding 0.6. These were found almost exclusively at

the upper end of phonon band gaps/lagoons in the dispersion spectrum (with the exception

of one resonant step mode at the Γ-point). A corresponding number of modes localised at

a step in MgO could be detected only at a cutoff value of 0.4. This tendency can be seen

with the naked eye in the displacement diagrams. (However the decision which modes to

draw necessitated an automated criterion, as otherwise 1338 normal modes would need to

be analysed by hand.) The aline attenuation parameter does not reach 0.27 in the CaF2 step

simulation supercell. This means that no step modes in CaF2 are predicted by the present

model.

Unlike in MgO, the ions at the step immediately below the upper terrace can partake

very strongly in certain step modes of KBr. In MgO, the largest displacements are reserved

for the ions at the boundary of the upper terrace. Step localised modes were found in

MgO and KBr that involve vibration along the step. Their counterparts are step modes

that involve motion around the step. This invites speculation that these are candidates for

energy transport along the step and funnelling of a vibrational excitation to the step. A

13

process involving energy exchange between two admolecules along the step is proposed and

may be considered with a molecular dynamics simulation in the future.

Acknowledgements

The authors are indebted to J. Harding and A.H. Harker for fruitful discussions on shell

model lattice dynamics. A. Markmann would like to thank the Physics and Astronomy

Department of University College London for his studentship and the Delbrucksche Fami-

lienstiftung for further financial support. J. L. Gavartin would like to thank the Leverhulme

trust for funding.

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fluorides, J. Phys. Cond. Mat. 1 (1989) SB189.

13 A. Jockisch, U. Schroder, F. W. de Wette, W. Kress, Relaxation and dynamics of the (111)

surface of the fluorides CaF2 and SrF2, J. Phys. Cond. Mat. 5 (1993) 5401.

14 A. Markmann, J. L. Gavartin, A. L. Shluger, Lattice dynamics simulation of ionic crystal

surfaces: I. Criteria for vibrational mode localisation and surface modes, in preparation.

15 V. E. Puchin, A. V. Puchina, M. Huisinga, M. Reichling, Theoretical modelling of steps on the

CaF2(111) surface, Journal of Physics: Condensed Matter 13 (2001) 2081.

16 N.-S. Park, M.-W. Kim, S. C. Langford, J. T. Dickinson, Atomic layer wear of single-crystal

calcite in aqueous solution using scanning force microscopy, J. Appl. Phys. 80 (1996) 2680.

17 Rather than using the degree of localisation to find it, it may be found according to its symmetry.

This is, however, outside the scope of our present work.

15

Figure captions:

Figure 1: Setup of MgO and KBr step mode calculations. Note that the unit cell is two

atoms deep into the plane of the paper and that there are four steps in each unit cell, two

on each side of the slab. In this example, the width of the unit cell is y = 4 simple cubic

supercells, while the number of layers in the thinner half is b = 3. In the actual calculations,

we have used y = 16 and b = 13.

Figure 2: Setup of CaF2 step calculation. The top view shows the rectangular unit cell

ions in black. A trilayer of ions outside the area enclosed by the red dashed lines can be cut

away to form steps. The side view shows a stepped surface with the positions of the ions

that have been cut away specified by the dashed grey circles.

Figure 3: Histograms of the attenuation parameters for MgO with step.

(a) Surface attenuation parameter aplane, (b) step attenuation parameter aline.

Figure 4: Dispersion spectrum of the MgO model system involving a step. Normal modes

localised at the step satisfying different cutoff values of the aline attenuation parameter are

marked by dots.

Figure 5: Histograms of the attenuation parameters for KBr with step.


Figure 7: Dispersion spectrum of the KBr model system involving a step. Normal modes


marked by dots.

Figure 6: Illustration of some step modes in the MgO(001) surface. (a) Longitudinal

optical mode along the step edge that is a candidate for energy transport along the step

edge. (b) Optical mode that is a candidate for funnelling energy to the step.

Figure 8: Illustration of some step modes in the KBr(001) surface. (a) Optical mode

localised at the the step edge. Note that the localisation is much stronger than in any MgO

step mode and that the ion below the step has the strongest participation, a characteristic not

seen in MgO. (b) KBr step mode at a somewhat relaxed cutoff of 0.4 (here 0.49) comparable

to the values achieved by MgO step modes. Here too, the ions next to the very step are

involved most strongly.

Figure 9: Histograms of the attenuation parameters for CaF2 with step.


16

Figure 10: Dispersion spectrum of the CaF2 model system involving a step. Normal modes


marked by dots.

Figure 11: Setup of (a) periodic supercell with a kink and (b) finite cube of MgO. Note

that although the unit cell shown in (a) presents the largest computationally viable size for

a periodic calculation and subsequent analysis, this unit cell size implies very short distances

between kinks and would not yield results usable for the detection of kink-localised normal

modes. Results of a normal mode calculation of an MgO cube as shown in (b) are therefore

presented in this section.

Figure 12: Point attenuation parameter apoint applied to the MgO cube.

(a) Histogram of the corner attenuation parameter apoint. It can be seen that the his-

togram is split into modes whose value is below and above 0.3. (b) Density of states (DOS)

of the cube (solid line) with frequencies of the normal modes whose apoint value exceeds 0.3

marked with dashed lines. The lower two corner modes are acoustic modes and triply and

doubly degenerate, respectively, due to symmetry. The higher modes are optical modes and

are discussed in more detail in the text.

Figure 13: Displacement diagrams of normal modes localised at the corners of an MgO

cube. (a) Generalised Rayleigh-type corner mode f = 2.97 THz, apoint = 0.32 (similar ones

exist at f = 3.28 THz and apoint = 0.31, (b) corner mode at f = 7.11 THz and apoint = 0.32

involving mostly the corner-neighbour magnesium ions in a rotational motion around the C3

axes, (c) longitudinal optical corner mode involving mostly the corner ions. Of the latter,

two different varieties exist localised mostly at the oxygen and the magnesium corners.

Their respective frequencies and attenuation parameter values are: Magnesium corner – f

= 18.4 THz, apoint = 0.45; oxygen – f = 19.2 THz, apoint = 0.31. Note that the magnesium

corner mode localisation exceeds the localisation of all other corner modes by more than

0.12.

Figure 14: Surface kink displacement, based on speculation due lattice dynamics results

on a finite cube of MgO.

Figure 15: Illustration of fictitious interaction process between an incoming vibrational

wave packet and a surface step in MgO or KBr. Besides transmitted and reflected wave

packets, vibrations along the step and around the step also overlap with the incoming wave

packet and hence remain after the interaction.

17

Figure 16: Illustration of fictitious interaction process between the vibrations of two

admolecules and the vibrational normal mode along the step. The second admolecule can

be indirectly excited vibrationally due to the coupling between the admolecules and the step

mode, even if it is not photosensitive.

18

2nd surface translational vector

two atoms deep

A

B

C

D

Figure 1: A. Markmann et al., ???

19

z x

y

CDCCDCCDCEDEEDEEDEFDFFDFFDFGDGGDGGDG HDHHDHHDHIDIIDIIDI

I

II

top view

side view

?

?

TT

1

2

1

2


20

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

MgO surf criterion

(a)

0

0.05

0.1

0.15

0.2

0.25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

MgO step criterion

(b)


21

0

5

10

15

20

25

30

frequ

ency

[TH

z]

0 0 0 k-point 0 .5 0

MgO step phonon modes - Comparison of Threshold Levels

MgO 15 layer dispersionaline > 0.30aline > 0.33aline > 0.40


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

KBr surf criterion

(a)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

KBr step criterion

(b)


22

z x

y

step edge

(a)

x

zy

(b)


0

1

2

3

4

5

6

7

frequ

ency

[TH

z]

0 0 0 k-point 0 .5 0

KBr step phonon modes - Comparison of Threshold Levels

KBr 15 layer dispersionaline > 0.40aline > 0.50aline > 0.60


23

x

zy

(a)

x

zy

(b)


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

CaF2 surf criterion

(a)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

CaF2 step criterion

(b)


24

0

2

4

6

8

10

12

14

frequ

ency

[TH

z]

0 0 0 k-point 0 .5 0

CaF2 step phonon modes - Comparison of Threshold Levels

CaF2 15 layer dispersionaline > 0.20aline > 0.22aline > 0.25



25

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

MgO corner criterion

(a)

0 5 10 15 20 25

inte

nsity

[arb

. uni

ts]

frequency [THz]

MgO cluster phonon modes

cube DOScorner crit 0.3

(b)


(a) (b) (c)


26

(a) (b)


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LKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKLKL

transmittedwave packetmotion

around step

(2) vibrations excited byincoming wave packet

reflectedwave packet

motion along step

wave packetincoming

(1) equilibrium geometry


27

MNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNM

ONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONOONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONONO

hω

(2) transport of vibrationalenergy along step

(1) photosensitivemolecule excited

(3) non−photosensitivemolecule excited


28

lattice dynamics simulation of ionic crystal surfaces: ii...

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