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Dispersion of surface phonons in a xenon monolayer physisorbed on the graphite basal plane
Surface
364
DISPERSION PHYSISORBED Maurizio Dqxumerm~
Saence 145 (1984) 364-370 North-Holland, Amsterdam
OF SURFACE PHONONS IN A XENON ON THE GRAPHITE BASAL PLANE *
MARCHESE dr F’mu,
and Gianni
lJ~~wrr.vrir dr Trwto
MONOLAYER
JACUCCI Pow.
Trrrlto.
Irtrl,~
and
Received
7 May 1984: accepted
for publication
7 June 1984
Molecular dynamics calculations have been used to study the effect of temperature on the dispersion of surface phonons in a registered xenon monolayer physisorbed on the graphite basal plane.
1. Introduction
Interest in the properties of xenon physisorbed on the graphite (Gr) basal plane continues unabated. For example, complementary experimental [l] and theoretical [2] studies have revealed a complicated melting behaviour of solid monolayers whose low temperature structure appears to be commensurate with the underlying graphite surfaces; domain walls playing an important role [3,4]. Recent theoretical work has focussed on the nature of adatom-surface and adatom-adatom [5] potentials. In particular the Debye-Waller factor derived from atomic beam data [6] and isosteric heat measurement [7] suggest a need to revise the Xe/Gr potential [8]. In the present article we examine the lattice dynamics of Xe overlayers using the molecular dynamics (MD) technique [9]. The advantage of such an approach over conventional lattice dynamics [lo] is that the MD calculations automatically include anharmonic effects. We have investigated the sensitivity of the overlayer phonon frequencies to variations in the temperature of the overlayer and to changes in the Xe/Gr potential; a realistic Xe interatomic potential is used in all our calculations [4]. The phonon dispersion curves are found to depend on the choice of Xe/Gr potential. * Issued as NRCC
0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
Unfortunately, neither the inelastic scattering of neutrons [12] have been applied to the Xe/Gr system so that refinement
of the adatom-surface
potentials
[ll] nor He atoms at present further
is not possible.
2. The models Our model of physisorbed triangular [lo].
The
potential
(fi
X fi
Xe
atoms
interact
with
[13] scaled to the correct
for Xe [14]. (Lattice that
Xe overlayer
this potential
experiment
[16].)
consists
R30 “) lattice in registry
dynamics yields
bulk
It is now
of Xe atoms arranged
with a rigid graphite
a realistic
potential;
actually
a Kr-Kr
well depth (282 K) and minimum
calculations phonon widely
on a
basal plane (4.36 A)
on solid Xe [15] have confirmed
frequencies
accepted
that
in excellent the
accord
underlying
with
graphite
substrate mediates the interactions between the physisorbed Xe atoms; we have therefore included this effect using parameters taken from the literature [5]. The adatom-surface potential was represented as a sum of Lennard-Jones (12-6) Xe-C potentials, characterised by the usual parameters c and u. However, in adatom-surface expansion
actual calculations, because of the weak variation of the potential across the basal plane, a truncated Fourier series
of the Xe/Gr
[17]. Many
potential
sets of potential
and/or interpret various these are listed in table Debye-Waller universal correct otherwise
factor
potential asymptotic
was employed
parameters
for physisorption
behaviour
unacceptable
to explain
experiments involving the Xe/Gr system, some 1. Model A is the most recent and was fitted
data [8] while model
model
rather than direct summation
(c, a) have been proposed
for
[8]. Model
large
B was deduced potentials
adatom-surface
on the basis of a
[5]. Model
C has the
separations
D is what can be regarded
set of parameters [17] and model E has been simulation studies of domain wall formation
of to
but
is
as a traditional
used in extensive and melting [2].
variation in potential parameters exhibited in table 1 probably reflects uncertainty of analysis but also the inadequacy of the Lennard-Jones functional form chosen for the Xe-C interaction [8].
computer The wide not only (12-6)
3. Phonons via molecular dynamics The models listed in table 1 have been used to study phonons propagating in a Xe monolayer. Since the graphite substrate is assumed to be rigid we are, in effect, studying the phonon modes of a monolayer in an external field. The modes in such a film can be characterised by a two-dimensional wave vector k = (k,, ky ) that is perpendicular to the surface normal n; the plane defined by k and n is called the sagittal plane [lo]. For a given k it is convenient to
classify the phonons using the dominant polarisation characteristic. Following ref. [lo] we label a mode SH (shear horizontal) when the displacements are or SP,, when the displacements are in normal to the sagittal plane, and SP, the sagittal plane, perpendicular The phonon atomic involved R30”)
response
displacement integrating triangular
or parallel
functions
patterns
generated
the equations lattice
100 ps, the phase space trajectory
Xe / Gr Ml3 Point
via the MD
of motion
of Xe atoms.
MONOLAYER
to k.
follow from a Fourier
decomposition
calculations.
of a 12 X 12 registered
Typically
of the
The latter (fi
x fi
the system was followed
being stored on magnetic
for
tape. Thermody-
[OOOll
sy ___ SG -.--.-_ SH ___~__ 3 -,
tu G
a FREOUENCY
MONOLAYER M,
10
l/cm
I
Xe / Gr Point
20
FREQUENCY Fig. 1. Phonon response functions the graphite basal plane in a fi models A and D of the text.
(00011
30
40
( l/cm F( k, w)
1
I
I
at the Brillouin zone M, point for Xe physisorbed
Xv’% R30°
structure.
Panels
(a) and (b) refer respectively
on to
M. Murchese
Ed 01. /
namic data generated from Monte Carlo calculations discuss the thermodynamic phonon response using the S(k,
w,=jm --oo
d t exp(iwt)
Dispersron
of surjuce phomm~ rn Xe OII gruphrfe
36-l
the phase space averages agreed well with previous using the same model [g]. Accordingly, we do not data further. In solids it is customary to study the van Hove function (~(k,
t) P(-k,
O)),
where p(k,
1)=x
exp[-ik.r,(t)],
and the summation runs over all the atoms in the monolayer whose positions at time t are given by q. However, since for the monolayer k is a two-dimensional vector, this is not the most useful way to proceed. Instead, in the spirit of the one-phonon approximation for bulk crystals we define new response functions: F(k,
w) =Jm d t exp(iwt) -02
(f(k,
t)f(-k,
O)),
where f(k,
t) =C
exp(-k.R,)
W(t),
and Y(SH)=(nxk).u,,
Iq(SP,)=(n.u,),
MONOLAYER MR
Xe / Gr Point
[OOOl)
3 r; LL
0
10
20
FREOUENCY
30
I
l/cm
Fig. 2. The effect of temperature on the SP, zone ,Y, point for model A of the text.
40
50
I p honon response function F( k. w) at the Brillouin
u, being the displacement from the mean position R,. In actual calculations the three response functions were evaluated using a direct method [9], F(k,
w)=
lim jf(k,
@)1*/t.
A typical set of response Changes effect
functions
in the surface-adatom
of temperature
4. Dispersion
is shown in fig. 1 for both models A and D. potential
is to soften
mainly
and broaden
affects
the SP,
the response
mode. The
functions
(fig. 2).
curves
The method outlined in the previous section has been used to generate phonon dispersion curves such as those shown in figs. 3 and 4. At low temperatures agreement
(see fig. 3) the peaks in the response
with lattice dynamical
calculations
functions
are in excellent
based on the same model [10,15].
Since our substrate is assumed rigid we do not observe the resonant coupling of the SP, branch to the graphite modes [lO,ll], hence this branch is virtually dispersion free. The M,-point frequency is however independent of this effect [lO,ll]. Moreover, since it is also sensitive to the choice of adatom-surface potential parameters, inelastic measurements using He-atoms [12] or neutrons [ll]
could in principle
help choose between
presently
available
models.
Unfor-
tunately, to date no such data have been reported. The phonon branches SP,, and SH depend more upon the Xe interatomic potential. corrugation
We have verified
by explicit
[17] nor the substrate
calculations
mediation
effects
that neither [5] contribute
the surface significantly
Fig. 3. Phonon response functions F(k, w) at 21 K for the direction 1. -t K, based upon model A of the text. The dashed cmve is based upon lattice dynamics calculations for the same model [15].
to these frequencies.
Although
increasing
the temperature
causes the response
functions to broaden (fig. 2) there is no dramatic change, possibly because we have artifically held the monolayer in registrary with the underlying graphite at a fixed lattice constant of 4.26 A. This restriction could be removed by using either large simulation systems [2] or a flat surface. However, in the absence of a reliable surface-adatom potential we did not consider such a study worthwhile.
The main effect caused by temperature
a small increase SP,
branch
in the monolayer
surface
separation,
in our work appears
to be
which in turn causes the
to soften.
5. Discussion We have physisorbed
calculated
phonon
on the graphite
response
basal
plane.
functions
for a monolayer
Our most important
finding
of Xe is that
explicit anharmonic effects appear to be quite small at low temperature and hence here lattice dynamics calculations should provide a reliable means of investigating
the phonons.
However,
the method
we have used is applicable
to
Fig. 4. Phonon dispersion curves for a registered Xe monolayer based upon model A of the text. The circles are the MD results at 21 K while the curves are lattice dynamical results for the same system [15]. Table 1 Potential parameters for Xe-C Ref
Model
c (K)
A B
107.0 103.6
3.40 3.36
C
83.2
3.36
D E
79.5 79.5
3.69 3.74
Fl f51 [51 [I71 PI
higher temperatures and to molecular systems where anharmonic affects will play a more important role. In future work we plan to examine such systems in some detail.
Acknowledgements This work was supported O’Shea
and Gianni
unpublished
Cardini
calculations
in part by NATO for helpful
grant 242.80.
discussion
We thank Seamus
and the free exchange
of
on the same system.
References [l] P.A. Heiney. R.J. Birgenau, G.S. Brown, P.M. Horn, D.E. Morton and P.W. Stephens, Phys. Rev. Letters 48 (1982) 104. [2] F.F. Abraham, Phys. Rev. Letters 50 (1983) 978; SW. Koch and F.F. Abraham, Phys. Rev. B27 (1983) 2964. [3] C.W. Mowforth, T. Rayment and R.K. Thomas, to be published. [4] B. Joos, B. Bergersen and M.L. Klein, Phys. Rev. B28 (1983) 7219. [5] G. Vidali, M.W. Cole and J.R. Klein, Phys. Rev. B28 (1983) 3064 S. Rauber, J.R. Klein, M.W. Cole and L.W. Bruch, Surface Sci. 123 (1982) 173; S. Rauber. J.R. Klein and M.W. Cole, Phys. Rev. B27 (1983) 1314. [6] T.H. Ellis, G. Stoles and U. Valbusa, Chem. Phys. Letters 94 (1983) 247. [7] J. Piper and J.A. Morrison, Chem. Phys. Letters 103 (1984) 323. [8] M.L. Klein, Y. Ozaki and S.F. O’Shea, J. Phys. Chem. 88 (1984) 1420. [9] J.P. Hansen and M.L. Klein, Phys. Rev. B13 (1976) 878; G. Jacucci and M.L. Klein, Solid State Commun. 32 (1979) 437. [lo] E. de Rouffignac, G.P. Alldredge and F.W. de Wette. Chem. Phys. Letters 69 (1980) 29; Phys. Rev. B24 (1981) 6050. [ll] H. Taub, K. Carneiro, J.K. KJems, L. Passe11 and J.P. McTague, Phys. Rev. B16 (1977) 4551. [12] B.F. Mason and B.R. Williams, Phys. Rev. Letters 46 (1981) 1138. [13] R.A. Aziz, Mol. Phys. 38 (1979) 177. [14] J.A. Barker, M.L. Klein and M.V. Bobetic, IBM J. Res. Develop. 20 (1976) 222. [15] G.G. Cardim and S.F. O’Shea, to be published. [16] N.A. Lurie, G. Shirane and J. Skalyo, Phys. Rev. 89 (1974) 5300. [17] W.A. Steele, Surface Sci. 73 (1973) 317.
364
DISPERSION PHYSISORBED Maurizio Dqxumerm~
Saence 145 (1984) 364-370 North-Holland, Amsterdam
OF SURFACE PHONONS IN A XENON ON THE GRAPHITE BASAL PLANE *
MARCHESE dr F’mu,
and Gianni
lJ~~wrr.vrir dr Trwto
MONOLAYER
JACUCCI Pow.
Trrrlto.
Irtrl,~
and
Received
7 May 1984: accepted
for publication
7 June 1984
Molecular dynamics calculations have been used to study the effect of temperature on the dispersion of surface phonons in a registered xenon monolayer physisorbed on the graphite basal plane.
1. Introduction
Interest in the properties of xenon physisorbed on the graphite (Gr) basal plane continues unabated. For example, complementary experimental [l] and theoretical [2] studies have revealed a complicated melting behaviour of solid monolayers whose low temperature structure appears to be commensurate with the underlying graphite surfaces; domain walls playing an important role [3,4]. Recent theoretical work has focussed on the nature of adatom-surface and adatom-adatom [5] potentials. In particular the Debye-Waller factor derived from atomic beam data [6] and isosteric heat measurement [7] suggest a need to revise the Xe/Gr potential [8]. In the present article we examine the lattice dynamics of Xe overlayers using the molecular dynamics (MD) technique [9]. The advantage of such an approach over conventional lattice dynamics [lo] is that the MD calculations automatically include anharmonic effects. We have investigated the sensitivity of the overlayer phonon frequencies to variations in the temperature of the overlayer and to changes in the Xe/Gr potential; a realistic Xe interatomic potential is used in all our calculations [4]. The phonon dispersion curves are found to depend on the choice of Xe/Gr potential. * Issued as NRCC
0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
Unfortunately, neither the inelastic scattering of neutrons [12] have been applied to the Xe/Gr system so that refinement
of the adatom-surface
potentials
[ll] nor He atoms at present further
is not possible.
2. The models Our model of physisorbed triangular [lo].
The
potential
(fi
X fi
Xe
atoms
interact
with
[13] scaled to the correct
for Xe [14]. (Lattice that
Xe overlayer
this potential
experiment
[16].)
consists
R30 “) lattice in registry
dynamics yields
bulk
It is now
of Xe atoms arranged
with a rigid graphite
a realistic
potential;
actually
a Kr-Kr
well depth (282 K) and minimum
calculations phonon widely
on a
basal plane (4.36 A)
on solid Xe [15] have confirmed
frequencies
accepted
that
in excellent the
accord
underlying
with
graphite
substrate mediates the interactions between the physisorbed Xe atoms; we have therefore included this effect using parameters taken from the literature [5]. The adatom-surface potential was represented as a sum of Lennard-Jones (12-6) Xe-C potentials, characterised by the usual parameters c and u. However, in adatom-surface expansion
actual calculations, because of the weak variation of the potential across the basal plane, a truncated Fourier series
of the Xe/Gr
[17]. Many
potential
sets of potential
and/or interpret various these are listed in table Debye-Waller universal correct otherwise
factor
potential asymptotic
was employed
parameters
for physisorption
behaviour
unacceptable
to explain
experiments involving the Xe/Gr system, some 1. Model A is the most recent and was fitted
data [8] while model
model
rather than direct summation
(c, a) have been proposed
for
[8]. Model
large
B was deduced potentials
adatom-surface
on the basis of a
[5]. Model
C has the
separations
D is what can be regarded
set of parameters [17] and model E has been simulation studies of domain wall formation
of to
but
is
as a traditional
used in extensive and melting [2].
variation in potential parameters exhibited in table 1 probably reflects uncertainty of analysis but also the inadequacy of the Lennard-Jones functional form chosen for the Xe-C interaction [8].
computer The wide not only (12-6)
3. Phonons via molecular dynamics The models listed in table 1 have been used to study phonons propagating in a Xe monolayer. Since the graphite substrate is assumed to be rigid we are, in effect, studying the phonon modes of a monolayer in an external field. The modes in such a film can be characterised by a two-dimensional wave vector k = (k,, ky ) that is perpendicular to the surface normal n; the plane defined by k and n is called the sagittal plane [lo]. For a given k it is convenient to
classify the phonons using the dominant polarisation characteristic. Following ref. [lo] we label a mode SH (shear horizontal) when the displacements are or SP,, when the displacements are in normal to the sagittal plane, and SP, the sagittal plane, perpendicular The phonon atomic involved R30”)
response
displacement integrating triangular
or parallel
functions
patterns
generated
the equations lattice
100 ps, the phase space trajectory
Xe / Gr Ml3 Point
via the MD
of motion
of Xe atoms.
MONOLAYER
to k.
follow from a Fourier
decomposition
calculations.
of a 12 X 12 registered
Typically
of the
The latter (fi
x fi
the system was followed
being stored on magnetic
for
tape. Thermody-
[OOOll
sy ___ SG -.--.-_ SH ___~__ 3 -,
tu G
a FREOUENCY
MONOLAYER M,
10
l/cm
I
Xe / Gr Point
20
FREQUENCY Fig. 1. Phonon response functions the graphite basal plane in a fi models A and D of the text.
(00011
30
40
( l/cm F( k, w)
1
I
I
at the Brillouin zone M, point for Xe physisorbed
Xv’% R30°
structure.
Panels
(a) and (b) refer respectively
on to
M. Murchese
Ed 01. /
namic data generated from Monte Carlo calculations discuss the thermodynamic phonon response using the S(k,
w,=jm --oo
d t exp(iwt)
Dispersron
of surjuce phomm~ rn Xe OII gruphrfe
36-l
the phase space averages agreed well with previous using the same model [g]. Accordingly, we do not data further. In solids it is customary to study the van Hove function (~(k,
t) P(-k,
O)),
where p(k,
1)=x
exp[-ik.r,(t)],
and the summation runs over all the atoms in the monolayer whose positions at time t are given by q. However, since for the monolayer k is a two-dimensional vector, this is not the most useful way to proceed. Instead, in the spirit of the one-phonon approximation for bulk crystals we define new response functions: F(k,
w) =Jm d t exp(iwt) -02
(f(k,
t)f(-k,
O)),
where f(k,
t) =C
exp(-k.R,)
W(t),
and Y(SH)=(nxk).u,,
Iq(SP,)=(n.u,),
MONOLAYER MR
Xe / Gr Point
[OOOl)
3 r; LL
0
10
20
FREOUENCY
30
I
l/cm
Fig. 2. The effect of temperature on the SP, zone ,Y, point for model A of the text.
40
50
I p honon response function F( k. w) at the Brillouin
u, being the displacement from the mean position R,. In actual calculations the three response functions were evaluated using a direct method [9], F(k,
w)=
lim jf(k,
@)1*/t.
A typical set of response Changes effect
functions
in the surface-adatom
of temperature
4. Dispersion
is shown in fig. 1 for both models A and D. potential
is to soften
mainly
and broaden
affects
the SP,
the response
mode. The
functions
(fig. 2).
curves
The method outlined in the previous section has been used to generate phonon dispersion curves such as those shown in figs. 3 and 4. At low temperatures agreement
(see fig. 3) the peaks in the response
with lattice dynamical
calculations
functions
are in excellent
based on the same model [10,15].
Since our substrate is assumed rigid we do not observe the resonant coupling of the SP, branch to the graphite modes [lO,ll], hence this branch is virtually dispersion free. The M,-point frequency is however independent of this effect [lO,ll]. Moreover, since it is also sensitive to the choice of adatom-surface potential parameters, inelastic measurements using He-atoms [12] or neutrons [ll]
could in principle
help choose between
presently
available
models.
Unfor-
tunately, to date no such data have been reported. The phonon branches SP,, and SH depend more upon the Xe interatomic potential. corrugation
We have verified
by explicit
[17] nor the substrate
calculations
mediation
effects
that neither [5] contribute
the surface significantly
Fig. 3. Phonon response functions F(k, w) at 21 K for the direction 1. -t K, based upon model A of the text. The dashed cmve is based upon lattice dynamics calculations for the same model [15].
to these frequencies.
Although
increasing
the temperature
causes the response
functions to broaden (fig. 2) there is no dramatic change, possibly because we have artifically held the monolayer in registrary with the underlying graphite at a fixed lattice constant of 4.26 A. This restriction could be removed by using either large simulation systems [2] or a flat surface. However, in the absence of a reliable surface-adatom potential we did not consider such a study worthwhile.
The main effect caused by temperature
a small increase SP,
branch
in the monolayer
surface
separation,
in our work appears
to be
which in turn causes the
to soften.
5. Discussion We have physisorbed
calculated
phonon
on the graphite
response
basal
plane.
functions
for a monolayer
Our most important
finding
of Xe is that
explicit anharmonic effects appear to be quite small at low temperature and hence here lattice dynamics calculations should provide a reliable means of investigating
the phonons.
However,
the method
we have used is applicable
to
Fig. 4. Phonon dispersion curves for a registered Xe monolayer based upon model A of the text. The circles are the MD results at 21 K while the curves are lattice dynamical results for the same system [15]. Table 1 Potential parameters for Xe-C Ref
Model
c (K)
A B
107.0 103.6
3.40 3.36
C
83.2
3.36
D E
79.5 79.5
3.69 3.74
Fl f51 [51 [I71 PI
higher temperatures and to molecular systems where anharmonic affects will play a more important role. In future work we plan to examine such systems in some detail.
Acknowledgements This work was supported O’Shea
and Gianni
unpublished
Cardini
calculations
in part by NATO for helpful
grant 242.80.
discussion
We thank Seamus
and the free exchange
of
on the same system.
References [l] P.A. Heiney. R.J. Birgenau, G.S. Brown, P.M. Horn, D.E. Morton and P.W. Stephens, Phys. Rev. Letters 48 (1982) 104. [2] F.F. Abraham, Phys. Rev. Letters 50 (1983) 978; SW. Koch and F.F. Abraham, Phys. Rev. B27 (1983) 2964. [3] C.W. Mowforth, T. Rayment and R.K. Thomas, to be published. [4] B. Joos, B. Bergersen and M.L. Klein, Phys. Rev. B28 (1983) 7219. [5] G. Vidali, M.W. Cole and J.R. Klein, Phys. Rev. B28 (1983) 3064 S. Rauber, J.R. Klein, M.W. Cole and L.W. Bruch, Surface Sci. 123 (1982) 173; S. Rauber. J.R. Klein and M.W. Cole, Phys. Rev. B27 (1983) 1314. [6] T.H. Ellis, G. Stoles and U. Valbusa, Chem. Phys. Letters 94 (1983) 247. [7] J. Piper and J.A. Morrison, Chem. Phys. Letters 103 (1984) 323. [8] M.L. Klein, Y. Ozaki and S.F. O’Shea, J. Phys. Chem. 88 (1984) 1420. [9] J.P. Hansen and M.L. Klein, Phys. Rev. B13 (1976) 878; G. Jacucci and M.L. Klein, Solid State Commun. 32 (1979) 437. [lo] E. de Rouffignac, G.P. Alldredge and F.W. de Wette. Chem. Phys. Letters 69 (1980) 29; Phys. Rev. B24 (1981) 6050. [ll] H. Taub, K. Carneiro, J.K. KJems, L. Passe11 and J.P. McTague, Phys. Rev. B16 (1977) 4551. [12] B.F. Mason and B.R. Williams, Phys. Rev. Letters 46 (1981) 1138. [13] R.A. Aziz, Mol. Phys. 38 (1979) 177. [14] J.A. Barker, M.L. Klein and M.V. Bobetic, IBM J. Res. Develop. 20 (1976) 222. [15] G.G. Cardim and S.F. O’Shea, to be published. [16] N.A. Lurie, G. Shirane and J. Skalyo, Phys. Rev. 89 (1974) 5300. [17] W.A. Steele, Surface Sci. 73 (1973) 317.