ARTICLE pubs.acs.org/JPCA
Properties of a Suggested Commensurate Monolayer Solid of Kr/NaCl(001) L. W. Bruch* Department of Physics, University of WisconsinMadison, Madison, Wisconsin 53706, United States ABSTRACT: It is shown that a commensurate square monolayer solid of Kr/ NaCl(001) can be stabilized with a model incorporating a rather large energy corrugation amplitude. Then a square bilayer is formed under a compression and preempts the formation of an incommensurate triangular monolayer lattice. The lattice dynamics of the commensurate monolayer may be complex, because the modeling admits the possibility that it has a (2 2) unit cell with four Kr atoms.
’ INTRODUCTION Rare gas solids present an almost bewildering array of possible structures when the intrinsic close-packed ordering is frustrated by external potentials such as arise in monolayer adsorption,13 and to firmly characterize the stable ones is quite difficult. Indeed, the identification of the densest packing for spheres in three dimensions (3D) was finally given a mathematical proof only in the past decade.4 In physical adsorption there is a competition between the effects of adatomadatom and adatomsubstrate potentials and many structures may occur,1,3 even when the topmost substrate layer is triangular close-packed. The geometrical frustration is greater when an intrinsically triangular monolayer rare gas solid is adsorbed on a square lattice, and there is an extensive body of work on the modeling of the phenomena.2 In this context, the monolayer solid of krypton on the (001) face of NaCl might be relatively simple because the 3D nearestneighbor spacing of Kr at low temperature is5 Lnn = 3.99 Å, and the low-temperature lattice constant of the square surface lattice of NaCl(001) is6 l = 3.96 Å. Experiments79 on Kr/NaCl(001) show a remarkable degree of contradiction, although there is agreement that the basic spatial arrangement of the monolayer Kr is a square lattice. Further, as will be shown here, modeling of the Kr/NaCl(001) monolayer solid depends on corrugation energy terms that are poorly known and that admit complex structures. The Kr monolayer is well-approximated as a near-classical monatomic lattice and therefore is simpler to analyze than the square lattice of the quantum solid10 of H2/NaCl(001). The goal of this work is to construct a model that reproduces many of the results of the Kr experiments and that highlights special features r 2011 American Chemical Society
of this system and then to make predictions of the lattice dynamics that might be tested in an experiment such inelastic helium atom scattering. Most modeling of rare gas solids relies on the transferability of the pair potential from the dilute 3D gas phase to dense phases in 3D or a monolayer.1 This was confirmed11 to give the major part of the monolayer forces for Ar, Kr, and Xe on Pt(111). The atomsubstrate potential is still a work-in-progress as far as fundamental calculations are concerned.3 Here the Kr/NaCl(001) monolayer is treated as a two-dimensional (2D) solid (atoms in a plane) and the two principal parameters of the Kr-NaCl interaction are the average energy V0 and the leading Fourier amplitude Vg0 of the spatially periodic potential. To reproduce the observed structures, the amplitude is |Vg| = 5 meV, which is a surprisingly large value.3 This and the value V0 = 80 meV are of a magnitude that should be accessible to electronic energy calculations. There are several remarkable features of the Kr/NaCl(001) experiments that become evident when the observations are compared against other physisorbed krypton layers. Besides the occurrence of the square lattice,79 there is the quite narrow chemical potential range of stability of the monolayer solid before the bilayer is formed.8 There also is the question of stability relative to the 2D triangular lattice and to uniaxial compression from the square lattice. Special Issue: J. Peter Toennies Festschrift Received: December 10, 2010 Revised: March 28, 2011 Published: April 06, 2011 6882
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measured the energy quantum ω^ of the perpendicular vibration of the Kr. Data for krypton monolayers and bilayers on several substrates are assembled in Table 1. The Kr/Cu(110) monolayer solid is an HOC lattice and Kr/NaCl(001) is a square lattice, but the other examples are all triangular lattices, including Kr/MgO(001). Kr/ NaCl(001) is remarkable for the narrow chemical potential range 7 meV of the monolayer compared to the monolayers of Kr/ Ag(111) and Kr/graphite where the range is 3050 meV.
’ MODEL CALCULATIONS
Figure 1. A schematic of the (1 1) and (1 2) monolayer lattices of krypton on the (001) face of NaCl. The small circles denote Cl ions and the large circles denote Kr atoms which are located atop Naþ ions before the relaxations indicated by the arrows. The p(1 1) commensurate monolayer would have the square unit cell on the left, without any relaxation. The (1 2) commensurate monolayer would have the rectangular unit cell on the right, with the relaxations indicated by the arrows. These are the unit cells proposed by Gerlach and by Budde et al., respectively. The ^x-axis corresponds to the [110] azimuth of the NaCl.15.
’ REVIEW OF EXPERIMENTS Three diffraction experiments report a square p(1 1)Kr/ NaCl(001) monolayer solid or one that is only slightly perturbed from the simple square. An experiment8 with low energy electron diffraction (LEED) observed a (1 2) Kr solid on thin epitaxial NaCl films grown on a Ge(100) substrate at temperatures near 45 K and a pressure temperature phase diagram for temperatures from 38 to 47 K was constructed. The unit cell is illustrated in Figure 1. The monolayer solid was succeeded by a square bilayer lattice at higher chemical potential. Budde et al.8 fitted integral heats of adsorption qi to the monolayer and bilayer condensation lines, with rather large uncertainties, and also reported isosteric heats (q1 = 133 ( 5 meV and q2 = 114 ( 10 meV) from analysis of thermal desorption spectra. The modeling here uses the chemical potential increment from monolayer to bilayer condensation, and that is better determined directly from the phase diagram at one temperature. Using the 3D gas pressure range at 44 K, the increment is approximately 7 meV. DeKieviet et al.7 observed a square monolayer lattice of Kr with nearest-neighbor distance Lnn = 4.02 ( 0.04 Å at 30 K using helium atom diffraction. This is about 1.6% larger than the lattice constant of NaCl and about 0.5% larger than the nearestneighbor spacing5 in 3D Kr at 30 K. They also reported a bilayer solid with a square lattice. In an analysis of the monolayer diffraction intensities, they fitted an overlayer distance of Kr relative to the NaCl of 3.4 Å. Gerlach9 observed the Kr/NaCl(001) with elastic and inelastic helium atom scattering at 47 K and concluded the Kr formed a simple (1 1) commensurate square lattice. [The unit cell is illustrated in Figure 1.] He was aware of the Budde et al. results, but did not find any evidence for a higher-order-commensurate (HOC) lattice in diffraction scans along three azimuths. He also
Overview. The calculations reported here are based on static potential energy minimizations for several trial monolayer lattices. Thus there is a considerable gap between the models and experiments that treat systems with thermal expansion5 (0.6% in nearest neighbor spacing from 0 to 40 K for 3D Kr) and zero point energy (3.0 meV in a 2D calculation12 for triangular lattices of Kr/ Ag(111)). Such model calculations have been informative in the past, as they give perspective on the experimental results and identify some of the quantitative surprises in the data. This is also likely to be true for the Kr/NaCl(001) system, where the phenomena seem to be manifestations of the geometric competition between the square lattice symmetry and the triangular lattice symmetry of the intrinsic monolayer packing. As a preliminary to the detailed modeling, here are some general considerations on what might be anticipated for Kr/NaCl(001). First, it is quite unlikely that there will be a condensation into a triangular monolayer lattice followed by compression to a square p(1 1) monolayer lattice. Under compression, the stable lattice is the one with the higher density (smaller area per atom) and the triangular lattice has such high packing efficiency that an initial triangular lattice on NaCl(001) would have to have a nearestneighbor spacing Lnn > 4.25 Å. Since most observed monolayer Kr triangular lattices have Lnn < 4.20 Å, this compression transition is not likely to occur. Second, the low temperature triangular monolayer solids of Kr on Ag(111) and Pt(111) have Lnn = 4.054.15 Å, and hence 510% denser lattices than the l = 3.96 Å square lattice on NaCl(001). Then a plausible scenario for compression of the p(1 1) commensurate lattice of Kr/NaCl(001) is a first-order transition to a (modulated) triangular lattice. It becomes a quantitative matter whether compressing the square monolayer solid leads to a square bilayer solid or to a triangular monolayer solid. For corrugation energies close to the stability threshold for the commensurate square lattice relative to the triangular lattice, the registry energies of the square lattice are such a big contribution that it is difficult to see how a square lattice with small positive misfit would be stable relative to both the commensurate square and the triangular lattice. The calculations do not show such a stability. An important scenario for the continuous compression from a commensurate triangular lattice to an incommensurate lattice is via uniaxial incommensurate (UIC) lattices.13 However the strong geometrical frustration here gives rise to the possibility that the transition under uniaxial compression is first order. A first-order transition is seen in these calculations when the energy corrugation is in the range that an increase of chemical potential drives a transition to incommensurate solid before the bilayer solid condenses. Methods. The calculations evaluate the static potential energy for lattices of Kr in the presence of the NaCl(001) surface. The Kr pair potentials are the HFD-B2 semiempirical potential fitted to 3D data16 supplemented by the McLachlan substrate-mediated 6883
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Table 1. Properties of Monolayer Kr Solidsa property
Kr/NaCl(001)
Kr/Ag(111)
Kr/Pt(111)
Kr/Cu(110)
Kr/graphite
Kr/MgO(001)
q1 (meV)
130b
151 ( 5
154
114 ( 5
172 ( 2c
134 ( 9d
q2 (meV)
b
118 ( 4
∼120
∼ 116d
ω^ (meV)
123
e
3.15
2.92
c
3.854.05
2.8
f
4.1
a
The low temperature sublimation energy of 3D krypton is 116 meV. Unless otherwise noted, the data are from reviews of Bruch et al.1,3 and Zeppenfeld.17 b From fitting the PT phase diagram of Budde et al.8 to chemical potentials at 44 K. c Values from Venables and Schabes-Retchkiman.19 The chemical potential increment from monolayer condensation to bilayer condensation in their data at 56 K is 51 meV, consistent with the value 52 meV at 70 K given in a phase diagram by Specht et al.20 d Monolayer data from Meichel et al.21 and Angot and Suzanne.22 q2q1 from the chemical potential increment δμ = 18 meV at 44 K in the PT phase diagram of Meichel.23 e Gerlach.9 f Cui et al.18
dispersion energy1 with coefficients Cs1 = 39.2 au and Cs2 = 13.5 au, estimated using the approximation of Rauber et al., and an effective distance zov = 1.8 Å to the image plane, based on an estimated KrNaCl distance.7 For the bilayer calculations, the van der Waals coefficient for second layer Kr atoms is C3 = 0.256 au, again evaluated using the Rauber approximation. The potential energy and forces arising from 36 neighboring Kr atoms are calculated; this treats five shells of neighbors in the triangular lattice (L5 = 3Lnn) and seven shells in the square lattice (L7 = (10)1/2Lnn). The interaction of a first layer Kr atom with the NaCl substrate is taken to be Vs ðx, yÞ ¼ V0 þ 2Vg0 ½cos g0 x þ cos g0 y
ð1Þ
where g0 = 2π/l is the leading reciprocal lattice vector of the substrate, the x,y coordinates are centered on a Cl ion, and Vg0 is the leading corrugation energy amplitude. V0 and Vg0 are parameters determined by fitting to the difference in the monolayer and bilayer condensation energies and the relative stability of proposed monolayer structures, as described in the next subsection. In place of Vg0, a frequency parameter is used qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ω ¼ pg0 2jVg0 j=m ð2Þ
)
where m is the atomic mass; this sets the scale of the frequency spectrum of monolayer lattice vibrations near the Brillouin zone center. For Kr mass and l = 3.96 Å, Ω = 1.0 meV corresponds to |Vg0| = 3.98 meV. For the p(1 1) lattice, Ω is equal to the Brillouin zone center gap ω0 . The following monolayer structures have been treated in the calculations: the (1 1) and (1 2) commensurate lattices illustrated in Figure 1, the (2 2) commensurate lattice illustrated in Figure 2, the intrinsic 2D triangular lattice, the average 2D triangular lattice with spatial modulation that arises in the second-order Novaco-McTague perturbation theory,14 and UIC lattices1,13 with compression or dilation along one axis (the ^x axis here). The UIC lattices are characterized by misfit of the average spacing along ^x, l/l = 1 ( m in a series of structures formed from HOC lattices with NX columns, m = 1/NX; values of NX in the range 8400 are used. The dynamical stability of the commensurate monolayer lattices is tested by solving for the frequencies of lattice normal modes, scanning wave vectors in the first Brillouin zone. For small values of Ω, the (1 1) lattice is dynamically unstable as noted by Budde et al.8 However, the (1 2) lattice they proposed is also dynamically unstable then. The (2 2) is dynamically stable and degenerates to the (1 1) lattice at large enough Ω, typically for Ω > 1.1 meV in these calculations. The lowest energy of the (1 2) or (2 2) lattice, for a given pair potentials and Ω, is determined by a search as a function of
Figure 2. A schematic of the (2 2) monolayer lattice of krypton on the (001) face of NaCl. The small circles denote Cl ions and the large circles denote Kr atoms that are located atop Na þ ions before the relaxations indicated by the arrows. In the model calculations, at small Ω the unit cell has four atoms with the relative displacements as shown. At large Ω, the displacements are zero and the lattice reduces to the p(1 1) lattice shown in Figure 1.
the displacement vectors δB = ((δx, ( δy) [four combinations for the (2 2) ]. For the HOC lattices, a force relaxation algorithm13 is used to reach energy minima in the 4NX position variables. Relative stability of Kr lattices is determined by the grand potential Φ of N atoms on specified adsorbing area A and chemical potential μ. The grand potential plays the role of a generalized pressure for commensurate lattices and in the static lattice approximation is Φ ¼ NðE μÞ
ð3Þ
with energy per atom E. The Vg0 in eq 1 is fit to make the square commensurate monolayer lattice [(1 1) or (2 2)] the lowest energy structure with energy E1(0) and N = Nc = A/l 2. Then the chemical potential at monolayer condensation is μ1 = E1(0). The chemical potential μ2 at condensation of the square commensurate bilayer is set by equality of the grand potentials: Φ2 ¼ 2Nc ½E2 ð0Þ μ2 ¼ Nc ½E1 ð0Þ μ2
ð4Þ
μ2 μ1 ¼ 2½E2 ð0Þ E1 ð0Þ
ð5Þ
and so
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Table 2. Lattice Properties with McLachlan Energy Termsa Ωb (meV) LΔ(NM)c (Å) EΔ(NM)c (meV) δx =
δye e
(Å)
E1(0) (meV) ÆVæ/4Vg
e
LΔ(NM,c)f (Å)
0.9
1.0
4.035
4.07
50.80
51.24
1.1
1.25
Ω (meV) 52d
53d
LΔ(NM) (Å)
0.43
0.38
0.0
0.0
0.0
52.12
54.52
57.45
63.06
74.02
1.0
1.1
1.25
4.005
4.015
56.61 0.434
56.93 0.38
58b 0.
59b 0.
0.0
E1(0) (meV)
56.76
59.18
62.18
67.79
78.75 1.0
0.822
1.0
1.0
3.98
3.96
3.92
ÆVæ/4Vg
0.772
0.826
1.0
1.0
LΔ(NM,c) (Å)
4.00
3.98
3.95
3.91
12.3
30.1
50.2
85.4
= 0.12 12.3
= 0.12 28.9
= 0.12 48.7
= 0.12 85.7
1.5
EΔ(NM) (meV) δx = δy (Å)
4.00
misfit (UIC)g δμ(UIC)g (meV)
1.0
0.9
0.772
δμ(NM) (meV) f
1.5
Table 3. Lattice Properties without McLachlan Energy Termsa
= 0.12 151
δμ(NM) (meV) misfit
1.3
18.9
39.1
73.8
= 0.12
= 0.12
= 0.12
= 0.12
1.0
17.6
38.0
74.8
140
= 0.12
h
E2(0) (meV)
89.3
90.5
91.9
94.8
100.2
δμ (meV)
V0I μ1I
(meV)
80.9
79.1
76.6
71.0
60.0
(meV)
133
134
134
134
134
E2(0) V0 (meV)
93.5 81.0
95.0 79.2
96.7 76.6
99.5 71.0
105.0 60.0
μ1 (meV)
138
138
139
139
139
a
)
The energy per atom and nearest-neighbor spacing of the minimum energy intrinsic triangular lattice, without the V0 term, are EΔ = 50.13 meV and LΔ = 4.011 Å. b In terms of the basic substrate reciprocal lattice vector g0, the adatom mass m, and the leading corrugation energy amplitude Vg0, Ω = g0;(2|Vg0|/m)1/2, and the Brillouin zone center gap for δx = δy is ω0 = Ω(cos g0δx)1/2. c Energy and average nearest neighbor spacing of the minimum energy modulated triangular lattice, using the NovacoMcTague perturbation theory, without the V0 term. d Estimate using the force relaxation energies at A = 13.714.1 Å2/atom; LΔ = 3.984.04 Å. e Displacement vectors and energy per atom of the (2 2) commensurate square lattice, without V0. The third line is the fraction of the total available registry energy in the optimized lattice. f Lattice constant and chemical potential increment for the modulated (NM) triangular lattice obtained by compression from the 0 using eq 6. g Misfit and chemical potential increment to create a monolayer UIC lattice from the commensurate square monolayer. h Energy per atom in the commensurate bilayer square lattice without the first layer average holding potential V0/2. I Values for the average monolayer holding potential V0 and chemical potential μ1 at monolayer condensation obtained from the energies in this table and using eq 5 with μ2μ1 = 7 meV.
A given value of μ2μ1 then determines V0. Another stability test compares the commensurate monolayer and a compressed monolayer with area per atom Af = fl 2. Then the increase in chemical potential to the threshold for this compressed monolayer is μðf Þ μ1 ¼
1 ½Eðf Þ E1 ð0Þ 1f
ð6Þ
A first order transition under compression occurs when E(f) has a concave dependence on the misfit m, and the threshold is at the misfit that makes μ(f) μ1 a minimum. Results. Results of the calculations for l = 3.96 Å and pair models that include or omit the McLachlan energy are presented in Table 2 and Table 3. Results of other calculations with l = 4.00 Å and l = 3.92 Å are not shown here. Those calculations were performed to explore effects of a possible larger repeat length for the NaCl epitaxially grown on Ge(100) and for the ca. 0.5% thermal expansion of krypton at 40 K relative to 0 K. The results are qualitatively similar to those in the tables. The LΔ and EΔ in the Table headings are for the intrinsic 2D triangular lattices with the HFD-B2 potential and including or omitting the McLachlan energy. The entries, LΔ(NM) and EΔ(NM), are the results of including the second-order Novaco-McTague (NM) energy14 that arises from density modulations driven by the Vg0 terms. For Kr/NaCl(001), the NM energy acts to dilate the lattice; the perturbation theory is unstable at Ω = 1.1 and drives an unphysically large expansion (the energy is still decreasing for Lnn = 4.20 Å). The
a
Identifications as in Table 2. The energy per atom and nearestneighbor spacing of the minimum energy intrinsic triangular lattice, without the V0 term, are EΔ = 56.05 meV and LΔ = 3.987 Å. b Estimate using the force relaxation energies at A = 13.714.1 Å2/atom.
NM energy at Ω = 0.9 and 1.0 meV is close to that resulting from the force relaxation at NX = 8 and 10, with areas per atom close to the optimized NM lattice. Therefore, the result of the force relaxation at those areas is used to estimate the NM energy for Ω = 1.1 and 1.25 meV. The commensurate square monolayer lattice becomes stable relative to the modulated triangular lattice at Ω g 0.85 meV. However the (1 1) commensurate lattice is unstable relative to the (2 2) lattice, with the displacements given in the Tables, for Ω < 1.1 meV. This is a reflection of the frustration of packing in a square lattice. The adatomadatom energy in the square cell is lowered at the expense of the registry energy by such displacements for Ω < 1.1 meV. It is evident both in the potential energy calculations and in the dynamical instability of the lattice under small amplitude (harmonic) vibrations. However, in the bilayer commensurate solid, in which second layer atoms are in 4-fold coordinated sites relative to the first layer, the first layer is reorganized to have zero or very small δB. The energy to create a wall in the UIC lattice with small positive misfit (0.52.0%) is positive for all Ω cases close to the threshold for stability of the commensurate square lattice including the calculations with l = 3.92 and 4.00 Å, i.e., changing the substrate length scale by (1%. This strongly suggests that there is no stable square lattice of Kr/NaCl(001) with small positive misfit. δμ(NM) is the increase in the chemical potential to create a monolayer triangular lattice calculated using eq 6, including the second-order NovacoMcTague energy, and LΔ(NM,c) is the corresponding average nearest-neighbor spacing in this compressed lattice. The instability of the NM energy under expansion of the lattice does not disrupt the compression calculation for Ω = 1.1 and 1.25 meV. These results show that the bilayer condenses before the transition to the triangular lattice for Ω larger than about 1 meV. δμ(UIC) is the increase in the chemical potential to create a UIC monolayer solid from the commensurate monolayer. The δμ(UIC) values are close to δμ(NM), and the aspect ratio of the optimal UIC solid cell is close to that of a triangular lattice. This is a fortuitous aid to the calculations because the UIC lattice usually is invoked for small nearly continuous changes from a commensurate lattice, and a 2D array of misfit ordinarily would be needed for the treatment of a first-order change. 6885
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frequencies at ω > 2 meV are mostly determined by the atomatom forces and the fact that they generally are larger than those observed11 for Kr/Pt(111) reflects the smaller nearest-neighbor spacing in the lattice, 3.96 Å for Kr/NaCl(001) versus 4.09 Å for Kr/Pt(111). The main differences with changing Ω are for the smallest frequencies. In the calculations that include the McLachlan energy terms, as in Figure 3, the smallest frequencies are 0.05 meV for Ω = 1.1 meV, 0.60 meV for Ω = 1.25 meV, and 0.63 meV for Ω = 1.0 meV. In the p(1 1) lattice for Ω g 1.1 meV, the polarizations of the modes are rigorously transverse or longitudinal for wave vectors along the ΓM(^x)-axis as shown in Figure 3. In the (2 2) lattices, the polarizations generally are mixed, even along the ΓM-axis.
Figure 3. Frequency spectrum of the p(1 1) commensurate monolayer solid of Kr/NaCl(001) for Ω = 1.1 meV, including the McLachlan energy terms. Frequencies (in meV) are shown as a function of wave vectorBq for two scans along a ΓM axis of the monolayer. The frequencies of transverse and longitudinal polarizations for Bq = (q,0) are denoted by filled circles and squares, respectively, and for Bq = (q,0.5g0) by filled triangles and diamonds, respectively. When the four branches on 0 < q/g0 < 0.5 are folded onto 0 < q/g0 < 0.25, they reproduce the main features of the eight-branch spectrum of the four-atom unit cell for Ω = 1.0 meV.
The experiments place a bound δμ > 7 meV because it is observed8 that the commensurate bilayer is formed before a transition to the incommensurate monolayer. Then Ω ≈ 1.01.1 meV is a range that reproduces the observed sequence of transitions. The value μ1 derived from fitting μ2μ1 = 7 meV using eq 5 is closer to the experimental q1 for the calculations that include the McLachlan energy. This is in accord with experience1,3 for rare gases on other substrates that including the McLachlan energy generally improves agreement of calculations with experiment. The fact that μ1 only depends weakly on Vg0 for the given μ2μ1 is closely related to the result15 that μ2 only depends weakly on the first layer holding potential when the lattice constant change at the monolayer to bilayer transition is small. The corrugation energy amplitude for Ω = 1.0 meV is |Vg0| = 4.0 meV and for Ω = 1.1 meV is |Vg0| = 4.8 meV. These values, for g0 = 1.59 Å1, are larger than the values3 2.6 meV for Xe/Cu(110) in a HOC lattice with g0 = 1.74 Å1 and 2.6 meV for Xe/Pt in a triangular commensurate lattice with g0 = 2.62 Å1. The stability requirements on Kr/NaCl(001) lead to a value for |Vg0| of at least 4 meV, which still admits the possibility of the (2 2) commensurate monolayer lattice. The diffraction pattern would be a way to discriminate between the p( 1) and (2 2) commensurate lattices because of the doubling of the unit cell dimensions in the latter. There would be extra diffraction spots for the (2 2) case and the pattern would be only slightly different (alternating extinctions along the ^x and ^y axes) from what might be observed for two perpendicular domains of the (1 2) structure proposed by Budde et al.8 Measuring the phonon spectra of the monolayer would also tighten the determination of |Vg0| because for Ω < 1.1 meV, the spectrum has eight branches, while the spectrum for the (1 1) lattice has two branches as shown for Ω = 1.1 meV in Figure 3. Figure 3 shows the phonon spectra for B q = (q,0) and also for B q= (q,π/l ) as these four branches folded onto 0 < qx < π/(2l ) are a good guide to the spectrum for the (2 2) cell at Ω = 1.0 meV. The
’ CONCLUSIONS A model has been constructed that reproduces the stability of a commensurate square lattice of Kr/NaCl(001) and the fact that the next structure under increase of chemical potential is a bilayer commensurate square lattice. Combining the observed chemical potential increase from monolayer condensation to bilayer condensation with the interaction model leads to a value for the chemical potential at monolayer condensation that is close to the experimental value and is insensitive to the value for the corrugation energy amplitude |Vg0|. A value of |Vg0| = 45 meV suffices to reproduce the observed stabilities. However, values at the lower end of this range, which are not yet excluded by the modeling, lead to a four-atom (2 2) commensurate cell for the monolayer solid. This is a manifestation of the fact that the registry potential of the NaCl(001) then is not large enough to completely overcome the intrinsic energy advantages of the triangular lattice with its 6-fold coordination compared to the 4-fold coordination of the square. Kr/NaCl(001) therefore is a good candidate for further study of the way in which the competition of adatomadatom and adatomsubstrate forces lead to complex structures. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT The author thanks Professor R. D. Diehl for helpful discussions of the experimental data. ’ REFERENCES (1) Bruch, L. W.; Cole, M. W.; Zaremba, E. Physical Adsorption: Forces and Phenomena; Oxford University Press: Oxford, 1997. (2) Patrykiejew, A.; Sokozowski, S.; Binder, K. Surf. Sci. Rep. 2000, 37, 207. (3) Bruch, L. W.; Diehl, R. D.; Venables, J. A. Rev. Mod. Phys. 2007, 79, 1381. (4) Hales, T. C. Discrete Comput. Geom. 2006, 36, 5. (5) Korpiun, P.; L€uscher, E. In Rare Gas Solids; Klein, M. L., Venables, J. A., Eds.; Academic: New York, 1977; Vol. II, Chapter 12, pp 729822. (6) Diederichs, J.; Schilling, J. S.; Herwig, K. W.; Yelon, W. B. J. Phys. Chem. Solids 1997, 58, 123and references contained therein. (7) DeKieviet, M. F. M.; Bahatt, D.; Scoles, G.; Vidali, G.; Karimi, M. Surf. Sci. 1996, 365, 789. (8) Budde, K.; Schimmelpfennig, J.; Eichmann, M.; Ernst, W.; Pfn€ur, H. Surf. Sci. 2001, 473, 71. 6886
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