3274
J. Phys. Chem. 1988, 92, 3274-3271
Simulation of Quantum Helium Films on Graphlte Jeremy Q. Broughton Materials Science Department.t State University of New York, Stony Brook, New York 1 1 794, and Naval Research Laboratory, Washington, D.C. 20375
and Farid F. Abraham* IBM Almaden Research Center, K33/801, San Jose, California 951 20-6099 (Received: June 18, 1987; In Final Form: August 17, 1987)
We have recently simulated the phases of monolayer films of )He adsorbed on graphite using the Feynman path-integral Monte Carlo method, realistic potential functions for the substrate-adsorbate and adsorbate-adsorbate interactions, and a three-dimensional geometry. We found the fluid, commensurate solid, incommensurate solid, and reentrant fluid phases to be in agreement with the experimental phase diagram. The microscopic structure of the reentrant fluid was observed to be a striped domain-wall liquid and is consistent with a recent experimental interpretation and theoretical model. In this paper, we describe. our earlier findings and expand the simulation study of the reentrant fluid phase. Also the consequences to the structure and energetics of helium films adsorbed on an unmodulated graphite substrate are reported.
We have recently initiated an investigation of the phases of quantum helium films adsorbed on graphite,’ the study being largely motivated by the He/graphite system’s rich phase diagram (ref 2-4 and Figure 1) and our interest in using the quantum Monte Carlo method to gain important information on the microstructure of the adsorbed phases. There has been significant progress in studying various quantum many-body problems at finite temperature by using computer s i m ~ l a t i o n . ~ The - ~ proceedings of the conference on Frontiers of Quantum Monte Carlo provides an up-to-date overview of this rapidly growing field.g We selected 3He in order to study the Fermi statistics problem which can plague attempts to obtain reasonable averages basic to the Monte Carlo sampling procedure. It was only after we had implemented a scheme for accounting for Fermi statistics did we learn from computer experimentation that exchange was not important at the temperatures and densities of our study. We will expand on our demonstration (1) that the phases of 3He adsorbed on graphite can be accurately simulated by the Feynman path-integral Monte Carlo method, realistic potential functions for the substrate-adsorbate and adsorbate-adsorbate interactions, and a three-dimensional geometry. By analyzing structural information, we have argued that the fluid, commensurate solid, incommensurate solid, and reentrant fluid (p) phases are found and are in agreement with the experimental phase diagram (see Figure 2). The microscopic structure of the reentrant fluid was observed to be a striped domain-wall liquid, in agreement with the experimental interpretation of Motteler4 and the striped helical Potts model calculation of Halpin-Healy and Kardar.Io In this paper, we describe our earlier findings and expand the simulation study of the reentrant fluid phase. Also the consequences for the structure of helium films when adsorbed on an unmodulated graphite substrate are reported and give valuable insights concerning important effects of the external graphite field other than providing a “platform” for creating a quasi two-dimensional film. But first we review the Feynman path-integral Monte Carlo method. In the Feynman path-integral representation, a single quantum particle is isomorphic to a classical cyclic polymer chain of M beads in which each bead j interacts with its neighboring beads j - 1 and j 1 through a harmonic force constant mM/h2B2and experiences a reduced external potential V(?(j))/M (the particle mass is m, and p is inverse temperature.) The harmonic coupling arises from the free particle propagatorfdescribing the quantum mechanical contribution of kinetic energy to the density matrix:
+
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0022-3654/88/2092-3274$01.50/0
This assumes periodic boundaries for a computational box length L. The quantum one-body partition function is
This representation is exact only in the limit of M going to infinity: 2 = 1imM+ Z,. For a given temperature and density, one has to determine empirically that M beyond which the thermodynamic properties do not effectively change. The lower the temperature, the larger M must be. The quantum N-body fermion partition function with periodic boundaries is Z =
(
M
n
$ ) M ~ ~ ~ d r ij =(I jdet ) A ( j + l j ) exp
MG+l)Ik,, =flrkG+l),rt0’)), 1 5 k , 15 N
(4)
Quantum indistinguishability (exchange) leads to a complicated N-M-body problem. ( 1 ) Abraham, F. F.; Broughton, J. Q.Phys. Reu. Lett. 1987, 59, 64. (2) Schick, M. In Phase Transitions in Surface Films;Dash, J. G., Ruvalds, J., Eds.; Plenum: New York; p 68. (3) Ecke, R. E.; Dash, J. G . Phys. Rev. E 1983, 828, 3738.
(4) Motteler, F. C. Ph.D. Dissertation, University of Washington, Seattle, WA, 1986.
(5) Alder, B. J.; Ceperley, D. M.; Pollock, E. L. Acc. Chem. Res. 1985, 18, 268.
(6) Ceperley, D. M.; Alder, B. J. Science 1986, 231, 5 5 5 . (7) Ceperley, D. M.; Pollock, E. L. Phys. Reu. Lett. 1986, 56, 3 5 1 . (8) Doll, J. D.; Freemam, D. L. Science 1986, 234, 1356. (9) “Proceedings of the Conference on Frontiers of Quantum Monte Carlo“, published as J . Stat. Phys. 1986, 43, No. 5 / 6 . (10) Halpin-Healy, T.;Kardar, M. Phys. Rev. E 1986, E34, 318.
0 1988 American Chemical Society
Simulation of Quantum Helium Films on Graphite
O.O5[ 0.04
,
, 3He
1
2
0
0
,;Jc+: 3
4
, 1
Temperature
2
F4H;
The Journal of Physical Chemistry, Vol. 92, No. 11, I988 3275
1
3
(K)
Figure 1. Phase diagram for ’He and 4He on graphitc2 The &phases were not a part of the original published diagrams and were taken from ref 4. The solid circles denote temperature-density locations were the Monte Carlo simulations were performed. C, IC, and F denote commensurate solid, incommensurate solid, and fluid, respectively.
A
S
ASP
Figure 3. Helium-graphite x-y surface potential energy dependence for a fixed height z above the graphite substrate.
functions W after and before is greater than a homogeneous random number r between 0 and 1:
> r,
IW,/W,l
Ir I1
(7)
The average value of X is given by G
C X ( i ) sign ( W ( i ) ) Figure 2. Schematic representation of the 4 3 X 43,30 registered solid (top) and of an incommensurate solid monolayer structure (bottom).
(X)=
i=l
(8)
C sign (W(i))
r=l
We have adopted a higher order correction to this “high-temperature approximation”. It takes the form of a simple modification to the potential energy El2
+ --
V’(F(j)) = V(70’))
We refer the reader to the papers of Takahashi and Imada1’-I3 for a detailed decription of the path-integral Monte Carlo method that we implemented. Unless the 3He atoms were treated as distinguishable, direct calculation of the determinant of free particle propagators was performed in evaluating the Monte Carlo weight function IWl for the spinless fermion system of N atoms14
W((ri0’)))=
(i n) M M
j= 1
P
M
det A ( j + l j ) exp( -M jCV’(rl(j), =1
..., r d ) ) )
where V’(?,(j),..., ?A)) is defined by eq 5 , and A u + l j ) is an N X N matrix of the one particle propagator matrix elements f(?&+l),?&)). The absolute value of Wis taken since the weight function in importance sampling should be positive. A Monte Carlo move is accepted if the “absolute value” of ratio of weight (11) Takahashi, M.; Imada, M. J . Phys. SOC.Jpn. 1984, 53, 963. (12) Takahashi, M.; Imada, M.J . Phys. SOC.Jpn. 1984, 53, 3765. (13) Imada, M.; Takahashi, M. J. Phys. SOC.Jpn. 1984, 53, 3770. (14) In ref 8, it is stated that the determinate evaluation scales as the cube power of the number of particles. However, in the Monte Carlo procedure,
only one row is changed for an attempted displacement, and an algorithm for the determinate evaluation has been devised that scales as the square of the number of particles (Nimrod Megiddo, IBM ARC, San Jose, CA, private communication).
For fermions, sign (W(i))can be minus one (-1). At high temperatures, the identity permutation is dominant and negative sign states are small, while at sufficiently low temperatures, all permutations have approximately equal probability and the ratio of negative to positive states approaches 50%. Two kinds of displacements of coordinates are adopted for importance sampling; a “microscopic” displacement of an individual bead and a “macroscopic” displacement of all of the beads in a cyclic polymer chain according to the recipe of Takahashi and Imada.g One Monte Carlo move is defined as N attempted macroscopic displacements, each one made after A4 trials of the microscopic displacements. Primitive displacement parameters were adjusted so that the acceptance ratios for microscopic and macroscopic displacements are approximately one-half. A similar approach for bosons would be impractical since the determinant in eq 6 becomes a perminant, and one must approach the boson problem in a different manner.’ In our Monte Carlo simulations, the number of atoms varied from 36 to 42, depending on the coverage of interest, chosen dimensions of the graphite substrate, and compatibility with periodic conditions for the initialized triangular lattice of a helium solid and the graphite lattice. Periodic boundary conditions were imposed at the four faces of the computational cell which pass through the sides of the basal plane at normal incidence to the surface. A reflecting wall was placed at the top of the computational box beyond the second layer height, but no atom was promoted to the second layer in any of the simulations. We adopted the Lennard-Jones 12-6 pair potential to represent the interaction between helium atoms and helium-carbon atoms, and the potential parameters are taken from Cole and Klein.Is Similar parameters have been shown to describe the phase diagram of (15) Cole, M. W.; Klein, J. R.Surf.Sci. 1983, 124, 547
Broughton and Abraham
3216 The Journal of Physical Chemistry, Vol. 92. No. 11. 1988
"T
a
COVERAGE OF UNITY T = 2K (DASHED1 T = UK ISOLIO I
a IS. R Figure 5. Helium radial distribution functions for the commensurate solid at 2.0 K and for the fluid of coverage of unity at 4.0 K.
1=2.5K COV=I.O IO
