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Phase transitions in diatomic molecular monolayers
517
Superlattices and Microstructures, Vol. 1, No.6, 1985
PHASE TRANSITIONS IN DIATOMIC MOLECULAR MONOLAYERS* S. D. Mahanti and S. Tang** Department of Physics and Astronomy, Michigan State University East Lansing, MI 48824 (Received 6 November 1985) Orientational order-disorder transitions in simple diatomic molecular monolayers (such as N2 and O2 on different substrates) have been studied using various theoretical techniques. The nature of such a phase transition and physical mechanisms driving it are discussed for two types of systems; one with fixed centers of mass (CM) and the other where the eM-lattice structure is allowed to change at the phase transition. The latter provides a simple example of a ferroelastic phase transitions in 2 dimensions.
I.
Introduction
Phase transitions in monolayers of monoand mUltiatomic systems adsorbed on clean surfaces have been of considerable experimental'-' and theoretical interest 6 -'O during the last decade. Multiatomic systems are particularly interesting because they exhibit phase transitions involving both translational and rotational degrees of freedom~ The nature of these transitions and the order in which they take place depends on the relative strengths of the anisotropic and isotropic parts of the intermolecular interaction. For example, in the weak anisotropy limit they can show simple orientational order-disorder (OOD) or ferroelastic transitions followed by the melting of a 2-dimensional (2D) 'isotropic sOlid.' On the other hand, for systems with strong anisotropy, one may expect to see the melting of an 'anisotropic 2D solid' to a 2D liquid-crystal phase which then undergoes a transition to a 2D isotropic liquid phase. In addition, molecular monolayers are examples of systems in which topological defects in the molecular orientation space (such as vortices and domain~walls) can coexist and interact with similar defects in the center of mass (eM) space (such as dislocations and disclinations). We have been interested during the last several years in the OOD transitions and associated thermo-elastic anomalies in 3D ionic molecular solids."k'· An important characteristic of these systems is that the nature of orientational order and the orderdisorder transition is influenced by the competition between the direct and lattice-
mediated intermolecular interaction. For 2D monolayers the periodic potential of the rigid substrate adds another dimension to the problem; if strong enough it can also influence the molecular orientation and the 2D lattice structure. Since in 2D systems fluctuation effects are strong, one expects to see important consequences of the above mentioned competition in molecular monolayers. With these ideas in mind we have studied the properties of simple 2D diatomic molecular monolayers using different theoretical techniques. Another interesting feature of these systems is that under suitable conditions of intermolecular and moleculesUbstrate interaction, the properties of the molecular overlayer can be mapped into an anisotropic planar rotor model.'~ One can use all the techniques and physical concepts developed for the isotropic 2D XY model to study the properties of these molecular systems.'s In this paper we shall summarize some of our recent results along with a discussion of related work of others in this area. In Sec. II, we give a discussion of the intermolecular interaction. Sec. III deals with the orientational order-disorder transition for systems With fixed centers of mass and IV presents the case where both molecular orientation and center of mass structure change at the transition. Finally in Sec. V we present a brief summary. II.
Intermolecular Interaction
For a proper theoretical understanding of the physical properties of molecular monolayers, we have to know the interaction potential
*This work was partially supported by the National Science Foundation, Grant No. DMR-81-17297. **Present address: Athens, GA 30602
Department of Physics, University of Georgia,
0749-6036/85/060517 + 11 $02.00/0
© 1985 Academic Press Inc. (London) Limited
Superlattices and Microstructures, Vol. 1, No.6, 1985
518 between the adsorbed molecule and the substrate and also the intermolecular interaction. 16 In general, handling both these potentials simultaneously in a theoretical study is difficult. For simplicity one makes some assumptions. For example, in the low density monolayer phase of N:> and 02 on graphite the substrate potential makes the molecule orient parallel to the substrate. In case of N the substrate provides a strong periodic poten'tial
y
which results in a 13 x 13 commensurate lattice structure of the molecular centers of mass. 17 In contrast, the commensurate potential is not strong enough for 02 and the lattice is incommensurate with the underlying 13 x 13 triangular lattice of the graphite substrate. Ie In our study of model diatomic systems, we have not explicitly included the substrate potential but take its effect into account by confining the molecular centers of mass (CM) and orientation to a 20 plane. In some cases (commensurate) CM's are fixed to a triangular lattice and in others (incommensurate) they are free to form their own lattice structure. Only the intermolecular interaction V is considered ij in great detail. The direct interaction potential between two ..homo~uclear molecules (i ,j) whose CM' s are at R~, R~ and whose molecular axis orientations are i an fi j , consist of two parts i.e. LJ Vij = Vij + VE (1 'J ij In Eq. (1) V~~ is the Lennard-Jones part which can be writ~en in terms of the atom-atom potential vaa
where
~....
..-+
and R - R + sdn , R1 ,= R1+ s'dn 1 , give the i the ~~oms of the t~o molecules i Ls positIon vectors (2d is the intermolecular distance). The electrostatic contribution comes mainly from the quadrupole moment Q, and is the only one that we willEconsider in this paper. The general form of V can be found in Ref. 19, but in the i1 paper we will consider the limiting case present where fi and fi are confined to the 20 (XY) j i plane. In the above limiting case if we expand Vi1 in terms of spherical harmonics and keep term~ up to 1=2, then we obtain Vij = V~j+ hij[cos2(ei~ ~ij) +
°ij[cos4(ei~ ~ij)
+
+
cos2(ej~ ~ij)]
cos4(ej~ ~ij)]
+ Ji{Os2(ei~ e ) + Kijcos2(e i + ej -2ljJij) j
(4)
X
Fig.
1
Two diatomic molecules with intermolecular separation Rand oriented along fi i and fi 1 directions. (A ,A ) and (If ,A ) :ire the atoms 4 1 2 associated with t~e two molecules.
where e i and e 1 are the molecul~r orientations and ~i.J 1.s the angle the vector Ri 1 makes wi th the x· axis (see Fig. 1). ·For the Lennard-Jones interaction, h i1 , Oil' J and Kij all are nonzero and depeWd upon theUparameter a=2d/R ij " In addition higher order spherical harmonIc terms are also present. For the pure quadrupole case h =D =O and K =35/3 J i1 • Thus depending ij i1 on the parameter i1 cr and thE!" strength of the quadrupole moment, different terms in the interaction potential will dominate. If we force the centers of mass to lie on either a square or a triangular lattice then in the total potential energy i.e. L Vij ' the first term V~ gives a const~irl?, the term proportional to i l vanishes and one is left with a rotational Iramiltonian H given by R
-1
HR = L Di .[cos4(e i - ~ij) J +
L [JijcOs2(ei~
ej )
+
+
cos4(ej~ ~ij)]
Ki .cos2(e i + e.- 2~ij)] J J
In the next section we will discuss in detail the nature of orientational order-disorder transitions for the above Hamiltonian. III.
Orientational Transition for a Fixed Lattice
A. Orientational Order-disorder Transition (OOOT) on a Fixed Triangular Lattice (Quadrupole Dominated Regime)
Superlattices and Microstructures, Vol. 1, No.6, 1985 Berlinsky and Harris (BH)'O made the first detailed study of the phase transition of a set of interacting quadrupoles on a triangular lattice using mean~field theory. They considered the general case where the molecules can orient in all directions. They, in addition, introduced a 2ingle-site crystal field of the form ~Vcf (3Cos e i L l) , where eiis the angle the molecular axis makes with the normal to the 20 XY-plane. In the limit Vc " - "', molecules orient parallel to the plane and their model reduces to the one given in Eq. (S) with Dij=O and Kij=3S/3Ji .• Their main results can be summarizea as fOlloJs: For large and negative Vc ' the ground state is a 2-sublattice herringbone(HB) structure. There are three equivalent orientations for the HB structure and each of these can sit on a lattice two ways thus making the ground state six-fold degenerate. This structure is quite close to the commensurate HB structure seen for N /graphite at temperature 2 below ~7K.'7 BH found that in the mean-field approximation this HB phase went through a continuous transition to an 00 phase. O'Shea and Klein" made Monte Carlo (MC) simulations of 6x6 and 12x12 clusters and found a similar transition. However the size of the systems studied were rather small and it was not possible to infer about the true nature of the 000 transition. An interesting feature of the above 000 transition is that the order parameter for the HB phase has three real components and it is predicted to lie in the universality class of the 3-component (n=3) Heisenberg model with face~type cubic anisotropy." In this case, although the mean-field theory gives a continuous transition, renormalization group (RG) arguments suggest that strong 20 fluctuations drive this transition first order. Careful MC work by Mouritsen and Berlinsky" on ~OO, 1600, 6~00 and 10,000 particle systems based on statistics ranging from SOO to SOOO MC steps/site (MCS/s) suggests that the transition is weakly first-order. This provides a good example of a fluctuation-driven first order transition" resulting from multiply degenerate ground states. In addition, the "dynamics" of the domain-walls are found to be quite slow near the phase transtion in accordance with the theoretical calculations by Safran" which showed that the domain sizes equilibriated as In(t) for n > d.l, where d is the spatial dimensionality (d-2,n=3). B. OODT on a Fixed Square Lattice (LennardJones Limit) In contrast to the triangular lattice where anisotropic intermolecular interaction gives rise to a ground state with high degeneracy (six), the square lattice is expected to give a simple ground state structure (doubly degenerate and n-l). Consequently the 000 transition is Ising-like and the dynamics near the phase
519 transition is non-logarithmic. To explore this question for a diatomic molecular monolayer we have studied the properties of a system of Lennard-Jones molecules using molecular dynamics (HO) simUlations.'· For comparison we also have constructed an effective spin Hamiltonian of the anisotropic planar rotor form (with single-site anisotropy terms) and studied the OODT using MC simulations.'7 The purpose behind the latter analysis was to use all the physical concepts developed for the planar rotor systems, namely to understand the roles and interplay of different topological defects such as vortices and domain-walls (strings) to understand the phase transitions in molecular systems. Of course the topological defects in the spin space have to be mapped into the molecular orientation space to see the true nature of the molecular arrangements. MD calculations'· were performed on a system of 100,~00 and 900 diatomic molecules with fixed mole£~lar density (constant area MD) p-S.S397xl0'·cm • Periodic boundary conditions were used to avoid surface or finite-size effects. The mass (M), internuclear separation (2d), the lattice constant (a) were taken to be -26 ~.6S17xl0 kg, 1.287S A , and ~.2787S A respectively. The atom-atom potential parameters (see Eq. 3) were 0-3.708 A. For computational convenience we measured energy in units of E, temperature in units of E/k and B time in units of t - o(m/c)1/2. MD calculat!~~s o a time step of 0.5xl0 were performed wlth s (0.005 picosecond) Which conserved energy up to 1 part in 1000. We give our main results below: The ground state is a 2-sublattice antiferro structure, the wave vector Qof the ordering is (lI/a)(I,l). In Fig. 2(a) and (b), we give the temperature dependence of the internal energy E* and the order parameter n which is given by the equation
. ..
q - < ~
I .1-1
iQ'R cos2a j e
j
>
(6)
i i ' is the CM position vector or tne jth morkcule. Both E* and n change continuously in the transition region and there is no evidence of hysterisis. The continuous transition is consistent with the prediction of an Ising-like transition based on symmetry. We do not find extremely slow relaxation processes above but near T , the transition temperature (which was seeg in the triangular lattice). A disordered system at T-l.1T was quenched by removing a certain amount of ki~etic energy. We found that the instantaneous temperature T~t) and the mean~~quare angUlar displacement e (t)=f (a (t)"'e (0) IN approached i i their equilibrium values in about 1-2 picoseconds which is quite fast (see Sec. IV for quench reSUlts in a slowly relaxing system). C. Monte Carlo Simulation of Anisotropic Planar Rotor Systems on a Square Lattice
520
Superlattices and Microstructures. Vol. 1. No.6. 1985
N=400
various theoretical techniques. For simplicity we consider a system where only nearest neighbor interactions are nonzero. For a square lattice, Wit in Eq. (5) takes values 0,n/2,n,3n/2 and the errective spin Hamiltonian is of the form
a
12.0
Hsp = 4Df
.. 8.0
e-c
+ KL(OixOjX- 0iyojy)
4.0
In Eq. (7) the classical spin variables are related to the molecular orientations by the relation
0.0
(COS29 , sin29 ) i i (8)
-4.0 1.0
H has the form of an XY model which has both srRgle~site and interaction anisotropies. The ground state of H can be obtained by using Luttinger-Tisza methgS 2 • and in Fig. 3 we show the phase diagram at T=O for J>O. For J 1.0 for AF1 and O/J > 0.5 and K/J < ~1 for AF2 respectively.
0.8
11
0.6 0.4 0.2 0.0 0
Fig. 2
(7)
r*
5
10
energy per molecule, E -E/c and (b) order parameter as f~nctions of reduced temperature T • kBT/c.
Instead of studying the thermodynamic properties of H at finite T in the entire parameter spaclf we have focussed on the limiting case where 0=0, and J,K satisfy the relation J
<~)Internal
To understand the physical nature of the excitations which drive the OOOT in these molecular monolayers, we have investigated the nature of the phase transition for the equivalent effective spin Hamiltonian using
I I
3
2
PHASE DIAGRAM FOR J> 0
I
I
$
:k1
I
I I
I
//
/"
1----------------------
//
-------------------~//
D/J ·1
·2
-3
~
II i
·5
·3
·2
·1
o
2
3
4
5
KlJ Fig. 3
Ground~state configurations for different values of O/J and K/J for
J>O. For J
521
Superlattices and Microstructures, Vol. 1, No.6, 1985 the single~site anisotropy in RG iterations thereby justifying the neglect of the latter in the study of the nature of the phase transition and the interplay of domain-wall and vortex excitations. Thermodynamic properties of a H wi th the above choice of parameters have b~~n studied using spin~wave theory at low T, Migdal Kadanof Renormalization Group (W~RG),'O Monte Carlo and Monte Carlo Renormalization Group (MCRG)" procedures. Our main results are summarized below. At low T, the orientational correlation function g(r)- can be calculated in a spin-wave approximation which in the thermodynamic limit gives (9)
where
. .. l-cosk'r 1-Y(COSk
• X
and the parameter Y-(J~K)/2(J.K). If K~O, then Y <1/2 and there is no singularity in the integrand of Eq., (10). In this case (11)
where the order par~~eter n can be written as an infinite series of the form _
kT
exp
[_ ~(1.1£)-2 '\ [(2n-l)!!]2 2n (12) 4J £ L 2n I I a n-l ..
where £-K/J and 1a-(1-1£)/(1.1£). In the limit £"0, the leading term in the series is - (l/~)ln(£) which gives k T/41TJ B n - UKI J I) (13) Thus we have n ; a exp(~CT) for K-J and a(K/J)DT for K-O, where C and D are constants. To get a general idea about the phase diagram we have used the MKRG procedure. 10 In contrast to some earlier works, the single~site field terms generated during the MKRG iterations were not moved along with the interaction bonds." The reliability of our procedure was checked by applying it to the single-site anisotropy model studied by Jose' et al." For our model we find as long as K~O (we start from K/J=O.OOOl), only after 10~20 iterations the renormalized Hamiltonian H" can be represented extremely well by a simple two~ parameter Hamiltonian of the form I
H - -J
L
x
coseicose j - A
L cos2e l •
i
•
Table I. The critical temperature T obtained by Migdal Kadanoff (MKRG) and Monte Carlo (MCRG) Renormallzation Group calcul~tions for three different. values of £=K/J. T is given by the relation T - k BTc /J(l+£).
£-K/J
MKRG T"
MCRG T"
0.01 0.10 1.00
1.02 1.18 1.32
1.04 1.20 1.34
COSk y ) (10)
n
infinity (T-O fixed point); for T > Tc ' SA iterates .to a fixed value while BJ"O. This suggests that the system Iterates to a noninteractlng "Ising" system. The transition temperature along with those calculated using MCRG procedure (see below) are given In the Table I.
(14)
There is a small correction of the type LA Cos(pB), p>2. There exists a temperature T~ such that for T < Tc ' both BJ and BA iterate to
Although MKRG results give a single Isinglike transition, to explore the true nature of the transition in more detail and to see the nature of topological excitations that drive this transition we have carried out MC simulation of (16x16) and (32x32) systems. The number of MC steps per spin were between 5000 and 8000 excepting near the critical temperature where we discarded about 11000 MCS!s and took averages over 12000 MCS!s. The temperature dependence of the heat capacity C and the spin susceptibility X indicated one phase transition for all three values of K/J-O.Ol ,0.1,1.0 studied. Both C and X peak at the same T suggesting an Ising-like transition and MCRG calculations support this picture.'7 To understand the nature of topological excitations that destroy the long range order (LRO) with increasing T, we have made a series of MC quench studies for K!J-O.l. In Fig. (4) we give our quench results starting from two initial temperatures T>T c and TT shows drastically different c behaviour. In a few MC steps most of the closely spaced VA pairs which are usually trapped inside domain-walls annihilate each other and the domain-walls get sharper. After a long MC "time" we find relatively long lived (MC "time">2000MCS/s) defects which consist of VA pairs connected by relatively sharp domain-walls (strings). The total magnetization is nearly zero after 2000 MCS/s and the system does not come to the T-O.l J/k equilibrium state. The B quench picture strongly supports the Ising character of the phase transition where the LRO is destroyed by the formation of domain walls
522
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OODT on a Nonrigid Lattice~a Model 2D Ferro-elastic Phase Transition
Until now we have been discussing the nature of phase transitions in systems where the centers of mass (CM) of the diatomic molecules are fixed. It is known that in molecular solids there is a strong coupling between orientational and translational degrees of freedom. This coupling gives rise to indirect intermolecular interaction which in some cases competes with the direct interaction in determining the ground state. The indirect interaction usually favors
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~: ~~ :b:~ ~5~cM~~~~o:;t~~e~~:~Ch) a ferro-rotational (F) order whereas the direct interaction favors an antiferro-rotational (AF) order." Depending on the dominance of one above the other one has different types of order and lattice structure. Simple examples in 3D are NaO and K0 2 • In the former the molecular orienta'tions are AF-type and in the latter they are F~type.·· In 2D, N on graphite in the lowdensity commensurate2 monolayer phase is an example of an AF structure'? (2 sublattice herringbone (HB) structure) and O2 on graphite in the monolayer o-phase is tnat of a F structure." For systems with AF ordering, the symmetry of the CM lattice does not change when orientational disorder sets in whereas those with the F ordering it does change. In latter
Superlattices and Microstructures, Vol. 1, No.6, 1985 case the orientationally ordered ferro-elastic phase undergoes a transition to the orientationally disordered para-elastic (plastic) phase. In this transition the CM lattice changes from a distorted triangular to triangular structure. We have studied'· the nature of 2D ferroelastic phase transition for a system which closely resembles the o-phase of 02 on graphite using a constant pressure MolecUlar Dynamics In our study we have (MD) method.'7'" considered only the Lennard-Jones interaction and ignored the quadrupole-quadrupole interaction. Furthermore the substrate effect has been included by demanding the 02 moleCUles to orient parallel to the substrate. The corrugation part of the substrate potential is known to be small for O/graphi te and has been dropped. Thus our system is a model anisotropic planar rotor system where the CM's of the rotors are free to move. The atom-atom Lennard-Jones parameters used here are &=54.34 ka , 0=3.05 A , the internuclear separation 2d-1.205 A and the mass of each atom m=15.99amu. These parameters are appropriate for the oxygen molecule. In the ground state we find that the molecules are parallel to each other and the CM of the molecules form a centered rectangular lattice (see Fig. 5) with a=3.332 A and b-8.054 A. These values differ by only a few percent from those obtained by Etters et al.'9 who used a pattern search program to find the ground state. If we_~~ the qUtdrupole interaction (Q • -0. 39x1 esu .cm ), the lattice parameters change by less than 0.3% and the ground state energy changes by 2%. Therefore we can put Q=O for this system without sacrificing much of the physics. Taking the
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coverage of 13 x 13 structure as 2he uni t which corresponds to 0.0636 molecules/A , our ground state has a coverage 1.112. The constant pressure MO developped by Anderson'7 and by Parrinello and Rahman" was used to study the finite T·properties. In applying this procedure to the multiatomic systems one has to evaluate the internal stress tensor P properly. In earlier studies of these syst~~s only the CM contribution to P was taken into account. This is reasonable whgX the CM-lattice symmetry does not change at the 000 transition. But for our system in which there is symmetry change we found it essential to include the rotational contribution to P • The details of the calculation of the interNNl stress tensor including both the CM and rotational contribution will be published in a separate paper.~· Here we only give the expressions for P for the planar rotor limit. The four elements~~f P are given by ~\I
• pCM _ p xx xx
L dgicos6isin6 i
(15.a)
+ Pyy _ pCM yy
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(15.b)
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(15.c)
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r
3.975 -4
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p=D.680 Fig. 5
Fer r 0 ~ e las tic (c e n t ere d rectangular/distorted triangular) and herringbone (triangular) structure of 02 molecules interacting via LennardJones potential and confined to orient in the 2-0 XY plane. All lengths are in A units.
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(15.d)
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524
Superlattices and Microstructures, Vol. 1, No.6, 1985
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(~ee text T =1=54.21K.
for
definition)
rotational motion and we have found that neglect of the rotational contribution to P makes the system rigid towards shear deform~~ions and increases the ferro-paraelastic transition tempera ture. MD equations were solved" by using a predictor-gorrector algorithim with accuracy up to O(6t). The cut~off distance for the potential i.e. the distance within which the intermolecular potential is calculated is Sa. The high accuracy of the algorithim coupled with a large cut-off distance enabled us to use a
All the three order Where 6 = L ai/N. parameters ~how sharp drops in the temperature interval 0.36 and 0.38 suggesting a first order transition. At low T the decrease in these order parameters with increasing T is small indicating a hard lattice (large librational energy gap). The energy and the density also show a discontinuous change in the transition region confirming the first-order nature of the transition. The change in the entropy per particle at the transition is 0.88 kB• Since for a free 2D rotor the total orientational
rather large time step 6taO.01102 a!m7E(- 0.02 picosecond) in our simulation studies even when fast librational motions were present. Periodic boundary conditions have been used and the system consists of 400 particles. Starting from T=O.12 (unit of temperature is elks ' e=54.34 k ) and zero external pressure the system is SIOW~y heated in temperature step of 6T=0.06. At T, -O.38 (=20.6K), the system undergoes a ferro-paraelastic transition Where the orientational LRO is lost and the lattice becomes triangular. At T2 =0.70 ( 38.0K), the orientationally disordered plastic phase melts to a 2D isotropic liqUid. We have calCUlated internal energy (E), density (p), radial distribution function (rdf), orientational order parameters (n 2 ,n6) and the lattice anisotropy parameter (YJ as functions of T. The results are briefly discussed below. In Fig. 6 , we give the T~dependence of Y,n 2and n6 which are respectively defined as
entropy is k In(2n)=1.83 k , i t seems that above the trnnsition tempera~ure the rotors are quasi-free. The detailed nature of the orientational and coupled translationalrotational dynamics are presently under investigation. We have also calculated the rotational self~diffusion constant De and different elastic constants both below (T a O.36) and above (T=0.38) the ferro~elastic transition temperature. The reSUlts are given in Table II. The fast
Y 1 n2 = N 1 n6 = N
(b/l3a - 1)
L
COS2(6i-6)
L
cos6(6i-6)
i
i
Table II. ~otational diffusion constant D (In units of rad ~ec.) and elastic constants (in units of e/a , where and are the constants in the Lennard-Jones potential) for temperature above (T-0.38) and below (T~0.36) the ferroelastic transition temperature.
De Cll C22 C12 C44
T - 0.36
T = 0.38
4.9 ±90.4 x10
1.49 ±,q.08 xlO
1.45 0.97 2.10 1.33
1.05 1.07 1.02 0.11
Superlattices and Microstructures, Vol. 1, No.6, 1985
525
4- 00 r - - - - - - - - - - - - - - - - - - - - - - - - - - ,
Reduced tempera lure is 0-36 Steps are from 12001 to 15000
3'00 c 0
u
c
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The radial distribution function of the centers of mass of the molecules at T-0.36 averaged over 3000 time
steps (60 picoseconds). The positions of the peaks indicate a distorted triangular lattice.
4·00r----------------------------, 3-50
3'00
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.;?
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Fig.8
The radial distribution function of the centers of mass of the molecules at T-0.38 averaged over 3000 time
steps (60 picoseconds). The positions of the peaks indicate a triangular lattice.
526
Superlattices and Microstructures. Vol. 1. No.6, 1985
rotational diffusion and small transverse elastic constants C 44 and C -C are characteristics of the orie~lat~o~allY disordered plastic phase. Finally, to see how the structural phase trans i t ion .takes place as a function of time we have calculated rdfs (radial distribution functions) at T=0.36 and 0.38 at different times after the system was heated from the nearby lower temperature. The shape of the first peak of the rdf was monitored as a function of time. For the triangular lattice there are six nearest neighbors (nn) which give a single peak (broadened by thermal vibrations) with weight sIx. For a distorted triangular lattIce this peak splits into two peaks • one coming from two nn's and the other from four next nn's. Thus a transition from a two-peak to a single-peak structure is a signature of the structural phase transition. The rdf at T-0.36 (Fig. 7) sho~s a sharp two-peak structure in the region 1< r < 2.65 (r is measured in units of 0) over the entire range of our MD run (300 ps). However as we heat the system to T-0.38, we find that after about 60 ps the two-peak structure distorts i.e. another peak forms in between the two peaks and after -300 ps, the system completely transforms to a single~peak (triangular lattice) structure. If we quench the system from T=0.36 to T=O.OI we recover the equilibrium values of the order parameters in a few picoseconds. However after quenching the system from T-0.38 to T-O.Ol we find that the order parameters are very small after several thousand time steps (20-50 ps); the recovery time is likely to be much larger than 50 ps. This picture is consistent With a first-order transition. The quench picture shows small ferro-elastic domains (clusters consisting of 7-20 molecules) oriented in three different but equivalent directions. In addition we also see localized defect clusters of herringbone structure. The nature and the dynamics of these clusters will be a subject of future investigation. V. Summary Overlayers of diatomic molecules exhibit a rich variety of phase transitions involving orientational and translational degrees of freedom. They provide simple experimental realizations of interesting phenomena such as fluctuation-driven first order phase transition, first~order transition in compressible anisotroPic planar-rotor systems. To extend our studies to the entire phase diagram in the p(coverage) vs T plane, we have to allow the molecules to orient away from the 2D plane. In addition, the effect of the corrugation of the SUbstrate potential (on the orientational orderdisorder and ferroelastic transition) needs a careful investigation. These are presently in progress. Acknowledgment--We thank Dr. R. K. Kalia and Dr. G. Kemeny for helpful discussions.
References 1.
2.
3.
4.
5. 6.
8.
9. 10. 11.
12. 13. 14. 15. 16.
18.
19. 20.
21.
22.
23.
Phase Transitions in Surface Film, ed. by J. G. Dash and J. P. Ruvalds, Plenum Press, N.Y. (1979). Ordering in Two Dimensions, ed. by S. K. Sinha. North Holland, Amsterdam (1980). M. Nielsen, J. P. McTague and W. Ellenson, Jour. de Physique C4, Supl. 9, 38. 1(1977)' M. Nielsen and J.-P. McTague,-Phys. Rev: B19, 3096(1979). ~F. Toney, R. D. Diehl and S. C. Fain, Phys. Rev. B27, 6413(1983); F. Y. Hansen et al., Phys. Rev. Lett. 53, 572(1985). See for example the session on 2~D Phase Transitions; Overlayers. BUll. Am. Phys. Soc 30, 333(1985). A. J7Berlinsky and A. B. Harris, Phys. Rev. Lett. 40, 1579(1978); c. R Fuselier N. S. GilliS-and J. C. Raich, Sol. St: Comm. 25, 747(1978). R. D-:- E t t e r s, R. Pan and V. Chandrasekharan. Phys. Rev. Lett. ~, 645 (1980). R. K. Kalia, P. Vashishta and S. D. Mahanti, Phys. Rev. Lett. 49, 676(1982). O. G. Mouritsen and A. J.-Serlinsky, Phys. Rev. Lett. 48, 181 (1982). S. Ostlund-and B. I. Halperin, Phys. Rev. B23. 335(1981). S:-D. Mahanti and G. Kemeny, Phys. Rev. B20, 2105(1979). O:-Sahu and S. D. Mahanti, Phys. Rev. B26, 2981 (1982). --s. D. Mahanti, Proc. of DAE Symposium 22C, 694 (1980). S. Tang, S. D. Mahanti and R. K. Kalia, Phys. Rev. B32, 3148 (1985). S. Tang andS. D. Mahanti, Phys. Rev. (to be published). W. A. Steele, The Interaction of Gases with Solid Surfaces, Pergamon Press, New York (1974). R. D. Diehl and S. C. Fain, Surf. Science l~~. 116 (1983); J. Talbot, D. J. Tildeseley and W. A. Steele, Mol. Phys. 51, 1331 (1984). -P. A. Heiney, P. W. Stephens. S. G. J. Mochrie, J. Akikatsu and R. J. Birgeneau, Surf. Science ~, 539 (1983); M. F. Toney. R. D. Diehl and S. C. Fain, Phys. Rev. B27. --6413 (1983). D. A. Goodings and M. Henkelman, Can. J. Phys. ~, 2898 (1971); P. V. Dunmore, Can. J. Phys. 55. 554 (1977). A. J. Berlinsky and A. B. Harris (see ref.6), the results of this paper, although written explicitly for H , are applicable 2 to N • 2 S. F. O'Shea and M. L. Klein, Chern. Phys. Lett. 66, 381 (1979). A. B. Harris and A. J. Berlinsky, Can. J. Phys. 57, 1852 (1979). O. G. Mouritsen and A. J. Berlinsky (see ref.9); O. G. Mouritsen, Computer Studies of )hase Transition and Critical Phenomena (Springer Verlag, He i delberg, 1984). Ch.5.
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Superlattices and Microstructures, Vol. 1, No.6, 1985 24.
25. 26. 27. 28. 29. 30. 31.
E. K. Riedel, Physica (Utrecht) 106A, 110 (1981); P. Pfeuty and G. Toulouse, In troduction to Renormalization Group and Critical Phenomena (Wiley, New York, 1977), Ch.9. s. A. Safran, Phys. Rev. Lett. 46, 1581 (1981); See also 1. M. Lifshitz, SOV. Phys. JETP 15, 939 (1962). R. K.~alia et.al (see ref.8). S. Tang and S. D. Mahanti (to be published) • J. M. Luttinger and L. Tisza, Phys. Rev. 70, 954 (1946); J. M. Luttinger, Phys. Rev. 8T, 1015 (1951). Jose', L.P. Kadanoff, S. Kirkpatrick and D. R. Nelson, Phys. Rev. B16, 1217 (1977). A. A. Migdal, Sov. PhYS. JETP 42, 743 (1976); L. P. Kadanoff, Ann. Phys 100, 359 (1976). ---S. H. Shenker and J. Tobochnik, Phys. Rev.
v:
32. 33. 34. 35. 36. 37. 38. 39.
40.
822, 4462 (1980); see also J. Tobochnik, Phys. Rev. 826, 6201 (1982). S. Tang, Ph. D thesis, Michigan State Universiry, East Lansing (1985). V. Jose' et.al (see ref. 29). D. Sahu and S. D. Mahanti (see ref. 12). S. D. Mahanti (see ref 11 and 13 ). S. Tang, S. D. Mahanti and R. K. Kalia (submitted to Phys. Rev. Lett.) H. C. Anderson, J. Chern. Phys. 72, 2384 (1980). -M. Parrinello and A. Rahman, Phys. Rev. Lett. 45, 1196 (1980); J. Appl. Phys. 52, 7182 (1~0); J. Chern Phys. 76, 2662 (1980). R. D. Etters et. al. (seeref.7); R. Pan, R. D. Etters, K. Kobashi and V. Chandrasekharan, J. Chern. Phys. 77, 1035 (1982). -S. Tang, S. D. Mahanti and R. K. Kalia (to be published).
Superlattices and Microstructures, Vol. 1, No.6, 1985
PHASE TRANSITIONS IN DIATOMIC MOLECULAR MONOLAYERS* S. D. Mahanti and S. Tang** Department of Physics and Astronomy, Michigan State University East Lansing, MI 48824 (Received 6 November 1985) Orientational order-disorder transitions in simple diatomic molecular monolayers (such as N2 and O2 on different substrates) have been studied using various theoretical techniques. The nature of such a phase transition and physical mechanisms driving it are discussed for two types of systems; one with fixed centers of mass (CM) and the other where the eM-lattice structure is allowed to change at the phase transition. The latter provides a simple example of a ferroelastic phase transitions in 2 dimensions.
I.
Introduction
Phase transitions in monolayers of monoand mUltiatomic systems adsorbed on clean surfaces have been of considerable experimental'-' and theoretical interest 6 -'O during the last decade. Multiatomic systems are particularly interesting because they exhibit phase transitions involving both translational and rotational degrees of freedom~ The nature of these transitions and the order in which they take place depends on the relative strengths of the anisotropic and isotropic parts of the intermolecular interaction. For example, in the weak anisotropy limit they can show simple orientational order-disorder (OOD) or ferroelastic transitions followed by the melting of a 2-dimensional (2D) 'isotropic sOlid.' On the other hand, for systems with strong anisotropy, one may expect to see the melting of an 'anisotropic 2D solid' to a 2D liquid-crystal phase which then undergoes a transition to a 2D isotropic liquid phase. In addition, molecular monolayers are examples of systems in which topological defects in the molecular orientation space (such as vortices and domain~walls) can coexist and interact with similar defects in the center of mass (eM) space (such as dislocations and disclinations). We have been interested during the last several years in the OOD transitions and associated thermo-elastic anomalies in 3D ionic molecular solids."k'· An important characteristic of these systems is that the nature of orientational order and the orderdisorder transition is influenced by the competition between the direct and lattice-
mediated intermolecular interaction. For 2D monolayers the periodic potential of the rigid substrate adds another dimension to the problem; if strong enough it can also influence the molecular orientation and the 2D lattice structure. Since in 2D systems fluctuation effects are strong, one expects to see important consequences of the above mentioned competition in molecular monolayers. With these ideas in mind we have studied the properties of simple 2D diatomic molecular monolayers using different theoretical techniques. Another interesting feature of these systems is that under suitable conditions of intermolecular and moleculesUbstrate interaction, the properties of the molecular overlayer can be mapped into an anisotropic planar rotor model.'~ One can use all the techniques and physical concepts developed for the isotropic 2D XY model to study the properties of these molecular systems.'s In this paper we shall summarize some of our recent results along with a discussion of related work of others in this area. In Sec. II, we give a discussion of the intermolecular interaction. Sec. III deals with the orientational order-disorder transition for systems With fixed centers of mass and IV presents the case where both molecular orientation and center of mass structure change at the transition. Finally in Sec. V we present a brief summary. II.
Intermolecular Interaction
For a proper theoretical understanding of the physical properties of molecular monolayers, we have to know the interaction potential
*This work was partially supported by the National Science Foundation, Grant No. DMR-81-17297. **Present address: Athens, GA 30602
Department of Physics, University of Georgia,
0749-6036/85/060517 + 11 $02.00/0
© 1985 Academic Press Inc. (London) Limited
Superlattices and Microstructures, Vol. 1, No.6, 1985
518 between the adsorbed molecule and the substrate and also the intermolecular interaction. 16 In general, handling both these potentials simultaneously in a theoretical study is difficult. For simplicity one makes some assumptions. For example, in the low density monolayer phase of N:> and 02 on graphite the substrate potential makes the molecule orient parallel to the substrate. In case of N the substrate provides a strong periodic poten'tial
y
which results in a 13 x 13 commensurate lattice structure of the molecular centers of mass. 17 In contrast, the commensurate potential is not strong enough for 02 and the lattice is incommensurate with the underlying 13 x 13 triangular lattice of the graphite substrate. Ie In our study of model diatomic systems, we have not explicitly included the substrate potential but take its effect into account by confining the molecular centers of mass (CM) and orientation to a 20 plane. In some cases (commensurate) CM's are fixed to a triangular lattice and in others (incommensurate) they are free to form their own lattice structure. Only the intermolecular interaction V is considered ij in great detail. The direct interaction potential between two ..homo~uclear molecules (i ,j) whose CM' s are at R~, R~ and whose molecular axis orientations are i an fi j , consist of two parts i.e. LJ Vij = Vij + VE (1 'J ij In Eq. (1) V~~ is the Lennard-Jones part which can be writ~en in terms of the atom-atom potential vaa
where
~....
..-+
and R - R + sdn , R1 ,= R1+ s'dn 1 , give the i the ~~oms of the t~o molecules i Ls positIon vectors (2d is the intermolecular distance). The electrostatic contribution comes mainly from the quadrupole moment Q, and is the only one that we willEconsider in this paper. The general form of V can be found in Ref. 19, but in the i1 paper we will consider the limiting case present where fi and fi are confined to the 20 (XY) j i plane. In the above limiting case if we expand Vi1 in terms of spherical harmonics and keep term~ up to 1=2, then we obtain Vij = V~j+ hij[cos2(ei~ ~ij) +
°ij[cos4(ei~ ~ij)
+
+
cos2(ej~ ~ij)]
cos4(ej~ ~ij)]
+ Ji{Os2(ei~ e ) + Kijcos2(e i + ej -2ljJij) j
(4)
X
Fig.
1
Two diatomic molecules with intermolecular separation Rand oriented along fi i and fi 1 directions. (A ,A ) and (If ,A ) :ire the atoms 4 1 2 associated with t~e two molecules.
where e i and e 1 are the molecul~r orientations and ~i.J 1.s the angle the vector Ri 1 makes wi th the x· axis (see Fig. 1). ·For the Lennard-Jones interaction, h i1 , Oil' J and Kij all are nonzero and depeWd upon theUparameter a=2d/R ij " In addition higher order spherical harmonIc terms are also present. For the pure quadrupole case h =D =O and K =35/3 J i1 • Thus depending ij i1 on the parameter i1 cr and thE!" strength of the quadrupole moment, different terms in the interaction potential will dominate. If we force the centers of mass to lie on either a square or a triangular lattice then in the total potential energy i.e. L Vij ' the first term V~ gives a const~irl?, the term proportional to i l vanishes and one is left with a rotational Iramiltonian H given by R
-1
HR = L Di .[cos4(e i - ~ij)
L [JijcOs2(ei~
ej )
+
+
cos4(ej~ ~ij)]
Ki .cos2(e i + e.- 2~ij)] J J
In the next section we will discuss in detail the nature of orientational order-disorder transitions for the above Hamiltonian. III.
Orientational Transition for a Fixed Lattice
A. Orientational Order-disorder Transition (OOOT) on a Fixed Triangular Lattice (Quadrupole Dominated Regime)
Superlattices and Microstructures, Vol. 1, No.6, 1985 Berlinsky and Harris (BH)'O made the first detailed study of the phase transition of a set of interacting quadrupoles on a triangular lattice using mean~field theory. They considered the general case where the molecules can orient in all directions. They, in addition, introduced a 2ingle-site crystal field of the form ~Vcf (3Cos e i L l) , where eiis the angle the molecular axis makes with the normal to the 20 XY-plane. In the limit Vc " - "', molecules orient parallel to the plane and their model reduces to the one given in Eq. (S) with Dij=O and Kij=3S/3Ji .• Their main results can be summarizea as fOlloJs: For large and negative Vc ' the ground state is a 2-sublattice herringbone(HB) structure. There are three equivalent orientations for the HB structure and each of these can sit on a lattice two ways thus making the ground state six-fold degenerate. This structure is quite close to the commensurate HB structure seen for N /graphite at temperature 2 below ~7K.'7 BH found that in the mean-field approximation this HB phase went through a continuous transition to an 00 phase. O'Shea and Klein" made Monte Carlo (MC) simulations of 6x6 and 12x12 clusters and found a similar transition. However the size of the systems studied were rather small and it was not possible to infer about the true nature of the 000 transition. An interesting feature of the above 000 transition is that the order parameter for the HB phase has three real components and it is predicted to lie in the universality class of the 3-component (n=3) Heisenberg model with face~type cubic anisotropy." In this case, although the mean-field theory gives a continuous transition, renormalization group (RG) arguments suggest that strong 20 fluctuations drive this transition first order. Careful MC work by Mouritsen and Berlinsky" on ~OO, 1600, 6~00 and 10,000 particle systems based on statistics ranging from SOO to SOOO MC steps/site (MCS/s) suggests that the transition is weakly first-order. This provides a good example of a fluctuation-driven first order transition" resulting from multiply degenerate ground states. In addition, the "dynamics" of the domain-walls are found to be quite slow near the phase transtion in accordance with the theoretical calculations by Safran" which showed that the domain sizes equilibriated as In(t) for n > d.l, where d is the spatial dimensionality (d-2,n=3). B. OODT on a Fixed Square Lattice (LennardJones Limit) In contrast to the triangular lattice where anisotropic intermolecular interaction gives rise to a ground state with high degeneracy (six), the square lattice is expected to give a simple ground state structure (doubly degenerate and n-l). Consequently the 000 transition is Ising-like and the dynamics near the phase
519 transition is non-logarithmic. To explore this question for a diatomic molecular monolayer we have studied the properties of a system of Lennard-Jones molecules using molecular dynamics (HO) simUlations.'· For comparison we also have constructed an effective spin Hamiltonian of the anisotropic planar rotor form (with single-site anisotropy terms) and studied the OODT using MC simulations.'7 The purpose behind the latter analysis was to use all the physical concepts developed for the planar rotor systems, namely to understand the roles and interplay of different topological defects such as vortices and domain-walls (strings) to understand the phase transitions in molecular systems. Of course the topological defects in the spin space have to be mapped into the molecular orientation space to see the true nature of the molecular arrangements. MD calculations'· were performed on a system of 100,~00 and 900 diatomic molecules with fixed mole£~lar density (constant area MD) p-S.S397xl0'·cm • Periodic boundary conditions were used to avoid surface or finite-size effects. The mass (M), internuclear separation (2d), the lattice constant (a) were taken to be -26 ~.6S17xl0 kg, 1.287S A , and ~.2787S A respectively. The atom-atom potential parameters (see Eq. 3) were 0-3.708 A. For computational convenience we measured energy in units of E, temperature in units of E/k and B time in units of t - o(m/c)1/2. MD calculat!~~s o a time step of 0.5xl0 were performed wlth s (0.005 picosecond) Which conserved energy up to 1 part in 1000. We give our main results below: The ground state is a 2-sublattice antiferro structure, the wave vector Qof the ordering is (lI/a)(I,l). In Fig. 2(a) and (b), we give the temperature dependence of the internal energy E* and the order parameter n which is given by the equation
. ..
q - < ~
I .1-1
iQ'R cos2a j e
j
>
(6)
i i ' is the CM position vector or tne jth morkcule. Both E* and n change continuously in the transition region and there is no evidence of hysterisis. The continuous transition is consistent with the prediction of an Ising-like transition based on symmetry. We do not find extremely slow relaxation processes above but near T , the transition temperature (which was seeg in the triangular lattice). A disordered system at T-l.1T was quenched by removing a certain amount of ki~etic energy. We found that the instantaneous temperature T~t) and the mean~~quare angUlar displacement e (t)=f (a (t)"'e (0) IN approached i i their equilibrium values in about 1-2 picoseconds which is quite fast (see Sec. IV for quench reSUlts in a slowly relaxing system). C. Monte Carlo Simulation of Anisotropic Planar Rotor Systems on a Square Lattice
520
Superlattices and Microstructures. Vol. 1. No.6. 1985
N=400
various theoretical techniques. For simplicity we consider a system where only nearest neighbor interactions are nonzero. For a square lattice, Wit in Eq. (5) takes values 0,n/2,n,3n/2 and the errective spin Hamiltonian is of the form
a
12.0
Hsp = 4Df
.. 8.0
e-c
+ KL(OixOjX- 0iyojy)
4.0
In Eq. (7) the classical spin variables are related to the molecular orientations by the relation
0.0
(COS29 , sin29 ) i i (8)
-4.0 1.0
H has the form of an XY model which has both srRgle~site and interaction anisotropies. The ground state of H can be obtained by using Luttinger-Tisza methgS 2 • and in Fig. 3 we show the phase diagram at T=O for J>O. For J
0.8
11
0.6 0.4 0.2 0.0 0
Fig. 2
(7)
r*
5
10
energy per molecule, E -E/c and (b) order parameter as f~nctions of reduced temperature T • kBT/c.
Instead of studying the thermodynamic properties of H at finite T in the entire parameter spaclf we have focussed on the limiting case where 0=0, and J,K satisfy the relation J
<~)Internal
To understand the physical nature of the excitations which drive the OOOT in these molecular monolayers, we have investigated the nature of the phase transition for the equivalent effective spin Hamiltonian using
I I
3
2
PHASE DIAGRAM FOR J> 0
I
I
$
:k1
I
I I
I
//
/"
1----------------------
//
-------------------~//
D/J ·1
·2
-3
~
II i
·5
·3
·2
·1
o
2
3
4
5
KlJ Fig. 3
Ground~state configurations for different values of O/J and K/J for
J>O. For J
521
Superlattices and Microstructures, Vol. 1, No.6, 1985 the single~site anisotropy in RG iterations thereby justifying the neglect of the latter in the study of the nature of the phase transition and the interplay of domain-wall and vortex excitations. Thermodynamic properties of a H wi th the above choice of parameters have b~~n studied using spin~wave theory at low T, Migdal Kadanof Renormalization Group (W~RG),'O Monte Carlo and Monte Carlo Renormalization Group (MCRG)" procedures. Our main results are summarized below. At low T, the orientational correlation function g(r)-
where
. .. l-cosk'r 1-Y(COSk
• X
and the parameter Y-(J~K)/2(J.K). If K~O, then Y <1/2 and there is no singularity in the integrand of Eq., (10). In this case (11)
where the order par~~eter n can be written as an infinite series of the form _
kT
exp
[_ ~(1.1£)-2 '\ [(2n-l)!!]2 2n (12) 4J £ L 2n I I a n-l ..
where £-K/J and 1a-(1-1£)/(1.1£). In the limit £"0, the leading term in the series is - (l/~)ln(£) which gives k T/41TJ B n - UKI J I) (13) Thus we have n ; a exp(~CT) for K-J and a(K/J)DT for K-O, where C and D are constants. To get a general idea about the phase diagram we have used the MKRG procedure. 10 In contrast to some earlier works, the single~site field terms generated during the MKRG iterations were not moved along with the interaction bonds." The reliability of our procedure was checked by applying it to the single-site anisotropy model studied by Jose' et al." For our model we find as long as K~O (we start from K/J=O.OOOl), only after 10~20 iterations the renormalized Hamiltonian H" can be represented extremely well by a simple two~ parameter Hamiltonian of the form I
H - -J
L
x
coseicose j - A
L cos2e l •
i
•
Table I. The critical temperature T obtained by Migdal Kadanoff (MKRG) and Monte Carlo (MCRG) Renormallzation Group calcul~tions for three different. values of £=K/J. T is given by the relation T - k BTc /J(l+£).
£-K/J
MKRG T"
MCRG T"
0.01 0.10 1.00
1.02 1.18 1.32
1.04 1.20 1.34
COSk y ) (10)
n
infinity (T-O fixed point); for T > Tc ' SA iterates .to a fixed value while BJ"O. This suggests that the system Iterates to a noninteractlng "Ising" system. The transition temperature along with those calculated using MCRG procedure (see below) are given In the Table I.
(14)
There is a small correction of the type LA Cos(pB), p>2. There exists a temperature T~ such that for T < Tc ' both BJ and BA iterate to
Although MKRG results give a single Isinglike transition, to explore the true nature of the transition in more detail and to see the nature of topological excitations that drive this transition we have carried out MC simulation of (16x16) and (32x32) systems. The number of MC steps per spin were between 5000 and 8000 excepting near the critical temperature where we discarded about 11000 MCS!s and took averages over 12000 MCS!s. The temperature dependence of the heat capacity C and the spin susceptibility X indicated one phase transition for all three values of K/J-O.Ol ,0.1,1.0 studied. Both C and X peak at the same T suggesting an Ising-like transition and MCRG calculations support this picture.'7 To understand the nature of topological excitations that destroy the long range order (LRO) with increasing T, we have made a series of MC quench studies for K!J-O.l. In Fig. (4) we give our quench results starting from two initial temperatures T>T c and T
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Monte Carlo quench studies from T~ to Tf (in units of J/k B ) for K/J=~.l; .-vortex and o=antivortex. a. Ti =1.1 < Tc (before quench)
and takes place at a temperature where the wall tension goes to zero. IV.
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t
OODT on a Nonrigid Lattice~a Model 2D Ferro-elastic Phase Transition
Until now we have been discussing the nature of phase transitions in systems where the centers of mass (CM) of the diatomic molecules are fixed. It is known that in molecular solids there is a strong coupling between orientational and translational degrees of freedom. This coupling gives rise to indirect intermolecular interaction which in some cases competes with the direct interaction in determining the ground state. The indirect interaction usually favors
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~: ~~ :b:~ ~5~cM~~~~o:;t~~e~~:~Ch) a ferro-rotational (F) order whereas the direct interaction favors an antiferro-rotational (AF) order." Depending on the dominance of one above the other one has different types of order and lattice structure. Simple examples in 3D are NaO and K0 2 • In the former the molecular orienta'tions are AF-type and in the latter they are F~type.·· In 2D, N on graphite in the lowdensity commensurate2 monolayer phase is an example of an AF structure'? (2 sublattice herringbone (HB) structure) and O2 on graphite in the monolayer o-phase is tnat of a F structure." For systems with AF ordering, the symmetry of the CM lattice does not change when orientational disorder sets in whereas those with the F ordering it does change. In latter
Superlattices and Microstructures, Vol. 1, No.6, 1985 case the orientationally ordered ferro-elastic phase undergoes a transition to the orientationally disordered para-elastic (plastic) phase. In this transition the CM lattice changes from a distorted triangular to triangular structure. We have studied'· the nature of 2D ferroelastic phase transition for a system which closely resembles the o-phase of 02 on graphite using a constant pressure MolecUlar Dynamics In our study we have (MD) method.'7'" considered only the Lennard-Jones interaction and ignored the quadrupole-quadrupole interaction. Furthermore the substrate effect has been included by demanding the 02 moleCUles to orient parallel to the substrate. The corrugation part of the substrate potential is known to be small for O/graphi te and has been dropped. Thus our system is a model anisotropic planar rotor system where the CM's of the rotors are free to move. The atom-atom Lennard-Jones parameters used here are &=54.34 ka , 0=3.05 A , the internuclear separation 2d-1.205 A and the mass of each atom m=15.99amu. These parameters are appropriate for the oxygen molecule. In the ground state we find that the molecules are parallel to each other and the CM of the molecules form a centered rectangular lattice (see Fig. 5) with a=3.332 A and b-8.054 A. These values differ by only a few percent from those obtained by Etters et al.'9 who used a pattern search program to find the ground state. If we_~~ the qUtdrupole interaction (Q • -0. 39x1 esu .cm ), the lattice parameters change by less than 0.3% and the ground state energy changes by 2%. Therefore we can put Q=O for this system without sacrificing much of the physics. Taking the
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coverage of 13 x 13 structure as 2he uni t which corresponds to 0.0636 molecules/A , our ground state has a coverage 1.112. The constant pressure MO developped by Anderson'7 and by Parrinello and Rahman" was used to study the finite T·properties. In applying this procedure to the multiatomic systems one has to evaluate the internal stress tensor P properly. In earlier studies of these syst~~s only the CM contribution to P was taken into account. This is reasonable whgX the CM-lattice symmetry does not change at the 000 transition. But for our system in which there is symmetry change we found it essential to include the rotational contribution to P • The details of the calculation of the interNNl stress tensor including both the CM and rotational contribution will be published in a separate paper.~· Here we only give the expressions for P for the planar rotor limit. The four elements~~f P are given by ~\I
• pCM _ p xx xx
L dgicos6isin6 i
(15.a)
+ Pyy _ pCM yy
L dgicos6isin6 i
(15.b)
_ Pxy • pCM xy
L dgi sin6 i sin6 i
(15.c)
i
i
i
r
3.975 -4
HERRING BONE E=·7.83
p=D.680 Fig. 5
Fer r 0 ~ e las tic (c e n t ere d rectangular/distorted triangular) and herringbone (triangular) structure of 02 molecules interacting via LennardJones potential and confined to orient in the 2-0 XY plane. All lengths are in A units.
PyX =
P~~ +
f dgicoseicose i
(15.d)
e e e e Where gi-Fi1~ F~2; F i1 and F i2 are the components of the rorces on the atoms 1 and 2 of the ith molecule produced by the rest of the system along a direction perpendic~r to the internuclear bond. In Eq. (15), p is the stress tensor associated with the ~~ motion. From the above equations we can clearly see that the hydrostatic pressure p=(P +P )/2 comes from the CM contributions onlf~ Kdwever the individual components of P pv do depend on the
524
Superlattices and Microstructures, Vol. 1, No.6, 1985
y
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Temperature (T *~kBT/e) variation of the orientational order parameters n2 and n 6 and strain order parameter 1
(~ee text T =1=54.21K.
for
definition)
rotational motion and we have found that neglect of the rotational contribution to P makes the system rigid towards shear deform~~ions and increases the ferro-paraelastic transition tempera ture. MD equations were solved" by using a predictor-gorrector algorithim with accuracy up to O(6t). The cut~off distance for the potential i.e. the distance within which the intermolecular potential is calculated is Sa. The high accuracy of the algorithim coupled with a large cut-off distance enabled us to use a
All the three order Where 6 = L ai/N. parameters ~how sharp drops in the temperature interval 0.36 and 0.38 suggesting a first order transition. At low T the decrease in these order parameters with increasing T is small indicating a hard lattice (large librational energy gap). The energy and the density also show a discontinuous change in the transition region confirming the first-order nature of the transition. The change in the entropy per particle at the transition is 0.88 kB• Since for a free 2D rotor the total orientational
rather large time step 6taO.01102 a!m7E(- 0.02 picosecond) in our simulation studies even when fast librational motions were present. Periodic boundary conditions have been used and the system consists of 400 particles. Starting from T=O.12 (unit of temperature is elks ' e=54.34 k ) and zero external pressure the system is SIOW~y heated in temperature step of 6T=0.06. At T, -O.38 (=20.6K), the system undergoes a ferro-paraelastic transition Where the orientational LRO is lost and the lattice becomes triangular. At T2 =0.70 ( 38.0K), the orientationally disordered plastic phase melts to a 2D isotropic liqUid. We have calCUlated internal energy (E), density (p), radial distribution function (rdf), orientational order parameters (n 2 ,n6) and the lattice anisotropy parameter (YJ as functions of T. The results are briefly discussed below. In Fig. 6 , we give the T~dependence of Y,n 2and n6 which are respectively defined as
entropy is k In(2n)=1.83 k , i t seems that above the trnnsition tempera~ure the rotors are quasi-free. The detailed nature of the orientational and coupled translationalrotational dynamics are presently under investigation. We have also calculated the rotational self~diffusion constant De and different elastic constants both below (T a O.36) and above (T=0.38) the ferro~elastic transition temperature. The reSUlts are given in Table II. The fast
Y 1 n2 = N 1 n6 = N
(b/l3a - 1)
L
COS2(6i-6)
L
cos6(6i-6)
i
i
Table II. ~otational diffusion constant D (In units of rad ~ec.) and elastic constants (in units of e/a , where and are the constants in the Lennard-Jones potential) for temperature above (T-0.38) and below (T~0.36) the ferroelastic transition temperature.
De Cll C22 C12 C44
T - 0.36
T = 0.38
4.9 ±90.4 x10
1.49 ±,q.08 xlO
1.45 0.97 2.10 1.33
1.05 1.07 1.02 0.11
Superlattices and Microstructures, Vol. 1, No.6, 1985
525
4- 00 r - - - - - - - - - - - - - - - - - - - - - - - - - - ,
Reduced tempera lure is 0-36 Steps are from 12001 to 15000
3'00 c 0
u
c
::
2'50
c
~
2-00
:2
I-50
~ '0
."
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a:
1-00
0-50
0-00
2-50 R-squore
Fig. 7
The radial distribution function of the centers of mass of the molecules at T-0.36 averaged over 3000 time
steps (60 picoseconds). The positions of the peaks indicate a distorted triangular lattice.
4·00r----------------------------, 3-50
3'00
Reduced lemperoMe is 0'38 Sleps ore from 12001 to 15000
c o
~
2-50
.;?
1-00
0-50
0-00
12-50
15-00
17·50
20-00 22-50
25-00
R-squore
Fig.8
The radial distribution function of the centers of mass of the molecules at T-0.38 averaged over 3000 time
steps (60 picoseconds). The positions of the peaks indicate a triangular lattice.
526
Superlattices and Microstructures. Vol. 1. No.6, 1985
rotational diffusion and small transverse elastic constants C 44 and C -C are characteristics of the orie~lat~o~allY disordered plastic phase. Finally, to see how the structural phase trans i t ion .takes place as a function of time we have calculated rdfs (radial distribution functions) at T=0.36 and 0.38 at different times after the system was heated from the nearby lower temperature. The shape of the first peak of the rdf was monitored as a function of time. For the triangular lattice there are six nearest neighbors (nn) which give a single peak (broadened by thermal vibrations) with weight sIx. For a distorted triangular lattIce this peak splits into two peaks • one coming from two nn's and the other from four next nn's. Thus a transition from a two-peak to a single-peak structure is a signature of the structural phase transition. The rdf at T-0.36 (Fig. 7) sho~s a sharp two-peak structure in the region 1< r < 2.65 (r is measured in units of 0) over the entire range of our MD run (300 ps). However as we heat the system to T-0.38, we find that after about 60 ps the two-peak structure distorts i.e. another peak forms in between the two peaks and after -300 ps, the system completely transforms to a single~peak (triangular lattice) structure. If we quench the system from T=0.36 to T=O.OI we recover the equilibrium values of the order parameters in a few picoseconds. However after quenching the system from T-0.38 to T-O.Ol we find that the order parameters are very small after several thousand time steps (20-50 ps); the recovery time is likely to be much larger than 50 ps. This picture is consistent With a first-order transition. The quench picture shows small ferro-elastic domains (clusters consisting of 7-20 molecules) oriented in three different but equivalent directions. In addition we also see localized defect clusters of herringbone structure. The nature and the dynamics of these clusters will be a subject of future investigation. V. Summary Overlayers of diatomic molecules exhibit a rich variety of phase transitions involving orientational and translational degrees of freedom. They provide simple experimental realizations of interesting phenomena such as fluctuation-driven first order phase transition, first~order transition in compressible anisotroPic planar-rotor systems. To extend our studies to the entire phase diagram in the p(coverage) vs T plane, we have to allow the molecules to orient away from the 2D plane. In addition, the effect of the corrugation of the SUbstrate potential (on the orientational orderdisorder and ferroelastic transition) needs a careful investigation. These are presently in progress. Acknowledgment--We thank Dr. R. K. Kalia and Dr. G. Kemeny for helpful discussions.
References 1.
2.
3.
4.
5. 6.
8.
9. 10. 11.
12. 13. 14. 15. 16.
18.
19. 20.
21.
22.
23.
Phase Transitions in Surface Film, ed. by J. G. Dash and J. P. Ruvalds, Plenum Press, N.Y. (1979). Ordering in Two Dimensions, ed. by S. K. Sinha. North Holland, Amsterdam (1980). M. Nielsen, J. P. McTague and W. Ellenson, Jour. de Physique C4, Supl. 9, 38. 1(1977)' M. Nielsen and J.-P. McTague,-Phys. Rev: B19, 3096(1979). ~F. Toney, R. D. Diehl and S. C. Fain, Phys. Rev. B27, 6413(1983); F. Y. Hansen et al., Phys. Rev. Lett. 53, 572(1985). See for example the session on 2~D Phase Transitions; Overlayers. BUll. Am. Phys. Soc 30, 333(1985). A. J7Berlinsky and A. B. Harris, Phys. Rev. Lett. 40, 1579(1978); c. R Fuselier N. S. GilliS-and J. C. Raich, Sol. St: Comm. 25, 747(1978). R. D-:- E t t e r s, R. Pan and V. Chandrasekharan. Phys. Rev. Lett. ~, 645 (1980). R. K. Kalia, P. Vashishta and S. D. Mahanti, Phys. Rev. Lett. 49, 676(1982). O. G. Mouritsen and A. J.-Serlinsky, Phys. Rev. Lett. 48, 181 (1982). S. Ostlund-and B. I. Halperin, Phys. Rev. B23. 335(1981). S:-D. Mahanti and G. Kemeny, Phys. Rev. B20, 2105(1979). O:-Sahu and S. D. Mahanti, Phys. Rev. B26, 2981 (1982). --s. D. Mahanti, Proc. of DAE Symposium 22C, 694 (1980). S. Tang, S. D. Mahanti and R. K. Kalia, Phys. Rev. B32, 3148 (1985). S. Tang andS. D. Mahanti, Phys. Rev. (to be published). W. A. Steele, The Interaction of Gases with Solid Surfaces, Pergamon Press, New York (1974). R. D. Diehl and S. C. Fain, Surf. Science l~~. 116 (1983); J. Talbot, D. J. Tildeseley and W. A. Steele, Mol. Phys. 51, 1331 (1984). -P. A. Heiney, P. W. Stephens. S. G. J. Mochrie, J. Akikatsu and R. J. Birgeneau, Surf. Science ~, 539 (1983); M. F. Toney. R. D. Diehl and S. C. Fain, Phys. Rev. B27. --6413 (1983). D. A. Goodings and M. Henkelman, Can. J. Phys. ~, 2898 (1971); P. V. Dunmore, Can. J. Phys. 55. 554 (1977). A. J. Berlinsky and A. B. Harris (see ref.6), the results of this paper, although written explicitly for H , are applicable 2 to N • 2 S. F. O'Shea and M. L. Klein, Chern. Phys. Lett. 66, 381 (1979). A. B. Harris and A. J. Berlinsky, Can. J. Phys. 57, 1852 (1979). O. G. Mouritsen and A. J. Berlinsky (see ref.9); O. G. Mouritsen, Computer Studies of )hase Transition and Critical Phenomena (Springer Verlag, He i delberg, 1984). Ch.5.
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Superlattices and Microstructures, Vol. 1, No.6, 1985 24.
25. 26. 27. 28. 29. 30. 31.
E. K. Riedel, Physica (Utrecht) 106A, 110 (1981); P. Pfeuty and G. Toulouse, In troduction to Renormalization Group and Critical Phenomena (Wiley, New York, 1977), Ch.9. s. A. Safran, Phys. Rev. Lett. 46, 1581 (1981); See also 1. M. Lifshitz, SOV. Phys. JETP 15, 939 (1962). R. K.~alia et.al (see ref.8). S. Tang and S. D. Mahanti (to be published) • J. M. Luttinger and L. Tisza, Phys. Rev. 70, 954 (1946); J. M. Luttinger, Phys. Rev. 8T, 1015 (1951). Jose', L.P. Kadanoff, S. Kirkpatrick and D. R. Nelson, Phys. Rev. B16, 1217 (1977). A. A. Migdal, Sov. PhYS. JETP 42, 743 (1976); L. P. Kadanoff, Ann. Phys 100, 359 (1976). ---S. H. Shenker and J. Tobochnik, Phys. Rev.
v:
32. 33. 34. 35. 36. 37. 38. 39.
40.
822, 4462 (1980); see also J. Tobochnik, Phys. Rev. 826, 6201 (1982). S. Tang, Ph. D thesis, Michigan State Universiry, East Lansing (1985). V. Jose' et.al (see ref. 29). D. Sahu and S. D. Mahanti (see ref. 12). S. D. Mahanti (see ref 11 and 13 ). S. Tang, S. D. Mahanti and R. K. Kalia (submitted to Phys. Rev. Lett.) H. C. Anderson, J. Chern. Phys. 72, 2384 (1980). -M. Parrinello and A. Rahman, Phys. Rev. Lett. 45, 1196 (1980); J. Appl. Phys. 52, 7182 (1~0); J. Chern Phys. 76, 2662 (1980). R. D. Etters et. al. (seeref.7); R. Pan, R. D. Etters, K. Kobashi and V. Chandrasekharan, J. Chern. Phys. 77, 1035 (1982). -S. Tang, S. D. Mahanti and R. K. Kalia (to be published).