Phase diagram of pinned lattices in the phase field crystal model

Abstract

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Phase diagram of pinned lattices in the phase field crystal
model
C. V. Achim1, M. Karttunen2, K.R. Elder3, E. Granato4, T. Ala-Nissila1,5and S.C.
Ying5
1Laboratory of Physics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 TKK, Finland
2Department of Applied Mathematics, The University of Western Ontario, London (ON), Canada
3Department of Physics, Oakland University, Rochester, Michigan, 48309-4487, USA
4Laborat´ orio Associado de Sensores e Materiais, Instituto Nacional de Pesquisas Espaciais, S˜ ao Jos´ e dos
Campos, SP Brazil
5Department of Physics, Brown University, Providence, R.I. 02912-1843, USA
E-mail: cva@fyslab.hut.fi
Abstract.
dimensional phase field crystal model for adsorbed layers. The model allows for both elastic and plastic
deformations on atomic and diffusive time-scales, and provides a continuum description of lattice systems,
such as adsorbed atomic layers or two-dimensional vortex lattices. Analytically, mode expansion analysis
and numerical minimization of the free energy are used to determine the ground states as a function of
the pinning potential and lattice mismatch parameter. The results show a rich phase diagram with several
different types of commensurate and incommensurate phases.
Westudythephasediagramandthecommensurate-incommensuratephasetransitionsofatwo-
1. Introduction
Many systems in nature possess two or more competing length scales. Such systems often exhibit
commensurate-incommensurate transitions, which are characterized by structural changes induced by
the competition between these scales [1]. Examples of such systems include spin and charge density
waves [2, 3], vortex lattices in superconducting films [4] and weakly adsorbed monolayers on a substrate
[5]. All these systems are characterized by an order parameter ψ (e.g. charge or density wave, density of
superconducting electrons, or mass density) which is modulated with a periodicity a which is usually
incommensurate with the underlying lattice (of periodicity b). The interaction with the substrate is
characterize by the coupling strength V0. In a 1D system, we have a commensurate phase if the ratio
between the average period ˜ a of the order parameter and b is a rational number, while in the opposite
case the phase is incommensurate with the underlying lattice.
In this study we extend the work started in Ref. [6], where we considered a 2D system described by
the phase field crystal (PFC) model [7, 8]. The effect of the underlying lattice is realized by a periodic
potential which is linearly coupled to the order parameter. Such a model should provide a suitable
continuum description of many lattice systems such as weakly adsorbed overlayers or 2D vortex lattices.
The commensurate incommensurate-transitions are induced by varying the pinning strength V0and the
periodicity of the potential b.
IVC-17/ICSS-13 and ICN+T2007
Journal of Physics: Conference Series 100 (2008) 072001
IOP Publishing
doi:10.1088/1742-6596/100/7/072001
c ? 2008 IOP Publishing Ltd
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2. Model
The system is described by the PFC free-energy for an adsorbed monolayer [6],
?
where ψ(? x) is a continuous field representing the number density of the adsorbed layer, and V (? x)
represents an effective potential due to the substrate. The constant F0sets the scale for the energies,
while r is an effective temperature which together with the average density¯ψ = (1/S)?d2? xψ(? x)
We consider a pinning potential V (? x) with square symmetry
F = F0
d2? x
?ψ(? x)
2
((r + (1 + ∇2)2)ψ(? x) +ψ(? x)4
4
+ ψ(? x)V (? x)
?
.
(1)
determines the phase diagram in the absence of a pinning potential [7, 8].
V (? x) = V0[cos(ksx) + cos(ksy)],
(2)
where ks= 2π/b is the wave vector of the pinning potential. The parameters r and¯ψ are chosen so that
intheabsenceofthepinningpotentialthesystemisahexagonalstatewithalatticeconstantatandawave
vector kt= 2π/(at
potential. The lattice mismatch between the adsorbate and substrate is defined as δm= (kt−ks)/kt(for
the free-energy given by Eq. (1) kt= 1). In Ref. [6] only δm> 0 was considered, and only the boundary
between incommensurate and commensurate phase was shown. In this research we extend our previous
work to δm< 0. Also in the phase diagram all the commensurate phases and the boundaries between
them are shown.
Numerically, the equilibrium configurations (minimum energy states) can be found by solving a
conserved equation of motion:
?(3)/2), i.e., r = −1/4 and¯ψ = −1/4, which is incommensurate with the pinning
∂ψ(? x,t)
∂t
= ∇2?((r + (1 + ∇2)2)ψ(? x,t) + ψ(? x,t)3+ V (? x)?.
(3)
Eq. (3) was solved on a 256×256 grid with a spacial discretization dx = π/4. Analytically, the state of
the system can be found by approximating the density with a sum of Fourier modes
?
ψ(? x) =
n,m
An,mexp(i?Gm,n· ? x) +¯ψ,
(4)
where?Gm,nare the relevant reciprocal lattice vectors. The Fourier modes correspond to all possible
different symmetries (hexagonal, square, etc.). The density given by Eq. (4) is inserted into the free-
energy expression. After integration F(V0,δm,Am,n) is then minimized with respect to the coefficients
Am,n.
3. Results and discussion
3.1. Numerical minimization method
As mentioned above, we find the equilibrium states numerically by solving the equation of motion given
by Eq. (3). We start with a hexagonal phase and vary the pinning strength V0from 0 to some maximum
value, and then we decrease V0back to zero. The equilibrium state for each V0and δmis the one with the
lowest energy. Several states were found to minimize the free-energy. For small values of the pinning
strength the system is in a hexagonal incommensurate (IC) phase for all mismatches (Fig. 1(a)). If the
pinning strength is large enough, the system will be in one of the commensurate phases. The (1 × 1)
phase is an exact match with the pinning potential (Fig. 1(b)).
The other phases are higher commensurate phases which exist only when the relevant reciprocal
lattice vectors are close to kt. One these phases is the c(2 × 2) phase in which every second site of
the lattice of the pinning potential is occupied [5] (Fig. 1(d)). The relevant reciprocal vectors have the
IVC-17/ICSS-13 and ICN+T2007
Journal of Physics: Conference Series 100 (2008) 072001
IOP Publishing
doi:10.1088/1742-6596/100/7/072001
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magnitude ks/√2. This state is favored for mismatch values close to 1 −√2 (Fig 2(a)). Another higher
commensurate phase is the (2 × 1) which is generated by a translation of the basis with the reciprocal
lattice vectors of the a c(2 × 1) lattice [5] (Fig. 1(c)). The structure factor (S(?k) =ˆψ(?k)ˆψ(−?k)) of
this phase contains a set of peaks corresponding to the (1 × 1) phase and a set of peaks located between
the first and second order peaks of the (1 × 1) phase. The magnitude of the reciprocal vector which
gives the position of this set of peaks is ks
the mismatch is close to 0.1 (Fig 2(b)). The (2√2 ×√2) phase is similar to the (2 × 1) phase. The
lattice is generated by a translation of the basis with vectors which are rotated 45◦with respect to the
pinning potential and the magnitudes of the vectors are 2√2b and√2b. The structure factor contains a
set of peaks corresponding to the c(2 × 2) phase, while the position of the additional peaks is given by a
vectors with magnitude ks
√5/2, which suggests that this phase is favored to exist when
√10/4. The phase is favored for mismatch values close to −0.27 (Fig 2(a)).
(a) (b) (c) (d) (e)
Figure 1. The phases that minimize the free-energy: a) hexagonal, b) square (1 × 1), c) square (2 × 1),
d) square c(2 × 2), and e) square 2√2 ×√2. The upper panels represent the density plotted in a
gray colormap and the corresponding lattice vectors, while the lower panels the structure factors and
the relevant reciprocal lattice vectors. The black contours in Figs. 1(c) and 1(e) show the bases which
generate the (2 × 1) and (2√2 ×√2) lattices.
The transitions between the different phases are found by investigating the positions and the heights
of the peaks in the structure factor. The transitions from IC to any of the commensurate phases and from
c(2 × 2) to (1 × 1) are discontinuous because these involve a change in the symmetry and/or lattice
constant, while the transitions from (2 × 1) to (1 × 1) and from (2√2 ×√2) to c(2 × 2) appear to be
continuous. In this case the magnitude of the additional peaks decrease continuously to zero when the
pinning strength is increased. A complete diagram of all these phases is shown in Fig. 2(a) for δm< 0
and in Fig. 2(b) for δm> 0.
3.2. Fourier expansion of the density
Intheanalyticalmethodweconsiderthedensitytobeasumofhexagonal(At(cos(qtx)cos(qty/?(3))−
qt= 2π/at. After the integration the free-energy is minimized with respect to the amplitudes At, As1,
and As2. The transition from IC phase to a commensurate phase occurs when At= 0. In this expansion
the transition from IC phase to commensurate phase is discontinuous. This approximation gives good
agreement with the numerical minimization for very small and very large values of the pinning strength.
The boundary between the IC phase and (1 × 1) phase is shown in Figs. (2(a) and 2(b)).
1
2cos(2qty/?(3))) and square modes, (As1cos(ksx) + cos(ksy)) + As2cos(ksx)cos(ksy)), where
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Journal of Physics: Conference Series 100 (2008) 072001
IOP Publishing
doi:10.1088/1742-6596/100/7/072001
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(a)(b)
Figure 2. The phase diagram in terms of pinning strength (V0) and mismatch δmfor a) δm< 0 and b)
δm> 0. The dashed lines in a) and b) show the analytical prediction for the boundary between the IC
phase and the (1x1) phase. The solid lines show the boundaries between the different phases found by
numerical minimization.
4. Conclusions
In this work we extended the research started in Ref. [6]. Detailed phase diagrams of the PFC model
in the presence of an external pinning potential are presented for positive and negative mismatch using
a numerical minimization of the free-energy. An analytical Fourier expansion of the density provides
with us with a qualitative estimate of the transition, but numerical simulations were needed to study
the other phases present as well as the phase diagram. The transitions between different phases where
found to be discontinuous for IC-(1×1) and c(2×2)-(1×1),while continuous for (2×1)-(1×1) and
(2√2 ×√2)-c(2 × 2). Additional work is required in order to determine the true nature of the different
phase transitions by adding thermal fluctuations. Thermal fluctions can be included by adding a noise
term in the equations of motion with zero mean and correlation chosen to insure thermal equilibrium at a
finite temperature [8] or by Monte Carlo methods [9]. Another particularly interesting application of the
model is to pinned driven systems such as in sliding friction of adsorbed layers [10, 11].
5. Acknowledgments
This work supported been supported by a joint fund under EU STREP 016447 MagDot and NSF (DMR-
0502737), the Academy of Finland (C.V.A.,T.A.N.), NSF (DMR-0413062 ) (K.R.E.) and NSERC of
Canada (M.K.).
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IVC-17/ICSS-13 and ICN+T2007
Journal of Physics: Conference Series 100 (2008) 072001
IOP Publishing
doi:10.1088/1742-6596/100/7/072001
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