Some Theorems on the Free Energies of Crystal Surfaces

Herring, Conyers

doi:10.1103/physrev.82.87

Cited by 1,495 publications

(752 citation statements)

References 16 publications

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“…(41) with the PWFRG results in §2. 4, we see that the nonuniversal shape exponents for the p-RSOS model agree with the values obtained using Eq. (41).…”

Section: Thermodynamic Expression For Non-gmpt Shape Exponentssupporting

confidence: 86%

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Non-universal equilibrium crystal shape results from sticky steps

Akutsu¹

2011

J. Phys.: Condens. Matter

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Abstract. The anisotropic surface free energy, Andreev surface free energy, and equilibrium crystal shape (ECS) z = z(x, y) are calculated numerically using a transfer matrix approach with the density matrix renormalization group (DMRG) method. The adopted surface model is a restricted solid-on-solid (RSOS) model with "sticky" steps, i.e., steps with a point-contact type attraction between them (p-RSOS model). By analyzing the results, we obtain a first-order shape transition on the ECS profile around the (111) facet; and on the curved surface near the (001) facet edge, we obtain shape exponents having values different from those of the universal GruberMullins-Pokrovsky-Talapov (GMPT) class. In order to elucidate the origin of the non-universal shape exponents, we calculate the slope dependence of the mean step height of "step droplets" (bound states of steps) n(p) using the Monte Carlo method, where p = (∂z/∂x, ∂z/∂y), and · represents the thermal average. Using the result of the |p| dependence of n(p) , we derive a |p|-expanded expression for the non-universal surface free energy f eff (p), which contains quadratic terms with respect to |p|. The first-order shape transition and the non-universal shape exponents obtained by the DMRG calculations are reproduced thermodynamically from the non-universal surface free energy f eff (p).

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“…(41) with the PWFRG results in §2. 4, we see that the nonuniversal shape exponents for the p-RSOS model agree with the values obtained using Eq. (41).…”

Section: Thermodynamic Expression For Non-gmpt Shape Exponentssupporting

confidence: 86%

“…(41) with the PWFRG results in §2. 4, we obtain βA eff (π/4) = (6.22 ± 0.06) × 10 −3 and βγ 1 (π/4) = 1.39 ± 0.03. This value of βγ 1 (π/4) agrees well with βγ(π/4) Ising (Eq.…”

Section: Thermodynamic Expression For Non-gmpt Shape Exponentsmentioning

confidence: 73%

Non-universal equilibrium crystal shape results from sticky steps

Akutsu¹

2011

J. Phys.: Condens. Matter

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“…19 Given a set of parameters of , , and 0 , the total film free energy G is calculated and plotted as the function of nominal feature depth, d, to evaluate the final equilibrium profile of the imprinted nanostructures (here, we assume ␥ 0 ϭ 1). In terms of thermodynamic stability, the equilibrium depth (or height) of imprinted periodic features can be determined by locating the local minimum point on GϪd curves.…”

Section: Resultsmentioning

confidence: 99%

Formation of Bandgap and Subbands in Graphene Nanomeshes with Sub-10 nm Ribbon Width Fabricated via Nanoimprint Lithography

et al. 2010

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O rganic-based photovoltaic cells (OPVs) are of great interest owing to their potential for low-cost solar energy conversion. 1 An important breakthrough for OPVs was the use of a heterojunction (HJ) structure, in which the difference of the energy levels of two materials (donor and acceptor) can lead to efficient dissociation of photogenerated excitons at the HJ interfaces. 1 Since then, tremendous efforts have been taken to optimize the carrier donor/acceptor (DA) interface morphology to improve the photogenerated exciton dissociation and consequently the overall power conversion efficiency. 2 One successful approach is to use a bulk heterojunction (BHJ) structure that can create dissociation centers everywhere within the active layer. 2 Typically, the formation of a BHJ structure can be achieved via self-assembly of nanostructured soft materials by spontaneous phase separation in a number of solution processed polymer/ fullerene systems, yet efficiency in these structures might be significantly reduced through unpredicted shunt paths and isolated islands of materials. 3 Nanoimprint lithography (NIL) offers a potential solution for producing well-defined interpenetrating networks at the DA interface and is compatible with roll-to-roll manufacturing for lowcost and high-throughput nanopatterning. 4,5 To efficiently harvest photogenerated excitons, densely packed nanoimprinted OPV structures with halfpitch smaller than 2 times that of the exciton diffusion length are needed (typically sub-20 nm regime). 6 Recent efforts in this field have been mainly focusing on polymeric PV materials. However, OPVs with small-molecular weight materials could also benefit from similar morphologies. In addition, small-molecular weight OPV materials provide additional advantages over polymers, such as higher chemical/thermal stability and higher material purity. 7 Previous work has shown relatively poor stability of imprinted nanostructures in smallmolecular compounds. 8Ϫ10 Problems arise due to pronounced surface diffusion and self-faceting and are exacerbated when features head toward the sub-20 nm regime. 11,12 These instabilities must be understood and overcome to achieve efficient nanostructured OPVs.Boron subphthalocyanines (SubPcs) are a class of photoactive small-molecularweight materials with unique physical properties. 13 A typical SubPc has a nonplanar pyramid-shaped structure, in which the boron atom is surrounded by three coupled

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“…is the second angular derivative of c s' ð/Þ appearing in Herring's relations [4]. The stiffness in three dimensions is similarly given by c s' þ r 2 c s' , where the Laplacian is applied to c s' ðh; /Þ.…”

mentioning

confidence: 99%

Growth directions in directionally solidified Al–Zn and Zn–Al alloys near eutectic composition

2008

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Growth directions and crystallographic orientations of solidification microstructures have been measured in Al-Zn alloy near the eutectic composition. Al dendrites in Al-92 wt.% Zn alloy were found to grow along the h1 1 0i directions while Zn dendrites in Al-96 and 98 wt.% Zn have h1 0 1 0i trunks. In the lamellar eutectic, a crystallographic relationship has been found between the dense plane of each phase, i.e. f1 1 1g fcc k f0 0 0 1g hcp , and the dense directions, i.e. h1 1 0i fcc kh1 2 1 0i hcp .

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Some Theorems on the Free Energies of Crystal Surfaces

Cited by 1,495 publications

References 16 publications

Non-universal equilibrium crystal shape results from sticky steps

Non-universal equilibrium crystal shape results from sticky steps

Formation of Bandgap and Subbands in Graphene Nanomeshes with Sub-10 nm Ribbon Width Fabricated via Nanoimprint Lithography

Growth directions in directionally solidified Al–Zn and Zn–Al alloys near eutectic composition

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