A method for the estimation of the step-step attractive energy from experiments

抄録

Faceted macrosteps formed at equilibrium are considered to be caused by step-step attraction [1]. However, the explicit value of the step-step attractive energy has not yet been obtained. This is because macrostep formation at equilibrium is one of the difficult problems arising from the surface roughness associated with the crystal morphology.

The Mermin-Wagner theorem states that the long-range order is destroyed by thermal fluctuations in low-dimensional systems with short-range forces, such as the crystal surface or interface. The thermal roughening transition is known as the topological order phase transition with the Berezinskii-Kosterlitz-Thouless (BKT) universality class [2]. In addition, the equilibrium crystal shape (ECS), which is the shape of a crystallite with the lowest surface free energy, causes the faceting transition just at the thermal roughening transition temperature T_R [3].

In our previous work for a lattice model with microscopic step-step attraction, which is caused by quantum mechanics, the macrostep is formed at equilibrium as a result of the first-order shape transition, i.e., two-surface coexistence [4]. To obtain the phase diagram (faceting diagram), we need to calculate the surface free energy in detail with high confidence in the surface entropy. Although the lattice model of the surfaces is on the square lattice, we calculated the surface free energy [5,6] with high accuracy using the product wave function renormalization group method [7], which is an extension of the tensor network method.

Using the phase diagram [5,6], we show how to estimate the step-step attractive energy [6]. As an example, the method is applied to the Si(113)+(114) surface, where the phase diagram was obtained experimentally [1]. The step-step attractive energy is then estimated to be approximately -123 meV [6].

References

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[2] van Beijeren, H. Exactly Solvable Model for the Roughening Transition of a Crystal Surface. Phys. Rev. Lett 1977, 38, 993—996.

[3] Rottman, C.; Wortis, M. Statistical mechanics of equilibrium crystal shapes: Interfacial phase diagrams and phase transitions. Phys. Rep., 1984, 103, 59—79.

[4] Akutsu, N. Thermal step bunching on the restricted solid-on-solid model with point contact inter-step attractions. Appl. Surf. Sci., 2009, 256, 1205—1209. ibid Non-universal equilibrium crystal shape results from sticky steps. J. Phys. Condens. Matte, 2011, 23, 485004, 1—17.

[5] Akutsu, N. Faceting diagram for sticky steps. AIP Adv., 6, 035301 (2016).

[6] Akutsu, N.; Akutsu, Y. Slope—Temperature Faceting Diagram for Macrosteps at Equilibrium. Sci. Rep., 2022, 12, 17037, 1—11.

[7] Nishino, T.; Okunishi, K. Product wave function renormalization group. J. Phys. Soc. Jpn., 1995, 64, 4084—4087. Hieida, Y.; Okunishi, K.; Akutsu, Y.

Magnetization process of a one-dimensional quantum antiferromagnet: The product-wave-function renormalization group approach. Phys. Lett. A,1997, 233, 464—470.