Abstract
1. Introduction
The fundamental thermodynamic theory of surfaces, initialized by the American scientist Josiah Willard Gibbs, is one of the most practical tools for the study of surface-related phenomena[
For surfaces of solids, especially metals and semiconductors, the surface energy is important in many related fields, determining the equilibrium shape of monocrystals, brittle fracture, or the rate of sintering. Wulff construction[
The anisotropy of surface free energy
Figure 1.(Color online) Workflow of Wulff construction: (I) draw a
where
Historically, among solids the surface energy of elemental crystals, mostly metals, was the earliest to be studied by researchers[
In the framework of DFT[
Before any algorithm is discussed, let us briefly have an overview on the history of the algorithm developments so as to have a general understanding of key advancements in the algorithm design of semiconductor surfaces and edges.
For non-polar surfaces, Feibelman, in 1983, used metallic Al and Mg crystal as examples to demonstrate the calculation of non-polar surface energies[
For polar surfaces, Chetty and Martin, in 1991, made the first attempt to calculate the absolute surface energy as an integral of local energy density[
For semi-polar surfaces, Li et al., in 2015, proposed a wedge scheme to find the energy difference between surfaces and a reference polar surface in the system of GaN[
For polar edges of 2D materials, Mukherjee et al., in 2011, presented a strategy with pure bare triangular clusters of h-BN in single size, each of which has only one type of exposed edge, to calculate the edge energies individually[
After the historical review, the algorithm designs for the calculation of different surface and edge energies are going to be discussed separately as follows.
Before diving into polar and semi-polar surfaces and polar edges, it is good to have a brief review on the non-polar surface or edges, and compare them with polar and semi-polar ones. For slab (ribbon) exposing non-polar surfaces (edges), the constituent elements are in the stoichiometric ratio within the whole structure. Therefore, there is no need to consider the chemical potential contribution from individual elements. Also, since the top and bottom surfaces are identical, it is possible to obtain the formation by assuming both surfaces have the same contribution to the total formation energy. This idea was first proposed by Feibelman in metallic materials[
where
where
where
For symmetric surface (edge for 2D lattice) pairs, the surface (edge) energy
where
Figure 2.(Color online) A slab created by cleaving a zinc-blende structure in (111) plane, grey and yellow atoms represent atom species A and B. Note the resultant upper and lower surface is of different termination.
2. Polar surfaces
At the earliest stage of polar surface study of semiconductors, fractionally charged pseudo hydrogen (pseudo-H) has been suggested to passivate one of the surfaces of the slab model so as to remove the charge transfer between surfaces[
Zhang and Wei[
Figure 3.(Color online) Wedge structure of size
A heterojunction supercell (Fig. 4) was also used to approximate unknown surface energies from known ones[
Figure 4.An illustration of a slab containing an interface and two passivated surfaces.
In Eq. (7), chemical potential has to be obtained from strained bulk simulation using the slab lattice constant to account for stress due to lattice mismatch. The heterojunction scheme is employed to study the (001) and (00
Figure 5.(Color online) A ZB(111)/WZ(001) heterojunction supercell consists of 6 WZ and ZB layers used in the calculation scheme of Tang
With the key assumption of localized energy contributions, the energy of an isolated crystal can be computed[
The passivated surface energy can then be systematically obtained by cancellation of bulk, edge and corner contributions. The transferability of such an estimation scheme mainly depends on 1) the ability of the proposed structure to capture the local bonding environment of the surface to be estimated, and 2) good size convergence of the proposed structure so that systematic cancellation of other contributions is possible. Zhang et al. proposed a tetrahedral cluster reproducing the same symmetry as the zinc-blende[
This estimation was shown to yield an acceptably accurate surface energy estimation[
Figure 6.(Color online) Tetrahedral cluster of zinc-blende structure of size
Calculation of 4 clusters of different sizes allows solutions for all unknowns (
With a fractional hydrogen[
Absolute interface energy is of equivalent physical interest including the determination of the wetting condition[
3. Semi-polar surfaces
Over the past few decades, even though the technologies in industrial application, like the quantum dot light-emitting diodes (LEDs)[
The definition of a semi-polar surface, with an example shown in Fig. 7, was initially made by Baker et al.[
Figure 7.(Color online) GaN crystal with 3 different types of surface cut. The semi-polar one is highlighted pink.
A comprehensive understanding of the absolute surface energies of all possible GaN surfaces is crucial to the estimation of equilibrium shape in the thermodynamic stability study, leading to important factors that can be used to modulate the crystallographic growth of GaN, which are regarded as the key issue in the realization of broadband and multi-color emission[
There is an early method proposed by Jindal and Shahedipour-Sandvik in 2009[
A 2D scheme of Wulff construction is applied to one of the cross-sections of GaN as indicated in Fig. 8. The length of
Figure 8.(Color online) Wulff construction of one of the 2D cross-sections of GaN. The yellow shaded area is a quarter of ECS in the cross-section. This strategy is from Ref. [
For this example,
Figure 9.(Color online) Workflow of finding the difference in crystal plane radii. Blue and black notations correspond to unrelaxed and relaxed surface structures respectively. This strategy is from Ref. [
However, based on the experimental observation from[
The acquisition of accurate energies of semi-polar surfaces is particularly difficult for three reasons: firstly, the conventional slab method cannot be used to deal with individual semi-polar surfaces resulting from the structural asymmetry; secondly, large computational input in the form of wedges, usually with high index planes, are involved which leads to high computational cost; thirdly, as semi-polar surfaces are sometimes of a step nature, it is not always feasible to passivate the bottom surfaces of slabs with pseudo-H atoms in the absence of significant unphysical charge transfer and steric effect, which deteriorates the result accuracy[
To overcome these mentioned difficulties, Zhang et al. introduced a fundamentally different algorithm in 2018, using GaN as an example[
Figure 10.(Color online) (a) and (b) are slabs with upper semi-polar surfaces of m- and a-family, respectively, and with bottom side cut into step-structure in which the non-polar and polar surfaces are passivated by either H or pseudo-H. These figures are adapted from Ref. [
According to Eq. (3),
where
where
However, taking the semi-polar surface (11
Figure 11.(Color online) Slab with a well being cut with width and depth as
where
Zhang has used a slab with both sides cut with a zigzag structure to implement the convergence test to give a residual error less than 1.5 meV/Å2, indicating the high accuracy of the method. This new algorithm to estimate the absolute energy of semi-polar surface is completely different from the traditional methods which are based on wedges and slabs. It is because this new method is applicable to an arbitrary surface as long as we can passivate the polar and non-polar planes at the bottom surface with zigzag structure.
Later on another Japanese group, Seta et al., published some literature in 2019 using both slabs and wedges to estimate the absolute energy of semi-polar surfaces[
Figure 12.(Color online) Cross section view of AlN triangular wedge with surface (
After calculating the energies of wedges in Fig. 12 for different sizes, the same procedures were repeated by interchanging the position of Al and N atoms. Therefore, the energy difference (here we use the notation
where
where
Seta has given one more improvement on the temperature dependence on the estimation of surface energies by including the translational, rotational and vibrational motion of atoms in gaseous phase into the chemical potential
where
Nearly at the same time, Akiyama, from the same Japanese group, proposed another algorithm to calculate the energy of polar and semi-polar surfaces simultaneously[
4. Polar edges of 2D materials
In the world of 2D materials, graphene is the system that was first discovered and the most intensively studied[
In recent years, the family of 2D materials has grown larger, including compounds such as hexagonal boron nitride (h-BN) that is an insulator, and molybdenum dichalcogenides (
To understand various equilibrium shapes in experiments, obtaining the energies of different edges are crucial. Due to the structural asymmetry of the nanoribbon of these compounds, as shown in Fig. 13, edges with polarities emerge. The direct calculation of polar edge energy is no longer achievable because the early method used in graphene can only estimate the average energies of two opposite zigzag edges in the ribbons[
Figure 13.(Color online) (a)/(c) and (b)/(d) are the (top view/side view) of h-BN and
In the past few years, several groups have developed their own methods to estimate the absolute energy of polar edges. Most of them are based on the creation of triangular nanoclusters terminated at the edges with the same polarity[
In all methods, polar edge energies can be obtained under different chemical potentials. The range of the chemical potentials can be obtained from a standard calculation of phase diagram of various secondary phases of the compound, which has been widely applied in the energy calculations of point defects, surfaces, and interfaces[
Besides chemical potential, passivation of edges is the key to the calculation of edge energy. For early methods, passivation was not taken into the consideration[
The first theoretical attempt was made by estimating the average energies of h-BN edges terminated by B and N[
where
Figure 14.(Color online) (a) The computational setup for triangular clusters with green dots as boron atoms and silver dots as nitrogen atoms. (b) The result of equilibrium shapes at different chemical potential ranges[
The results suggest that it is possible for armchair-edged hexagon to exist at the mid-range of chemical potential. In addition, other literature gives the same computational prediction on the stability of the armchair edge[
The bare triangular cluster was reported to contain corner distortions and inter-edge couplings[
Another method proposed by Cao et al. to find the equilibrium shape of MoS2[
Figure 15.(Color online) (a) The
where
where
Figure 16.(Color online) Equilibrium shape of
From the Wulff construction, the S-terminated triangular shape can be observed in an S-rich condition in which the shape matches the experimental results[
Before entering the last example, it is good to mention that the method of Cao's example is suitable for obtaining a fast calculation. Yet, there is another method proposed by Zhang et al.[
Figure 17.(Color online) B, N and H atoms are denoted by pink, blue and white spheres respectively. (a) Passivated and unpassivated zigzag and armchair edges. (b) Reconstruction of seven- and five- rings on the ZZN and ZZB edges, respectively. (c) Ribbon of bottom zigzag edged passivated with hydrogen and arbitrary configuration on the upper zigzag edge. (d) N-terminated passivated triangular cluster of size
where
Therefore, instead of a direct calculation of edge energy from the bare triangular cluster, the chemical potentials of passivating hydrogens have to be first estimated from the passivated cluster. The reason for the more reliable calculation of the chemical potential of passivating hydrogen than the edge energy of the bare triangle is that the passivation helps to reduce corner distortion and the unphysical charge transfer[
where m is the cluster size and
where
There are three red-colored parameters to be fitted
Figure 18.(Color online) Total energy of H-passivated triangular clusters with different size (
After the estimation of hydrogen chemical potential, the half-passivated ribbon with arbitrary configuration on the upper edge (Fig. 17(c)) can be calculated to obtain the absolute energy of the particular polar edge by Eq. (22). Also, Zhang has proposed a self-consistency check to ensure the accuracy of the algorithm by calculating the residual error Er. Er can be calculated by Eq. (26) after the calculation of total energy of both sides passivated ribbon Ep. Zhang has shown the error is reduced from 3.43% to 0.12% when compared with the bare triangular method.
After that, Zhang had shown in Fig. 3 of their literature[
where
to obtain
Figure 19.(Color online) Equilibrium shapes of h-BN nanocluster under different chemical potentials at 1300 K, consisting of H-passivated edges. Yellow, green and black lines are of ZZBH, ZZNH and ARMH edges respectively. The figure is adapted from Ref. [
After the discussion of several methods of calculating the polar edge energy, the last one proposed by Zhang, is able to capture more physical pictures and also gives the highest accuracy because it includes the temperature effect and passivation which leads to stabilization effect to all type of edges. It is also capable of revealing the important role played by hydrogen atoms in the growth of 2D h-BN monolayer.
5. Conclusion
We have reviewed some important historical algorithms on the assessment of surface and edge stability of various semiconducting compounds. The key concept for a successful algorithm is to eliminate the long range charge transfer and interaction of different surfaces or edges by passivating dangling bonds and mimicking the electronic environment of the desired surfaces or edges. In addition, not all passivation can yield a reliable result because an electron counting model has to be satisfied and steric effects should be avoided. To estimate the localized steric effects, it is possible to perform further simulation that can mimic the stressed local configuration. Still, further investigations of quasi-2D structures are highly important, yet largely missing because they lack effective passivation schemes on the edges. With all the technological advancements, we can safely conclude that a highly accurate algorithm combining a reasonable analysis of passivation and temperature effects can have strong predictive power in the equilibrium shape under various growth conditions and the dawn of a highly effective collaboration between theoreticians and experimentalists may largely improve the field of crystal growth and device fabrication of semiconductors.
Acknowledgements
The research is supported by HKRGC, GRF with the Project Codes of 14307219, 14307018, 14301318, and 14319416; and by direct grant from CUHK. .
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