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Journal of Semiconductors
Vol. 41, Issue 6, 061101 (2020)

A brief review of formation energies calculation of surfaces and edges in semiconductors

Chuen-Keung Sin, Jingzhao Zhang, Kinfai Tse, and Junyi Zhu

Author Affiliations

The Chinese University of Hong Kong, Hong Kong, China

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DOI: 10.1088/1674-4926/41/6/061101 Cite this Article

Chuen-Keung Sin, Jingzhao Zhang, Kinfai Tse, Junyi Zhu. A brief review of formation energies calculation of surfaces and edges in semiconductors[J]. Journal of Semiconductors, 2020, 41(6): 061101 Copy Citation Text

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Abstract

To have a high quality experimental growth of crystals, understanding the equilibrium crystal shape (ECS) in different thermodynamic growth conditions is important. The factor governing the ECS is usually the absolute surface formation energies for surfaces (or edges in 2D) in different orientations. Therefore, it is necessary to obtain an accurate value of these energies in order to give a good explanation for the observation in growth experiment. Historically, there have been different approaches proposed to solve this problem. This paper is going to review these representative literatures and discuss the pitfalls and advantages of different methods.

$ r({{h}}) = \min \limits_{ {{m}}}\frac{\gamma({{m}})}{{{m}}\cdot{{h}}}, $

(1)

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$ \gamma = \frac{1}{2a}(E_{\rm{slab}}-nE_{\rm{bulk}}), $

(2)

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$ \mu_{\rm{A}}+\mu_{\rm{B}} = E_{\rm{AB}} = E_{\rm{A}}+E_{\rm{B}}+\Delta H_{\rm{f}}, $

(3)

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$ E_\alpha+\Delta H_{\rm{f}} \leqslant\mu_\alpha\leqslant E_\alpha, $

(4)

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$ \Delta H_{\rm{f}} \leqslant \Delta\mu_{\alpha}\leqslant 0. $

(5)

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$ \gamma = \frac{1}{2a}(E_{\rm{slab}}-n_{\rm{A}}\mu_{\rm{A}}-n_{\rm{B}}\mu_{\rm{B}}), $

(6)

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$ E_{\rm{slab}} = E_{{\rm{surface}}\ {\rm{A}}}+E_{{\rm{surface}}\ {\rm{B}}}+E_{\rm{interface}}+\sum\limits_{ i}n_i \mu_{ i,{\rm{strained}}}. $

(7)

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$\begin{array}{l} E_{{\rm{tot}},{\rm{AB}}} = n_{\rm{A}}\mu_{\rm{A}} + n_{\rm{B}}\mu_{\rm{B}} + \displaystyle\sum\limits_{\rm{surfaces}}\sigma_{\rm{surfaces}} \\ \quad\quad\quad\quad+\displaystyle\sum\limits_{\rm{edges}}\sigma_{\rm{edges}} + \displaystyle\sum\limits_{\rm{corners}}\sigma_{\rm{corners}}. \end{array} $

(8)

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$ E(\rm{AH}_4) = 4\mu_{{\rm{H}}_{\rm{A}}} + \mu_{\rm{A}}. $

(9)

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$ \begin{array}{l} E_{\rm{tot}}(n) = \dfrac{1}{6}n(n+1)(n+2)\mu_{\rm{A}} + \dfrac{1}{6}\ n(n-1)(n+1)(E_{\rm{AB}}-\mu_{\rm{A}}) \\ \;\;\;\;\;\;\;\;\;\;\;\;\; +\, 2(n\!-\!2)(n\!-\!3)\mu_{{\rm{H}}_{\rm{A}}}^{\rm{surface}} \!+\! 12(n\!-\!2)\mu_{{\rm{H}}_{\rm{A}}}^{\rm{edge}} \!+\! 12\mu_{{\rm{H}}_{\rm{A}}}^{\rm{corner}}\!. \end{array} $

(10)

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$ \mu_{\rm{Ga}}+\mu_{\rm{N}} = E_{\rm{GaN}} = E_{\rm{Ga}} + E_{{\rm{N}}_2} + \Delta H_{\rm{f}}({\rm{GaN}}), $

(11)

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$ \begin{array}{l} \gamma = \dfrac{1}{A}\{E_{\rm{slab}}-n_{\rm{Ga}}[E_{\rm{Ga}}+\Delta H_{\rm{f}}({\rm{GaN}})]-n_{\rm{N}} E_{{\rm{N}}_2} \\ \quad\quad-\;(n_{\rm{N}} - n_{\rm{Ga}})\Delta\mu_{\rm{N}} - \displaystyle\sum\mu_{{\rm{H}}_{\rm{Ga}}} - \displaystyle\sum\mu_{{\rm{H}}_{\rm{N}}}\}, \end{array}$

(12)

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$ \begin{array}{l} \mu_{{\rm{H}}_{\rm{Ga}}}^{\rm{steric}} = \dfrac{1}{n_{{\rm{H}}_{\rm{Ga}}}^{\rm{steric}}}\Big[E_{\rm{slab}}-n_{\rm{Ga}}(E_{\rm{Ga}}+\Delta H_{\rm{f}}({\rm{GaN}}))\\ \quad\quad\quad\quad-\;n_{\rm{N}} E_{{\rm{N}}_2} - (n_{\rm{N}} - n_{\rm{Ga}})\Delta\mu_{\rm{N}} - \displaystyle\sum\mu_{{\rm{H}}_{\rm{Ga}}} - \displaystyle\sum\mu_{{\rm{H}}_{\rm{N}}}\Big], \end{array} $

(13)

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$ \begin{array}{l} \gamma = \dfrac{1}{A}\Big[E_{\rm{slab}}- n_{\rm{Ga}}(E_{\rm{Ga}}+\Delta H_{\rm{f}}({\rm{GaN}}))\\ \quad\quad-\;n_{\rm{N}} E_{{\rm{N}}_2} - (n_{\rm{N}} - n_{\rm{Ga}})\Delta\mu_{\rm{N}} - \displaystyle\sum\mu_{{\rm{H}}_{\rm{Ga}}} \\ \quad\quad-\displaystyle\sum\mu_{{\rm{H}}_{\rm{N}}}-n_{{\rm{H}}_{\rm{Ga}}}^{\rm{steric}}\mu_{{\rm{H}}_{\rm{Ga}}}^{\rm{steric}}-n_{{\rm{H}}_{\rm{N}}}^{\rm{steric}}\mu_{{\rm{H}}_{\rm{N}}}^{\rm{steric}}\Big]. \end{array}$

(14)

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$ \begin{array}{l} \sigma_{\rm{pass}}^{11\overline{2}2}-\sigma_{\rm{pass}}^{\overline{11}2\overline{2}} = \dfrac{1}{4A^{11\overline{2}2}}\{ [E_{\rm{wedge}}^{11\overline{2}2}({\rm{large}})-E_{\rm{wedge}}^{\overline{11}2\overline{2}}({\rm{large}})]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-[E_{\rm{wedge}}^{11\overline{2}2}({\rm{small}})-E_{\rm{wedge}}^{\overline{11}2\overline{2}}({\rm{small}})]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;- 4[A^{ 000\overline{1}}\sigma_{\rm{pass}}^{ 000\overline{1}}-A^{\rm{0001}}\sigma_{\rm{pass}}^{\rm{0001}}]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+2[\mu_{\rm{N}}-\mu_{{\rm{A}}l}]\}, \end{array} $

(15)

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$ \sigma_{\rm{pass}}^{11\overline{2}2} = \frac{1}{A^{11\overline{2}2}}[E_{\rm{total}}^{11\overline{2}2}-\mu_{{\rm{A}}l}(n_{{\rm{A}}l}-n_{\rm{N}})-E_{\rm{AlN}}^{\rm{bulk}}n_{\rm{N}} - \mu_{\rm{H}} n_{\rm{H}} - A^{11\overline{2}2}\sigma_{\rm{pass}}^{\overline{11}2\overline{2}}], $

(16)

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$ \mu_{\rm{gas}} = -k_{\rm{B}} T \ln{\frac{gk_{\rm{B}} T}{p}\times \xi_{\rm{trans}} \xi_{\rm{rot}} \xi_{\rm{vib}}}, $

(17)

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$ \gamma_{\rm{ZB}}(\mu) = \frac{E_{{\rm{B-}}{\rm{edged}}\ {\rm{cluster}}}-n_{\rm{BN}}\mu_{\rm{BN}}-n_{\rm{edge}}\mu_{\rm{B}}}{3n_{\rm{edge}}}, $

(18)

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$ \gamma_{\rm{ZN}}(\mu) = \frac{E_{{\rm{N-}}{\rm{edged}}\ {\rm{cluster}}}-n_{\rm{BN}}\mu_{\rm{BN}}-n_{\rm{edge}}\mu_{\rm{N}}}{3n_{\rm{edge}}}, $

(19)

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$ \gamma = E_{\rm{Z}} + E_{\rm{ZPE}} - n_{\rm{Mo}}\mu_{\rm{Mo}} - n_{\rm{S}}\mu_{\rm{S}} = 3l\gamma_{\rm{Z}} + 3\gamma_{\rm{V}}, $

(20)

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$ \begin{array}{l} \;\;\;\; \gamma_{\rm{Z}} = (\Delta E_{\rm{Z}}+\Delta E_{\rm{ZPE}}-\Delta n_{\rm{Mo}}\mu_{{\rm{MoS}}_2}-\Delta n\mu_{\rm{S}})/3(l_1-l_2),\\ \;\;\;\; \Delta n_{\rm{Mo}} = n_{\rm{Mo}}(l_1)-n_{\rm{Mo}}(l_2),\\ \;\;\;\;\Delta n = n_{\rm{S}}(l_1) - n_{\rm{S}}(l_2)-2[n_{\rm{Mo}}(l_1)-n_{\rm{Mo}}(l_2)], \end{array} $

(21)

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$ \gamma_{\rm{edge}} = \frac{1}{l}\left(E_{\rm{tot}}-n_{\rm{B}}\mu_{\rm{B}} - n_{\rm{N}}\mu_{\rm{N}} - n_{{\rm{H}}_{\rm{N}}}\mu_{{\rm{H}}_{\rm{N}}}-\sum\limits_{{i}}n_i\mu_i\right), $

(22)

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$ E_{\rm{tot}}^{\rm{cluster}} = \frac{m^2+m}{2}\mu_{\rm{N}} + \frac{m^2-m}{2}\mu_{\rm{B}} + (3m-6)\mu_{{\rm{H}}_{\rm{N}}} + 6\mu_{{\rm{H}}_{\rm{N}}}^{\rm{corner}}, $

(23)

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$ \mu_{\rm{B}}+\mu_{\rm{N}} = E_{{{\rm{h}}\rm{-}{\rm{BN}}}} = E_{\rm{B}} + E_{\rm{N}} + \Delta H_{{{\rm{h}}\rm{-}{\rm{BN}}}}, $

(24)

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$ E_{\rm{tot}}^{\rm{cluster}} \!=\! m^2\! \left(\!\frac{{E_{{{\rm{h}}-{\rm{BN}}}}}}{2}\!\right)+m\left(\!\mu_{\rm{N}}-\frac{{E_{{{\rm{h}}\!-\!{\rm{BN}}}}}}{2}+3{\mu_{{\rm{H}}_{\rm{N}}}} \!\right)+6({\mu_{{\rm{H}}_{\rm{N}}}^{\rm{corner}}}-{\mu_{{\rm{H}}_{\rm{N}}}}). $

(25)

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$ E_r = \frac{1}{2l}(E_{\rm{p}}-n_{\rm{N}}\mu_{\rm{N}}-n_{\rm{B}}\mu_{\rm{B}}-n_{{\rm{H}}_{\rm{N}}}\mu_{{\rm{H}}_{\rm{N}}}-n_{{\rm{H}}_{\rm{B}}}\mu_{{\rm{H}}_{\rm{B}}}). $

(26)

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$ \mu_{\rm{H}} = \frac{1}{2}[E_{{\rm{H}}_2}+2\Delta\mu_{\rm{H}}(T,p)], $

(27)

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$ \Delta\mu_{\rm{H}}(T,p) = \frac{G}{2N} $

(28)

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Chuen-Keung Sin, Jingzhao Zhang, Kinfai Tse, Junyi Zhu. A brief review of formation energies calculation of surfaces and edges in semiconductors[J]. Journal of Semiconductors, 2020, 41(6): 061101

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