Average binding energy Eb[AB]=(NA∗EA+NB∗EB−Etot[AB])/(NA+NB) where EA and EB are the energy of free atom in vacuum for A and B.
[1,2]Curvature energy for nanotubes Etube=E2D−Ecurv=E2D−A/R2
2 Formation Energy
[3]General trend in formation energy for QD: ε(n)−ε(∞)∝const/n
2.1 formation energy of a defect or impurity
[4]The formation energy of a defect or impurity X in charge state q is defined as Ef[Xq]=Etot[Xq]−Etot[GaN,bulk]−i∑niμi+q[EF+Eν+ΔV]Etot[X] is the total energy derived from a supercell calculation with one impurity or defect X in the cell, and Etot[GaN,bulk] is the total energy for the equivalent supercell containing only bulk GaN. ni indicates the number of atoms of type i (host atoms or impurity atoms) that have been added to (ni>0) or removed from (ni<0) the supercell when the defect or impurity is created, and the μi are the corresponding chemical potentials of these species. E~F ~ is the Fermi level, referenced to the valence-band maximum in the bulk. Due to the choice of this reference, we need to explicitly put in the energy of the bulk valence-band maximum, Eν, in our expressions for formation energies of charged states. As discussed in Sec. II C, we also need to add a correction term DV, to align the reference potential in our defect supercell with that in the bulk.
extreme N-rich conditions: μN=μN[N2],μGamin=Etot[GaN]−μN[N2] The total energy of GaN can also be expressed as Etot[GaN]=μGa[bulk]+μN[N2]+ΔHf[GaN] where ΔHf[GaN] is the enthalpy of formation, which is negative for a stable compound.
2.2 Chemical Potential
The chemical potentials can in principle be related to partial pressures, using standard thermodynamic expressions. For instance, the chemical potential for hydrogen atoms in a gas of H2 molecules is given by 2μH=EH2+kBTln(kBTpVQ−lnZrot−lnZvib) where EH2 is the energy of an H2 molecule, kB is the Boltzmann constant, T is the temperature, and p is the pressure. VQ=(h2/2πmkBT)3/2 is the quantum volume, and Zrot and Zvib are the rotational and vibrational partition functions.
2.3 Helmholtz free energy[5]
From the computed phonon spectrum of the sheet configurations, the temperature-dependent vibrational Helmholtz free energy and entropy can be calculated in the harmonic approximation21. F(V,T), the free energy of a crystal, can be written as a sum of E(V), the energy of the static lattice at the equilibrium configuration, and Fvib(V,T), the vibrational free energy associated with the modes, ω(q). F=E+Fvib=E+q∑2ℏω(q)−ST Here, S is entropy, ∑q2ℏω(q) is the summation of contributions of the zero-point energy from each phonon mode, and T is the temperature.
Considering the first-order corrections to be small, S can be calculated20 as follows: S=−kBq∑ln(1−exp(−ℏω(q)/kBT)) The lattice heat capacity per unit cell at constant volume can be defined as follows: CV(T)=q∑cv(q)=kBq∑(ℏω(q)/2KBT)2/sinh2(ℏω(q)/2KBT)
2.4 Formation Valley[6]
Fig. 1:
Formation energies per atom (Ef) for 2D TiBx-multilayers. Ef=(ETiBx−ETi−xEB)/(x+1) from alloy, bulk Ti and α sheet.
2.5 Surface Energy
Thick engough,surface energy γ vs. thickness convergence test needed
γ=2AEslab−N∗Ebulk, where Eslab is the total energy of surface slab obtained using density functional theory. N is the number of atoms in the surface slab. Ebulk is the bulk energy per atom. A is the surface area. For a slab, we have two surfaces and they are of the same type, which is reflected by the number 2 in the denominator. To guarantee this, we need to create the slab carefully to make sure that the upper and lower surfaces are of the same type.
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