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To quantitatively evaluate the formation energies of Cu, Zn, Sn, and S vacancies in kesterite-type Cu2ZnSnS4 (CZTS), first-principles pseudopotential calculations using plane-wave basis functions were performed. The formation energies of neutral Cu, Zn, Sn, and S vacancies were calculated as a function of the atomic chemical potentials of constituent elements. We compared the vacancy formation in the In-free photovoltaic semiconductor CZTS with those of Cu2ZnSnSe4 (CZTSe) and CuInSe2 (CIS). The obtained results were as follows. (1) Under the Cu-poor and Zn-rich condition, the formation energy of the Cu vacancy was generally smaller than those of the Zn, Sn and S vacancies in CZTS, as is the case for CZTSe. (2) The formation energies of Cu, Zn, and Sn vacancies in CZTS were larger than those in CZTSe. On the other hand, the formation energy of the S vacancy is smaller than that of the Se vacancy in CZTSe. (3) Under the Cu-poor and Zn-rich condition, the formation energies of the Cu vacancy in CZTS and CZTSe are much larger than that in CIS. These results indicate that in kesterite-type CZTS and CZTSe, the Cu vacancy is easily formed under Cu-poor, Zn-rich, and S(Se)-rich condition, but it is more difficult than that in CIS.

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We have theoretically evaluated phase stability and electronic structure of Cu2ZnSiSe4 and Cu2ZnGeSe4 and compared the results with those of Cu2ZnSnSe4. The enthalpies of formation for kesterite (KS), stannite (ST), and wurtz-stannite (WST) phases of Cu2ZnSiSe4, Cu2ZnGeSe4, and Cu2ZnSnSe4 (CZTSe) were calculated by first-principles calculations. In these three compounds, the KS phase is more stable than the ST and WST phases. The theoretical band gaps of KS-type Cu2ZnSiSe4 (1.48 eV) and Cu2ZnGeSe4 (1.10 eV) are wider than that of KS-type Cu2ZnSnSe4 (0.63 eV). The valence band maximum (VBM) of KS-type Cu2ZnIVSe4 consists of antibonding orbital of Cu 3d and Se 4p, while the conduction band minimum (CBM) consist of antibonding orbital of IV ns and Se 4p. The VBMs of Cu 3d + Se 4p in Cu2ZnSiSe4 and Cu2ZnGeSe4 are similar to that in Cu2ZnSnSe4. Therefore, the energy levels of VBMs in Cu2ZnIVSe4 (IV = Si, Ge) do not change so much compared with that of CZTSe. On the other hand, the energy levels of CBMs of IV ns + Se 4p in Cu2ZnSiSe4 and Cu2ZnGeSe4 become higher than that in Cu2ZnSnSe4. These trends in the electronic structures are explained by the schematic molecular orbital diagrams of tetrahedral CuSe4 7-, ZnSe4 6-, and IVSe4 4- (IV = Si, Ge, Sn) clusters.

We apply the Jeffrey-Kirwan method to compute the multiple integrals for the BCD type Nekrasov partition functions of four dimensional N = 2 supersymmetric gauge theories. We construct a graphical distinction rule to determine which poles are surrounded by their integration cycles. We compute the instanton correction of the "Sp(0)" pure super-Yang-Mills theory and find that Z Sp(0) k