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We performed first-principles calculations to investigate the electronic structures and magnetic properties of neutral and charged Ga vacancies of 3C-GaN. Our results show that the neutral, −1 charged, and −2 charged Ga vacancies can introduce the magnetic moments of 3μ B , 2μ B , and 1μ B , respectively. We found that both −1 charged Ga vacancies and −2 charged Ga vacancies couple ferromagnetically, while the −1 charged Ga vacancies have lower formation energy compared with the −2 charged ones, which may be responsible for the unexpected ferromagnetism in the system. However, the neutral Ga vacancies couple antiferromagnetically, which cannot contribute to the ferromagnetism.

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Using the semi-classical (Thomas–Fermi) approximation, the thermodynamic properties of ideal Fermi gases in a harmonic potential in an n-dimensional space are studied under the generalized uncertainty principle (GUP). The mean particle number, internal energy, heat capacity and other thermodynamic variables of the Fermi system are calculated analytically. Then, analytical expressions of the mean particle number, internal energy, heat capacity, chemical potential, Fermi energy, ground state energy and amendments of the GUP are obtained at low temperatures. The influence of both the GUP and the harmonic potential on the thermodynamic properties of a copper-electron gas and other systems with higher electron densities are studied numerically at low temperatures. We find: (1) When the GUP is considered, the influence of the harmonic potential is very much larger, and the amendments produced by the GUP increase by eight to nine orders of magnitude compared to when no external potential is applied to the electron gas. (2) The larger the particle density, or the smaller the particle masses, the bigger the influence of the GUP. (3) The effect of the GUP increases with the increase in the spatial dimensions. (4) The amendments of the chemical potential, Fermi energy and ground state energy increase with an increase in temperature, while the heat capacity decreases. TF0 is the Fermi temperature of the ideal Fermi system in a harmonic potential. When the temperature is lower than a certain value (0.22 times TF0 for the copper-electron gas, and this value decreases with increasing electron density), the amendment to the internal energy is positive, however, the amendment decreases with increasing temperature. When the temperature increases to the value, the amendment is zero, and when the temperature is higher than the value, the amendment to the internal energy is negative and the absolute value of the amendment increases with increasing temperature. (5) When electron density is greater than or equal to 1037 m−3, the influence of the GUP becomes the dominant factor affecting the thermodynamic properties of the system.