Surface correction to Landau diamagnetism

For a semi-infinite free-electron model with a magnetic field applied perpendicular to the surface, we show that ${\chi }_d^s=-\frac{{\chi }_p^s}{3}$, where ${\chi }_d^s$ and ${\chi }_p^s$ are, respectively, the surface corrections to the Landau diamagnetism and the Pauli paramagnetism. The total surface contribution to the susceptibility is ${\chi }^s=\left(\frac{2}{3}\right)\left(\frac{{\mu }_B^2}{{\pi }^2}\right)A\left[\gamma \right(k_F)-\frac{\pi }{4}]$, where $\gamma \left(k_F\right)$ is the phase shift for $k_z=k_F$. Since $〈\gamma 〉=\frac{\pi }{4}$ and $\gamma \left(k_F\right)>\mathrm{}〈\gamma 〉$, the surface contributions to the magnetic susceptibility and electronic heat capacity are positive.