Surface correction to Landau diamagnetism

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

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