Note on de Haas-van Alphen diamagnetism in thin, free-electron films

We revisit the problem of de Haas-van Alphen (dHvA) diamagnetic susceptibility oscillations in a thin, free-electron film trapped in a synthetic harmonic potential well. A treatment of this phenomenon at zero temperature was announced many years ago by Childers and Pincus (designated hereafter as CP), and we traverse initially much the same ground, but from a slightly different analytic perspective. That difference hinges around our use, in calculating the Helmholtz free energy F, of an inverse Laplace transform, Bromwich-type contour integral representation for the sharp distribution cutoff at Fermi level μ. The contour integral permits closed-form summation all at once over the discrete orbital Landau energy levels transverse to the magnetic field, and the energy associated with the in-plane canonical momenta ℏ k x and ℏ k z. Following such summation/integration, pole/residue pairs appear in the plane of complex transform variable s, a fourth-order pole at origin s = 0, and an infinite ladder, both up and down, of simple poles along the imaginary axis. The residue sum from the infinite pole ladder automatically engenders a Fourier series with period one in dimensionless variable μ/ ℏ ω (with effective angular frequency ω suitably defined), series which admits closed-form summation as a cubic polynomial within any given periodicity slot. Such periodicity corresponds to Landau levels slipping sequentially beneath Fermi level μ as the ambient magnetic field H declines in strength, and is manifested by the dHvA pulsations in diamagnetic susceptibility. The coëxisting steady contribution from the pole at origin has a similar cubic structure but is opposite in sign, inducing a competition whose outcome is a net magnetization that is merely quadratic in any given periodicity slot, modulated by a slow amplitude growth. Apart from some minor notes of passing discord, these simple algebraic structures confirm most of the CP formulae, and their graphic display reveals a numerically faithful portrait of the oscillatory dHvA diamagnetic susceptibility phenomenon. The calculations on view have a merely proof-of-principle aim, with no pretense at all of being exhaustive. The zero-temperature results hold moreover the key to the entire panorama of finite-temperature thermodynamics with T > 0. Indeed, thanks to the elegant work of Sondheimer and Wilson, one can promote the classical, Maxwell-Boltzmann partition function Z ( s ), via an inverse Laplace transform of its ratio to s 2, directly into the required, Fermi-Dirac Helmholtz free energy F at finite temperature T > 0. While the underlying cubic polynomial commonality continues to bestow decisive algebraic advantages, the evolving formulae are naturally more turgid than their zero-temperature counterparts. Nevertheless we do retain control over them by exhibiting their retrenchment into precisely these antecedents. So fortified, we undertake what is at once both a drastic and yet a simple-minded step of successive approximation, a step which clears the path toward numerical evaluation of the finite-temperature diamagnetic susceptibility. We are rewarded finally with a persistent dHvA periodicity imprint, but with its peaks increasingly flattened and its valleys filled in response to temperature rise, all as one would expect on physical grounds.