Single-Scale Magnetoelastic Landau Quantization: Thermodynamics, Quantum Oscillations, and Metrology

TL;DR

This work develops a unified, single-scale description of thermodynamics and quantum oscillations for electrons in a medium with a uniform screw-dislocation density under a magnetic field. By mapping the transverse spectrum to an elastic-Landau ladder with spacing $\hbar|\\omega_{eff}|$ and defining $\omega_{eff}=\\omega_c+\\omega_{cl}$, all observables are expressed through the universal kernel $x=\\hbar|\\omega_{eff}|/(2k_B T)$, enabling phase-coherent interpolation between 2D-like quantization and 3D behavior. The theory predicts a compensated-field line, a Schottky-type heat-capacity signature, torsion-induced shifts in de Haas-van Alphen and Shubnikov-de Haas oscillations, and rigid shifts of IQHE plateaus via the Středa relation, which together form a metrological loop to extract dislocation density from a single field sweep. Finite-size and weak-disorder corrections preserve the kernel structure and yield mesoscopic caloric fingerprints, offering practical pathways for on-chip calorimetry, torque magnetometry, and strain-engineered devices. The results establish a compact design rule around the single scale $\hbar|\\omega_{eff}|$ for metrology, caloritronics, and magnetoelastic transduction in defect-engineered quantum materials.

Abstract

We develop a unified, single-scale description of thermodynamics and quantum oscillations in electronic systems with a uniform areal density of screw dislocations under a uniform magnetic field. A single tunable gap, $\hbar|ω_{eff}|$ with $ω_{eff}=ω_{c}+ω_{cl}$, organizes all equilibrium observables obtained from a compact harmonic-oscillator partition function: free energy, internal energy, entropy, heat capacity, magnetization, magnetic susceptibility, and magnetocaloric responses collapse onto universal hyperbolic kernels in $x=\hbar|ω_{eff}|/(2k_{B}T)$. We identify a compensated-field regime where the transverse gap closes and the heat capacity reaches an equipartition plateau, providing a sharp signature of magnetoelastic interference. In transport and torque, the same scale rigidly shifts the Hall fan and compresses the $1/B$ period of de Haas-van Alphen and Shubnikov-de Haas oscillations when expressed in $1/B_{eff}$, enabling a phase-unwarping protocol that metrologizes the dislocation density from a single field sweep. In mesoscopic samples, boundary corrections to the Landau degeneracy generate finite-size calorimetric oscillations that diagnose the effective magnetic length. Moderate disorder and weak interactions preserve the kernel structure while smoothing amplitudes. We outline an experimental roadmap combining on-chip calorimetry, torque magnetometry, and transport, and discuss device-level opportunities in caloritronics and strain engineering, magnetocaloric microcooling, magnetoelastic heat switching, and dilatometric transduction, where the single scale $\hbar|ω_{eff}|$ enables rational design and optimization.

Figures (23)

• Figure 1: Thermodynamic properties of the system as a function of temperature for a fixed torsion parameter ($\Omega = 1.5 \times 10^7\, \mathrm{m^{-1}}$) and different values of the external magnetic field, $B$. (a) The internal energy $U$ increases with $B$ due to the widening of the energy gap. (b) The heat capacity $C_V$ shows a Schottky-like peak that shifts to higher temperatures as $B$ increases, indicating a larger energy spacing. (c) The entropy $S$ rises more slowly for stronger magnetic fields, as fewer states become less thermally accessible. (d) The Helmholtz free energy $A$ is higher for larger $B$ at low temperatures, reflecting the dominance of the internal energy contribution.
• Figure 2: Thermodynamic properties of the system as a function of temperature for a fixed magnetic field ($B = 5.0 \, \mathrm{T}$) and different values of the torsion parameter, $\Omega$. (a) The internal energy $U$ increases with torsion density. (b) The Schottky-like peak of the heat capacity $C_V$ shifts to higher temperatures for larger $\Omega$, indicating an increase in the effective energy gap. (c) The entropy $S$ is lower for higher torsion densities at a given temperature. (d) The Helmholtz free energy $A$ increases with $\Omega$, as the internal energy contribution dominates in this temperature range.
• Figure 3: Magnetization per particle as a function of temperature for a fixed field $B = 5\,$T and four torsion densities $\Omega = 0$, $0.5$, $1.0$ and $1.5 \times 10^{7}\,$m$^{-1}$, shown in black, red, blue, green, respectively. The curves are in units of the Bohr magneton, $M/\mu_{B}$, and correspond to Eq. \ref{['eq:M_theory']} with $k = 1\times10^{9}\,$m$^{-1}$. As $T\to0$ all curves converge to the universal Landau-diamagnetic value $M_{0}= -\hbar e/2\mu \simeq -\,\mu_{B}$. As $T$ increases and $k_{B}T\gg\hbar|\omega_{\text{eff}}|$, the leading $T/|\omega_{\text{eff}}|$ terms in Eq. \ref{['eq:M_theory']} cancel, leaving the next-order contribution $M\simeq -\,\mathrm{sgn}(\omega_{\text{eff}})\,(e/\mu)\,\hbar^{2}|\omega_{\text{eff}}|/(12\,k_{B}T)$. Increasing $\Omega$ enhances $|\omega_{\text{eff}}|$ and thus reduces the temperature-induced decay of $|M|$, meaning that torsion delays the loss of diamagnetic saturation.
• Figure 4: Normalized magnetic susceptibility $\chi/\chi_{0}$ as function of temperature, obtained from Eq. \ref{['eq:chi_theory']} using the same parameters of Fig. \ref{['fig:magnetisation_vs_Omega']}. The normalization $\chi_{0}(T)= (\hbar e/2\mu)^{2}/(k_{B}T)$ removes the trivial Curie factor. With the corrected susceptibility, $\frac{\chi}{\chi_0}= \csch^{2}x \;-\; \frac{1}{x^{2}},\qquad x=\frac{\hbar|\omega_{\text{eff}}|}{2k_{B}T},$ each curve starts from $\chi=0$ (ground-state rigidity) and, for $x \ll 1$ (high $T$), approaches the constant limit $\chi/\chi_0\to -\,1/3$, consistent with the diamagnetic high-$T$ behavior. Increasing $\Omega$ enlarges the effective gap, shifting the onset of strong susceptibility to higher $T$ and flattening the low-$T$ curvature.
• Figure 5: Torsional conjugate per particle, $\Pi_{\Omega}(T)$, for a fixed magnetic field ($B = 5\,\mathrm{T}$) and four torsion densities ($\Omega = 0$, $0.5$, $1.0$, $1.5 \times 10^{7}\,\mathrm{m^{-1}}$) shown in black, red, blue, and green, respectively. At $T \to 0$, all curves converge to $\Pi_{\Omega}^{0}=-\,\hbar^{2}k/\mu$, according to Eq. \ref{['eq:Pi_Omega']} with $\mathrm{sgn}(\omega_{\text{eff}})=+1$. For $k_{B}T>\hbar|\omega_{\text{eff}}|$, the $\coth$ term gives $|\Pi_{\Omega}|\propto T/|\omega_{\text{eff}}|$, so larger $\Omega$ (corresponding to larger $|\omega_{\text{eff}}|$) delays the onset of the increase.