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Double-Peak Specific Heat Anomaly in Quantum Oscillation

Kohama Group

Quantum oscillation phenomenon is an essential tool for understanding the electronic structure of a metal. The origin of the quantum oscillation is the Landau quantization of the carrier motion, which gives rise to a series of quantized singularities in the density of states (DOS) that cross the Fermi level. The Lifshitz–Kosevich (LK) theory has been widely used to describe the behavior of the quantum oscillation, notably to extract parameters such as the effective mass and Landé g-factor. Although the theory is remarkably successful in the most of metals over a wide range of magnetic fields and temperatures, there is growing evidence to suggest that experiment often deviates from the predicted LK behavior [1]. Although the oscillatory magnetoresistance, magnetization, and thermopower exhibit a clear departure from LK theory at the high-magnetic field limit called the quantum limit, the oscillatory behavior of the specific heat (Cp) in the quantum limit has yet to be fully explored. In this research, we have measured Cp of graphite as a function of the magnetic field and demonstrated that the crossing of a single spin Landau level and the Fermi energy gives rise to a double-peak structure in Cp, in striking contrast to the single-peak expected from LK theory.

When a Landau level crosses the Fermi energy, the occupation of the Landau level changes rapidly, inducing large changes in the entropy of the system, which can be probed using thermodynamic measurements, such as magnetocaloric effect (MCE) and Cp. To follow the evolution of the entropy in a metal sample, we show the MCE trace (1/T) of the natural graphite as a function of the magnetic field taken at an initial temperature of 0.7 K (Fig. 1a). The entropy is proportional to the logarithm of the number of states within the Fermi edge, and therefore shows a maximum when a Landau level is located at the Fermi level, resulting in a series of well-defined single peaks labeled as N±e=h (see Fig. 1a). Here, N is the Landau index, e/h indicates if the Landau level originates from the electron or hole pocket, and ± indicate the spin up/down levels. For better comparison, Fig. 1b shows background removed magnetoresistance ΔRxx at 0.5 K. These results are in stark contrast to the electronic specific heat divided by temperature Cel/T which is proportional to the temperature derivative of entropy (Fig. 1c). Crucially, when low-index Landau levels (Ne/h < 3) cross the Fermi energy, Cel/T exhibits a series of double-peak structure, as indicated by the double arrows in Fig. 1c.

In order to elucidate the origin of the double-peak structure, it is necessary to consider the exact expression for the specific heat. For electronic quasiparticles, Cel/T is given by, Cel/T = k2B-∞D(E)(-x2dF(x)dx)dx,   (Eq.1)
where F(x) = 1/(1 + ex), x = E/kBT and kB is the Boltzmann constant. The specific heat depends on the convolution of the Landau level DOS D(E) and a kernel term −x2dF(x)/dx, which involves the first derivative of the Fermi-Dirac distribution function. Importantly, the double-peak structure in Cel/T originates from the temperature-dependent splitting of the double maxima in the kernel term -x2dF(x)/dx. In Ref. 2, we show that there is a quantitative agreement between the observed dble-peak structure and calculated Cel/T curve.

In summary, we demonstrate that, as the quantum limit is approached in high-quality graphite, Landau levels crossing the Fermi energy give rise to single features in MCE and magnetoresistance, while simultaneously a novel double-peak structure is observed in the specific heat Cel/T. The calculation based on the exact form of the free electron expression successfully reproduces the double-peak structure in specific heat. The specific heat, which depends on an integral involving the kernel term, represents a spectroscopic tuning fork of width 4.8 kBT that can be tuned at will to resonance.


References
  • [1] A. E. Datars and J. E. Sipe, Phys. Rev. B 51, 4312 (1995).
  • [2] Z. Yang, B. Fauqué, T. Nomura, T. Shitaokoshi, S. Kim, D. Chowdhury, Z. Pribulová, J. Kačmarčík, A. Pourret, G. Knebel, D. Aoki, T. Klein, D. K. Maude, C. Marcenat, and Y. Kohama, Nature Communications 14 7006 (2023).
Authors
  • Y. Kohama and Z. Yang