Draft:Lifshitz–Kosevich Formula

Lifshitz–Kosevich Formula

The Lifshitz–Kosevich formula is a fundamental result in the study of quantum oscillations, particularly in systems with a Fermi surface subject to an external magnetic field. It describes the oscillatory behavior of physical quantities, such as the density of states or magnetization, as a function of the inverse magnetic field (1/B). The formula is named after the Soviet physicists Ilya Lifshitz and Alexander Kosevich, who first derived it in 1954.

Mathematical Formulation

The Lifshitz–Kosevich formula for the oscillatory part of the magnetization $M$ is given by: $M\propto \frac{\lambda T}{\sinh (\lambda T)}\exp (-\frac{\lambda T_D}{B})\cos (2\pi \frac{F}{B}+\varphi )$

where:

• $B$ is the external magnetic field.
• $T$ is the temperature.
• $T_D$ is the Dingle temperature, which accounts for the effect of impurities and scattering.
• $F$ is the Onsager frequency, which is proportional to the extremal cross-section of the Fermi surface.
• $\varphi$ is a phase shift determined by the Berry phase and the Landau level quantization.
• $\lambda =2{\pi }^2{k}_B{m}^{*}/\hslash e$, where $m^{*}$ is the effective mass of the quasiparticle, $k_B$ is the Boltzmann constant, and $\hslash$ is the reduced Planck constant.

The key features of the Lifshitz–Kosevich formula include the temperature dependence via the term $\frac{\lambda T}{\sinh (\lambda T)}$, which reflects the thermal broadening of the Landau levels, and the Dingle factor $\exp (-\frac{\lambda T_D}{B})$, which accounts for impurity scattering effects.

Onsager Relation

The Onsager relation provides a link between the frequency $F$ of quantum oscillations and the extremal cross-sectional area $A$ of the Fermi surface. The relationship is given by: $F=\frac{\hslash }{2\pi e}A$

where:

• $A$ is the cross-sectional area of the Fermi surface in momentum space perpendicular to the magnetic field $B$.
• $F$ is the oscillation frequency observed in the de Haas–van Alphen or Shubnikov–de Haas effects.
• $e$ is the elementary charge.
• $\hslash$ is the reduced Planck constant.

This relation allows experimentalists to deduce the shape and size of the Fermi surface by measuring oscillation frequencies. By analyzing the oscillation frequencies, one can extract information about the electronic structure of materials.

Physical Interpretation

The oscillatory nature of the Lifshitz–Kosevich formula is a hallmark of quantum oscillations, such as the de Haas–van Alphen effect and Shubnikov–de Haas effect. These oscillations occur due to the quantization of cyclotron orbits in the presence of a magnetic field, which affects the density of states at the Fermi surface.

The frequency $F$ of these oscillations is linked to the extremal cross-sectional area $A$ of the Fermi surface via the Onsager relation: $F=\frac{\hslash }{2\pi e}A$

This relationship allows experimentalists to extract information about the Fermi surface geometry from measurements of magnetization or resistivity oscillations.

Applications

The Lifshitz–Kosevich formula is widely used in condensed matter physics to investigate the properties of materials with complex Fermi surfaces. By analyzing the oscillatory behavior in the de Haas–van Alphen and Shubnikov–de Haas effects, researchers can deduce properties such as:

• The shape and size of the Fermi surface.
• The effective mass of charge carriers.
• The Dingle temperature, which provides information on impurity scattering.

Such information is crucial in the study of materials like topological insulators, heavy fermion systems, and strongly correlated electron systems.

Historical Context

The formula was first derived in 1954 by I. M. Lifshitz and A. M. Kosevich in the context of density of states oscillations in metals at low temperatures under a magnetic field. The derivation relies on the concept of quantized cyclotron orbits and the energy levels known as Landau levels. Their work laid the foundation for modern studies of quantum oscillations in metals and semiconductors. The method has since been extended to analyze other quantum oscillatory phenomena, such as the quantum Hall effect and the properties of topological materials.

See Also

• de Haas–van Alphen effect
• Shubnikov–de Haas effect
• Fermi surface
• Berry phase
• Quantum oscillation