Landau levels
In quantum mechanics, the energies of cyclotron orbits of charged particles in a uniform magnetic field are quantized to discrete values, thus known as Landau levels.
Under strong magnetic fields, Landau quantization leads to oscillations in electronic properties of materials as a function of the applied magnetic field known as the De Haas–Van Alphen and Shubnikov–de Haas effects.
Landau quantization is a key ingredient in explanation of the integer quantum Hall effect.
Consider a system of non-interacting particles with charge q and spin S confined to an area A = LxLy in the x-y plane.
In SI units, the Hamiltonian of this system (here, the effects of spin are neglected) is
There is some gauge freedom in the choice of vector potential for a given magnetic field.
The Hamiltonian is gauge invariant, which means that adding the gradient of a scalar field to A changes the overall phase of the wave function by an amount corresponding to the scalar field.
From the possible solutions for A, a gauge fixing introduced by Lev Landau is often used for charged particles in a constant magnetic field.
This is exactly the Hamiltonian for the quantum harmonic oscillator, except with the minimum of the potential shifted in coordinate space by
The energies of this system are thus identical to those of the standard quantum harmonic oscillator,[4]
However, by the symmetry of the system, there is no physical quantity which distinguishes these coordinates.
Strictly speaking, using the standard solution of the harmonic oscillator is only valid for systems unbounded in the
is finite, boundary conditions in that direction give rise to non-standard quantization conditions on the magnetic field, involving (in principle) both solutions to the Hermite equation.
As the magnetic field is increased, more and more electrons can fit into a given Landau level.
The occupation of the highest Landau level ranges from completely full to entirely empty, leading to oscillations in various electronic properties (see De Haas–Van Alphen effect and Shubnikov–de Haas effect).
Zeeman splitting has a significant effect on the Landau levels because their energy scales are the same,
-direction, which is a relevant experimental situation — found in two-dimensional electron gases, for instance.
This term then fills in the separation in energy of the different Landau levels, blurring the effect of the quantization.
Each Landau level has degenerate orbitals labeled by the quantum numbers
An electron following Dirac equation under a constant magnetic field, can be analytically solved.
where c is the speed of light, the sign depends on the particle-antiparticle component and ν is a non-negative integer.
Due to spin, all levels are degenerate except for the ground state at ν = 0.
The massless 2D case can be simulated in single-layer materials like graphene near the Dirac cones, where the eigenergies are given by[8]
The Fermi gas (an ensemble of non-interacting fermions) is part of the basis for understanding of the thermodynamic properties of metals.
Landau also noticed that the susceptibility oscillates with high frequency for large magnetic fields,[9] this physical phenomenon is known as the De Haas–Van Alphen effect.
The tight binding energy spectrum of charged particles in a two dimensional infinite lattice is known to be self-similar and fractal, as demonstrated in Hofstadter's butterfly.
For an integer ratio of the magnetic flux quantum and the magnetic flux through a lattice cell, one recovers the Landau levels for large integers.
[10] The energy spectrum of the semiconductor in a strong magnetic field forms Landau levels that can be labeled by integer indices.
In addition, the Hall resistivity also exhibits discrete levels labeled by an integer ν.
The fact that these two quantities are related can be shown in different ways, but most easily can be seen from Drude model: the Hall conductivity depends on the electron density n as
