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. These levels are degenerate, with the number of electrons per level directly proportional to the strength of the applied magnetic field. It is named after the Soviet physicist Lev Landau who developed the theory in 1930.

Landau quantization contributes towards magnetic susceptibility of metals, known as Landau diamagnetism. 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.

Derivation

Consider a system of non-interacting particles with charge and spin confined to an area in the along the -axis. In SI units, the Hamiltonian is the canonical momentum operator is the operator for the electromagnetic vector potential (in position space).

The vector potential is related to the magnetic field

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 changes the overall phase of the wave function by an amount corresponding to the scalar field. But physical properties are not influenced by the specific choice of gauge.

In the Landau gauge

From the possible solutions for A, a gauge fixing introduced by Lev Landau is often used for charged particles in a constant magnetic field.

is a possible solution in the Landau gauge (not to be mixed up with the R_\xi gauge).

does not appear in the Hamiltonian and only the z-momentum appears in the kinetic energy, this motion along the z-direction is a free motion.

The Hamiltonian can also be written more simply by noting that the cyclotron frequency This is exactly the Hamiltonian for the quantum harmonic oscillator.

To find the energies, note that translating the harmonic oscillator potential does not affect the energies. The energies of this system are thus identical to those of the standard quantum harmonic oscillator, is called a Landau level.

is the n-th state for the quantum harmonic oscilator. In sum, the state of the electron is characterized by the quantum numbers.

In the symmetric gauge

The derivation treated and y as asymmetric. However, by the symmetry of the system, there is no physical quantity which distinguishes these coordinates. The same result could have been obtained with an appropriate interchange of and .

.

where we reintroduced the units back.

is the eigenvalue.

by one unit. The analogy to quantum harmonic oscillator

.

.

Degeneracy of the Landau levels

In the Landau gauge

The effects of Landau levels may only be observed when the mean thermal energy is smaller than the energy level, meaning low temperatures and strong magnetic fields.

can be simply written as a ratio of fluxes is the fundamental magnetic flux quantum).

, two available states for each flux quantum that penetrates the system.

The above gives only a rough idea of the effects of finite-size geometry. Strictly speaking, using the standard solution 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. The filling of these levels with many electrons is still an active area of research.

In general, Landau levels are observed in electronic systems. 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).

If Zeeman splitting. However, the Fermi energy and ground state energy stay roughly the same in a system with many filled levels, since pairs of split energy levels cancel each other out when summed.

-plane, perpendicular to the magnetic field, is still quantized.

In the symmetric gauge

in symmetric gauge. The degeneracy per unit area is the same in each Landau level.

The z component of angular momentum

th Landau level. However, it may be arbitrarily large, which is necessary to obtain the infinite degeneracy (or finite degeneracy per unit area) exhibited by the system.

Relativistic case

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 .

The massless 2D case can be simulated in single-layer materials like graphene near the Dirac cones, where the eigenergies are given by where the speed of light has to be replaced with the Fermi speed vF of the material and the minus sign corresponds to electron holes.

Magnetic susceptibility of a Fermi gas

The Fermi gas (an ensemble of non-interacting fermions) is part of the basis for understanding of the thermodynamic properties of metals. In 1930 Landau derived an estimate for the magnetic susceptibility of a Fermi gas, known as Landau susceptibility, which is constant for small magnetic fields. Landau also noticed that the susceptibility oscillates with high frequency for large magnetic fields, this physical phenomenon is known as the De Haas–Van Alphen effect.

Two-dimensional lattice

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.

Integer quantum Hall effect

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 . 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 as

Since the resistivity plateau is given by

the required density is

which is exactly the density required to fill the Landau level. The gap between different Landau levels along with large degeneracy of each level renders the resistivity quantized.

See also

• Laughlin wavefunction

References

External links

Further reading

• Landau, L. D.; and Lifschitz, E. M.; (1977). Quantum Mechanics: Non-relativistic Theory. Course of Theoretical Physics. Vol. 3 (3rd ed. London: Pergamon Press). .