In quantum mechanics, the energies of cyclotron orbits of charged particles in a uniform magnetic field are quantized to discrete values, called 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.[1]

Landau quantization contributes to the 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 to explain the integer quantum Hall effect.

Derivation edit

 
Diagram of a cyclotron orbit of a particle with speed v, which is the classical trajectory of a charged particle (here positive charge) under a uniform magnetic field B. The Landau quantization refers to a quantum charged particle under a uniform magnetic field.

Consider a system of non-interacting particles with charge q and spin S confined to an area A = LxLy in the x-y plane. Apply a uniform magnetic field   along the z-axis. In SI units, the Hamiltonian of this system (here, the effects of spin are neglected) is

 
Here,   is the canonical momentum operator and   is the operator for the electromagnetic vector potential   (in position space  ).

The vector potential is related to the magnetic field by  

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. But physical properties are not influenced by the specific choice of gauge.

In the Landau gauge edit

From the possible solutions for A, a gauge fixing introduced by Lev Landau is often used for charged particles in a constant magnetic field.[2]

When   then   is a possible solution[3] in the Landau gauge.

In this gauge, the Hamiltonian is

 
The operator   commutes with this Hamiltonian, since the operator ŷ is absent by the choice of gauge. Thus the operator   can be replaced by its eigenvalue ħky. Since   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 is ωc = qB/m, giving

 
This is exactly the Hamiltonian for the quantum harmonic oscillator, except with the minimum of the potential shifted in coordinate space by x0 = ħky/c .

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,[4]

 
The energy does not depend on the quantum number ky, so there will be a finite number of degeneracies (If the particle is placed in an unconfined space, this degeneracy will correspond to a continuous sequence of  ). The value of   is continuous if the particle is unconfined in the z-direction and discrete if the particle is bounded in the z-direction also. Each set of wave functions with the same value of n is called a Landau level.

For the wave functions, recall that   commutes with the Hamiltonian. Then the wave function factors into a product of momentum eigenstates in the y direction and harmonic oscillator eigenstates   shifted by an amount x0 in the x direction:

 
where  . In sum, the state of the electron is characterized by the quantum numbers, n, ky and kz.

In the symmetric gauge edit

The derivation treated x 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 x and y.

A more adequate choice of gauge, is the symmetric gauge, which refers to the choice

 

In terms of dimensionless lengths and energies, the Hamiltonian can be expressed as

 

The correct units can be restored by introducing factors of   and  .

Consider operators

 

These operators follow certain commutation relations

 

In terms of above operators the Hamiltonian can be written as

 
where we reintroduced the units back.

The Landau level index   is the eigenvalue of the operator  .

The application of   increases   by one unit while preserving  , whereas   application simultaneously increase   and decreases   by one unit. The analogy to quantum harmonic oscillator provides solutions

 
where
 
and
 

One may verify that the above states correspond to choosing wavefunctions proportional to

 
where  .

In particular, the lowest Landau level   consists of arbitrary analytic functions multiplying a Gaussian,  .

Degeneracy of the Landau levels edit

In the Landau gauge edit

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

Each Landau level is degenerate because of the second quantum number ky, which can take the values

 
where N is an integer. The allowed values of N are further restricted by the condition that the center of force of the oscillator, x0, must physically lie within the system, 0 ≤ x0 < Lx. This gives the following range for N,
 

For particles with charge q = Ze, the upper bound on N can be simply written as a ratio of fluxes,

 
where Φ0 = h/e is the fundamental magnetic flux quantum and Φ = BA is the flux through the system (with area A = LxLy).

Thus, for particles with spin S, the maximum number D of particles per Landau level is

 
which for electrons (where Z = 1 and S = 1/2) gives D = 2Φ/Φ0, 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 of the harmonic oscillator is only valid for systems unbounded in the x-direction (infinite strips). If the size Lx 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[5] 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 is included, each Landau level splits into a pair, one for spin up electrons and the other for spin down electrons. Then the occupation of each spin Landau level is just the ratio of fluxes D = Φ/Φ0. Zeeman splitting has a significant effect on the Landau levels because their energy scales are the same, 2μBB = ħωc. 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.

Moreover, the above derivation in the Landau gauge assumed an electron confined in the z-direction, which is a relevant experimental situation — found in two-dimensional electron gases, for instance. Still, this assumption is not essential for the results. If electrons are free to move along the z direction, the wave function acquires an additional multiplicative term exp(ikzz); the energy corresponding to this free motion, (ħ kz)2/(2m), is added to the E discussed. This term then fills in the separation in energy of the different Landau levels, blurring the effect of the quantization. Nevertheless, the motion in the x-y-plane, perpendicular to the magnetic field, is still quantized.

In the symmetric gauge edit

Each Landau level has degenerate orbitals labeled by the quantum numbers   in symmetric gauge. The degeneracy per unit area is the same in each Landau level.

The z component of angular momentum is

 

Exploiting the property   we chose eigenfunctions which diagonalize   and  , The eigenvalue of   is denoted by  , where it is clear that   in the  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 edit

 
Landau levels in graphene. Charge carriers in graphene behave as relativistic massless Dirac particles.

An electron following Dirac equation under a constant magnetic field, can be analytically solved.[6][7] The energies are given by

 

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]

 
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 edit

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,[9] this physical phenomenon is known as the De Haas–Van Alphen effect.

Two-dimensional lattice edit

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]

Integer quantum Hall effect edit

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

 

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 edit

References edit

  1. ^ Landau, L. (1930). "Diamagnetismus der Metalle" [Diamagnetism of Metals]. Zeitschrift für Physik (in German). 64 (9–10). Springer Science and Business Media LLC: 629–637. Bibcode:1930ZPhy...64..629L. doi:10.1007/bf01397213. ISSN 1434-6001. S2CID 123206025.
  2. ^ "Charge in Magnetic Field" (PDF). courses.physics.illinois.edu. Retrieved 11 March 2023.
  3. ^ An equally correct solution in the Landau gauge would be:  .
  4. ^ Landau, L. D.; Lifshitz, E. M. (1977). Quantum mechanics : non-relativistic theory (3rd ed.). Amsterdam: Butterworth Heinemann. pp. 424–426. ISBN 978-0-7506-3539-4. OCLC 846962062.
  5. ^ Mikhailov, S. A. (2001). "A new approach to the ground state of quantum Hall systems. Basic principles". Physica B: Condensed Matter. 299 (1–2): 6–31. arXiv:cond-mat/0008227. Bibcode:2001PhyB..299....6M. doi:10.1016/S0921-4526(00)00769-9. S2CID 118500817.
  6. ^ Rabi, I. I. (1928). "Das freie Elektron im homogenen Magnetfeld nach der Diracschen Theorie". Zeitschrift für Physik (in German). 49 (7–8): 507–511. Bibcode:1928ZPhy...49..507R. doi:10.1007/BF01333634. ISSN 1434-6001. S2CID 121121095.
  7. ^ Berestetskii, V. B.; Pitaevskii, L. P.; Lifshitz, E. M. (2012-12-02). Quantum Electrodynamics: Volume 4. Elsevier. ISBN 978-0-08-050346-2.
  8. ^ Yin, Long-Jing; Bai, Ke-Ke; Wang, Wen-Xiao; Li, Si-Yu; Zhang, Yu; He, Lin (2017). "Landau quantization of Dirac fermions in graphene and its multilayers". Frontiers of Physics. 12 (4): 127208. arXiv:1703.04241. Bibcode:2017FrPhy..12l7208Y. doi:10.1007/s11467-017-0655-0. ISSN 2095-0462.
  9. ^ Landau, L. D.; Lifshitz, E. M. (22 October 2013). Statistical Physics: Volume 5. Elsevier. p. 177. ISBN 978-0-08-057046-4.
  10. ^ Analytis, James G.; Blundell, Stephen J.; Ardavan, Arzhang (May 2004). "Landau levels, molecular orbitals, and the Hofstadter butterfly in finite systems". American Journal of Physics. 72 (5): 613–618. Bibcode:2004AmJPh..72..613A. doi:10.1119/1.1615568. ISSN 0002-9505.

External links edit

Further reading edit

  • Landau, L. D.; and Lifschitz, E. M.; (1977). Quantum Mechanics: Non-relativistic Theory. Course of Theoretical Physics. Vol. 3 (3rd ed. London: Pergamon Press). ISBN 0750635398.