Landau level

Allowed states of particles in transverse momentum space and the classic spiral path of a particle in spatial space

The Landau levels (after Lew Dawidowitsch Landau ) represent a quantification of the energy of charged particles that move in homogeneous magnetic fields . It can be shown that the energy of a charged particle of mass (e.g. an electron ) and charge moving parallel to a magnetic field in the direction is as follows: ${\displaystyle m}$ ${\displaystyle e}$ ${\displaystyle B}$ ${\displaystyle z}$

${\displaystyle E(n,p_{z})=\hbar \omega _{c}\left(n+{\frac {1}{2}}\right)+{\frac {p_{z}^{2}}{2m}},\ \ \ \ \ n\in \mathbb {N} _{0},p_{z}\in \mathbb {R} .}$

It is the (unquantized) momentum of the particle in direction, the cyclotron frequency , and the reduced Planck's constant . If the charged particle also has a spin , this leads to an additional splitting of the levels according to the quantum number for the component (= magnetic field direction) of the spin: ${\displaystyle p_{z}}$ ${\displaystyle z}$ ${\displaystyle \omega _{c}=e\cdot B/m}$ ${\displaystyle \hbar }$ ${\displaystyle \sigma _{z}}$ ${\displaystyle z}$

${\displaystyle E(n,p_{z},\sigma _{z})=\hbar \omega _{c}\left(n+{\frac {1}{2}}\right)+{\frac {p_{z}^{2}}{2m}}-{\frac {e\hbar }{2m}}\cdot \sigma _{z}\;B}$

This means that (as indicated on the right in the figure) only certain particle trajectories are allowed, which are characterized by the two quantum numbers and (and possibly the spin ). One can also imagine the movement in such a way that the particle spreads freely longitudinally and transversely (radially) executes a harmonic oscillation movement (see harmonic oscillator (quantum mechanics) ). Overall, this corresponds to a helical path around the magnetic field lines. In the transversal momentum space (only - component) the movement is limited to a circle for each quantum number , in the 3-dimensional momentum space the states are therefore on cylinders (Landau cylinder). ${\displaystyle p_{z}}$ ${\displaystyle n}$ ${\displaystyle \sigma _{z}}$ ${\displaystyle p_{x}}$ ${\displaystyle ,p_{y}}$ ${\displaystyle n}$

The division into Landau levels can be measured in solid-state physics , for example ( De Haas van Alphen effect ). There the transverse impulses are quantized due to the crystal lattice. It can then be shown that there are exactly the same number of states on each Landau cylinder.

Theoretical derivation using the Schrödinger equation

The derivation shown here is based on the references and the original work.

Requirements and task

Consider a simple situation: A particle of mass and charge is in a homogeneous magnetic field that has only one component in the direction. This field can also be represented by the following vector potential : ${\displaystyle m}$ ${\displaystyle q}$ ${\displaystyle {\vec {B}}({\vec {r}})=\left(0,0,B\right)}$ ${\displaystyle z}$ ${\displaystyle {\vec {A}}({\vec {r}})}$

${\displaystyle {\vec {A}}({\vec {r}})=B\cdot {\begin{pmatrix}0\\x\\0\end{pmatrix}}.}$

One can easily show that this results in the above magnetic field again. ${\displaystyle {\vec {B}}={\vec {\nabla }}\times {\vec {A}}}$

One then obtains the (initially still classical) Hamilton function of this system:

${\displaystyle H({\vec {r}},{\vec {p}})={\frac {1}{2m}}\left[{\vec {p}}-q\cdot {\vec {A}}({\vec {r}})\right]^{2}={\frac {1}{2m}}\left[p_{x}^{2}+(p_{y}-qBx)^{2}+p_{z}^{2}\right]={\frac {1}{2}}m{\vec {V}}^{2}.}$

By replacing the position and momentum variables with the corresponding quantum mechanical operators (→ correspondence principle ), one obtains the Hamiltonian of the system. In the last part of the equation above, a speed (in the Hamilton operator a "speed operator") was defined which has the following form:

${\displaystyle {\vec {V}}={\frac {1}{m}}\left[{\vec {p}}-q\cdot {\vec {A}}({\vec {r}})\right]={\frac {1}{m}}{\begin{pmatrix}p_{x}\\p_{y}-q\cdot B\cdot x\\p_{z}\end{pmatrix}}.}$

From the classical treatment, we know that the solution of the problem a helical motion ( Helix movement , see figure above) in is direction. That is why it makes sense (which will also be shown in the later calculations), the following division of the Hamilton operator into a longitudinal (along the direction of the magnetic field) and a transverse part (in the classic view, a rotary movement takes place in this plane which leads to a screw movement): ${\displaystyle z}$

${\displaystyle {\hat {H}}={\hat {H}}_{\bot }+{\hat {H}}_{\|},\ \ \ \ \ {\mbox{mit:}}\ \ \ {\hat {H}}_{\bot }={\frac {m}{2}}\left({\hat {V}}_{x}^{2}+{\hat {V}}_{y}^{2}\right),\ \ \ {\hat {H}}_{\|}={\frac {m}{2}}{\hat {V}}_{z}^{2}.}$

The following commutation relation is obtained for the "speed operator" : ${\displaystyle {\hat {\vec {V}}}}$

${\displaystyle \left[{\hat {V}}_{x},{\hat {V}}_{y}\right]={\frac {1}{m^{2}}}\left(\left[{\hat {P}}_{x},{\hat {P}}_{y}\right]-qB\cdot \left[{\hat {P}}_{x},{\hat {X}}\right]\right)=i\hbar {\frac {qB}{m^{2}}}=-i\cdot {\frac {\hbar \omega _{c}}{m}}.}$

The cyclotron frequency was used for this. Furthermore, one can see in the definition of easy that ${\displaystyle \omega _{c}=-{\frac {q\cdot B}{m}}}$ ${\displaystyle {\hat {\vec {V}}}}$

${\displaystyle \left[{\hat {V}}_{x},{\hat {V}}_{z}\right]=\left[{\hat {V}}_{y},{\hat {V}}_{z}\right]=0.}$

This also interchanges and with each other and there is a basis of common eigenvectors for and . ${\displaystyle {\hat {H}}_{\bot }}$ ${\displaystyle {\hat {H}}_{\|}}$ ${\displaystyle {\hat {H}}_{\bot }}$ ${\displaystyle {\hat {H}}_{\|}}$

Eigenvalues ​​of H _||

The following exchange relation applies:

${\displaystyle m\cdot \left[{\hat {Z}},{\hat {V}}_{z}\right]=\left[{\hat {Z}},{\hat {P}}_{z}\right]=i\hbar .}$

A theorem about operators that swap according to the above relation (that is, swap like the canonical position and momentum operators) is applicable and we can conclude that has a continuous spectrum of eigenvalues . Furthermore, all eigenvectors of are also eigenvectors to . The energy eigenvalues of can thus be written in the following form: ${\displaystyle {\hat {V}}_{z}}$ ${\displaystyle v_{z}}$ ${\displaystyle {\hat {V}}_{z}}$ ${\displaystyle {\hat {H}}_{\|}}$ ${\displaystyle E_{\|}}$ ${\displaystyle {\hat {H}}_{\|}}$

${\displaystyle E_{\|}={\frac {m}{2}}v_{z}^{2}.}$

Thus, in analogy to classical mechanics, describes the free propagation of a particle in the direction. ${\displaystyle {\hat {H}}_{\|}}$ ${\displaystyle z}$

Eigenvalues ​​of H _⊥

In order to obtain the energy eigenvalues ​​of (and thus the so-called Landau levels ), one introduces the following operators with their commutation relation: ${\displaystyle H_{\bot }}$

${\displaystyle {\hat {Q}}={\sqrt {\frac {m}{\hbar \omega _{c}}}}\cdot {\hat {V}}_{y},\ \ \ \ \ \ {\hat {S}}={\sqrt {\frac {m}{\hbar \omega _{c}}}}\cdot {\hat {V}}_{x},\ \ \ \ \ \left[{\hat {Q}},{\hat {S}}\right]={\frac {m}{\hbar \omega _{c}}}\cdot \left[{\hat {V}}_{y},{\hat {V}}_{x}\right]=i.}$

It then has the form of a quantum harmonic oscillator that oscillates with the cyclotron frequency ω _c . ${\displaystyle {\hat {H}}_{\bot }}$

${\displaystyle {\hat {H}}_{\bot }={\frac {\hbar \omega _{c}}{2}}\left({\hat {Q}}^{2}+{\hat {S}}^{2}\right)}$

The energy eigenvalues ​​of are therefore ${\displaystyle H_{\bot }}$

${\displaystyle E_{\bot }(n)=\left(n+{\frac {1}{2}}\right)\hbar \omega _{c},\ \ \ \ \ n\in \mathbb {N} _{0}.}$

Eigenvalues ​​of H

The total energy results from the sum of the natural energies of and : ${\displaystyle {\hat {H}}_{\|}}$ ${\displaystyle {\hat {H}}_{\bot }}$

${\displaystyle E(n,v_{z})=\left(n+{\frac {1}{2}}\right)\hbar \omega _{c}+{\frac {1}{2}}\;m\;v_{z}^{2},\ \ \ \ \ n\in \mathbb {N} _{0},v_{z}\in \mathbb {R} .}$

These levels are known as the Landau levels. They are infinitely degenerate due to the continuous speed spectrum .

Depending on the applied magnetic field, different level distances are obtained for a fixed speed : ${\displaystyle v_{z}}$

${\displaystyle \Delta E=\hbar \omega _{c}={\frac {q\cdot B\cdot \hbar }{m}}.}$

additional

It can be shown that the degeneracy of the Landau levels proportional to the magnetic flux density is . With the above knowledge that the level differences are also proportional , one can explain the oscillations that occur in the De Haas van Alphen effect in physical quantities that depend on the density of states: If the magnetic field is increased, the energy of the increases Landa levels rise while at the same time their degeneration increases. Electrons will therefore migrate to a lower level. Therefore, if the top Landau level initially occupied (i.e. the former Fermi level) has been completely emptied, the next lower Landau level suddenly becomes the Fermi level. ${\displaystyle D}$ ${\displaystyle D\propto B}$ ${\displaystyle \Delta E=\hbar \omega _{c}={\frac {e\cdot B\cdot \hbar }{m}}}$ ${\displaystyle B}$

literature

• L. Landau : Diamagnetism of Metals . In: Journal of Physics . tape 64 , no. 9-10 , September 1930, ISSN  1434-6001 , pp. 629-637 , doi : 10.1007 / BF01397213 ( online ).
• LD Landau, EM Lifschitz : Quantum Mechanics: Non-relativistic theory 3rd edition, Pergamon Press, Oxford, 1977, pp. 455ff
• Claude Cohen-Tannoudji , Bernard Diu, Franck Laloë: Quantum Mechanics 1st 3rd edition. Walter de Gruyter, Berlin 2005, ISBN 3-11-013592-2 , p. 700.
• Charles Kittel: Introduction to Solid State Physics. Oldenbourg, Munich 2005, ISBN 3-486-57723-9 .

Individual evidence

1. ↑ ^{a b} L. Landau : Diamagnetism of Metals . In: Journal of Physics . tape 64 , no. 9-10 , September 1930, ISSN  1434-6001 , pp. 629-637 , doi : 10.1007 / BF01397213 .
2. ↑ LD Landau, EM Lifschitz : Quantum Mechanics: Non-relativistic theory 3rd edition, Pergamon Press, Oxford, 1977, pp. 455ff
3. ^ Claude Cohen-Tannoudji , Bernard Diu, Franck Laloë: Quantum Mechanics 1st 3rd edition. Walter de Gruyter, Berlin 2005, ISBN 3-11-013592-2 .
4. ↑ Kittel, Solid Body Physics, Edition 9, p. 286.
5. ^ Kittel, Solid Body Physics, Edition 9, p. 287.