What is the physical meaning of magnetic length $\ell_B=\frac{\hbar c}{e B}$ in 2D electron system under magnetic field? When $\ell_B \longrightarrow a$, where $a$ is the lattice constant, does that mean the Landau sub-band is nearly flat?
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3 Answers
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First of all, the expression for the magnetic length that you give is wrong: there is a square root missing: $l_B=\sqrt{\frac{\hbar c}{eB}}$.
Secondly, to understand the meaning, you don't really need to think about lattices or phases of the electron wavefunction as the previous answers would have it. Instead, begin by thinking about a motion of a classical charged particle in a magnetic field. If it has an initial velocity in the direction perpendicular to the field, it will move in a circle whose radius is found from the second Newton's law: $m v^2/R = \frac{e}{c}vB \Rightarrow R=\frac{mvc}{eB}$ - the so-called Larmor radius.
Now let's ask ourselves, what would be the smallest radius allowed by the uncertainty principle? After all, the Larmor radius $R\propto mv=p$, but the uncertainty principle states that the two cannot be arbitrarily small simultaneously. In fact, the uncertainties in the particle's position and momentum are bounded by $\Delta x \Delta p \geq \hbar/2$. For a particle moving around a circle, $\Delta x = 2R$ while $\Delta p = 2mv$. In order to minimise the uncertainty we should replace $mv \to \hbar/R$ in the expression for the Larmor radius (I am ignoring the factor of 8 here - but we are after the physical meaning, not the exact numbers here). This gives $R^2=\frac{\hbar c}{eB}$ - which is the expression for the magnetic length! In other words, the magnetic length $l_B$ has the physical meaning of the smallest size of a circular orbit in a magnetic field which is allowed by the uncertainty principle.
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The physical meaning is the length of electron trajectory along which this electron gains phase factor comparable with $2\pi$ from the magnetic field. Normally, it is rather large. When is is as small as lattice constant, it means that the magnetic field is comparable with electric field in the atom which is rather rare situation.
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There is a deeper meaning of the magnetic length $l_B$. It is the lattice constant of a two-dimensional (2D) artificial structure. Let's consider a particle moving in 2D plane under a perpendicular uniform magnetic field of strength $B$. In classical mechanics, the system is obviously translationally invariant under any translation. However, in quantum mechanics, it is not so, because what enters the Schrodinger equation is the corresponding vector potential, which breaks this symmetry. Nevertheless, it is still possible to define an artificial lattice with primitive vectors $\vec{a}$ and $\vec{b}$, so that the translation by any $\vec{R} = m\vec{a} + n\vec{b}$ remains a symmetry, provided the area of the unit cell is $|\vec{a}\times\vec{b}|=l^2_B$. This means that, $l_B$ measures the lattice constant of this artificial lattice.
For more details, please see the book The Geometric Phase in Quantum Systems by A. Bohm et al. Also, this paper is very useful: PRL49:405(1982).
