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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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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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