Bloch oscillations

Derivation

The one-dimensional equation of motion for an electron in a constant electric field E is:

ℏ d k d t = − e E ,

which has the solution

k ( t ) = k ( 0 ) − e E ℏ t .

The velocity v of the electron is given by

v ( k ) = 1 ℏ d E d k ,

where E ( k ) denotes the dispersion relation for the given energy band. Suppose that the latter has the (tight-binding) form

E ( k ) = A cos ⁡ a k ,

where a is the lattice parameter and A is a constant. Then v(k) is given by

v ( k ) = 1 ℏ d E d k = − A a ℏ sin ⁡ a k ,

and the electron position x by

x ( t ) = ∫ 0 t v ( k ( t ′ ) ) d t ′ = x ( 0 ) − A e E cos ⁡ ( a e E ℏ t ) .

This shows that the electron oscillates in real space. The angular frequency of the oscillations is given by ω B = a e | E | / ℏ .

Discovery and experimental realizations

Bloch oscillations were predicted by Nobel laureate Leo Esaki in 1970. However, they were not experimentally observed for long as in natural solid state bodies ω B is even with very high electric field strengths not sufficiently large to allow for full oscillations of charge carriers within the diffraction and tunneling times due to relatively small lattice periods. The development in semiconductor technology has recently led to the fabrication of structures with sufficiently super lattice periods that are now sufficiently large, based on artificial semiconductors. The oscillation period in those structures is smaller than the diffraction time of the electrons, hence more oscillations can be observed in a time window below the diffraction time. For the first time the experimental observation of Bloch oscillations in such super lattices at very low temperatures was shown by Jochen Feldmann and Karl Leo in 1992. Other realizations were