.

Bloch oscillation is a phenomenon from solid state physics. It describes the oscillation of a particle (e.g. an electron) confined in a periodic potential when a constant force is acting on it. It was first pointed out by Bloch and Zener while studying the electrical properties of crystals. In particular, they predicted that the motion of electrons in a perfect crystal under the action of a constant electric field would be oscillatory instead of uniform. While in natural crystals this phenomenon is extremely hard to observe due to the scattering of electrons by lattice defects, it has been observed in semiconductor superlattices and in different physical systems such as cold atoms in an optical potential and ultrasmall Josephson junctions.

Derivation

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

\( \hbar \frac{dk}{dt}=-eE , \)

which has the solution

\( k(t)=k(0) - \frac{eE}{\hbar} t . \)

The velocity v of the electron is given by

\(v(k)=\frac{1}{\hbar}\frac{d\mathcal{E}}{dk} ,\)

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

\( \mathcal{E}(k)= A \cos{ak} , \)

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

\( v(k)=\frac{1}{\hbar}\frac{d\mathcal{E}}{dk}=-\frac{Aa}{\hbar}\sin{ak} \) ,

and the electron position x by

\( x(t)=\int{v(k(t))}{dt}= -\frac{A}{eE}\cos\left({\frac{aeE}{\hbar}t}\right) \) .

This shows that the electron oscillates in real space. The frequency of the oscillations is given by \( \omega_B = ae|E|/\hbar \).

.

Bloch oscillations