Bloch Tight-Binding Oscillations

Worked Example

Here we manually trace one step of the particle's evolution from k₀ = 0 to k(t) = 1 under F = 1.

1. Initialize the starting parameters: Hopping amplitude J = 1, force F = 1, and initial momentum k₀ = 0.
2. Calculate the momentum at time t = 1 using the force relation: k(t) = k₀ + F · t = 0 + 1 · 1 = 1.
3. Evaluate the resulting band energy from the dispersion relation: E = -2J cos(k) = -2(1)cos(1) = -1.08.
4. Calculate the corresponding group velocity: v = 2J sin(k) = 2(1)sin(1) = 1.68.
5. Determine the spatial displacement: x - x₀ = (2J / F) [cos(k₀) - cos(k₀ + F t)] = (2 / 1)[1 - cos(1)] = 0.92.

Experiment

Using the Bloch Tight-Binding Oscillations module, adjust the 'Force' control to 1, and watch how the 'Displacement' and 'Momentum phase' variables evolve with the 'Time' control. This demonstrates real-space oscillations instead of unbound acceleration.

1. Set the initial state by keeping Hopping at 1, Force at 1, Time at 0, and Initial momentum at 0. Observe the zero displacement and ground state energy E = -2.
2. Gradually advance the Time control. Notice how the particle accelerates initially but its velocity maxes out and then begins to drop as it climbs the energy band.
3. At Time π, observe that the group velocity drops to 0, representing the classical turning point in real space despite a constant forward force.
4. Reset Force to 0 and observe the uniform non-oscillatory motion, showing how the lack of force leads to ballistic transport at constant velocity.

Limitations

This semiclassical model strictly assumes isolated single-band dynamics and ignores interband Zener tunneling and scattering events. In a real crystal, sufficiently strong electric fields lead to breakdown.

References

For a comprehensive overview of Bloch oscillations and related semiclassical quantum dynamics, consult Graefe's Lisbon 2020 online seminar slides.

• Graefe Seminar Slides (2020) - Details the exact solutions to Bloch dynamics and standard tight-binding models.