1. Bloch Oscillations Alan Wu April 14, 2009 Physics 138

2. Outline Phenomenon Description Semi-Classical Derivation Wannier-Stark States Implications and Applications –Terahertz Oscillations –Bloch Oscillation Transistors

3. Bloch Oscillation Phenomenon Described by Bloch (1928) Imagine a particle in a periodic potential acted on by a constant force. Example: electrons in crystal lattice exposed to constant electric field Classically, we expect Ohmic behavior

4. Bloch Oscillation Phenomenon But quantum mechanics predicts that the particle will undergo an oscillation The periodicity causes the group velocity of the wavefunction to oscillate Ohmic behavior results from scattering

5. Bloch Oscillation Frequency Use invariance: shift 1 period d and shift energy ΔE Phase shift now Corresponding frequency is

6. Semi-Classical Derivation Schrodinger’s Equation can be transformed into the form: Known as the Acceleration Theorem, since it describes change in momentum Like classical relation between momentum and force

7. K-Space in a Lattice Potential periodicity in real space => periodicity in k-space Also known as a reciprocal lattice

8. The Brillouin Zone Brillouin Zone: a basic cell in the reciprocal lattice The dispersion relation gives an oscillating k within this zone

9. Kronig Penney Model The Kronig Penney model for a lattice can be used to find the potential in k-space. Source: http://fermi.la.asu.edu/schmidt/applets/kp/plugkp.html

10. Dispersion in lattice

11. Wannier-Stark Resonance States At each well, a series of energies are available, much like that of a harmonic resonator. These states form what is known as a Wannier-Stark energy ladder.

12. Tight-binding model Consider just interactions between neighboring wells (known as Wannier representation) Also have energy difference from constant force

13. Experimental Confirmations Bloch oscillations have been observed in semiconductor lattices Shining a laser will excite the Wannier Stark states, which then oscillate. These oscillations can be measured

14. Terahertz Radiation Changing the electric field allows for a tunable radiation source. Can get frequencies in the terahertz

15. Bloch Oscillation Transistors Bloch oscillations can control Josephson Junctions Act much like bipolar transistors

16. Conclusion Bloch oscillations are just another strange quantum phenomenon They can be used for frequencies in the terahertz range Bloch oscillator transistors are an interesting way of amplifying signals