Set of supplementary notes.

More documents

Recommendations

Info

82 CHAPTER 5. BANDSTRUCTURE OF REAL MATERIALS Figure 5.9: Band energy E(k) (solid line) and group velocity v(k) dashed line in a simple 1D band. A wavepacket progressing its crystal momentum according to (5.17) accelerates as k increases from zero, and then slows and reverses direction as k approaches the zone boundary. so that the wave packet of electrons oscillates up and down the energy surface. It we start from the minimum of the band, then the group velocity grows linearly in time as for a free electron accelerating (though with a mass different from the free electron mass). However, on approaching the zone boundary, the group velocity slows - the acceleration of the particle is opposite to the applied force. What is actually happening is buried within the semiclassical model via the dispersion ɛ(k): as the wavepacket approaches the Brillouin zone boundary, real momentum (not crystal momentum k) is transferred to the lattice, so that on reaching the zone boundary the particle is Bragg-reflected. Thus a DC electric field may be used - in principle - to generate an AC electrical current. All attempts to observe these Bloch oscillations in conventional solids has so far failed. The reason is that in practice it is impossible to have wavepackets reach such large values of momentum as π/a due to scattering from impurities and phonons in the solid. We will incorporate scattering processes in the theory in a moment. It turns out however, that one can make artificial periodic potentials in a semiconductor superlattice. The details of this process will be discussed later, but for our purposes the net effect is to produce a square well potential that is periodic with a periodicity that can be much longer than the atom spacing. The corresponding momentum at the zone boundary is now much smaller, so the wavepacket does not have to be excited to such high velocities. The signature of the Bloch oscillations is microwave radiation produced by the oscillating charge - at a frequency that is proportional to the DC electrical field.
5.3. SEMICLASSICAL DYNAMICS 83 Figure 5.10: Schematic diagram of energy versus space of the conduction and electron bands in a periodic heterostructure lattice. The tilting is produced by the applied electric field. The levels shown form what is called a Wannier-Stark ladder for electron wavepackets made by excitation from the valence band in one quantum well, either vertically (n=0) or to neighbouring (n = ±1) or next-neighbouring n = ±2 wells of the electron lattice. In the experiment, electrons (and holes) are excited optically by a short-pulse laser whose frequency is just above the band gap of the semiconductor (i.e. a few ×10 15 Hz). The electrical radiation (on a time scale of picoseconds) is monitored as a function of time and for different DC electrical biases, shown on the left panel. The spectral content is then determined by taking a fourier transform of the wavepackets (right panel); at large negative voltages one sees a peak at a frequency that increases with increasing bias. The device is not symmetric, and therefore has an offset voltage of about -2.4 V before the Bloch oscillation regime is reached. [From Waschke et al. Physical Review Letters 70, 3319 (1993).] Approximations and justification for the semiclassical model A full justification of the semiclassical model is not straightforward and we will not go into that here. [See Kittel, Appendix E, and for a more formal treatment J. Zak, Physical Review 168 686 (1968)] • Note that at least the semiclassical picture takes note of the fact that the Bloch states are stationary eigenstates of the full periodic potential of the lattice, and so there are no collisions with the ions. • We must be actually describing the motion of a wavepacket ψ n (r, t) = ∑ k ′ g(k − k ′ )ψ nk ′(r, t) exp [−iɛ n (k ′ )t/h¯] where g(k) → 0 if |k| > ∆k (5.18) The wavepacket is described by a function g(k) that is sharply peaked, of width ∆k, say. ∆k ≪ 1/a, with a the lattice constant (otherwise the packet will disperse strongly). Clearly
Page 1:
Quantum Condensed Matter Physics Pa
Page 4 and 5:
4 CONTENTS 5.1 Bands and Brillouin
Page 6 and 7:
6 CONTENTS also serves to give impo
Page 8 and 9:
8 CONTENTS
Page 10 and 11:
10 CHAPTER 1. CLASSICAL MODELS FOR
Page 12 and 13:
Page 14 and 15:
Page 16 and 17:
Page 18 and 19:
Page 20 and 21:
Page 22 and 23:
Page 24 and 25:
24 CHAPTER 2. SOMMERFELD THEORY Fig
Page 26 and 27:
26 CHAPTER 2. SOMMERFELD THEORY 2.4
Page 28 and 29:
28 CHAPTER 2. SOMMERFELD THEORY Res
Page 30 and 31:
30 CHAPTER 2. SOMMERFELD THEORY Thi
Page 32 and 33: 32 CHAPTER 2. SOMMERFELD THEORY
Page 34 and 35: 34 CHAPTER 3. FROM ATOMS TO SOLIDS
Page 56 and 57: 56 CHAPTER 4. ELECTRONIC STRUCTURE
Page 74 and 75: 74 CHAPTER 5. BANDSTRUCTURE OF REAL
Page 88 and 89: 88 CHAPTER 6. EXPERIMENTAL PROBES O
Page 100 and 101: 100 CHAPTER 7. SEMICONDUCTORS Figur
Page 102 and 103: 102 CHAPTER 7. SEMICONDUCTORS is fo
Page 104 and 105: 104 CHAPTER 7. SEMICONDUCTORS Figur
Page 106 and 107: 106 CHAPTER 8. SEMICONDUCTOR DEVICE
Page 116 and 117: ecessitates variances in circuit th
Page 118 and 119: nc... Freescale Semiconductor, I
Page 124 and 125: 124 CHAPTER 9. ELECTRONIC INSTABILI
Page 132 and 133:
132 CHAPTER 9. ELECTRONIC INSTABILI
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
146 CHAPTER 10. FERMI LIQUID THEORY
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158:
show all

Set of supplementary notes.

Create successful ePaper yourself

Delete template?

Save as template?