1	SemiconductorCrystals Semiconductors: have electrical resistivities in the range of .01 to 10**9 ohm-cm at room temperature
2	Carrier concentrations: metals, semi-metals and semiconductors
3	Valence and Conduction Bands (very pure crystals) The conduction band is vacant at absolute zero and is separated by an energy gap Eg from the filled valence band The lowest point in the conduction band is called the conduction band edge; the high point in the valence band is called the valence band edge. The band gap is the difference between these two band edges.
4	Band structure: intrinsic conductivity in semiconductor
5	Role of temperature Thermal energy excites electrons from the valence band to the conduction band. The thermally excited electrons in the conduction band then contribute to the electrical conductivity of the semiconductor. The vacant orbitals or holes left behind in the valence band also contribute to the electrical conductivity!
6	Intrinsic carrier concentration and intrinsic conductivity These are largely controlled by the ratio of the band gap to the thermal energy. If this ratio is large, there will be few electrons excited. Therefore the concentration of the carriers will be low, which means the conductivity will also be small.
7	Temperature dependence of carriers: Germanium
8	Temperature Dependence of Electron Concentration: Silicon
9	Energy gap between valence and conduction bands
10	How does one measure the band gap? The best way to measure the value of the band gap for a semiconductor is via optical absorption. The threshold of continuous optical absorption at a frequency determines the band gap via
11	Direct absorption In a direct absorption process the crystal absorbs a photon with the creation of an electron and hole.
12	Direct absorption
13	Optical Absorption in InSb: Direct Transition
14	Indirect absorption process In this case the minimum energy gap involves electrons and holes that are separated by a non-negligible wavevector Hence one cannot have a direct photon transition at the minimum gap, as one cannot safisfy the requirement of conservation of “momentum” (or wavevector).
15	Indirect absorption
16	Indirect absorption process
17	Conservation laws
18	Equation of motion for k Consider a wave packet made of up wave functions near a particular wavevector k
19	Apply an external electric field E Consider the work done on the electron by an electric field E in a small time interval
20	Rate of change of k in a crystal Equating these two equations for energy:
21	Equation of motion Note that an electron in a crystal is subject to forces from the crystal (e.g. interactions with the ions), in addition to the external electromagnetic force F.
22	Effect of magnetic field on electron Consider an electron of group velocity v in a constant magnetic field B
23	Electron motion in constant B field The gradient of the energy is normal to the energy surface. The rate of change of k is in a direction normal to this gradient and to the magnetic field. Therefore, an electron moves in k space under the influence of a constant B field on a surface of constant energy.
24	An alternative derivation of equation of motion for k
25	Expectation value of momentum of electron in one dimension s
26	Generalization to three dimensions Consider the Bloch energy eigenfunction for an electron in three dimensions with wavevector k
27	Expectation value of momentum of electron in three dimensions
28	Transfer of momentum between electron and lattice Suppose we now apply an external force to the electron, changing its state from k to k + k We will assume the crystal is an insulator and Electrostatically neutral except for a single electron in the state k of a single band. The total impulse J given to the crystal system by a force F acting during a time interval t is J=F t
29	Momentum change The impulse will equal the total change in momentum of the system, namely the change of momentum of the lattice plus electron
30	Change in crystal lattice momentum Next, we need to calculate what the change in the lattice momentum is that results from the change in the state of the electron, due to the external force F We first note that an electron that is reflected by the lattice transfers momentum to the lattice, as we have seen earlier.
31	Physical argument Namely, if an incident electron with a plane wave component of momentum hk is reflected with momentum h(k+G), then the lattice gains the momentum –hG. If we extend this argument to a wavepacket, then there will be a contribution to the lattice momentum from each individual component of the wavepacket.
32	Lattice momentum change
33	Equation of motion for k Therefore the total change in momentum of the system is
34	Schematic picture of physics described by semiclassical model
35	Response to External E and B fields The semiclassical model describes the response of the electrons to external E and B fields that vary slowly over the dimensions of the wave packet describing the electron (and therefore slowly over a few primitive cells) Such E and B fields give rise to ordinary classical forces in the equations describing the position and wave vector of the wave packet However, the periodic potential of the lattice varies over dimensions that are small compared with the spread of the wave packet and cannot be treated classically.
36	Semiclassical Model The semiclassical model predicts how (in the absence of collisions) the position r and the wave vector k of each electron evolve in the presence of externally applied electric and magnetic fields. It assumes a knowledge of the band structure of the material, i.e. of energy eigenvalues as a function of k. We often label each band of the energy by a subscript integer n.
37	Semiclassical model (continued) The band index n is a constant of the motion. The semiclassical model ignores transitions between bands. In thermal equilibrium the average occupation number of the electrons is given by the Fermi-Dirac distribution function
38	Semiclassical: The equations of motion for position and wave vector 3. Given an energy band with band index n the equations of motion are
39	A filled band cannot contribute to electric or thermal currents A small phase space element dk about k contributes a term –e v(k) dk to the electrical current, where v(k) is the group velocity. Thus the total contribution over the band is
40	Only partially filled bands contribute to electrical current But one can prove that the integral over any primitive cell of the gradient of a periodic function must vanish. Therefore for a completely filled band the integral for both the electric and energy currents must vanish, as the energy eigenvalue is periodic in the reciprocal lattice.
41	Holes: 1 Since electrons in a volume element dk about k contribute a term proportional to -e v(k)dk to the current density, the contribution of all electrons in a given band is
42	The current from a fully occupied band is zero. Therefore
43	Holes: 2 The unoccupied levels in a band evolve in time under the influence of external applied fields precisely as if they were occupied by real electrons of charge –e. The reason is that given the values of k and r at t=0, the semiclassical equations of motion (six first-order differential eqns in 6 variables) uniquely determine their subsequent orbit in the presence of E and B. We can therefore separate the orbits into unoccupied or occupied orbits, depending on their states at t=0. (No two distinct orbits can intersect.)
44	Distinguish between electron and hole orbits
45	Wavevector of a hole
46	Equation of motion of holes
47	Hole band
48	Motion of electrons and holes
49	Motion of holes and electrons
50	Cyclotron resonance experiment
51	Band edge structure: direct gap semiconductor (simplified)
52	Effective masses: electrons and holes
53	Band structure of Germanium
54	Energy scale: (thermal energy much less than band gap energy)
55	Chemical Potential
56	Negative effective mass near a Brillouin zone boundary
57	Optical absorption: Indirect transition