1. 3.1.4 Direct and Indirect Semiconductors
the real band structure in 3D is calculated with various numerical methods
the lattice is assumed to be a perfectly periodic lattice
the space dependent wave function is assumed to be a plane wave:
the allowed energies E are plotted vs k.
k is called the wave vector, or propagation constant
- for electron transition, both E and p must be conserved.
See Appendix III for more data on semiconductor materials

2. Figure 3—5
Direct and indirect electron transitions in semiconductors: (a) direct transition with accompanying photon emission; (b) indirect transition via a defect level.
Direct semiconductors are suitable for making light-emitting devices, whereas the indirect semiconductors are not.
- A semiconductor is indirect if the maximum of the valence band and the minimum of the conduction band do not have the same k value
- A semiconductor is direct if the maximum of the valence band and the minimum of the conduction band do have the same k value

3. 3.1.5 Variation of Energy Bands with Alloy Composition
We can tailor the band gap, and hence the wavelength of emitted photons in LEDs.
Figure 3—6
Variation of direct and indirect conduction bands in AIGaAs as a function of composition: (a) the ( E,k) diagram for GaAs, showing three minima in the conduction band; (b) AIAs band diagram; (c) positions of the three conduction band minima in AI x Ga 1- x As as x varies over the range of compositions from GaAs ( x = 0) to AIAs ( x = 1). The smallest band gap, E g (shown in color), follows the direct band to x = 0.38, and then follows the indirect X band.

4. 3.2 Charge Carriers in Semiconductors
3.2.1 Electrons and Holes
Ec the bottom of the conduction band
Ev the top of the valence band
EHP electron-hole pair
At 0K, a semiconductor is an insulator with no free charge carriers
At T > 0K, some electrons in the valence band are excited to the conduction band
The electrons in the conduction band are free to move about via many available states
An empty state in the valence band is referred as a hole

5. The total current in a volume with N electrons
The concept of hole
The total current in a volume with N electrons
The total current with the jth electron missing
A valence band (E vs k ) diagram
with all states filled
The net result: a positive charge moving with velocity vj
A hole is an imaginary positive charge moving in the valence band
The energy of a hole increases downward in a normal band diagram
The total current flow in a semiconductor is the sum of electron current and hole current

6. Figure 3—8
A valence band with all states filled, including states j and j’, marked for discussion. The jth electron with wave vector kj is matched by an electron at j’ with the opposite wave vector -kj . There is no net current in the band unless an electron is removed. For example, if the jth electron is removed, the motion of the electron at j’ is no longer compensated.

7. Figure 3—9
Superimposition of the ( E,k) band structure on the E-versus-position simplified band diagram for a semiconductor in an electric field. Electron energies increase going up, while hole energies increase going down. Similarly, electron and hole wave vectors point in opposite directions and these charge carriers move opposite to each other, as shown.

8. 3.2.2 Effective Mass
—The effective mass of an electron in a band with a given (E, k) relationship is defined as
(3-3)
For free electrons,
 m* = m
The effective mass is inversely proportional to the curvature of the band
The electrons near the top of the valence band have negative effective mass
In general m* is different in each direction and is a tensor; appropriate averages are
needed for various calculation purposes (e.g. density of state effective mass vs
conductivity effective mass, section 3.4.1)
The introduction of m* will simplify calculations
electron effective mass is denoted by mn*
hole effective mass is denoted by mp*

9. Figure 3—10
Realistic bandstructures in semiconductors: (a) Conduction and valence bands in Si and GaAs along [111] and [100]; (b) ellipsoidal constant energy surface for Si, near the 6 conduction band minima along the X directions. (From Chelikowsky and Cohen, Phys. Rev. B14, 556, 1976).

10. 3.2.3 Intrinsic Semiconductor
—a perfect semiconductor crystal with no impurities or lattice defects
n  conduction band electron concentration
(electrons per cm3)
p  valence band hole concentration
n=p=ni
(3-6)
ri  recombination rate of EHP; gi  generation rate
n0 , p0  concentrations at equilibrium; r  constant
(3-7)
ri=rn0p0 = rni2 =gi
EHP generation in an
intrinsic semiconductor

11. Figure 3—11
Electron–hole pairs in the covalent bonding model of the Si crystal.

12. 3.2.4 Extrinsic Semiconductor
 a doped semiconductor crystal whose equilibrium carrier concentrations n0 and
p0 are different from the intrinsic carrier concentration ni
The consequences of doping
new donor or acceptor levels are created in the band gap
conductivities can be vastly increased
(n0 or p0 >> ni )
semiconductor becomes either n-type or p-type
(either n0 >> p0 or p0 >> n0 )
For Si and Ge
Group V elements such as As, P, Sb are donor impurities
Group III elements such as B, Al, Ga and In are acceptor impurities

13. Figure 3—12
Energy band model and chemical bond model of dopants in semiconductors: (a) donation of electrons from donor level to conduction band; (b) acceptance of valence band electrons by an acceptor level, and the resulting creation of holes; (c) donor and acceptor atoms in the covalent bonding model of a Si crystal.

14. The donor binding energy for GaAs—an example
From Bohr model, the ground state energy of an “extra” electron of the donor is
(3-8)
Compare with the room temperature (300K) thermal energy E=kT≈26meV
All donor electrons are freed to the conduction band (ionized)
Compare with the intrinsic carrier concentration in GaAs (ni=1.1 x 106 /cm3)
We will have an increase in conduction electron concentration by 1010 if we dope
GaAs with 1016 S atoms/cm3

15. Figure 3—13
Energy band discontinuities for a thin layer of GaAs sandwiched between layers of wider band gap AIGaAs. In this case, the GaAs region is so thin that quantum states are formed in the valence and conduction bands. Electrons in the GaAs conduction band reside on “particle in a potential well” states such a E 1 shown here, rather than in the usual conduction band states. Holes in the quantum well occupy similar discrete states, such a E h .