1. Semiconductor crystals
M.C. Chang
Dept of Phys
Semiconductor crystals
Overview
The concept of hole and effective mass
Intrinsic semiconductor
Doped semiconductor

2. Families of semiconductors
Elements
Compounds
Bonding becomes more ionic

3. 2 overlapping hcp lattices
Basic properties
(at 300K) Ge Si GaAs GaN
energy gap (eV) 0.67 (i) (i) (d) 3.39 (d)
lattice type Diamond Diamond Zincblend Wurtzite
2 overlapping hcp lattices
Semiconductor is insulator at 0 K, but because of its smaller energy gap (insulator diamond = 5.4 eV), electrons can be thermally excited to the conduction band (and transport current) easily.
Si-based device can endure higher working temperature than Ge-based (75 oC) device ( Si has a larger band gap) For some interesting history of semiconductor industry, see
矽晶之火, by M.Riordan and L.Hoddeson.

4. Direct band gap (GaAs, GaN…) Indirect band gap (Si, Ge…)
=Eg
=Eg+ 
Direct band gap semiconductor can emit light efficiently
(1μm=1.24 eV)

5. A filled band does not carry current (Peierls, 1929)
Electric current density
For crystals with inversion symmetry,
εn(k)= εn(-k)
→ electrons with momenta ħk and -ħk have opposite velocities
→ no net current in equilibrium
A nearly-filled band
∴ unoccupied states behave as +e charge carriers

6. h k The concept of hole (Peierls, 1929)
If an electron of wavevector ke is missing, then k= -ke. Alternatively speaking, a hole with wavevector kh is produced (and kh= -ke).
The lower in energy the missing electron lies, the higher the energy of the whole system. If the energy of a filled valence band is set to zero, then h
The missing electron k

7. important
Effective mass
Near the bottom of a conduction band, the energy dispersion is approximately parabolic,
Effective mass matrix
The electron near band bottom is like a free electron (with m*).
For a spherical FS, m*ij=m*δij, only one m* is enough.
In general, electron in a flatter band has a larger m*.
Negative effective mass:
If ε(k) is (e.g. top of valence band), then m*<0
 electron (-e) with negative m* = hole (+e) with positive m*.

8. For ellipsoidal FS, there can be at most three different m. s Eg
For ellipsoidal FS, there can be at most three different m*s Eg. the FS of Si is made of six identical ellipsoidal pockets. L T
For Si, εg = 1.1 eV, mL = 0.9 m, mT = 0.2 m
(It’s more difficult for the electron to move along L The mass is larger because the band is flatter along that direction.)

9. More band structures and Fermi surfaces
Common features
Some useful parameters
mL /mT mHH/mLH Δ
Si 0.91/ / eV
GaAs / eV

10. Overview
The concept of hole and effective mass
Intrinsic semiconductor (no doping)
Doped semiconductor

11. DOS and carrier density
DOS for free electron (ch 6)
DOS (per V) for semicond
Fermi distribution
carrier density
Top of valence band is set as ε=0

12. Consider the “non-degenerate” case: For electrons,ε-μ >> kBT
For holes, μ- ε>> kBT
electron density in conduction band:
hole density in valence band:
Valid even with doping

13. Carrier density and energy gap
In intrinsic semiconductor,
Chemical potential
The density of intrinsic carriers depends only on the energy gap.
The 2nd term is very small because kBT<<EG
(For Si, the atom density is 5×1022cm-3.)

14. Extrinsic carriers by doping

15. Impurity level: Bohr atom model
The ionized impurity atom has a hydrogen-like potential,
(m → me, ε0 → ε)
Dielectric constant of Si = 11.7 (Ge=15.8, GaAs=13.13),
Effective mass for Si = 0.2 m.
Therefore, the donor ionization energy = 20 meV.
Bohr radius of the donor electron:
For Si, it's about 50 A (justifies the use of a constant ε).

16. Conduction band
Valence band
Impurity levels in Si
Energy-band point of view

17. Light emitting diode

19. Solid state lighting
From S. Nakamura’s slide

20. The invention of blue-light LED
GaN can emit blue light because of its large band gap (3.4 eV).
First blue-light LED: Shuji Nakamura 1989.
First blue-light laser: Shuji Nakamura 1997.
中村修二
Before
2006
After
See a nice “Interview with Nakamura”: Scientific American, July, 2000

21. UV Water Purifier Blade runner (1982) LED-backlit LCD TV
Kerosene lighting and firewood are used by 1/3 of the world; they cause countless fires and are very inefficient.
UV Water Purifier
LED-backlit LCD TV
Blade runner (1982)