1. Semiconductor crystals
Overview
The concept of hole and effective mass
Intrinsic semiconductor
Doped semiconductor
p-n junction
Refs for history:
半導體的故事, by 李雅明
矽晶之火, by M.Riordan and L.Hoddeson
M.C. Chang
Dept of Phys

2. Elements Compounds Bonding becomes more ionic (at 300K) Ge Si GaAs GaN
energy gap (eV) 0.67 (i) (i) (d) 3.39 (d)
lattice type Diamond Diamond Zincblend
Semiconductor is insulator at 0 K, but because of its smaller energy gap (diamond = 5.4 eV), electrons can be thermally excited to the conduction band more easily.
Si-based device can endure higher working temperature than Ge-based (75 oC) device (Si has a larger band gap) For history of semiconductor industry, see 矽晶之火

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

4. 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
發明前
中村秀二
發明後
See Interview with Nakamura: Scientific American, July, 2000

6. Overview
The concept of hole and effective mass
Intrinsic semiconductor
Doped semiconductor
p-n junction

7. A filled band does not carry current (Peierls, 1929)
Electric current density
For crystals with inversion symmetry,
n(k)= n(-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

8. The concept of holes (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 k
The missing electron

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

10. 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, Eg = 1.1 eV, mL = 0.9 m, mT = 0.2 m
(it’s more difficult for the electron to move along the L direction because the band is flatter along that direction)

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

12. Overview
The concept of hole and effective mass
Intrinsic semiconductor
Doped semiconductor
p-n junction

13. Density of states and carrier density
At room temperature, ε-μ >> kBT， electron density in conduction band:
Top of valence band is set as E=0
hole density in valence band:

14. Carrier density and energy gap Chemical potential
In intrinsic semiconductor,
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.)

15. Extrinsic carriers by doping

16. Impurity level: Bohr atom model
The ionized impurity atom has hydrogen-like potential,
(m  m*, 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)

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

18. The law of mass action
When the doping density >> ni, we can simply ignore ni (for majority carrier). Then the density of minority carrier is determined by the law of mass-action.

19. Change of chemical potential

20. Overview
The concept of hole and effective mass
Intrinsic semiconductor
Doped semiconductor
p-n junction (extra)

21. Equilibrium
B.G. Streetman, Solid State Electronic Devices

22. Equilibrium With external bias Rectification
B.G. Streetman, Solid State Electronic Devices