Chapter 3 - Carriers in Semiconductor
ECE 4339
Han Q. Le
(copyrighted) U. of Houston
Part 3
0. Physical constants or frequently used formulas
8. App: Carrier concentration and distribution at thermal equilibrium
8.1 Calculations for some semiconductors
App Demo: Carrier Concentration and Distribution at Thermal Equilibrium
8.2 Example 1: GaAs
Refer to this website (or you can find other
similar websites) to look up for semiconductor properties:
http://www.ioffe.ru/SVA/NSM/Semicond/
Find the density of state Nc, Nv, and find the
intrinsic carrier concentration 
for GaAs at 300 K. Plot 
from 100 K to 300 K. Plot and compare 
for GaAs with Si
8.1.1 Find Nc:
(Tutorial in Mathematica)
Hence: 
8.1.2 Find Nv (see exercise/HW)
8.1.3 Find ni (see exercise/HW)
8.1.4 Plot intrinsic carrier concentration (see exercise/HW)
Neglecting temperature dependence of the band gap (see app for T-dependence). Below is the answer.
8.3 Comparing intrinsic GaAs and Si
8.3.1 Calculation for silicon:
For Si, we can use the same formulas, but with a modification for the Si conduction band peculiarity:
From: http://www.mtmi.vu.lt/pfk/funkc_dariniai/quant_mech/bands.htm
It has 6 valleys: we call it 6-fold degeneracy.
Also, the energy surface is not spherical: it has different
effective mass in different direction. A good approximation of the
spatially averaged mass is:
Alternatively, we can define an DOS effective mass that takes into account anisotropy and degeneracy:
8.3.2 Comparison
8.4 Example 2: p-dope and n-dope
A piece of Si is doped with boron up to 
/
.
Assume that all boron atoms are ionized at 300 K.
Is it p-type or n-type?
http://demonstrations.wolfram.com/DopedSiliconSemiconductors/
Contributed by: S. M. Blinder
What is the free electron concentration? (find 
for boron-doped Si )
Since 
>> 
, approximate
= 
= 10^16 ;
Alternative answer: by looking up, the value of 
for Si at 300 K is 
8.5 Example 3: Find po for Sb-doped Ge (see exercise/HW)
8.5.1 Solving for carrier concentrations
A piece of Ge is doped with Sb at 
.
Is it n-type or p-type?
Assume that acceptor doping is insignificant, what is the electron
concentration?
Since 
is unspecified, it is negligible: 0
Condition of charge neutrality:
Equilibrium:
How many Eqs. do we have? How many unknown?
8.5.2 Plot electron concentration as a function of doping (see exercise/HW)
Plot the electron concentration calculated above as a function of doping from 10^12 to 5*10^14 at 300 K
9. Fermi level across different materials
9.1 Overview of concept
9.2 Example (see exercise/HW)
A Te-doped GaAs with 
density layer is epitaxially grown on a Mg-doped GaAs substrate
with 
concentration. Calculate and draw the relative band diagram of the
2 layers.
