1. Lecture #5 OUTLINE Intrinsic Fermi level Determination of EF
Degenerately doped semiconductor
Carrier properties
Carrier drift
Read: Sections 2.5, 3.1

2. Intrinsic Fermi Level, Ei
To find EF for an intrinsic semiconductor, use the fact that n = p:
EE130 Lecture 5, Slide 2

3. n(ni, Ei) and p(ni, Ei)
In an intrinsic semiconductor, n = p = ni and EF = Ei :
EE130 Lecture 5, Slide 3

4. Example: Energy-band diagram
Question: Where is EF for n = 1017 cm-3 ?
EE130 Lecture 5, Slide 4

5. Dopant Ionization Consider a phosphorus-doped Si sample at 300K with
ND = 1017 cm-3. What fraction of the donors are not ionized?
Answer: Suppose all of the donor atoms are ionized.
Then
Probability of non-ionization 
EE130 Lecture 5, Slide 5

6. Nondegenerately Doped Semiconductor
Recall that the expressions for n and p were derived using the Boltzmann approximation, i.e. we assumed
The semiconductor is said to be nondegenerately doped in this case. Ec
3kT
EF in this range
3kT Ev
EE130 Lecture 5, Slide 6

7. Degenerately Doped Semiconductor
If a semiconductor is very heavily doped, the Boltzmann approximation is not valid.
In Si at T=300K: Ec-EF < 3kT if ND > 1.6x1018 cm-3
EF-Ev < 3kT if NA > 9.1x1017 cm-3
The semiconductor is said to be degenerately doped in this case.
Terminology:
“n+”  degenerately n-type doped. EF  Ec
“p+”  degenerately p-type doped. EF  Ev
EE130 Lecture 5, Slide 7

8. Band Gap Narrowing
If the dopant concentration is a significant fraction of the silicon atomic density, the energy-band structure is perturbed  the band gap is reduced by DEG :
N = 1018 cm-3: DEG = 35 meV
N = 1019 cm-3: DEG = 75 meV
EE130 Lecture 5, Slide 8

9. Mobile Charge Carriers in Semiconductors
Three primary types of carrier action occur inside a semiconductor:
Drift: charged particle motion under the influence of an electric field.
Diffusion: particle motion due to concentration gradient or temperature gradient.
Recombination-generation (R-G)
EE130 Lecture 5, Slide 9

10. Electrons as Moving Particles
In vacuum
In semiconductor
F = (-q)E = moa
F = (-q)E = mn*a
where
mn* is the electron effective mass
EE130 Lecture 5, Slide 10

11. Carrier Effective Mass
In an electric field, E, an electron or a hole accelerates:
Electron and hole conductivity effective masses:
electrons
holes * *
EE130 Lecture 5, Slide 11

12. Thermal Velocity Average electron kinetic energy
EE130 Lecture 5, Slide 12

13. Carrier Scattering
Mobile electrons and atoms in the Si lattice are always in random thermal motion.
Electrons make frequent collisions with the vibrating atoms
“lattice scattering” or “phonon scattering”
increases with increasing temperature
Average velocity of thermal motion for electrons: ~ K
Other scattering mechanisms:
deflection by ionized impurity atoms
deflection due to Coulombic force between carriers
“carrier-carrier scattering”
only significant at high carrier concentrations
The net current in any direction is zero, if no electric field is applied. 1 2 3 4 5
electron
EE130 Lecture 5, Slide 13

14. Carrier Drift
When an electric field (e.g. due to an externally applied voltage) is applied to a semiconductor, mobile charge-carriers will be accelerated by the electrostatic force. This force superimposes on the random motion of electrons: 1 2 3 4 5
electron E
Electrons drift in the direction opposite to the electric field
 current flows
Because of scattering, electrons in a semiconductor do not achieve constant acceleration. However, they can be viewed as quasi-classical particles moving at a constant average drift velocity vd
EE130 Lecture 5, Slide 14

15. Electron Momentum (-q)Etmn
With every collision, the electron loses momentum
Between collisions, the electron gains momentum
(-q)Etmn
tmn is the average time between electron scattering events
EE130 Lecture 5, Slide 15

16. Carrier Mobility mn*vd = (-q)Etmn |vd| = qEtmn / mn* = mn E
n  [qtmn / mn*] is the electron mobility
|vd| = qEtmp / mp*  mp E
Similarly, for holes:
p  [qtmp / mp*] is the hole mobility
EE130 Lecture 5, Slide 16

17. Electron and Hole Mobilities
 has the dimensions of v/E :
Electron and hole mobilities of selected intrinsic semiconductors (T=300K)
EE130 Lecture 5, Slide 17

18. Example: Drift Velocity Calculation
a) Find the hole drift velocity in an intrinsic Si sample for E = V/cm.
b) What is the average hole scattering time?
Solution: a) b)
vd = mn E
EE130 Lecture 5, Slide 18

19. Mean Free Path Average distance traveled between collisions
EE130 Lecture 5, Slide 19

20. Summary The intrinsic Fermi level, Ei, is located near midgap
Carrier concentrations can be expressed as functions of Ei and intrinsic carrier concentration, ni :
In a degenerately doped semiconductor, EF is located very near to the band edge
Electrons and holes can be considered as quasi-classical particles with effective mass m*
In the presence of an electric field e, carriers move with average drift velocity , where m is the carrier mobility
EE130 Lecture 5, Slide 20