2. Potential vs. Kinetic Energy
Chapter 3
Carrier Action
Potential vs. Kinetic Energy
Electron kinetic energy
Increasing electron energy Ec Ev
Hole kinetic energy
Increasing hole energy
Ec represents the electron potential energy:

3. Chapter 3
Carrier Action
Band Bending
Until now, Ec and Ev have always been drawn to be independent of the position.
When an electric field E exists inside a material, the band energies become a function of position. Ec Ev E x
Variation of Ec with position is called “band bending”

4. Chapter 3
Carrier Action
Band Bending
The potential energy of a particle with charge –q is related to the electrostatic potential V(x):
Since Ec, Ev, and Ei differ only by an additive constant

5. Chapter 3
Carrier Action
Diffusion
Particles diffuse from regions of higher concentration to regions of lower concentration region, due to random thermal motion (Brownian Motion).

6. Chapter 3
Carrier Action
1-D Diffusion Example
Thermal motion causes particles to move into an adjacent compartment every τ seconds.

7. Diffusion Currents n x e- p x h+
Chapter 3
Carrier Action
Diffusion Currents n x e- p x h+
Electron flow
Hole flow
Current flow
Current flow
D is the diffusion coefficient [cm2/sec]

8. Total Currents Drift current flows when an electric field is applied.
Chapter 3
Carrier Action
Total Currents
Drift current flows when an electric field is applied.
Diffusion current flows when a gradient of carrier concentration exist.

9. Current Flow Under Equilibrium Conditions
Chapter 3
Carrier Action
Current Flow Under Equilibrium Conditions
In equilibrium, there is no net flow of electrons or :
The drift and diffusion current components must balance each other exactly.
A built-in electric field of ionized atoms exists, such that the drift current exactly cancels out the diffusion current due to the concentration gradient.

10. Current Flow Under Equilibrium Conditions
Chapter 3
Carrier Action
Current Flow Under Equilibrium Conditions
Consider a piece of non-uniformly doped semiconductor:
n-type semiconductor
Decreasing donor
concentration
Ev(x)
Ec(x) EF
Under equilibrium, EF inside a material or a group of materials in intimate contact is not a function of position

11. Einstein Relationship between D and m
Chapter 3
Carrier Action
Einstein Relationship between D and m
But, under equilibrium conditions, JN = 0 and JP = 0
Similarly,
Einstein Relationship
Further proof can show that the Einstein Relationship is valid for a non-degenerate semiconductor, both in equilibrium and non-equilibrium conditions.

12. Example: Diffusion Coefficient
Chapter 3
Carrier Action
Example: Diffusion Coefficient
What is the hole diffusion coefficient in a sample of silicon at 300 K with p = 410 cm2 / V.s ?
Remark: kT/q = mV at room temperature

13. Recombination–Generation
Chapter 3
Carrier Action
Recombination–Generation
Recombination: a process by which conduction electrons and holes are annihilated in pairs.
Generation: a process by which conduction electrons and holes are created in pairs.
Generation and recombination processes act to change the carrier concentrations, and thereby indirectly affect current flow.

14. Generation Processes Band-to-Band R–G Center Impact Ionization
Chapter 3
Carrier Action
Generation Processes
Band-to-Band
R–G Center
Impact Ionization
Release of energy
ET: trap energy level
Due to lattice defects or unintentional impurities
Also called indirect generation EG
Only occurs in the presence of large E

15. Recombination Processes
Chapter 3
Carrier Action
Recombination Processes
Band-to-Band
R–G Center
Auger
Collision
Rate is limited by minority carrier trapping
Primary recombination way for Si
Occurs in heavily doped material

16. Direct and Indirect Semiconductors
Chapter 3
Carrier Action
Direct and Indirect Semiconductors Ev Ec
E-k Diagrams Ec Ev
Phonon
Photon
Photon
GaAs, GaN
(direct semiconductors)
Si, Ge
(indirect semiconductors)
Little change in momentum is required for recombination
Momentum is conserved by photon (light) emission
Large change in momentum is required for recombination
Momentum is conserved by mainly phonon (vibration) emission + photon emission

17. Excess Carrier Concentrations
Chapter 3
Carrier Action
Excess Carrier Concentrations
Values under arbitrary conditions
Deviation from
equilibrium values
Equilibrium values
Positive deviation corresponds to a carrier excess, while negative deviations corresponds to a carrier deficit.
Charge neutrality condition:

18. “Low-Level Injection”
Chapter 3
Carrier Action
“Low-Level Injection”
Often, the disturbance from equilibrium is small, such that the majority carrier concentration is not affected significantly:
For an n-type material
For a p-type material
Low-level injection condition
However, the minority carrier concentration can be significantly affected.

19. Indirect Recombination Rate
Chapter 3
Carrier Action
Indirect Recombination Rate
Suppose excess carriers are introduced into an n-type Si sample by shining light onto it. At time t = 0, the light is turned off. How does p vary with time t > 0?
Consider the rate of hole recombination:
NT : number of R–G centers/cm3
Cp : hole capture coefficient
In the midst of relaxing back to the equilibrium condition, the hole generation rate is small and is taken to be approximately equal to its equilibrium value:

20. Indirect Recombination Rate
Chapter 3
Carrier Action
Indirect Recombination Rate
The net rate of change in p is therefore:
where
For holes in n-type material
Similarly,
where
For electrons in p-type material

21. Minority Carrier Lifetime
Chapter 3
Carrier Action
Minority Carrier Lifetime
The minority carrier lifetime τ is the average time for excess minority carriers to “survive” in a sea of majority carriers.
The value of τ ranges from 1 ns to 1 ms in Si and depends on the density of metallic impurities and the density of crystalline defects.
The deep traps originated from impurity and defects capture electrons or holes to facilitate recombination and are called recombination-generation centers.

22. Example: Photoconductor
Chapter 3
Carrier Action
Example: Photoconductor
Consider a sample of Si doped with 1016 cm–3 Boron, with recombination lifetime 1 μs. It is exposed continuously to light, such that electron-hole pairs are generated throughout the sample at the rate of 1020 per cm3 per second, i.e. the generation rate GL = 1020/cm3/s
a) What are p0 and n0?
b) What are Δn and Δp?
Hint: In steady-state, generation rate equals recombination rate

23. Example: Photoconductor
Chapter 3
Carrier Action
Example: Photoconductor
Consider a sample of Si at 300 K doped with 1016 cm–3 Boron, with recombination lifetime 1 μs. It is exposed continuously to light, such that electron-hole pairs are generated throughout the sample at the rate of 1020 per cm3 per second, i.e. the generation rate GL = 1020/cm3/s.
c) What are p and n?
d) What are np product?
Note: The np product can be very different from ni2 in case of perturbed/agitated semiconductor

24. Net Recombination Rate (General Case)
Chapter 3
Carrier Action
Net Recombination Rate (General Case)
For arbitrary injection levels and both carrier types in a non-degenerate semiconductor, the net rate of carrier recombination is:
where
ET : energy level of R–G center

25. Chapter 2
Carrier Action
Homework 3
1. (4.27)
Problem 3.12, from (a) until (f), for Figure P3.12(a) and Figure P3.12(f), Pierret’s “Semiconductor Device Fundamentals”.
2. (5.28)
The electron concentration in silicon at T = 300 K is given by
where x is measured in μm and is limited to 0 ≤ x ≤ 25 μm. The electron diffusion coefficient is DN = 25 cm2/s and the electron mobility is μn = 960 cm2/(Vs). The total electron current density through the semiconductor is constant and equal to JN = –40 A/cm2. The electron current has both diffusion and drift current components.
Determine the electric field as a function of x which must exist in the semiconductor. Sketch the function.
Deadline: 10 February 2011, at 07:30.