1. Einstein Relation—the relation between  and D
At equilibrium, if there is a diffusion current (e.g., due to doping gradient or band gap
variation), it must be balanced by a built-in field to give zero net current flow,
(4-27)
Use
Eq.(4-27) becomes:
(4-28)
Because
And that EF is constant at equilibrium, Eq. (4-28) reduces to:
Einstein relation
(4-29)
Valid for either carrier type!

2. Example 4.5
An intrinsic Si sample is doped with donors from one side such that Nd = N0e-ax
a. Find an expression for E(x) at equilibrium over the range for which Nd » ni
b. Find an expression for E(x) when a = 1 m-1
c. Sketch a band diagram and indicate the direction of E(x)

3. 4.4.3 Diffusion and Recombination; The Continuity Equation
in our discussions of Diffusion, we have neglected Recombination.
Recombination can cause significant variation in the carrier distribution
Figure 4—16
Current entering and leaving a volume ∆xA.

4. Consider both diffusion and recombination, we have:
Continuity Equations
(4-31b)
If the current is strictly by diffusion:
(4-33a)
Diffusion Equations
(4-33b)

5. A pulse of electrons in a semiconductor:
spreads out by diffusion, and disappears by recombination
to find n(x,t) we would start with:

6. 4.4.4 Steady State Carrier Injection; Diffusion Length
In the steady state case the diffusion equations become
(4-34a)
Steady State Diffusion Equations
(4-34b)
Diffusion Length
Let us assume that excess holes are injected into a semi-infinite semiconductor bar at x=0,
and that at the injection point p(x=0) = p is constant. The solution to Eq. (4-34b)
has the form:
(4-35)

7. We expect p decay to zero for large x due to recombination, therefore C1=0. Similarly, the
condition p = p at x=0 gives C2=p, and the solution is:
(4-36)
The physical significance of Lp (or Ln)
p = (1/e) p at x = Lp (2) Lp is the average distance a hole diffuses before
recombining. <x>=Lp
(4-40a)
(4-40b)

8. Figure 4—17
Injection of holes at x = 0, giving a steady state hole distribution p(x) and a resulting diffusion current density Jp(x).

9. 4.4.6 Gradients in the Quasi-Fermi Levels
- At equilibrium, there is no gradient in the Fermi level EF
In the steady state, there is a gradient in the quasi-Fermi level due to drift and diffusion
The total electron current becomes:
(4-51)
(4-52a)
Modified Ohm’s Law
(4-52b)

10. first performed in 1951at Bell Telephone Laboratories
4.4.5 The Haynes-Shockley Experiment (Please read Section 4.4.5)
classic experiment demonstrating the drift and diffusion of minority carriers
first performed in 1951at Bell Telephone Laboratories
independent measurement of the minority carrier mobility and diffusion coefficient
Figure 4—18
Drift and diffusion of a hole pulse in an n-type bar: (a) sample geometry; (b) position and shape of the pulse for several times during its drift down the bar.

11. Figure 4—19
Calculation of Dp from the shape of the p distribution after time td. No drift or recombination is included

12. Figure 4—20
The Haynes–Shockley experiment: (a) circuit schematic; (b) typical trace on the oscilloscope screen.

13. Chapter 5 Junctions
p-n junctions
metal-semiconductor junctions
heterojunctions
Junctions
Fabrication
Equilibrium conditions
Biased junctions; steady state conditions
Reverse bias breakdown
Transient and AC conditions
Deviations from simple theory
p-n junctions
Strong qualitative understanding of the properties of p-n junctions
Know how to use the mathematics of p-n junctions to make
calculations
Goals

14. 5.1 Fabrication of p-n junctions
Major process steps (Please read Section 5.1)
Thermal Oxidation
Diffusion
Rapid Thermal Processing
Ion Implantation
Chemical Vapor Deposition (CVD)
Photolithography
Etching
Metallization
Crystal Growth and Wafer Preparation (Chap.1)
Process Simulation
Process Integration

15. Figure 5—10
Simplified description of steps in the fabrication of p-n junctions. For simplicity, only four diodes per wafer are shown, and the relative thicknesses of the oxide, PR, and the Al layers are exaggerated.

16. 5.2 Equilibrium Conditions
What will happen if we bring a p-type semiconductor and a n-type semiconductor together
to form a junction?
Based on knowledge gained from previous chapters, we expect:
Initially, current will flow due to diffusion
No net current can flow across the junction at equilibrium
An internal electric field E will build up as a result of
uncompensated donor ions (Nd+) and acceptor ions (Na+)
The electric field gives rise to a “contact potential” V0
across the junction E=-dV(x)/dx
The Fermi levels will be aligned at equilibrium