1. PN Junction: Physical and Mathematical Description of Operation Dragica Vasileska Professor Arizona State University
PN Junctions – General Considerations
Ideal Current-Voltage Characteristics
Generation and Recombination Currents
Breakdown Mechanisms
AC-Analysis and Diode Switching

2. Linearly-graded junction
1. PN-junctions - General Consideration:
PN-junction is a two terminal device.
Based on the doping profile, PN-junctions can be separated into two major categories:
- step junctions
- linearly-graded junctions
p-side
n-side
p-side
n-side
Step junction
Linearly-graded junction

3. (A) Equilibrium analysis of step junctions (a) Built-in voltage Vbi:
(b) Majority- minority carrier
relationship:
-qNA
qND + -
p-side
n-side

4. (c) Depletion region width:
 Solve 1D Poisson equation using depletion charge approximation, subject to the following boundary condi- tions:
p-side:
n-side:
 Use the continuity of the two solutions at x=0, and charge neutrality, to obtain the expression for the depletion region width W:

5. (d) Maximum electric field:
The maximum electric field, which occurs at the metallurgical junction, is given by:
(e) Carrier concentration variation:

6. (f) Analytical vs. numerical data

8. (g) Depletion layer capacitance:
 Consider a p+n, or one-sided junction, for which:
 The depletion layer capacitance is calculated using:
Measurement setup: W dW ~ V
Reverse
bias
Forward bias
vac

9. (B) Equilibrium analysis of linearly-graded junction:
(a) Depletion layer width:
(c) Maximum electric field:
(d) Depletion layer capacitance:
Based on accurate numerical simulations, the depletion layer capacitance can be more accurately calculated if Vbi is replaced by the gradient voltage Vg:

10. (2) Ideal Current-Voltage Characteristics:
Assumptions:
Abrupt depletion layer approximation
Low-level injection  injected minority carrier density much smaller than the majority carrier density
No generation-recombination within the space-charge region (SCR)
(a) Depletion layer:

11. (b) Quasi-neutral regions:
Using minority carrier continuity equations, one arrives at the following expressions for the excess hole and electron densities in the quasi-neutral regions:
Forward bias
Reverse bias
Space-charge
region W

12. No SCR generation/recombination
Corresponding minority-carriers diffusion current densities are:
No SCR generation/recombination
Shockley model

13. (c) Total current density:
Total current equals the sum of the minority carrier diffu-sion currents defined at the edges of the SCR:
Reverse saturation current IS: I V Ge Si
GaAs

14. (d) Origin of the current flow:
Forward bias:
Reverse bias: Ln Lp
Reverse saturation current is due to minority carriers being collected over a distance on the order of the diffusion length.

15. (e) Majority carriers current:
Consider a forward-biased diode under low-level injection conditions:
Total hole current in the quasi-neutral regions:
Quasi-neutrality requires:
This leads to:

16. Electron drift current in the quasi-neutral region:

17. (f) Limitations of the Shockley model:
The simplified Shockley model accurately describes IV-characteristics of Ge diodes at low current densities.
For Si and Ge diodes, one needs to take into account several important non-ideal effects, such as:
 Generation and recombination of carriers within the depletion region.
 Series resistance effects due to voltage drop in the quasi-neutral regions.
 Junction breakdown at large reverse biases due to tun-neling and impact ionization effects.

18. (3) Generation and Recombination Currents
 Continuity equation for holes:
 Steady-state and no light genera-
tion process:
Space-charge region recombination current:

19. Reverse-bias conditions:
Concentrations n and p are negligible in the depletion region:
Space-charge region current is actually generation current:
Total reverse-saturation current:
Generation lifetime

20. Generation current dominates when ni is small, which is always the case for Si and GaAs diodes.
I (log-scale) W
V (log-scale)
IV-characteristics
under reverse bias conditions
Generated carriers are swept away from the depletion region.

21. Recombination lifetime
Forward-bias conditions:
Concentrations n and p are large in the depletion region:
Condition for maximum recombination rate:
Estimate of the recombination current:
Recombination lifetime

22. Exact expression for the recombination current:
Corrections to the model:
Total forward current:
  ideality factor. Deviations of  from unity represent an important measure for the recombination current.

23. Importance of recombination effects:
Low voltages, small ni  recombination current dominates
Large voltages  diffusion current dominates
log(I) V

24. (4) Breakdown Mechanisms
Junction breakdown can be due to:
 tunneling breakdown
 avalanche breakdown
One can determine which mechanism is responsible for the breakdown based on the value of the breakdown voltage VBD :
 VBD < 4Eg/q  tunneling breakdown
 VBD > 6Eg/q  avalanche breakdown
 4Eg/q < VBD < 6Eg/q  both tunneling and
avalanche mechanisms are responsible

25. Tunneling breakdown:
Tunneling breakdown occurs in heavily-doped pn-junctions in which the depletion region width W is about 10 nm.
Zero-bias band diagram:
Forward-bias band diagram:
EFn EF
EFp EC EC EV EV W W

26. Reverse-bias band diagram:
Tunneling current (obtained by using WKB approximation):
Fcr  average electric field in
the junction
The critical voltage for tunneling breakdown, VBR, is estimated from:
With T, Eg and It .
EFp
EFn EC EV

27. Avalanche breakdown:
Most important mechanism in junction breakdown, i.e. it imposes an upper limit on the reverse bias for most diodes.
Impact ionization is characterized by ionization rates an and ap, defined as probabilities for impact ionization per unit length, i.e. how many electron-hole pairs have been generated per particle per unit length:
- Ei  critical energy for impact ionization to occur
- Fcr  critical electric field
- l  mean-free path for carriers

28. Generation of the excess electron-hole
Avalanche mechanism:
EFp
EFn EC EV
Generation of the excess electron-hole
pairs is due to impact ionization.
Expanded view of the
depletion region

29. Description of the avalanche process:dx
Impact ionization initiated by electrons.
Multiplication factors for
electrons and holes: dx
Impact ionization initiated by holes.

30. The breakdown condition does not depend on which
Breakdown voltage  voltage for which the multiplication rates Mn and Mp become infinite. For this purpose, one needs to express Mn and Mp in terms of an and ap:
The breakdown condition does not depend on which
type of carrier initiated the process.

31. Limiting cases:
(a) an=ap (semiconductor with equal ionization rates):
(b) an>>ap (impact ionization dominated by one carrier):

32. Breakdown voltages: (a) Step p+n-junction
For one sided junction we can make the following approximation:
Voltage drop across the depletion region on the n-side:
Maximum electric field:
Empirical expression for the breakdown voltage VBD:

33. (b) Step p+-n-n+ junction
Extension of the n-layer large:
Extension of the n-layer small:
Final expression for the punch-through voltage VP:

34. Width of the n-layer W1 increases
Doping-dependence of the breakdown voltage VBD:
Temperature dependence:
As temperature increases, lattice scattering increases which makes impact ionization less probable. As a result of this, the breakdown voltage increases.
Width of the n-layer W1 increases
Log-scale
p+-n-n+
One-sided abrupt
junction
Tunneling breakdown

35. (c) Plane vs. planar or cylindrical junction Plane junction:
Planar junction:
Maximum electric field:
Except for surface effects, this is an
ideal junction. p+ n
Maximum electric field:
The smaller the radius rj, the larger
the electric field crowding. rj p+ W n

36. (5) AC-Analysis and Diode Switching
(a) Diffusion capacitance and small-signal equivalent
circuit
This is capacitance related to the change of the minority carriers. It is important (even becomes dominant) under forward bias conditions.
The diffusion capacitance is obtained from the device impedance, and using the continuity equation for minority carriers:
Applied voltages, currents and solution for Dpn:

37. Equation for pn1(x):
Boundary conditions:
Final expression for pn1(x):

38. Small-signal hole current:
Low-frequency limit for the admittance Y:
RC-constant:
The characteristic time constant is on
the order of the minority carriers lifetime.

39. Equivalent circuit model for forward bias:
Bias dependence:

40. Qp(t) = excess hole charge
(b) Diode switching
For switching applications, the transition from forward bias to reverse bias must be nearly abrupt and the transit time short.
Diode turn-on and turn-off characteristics can be obtained from the solution of the continuity equations:
Qp(t) = excess hole charge
Valid for p+n diode

41. Final expression for the excess hole charge:
Diode turn-on:
For t<0, the switch is open, and the excess hole charge is:
At t=0, the switch closes, and we have the following boundary condition: p+ n
t=0 IF
Final expression for the excess hole charge:

42. Graphical representation:
Steady state value for the bias across the diode:
t increasing
Slope almost constant

43. The current through the diode until time t1 is:
Diode turn-off:
For t<0, the switch is in position 1, and a steady-state situation is established:
At t=0, the switch is moved to position 2, and up until time t=t1 we have:
The current through the diode until time t1 is: p+ n
t=0 VF R VR 1 2

44. To solve exactly this problem and find diode switching time, is a rather difficult task. To simplify the problem, we make the crucial assumption that IR remains constant even beyond t1.
The differential equation to be solved and the initial condition are, thus, of the form:
This gives the following final solution:
Diode switching time:

45. Graphical representation:
Slope almost
constant
t=0
t=ts
ttrr
ts  switching time
trr  reverse recovery time