1. PN Junction Diode: I-V Characteristics
Chapter 6.
PN Junction Diode: I-V Characteristics
Sung June Kim

2. Contents Qualitative Derivation Quantitative Solution Strategy
Quasineutral Region Considerations
Depletion Region Considerations
Boundary Conditions

3. Qualitative Derivation
The Ideal Diode Equation
The I-V characteristics of the ideal diode are modeled by the ideal diode equation  qualitative and quantitative derivation
Qualitative Derivation
Equilibrium situation
balance
high-energy carrier
drift
diffusion
potential
hill

4. Forward bias situation
 a lowering of the potential hill
The same number of minority
carriers are being swept
More majority carriers can surmount the hill  IN and IP  I (pn)
The number of carriers that have sufficient energy to surmount the barrier goes up exponentially with VA  exponential increase of the forward current

5. Reverse bias situation
 an increase of the potential hill
The barrier increase reduces the majority carrier diffusion to a negligible level
The p-side electrons and n-side holes can wander into the depletion region and be swept to the other side  reverse I (np)
Being associated with minority carriers, the reverse bias current is expected to be extremely small

6. Reverse current is expected to saturate(bias independent)
The minority carrier drift currents are not affected by the height of the hill (The situation is similar to a waterfall)
Reverse current is expected to saturate(bias independent)
If the reverse bias saturation current is taken to be –I0, the overall I-V dependence is
Rectification
두번째 문장에 그러므로 리버스 커런트는 바이어스 인디펜던트하다고 넣으면 자연스럽지 않을까
I-V characteristic

7. ohmic
minority
excess majority carriers  local
ohmic
excess majority carriers  local
recombination
Excess carriers move to the contact with a relaxation time  greatly fast
minority
Depletion region : electrons and holes
p-region (far) : holes
n-region (far) : electrons
Current component

8. Quantitative Solution Strategy Basic assumptions
Steady state conditions
A nondegenerately doped step junction
One-dimensional
Low-level injection
GL=0
and low-level injection
 minority carrier diffusion equations

9. Quasineutral Region Considerations
Since and dn0/dx=dp0/dx=0 in the quasineutral regions
We can only determine JN(x) in the quasineutral p-region and JP(x) in the quasineutral n-region

10. ∴ Depletion Region Considerations
Suppose that thermal recombination-generation is negligible throughout the depletion region;
 JN and JP are constants inside the depletion region ∴

11. At the Depletion Region Edges
Boundary Conditions
At the Ohmic Contacts
The ideal diode is usually taken to be a “wide-base” diode
The contacts may effectively be viewed as being positioned at x=
At the Depletion Region Edges
Under nonequilibrium conditions:

12. If the equal sign is assumed to
hold throughout the depletion region
: law of the junction

13. Evaluating the equation at the p-edge
Similarly,

14. Derivation Proper
The origin of coordinates is shifted to the n-edge of the depletion region
Boundary conditions
The general solution

15. A2  0 because exp(x’/Lp)   as x’  
With , A1=pn(x’=0)
On the p-side of the junction with the x’’-coordinate.

16. The current densities at the depletion region edges,
Ideal diode equation or Shockley equation

17. Junction Theory
1-D general continuity equation

18. In the depletion region, steady state, ignoring last two terms,
and
∴ inside depletion region

19. In the quasineutral region, E = 0 and diffusion is dominant,
Constant through the PN junction
To get continuity equation in quasineutral regions,
Solutions.

20. Under non-equilibrium condition

21. With slight shift of coordinates,

22. (recap)diode I-V characteristic (The Diode Equation)
Diode current is determined by the diffusion current of the minority carriers
Both hole current and electron current should be considered
They then are added up
This is the Diode Equation

23. (recap)Reverse Saturation Current
After defining Io(reverse saturation current), the well known J-V or I-V relation is established

24. Examination of Results
Ideal I-V
1. For forward biasing greater than a few kT/q,
2. For reverse biases greater than a few kT/q,

25. The Saturation Current
The current depends on doping of the LIGHTLY doped region.

26. Carrier currents
The total current density is constant
The majority-carrier current densities are obtained by graphically subtracting the minority-carrier current densities from the total current density

27. Carrier concentrations
Forward biasing increases the concentration
Reverse decreases
Under the low-level injection, the majority carrier concentrations in these regions are everywhere approximately equal to their equilibrium values

28. Under reverse biasing the depletion region acts like a “sink” for minority carriers
Larger reverse biases have little effect
NA > ND

29. new

30. Trends of diffusion and drift currents
new
Trends of diffusion and drift currents
Separation of the energy band
Diffusion current is strongly dependent on the potential barrier changing to bias.
Drift current of minority carriers do not change much because they are limited in number.
Therefore the total current is mostly diffusion current in forward bias, and mostly drift current of minority carriers (called generation current) in reverse direction.
I-V characteristic of a p-n junction

31. Charge Control Model(1)

32. Charge Control Model (2)
Suppose that Ip(xn=0) is a supplying current to maintain the condition for every
Result is same as charge control model (1) (from slope of minority carrier distribution)
can be calculated in the same way

33. (recap) Total current (1)
“The summation of current is constant while current component is changing”
Fig 5-17
Electron and hole components of current in a forward-biased p-n junction. In this example, we have a higher injected minority hole current on the n-side than electron current on the p side because we have a lower n doping than p doping.
Drift of majority carrier
Diffusion of minority carrier

34. Total current(2) I = Ip(xn=0) – In(xp=0)
No recombination in W ( Ip(xn=0) and In(xp=0) are constant )
Ip(xn) is diffusion current decreasing exponentially ( Ip(xn) is proportioned to δp(xn) )
In(xn) is drift current which supplies hole for p-area and electron for n=area by recombination
In(xp) = I – Ip(xn)
the electric field of neutral region is very small compared with the field of pn junction area

35. (recap)Reverse Biased pn junction(1)
Fig 5-18
Reverse-biased p-n junction: (a) minority carrier distribution near the reverse-biased junction; (b) variation of the quasi-Fermi levels

36. (recap)Reverse Biased pn junction(2)
For Vr>> kT/q
Minority carrier extraction
Quasi Fermi Level widens

37. Breakdown Deviations from the Ideal Ideal Theory versus Experiment
I-V characteristic derived from a Si diode
A large reverse-bias current flows when the reverse voltage exceeds a certain value
Breakdown

38. Reverse Bias Breakdown

39. VBR tends to increase with band gap of the semiconductor and the doping on the lightly doped side of the junction
(NB is the doping on the lightly doped side of the junction )

40. Avalanche Breakdown Impact ionization. Carrier multiplication.
Electron-hole pairs created by impact ionization:
(a) band diagram of a p-n junction in reverse bias showing(primary) electron gaining kinetic energy in the field of the depletion region, and creating a (secondary) electron-hole pair by impact ionization, the primary electron losing most of its kinetic energy in the process;
(b) a single ionizing collision by an incoming electron in the depletion region of the junction;
(c) primary, secondary and tertiary collisions.

41. VBR tends to increase with band gap of the semiconductor and the doping on the lightly doped side of the junction
Carrier multiplication model(mutiplication factor M)
empirical fit to experimental data,
M is used to correct the ideal diode equation to account for avalanching and carrier multiplication
In other words, breakdown occurs when the electric field in the depletion region reaches some critical value
Then, when,
Electric field is independent of doping. So,

42. Zener Process The particle energy remains constant during the process.
Tunneling
The particle energy remains constant during the process.
The particle and the barrier are not damaged.
(1) There must be filled states on one side and empty states on the other side at the same energy.
(2) d must be very thin.(d < 10-6 cm)

43. Reverse bias↑ ⇒ # of filled valence electrons placed opposite empty conduction-band states↑ ⇒ current↑

44. Zener Breakdown By Tunneling.(decrease of d in reverse bias)
highly doped p+n+ junction.

45. Use of zener diode as voltage regulator

46. Ec Ef Ev VR IR
VR = 0 V (Equilibrium)

47. Ec Ef Ev IR Ec Ef Ev VR h+ e-
VR = 0 V
VR < 0 V

48. VR << 0 V (Zener Breakdown, Tunneling)Ec Ef Ev IR Ec Ef Ev e- e- e- e- e- VR
VR << 0 V (Zener Breakdown, Tunneling)

49. Ideality factor n
The recombination current is complicated by the fact that recombination rate, which depends on the carrier concentrations, varies with position within depletion region.
The diode equation can be modified by including the parameter n :
n varies between 1 and 2, depending on the material , temperature and voltage

50. Derivation from simple theory
Forward and reverse current-voltage characteristics plotted on semi-log scales, with current normalized with respect to saturation current Io; (a) the ideal forward characteristic is an exponential with an ideality factor n=1 (dashed straight line on log-linear plot). The actual forward characteristics of a typical diode(colored line) have four regimes of operation; (b) ideal reverse characteristic (dashed line) is a voltage-independent current = -Io. Actual leakage characteristics(colored line) are higher due to generation in the depletion region, and show breakdown at high voltages.
qv/nkT 에서 n을 설명하는 슬라이드가 필요. 중간에 또 복잡한 식 이거 설명 필요

51. The R-G Current
A current far in excess of that predicted by the ideal theory exists at small forward bias and all reverse biases.
← thermal recombination-generation in the depletion region
Reverse biasing  carrier concentration in depletion region are reduced below their equilibrium values  lead to the thermal generation
Forward biasing  carrier concentration increase above their equilibrium values  carrier recombination

52. In steady state, net R-G rate is the same for electrons and holes
In depletion region, the general R-G relationship is used
For reverse biases greater than a few kT/q, carrier concentration is negligible(n→0, p→0)

53. Recombination mechanism(optional)rn gn Et Et Et Et rp gn Ev Ev Ev Ev
rn: electron capture rate
gn: electron emission rate
rp: hole capture rate
gp: hole emission rate
(The probability that an electron fills trap)
(The emission coefficient)
(steady state)

54. Then,
(steady state)

55. For forward biases, the carrier concentrations cannot be neglected.
We merely note that IR-G is expected to vary roughly as exp(qVA/ηkT). Typically η is expected close to 2.
Then combined forward and reverse bias dependence is approximately described by below
Diffusion current by ideal diode equation,
And total current is

56. In room temperature,
and IR-G current dominates at reverse and small forward biases.
With increasing forward biases, IDIFF increases more rapidly
Because while , the relative weight of the two component varies from semiconductor to semiconductor
Also, the reverse bias diffusion component of the current will increase at a faster rate with increasing temperature

57. VA  Vbi high-current phenomena
High level Injection
Minority and majority carrier concentrations adjacent to the depletion region are perturbed
The majority carrier concentration must increase to maintain approximate charge neutrality
An analysis of high-level injection leads to a predicted current varying roughly as exp(q/2kT)

58. VA  Vbi high-current phenomena
Large current : voltage drop in quasineutral region and high level injection
Series Resistance
Quaseneutral region have an inherent resistance RS
(when I is small, we can ignore IRA, then VJ = VA)
We can rewrite I-VA relationship

60. Narrow-Base Diode Current Derivation x’c = xc - xn and LP > x’c
minority carrier concentration at a contact a finite distance from the depletion region edge depends on the R-G rate at the contact
Ohmic contact, R-G rate is high and the minority carrier concentration is maintained near its equilibrium value

61. Paralleling the derivation of the ideal diode equation,
General solution is
Applying boundary condition,
Then,
Finally,

62. Narrow-Base Diode Limiting Cases If x’c →∞ , or x’c /LP >>1,
The perturbed carrier concentration becomes a linear function of position. This is a direct consequence of negligible thermal R-G in a region much shorter than a diffusion length.
This observation is justification for henceforth neglecting the thermal R-G term in the minority carrier diffusion equation when the quasineutral width is small compared to a diffusion length
37쪽도 마저

63. Narrow-Base Diode Punch-Through This involves the reverse-bias current
x’c /LP <<1,
The width of the quasineutral region x’c decreases with increasing reverse bias because of the growing depletion width
IDIFF (VA<0) does not saturate ( )
xc is sufficiently small, x’c →0
The situation where an entire device region becomes depleted is reffered to as punch through

64. Piecewise-linear approximations of junction diode characteristics
6.3.2 , 예제 6.9 등등 넣을거 많이 있음.

65. Wise choices large Eg small ni, small Io, large Eo, large Vbr
Low doping large Vbr
In this time, watch out for punchthrough (In Punchthrough, entire device region is depleted.)
guard ring to prevent premature breakdown
Contact Resistance decreases (From terminal to n+ doping)
p+-n-n+ structure (forward resistance decreases)

66. Beveled edge and guard ring to prevent edge breakdown under reverse bias: (a) diode with beveled edge; (b) closeup view of edge, showing reduction of depletion region near the bevel; (c) guard ring

67. Sung June Kim kimsj@snu.ac.kr http://nanobio.snu.ac.kr
Chapter 7.
PN Junction Diode : C-V
Sung June Kim

68. Contents
Reverse-Bias Junction Capacitance

69. Reverse-Bias Junction Capacitance
When reverse biased, the pn junction diode becomes functionally equivalent to a capacitor

70. A small sinusoidal voltage (va) is superimposed on the dc bias
Total voltage drop becomes VA+va
The overall effect of the a.c. signal may be viewed as a small oscillation of the depletion width and an associated  oscillation
reverse
forward
reverse

71. Effectively, it looks like plus and minus charges are being alternately added and subtracted from two planes separated by a width W.
junction or
depletion-layer capacitance
C-V Relationships

72. The junction depletion capacitance is