Figure 1: The default pn-junction device geometry we use in our analyses, including two alternative coordinate systems for convenience.
An overview of notation conventions we use in class in general and also in this document.
n
0
, p
0
A subscript ”
0
” for carrier concentrations indicates equilibrium values.
n
, p n 0 p
Subscripts ” n
” and ” p
” indicates carrier concentrations in n-type or p-type material.
Net hole concentration at equilibrium in n-type material.
Net electron concentration at equilibrium in p-type material.
n p 0 p p 0 n n 0
δn , δp
Net hole concentration at equilibrium in p-type material.
Net electron concentration at equilibrium in n-type material.
Excess carrier concentrations, such that net carrier concentrations are given by n = δn + n
0 and p = δp + p
0
.
The built-in potential for this junction is
1

V
0
= kT q ln(
N
A
N
D n
2 i
) , (1) which can be derived either from the band diagram with the Fermi energy constant throughout the device, or from equating the drift current through the junction equal to the diffusion current, since there should be no net current flow in equilibrium.
Using depletion approximation (i.e. the boundaries of the depletion region are abrupt, within which only the ionized donor (n-side) or acceptor (p-side) charges are present) and the assumption that the bulk regions (the regions outside the depletion region) are electrically neutral, we can calculate the depletion region width under no bias as
W d
= s
2
Si
V
0 q
(
N
A
+ N
D
N
A
N
D
) (2)
The penetration of the depletion region into the p- and n-sides are inversely proportional to the doping in the relevant region: x p 0
N
D
= (3) x n 0
N
A
Since x p 0
+ x n 0
= W d
, x p 0
= s
2
Si
V
0 q
N
D
(
N
A
( N
D
+ N
A
)
) and x n 0
= s
2
Si
V
0 q
(
N
D
( N
N
A
D
+ N
A
)
) (4)
Under external bias V
A
, applied between the p- and n-sides (i.e.
V
A
= V p
− V n by definition), the net potential drop across the depletion region becomes V
0
− V
A
. Under reverse bias (i.e. p-side biased more negatively with respect to n-side), V
A
< 0 so the drop across the junction is larger than the built-in potential. The depletion region also widens in that case and narrows under forward bias, when V
A
> 0:
W d
( V
A
) = s
2
Si
( V
0
− V
A
)
( q
N
A
+ N
D
)
N
A
N
D
(5)
• No external sources of carrier generation (such as illumination) present.
• The depletion approximation holds.
• We have an abrupt step junction between two uniformly-doped p- and n-type samples of the same semiconductor material.
• Steady-state: All ∂/∂t terms in our equations will be zero.
• No generation or recombination takes place in the depletion region.
2

• In the bulk (quasi-neutral) regions, low-level injection holds.
• The minority carriers experience zero electric field in the bulk regions, hence contribute only through diffusion current.
The set of equations we will solve in either bulk region are called the drift-diffusion equations , and consist of the diffusion current equation and the charge-continuity equation.
In the n-bulk region, where δp n is the excess hole concentration:
D p d
2
δp n dx 2 n
−
δp
τ p n
= 0
J p
≈ − qD p dδp n dx p n
= p n 0
+ δp n
(6)
(7)
(8)
Similarly in the p-bulk region:
D n d
2
δn p dx 2 p
−
δn
τ p p
= 0
J n
≈ qD n dδn p dx n p
= n p 0
+ δn p
(9)
(10)
(11)
The steps we will follow are
1. Solve 6 in the n-bulk for δp n
.
2. Put the result in 7 to get J p on the n-side.
3. Current through the diode must be constant everywhere, so J n
+ J p
=constant= J ⇒ J n
=
J − J p in the n-side.
4. Solve 9 in the p-bulk for δn p
.
5. Put the result in 10 to get J n on the p-side.
6. Similarly, J p
( x ) = J − J n
( x ).
In the depletion region, since we assume no generation or recombination, not only the total current but the individual electron and hole currents also remain constant. Therefore:
J p
( depl.reg.
) = J p
( − x p 0
) = J p
( x n 0
) = − qD p dδp n dx
J n
( depl.reg.
) = J n
( x n 0
) = J n
( − x p 0
) = qD n dδn p dx
+ − − − − − − − −− − − − − − − − − − − − − − − − − − − − − − − −
J total
= = J total
(12)
3

To solve 6 and 9 we need boundary conditions at x = x n 0 and x = − x p 0
. We derived the following result in the class while calculating the built-in potential across the junction: n p
( − x p 0
) = n p 0 e
( qVA kT
)
= n
2 i p p 0 e
( qVA kT
)
(13) where n p 0 is the electron concentration in the p-site at equilibrium . Similarly p n
( x n 0
) = p n 0 e
( qVA kT
)
= n
2 i p p 0 e
( qVA kT
)
(14)
This sets the boundary conditions at the depletion-region edges of the bulk regions; at those edges the excess minority carrier concentrations become
δn p
( − x p 0
) = n p 0 e
( qVA kT
)
− n p 0
= n p 0
( e
( qVA kT
)
− 1)
δp n
( x n 0
) = p n 0 e
( qVA kT
) − p n 0
= p n 0
( e
( qVA kT
) − 1) (15)
Far from the edges at either end of the sample, assuming the bulk regions are longer than the diffusion lengths ( W n
> L n and W p
> L p we can assume the excess minority carrier concentrations go to zero and the minority carrier concentrations converge to their equilibrium values. We can consider those far edges effectively infinity:
δn p
( −∞ ) = 0 and δp n
( ∞ ) = 0 (16)
Eqns. 7 and 8 imply that at the edges of the depletion region, J n
( − x p 0
) and J p
( x n 0
) are controlled respectively by the slopes of δn p and δp n at those edges.
For the next part of derivation we are shifting to the second coordinate system shown in the figure: x n and x p coordinates. Rewriting Eqn. 6 in the x p coordinate:
D p d 2 δp n
( x n
) dx 2 n
⇒
−
δp n
( x n
)
τ p d
2
δp n
( x n
) dx 2 n
= 0
=
δp n
( x n
)
D p
τ p
=
δp n
( x n
)
L 2 p
A general solution to this equation is
δp n
( x n
) = A
1 e xn
Lp + A
2 e
− xn
Lp
Since we want δp n
( x n
→ ∞ ) = 0, A
1
= 0 and
δp n
( x n
) = A
2 e
− xn
Lp
To find A
2 we use the boundary condition at the depletion region edge, where x n
= 0:
δp n
( x n
= 0) = p n 0
( e
( qVA kT
)
− 1) ⇒ A
2
= p n 0
( e
( qVA kT
)
− 1)
(17)
(18)
(19)
4

This gives us the injection of excess holes into the n-bulk side as a function of the applied voltage V
A
:
δp n
( x n
) = p n 0
( e
( qVA kT
)
− 1) e
− xn
Lp (20)
Thus the net hole density in the n-bulk region is p n
( x n
) = p n 0
+ p n 0
( e
( qVA kT
)
− 1) e
− xn
Ln (21)
The V
A dependency makes the ( e
( qVA kT
− 1) term positive for forward bias and negative for reverse bias. So for positive bias, excess holes are injected into the n-type bulk region, and for negative bias, holes are taken away at the depletion region edge to the point where the concentration is even lower than the equilibrium minority carrier concentration. The figure below shows the net minority carrier concentration in the n-bulk under positive and negative bias.
Using this in 7 we can get the hole current at the depletion region edge in the n-bulk:
J p
( x n
) = qD p p n 0
( e qVA kT
L p
− 1) e
− xn
Lp
Similarly, looking at the minority carrier concentration in the p-region we get
J p
( x n
= 0) = qD p
L p p n 0
( e qVA kT
− 1) and J n
( x p
= 0) = qD n
L n n p 0
( e qVA kT
− 1)
(22)
δn p
( x n
) = n p 0
( e qVA kT
− 1) e
− xp
Ln (23) and
J n
( x p
) = qD n n p 0
( e qVA kT
L n
− 1) e
− xp
Ln
This lets us calculate the minority carrier currents at either edge of the depletion region:
(24)
(25)
5

With the ”no generation or recombination in the depletion region” assumption, we consider these electron and hole currents to stay the same throughout within the depletion region as they are at the edges in question. Also, current continuity implies the total current, which is the sum of electron and hole currents, should be constant everywhere. So as we go away from the depletion region into either bulk region, while the minority currents decay according to Eqn. 24 and 25, the majority currents increase to compensate and keep the net current constant at every point through the diode.
The overall current density is given by
J net
= J n
+ J p
= q [
D p
L p p n 0
+
D n n p 0
]( e qVA kT
L n
− 1)
If the diode cross-section area is a constant A , the diode current is then
(26)
I d
( V
A
) = AJ net
= qA [
D p
L p p n 0
+
D n n p 0
]( e qVA kT
L n
− 1) = I
0
( e qVA kT
− 1) (27) where I
0 is the reverse saturation current .
The figure below shows the electron (dashed line) and hole (dash-dot line) currents and the net current (solid line) throughout the device under forward bias.
Note that the more lightly doped side will have a higher equilibrium concentration of minority carriers and demonstrate a higher minority current component at the relevant depletion region edge.
Thus, for instance, in the figure above the n-side doping has to be lower than the p-side doping for the currents to have the relative magnitudes depicted. In an example p
+ n-diode, the holes govern the current.
Under reverse bias, the thermally-generated minority carriers within a diffusion length of the depletion region can diffuse to the edge of the depletion region, join the drift current within the depletion region and contribute to the reverse saturation current.
6