p-n junction under forward bias
q (Vbi- VR)
q VR
q VF
=qV
qV
bi o qV
qVoq(V
- qV-V )
bi
F
V>0: the bias is called POSITIVE or Forward,
if the polarity is opposite to the built-in barrier.
Typical notation: VF. Forward bias REDUCES the potential barrier
V<0: the bias is called NEGATIVE or Reverse
if the polarity is the same as the built-in barrier.
Typical notation: VR. Reverse bias INCREASES the potential barrier
1
Electron injection
q (Vbi- VR)
q VR
q VF
=qV
qV
bi o qV
qVoq(V
- qV-V )
bi
F
V =0; The electron concentration on the p-side: n p = nn e
−
qVbi
kT
Also, n p = n p0 (equilibrium electron concentration on the p-side)
because the junction is in equilibrium
V >0; New barrier height = qVbi - qV.
n p = nn e
−
qVbi − qV
kT
= n p0
qV
e kT
2
Hole injection
In the equilibrium
pn0 − qVbi / kT
=e
p p0
(1)
Under forward bias VF, the barrier decreases by qVF:
pn
= e− q(Vbi −V ) / kT
p p0
(2)
Dividing (2) by (1):
pn
= e qV / kT
pn 0
(3)
pn = pn0 eqV / kT
The concentration of injected holes exponentially increases with the forward voltage
3
Summary of carrier injection under forward bias
Injected electron and hole concentrations:
=qVo qV
n p = n p 0× eqV / kT
qVo - qV
-xp0
xn0
pn = pn0× eqV / kT
The change in the electron concentration in the p-region:
∆n p = n p − n p0 = n p0 (eqV / kT − 1)
The change in the hole concentration in the n-region:
∆pn = pn − pn0 = pn0 (eqV / kT − 1)
4
Carrier injection and
forward current mechanism in p-n junction
=qV
qV
bi o
qV
q VF
qVo - qV
Forward bias is applied: VF>0.
As compared to equilibrium conditions, much more electrons can overcome the barrier.
Electrons are injected into the p-side.
Injected electrons recombine with holes.
To compensate the loss, more electrons and holes are pulled from the n- and p-contacts
correspondingly.
5
Injected carrier spatial distribution (1)
=qVo qV
Note that the carrier injection
takes place OUTSIDE the spacecharge region of the p-n junction.
qVo - qV
-xp0
xn0
We assume that the space charge
region is thin so that hot carriers
quickly drift through the
depletion region due to high
initial velocity.
Outside the depletion region the
field is low. Therefore the
electron and hole transport can
only be due to diffusion.
6
Injected carrier spatial distribution (2)
The injected carriers diffuse into n- and p-portions of the junctions and
RECOMBINE along with DIFFUSION;
SpaceCharge
region
SpaceCharge
region
7
Injected carrier spatial distribution (3)
The steady-state distribution of the injected holes can be obtained from the
following balance equation:
∂p
={Income} −{Outcome} = 0 , or
∂t
pn − pn0
∂ 2 pn
−
+ Dp
=0
2
τp
∂x
The solution is:
δ pn ( x ) = δ pn (0) e
− x / Lp
where
Lp = τ p D p
is the diffusion length of the holes.
8
Injected carrier spatial distribution (4)
This solution describes non-uniform spatial distribution of the injected carriers:
δ pn ( x ) = δ pn (0) e
− x / Lp
The solution applies to the
regions that begin at the
space-charge region edges.
Thus, pn(0) is measured at x
= xn
9
Injected carrier voltage and distance dependences
As a function of distance from the depletion region edges, the injected carrier
concentration can be found as
n p ( x p ) ≈ n p0× eqV / kT × e
pn ( xn ) ≈ pn0× eqV / kT × e
− x p / Ln
(for np >> np0)
− xn / L p
(for pn>> pn0)
Injected (excessive) electron and hole concentrations:
∆n p ( x p ) = n p 0 × (eqV / kT − 1) × e
∆pn ( xn ) = pn0 (eqV / kT − 1) ⋅ e
xp
xn
− x p / Ln
− xn / L p
xn and xp are the absolute distances
counted from the edges of the space
charge regions on the n- and p- sides
correspondingly.
We assume that xn0 << Ln, xp0 << Lp
10
Recombination current of the forward biased p-n junction
Injected (excessive) electron and hole concentrations in the p-n junction:
∆n p ( x p ) = n p 0 × (eqV / kT − 1) × e
∆pn ( xn ) = pn0 (eqV / kT − 1) ⋅ e
− x p / Ln
− xn / L p
Total excessive injected charges (A=junction area):
(
)
∞
∆Q p = q×A×∫ ∆pn ( x ) dx = q×A×pn0 × ( eqV / kT − 1)×L p
0
∞
∆Qn = q×A×∫ ∆n p ( x ) dx = q×A×n p 0 × eqV / kT − 1 ×Ln
0
The charge ∆Qn dissolves at the rate of ∆Qn/τn
The charge ∆Qp dissolves at the rate of ∆Qp/τp
The current supplied from the contacts
to compensate the recombination:
I =∆Qn/τn+ ∆Qp/τp
Total injected
electrons
per unit area
Total injected
holes
per unit area
11
I
Recombination current of the forward biased p-n junction (2)
(
)
(
∆Qn = q×A×n p 0 × eqV / kT − 1 ×Ln
)
∆Q p = q×A×pn0 × eqV / kT − 1 ×L p
The compensating current:
I =∆Qn/τn+ ∆Qp/τp
(
)
(
)
I = q×A×n p 0 × eqV / kT − 1 ×Ln / τ n + q×A×pn0 × eqV / kT − 1 ×L p / τ p
Electron current
(
I = q×A× e
qV / kT
Hole current
⎛
Lp
Ln
− 1 × ⎜ n p 0 × + pn0 ×
⎜
τn
τp
⎝
)
Ln, p = Dn, p τ n, p
(
I = q×A× e
qV / kT
>
τ n, p = L2n, p / Dn, p
⎛
Dp
Dn
− 1 ×⎜ n p0 ×
+ pn0 ×
⎜
Ln
Lp
⎝
)
⎞
⎟
⎟
⎠
⎞
⎟
⎟
⎠
12
Total p-n junction current
(
I = q×A× e
qV / kT
⎛
Dp
Dn
− 1 ×⎜ n p0 ×
+ pn0 ×
⎜
Ln
Lp
⎝
)
⎞
⎟
⎟
⎠
This current can be re-written as:
⎡ ⎛ qV ⎞ ⎤
I = IS ⎢exp ⎜ ⎟ −1⎥
⎣ ⎝ kT ⎠ ⎦
where
⎛ Dp pn0 Dnn p0 ⎞
⎟
+
IS = q A⎜
⎜ Lp
Ln ⎟⎠
⎝
13
P-n junction current – voltage characteristic (I-V)
⎡ ⎛ qV
I = I S ⎢exp⎜
⎣ ⎝ kT
⎞ ⎤
⎟ − 1⎥
⎠ ⎦
Forward current
14
p-n junction under reverse bias
q (Vbi- VR)
=qV
qV
bi o
q VR
qV
F
qVo - qV
V<0, the reverse bias. The external voltage increases the built-in barrier.
As compared to the equilibrium, even less electrons and holes can overcome the barrier
15
The concentrations of holes in the n-region and electrons
in the p-region:
pn = pn0 eqV / kT << pn0
n p = n p 0 eqV / kT << n p 0
(V < 0)
(V < 0)
The concentrations of injected electrons and holes
exponentially decrease with the reverse voltage
16
Reverse current of the p-n junction
Using the same expression as for the forward current
⎡ ⎛ qV ⎞ ⎤
I = IS ⎢exp ⎜ ⎟ −1⎥
⎣ ⎝ kT ⎠ ⎦
note that
⎛ qV ⎞
exp ⎜ ⎟ →0
⎝ kT ⎠
when V<0 and |V| >> kT/q
Note that kT/q = 0.026 V.
Typical reverse bias, |VR| >> kT/q
From this:
I ≈− IS
⎛ Dp pn0 Dnn p0 ⎞
⎟
IS = q A⎜
+
⎜ Lp
⎟
L
n
⎝
⎠
17
Reverse current of the p-n junction
⎡ ⎛ qV
I = I S ⎢exp⎜
⎣ ⎝ kT
⎞ ⎤
⎟ − 1⎥
⎠ ⎦
I ≈− IS
at negative V
Reverse current
18
The origin of the p-n junction reverse current
⎡ ⎛ qV
I = I S ⎢exp⎜
⎣ ⎝ kT
⎞ ⎤
⎟ − 1⎥
⎠ ⎦
IS
1.
2.
3.
Zero bias: the thermo-generated current of minority carriers is compensated by the
diffusion of “hot” carriers.
Positive (forward) bias: thermo generated current the same as at zero bias; the
diffusion current increases due to lower barrier. Total current ~ the diffusion current
Negative (reverse) bias: thermo generated current the same as at zero bias; the
diffusion current decreases due to higher barrier.
Total current ~ the thermo-generated current
19
Electron – hole flow balance in p-n junction
20