Diode Current Equation

1ENT201-Electronic Devices Instructor- V. R. GuptaLecture No. 10
ENT201-Electronic Devices
LectureNo.10
Unit-1
*
Quantitative Theory of the PN-Diode Currents
- Diode Current Equation
V. R. Gupta
Assistant Professor
Department of Electronics Engineering
Shri Ramdeobaba College of Engineering and Management, Nagpur.

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Quantitative Theory of the pn-Diode Currents
• To derive the expression for the total current as a function of
the applied voltage (the volt –ampere characteristics), let us
assume that the depletion width is zero.
• When the forward bias is applied to a diode, holes are
injected from the p-side into the n-side.
• Thus the concentration pn of holes in the n-side is increased
above its thermal equilibrium value pno
• Where, Lp is the diffusion length for holes in n-type
semiconductor
𝑝′
𝑛(𝑥) = 𝑝 𝑛(𝑥) − 𝑝 𝑛𝑜 = 𝑝′
𝑛(𝑜)𝑒− Τ𝑥 𝐿 𝑝
∴ 𝑝 𝑛(𝑥) = 𝑝 𝑛𝑜 + 𝑝′
𝑛(0)𝑒− Τ𝑥 𝐿 𝑝

• The injected or excess
concentration at x=0 is
• The diffusion hole current in n-
side is given by
𝑝′
𝑛(𝑥) = 𝑝𝑛(𝑥)− 𝑝𝑛𝑜 = 𝑝′
𝑛(𝑜)𝑒− Τ𝑥 𝐿 𝑝
∴ 𝑝𝑛(𝑥) = 𝑝𝑛𝑜 + 𝑝′
𝑛(0)𝑒− Τ𝑥 𝐿 𝑝
∴ 𝑝′
𝑛 0 = 𝑝𝑛 0 − 𝑝𝑛𝑜
൯𝒑′
𝒏
(𝟎
𝐼 𝑝𝑛 = −𝐴𝑞𝐷 𝑝
𝑑𝑝 𝑛
𝑑𝑥
since
𝑑𝑝𝑛
𝑑𝑥
=
൯𝑝′
𝑛
(0
𝐿𝑝
𝑒− Τ𝑥 𝐿 𝑝𝐼 𝑝𝑛(𝑥) =
൯𝐴𝑞𝐷 𝑝 𝑝′
𝑛
(0
𝐿 𝑝
𝑒− Τ𝑥 𝐿 𝑝

𝐼 𝑝𝑛(𝑥) =
൯𝐴𝑞𝐷 𝑝 𝑝′
𝑛
(0
𝐿 𝑝
𝑒− Τ𝑥 𝐿 𝑝
• From the above equation it is evident that the injected hole
current decreases exponentially with distance.
• Similarly
• Since the injected concentration is a function of voltage applied
across the pn-diode, thus we can say that Ipn depends upon the
applied voltage V.
𝐼 𝑛𝑝(𝑥) =
൯𝐴𝑞𝐷 𝑛 𝑛′
𝑝(0
𝐿 𝑛
𝑒 Τ𝑥 𝐿 𝑛

• Let us now find the dependence of ൯𝑝′
𝑛(0 upon V.
• From the Boltzmann relationship of kinetic gas theory:
• where
– pp and pn are hole concentrations at the edges of the space charge region
in p and n material respectively.
– VB is the barrier potential across the depletion layer
• Now, consider a pn-junction biased in a forward direction by an
applied voltage V.
• Then the barrier voltage VB is decreased from its equilibrium
value Vo by the amount V, or VB = Vo –V.
𝑝 𝑝 = 𝑝 𝑛 𝑒 Τ𝑉B 𝑉 𝑇

• The hole concentration throughout the p region is constant and
equal to the thermal equilibrium value 𝑝 𝑝 = 𝑝 𝑝𝑜
• The hole concentration varies with distance into the n-side as
shown in figure below
• At the edge of depletion layer
• i.e. at x=0
• Therefore, the Boltzmann relationship
for this case is
൯𝒑′
𝒏(𝟎)𝑝 𝑛 = 𝑝 𝑛(0
𝑝 𝑝(0) = 𝑝 𝑛(0) exp ൫𝑉𝑜 − 𝑉 Τ) 𝑉𝑇
∴ 𝑝 𝑛(0) = 𝑝 𝑛(0)𝑒 Τ𝑉 𝑉 𝑇
Refer next
slide to know
how this
equation has
come

• We know that, in an unbiased pn-junction diode, under equilibrium
condition
• Therefore by combining the equation obtained in previous slide with
the above equation, we obtain
𝑝 𝑝𝑜 = 𝑝 𝑛𝑜exp
𝑉0
𝑉𝑇
𝑝 𝑛(0)exp
𝑉0 − 𝑉
𝑉𝑇
= 𝑝 𝑛𝑜exp
𝑉0
𝑉𝑇
𝑝 𝑛(0)exp
𝑉0
𝑉𝑇
exp −
𝑉
𝑉𝑇
= 𝑝 𝑛𝑜exp
𝑉0
𝑉𝑇
𝑝 𝑛(0)exp −
𝑉
𝑉𝑇
= 𝑝 𝑛𝑜
𝑝 𝑛(0) = 𝑝 𝑛𝑜exp
𝑉
𝑉𝑇
This boundary
condition is called the
Law of the junction

• Thus, the hole concentration ൯𝑝′
𝑛(0 injected into the n-side at the
junction is obtained as given below:
• Thus the hole current crossing the junction into n-side, with x=0, is
given by
• Similarly, the electron current crossing the junction into the p-side, with
x=0, is given by
𝑝′
𝑛(0) = 𝑝 𝑛𝑜 exp( Τ𝑉 𝑉𝑇) − 1
𝐼 𝑝𝑛(0) =
𝐴𝑞𝐷 𝑝 𝑝 𝑛𝑜
𝐿 𝑝
exp( Τ𝑉 𝑉𝑇) − 1
𝐼 𝑛𝑝(0) =
𝐴𝑞𝐷 𝑛 𝑛 𝑝𝑜
𝐿 𝑛

• Finally, the total current I is the sum of 𝐼 𝑝𝑛 0 𝑎𝑛𝑑 𝐼 𝑛𝑝(0).
• Therefore
• Where,
• Io is the reverse saturation current.
൯𝐼 = 𝐼 𝑝𝑛(0) + 𝐼 𝑛𝑝(0
𝐼 =
𝐿 𝑝
+
𝐿 𝑛
𝐼 = 𝐼0 exp( Τ𝑉 𝑉𝑇) − 1
𝐼0 =
𝐿 𝑝
+
𝐿 𝑛

• Note that
– Throughout this discussion, we have neglected carrier generation
and recombination in the space-charge region.
– Such an assumption is valid for a germanium diode, but not for a
silicon device.
– If we consider the carrier generation and recombination in the space-
charge region, the general equation of the diode current is
approximately given by
– Where, For silicon η = 2 and for Germanium η=1.
𝑰 = 𝑰 𝟎 𝒆 Τ𝑽 𝜼𝑽 𝑻 − 𝟏

Figure: The minority (solid) and the majority (dashed) currents vs
distance in a PN-diode. It is assumed that no recombination takes
place in the very narrow depletion region.

Reading Assignment
• Chapter-3 of Milliman's Integrated Electronics
(2nd edition) Book.
• section 3.3.
• AND
• Chapter-5 of Milliman's Electronic Devices and Circuits
(3rd or 4th edition) Book.
• section 5.4 and 5.5.

Diode Current Equation

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