Lecture 4: pn Junctions IV Characteristics

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Winter 2021 - John Labram
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• Total of 25 marks.
• Due Thursday 21st January at the start of the lecture
(08:30am).
• Each homework contributes 15% of overall grade.
• I will return it one week later (January 28th).
• Homework 1 consists of content covered in Lectures 1, 2, 3, 4.
• Email it to me at [email protected].
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• You can download the homework from the Canvas or the
course website:
• http://classes.engr.oregonstate.edu/eecs/winter2021/ece61
5/homework.html
• In addition to the pdf document itself, there will be some data
(.csv file) you will need to download.
• You will be required to do some simple processing of this
data.
• I will hold and office hours over Zoom on Monday from 1pm
to 2pm.
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Last Time • We looked at a few generalizations to pn junctions.

0

−

Diode ~
VDC
VAC
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• Carrier Concentrations in Non-Equilibrium.
Qualitative Description of IV
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h+ h+ h+ h+ h+
h+ h+ h+ h+
e- e- e- e- e-
e- e- e- e-
h+ h+ h+ h+ h+
h+ h+ h+ h+
e- e- e- e- e-
e- e- e- e-
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carefully.
(Fermi-Dirac or Maxwell-Boltzmann distributions).
energetic to surmount the potential barrier or field
which pushes them away from the junction.
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Diffusion Current • Some of the energetic carriers surmount the barrier
and transport (against the field) to the opposite side
of the space charge region, where they become
minority carriers (minority carrier injection).
• Within a few diffusion lengths of the junction, they
recombine. This type of current is known as diffusion
current and depends strongly on the barrier height.
• Diffusion current involves injection and
recombination. Although it is almost always referred
to as diffusion current in the literature, denoting it as
injection current can lead to less confusion.
9
10
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+
+
+
+
+
+
-
-
-
-
-
-
e- e- e- e- e-
e- e- e- e-
h+ h+ h+ h+ h+
h+ h+ h+ h+
Drift Current • Drift current depends on minority carriers.
• The field in the space charge region (SCR) is such that
any minority carrier which reaches the edge of the
SCR is accelerated across the SCR by the field (true at
all applied biases).
mechanism for this type of current is the extraction of
minority carriers near the SCR by diffusion to the
edge of the SCR and drift across the SCR.
11
12
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Drift Current • The magnitude of the drift current depends on the
generation rate of minority carriers, which is not
affected by the field. Thus the drift current is
approximately constant, independent of the field.
• Drift current involves extraction and drift. Although
this kind of current is invariably referred to as the
drift current in the literature, denoting it as extraction
current can lead to less confusion.
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How are Carriers Replenished?
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Summary • PN diode current is controlled by the majority carrier
population density / energy.
barrier so that diffusion current dominates
• Forward bias:
• Reverse bias:
• In all bias cases, the thermal drift current remains the
same.
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Ideal I-V Characteristics • We are going to derive the ideal diode or Shockley
Equation that quantitatively describes the flow of
current through a biased pn junction.
= 0 − 1 (1)
• Describes the current that flows in a diode as a
function of voltages applied to it.
• Notice the exponential relationship.
Current density
Reverse saturation
current density
Applied voltage
Thermal energy
Shockley ≠ Schottky • Walter Schottky.
• William Shockley.
• American. 1910-1989.
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are going to split our
device into three
boundaries are
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Carrier Concentrations • Let’s first evaluate hole and electron concentrations
as a function of position in equilibrium: ≠ 0 = 0 = 0
I II III
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A Reminder on Notation
• 0 is hole carrier density on the p-side of the
junction.
• The full-size indicates this is hole density.
• The subscript indicates we are on side of the
junction.
No bias applied
• I.e. 0 is the hole carrier density on the p-side of
the junction in equilibrium.
i.e. a long way from the junction
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• We use the following notation for minority carriers:
• 0 is the electron concentration on the p-side, of
the junction in equilibrium.
• 0 is the hole concentration on the n-side, of the
junction in equilibrium.
concentrations using the law of mass action:
00 = 00 = 2
21
22
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equilibrium: ≠ 0 = 0 = 0
I II III
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Non-Equilibrium Carrier Concentrations • What happens when we apply a bias?
• Forward bias:
equilibrium values.
• Reverse bias:
equilibrium values.
the edges of the depletion region, dying out after
several diffusion lengths, & , and minority
carriers approach equilibrium values
Carrier Concentrations: Forward Bias • In this case the minority carrier concentration is
greater than equilibrium. p-type
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Carrier Concentrations: Forward Bias • In this case the minority carrier concentration is
greater than equilibrium. p-type
of minority carriers in
the junction
Carrier Concentrations: Forward Bias • What does our carrier concentration diagram look
like?
−
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Carrier Concentrations: Reverse Bias • In this case the minority carrier concentration is
decreased below equilibrium.
few reduces
Carrier Concentrations: Reverse Bias • What does our carrier concentration diagram look
like?
−
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depletion region.
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Strategy • To derive the Shockley Equation we are going to do
the following:
neutral region of the pn-junction.
2. Determine boundary conditions at the edge of
the depletion region.
density as function of position.
4. Evaluate hole current density in n-side () and
the electron current density in p-side ().
5. Add hole and electron currents: + .
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Assumptions • To derive the Shockley Equation we have to make the
following assumptions:
region and the bulk of the respective
semiconductors is abrupt. n-typep-type
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semiconductors is abrupt.
2. The Boltzmann Approximation is valid.
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5
0.0
0.2
0.4
0.6
0.8
1.0
=
semiconductors is abrupt.
3. Injected minority carrier densities are low
compared to majority carrier densities.
4. No generation-recombination current exists
inside the depletion layer, and the electron and
hole currents are constant throughout the
depletion layer.
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Fick’s 2nd Law • We are going to use a more general version of Fick’s
2nd Law in our derivation.
• Fick’s 2nd Law is also called the heat equation.
• Describes the diffusion of one species into another as
a function of time and space.
,
diode is 1D
,
37
38
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39
40
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Continuity Equation • If you want to know more about this equation, I have
some notes online from a previous class (ECE/ChE
611 – Lecture 11):
Equations (see next slide).
• We will just use the Continuity Equations, but more
details are available online.
Physics.
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=
+

in equilibrium

= −
−

• : electron mobility.
• : hole mobility.
(4)
(5)
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Steady State • We will approximate the system as being in steady-
state.

FR
• I.e. the system has had time to equilibrate and the
current is not changing with time.
• Even if we are sweeping voltage
or pulsing the device, we know
that generation, recombination
• In steady-state we say:
=
Region III. n-typep-type
• In Regions I and III we assume
that these sides remain ≈ neutral in steady state.
• If we add a hole to the n-
side, we need an electron
to compensate.

=
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=
+

(4)
(5)
• Recall, if we add a hole to the n-side, we need an
electron to compensate.

+

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Continuity Equations
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−
−
Continuity Equations • Gather terms:
− + 2 2
− +
+
2 2
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=
+
• Now let’s look at the other fraction in (14):
(15)
=
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=
+
− −
+
− − −
+
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Continuity Equations • We make the assumption that low-level injection is
taking place:
• In this case we say:
= +
+
mobilities are similar):

− − 0

+ 2 2
= − 0
55
56
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Sze & Ng p92
Fermi Levels: and .
57
58
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Quasi Fermi Levels • When we apply a voltage the Fermi level is no longer
constant throughout the pn junction.
• We wish to quantify the splitting of the Fermi levels.
• Start by describing carrier by Boltzmann Distribution:
= exp −
= exp −

Quasi Fermi Levels • Re-arranging:
= + ln
(24)
(25)
dependent on separation of QFLs at that position:
= 2 exp
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Quasi Fermi Levels • By definition, the splitting of the Fermi Levels is due
to the applied voltage. Hence we can say:
= − (27)
• To evaluate boundary conditions, we have to
consider the edges of the depletion region: = − and = .
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between regions I and II. n-typep-type

• I.e. = −.

61
62
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Charge Density at Boundary

(−) =
2
≈ 0

63
64
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between II and III. n-typep-type

• I.e. = .
(30)
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Continuity Equations • Let’s return to our continuity Equation in the n-region
− − 0
−

− − 0
+
2 2
= 0 (31)
0 ≠ 0
• It is slightly easier to talk about the perturbation. Let:
= − 0
• Since 0 is a constant, we can say:
2
• I.e. 2
67
68
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• We look up the solution and check it works.
• So we can write (33) as:
=
(34)
• Try:
(36)
− +
(36)
• Compare with the ODE:
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− +
(36)
• This is the general form, but we don’t know what or are.
• To get these parameters we need to apply some
boundary conditions.
= − 0
• So from (37) we can say:
→ ∞ = 0
→ ∞ = −∞ +
∞ = 0
In reality ≠ ∞, it is
just very large compared
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=
• To do this we can apply the boundary condition we
derived earlier:
(30)
= 0 exp
− 0

=
− = 0 exp
= = 0
− 1 exp −
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− 1 exp −
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Current Density • We are almost there now....
• We just need to evaluate the current that flows in our
pn-junction.
• Let’s talk about current density, so we don’t need to
worry about device area:
Current (A)
• The total current that flows is due to electrons (in the
p-region) and holes (in the n-region):
= + (40)Current density (A/cm2)
Electron current
Current Density • In lecture 2 we showed that the current density is due
to drift and diffusion currents:
= ,drift + ,diff (41)
• is hole current density due to drift.
• is the magnitude of the fundamental unit of charge.
• is the mobility of holes.
• is the hole number density in the n-side.
• is the electric field strength.
(42)
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78
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,diff =
• is hole current density due to diffusion.
• is the magnitude of the fundamental unit of charge.
• is the diffusion coefficient of holes.
• is the hole number density in the n-side.
• is the diffusion direction.
(42)
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• So we can say:
• We are evaluating the
i.e. = 0.
• Therefore:
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Current Density • Using the solution to the ODE (equation 38) we find
at = :
= = 0
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= 0 exp
• Where we define the reverse saturation current:
0 = 0
+ 0
(46)
0 =
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region and quasi-neutral regions, we derived the
Ideal Diode Equation (the Shockley Equation):
= 0 exp
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Lecture 4: pn Junctions IV Characteristics