Part III- Device Operations
Eric Polizzi
UMass, Amherst
ECE-344
http://www.ecs.umass.edu/∼polizzi
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
1
I.4 Biased Junction
An external bias, VA , is now applied to the junction.
Assumption: the drop of potential across the quasi-neutral region is
negligible. Therefore the potential drops across the transition region.
Figure 5.10
The voltage drop (i.e. new potential barrier) is
V (xN ) − V (−xP ) = Vbi − VA
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
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I.4.a Definition
All the expressions we derived so far for ε, V , xN , xP and W , stay
identical but Vbi must be replaced by Vbi − VA . 5.32 to 5.38
Remark
If VA > 0, we get a Forward bias, and the dimension of the
transition region decreases (i.e. xN , xP , and W all decrease)
If VA < 0, we get a Reverse bias, and the dimension of the
transition region increases (i.e. xN , xP , and W all increase)
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
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I.4.b Forward bias VA > 0
Figure 5.11
At non-equilibrium, EF is not unique and constant anymore and one
defines the quasi-Fermi-level for e− and h+ , such that
FN − FP = qVA
Figure 5.12
The forward bias decreases the potential barrier for e− and h+ , and
diffusion and drift forces are not longer equal with opposite signs.
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
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I.4.b Forward bias VA > 0
As a result:
e− can flow from N → P
h+ can flow from P → N
Figure 6.1
Diffusion processes>Drift phenomena
The total current is going from P to N, and it is expected to increase
exponentially with VA .
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
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I.4.c Reverse bias VA < 0
Figure 5.11
At non-equilibrium, EF is not unique and constant anymore and one
defines the quasi-Fermi-level for e− and h+ , such that
FP − FN = qVA
Figure 5.12
The reverse bias increases the potential barrier for e− and h+ , and
diffusion and drift forces are not longer equal with opposite signs.
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
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I.4.b Reverse bias VA < 0
As a result:
The Electric field increases
The carrier diffusion is stopped
Figure 6.1
Diffusion processes<Drift phenomena
The total current is going from N to
P, however, it is expected to be very
small since only few carriers available (involves only minority carriers). The P-N behaves then like a
diode rectifying current flow
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
7
I.5 I-V Characteristics
So far, using the depletion approximation and an applied bias VA :
For obtaining the current, we can make use of (A cross sectional area):
I = AJ
6.2;
J = Jn (x) + Jp (x)
6.3
with
dn
Jn (x) = qnµn ε +qDn
dx
6.4
Jp (x) = qpµp ε −qDp dp
dx
Remark: The Total current is independent of the position x.
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
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I.5.a General Considerations
If n and p were available (or the electrostatics potential V (x)) we
would be able to calculate the current. This could be done
numerically.
In this course, however, we propose to derive an approximate
analytical expression for the current.
Since it could be difficult to derive the current for the majority
carriers (e.g. n for the N-side and p for the P-side), we propose to
concentrate first on the minority carriers (e.g. p for the N-side and
n for the P-side).
Using the depletion approximation for the N-side and P-side (with
ε = 0), it comes 6.6
dp
if x ≥ xN 6.6b
dx
dn
P-side: Jn (x) = qDn
if x ≤ −xP 6.6a
dx
N-side: Jp (x) = −qDp
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
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I.5.a General Considerations
The following minority carrier diffusion equations are obtained by
replacing the expression of the current into the continuity equation
and by assuming: (i) steady state, (ii) no electric field, (iii) SRH
mechanism, (iv) no extrinsic generation rate:
d 2 pN
pN (x) − pN0
−
= 0 if x ≥ xN 6.5b
dx 2
τp
d 2 nP
nP (x) − nP0
P-side: Dn
−
= 0 if x ≤ −xP 6.5a
dx 2
τn
+
where pN means density of h in the N-side, etc.
N-side: Dp
Let us try to solve for nP in order to obtain Jn for −∞ < x ≤ −xP
(e.g. the diode is infinitely long). We need then two boundary
conditions:
at x → ∞, nP (−∞) = nP0 , because of thermodynamic equilibrium
far from the junction.
at x = −xP , nP (−xP ) =???
We must obtain the above boundary condition in order to solve the
electron diffusion equation for the P-side.
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
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I.5.a General Considerations
For non-equilibrium situation, we have seen that (3.72)
n(x) = ni exp(β(Fn − Ei (x)))
p(x) = ni exp(β(Ei (x) − Fp )))
where Fn and Fp are imref.
By multiplying the above equation we obtain
np = ni2 exp(β(Fn − Fp ))
6.11
which is valid in the entire diode under non-equilibrium condition.
Remark: if Fn = Fp = EF (e.g. equilibrium) we get np = ni2 .
In the transition region, we will assume a monotonic variation for
the imref, such that Fn − Fp = qVA
Figure 6.4b
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
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I.5.a General Considerations
It comes in the transition region as well (−xP ≤ x ≤ xN ):
np = ni2 exp(qβVA )
6.12
Finally, at x = −xP , assuming p(−xP ) = NA and with
ni2 /NA = nP0 ’:
nP (−xP ) = nP0 exp(qβVA )
6.14
√
Denoting Ln the diffusion length such that Ln = Dn τn
6.22
the general form of the solution 6.5b can be put into this form:
−x
x
nP (x) = nP0 + A exp(
) + B exp( )
Ln
Ln
using the boundary conditions:
nP (−∞) = nP0 leads to A = 0
xP
) leads to
Ln
xP
B = nP0 (exp(qβVA ) − 1) exp( )
Ln
nP (−xP ) = nP0 exp(qβVA ) = nP0 + B exp(−
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
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I.5.a General Considerations
It comes:
nP (x) = nP0 + nP0 (exp(qβVA ) − 1) exp(
x + xP
)
Ln
and then using 6.6a for x ≤ −xP
Jn (x) = qnp0
Dn
x + xP
(exp(qβVA ) − 1) exp(
)
Ln
Ln
6.27a
Similarly we can obtain for the minority holes in the N-side:
pN (x) = pN0 + pN0 (exp(qβVA ) − 1) exp(−
x − xN
)
Lp
and then using 6.6b for x ≥ xN
Jp (x) = qpN0
Eric Polizzi (UMass, Amherst)
Dp
x − xn
(exp(qβVA ) − 1) exp(−
)
Lp
Lp
Part III- Device Operations
6.27b
ECE-344
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I.5.a General Considerations
Finally we have obtained the minority current on both the N and P
sides i.e.
Remarks:
Jp (x) decreases if x increases (resp. Jn (x) increases if x
decreases). This is because of the recombination process, here
the h+ current is transformed into an e− current. So we can guess
why the total current is independent of the position x.
To calculate the total current, apriori one would need to know
either Jp in the P-type or Jn in the N-region. But these currents
involve majority carriers and are difficult to derive.
Therefore, in practice, the total current will obtain using the ideal
diode approximation.
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
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I.5.b Ideal Diode Approximation
Using the continuity equation inside the depletion region, we get:
0=
1 d
Jn − Un
q dx
and
0=−
1 d
Jp − Up
q dx
6.7
Ideal diode approximation means “No Generation/Recombination in
the depletion region” i.e.
Un = Up = 0
i.e. Jn and Jp are constant if −xP ≤ x ≤ xN , therefore
Jn (−xP ≤ x ≤ xN ) = Jn (−xP )
6.8
Jp (−xP ≤ x ≤ xN ) = Jp (xN )
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
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I.5.b Ideal Diode Approximation
As a result, there is now an overlap between Jn and Jp :
The total current can be obtained by:
J = Jn (−xP ) + Jp (xN )
6.9
which leads to the Shockley’s equation:
J = J0 (exp(qβVA ) − 1)
with J0 the saturation current density:
Dp
Dn
J0 = q
nP +
pN
Ln 0
Lp 0
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
6.29
6.30
ECE-344
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I.5.b Ideal Diode Approximation
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
17
I.5.c Deviation from ideality
See Figure 6.9, Figure 6.10a, Figure 6.10b
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
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I.5.c Deviation from ideality
Observations from the graphs:
Reverse bias (VA < 0): The current does not saturate but
continue to slightly increase, while finally, we get a “breakdown”
for large reverse bias.
Forward bias (VA > 0): “Ideal characteristics” are obtained
between 0.35 − 0.7V ; for small VA the slope is different than
q/(kB T ) and equal to q/(2kB T ); for large bias (VA → Vbi ) the
slope decreases progressively i.e. “slope over”
In the following, we propose to investigate the effect of:
R-G current in the depletion region
High reverse bias which leads to junction breakdown
Large forward bias which involves high-level injection carriers.
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
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I.5.c Deviation from ideality
The R-G current: we now consider Generation/Recombination in the
depletion region, and we known that these processes are proportional
to np − ni2 . We previously established that
np = ni2 exp(qβVA )
If VA > 0, np > ni2 , and recombination can be observed in the transition region. R-G current will then
decrease “the ideal current”.
6.12
Figure 6.15b
If VA < 0, np < ni2 , and generation will take place in the transition region. R-G current will then be
added to the “the ideal current”.
Figure 6.15a
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
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I.5.c Deviation from ideality
In contrast to the ideal case Un and Up are not equal to zero, and we
have from the continuity equation at the steady state:
d
d
Jn = − Jp = qU
dx
dx
Integrating from −xP to xN , it comes:
Z xN
Jn (xN ) = Jn (−xP ) + q
U(x)dx
−xP
the net current is:
J = Jp (xN ) + Jn (xN ) = Jp (xn ) + Jn (−xP ) + JG/R = Jideal + JG/R
6.47
and where JG/R denotes the integration on the intrinsic G/R; using
SRH one can derive the following expression:
JG/R = q
ni W
exp(qβVA ) − 1
√
2τ0 1 + qβ(Vbi − VA ) τn τp exp(qβVA /2)
2τ0
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
6.45
ECE-344
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I.5.c Deviation from ideality
As a result:
If VA > 0 (forward bias):
for small bias, the I-V curve follows
exp(
qVA
)
2kB T
which is characteristics of a recombination dominated current
for larger bias, the I-V curve follows
exp(
qVA
)
kB T
where the diffusion current takes over and completely overshadows
the recombination current
If VA > 0 (reverse bias): JG/R is added to jideal so the current
keeps increasing slowly with VA decreasing.
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
22
I.5.c Deviation from ideality
Junction breakdown: If VA < 0 and strong reverse bias, the electric
field near the junction can reach very high values.
When an e− is accelerated to high electric field, its energy can be
equal or larger than the energy bandgap. This energy can be released
(via collision) while creating an e− − h+ pair. So instead of having one
e− with high kinetic energy, we get two e− and one h+ subject to the
same high electric field.
It results an avalanche multiplication phenomenon. Figure 6.12
At some point, a sudden increase of current will be observed leading
to the breakdown
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
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I.5.c Deviation from ideality
High-Level Injection Carriers: If VA > 0 and large (VA → Vbi ), the
ideal diode model begins to fail.
For low-level injection, the changes in majority carriers at x = −xP and
x = xN become non-negligible, and the associated boundary
conditions for the minority carriers used for the derivation of the ideal
model, are not valid anymore.
Figure 6.17
The new current is expected instead to vary as ∼ exp(qβ/2)
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
24
I.5.c Deviation from ideality
Summary:
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
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I.5.d Zener diodes
If N-type and P-type are heavily doped:
The width of the transition region is very small.
In reverse bias regime, the energy bandgap is equivalent to a
potential barrier (potential wall) in quantum mechanics.
In reverse bias regime, e− in the valence band on the P-side, can
then directly tunnel through the bandgap to the conduction band
on the N-side. (i.e. Quantum tunneling)
which is equivalent to
Fig. 6.14
Fig. 6.13
As a result, he voltage breakdown can be accurately controlled while
adjusting the doping concentration. These diodes are called Zener
diodes used as voltage reference.
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
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I.5.e Narrow-base diodes
Previously we assumed that the length of the P and N regions
were much greater than the diffusion length of the minority
carriers (i.e. Ln and Lp ).
The term short-base or narrow-base comes from bipolar
transistors that consist in two P − N junctions PNP or NPN where
the central region is called ’base’
For short-base diode, the previous assumption for calculating the
minority carriers in the region pN (x → ∞) = pN0 must be replaced
by pN (b) = pN0
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
27
I.5.e Narrow-base diodes
As a result, the minority holes will not be able to recombine in the narrow
base (i.e. since b ∼ Lp ) and the current Jp can be assumed constant.
If Jp is constant, from Jp = −qDp (dp/dx) we can extract
p(x) = −
Jp x
+c
qDp
0≤x ≤b
and p(x) linear
at xN we know that p(xN ) = pN0 exp(qβVA ), it comes
Jp xN
c = pN0 exp(qβVA ) +
qDp
and then
p(x) = −
Jp
(x − xN ) + pN0 exp(qβVA )
qDp
Since pN (b) = pN0 , one can get an expression for Jp
Jp =
Eric Polizzi (UMass, Amherst)
qDp pN0
(exp(qβVA ) − 1)
b − xNPart III- Device Operations
6.68, 6.69
ECE-344
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I.6 P-N Junction Capacitance
So far, we considered steady state characteristics, however, it is also
important to determine how quickly the device can adjust to a new bias
condition
Capacitance is a measure of charged stored per unit of charge of
voltage
As a result, if capacitance is large, more charges must be moved
in or out, and for a fixed current, more time is then needed to
complete the process
Capacitance calculations help to estimate the ’time response’ of
the device
In P-N junction, there exist two major capacitance:
The depletion capacitance which is capacitance associated with
the charges which must be moved in or out the depletion region
The diffusion capacitance due to minority carriers under forward
bias
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
29
I.6.a Depletion capacitance
We know that the size of the depletion region (i.e. xN and xP ) depends
on VA
The total charge of the fixed, ionized impurities in each depletion region
is (absolute value):
Q = qND xN = qNA xP
where the cross section area A is set to 1 and we consider a step
junction.
If we associate xN0 and xP0 with a given potential VA , and xN and xP with
a given potential VA + ∆VA (∆VA > 0), the variation of the charges in the
depletion region is represented as follows:
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
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I.6.a Depletion capacitance
The capacitance is defined as:
dQ CT = dVA It comes for the step junction:
dxN CT = qNd dVA and after calculations, one can show:
CT =
xN + xP
The result is then equivalent to the capacitance of parallel plate
capacitor with a dielectric permittivity .
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
31
I.6.b Diffusion capacitance
Looking at the h+ density in the N-type region pN (x) i.e.
pN (x) = pN0 + pN0 (exp(qβVA ) − 1) exp(−
x − xN
)
Lp
we defined the excess hole density as ∆pN (x) = pN (x) − pN0 , and
with variation of bias potential VA , it comes:
The total charge density is then
Z ∞
∆pN (x) = qpN0 Lp exp(qβVA ) − 1
Q=q
xN
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
32
I.6.b Diffusion capacitance
The diffusion capacitance for the h+ injected into the N-side:
dQ
CDp =
= q 2 Lp βpN0 exp(qβVA )
dVA
which is actually equivalent to
CDp = qτp βJp (xN )
Similarly the diffusion capacitance for the e− injected into the
P-side is
CDn = qτn βJn (−xP )
The total diffusion capacitance needs to account for e− and h+ in
parallel:
CD = CDn + CDp = qβ {τp Jp (xN ) + τn Jn (−xP )}
Eric Polizzi (UMass, Amherst)
Part III- Device Operations
ECE-344
33
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