The p-n Junction (The Diode)
•
•
Derivation of ideal diode equation covered in the SMA Device Course
Development here introduces the fundamental materials concepts
p-type material in equilibrium
n-type material in equilibrium
p~Na
n~Nd
n~ni2/Na
p~ni2/Nd
⎛N
E F = −kbT ln⎜⎜ a
⎝ NV
⎛N
E F = E g + kbT ln⎜⎜ d
⎝ NC
⎞
⎟⎟
⎠
⎞
⎟⎟
⎠
Ec
Ec
EF
EF
Ev
Ev
What happens when you join these together?
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©1999 E.A. Fitzgerald
Joining p and n
p
n
Ec
EF
Ev
Carriers flow under driving force of diffusion until EF is flat
-
-
+
+ +
+
Holes diffuse
Electrons diffuse
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©1999 E.A. Fitzgerald
-
-
-
-
-
+
+
+
+ + + +
+
Holes diffuse
Electrons diffuse
-
- - - - -
+
+ +
+
E
+
+ +
+
An electric field forms due to the fixed nuclei in the lattice from the dopants
Therefore, a steady-state balance is achieved where diffusive flux of the carriers is balanced by the drift flux
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©1999 E.A. Fitzgerald
-
- - - - -
+
+ +
+
xp
+
+ +
+
Metallurgical junction
xn
W: depletion or space charge width
ρ
N d xn = N a x p
ρ (x)
E=∫
dx
ε
V = ∫ E(x)dx
©1999 E.A. Fitzgerald
E
Vbi
V
xp =
Nd
2ε r ε oVbi
e
Na (Nd + Na )
xn =
Na
2ε r ε oVbi
e
Nd (Nd + Na )
W=
2ε r ε oVbi N a + N d
e
Nd Na
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What is the built-in voltage Vbi?
p
n
np
eVbi
nn
pp
Ec
EF
eVbi
Ev
pn
eVbi=EFn-EFp
E Fp
⎛ p
= −k bT ln⎜⎜
⎝ NV
⎞
⎛N
⎟⎟ = −k b T ln⎜⎜ a
⎠
⎝ NV
∴Vbi =
⎞
⎟⎟
⎠
E Fn
⎛ p
= −kbT ln⎜⎜ n
⎝ NV
k bT ⎛ N a N d
ln⎜⎜
2
e
⎝ ni
⎞
⎛ ni2
⎟⎟ = −kbT ln⎜⎜
⎠
⎝ NV N d
⎞
⎟⎟
⎠
⎞
⎟
⎟
⎠
We can also re-write these to show that eVbi is the barrier to minority carrier injection:
pn = p p e
−eVbi
k bT
n p = nn e
−eVbi
k bT
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©1999 E.A. Fitzgerald
Qualitative Effect of Bias
• Applying a potential to the ends of a diode does NOT increase current through
drift
• The applied voltage upsets the steady-state balance between drift and
diffusion, which can unleash the flow of diffusion current
• “Minority carrier device”
np
eVbi
nn
pp
eVbi
pn
n p = nn e
−e(Vbi −Va )
kbT
Ec
EF
pn = p p e
+eVa
-eVa
Ev
−e(Vbi−Va )
k bT
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©1999 E.A. Fitzgerald
Qualitative Effect of Bias
• Forward bias (+ to p, - to n) decreases depletion region, increases diffusion
current exponentially
• Reverse bias (- to p, + to n) increases depletion region, and no current flows
ideally
Forward Bias
Reverse Bias
np
np
eVbi-e|Va|
nn
Ec
EF
nn
Ec
eVbi-e|Va|
pp
pp
Ev
pn
eVbi+e|Va|
EF
eVbi+e|Va|
Ev
pn
+
Va
I
V=f(I)
qV
a
⎞
⎛ qV
⎞
⎛ De ni2 Dh ni2 ⎞⎛⎜ kbTa
k
T
b
⎟
⎜
⎟ e −1 = J o e −1⎟
J = q⎜⎜
+
⎟
⎟
⎜
⎟
⎝ Le N a Lh N d ⎠⎜⎝
⎠
⎝
⎠
Di
μi
©1999 E.A. Fitzgerald
=
k bT
q
Rectification,
Non-linear, Non-Ohmic
Linear,
Ohmic
V
Li = Diτ i
7
V=IR
Other means to create internal potentials: Heterojunctions
• Different semiconductor materials have different band gaps and electron
affinity/work functions
• Internal fields from doping p-n must be superimposed on these effects:
Poisson Solver (dE/dx=V=ρ/ε)
Vacuum level
ϕ1
ϕ2
EF
Thin films
©1999 E.A. Fitzgerald
Substrate
Eg1
Eg2
Potential barriers for holes
and electrons can be created
inside the material
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Quantum Wells
EC
n=3
hν
n=2
n=1
EV
L
If we approximate well as having infinite potential boundaries:
k=
nπ
L
for standing waves in the potential well
h 2k 2
h2n2
E=
=
2m* 8m* L2
We can modify electronic
transitions through quantum wells
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©1999 E.A. Fitzgerald