Lecture 02
Diodes

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topics
•
•
•
•

Semiconductor physics
Diode forward characteristic
Diode reverse characteristic
Special diodes

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Solid-state material
• Insulator (SiO2)
– Strong covalent bonds, no free electron
−8
−1
σ
<
10
(
Ω
−
cm
)
– Conductivity
1eV ≡ 1.6 ×10 −19 Joule
– Energy gap Eg ≥ 3eV
eV: eletron voltage
• Semi-conductor (Si, Ge)
– Conductivity 10 −8 < σ < 103 (Ω − cm) −1
– Energy gap 0 < Eg < 3eV

• Conductor (Cu, Ag)
– Weak covalent bonds, many free electrons
– Conductivity σ > 103 (Ω − cm) −1
– Energy gap Eg = 0eV
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Energy gap

Energy level
high energy
Infinite many atoms
Low energy

conduction band
Energy-level gap

valance band

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One atom
Level 3 : high energy
Level 1 : low energy

electrons


conduction band
conduction band

conduction band
Energy gap overlap
valance band

Energy-level gap
valance band

Energy-level gap
valance band

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Famous Semi-conductors

• Element semi-conductor
– Si
• Cheap
• Energy gap > Ge Æ small leakage
current
• Stable oxide

– Ge
• First transistor
• Small energy gap
• Unstable oxide
• Compound semi-conductor
– GaAs

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Intrinsic semiconductor (silicon)
At very low temperature
ion

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Intrinsic semiconductor at room temperature
Mass-action law:


n = p = ni ⇒ np = ni2
3 − KTG

ni = BT e
2

E

For silicon semi-conductor
Material parameter

B = 5.4 ×1031
EG = 1.12eV
Boltzmann’s constant

k = 8.62 × 10 −5 eV K
At room temperature
Free electrons and holes generated
by thermal ionization, so the
concentration is same n = p = ni

T ≈ 300 K
ni ≈ 1.45 ×1010 carriers

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cm3

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Hole and electrons moving
1. drift
Drift velocity

v = μE

Resistivity

R=ρ l

E=

E : electric field strength (V/cm)
μ : mobility of hole/electron (cm2/V-sec)

l

A
Charge density ρ ≡ nq ( Ω − cm )
conductivity

A

e


e

J = σE
J n = qnμ n E
J drift = q ( pμ p + nμ n ) E

e

Nq Nqv
=
T
L
I
Nqv
J= ⇒J=
A
AL
N
n≡
⇒ J = nqv
LA
I≡

σ ≡ nq μ

J p = qpμ p E

f
q


For intrinsic silicon :
2
cm
μ p = 480
V ⋅s
2
cm
μ n = 1350

Electron density

V ⋅s
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2. diffusion
For intrinsic silicon :
q = 1.6 × 10 C
2
cm
D p = 12
J : current density
s
2
q : electron charge

cm
Dn = 34
D : diffusivity of hole/electron
s
−19

dp
J p = −qD p
dx
dn
J n = qDn
dx

p

n

electron diffusion

hole diffusion
Jp

Jn

x

x

The conventional current direction is the positive charge flow direction
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Currents in semi-conductor = drift current + diffusion current

dp
J p = pqμ p E − qD p
dx
dn
J n = nqμ n E + qDn
dx
r
E

r
E
hole

e−

+

Jn

Jp


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Einstein relationship : relationship between drift current and diffusion current

Dn

μn

=

Dp

μp

= VT

At room temperature (20oC)VT

kT
VT ≡
q
k = 1.38 × 10

VT thermal voltage

溫度伏特當量

≈ 25mV

Boltzmann’s constant

− 23 joules
kelvin

T = 273+ oC

T = 20o C → 293o K ⇒ VT ≈ 25mV
T = 27 o C → 300o K ⇒ VT ≈ 26mV

q = 1.6 × 10 −19 coulomb
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Current in solid-state
material

• Insulator

JT = 0


• Semi-conductor
JT = J p + J n

dp
J p = pqμ p E − qD p
dx
dn
J n = nqμ n E + qDn
dx

• Conductor
J T = J n = nqμ n E
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N-type impure semiconductor (donor)
Majority carriers are negative charges : electrons

N D : Donor atoms concentration
nno : Free electrons concentration

In thermal equilibrium

nno ≈ N D


n-type
from semiconductor physics
Mass-action law
donor atom

nno pno = ni2

ni2
pno ≈
ND

+
Doped +5 atoms: Sb, P, As
銻,磷,砷

electron
donor ion (positive ion)

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P-type impure semiconductor (acceptor)
Majority carriers are positive charges : holes

N A : Acceptor atoms concentration

p po : Free holes concentration

In thermal equilibrium

p po ≈ N A
from semiconductor physics
acceptor atom

Doped +3 atoms: B, Ga, In,
硼,鎵,銦

hole

p po n po = ni2
ni2
n po ≈
NA

acceptor ion (negative ion)
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+3
+3


+5
+3

+5
+5
圖片來自;www.rvweb.fr/.../electricite/partie3_4.php

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≈ 0.5μm
Depletion region

圖片來自;www.rvweb.fr/.../electricite/partie3_4.php

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PN Junction under open-circuit
Space charge


(≈ 0.5μm)

I =0

I D : Diffusion current
I S : Drift current
∴ I = 0 → ID = IS
Contact difference of potential
N N
Vo = VT ln( A 2 D )
ni

V0

For silicon at room temperature

Vo = 0.6 ~ 0.8V

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dp
dx
dn

J n = nqμ n E + qDn
dx
dp
J p = 0 ⇒ pqμ p E = qD p
dx
D p dp
E=
pμ p dx
J p = pqμ p E − qD p

Q

Dp

μp

=

Dn

μn

= VT

VT dp
p dx
dV
QE = −
dx
dp

⇒ dV = −VT
p
⇒E=

p1

p2

V1

V2

∫ dV = ∫ − VT

1
dp
p

p1
→ V2 − V1 = −VT (ln p2 − ln p1 ) = VT ln
p2
V21
Boltzmann equation
Mass-action law
VT
⇒ p1 = p2 e
2
J n = 0 ⇒ n1 = n2 e

Vo =V 21= VT ln


−V21
VT

p1
p2

N AND
Vo = VT ln
n 2i

n1 p1 = n2 p2 = ni

p

n

p1 = p po = N A
p2 = pn 0 =

ni2

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ND


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Depletion region

Gauss’s Law

∇ • E ( x) =

Charge density

Field strength

Wn

E ( x) = ∫

qN D

− Wp

−W p

Wn

− qN A

E (x)

電場的散度等於體電荷密度
(volume charge density)

除以真空容電率

qN AW p = qN DWn

ρ (x)

− Wp

Wn

V = −∫

−W p

Wn

V (x)

V0

ε

V0 = − ∫

0

−W p

=


qN AW p2

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qN AW p

E0 = −

E0

ρ ( x)
ε

2ε

ρ ( x)
dx
ε

E ( x)dx

=−

qN DWn

ε

Wn


E ( x)dx − ∫ E ( x)dx
0

qN DWn2
+
2ε
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Depletion width

W = W p + Wn ⇒ W =
⇒ Wn =

ND
Wn + Wn
NA

NA
ND
W ⇒ Wp =
W
N A + ND
N A + ND

qN AW p2

qN DWn2

V0 =
+
2ε
2ε
q N AND
V0 =
(
)W 2
2ε N A + N D

Wp
Wn
Charge equality
qW p AN A = qWn AN D
⇒

Wn N A
=
Wp N D

⇒ Wn N D = W p N A

Wdep

2ε s 1
1
= Wn + W p =
(
+
)V0

q N A ND

ε s : Silicon Electrical permirrivity
For silicon

容電係數

ε s = 1.04 × 10 −12 F cm
Wdep = 0.1μm ~ 1μm
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PN Junction reverse bias

IS > ID


IS − ID = I

q J = q N = qN DWn A
N AND
qJ = q
AWdep
N A + ND
Wdep = Wn + W p =

Cj =

Junction capacitance
dq J
Cj =
dVR
⇒ Cj =

VR =VQ

εs A

VR = 0

C jo = A

2ε s 1
1
(
+

)(V0 + VR )
q N A ND

C jo

varactor

1 + VVRo

qε s N A N D 1
(
)
2 N A + N D V0

Wdep
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Minority carriers distribution in forward bias (the story about reverse saturation current)
electrons

p1 = p2 e

V21
VT

page18

→ p n ( xn ) = p n 0 e

V
VT

holes

pn ( x) = pn 0 + [ pn ( xn ) − pn 0 ]e
Q J p = − qDP
⇒ Jp = q

Dp
Lp

− ( x − xn )

LP


dp
dx
pn 0 (e

V
VT

− 1)e

− ( x − xn )

LP

Total loss by recombination
= External electric field inject electrons

Inject holes from P region to diffuse away from the
junction into the N region and disappear by
recombination

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