Ch19
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Review of Basic Semiconductor Physics
Copyright © by John Wiley & Sons 2003
Semiconductor Physics - 1
Current Flow and Conductivity • Charge in volume Aδ x = δ Q = q n A δ x = q n A vδ t • Current density J = (δ Q/δ t)A1 =qnv • Metals - gold, platinum, silver, copper, etc. • n = 1023 cm-3 ;
σ = 107 mhos-cm
• Insulators - silicon dioxide, silicon nitride, aluminum oxide • n < 103 cm-3 ;
σ < 10-10 mhos-cm
• Semiconductors - silicon, gallium arsenide, diamond, etc. • 108 < n <1019 cm-3 ; 10-10 < Copyright © by John Wiley & Sons 2003
σ < 104 mhos-cm Semiconductor Physics - 2
Thermal Ionization
• Si atoms have thermal vibrations about equilibrium point.
• Small percentage of Si atoms have large enough vibrational energy to break covalent bond and liberate an electron.
Copyright © by John Wiley & Sons 2003
Semiconductor Physics - 3
Electrons and Holes -
• T3 > T 2 > T 1
A
t = T 1
• Density of free electrons = n : Density of free holes = p • p = n = ni(T) = intrinsic carrier density.
-
A
• ni2(T) = C exp(-qEg/(kT )) = 1020 cm-6 at 300 °K
+
B
generation of B
t = T 2
• T = temp in °K • k = 1.4x10-23 joules/ °K • Eg = energy gap = 1.1 eV in silicon -19
• q = 1.6x10 coulombs Copyright © by John Wiley & Sons 2003
recom bination of B
-
A
apparent m ovem ent of "H ole"
t= T 3
Semiconductor Physics - 4
Doped Semiconductors • Extrinsic (doped) semiconductors:p = po ≠ n = no ≠ ni • Carrier density estimates: • Law of mass action nopo = ni2(T) • Charge neutrality Na + no = Nd + po • P-type silicon with Na >> ni: po ≈ Na, no ≈ ni2/ Na
Copyright © by John Wiley & Sons 2003
• N-type silicon with Nd >> ni: no ≈ Nd, po ≈ ni2/ Nd
Semiconductor Physics - 5
Nonequilibrium and Recombination • Thermal Equilibrium - Carrier generation = Carrier recombination • n = no and p = po
• Nonequilibrium - n > no and p > po • n = no + δ n and p = no + δ n ; δ n = excess carrier density • Excess holes and excess electrons created in equal numbers by breaking of covalent bonds • Generation mechanisms -light (photoelectric effect), injection, impact ionization
• Recombination - removal of excess holes and electrons • Mechanisms - free electron captured by empty covalent bond (hole) or trapped by impurity or crystal imperfection • Rate equation:
d(δ n)/dt
=
- δ n/τ
• Solution δ n = δ n (0) e -t /τ Copyright © by John Wiley & Sons 2003
Semiconductor Physics - 6
Carrier Lifetimes • τ
= excess carrier lifetime
• Usually assumed to be constant. Changes in two important situations. • τ increases with temperature T • τ decreases at large excess carrier densities ; τ = τ o/[1 + (δ n/nb)2 ]
• Control of carrier lifetime values. • Switching time-on state loss tradeoff mandates good lifetime control. • Control via use of impurities such as gold - lifetime killers. • Control via electron irradiation - more uniform and better control.
Copyright © by John Wiley & Sons 2003
Semiconductor Physics - 7
Current Flow Drift
• Jdrift = q µn n E + q p µp E • µn = 1500 cm2/V-sec for silicon at room temp. and Nd < 1015 cm-3 • µp = 500 cm2/V-sec for silicon at room temp. and Na < 1015 cm-3
Copyright © by John Wiley & Sons 2003
Diffusion
• Jdiff = Jn + Jp = q Dndn/dx - q Dp dp/dx • Dn/µ n = Dp/µ p = kT/q ; Einstein relation • D = diffusion constant, µ = carrier mobility • Total current density J = Jdrift + Jdiff
Semiconductor Physics - 8
PN Junction metallurgical junction
P
N A
N
N
N A
D
NA
N
D
NA
x
-
N
D
Step (abrupt) junction Copyright © by John Wiley & Sons 2003
x
-
N
D
Linearly graded junction
Semiconductor Physics - 9
Formation of Space Charge Layer metallurgical junction
• Diffusing electrons and holes leave the region near metallurgical junction depleted of free carriers (depletion region).
x
ionized acceptors
ionized donors
+ +
P -
• Exposed ionized impurities form space charge layer.
Diffusing electrons
-
• Electric field due to space charge opposes diffusion.
+ + + +
N +
+
Electric field opposing diffusion Diffusing holes
+
space charge layer width = W Copyright © by John Wiley & Sons 2003
Semiconductor Physics - 10
Quantitative Description of Space Charge Region ρ
• Assume step junction. d2 Φ ρ = −ε δξ2
qN
-x p
ρ = −θ Να ; ξ < 0 ρ = θ Νδ ; ξ > 0
Ε(ξ ) =
d
-qN a
ω
Ε
dΦ = − Ε(ξ ) δξ θ Να(ξ +ξ π )
x
; − ξπ < ξ < 0 ε θ Νδ (ξ −ξ ν ) Ε(ξ ) = ; 0< ξ < ξν ε
Ε µαξ
xn Φc = -
Φ
ó E(x)dx õ
- xp
qNax p2Ê+ÊqNdx n2 Fc =2e Copyright © by John Wiley & Sons 2003
x
xn
x Φχ δεπλετιον λαψερ
Semiconductor Physics - 11
Contact (Built-in, Junction) Potential ¥ In thermal equilibrium Jn = q µ n n
dΦ δν + θ ∆ν =0 δξ δξ
Φ(ξν) ∞ Σεπαρατε ϖαριαβλεσ ανδ ιντεγρατε ;
⌠ ⌡δΦ
Φ(ξπ)
∆ν =− ∝ν
ν(ξν)
⌠δν ⌡ν
ν(ξπ)
κΤ ΝαΝδ ∞ Φ(ξν) − Φ(ξπ) = Φχ = λν ; Φχ = χονταχτ ποτεντιαλ θ 2 ν ι ∞ Εξαµ πλε ∞ ∞ ∞
Ρ οοµ τεµ περατυρε κΤ/θ = 0.025 ες Να = Νδ = 1016 χµ −3 ; νι2 = 1020 χµ −6 Φχ = 0.72 ες
Copyright © by John Wiley & Sons 2003
Semiconductor Physics - 12
Reverse-Biased Step Junction ¥ Starting equations ¥ W(V) = xn(V) + xp(V)
V
θ Ναξ π 2 ⊇+ ⊇θ Νδ ξ ν 2 ¥ V + Φχ = − 2ε
P
∞ Χηαργ ε νε υτραλιτψ θ Ναξ π = θ Νδ ξ ν
+
+ + ++ +
N
Wo W(V)
Φ
∞ Σολϖε ε θ υατιονσ σιµ υλτανε ουσλψ ∞ Ω (ς ) = Ω ο
∞ Ωο =
1 + ς / Φχ
2 εΦχ( Να+ Νδ ) θ ΝαΝδ 2 Φχ
∞ Εµ αξ = Ωο
1 ⊇+ ⊇ς / Φχ
Copyright © by John Wiley & Sons 2003
Φ
x χ
Φ + ς χ
−ξ π( ς )
ξ ν( ς )
Semiconductor Physics - 13
Forward-Biased PN Junction
• Forward bias favors diffusion over drift.
• Excess minority carrier injection into both p and n drift regions.
• Minority carrier diffusion lengths. • Ln = [Dnτ n]0.5 • Lp = [Dpτ p]0.5
Copyright © by John Wiley & Sons 2003
Semiconductor Physics - 14
Ideal PN Junction I-V Characteristics • Excess carriers in drift regions recombined and thus more must be constantly injected if the distributions np(x) and pn(x) are to be maintained. • Constant injection of electrons and holes results in a current density J given by Qn Θπ Λπ Λν 2 J = τ + τ = θ νι + ν π Ν τ Ν τ αν δ π ϑ=
ϑσ
θς ε ξπ( κΤ )⊇−⊇1
Λπ Λν 2 ; ϑσ = θ ν ι + Νδ τπ Νατν i
J
J
θς ε ξπ( κΤ ) ⊇−⊇1
v - J s v forward bias
Copyright © by John Wiley & Sons 2003
reverse bias
combined characteristic
v Semiconductor Physics - 15
Reverse Saturation Current • Carrier density gradient immediately adjacent to depletion region causes reverse saturation current to flow via diffusion.
+
V
+
+ ++
P
N
Wo
• Js independent of reverse voltage V because carrier density gradient unaffected by applied voltage.
W(V)
n po
p
no
n p(x)
+ p (x) n
x Electric field, Copyright © by John Wiley & Sons 2003
J
• Js extremely temperature sensitivity because of dependence on ni2(T.)
s Semiconductor Physics - 16
Impact Ionization • E ≥ EBD ; free electron can acquire sufficient from the field between lattice collisions (tc ≈ 10-12 sec) to break covalent bond. • Energy = 0.5mv2 = q Eg ; v = q EBD tc ¥ Solving for EBD gives EBD =
2ÊEgÊm qÊtc2
Si -
-
Si -
Electric field E
Si -
¥ Numerical evaluation ¥ m = 10-27 grams, Eg = 1.1 eV, tc = 10-12 sec. ¥ EBD = ¥
(2)Ê(1.1)Ê(1027) = 3x105 V/cm -19 -24 (1.6x10 )Ê(10 )
Experimental estimates are 2-3.5x105 V/cm
Copyright © by John Wiley & Sons 2003
Semiconductor Physics - 17
Copyright © by John Wiley & Sons 2003
Semiconductor Physics - 1
Current Flow and Conductivity • Charge in volume Aδ x = δ Q = q n A δ x = q n A vδ t • Current density J = (δ Q/δ t)A1 =qnv • Metals - gold, platinum, silver, copper, etc. • n = 1023 cm-3 ;
σ = 107 mhos-cm
• Insulators - silicon dioxide, silicon nitride, aluminum oxide • n < 103 cm-3 ;
σ < 10-10 mhos-cm
• Semiconductors - silicon, gallium arsenide, diamond, etc. • 108 < n <1019 cm-3 ; 10-10 < Copyright © by John Wiley & Sons 2003
σ < 104 mhos-cm Semiconductor Physics - 2
Thermal Ionization
• Si atoms have thermal vibrations about equilibrium point.
• Small percentage of Si atoms have large enough vibrational energy to break covalent bond and liberate an electron.
Copyright © by John Wiley & Sons 2003
Semiconductor Physics - 3
Electrons and Holes -
• T3 > T 2 > T 1
A
t = T 1
• Density of free electrons = n : Density of free holes = p • p = n = ni(T) = intrinsic carrier density.
-
A
• ni2(T) = C exp(-qEg/(kT )) = 1020 cm-6 at 300 °K
+
B
generation of B
t = T 2
• T = temp in °K • k = 1.4x10-23 joules/ °K • Eg = energy gap = 1.1 eV in silicon -19
• q = 1.6x10 coulombs Copyright © by John Wiley & Sons 2003
recom bination of B
-
A
apparent m ovem ent of "H ole"
t= T 3
Semiconductor Physics - 4
Doped Semiconductors • Extrinsic (doped) semiconductors:p = po ≠ n = no ≠ ni • Carrier density estimates: • Law of mass action nopo = ni2(T) • Charge neutrality Na + no = Nd + po • P-type silicon with Na >> ni: po ≈ Na, no ≈ ni2/ Na
Copyright © by John Wiley & Sons 2003
• N-type silicon with Nd >> ni: no ≈ Nd, po ≈ ni2/ Nd
Semiconductor Physics - 5
Nonequilibrium and Recombination • Thermal Equilibrium - Carrier generation = Carrier recombination • n = no and p = po
• Nonequilibrium - n > no and p > po • n = no + δ n and p = no + δ n ; δ n = excess carrier density • Excess holes and excess electrons created in equal numbers by breaking of covalent bonds • Generation mechanisms -light (photoelectric effect), injection, impact ionization
• Recombination - removal of excess holes and electrons • Mechanisms - free electron captured by empty covalent bond (hole) or trapped by impurity or crystal imperfection • Rate equation:
d(δ n)/dt
=
- δ n/τ
• Solution δ n = δ n (0) e -t /τ Copyright © by John Wiley & Sons 2003
Semiconductor Physics - 6
Carrier Lifetimes • τ
= excess carrier lifetime
• Usually assumed to be constant. Changes in two important situations. • τ increases with temperature T • τ decreases at large excess carrier densities ; τ = τ o/[1 + (δ n/nb)2 ]
• Control of carrier lifetime values. • Switching time-on state loss tradeoff mandates good lifetime control. • Control via use of impurities such as gold - lifetime killers. • Control via electron irradiation - more uniform and better control.
Copyright © by John Wiley & Sons 2003
Semiconductor Physics - 7
Current Flow Drift
• Jdrift = q µn n E + q p µp E • µn = 1500 cm2/V-sec for silicon at room temp. and Nd < 1015 cm-3 • µp = 500 cm2/V-sec for silicon at room temp. and Na < 1015 cm-3
Copyright © by John Wiley & Sons 2003
Diffusion
• Jdiff = Jn + Jp = q Dndn/dx - q Dp dp/dx • Dn/µ n = Dp/µ p = kT/q ; Einstein relation • D = diffusion constant, µ = carrier mobility • Total current density J = Jdrift + Jdiff
Semiconductor Physics - 8
PN Junction metallurgical junction
P
N A
N
N
N A
D
NA
N
D
NA
x
-
N
D
Step (abrupt) junction Copyright © by John Wiley & Sons 2003
x
-
N
D
Linearly graded junction
Semiconductor Physics - 9
Formation of Space Charge Layer metallurgical junction
• Diffusing electrons and holes leave the region near metallurgical junction depleted of free carriers (depletion region).
x
ionized acceptors
ionized donors
+ +
P -
• Exposed ionized impurities form space charge layer.
Diffusing electrons
-
• Electric field due to space charge opposes diffusion.
+ + + +
N +
+
Electric field opposing diffusion Diffusing holes
+
space charge layer width = W Copyright © by John Wiley & Sons 2003
Semiconductor Physics - 10
Quantitative Description of Space Charge Region ρ
• Assume step junction. d2 Φ ρ = −ε δξ2
qN
-x p
ρ = −θ Να ; ξ < 0 ρ = θ Νδ ; ξ > 0
Ε(ξ ) =
d
-qN a
ω
Ε
dΦ = − Ε(ξ ) δξ θ Να(ξ +ξ π )
x
; − ξπ < ξ < 0 ε θ Νδ (ξ −ξ ν ) Ε(ξ ) = ; 0< ξ < ξν ε
Ε µαξ
xn Φc = -
Φ
ó E(x)dx õ
- xp
qNax p2Ê+ÊqNdx n2 Fc =2e Copyright © by John Wiley & Sons 2003
x
xn
x Φχ δεπλετιον λαψερ
Semiconductor Physics - 11
Contact (Built-in, Junction) Potential ¥ In thermal equilibrium Jn = q µ n n
dΦ δν + θ ∆ν =0 δξ δξ
Φ(ξν) ∞ Σεπαρατε ϖαριαβλεσ ανδ ιντεγρατε ;
⌠ ⌡δΦ
Φ(ξπ)
∆ν =− ∝ν
ν(ξν)
⌠δν ⌡ν
ν(ξπ)
κΤ ΝαΝδ ∞ Φ(ξν) − Φ(ξπ) = Φχ = λν ; Φχ = χονταχτ ποτεντιαλ θ 2 ν ι ∞ Εξαµ πλε ∞ ∞ ∞
Ρ οοµ τεµ περατυρε κΤ/θ = 0.025 ες Να = Νδ = 1016 χµ −3 ; νι2 = 1020 χµ −6 Φχ = 0.72 ες
Copyright © by John Wiley & Sons 2003
Semiconductor Physics - 12
Reverse-Biased Step Junction ¥ Starting equations ¥ W(V) = xn(V) + xp(V)
V
θ Ναξ π 2 ⊇+ ⊇θ Νδ ξ ν 2 ¥ V + Φχ = − 2ε
P
∞ Χηαργ ε νε υτραλιτψ θ Ναξ π = θ Νδ ξ ν
+
+ + ++ +
N
Wo W(V)
Φ
∞ Σολϖε ε θ υατιονσ σιµ υλτανε ουσλψ ∞ Ω (ς ) = Ω ο
∞ Ωο =
1 + ς / Φχ
2 εΦχ( Να+ Νδ ) θ ΝαΝδ 2 Φχ
∞ Εµ αξ = Ωο
1 ⊇+ ⊇ς / Φχ
Copyright © by John Wiley & Sons 2003
Φ
x χ
Φ + ς χ
−ξ π( ς )
ξ ν( ς )
Semiconductor Physics - 13
Forward-Biased PN Junction
• Forward bias favors diffusion over drift.
• Excess minority carrier injection into both p and n drift regions.
• Minority carrier diffusion lengths. • Ln = [Dnτ n]0.5 • Lp = [Dpτ p]0.5
Copyright © by John Wiley & Sons 2003
Semiconductor Physics - 14
Ideal PN Junction I-V Characteristics • Excess carriers in drift regions recombined and thus more must be constantly injected if the distributions np(x) and pn(x) are to be maintained. • Constant injection of electrons and holes results in a current density J given by Qn Θπ Λπ Λν 2 J = τ + τ = θ νι + ν π Ν τ Ν τ αν δ π ϑ=
ϑσ
θς ε ξπ( κΤ )⊇−⊇1
Λπ Λν 2 ; ϑσ = θ ν ι + Νδ τπ Νατν i
J
J
θς ε ξπ( κΤ ) ⊇−⊇1
v - J s v forward bias
Copyright © by John Wiley & Sons 2003
reverse bias
combined characteristic
v Semiconductor Physics - 15
Reverse Saturation Current • Carrier density gradient immediately adjacent to depletion region causes reverse saturation current to flow via diffusion.
+
V
+
+ ++
P
N
Wo
• Js independent of reverse voltage V because carrier density gradient unaffected by applied voltage.
W(V)
n po
p
no
n p(x)
+ p (x) n
x Electric field, Copyright © by John Wiley & Sons 2003
J
• Js extremely temperature sensitivity because of dependence on ni2(T.)
s Semiconductor Physics - 16
Impact Ionization • E ≥ EBD ; free electron can acquire sufficient from the field between lattice collisions (tc ≈ 10-12 sec) to break covalent bond. • Energy = 0.5mv2 = q Eg ; v = q EBD tc ¥ Solving for EBD gives EBD =
2ÊEgÊm qÊtc2
Si -
-
Si -
Electric field E
Si -
¥ Numerical evaluation ¥ m = 10-27 grams, Eg = 1.1 eV, tc = 10-12 sec. ¥ EBD = ¥
(2)Ê(1.1)Ê(1027) = 3x105 V/cm -19 -24 (1.6x10 )Ê(10 )
Experimental estimates are 2-3.5x105 V/cm
Copyright © by John Wiley & Sons 2003
Semiconductor Physics - 17
