A.2 Semiconductor Equations - Electrical, Computer...
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A.2 Semiconductor Equations Semiconductor carrier concentrations Thermal equilibrium n = N c e -(E c -E f )/kT = n i e (E f -E i )/kT p = N v e -(E f -E v )/kT = n i e (E i -E f )/kT N c = 2 [ 2π m e * kT h 2 ] 3/2 N v = 2 [ 2π m h * kT h 2 ] 3/2 n p = n i 2 = N c N v e -E g /kT f(E) = 1 1 + e (E-E f )/kT n = N d -N a 2 + (N d -N a ) 2 4 + n i 2 p = N a -N d 2 + (N a -N d ) 2 4 + n i 2 Not in thermal equilibrium n = N c e -(E c -F n )/kT = n i e (F n -E i )/kT p = N v e -(F p -E v )/kT = n i e (E i -F p )/kT p n = n i 2 e (F n -F p )/kT = N c N v e (-E g /2+F n -F p )/kT Field, potential and energy q = - q dφ dx = dE c dx = dE v dx = dE i dx q = - q ∇φ = ∇E c = ∇E v = ∇E i d dx = ρ ε s = q ε s (p - n + N d + - N a - ) ∇. ∇. = ρ ε s = q ε s (p - n + N d + - N a - ) ⌡ ⌠ dA = Q ε s Poisson's equation ∇ 2 φ = ∂ 2 φ ∂x 2 + ∂ 2 φ ∂y 2 + ∂ 2 φ ∂z 2 = -ρ ε s cylindrical coordinates: 1 r ∂ ∂r [r ∂φ ∂r ] + 1 r 2 ∂ 2 φ ∂θ 2 + ∂ 2 φ ∂z 2 = -ρ ε s spherical coordinates: 1 r 2 ∂ ∂r [r 2 ∂φ ∂r ] + 1 r 2 sinθ ∂ ∂θ [r 2 sinθ ∂φ ∂θ ] + 1 r 2 sinθ ∂ 2 φ ∂ϕ 2 = -ρ ε s Current relations including drift and diffusion Principles of Electronic Devices A2.1 © Bart J. Van Zeghbroeck 1996
Transcript of A.2 Semiconductor Equations - Electrical, Computer...
A.2 Semiconductor Equations
Semiconductor carrier concentrations
Thermal equilibrium
n = Nc e-(Ec-Ef)/kT = ni e(Ef-Ei)/kT p = Nv e-(Ef-Ev)/kT = ni e(Ei-Ef)/kT
Nc = 2 [2π me* kT
h2 ]3/2 Nv = 2 [2π mh* kT
h2 ]3/2
n p = ni2 = NcNv e-Eg/kT f(E) = 1
1 + e(E-Ef)/kT
n = Nd-Na
2 + (Nd-Na)2
4 + ni2 p = Na-Nd
2 + (Na-Nd)2
4 + ni2
Not in thermal equilibrium
n = Nc e-(Ec-Fn)/kT = ni e(Fn-Ei)/kT p = Nv e-(Fp-Ev)/kT = ni e(Ei-Fp)/kT
p n = ni2 e(Fn-Fp)/kT= NcNv e(-Eg/2+Fn-Fp)/kT
Field, potential and energy
q = - q dφdx =
dEcdx =
dEvdx =
dEidx q = - q ∇∇φ = ∇∇Ec = ∇∇Ev = ∇∇Ei
ddx =
ρεs
= qεs
(p - n + Nd+ - Na-) ∇.∇. = ρεs
= qεs
(p - n + Nd+ - Na-)
⌡⌠ dA = Qεs
Poisson's equation
∇∇2φ = ∂2φ∂x2 +
∂2φ∂y2 +
∂2φ∂z2 =
-ρεs
cylindrical coordinates: 1r
∂∂r
[r ∂φ∂r
] + 1r2
∂2φ∂θ2 +
∂2φ∂z2 =
-ρεs
spherical coordinates: 1r2
∂∂r
[r2 ∂φ∂r
] + 1
r2sinθ ∂
∂θ [r2sinθ ∂φ∂θ] +
1r2sinθ
∂2φ∂ϕ2 =
-ρεs
Current relations including drift and diffusion
Principles of Electronic Devices A2.1 © Bart J. Van Zeghbroeck 1996
Jn = q n µn + q Dn ∂n∂x
Jn = q n µn + q Dn ∇∇n Dn = µn Vt
Jp = q p µp - q Dp ∂p∂x
Jp = q p µp - q Dp ∇∇n Dp = µp Vt
Minority carrier rate (continuity) equations
∂np∂t
= 1q
∂Jn∂x
+ (Gn-Rn) ∂pn∂t
= -1q
∂Jp∂x
+ (Gp-Rp)
∂np∂t
= 1q ∇.∇.Jp -
(np - np0)τn
+ Gn ∂pn∂t
= -1q ∇.∇.Jn -
(pn - pn0)τp
+ Gp
Gn = Gp = generation due to light, avalanche multiplication, etcetera.
Rn = (np - np0)
τn Rp =
(pn - pn0)τp
Generation-Recombination equations
Gn = ∂n∂t
light = Gp = ∂p∂t
light = α q Popt(x)A Eph
Ub-b = b (np - ni2)
USHR = (pn - ni2)
[n + p + 2 ni cosh((Ei-Et)/kT)] τ0 , with
1τ0
= Nt vth σ
Us = s p' with s = Nst vth σ
UA = Γn n (np - ni2) + Γp p (np - ni2)
Diffusion equations
0 = Dn d2ndx2 -
np - np0τn
, x < -xp , Dn = µn Vt
0 = Dp d2pdx2 -
pn - pn0τp
, x > xn , Dp = µp Vt
0 = Dn ∇∇2n - (np - np0)
τn0 = Dp ∇∇2p -
(pn - pn0)τp
Metal-semiconductor junction
Principles of Electronic Devices A2.2 © Bart J. Van Zeghbroeck 1996
φi - Va = ΦM - χ - 1q(Ec - Efn) -Va =
q Nd xn2
2εs (n-type)
φi - Va = χ + 1q(Ec - Efp) - ΦM -Va =
q Na xp2
2εs (p-type)
p-n junction relations (abrupt junction)
φi - Va = q Nd xn2/2εs + q Na xp2/2εs
max = q Nd xn/εs = q Na xp/εs
φi = Vt ln(NaNd
ni2) xd =
2εsq (
1Na
+ 1
Nd) (φi-Va)
xn = 2εsq
NaNd
1
Na+Nd (φi-Va) xp =
2εsq
NdNa
1
Na+Nd (φi-Va)
J = J0 (eVa/Vt - 1), J0 = q ni2 (Dp
NdWn' + Dn
NaWp') (ideal current for a short diode)
np(-xp) = np0 eVa/Vt pn(-xn) = pn0 eVa/Vt
Jt = q (Dp pn0
Lp +
Dnnp0Ln
+ xd ni2 b) (eVa/Vt - 1) + q x' ni2τ0
(eVa/2Vt - 1)
Principles of Electronic Devices A2.3 © Bart J. Van Zeghbroeck 1996
BJT relations (npn with short emitter, base and collector)
IE = a11 (eVBE/Vt - 1) + a12 (eVBC/Vt - 1)
IC = a21 (eVBE/Vt - 1) + a22 (eVBC/Vt - 1)
where for short quasi-neutral regions (short compared to the diffusion length)
a11 = -IES = - q A ni2 (DE
NEwE +
DBNBwB
)
a12 = a21 = αF IES = αR ICS = - q A ni2 (DB
NBwB)
a22 = -ICS = - q A ni2 (DC
NCwC +
DBNBwB
)
α = 1
1 + wBNBDEwENEDB
αT = 1 - wB
2
2DBτB
IrB = q A ni2wB
2NBτB (eVBE/2Vt - 1)
Principles of Electronic Devices A2.4 © Bart J. Van Zeghbroeck 1996
Miscellaneous Fundamental Relations
F = Q ( + v x B) , Q = charge of the particle
C = dQdV =
εsxd
F = m* dvdt =
h2π
dkdt
1m* =
4π2
h2 ∂2E∂k2
vg = 2πh
∂E∂k
∇2 Ψ + 8π2 m*
h2 (E - V(x)) Ψ = 0
∇2 φ = − ρε
Principles of Electronic Devices A2.5 © Bart J. Van Zeghbroeck 1996
