STEP 1 · Varshni Bandgap Equation
Eg(T) = Eg0 − α·T² / (T + β)
Material: —
Eg0 = — eV
α = — eV/K
β = — K
Varshni (1967): empirical fit to phonon-induced bandgap narrowing. α captures electron-phonon coupling strength; β ≈ Debye temperature.
STEP 2 · Eg at Current Temperature
Eg(300K) = —
− —·T²/(T+β)
= — eV
ΔEg vs 300K = — eV
STEP 3 · Effective Density of States
Nc(T) = Nc300·(T/300)^(3/2)
Nv(T) = Nv300·(T/300)^(3/2)
At T=300K:
Nc = — cm⁻³
Nv = — cm⁻³
T^(3/2) from 3D density-of-states integral over k-space. Effective masses are weakly T-dependent; this model uses fixed m*.
STEP 4 · Intrinsic Carrier Concentration
nᵢ(T) = √(Nc·Nv)·exp(−Eg(T)/2kT)
exp argument = −Eg(T)/2kT
= −— / (2×—)
= —
nᵢ(300K) = — cm⁻³
STEP 5 · Temperature-Dependent Mobility
μp(T) = μp300·(T/300)^(−γp)
μn(T) = μn300·(T/300)^(−γn)
Lattice scattering: γ ≈ 2.3 (Si)
At T=300K:
μp = — cm²/V·s
μn = — cm²/V·s
At low T, impurity scattering (∝T^+1.5) dominates — mobility rises. At high T, lattice (phonon) scattering (∝T^−2.3) dominates — mobility falls. This model uses lattice-dominated regime.
STEP 6 · Permittivity
εs = εr·ε₀
εr(—) = —
εs = — F/cm
(weakly T-dependent; treated as const here)
STEP 1 · kT/q at Temperature T
kT/q = (1.381×10⁻²³ × 300) / 1.602×10⁻¹⁹
= 0.02585 V
STEP 2 · p-side Fermi Level
E_F − Eᵢ = −(kT/q)·ln(NA/nᵢ)
= −—·ln(—)
= — eV below Eᵢ
STEP 3 · n-side Fermi Level
E_F − Eᵢ = +(kT/q)·ln(ND/nᵢ)
= +—·ln(—)
= — eV above Eᵢ
STEP 4 · Built-in Potential
φ₀ = (kT/q)·ln(NA·ND/nᵢ²)
= |E_Fp| + E_Fn [eV]
= — V
Note: φ₀ ↓ as T↑ because nᵢ grows exponentially
STEP 5 · Bias & Quasi-Fermi Split
φ_eff = φ₀ − Vₐ = — V
ΔEF = q·Vₐ = — eV
STEP 1 · Charge Neutrality
NA·xp = ND·xn → xp/xn = ND/NA
STEP 2 · Abrupt W(T,Vₐ)
W = √[ 2εs(φ₀−Vₐ)/q·(1/NA+1/ND) ]
εs = — F/cm
φ₀−Vₐ = — V
W = — μm
STEP 3 · Individual Widths
xp = W·ND/(NA+ND) = — μm
xn = W·NA/(NA+ND) = — μm
STEP 4 · Linear Grade W ∝ (φ₀−Va)^(1/3)
W_lin = [12εs(φ₀−Va)/(q·a)]^(1/3)
Symmetric: xp = xn = W/2
STEP 5 · Gaussian Junction Depth
xⱼ = Rp·√[2·ln(Npeak/ND)]
= — nm
STEP 6 · Charge & Field
Q_dep = q·NA·xp·A = — C
ξ_max = −q·NA·xp/εs = — V/cm
Cⱼ = εs·A/W = — F
STEP 1 · ODE (holes in n-side)
d²Δp/dx² − q·vth·σ·NT/(μp·kT)·Δp = 0
= d²Δp/dx² − Δp/Lp² = 0
STEP 2 · Lp with T-dependent μp
μp(T) = — cm²/V·s
Dp = μp·kT/q = — cm²/s
Lp = √[ μp·kT/(q·vth·σ·NT) ]
= √[ — ]
= — μm
STEP 3 · Ln
μn(T) = — cm²/V·s
Ln = √[ μn·kT/(q·vth·σ·NT) ]
= — μm
STEP 4 · Minority Equilibrium Conc.
pn0 = nᵢ²(T)/ND = — cm⁻³
np0 = nᵢ²(T)/NA = — cm⁻³
Note: nᵢ² grows ∝exp(−Eg/kT) with T
→ pn0, np0 increase strongly with T
STEP 5 · Full Minority Carrier Profile
Δp(x) = pn0·[exp(Va/—)−1]
· exp(−x/— μm)
Δn(x) = np0·[exp(Va/—)−1]
· exp(+x/— μm)
STEP 1 · Saturation Current Density
J₀ = q·Dp·pn0/Lp + q·Dn·np0/Ln
= q·pn0·√(μp·kT·vth·σ·NT/q)
+ q·np0·√(μn·kT·vth·σ·NT/q)
J₀ = — A/cm²
STEP 2 · I₀ and Shockley
I₀ = J₀·A = — A
I = I₀·(exp(Vₐ/kTq) − 1)
= I₀·(exp(Vₐ/—) − 1)
At Vₐ=—V:
I = — A
STEP 3 · Temperature Sensitivity of I₀
I₀ ∝ nᵢ²(T) ∝ exp(−Eg(T)/kT)
dI₀/dT: dominated by nᵢ²
→ I₀ doubles every ~8K near 300K (Si)
→ Effect stronger in narrow-gap (Ge,InAs)
→ Weaker in wide-gap (GaN, SiC)
STEP 4 · Capacitance
Cⱼ = εs·A/W = — F
1/Cⱼ² vs Vₐ → slope gives q·εs·NA·ND / (2·A²·(NA+ND))
THREE-PROFILE COMPARISON AT CURRENT Vₐ, T
Saturation Current I₀ (A)
Profile W-Voltage Scaling Law
ABRUPT: W ∝ (φ₀−Va)^(1/2)
LINEAR: W ∝ (φ₀−Va)^(1/3)
GAUSS: locally ≈ linear near xⱼ
Temperature effect on all three:
φ₀ ↓ as T↑ → W shrinks with T
nᵢ ↑ as T↑ → I₀ rises exponentially