Semiconductor Physics: E-k Diagrams, Effective Mass & Diodes

Quantum Mechanics, E-k Diagrams & Effective Mass

• Schrödinger Equation: H * Ψ = E * Ψ, where H = V - (ħ^2 * ∇^2) / (2 * m).

  • 1D Infinite Well / H-Atom: Gives discrete energy states (E_n = -13.6 eV / n^2 for H-atom).

  • Crystal Lattice (spacing a): Multi-atom electron interactions split discrete states into a continuum of states (Bands: Ec, Ev) separated by a forbidden Bandgap (Eg).

• E-k Diagram (Reciprocal Space): Repeats every 2π / a; First Brillouin Zone is -π / a to +π / a.

  • 3D Parabolic Band: E(k) = E_min/max + (ħ^2 * k^2) / (2 * m*)

  • Effective Mass Derivation: Take 2nd derivative d^2E / dk^2 = ħ^2 / m* ⇒ m* = ħ^2 / (d^2E / dk^2)

  • Mobility & Current: μ = (q * τ) / m*, v = μ * E, and J = I / A = n * e * v = n * e * μ * E. (Sharper curve → larger d^2E/dk^2 → smaller m* → higher μ & speed!)

• Direct vs. Indirect Bandgap:

  • Direct (GaAs, GaN): Ec minimum and Ev maximum at same k → emits photons easily (LEDs/Lasers).

  • Indirect (Si): Ec min and Ev max at different k → requires phonon-assisted transition.

• At T = 0 K: Valence band is completely full, conduction band is completely empty → semiconductor acts like an insulator because responding to an E-field requires availability of empty states + energy.

Carrier Concentration & Intrinsic Fermi Level Derivations

• General Integral: n = ∫ from Ec to ∞ of [ N(E) * f(E) dE ], where:

  • Density of States: N(E) = (√2 / π^2) * (m^(3/2) * (E - Ec)^(1/2)) / ħ^3

  • Fermi-Dirac Probability: f(E) = 1 / (1 + exp((E - Ef) / (kB * T)))

• Boltzmann Approximation (Non-degenerate, n << Nc):

  • n = Nc * exp( -(Ec - Ef) / (kB * T) )

  • p = Nv * exp( -(Ef - Ev) / (kB * T) )

• Mass-Action Law Derivation (n * p = ni^2):

  • n * p = Nc * Nv * exp( -(Ec - Ev) / (kB * T) ) = Nc * Nv * exp( -Eg / (kB * T) ) = ni^2

  • ni = √(Nc * Nv) * exp( -Eg / (2 * kB * T) )

• Intrinsic Fermi Level (Ef when n = p = ni):

  • Set Nc * exp( -(Ec - Ef) / (kB * T) ) = √(Nc * Nv) * exp( -Eg / (2 * kB * T) )

  • Ef = (Ec - Eg/2) + (1/2) * kB * T * ln(Nv / Nc)

  • If Nc = Nv ⇒ Ef = Eg / 2 (exact mid-gap at all T!).

Doping, Ionization Rules & Charge Neutrality

• Thermal Voltage at 300 K: (kB * T) / q = 0.026 eV = 26 meV.

• Rule of Thumb for Ionization:

  • Shallow / Full Ionization (|Ec - Ed| < 2·kB·T/q or |Ea - Ev| < 2·kB·T/q):

    • n-type: n = Nd+ = Nd, p_n = ni^2 / Nd, Ef = Ec - kB * T * ln(Nc / Nd)

    • p-type: p = Na- = Na, n_p = ni^2 / Na, Ef = Ev + kB * T * ln(Nv / Na)

  • Deep / Partial Ionization (|Ec - Ed| > 2·kB·T/q or |Ea - Ev| > 2·kB·T/q):

    • n_c = ni + Σ[ Nd_i * exp( -|Ec - Ed_i| / (kB * T) ) ]

    • p_v = ni + Σ[ Na_i * exp( -|Ea_i - Ev| / (kB * T) ) ]

• Charge Neutrality & Compensation Concepts:

  • Doped semiconductors are electrically neutral overall: n + Na- = p + Nd+.

  • Why donors donate to Ec instead of Ev: Ev is lower energy (favorable) but has no empty states; Ec has an abundance of states and only needs small thermal energy |Ec - Ed|.

  • Compensation Rule (p. 16): No matter how shallow donors are, if there are empty states below Ed (like acceptors Ea), donor electrons fall into those states first!

  • Ultra-High Temperature (ni >> Nd): n ~ ni, so Ef returns to intrinsic mid-gap (Eg / 2).

P-N Junction Electrostatics & Capacitance Derivation

• 1. System Charge Neutrality in Depletion Region (Region II has fixed ions only, n = p = 0):

  • Na- * Wp = Nd+ * Wn

• 2. Electric Field via Gauss's Law (E(x) = (1 / (ε0 * εr)) * ∫[ ρ(x) dx ]):

  • Neutral Regions I & III: E = 0 and V = 0 because mobile carriers balance fixed dopant ions.

  • p-side (-Wp < x < 0): slope = (q * Na-) / (ε0 * εr). n-side (0 < x < Wn): slope = -(q * Nd+) / (ε0 * εr).

  • Peak E-Field (at x = 0): |E_peak| = (q * Na- * Wp) / (ε0 * εr) = (q * Nd+ * Wn) / (ε0 * εr)

• 3. Built-in Potential (Vbi = Area of E-field triangle = Fermi level difference):

  • Vbi = Ef_n - Ef_p = Eg - φ_n - φ_p = (1/2) * |E_peak| * (Wp + Wn)

  • Vbi = (q * Na * Wp^2) / (2 * ε0 * εr) + (q * Nd * Wn^2) / (2 * ε0 * εr)

  • Under Bias: Replace Vbi with Vbi - V_app (Forward) or Vbi + |V_R| (Reverse).

  • One-Sided Junction (Na >> Nd): Wp << Wn (ignore Wp), so Vbi + |V_R| ~ (q * Nd * Wn^2) / (2 * ε0 * εr).

• 4. Junction Capacitance Derivation:

  • Apply small dV ⇒ adds charge dQ_p = q * Na * dWp = dQ_n = q * Nd * dWn.

  • dV = V_new - V_old = [q·Na / (2·ε0·εr)] · [(Wp + dWp)^2 - Wp^2] + [q·Nd / (2·ε0·εr)] · [(Wn + dWn)^2 - Wn^2]

  • Drop tiny dW^2 terms ⇒ dV = [q·Na·dWp / (ε0·εr)]·Wp + [q·Nd·dWn / (ε0·εr)]·Wn = dQ_p · (Wp + Wn) / (ε0 · εr)

  • C = dQ_p / dV = (ε0 · εr) / (Wp + Wn) (per unit area)

• General Depletion Width Formulas (Na ≠ Nd):

  • Substitute Wp = (Nd / Na) * Wn into Vbi = [q / (2 · ε0 · εr)] · (Na · Wp^2 + Nd · Wn^2):

  • Vbi = [q · Nd · Wn^2 / (2 · ε0 · εr)] · (1 + Nd / Na)

  • Wn = √( [2 · ε0 · εr · Vbi] / [q · Nd · (1 + Nd / Na)] )

  • Wp = √( [2 · ε0 · εr · Vbi] / [q · Na · (1 + Na / Nd)] )

P-N Junction Transport & Law of the Junction

• Zero-Bias Equilibrium (V_app = 0, J_net = 0):

  • Majority carriers face barrier Vbi (diffusion); minority carriers see no barrier (drift).

  • Electron flows balance: n_p0 = n_n0 * exp( -q * Vbi / (kB * T) ) = ni^2 / Na

  • Hole flows balance: p_n0 = p_p0 * exp( -q * Vbi / (kB * T) ) = ni^2 / Nd

• Law of the Junction (Boundary Conditions at Depletion Edges -Wp and +Wn under Bias V):

  • Forward bias +V lowers barrier to Vbi - V, splitting Ef into Quasi-Fermi Levels Ef_n and Ef_p where p * n = ni^2 * exp( (Ef_n - Ef_p) / (kB * T) ):

  • At p-side edge (x = -Wp): n_p(-Wp) = n_n0 * exp( -q(Vbi - V) / (kB·T) ) = n_p0 * exp( q·V / (kB·T) )

  • At n-side edge (x = +Wn): p_n(Wn) = p_p0 * exp( -q(Vbi - V) / (kB·T) ) = p_n0 * exp( q·V / (kB·T) )

  • Far from junction (x = ±∞): n_p(-∞) = n_p0 and p_n(+∞) = p_n0.

Current Continuity & Long-Base Diode Derivation

• 1. Current Continuity Equation (Conservation of Particles):

  • d(J_drift + J_diff) / dx = dJ_rec / dx

  • Where J_drift = q · μn · n · E(x) and J_diff = q · Dn · dn_p(x)/dx.

  • In neutral p/n regions outside depletion zone, E(x) = 0 and dE/dx = 0 ⇒ J_drift = 0 (Pure Diffusion!)

• 2. Differential Equation with Recombination (Ln, Lp << Width):

  • Dn · (d²n_p(x) / dx²) = (n_p(x) - n_p0) / τn ⇒ d²n_p(x) / dx² = (n_p(x) - n_p0) / Ln²

  • Diffusion Lengths: Ln = √(Dn · τn) and Lp = √(Dp · τp)

• 3. Exponential Carrier Profile & Ideal Diode Current (J = Jn + Jp):

  • Solution: n_p(x) - n_p0 = n_p0 · [ exp(q·V / (kB·T)) - 1 ] · exp( -(x - Edge) / Ln )

  • Evaluate Jn = q · Dn · dn_p/dx at x = Edge: Jn = (q · Dn · n_p0 / Ln) · [ exp(q·V / (kB·T)) - 1 ]

  • Evaluate Jp at x = Edge: Jp = (q · Dp · p_n0 / Lp) · [ exp(q·V / (kB·T)) - 1 ]

  • Total Current (J = Jn + Jp is constant everywhere so charge dq/dt doesn't build up): J = J0 · [ exp(q·V / (kB·T)) - 1 ] where J0 = (q · Dn · n_p0 / Ln) + (q · Dp · p_n0 / Lp)

  • Reverse Leakage & Bandgap (V = -V_R, exp(-q·V_R/kBT) ~ 0): If p_n0 >> n_p0: J0 ~ (q · Dp / Lp) · (ni^2 / Nd) = (q · Dp / (Lp · Nd)) · Nc · Nv · exp( -Eg / (kB · T) )

Short-Base Diode Derivation (Xp, Xn << Lp, Ln)

• 1. No Recombination in Neutral Regions:

  • Because region widths Xp and Xn are much shorter than Ln and Lp, minority carriers hit the Ohmic contact (the SINK) before recombining (R = 0)!

• 2. Linear Carrier Profile & Current Equations:

  • Continuity Eq with R = 0 and E = 0: d(J_diff) / dx = q · Dn · (d²n(x) / dx²) = 0

  • Solution is a straight line: n(x) = a·x + b

  • Boundary conditions: n_p0 · exp(q·V / (kB·T)) at depletion edge, dropping linearly to n_p0 at Ohmic contact (distance Xp away).

  • Replace Ln, Lp with Xp, Xn: Jn = (q · Dn · n_p0 / Xp) · [ exp(q·V / (kB·T)) - 1 ] and Jp = (q · Dp · p_n0 / Xn) · [ exp(q·V / (kB·T)) - 1 ]

P-I-N Diode, Schottky Diode & Ohmic Contact

• P-I-N Diode (Intrinsic i-layer of width Wi between p and n):

  • No fixed charge in i-region (ρ = 0) ⇒ Constant E-field across i-region (Trapezoid or Rectangle E-field profile).

  • Same Vbi as p-n diode, but |E_peak| = (Vbi + |V_R|) / Wi (for heavily doped ends where Ef = Ec and Ef = Ev).

  • Advantages: Lower peak E-field (prevents breakdown in power devices) + wider depletion width (great for solar cells).

• Schottky Diode (Metal-Semiconductor Junction):

  • Metal has sheet of charge at surface (W_metal = 0); all depletion Wn is in n-type semiconductor.

  • Barrier Height: q · φBn = q · φm - q · χ (Metal work function minus semiconductor electron affinity).

  • Built-in Voltage: Vbi = φBn - (Ec - Ef)n = (q · Nd · Wn^2) / (2 · ε0 · εr)

  • Thermionic Emission Current: J = A* · T² · exp( -q · φBn / (kB · T) ) · [ exp(q·V / (kB·T)) - 1 ] (A* = Richardson's constant).

  • Advantages over p-n diode: Majority carrier device → no recombination (fast switching response) + tunable barrier height (by choosing different metals).

• Ohmic Contact: Schottky junction with low barrier height + very high doping (n++) (thin Wn) so electrons easily tunnel both ways → obeys V = I · Rc.