The MOS Capacitor: Accumulation, Depletion, Inversion & C–V

The foundation of every MOSFET. Learn the metal–oxide–semiconductor structure, oxide capacitance, work-function difference φ_ms, flat-band voltage and oxide charges (fixed, mobile, trapped, interface traps), the band diagrams and charge distributions in accumulation, depletion, weak and strong inversion, surface potential and the 2φ_F inversion criterion, maximum depletion width, threshold voltage V_T = V_FB + 2φ_F + Q_d/C_ox, high- and low-frequency C–V curves, deep depletion, and extracting oxide thickness, doping, V_FB and interface-trap density from C–V — with Python computing C_ox, φ_F, V_FB, W_max, V_T and normalised C–V values.

Stack a gate electrode on a thin insulator on silicon and you have a MOS capacitor. Varying the gate voltage attracts majority carriers (accumulation), pushes them away (depletion), or pulls minority carriers to the surface to form an inversion layer — the channel of a MOSFET.

Understanding the MOS capacitor first makes MOSFET threshold voltage, body effect, subthreshold behaviour and C–V characterisation straightforward.

An analogy

The MOS capacitor is like a magnet held above a tray of mixed iron and copper filings (majority and minority carriers). A weak push clears the iron away (depletion); a strong pull of the opposite sign gathers a sheet of the other filings at the surface (inversion).

Flat-band and oxide charges

With no charges and equal work functions the bands would be flat at zero gate bias. Real structures need V_FB = φ_ms − Q_f/C_ox to flatten them. Charges: fixed oxide charge Q_f (near interface), mobile ions (Na⁺ — shift V_T with temperature/bias), oxide-trapped charge (from radiation, hot carriers) and interface traps D_it (charge that depends on surface potential, stretching the C–V curve and degrading subthreshold swing).

Inversion and threshold

Threshold components:

C–V characteristics

Low-frequency (quasi-static) C–V returns to C_ox in inversion because minority carriers can follow the signal; high-frequency (1 MHz) C–V stays at C_min because thermal generation is too slow. A fast ramp produces deep depletion (no inversion layer, W > W_max). In MOSFET-connected measurements the source/drain supply electrons, so even 'HF' curves show LF behaviour.

Extraction from C–V

t_ox from C_ox in accumulation (with quantum-mechanical corrections for thin oxides: equivalent oxide thickness, EOT); doping profile from the 1/C² slope in depletion; V_FB from the flat-band capacitance; D_it from comparing HF and LF curves or conductance method; mobile ions by bias-temperature stress shifts. High-k dielectrics are characterised by EOT = t_highk·(3.9/k).

Formulas

Worked example: Oxide capacitance

Given: t_ox 2 nm SiO₂

Find: C_ox.

Answer: 1.73 µF/cm²

Worked example: EOT

Given: 3 nm HfO₂ (k = 22)

Find: EOT.

Answer: 0.53 nm

Worked example: Flat-band shift from charge

Given: Q_f = 5 × 10¹¹ q/cm², C_ox 0.69 µF/cm²

Find: ΔV_FB.

Answer: −0.116 V

Three regimes (p-type substrate)

  V_G < V_FB  Accumulation: holes pile up at surface     C ≈ C_ox
              bands bend up at surface
  V_FB < V_G < V_T  Depletion: holes pushed away, −qN_A ions   C = C_ox ∥series C_dep (falls)
              W grows as √ψ_s
  V_G > V_T   Strong inversion: electron layer (channel) at ψ_s = 2φ_F
              W stays at W_max;  LF: C → C_ox, HF: C = C_min, fast ramp: deep depletion

Positive gate voltage on p-type silicon first depletes the surface (exposing acceptor ions), then, when the surface potential reaches 2φ_F, the electron density at the surface equals the bulk hole density: strong inversion. Additional gate charge is balanced by inversion electrons, so the depletion width stops growing.

Flat-band, threshold and C–V numbers

import math
# MOS capacitor: oxide capacitance, flat-band, threshold, depletion width and C-V (p-type substrate, n+ poly gate)
q, eps_si, eps_ox, VT, ni = 1.602e-19, 11.7 * 8.854e-14, 3.9 * 8.854e-14, 0.02585, 9.65e9
Na, tox = 1e17, 5e-7                        # 5 nm oxide
Cox = eps_ox / tox
phiF = VT * math.log(Na / ni)
phi_ms = -0.56 - phiF                       # n+ poly on p-Si
Qf = 1e10 * q                               # fixed oxide charge
VFB = phi_ms - Qf / Cox
Wmax = math.sqrt(2 * eps_si * 2 * phiF / (q * Na))
Qdmax = q * Na * Wmax
VT_th = VFB + 2 * phiF + Qdmax / Cox
print(f"Cox = {Cox*1e6:.3f} uF/cm^2, phiF = {phiF:.3f} V, phi_ms = {phi_ms:.3f} V")
print(f"VFB = {VFB:.3f} V, Wdmax = {Wmax*1e7:.1f} nm, Qd = {Qdmax*1e6:.3f} uC/cm^2, VT = {VT_th:.3f} V")
Cmin = 1 / (1 / Cox + Wmax / eps_si)
print(f"high-frequency C-V: accumulation Cox, inversion minimum Cmin/Cox = {Cmin/Cox:.3f}")
for psi in (0.1, 0.3, 0.6):
    W = math.sqrt(2 * eps_si * psi / (q * Na))
    print(f"depletion with surface potential {psi:.1f} V: W = {W*1e7:.1f} nm, C/Cox = {1/(1+Cox*W/eps_si):.3f}")
# Cox = 0.691 uF/cm^2, phiF = 0.418 V, phi_ms = -0.978 V
# VFB = -0.980 V, Wdmax = 103.9 nm, Qd = 0.166 uC/cm^2, VT = 0.096 V
# high-frequency C-V: accumulation Cox, inversion minimum Cmin/Cox = 0.126
# depletion with surface potential 0.1 V: W = 36.0 nm, C/Cox = 0.294
# depletion with surface potential 0.3 V: W = 62.3 nm, C/Cox = 0.194
# depletion with surface potential 0.6 V: W = 88.1 nm, C/Cox = 0.146

A 5 nm oxide gives C_ox = 0.69 µF/cm². With 10¹⁷ doping, φ_F = 0.42 V, W_max = 104 nm and — with an n⁺ poly gate (φ_ms ≈ −0.98 V) — the threshold is only ~0.1 V: real NMOS devices add channel implants to set V_T. In the high-frequency C–V curve the capacitance falls to 13% of C_ox in inversion; as the surface depletes, C/C_ox drops from 0.29 to 0.15.

Common mistakes

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Key takeaways

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