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Microelectronics models

Derive carrier, junction, transistor, small-signal, and noise models with twenty numerical device and circuit examples.

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Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

1. Carrier statistics and conduction

Definitions & inputs. n,p are electron/hole densities, ni intrinsic density, μn,μp mobilities; q=1.602176634×10⁻¹⁹ C.

  1. Equilibrium mass action relates carrier populations.

    np=ni2np=n_i^2

  2. Add electron and hole conductivity, then apply uniform-bar geometry.

    σ=q(nμn+pμp),R=ℓ/(σA)\sigma=q(n\mu_n+p\mu_p),\quad R=\ell/(\sigma A)

  3. Thermal voltage sets diffusion and diode voltage scales.

    VT=kBT/qV_T=k_BT/q

Interpretation. Mobility and intrinsic density depend strongly on material and temperature.

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2. Junction electrostatics and current

Definitions & inputs. NA,ND are acceptor/donor densities; εs permittivity; VR positive reverse bias; IS saturation current; η ideality factor.

  1. Fermi-level alignment creates a built-in potential.

    Vbi=VTln⁡(NAND/ni2)V_{bi}=V_T\ln(N_AN_D/n_i^2)

  2. Integrate Poisson’s equation in depleted regions and enforce charge neutrality.

    W=2ϵsq(1/NA+1/ND)(Vbi+VR)W=\sqrt{\frac{2\epsilon_s}{q}(1/N_A+1/N_D)(V_{bi}+V_R)}

  3. Diffusion current is exponential; differentiation yields incremental resistance well above leakage.

    I=IS(eV/(ηVT)−1),rd≃ηVT/II=I_S(e^{V/(\eta V_T)}-1),\quad r_d\simeq\eta V_T/I

Interpretation. Breakdown, tunneling, parasitics, and heating are outside this diode law.

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3. Long-channel MOS and small-signal gain

Definitions & inputs. Cox is oxide capacitance per area, β=μCoxW/L, Vov=VGS−Vth, λ channel-length-modulation coefficient.

  1. Gate electrostatics set the local inversion charge.

    Cox=ϵox/tox,Qinv≃−Cox(VGS−Vth−V(x))C_{ox}=\epsilon_{ox}/t_{ox},\quad Q_{\rm inv}\simeq-C_{ox}(V_{GS}-V_{th}-V(x))

  2. Integrate drift current along the channel before pinch-off.

    ID=β[(Vov)VDS−VDS2/2]I_D=\beta[(V_{ov})V_{DS}-V_{DS}^2/2]

  3. At VDS≥Vov use saturation, then differentiate at the bias point.

    ID,sat≃12βVov2,gm=βVov=2ID/Vov,ro≃1/(λID)I_{D,\rm sat}\simeq\tfrac12\beta V_{ov}^2,\quad g_m=\beta V_{ov}=2I_D/V_{ov},\quad r_o\simeq1/(\lambda I_D)

Interpretation. Bias conditions must be checked before selecting a region equation.

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4. Time response, sampling, and noise

Definitions & inputs. R,C are lumped resistance/capacitance; B bandwidth; T absolute temperature.

  1. Kirchhoff’s law gives a first-order charge response.

    RC dV/dt+V=Vin⇒τ=RCRC\,dV/dt+V=V_{in}\Rightarrow\tau=RC

  2. Thermal resistor noise and sampled capacitor noise have different circuit interpretations.

    vn2‾=4kBTRB,σV,kT/C=kBT/C\overline{v_n^2}=4k_BTRB,\quad\sigma_{V,kT/C}=\sqrt{k_BT/C}

  3. The first-order low-pass magnitude falls by 3 dB at this frequency.

    fc=(2πRC)−1f_c=(2\pi RC)^{-1}

Interpretation. Noise bandwidth may differ from a circuit’s −3 dB bandwidth.

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Graphical worked example

Ideal Shockley diode: IS=1 pA, ideality one, VT=25.85 mV; series resistance and breakdown excluded. X axis: Forward junction voltage (V). Y axis: Ideal diode current (mA).
Ideal Shockley diode: IS=1 pA, ideality one, VT=25.85 mV; series resistance and breakdown excluded. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Thermal voltage

Definitions & inputs. T=300 K, kB=1.380649×10⁻²³ J/K.

  1. Choose the governing model and isolate the requested quantity.

    VT=kBT/qV_T=k_BT/q

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    VT=(1.380649×10−23)(300)/(1.602176634×10−19)V_T=(1.380649\times10^{-23})(300)/(1.602176634\times10^{-19})

  3. Evaluate the expression; the result uses the units shown.

    Result=0.025852 V\mathrm{Result}=0.025852\ {\rm V}

Interpretation. About 25.85 mV sets a common room-temperature scale.

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Example 02. Minority carriers

Definitions & inputs. n=10²² m⁻³, ni=10¹⁶ m⁻³.

  1. Choose the governing model and isolate the requested quantity.

    p=ni2/np=n_i^2/n

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    p=1032/1022p=10^{32}/10^{22}

  3. Evaluate the expression; the result uses the units shown.

    Result=1×1010 m−3\mathrm{Result}=1\times10^{10}\ {\rm m}^{-3}

Interpretation. Majority doping suppresses equilibrium minority density.

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Example 03. Electron conductivity

Definitions & inputs. n=10²² m⁻³, μn=0.1 m²/(V s), hole contribution negligible.

  1. Choose the governing model and isolate the requested quantity.

    σ=qnμn\sigma=qn\mu_n

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    σ=(1.602176634×10−19)(1022)(0.1)\sigma=(1.602176634\times10^{-19})(10^{22})(0.1)

  3. Evaluate the expression; the result uses the units shown.

    Result=160.2177 S m−1\mathrm{Result}=160.2177\ {\rm S\,m}^{-1}

Interpretation. Mobility is a stated model input.

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Example 04. Resistor geometry

Definitions & inputs. Resistivity ρ=10⁻⁴ Ωm, length 10 µm, cross section 1 µm².

  1. Choose the governing model and isolate the requested quantity.

    R=ρℓ/AR=\rho\ell/A

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    R=10−4(10−5)/10−12R=10^{-4}(10^{-5})/10^{-12}

  3. Evaluate the expression; the result uses the units shown.

    Result=1000 Ω\mathrm{Result}=1000\ \Omega

Interpretation. Aspect ratio controls resistance for fixed material.

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Example 05. Built-in voltage

Definitions & inputs. VT=0.02585 V, NA=ND=10²³ m⁻³, ni=10¹⁶ m⁻³.

  1. Choose the governing model and isolate the requested quantity.

    Vbi=VTln⁡(NAND/ni2)V_{bi}=V_T\ln(N_AN_D/n_i^2)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Vbi=0.02585ln⁡(1014)V_{bi}=0.02585\ln(10^{14})

  3. Evaluate the expression; the result uses the units shown.

    Result=0.8333055 V\mathrm{Result}=0.8333055\ {\rm V}

Interpretation. This internal equilibrium potential is not an externally extractable battery voltage.

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Example 06. Depletion width

Definitions & inputs. εs=1.04×10⁻¹⁰ F/m, NA=ND=10²² m⁻³, Vbi+VR=1 V.

  1. Choose the governing model and isolate the requested quantity.

    W=(2ϵs/q)(1/NA+1/ND)VW=\sqrt{(2\epsilon_s/q)(1/N_A+1/N_D)V}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    W=(2⋅1.04×10−10/q)(2×10−22)W=\sqrt{(2\cdot1.04\times10^{-10}/q)(2\times10^{-22})}

  3. Evaluate the expression; the result uses the units shown.

    Result=5.095555×10−7 m\mathrm{Result}=5.095555\times10^{-7}\ {\rm m}

Interpretation. The result includes both depleted sides.

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Example 07. Forward diode current

Definitions & inputs. IS=1 pA, V=0.5 V, η=1, VT=0.02585 V.

  1. Choose the governing model and isolate the requested quantity.

    I=IS(eV/VT−1)I=I_S(e^{V/V_T}-1)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    I=10−12(e0.5/0.02585−1)I=10^{-12}(e^{0.5/0.02585}-1)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.0002513507 A\mathrm{Result}=0.0002513507\ {\rm A}

Interpretation. Series resistance can limit actual current at larger forward bias.

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Example 08. Diode incremental resistance

Definitions & inputs. I=1 mA, η=1, VT=25.85 mV.

  1. Choose the governing model and isolate the requested quantity.

    rd=ηVT/Ir_d=\eta V_T/I

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    rd=0.02585/0.001r_d=0.02585/0.001

  3. Evaluate the expression; the result uses the units shown.

    Result=25.85 Ω\mathrm{Result}=25.85\ \Omega

Interpretation. This is local slope resistance, not V/I.

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Example 09. Oxide capacitance density

Definitions & inputs. εox=3.45×10⁻¹¹ F/m; tox=10 nm.

  1. Choose the governing model and isolate the requested quantity.

    Cox=ϵox/toxC_{ox}=\epsilon_{ox}/t_{ox}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Cox=3.45×10−11/10−8C_{ox}=3.45\times10^{-11}/10^{-8}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.00345 F m−2\mathrm{Result}=0.00345\ {\rm F\,m}^{-2}

Interpretation. Thinner ideal dielectric increases capacitance density.

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Example 10. Gate capacitance

Definitions & inputs. Cox=0.00345 F/m²; gate area 1 µm².

  1. Choose the governing model and isolate the requested quantity.

    Cg=CoxWLC_g=C_{ox}WL

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Cg=0.00345(10−12)C_g=0.00345(10^{-12})

  3. Evaluate the expression; the result uses the units shown.

    Result=3.45×10−15 F\mathrm{Result}=3.45\times10^{-15}\ {\rm F}

Interpretation. Fringing and overlap would add capacitance.

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Example 11. MOS saturation current

Definitions & inputs. β=1 mA/V²; Vov=0.2 V.

  1. Choose the governing model and isolate the requested quantity.

    ID=12βVov2I_D=\tfrac12\beta V_{ov}^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    ID=0.5(10−3)(0.2)2I_D=0.5(10^{-3})(0.2)^2

  3. Evaluate the expression; the result uses the units shown.

    Result=2×10−5 A\mathrm{Result}=2\times10^{-5}\ {\rm A}

Interpretation. Check VDS≥0.2 V before using this result.

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Example 12. MOS triode current

Definitions & inputs. β=1 mA/V²; Vov=0.2 V; VDS=0.05 V.

  1. Choose the governing model and isolate the requested quantity.

    ID=β(VovVDS−VDS2/2)I_D=\beta(V_{ov}V_{DS}-V_{DS}^2/2)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    ID=10−3(0.2⋅0.05−0.052/2)I_D=10^{-3}(0.2\cdot0.05-0.05^2/2)

  3. Evaluate the expression; the result uses the units shown.

    Result=8.75×10−6 A\mathrm{Result}=8.75\times10^{-6}\ {\rm A}

Interpretation. VDS<Vov places the device in this model’s triode region.

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Example 13. Transconductance

Definitions & inputs. ID=20 µA; Vov=0.2 V.

  1. Choose the governing model and isolate the requested quantity.

    gm=2ID/Vovg_m=2I_D/V_{ov}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    gm=2(20×10−6)/0.2g_m=2(20\times10^{-6})/0.2

  3. Evaluate the expression; the result uses the units shown.

    Result=0.0002 S\mathrm{Result}=0.0002\ {\rm S}

Interpretation. An incremental gate voltage modulates drain current through gm.

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Example 14. Output resistance

Definitions & inputs. λ=0.1 V⁻¹; ID=20 µA.

  1. Choose the governing model and isolate the requested quantity.

    ro≃1/(λID)r_o\simeq1/(\lambda I_D)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    ro=1/(0.1⋅20×10−6)r_o=1/(0.1\cdot20\times10^{-6})

  3. Evaluate the expression; the result uses the units shown.

    Result=500000 Ω\mathrm{Result}=500000\ \Omega

Interpretation. Channel-length modulation makes saturation current voltage-dependent.

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Example 15. Common-source gain

Definitions & inputs. gm=0.2 mS; RD=10 kΩ; ro≫RD.

  1. Choose the governing model and isolate the requested quantity.

    Av≃−gmRDA_v\simeq-g_mR_D

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Av=−(0.0002)(10000)A_v=-(0.0002)(10000)

  3. Evaluate the expression; the result uses the units shown.

    Result=−2 \mathrm{Result}=-2\

Interpretation. The negative sign indicates inversion.

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Example 16. RC time constant

Definitions & inputs. R=10 kΩ, C=10 pF.

  1. Choose the governing model and isolate the requested quantity.

    τ=RC\tau=RC

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    τ=104(10−11)\tau=10^4(10^{-11})

  3. Evaluate the expression; the result uses the units shown.

    Result=1×10−7 s\mathrm{Result}=1\times10^{-7}\ {\rm s}

Interpretation. This is 100 ns.

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Example 17. RC cutoff

Definitions & inputs. R=10 kΩ, C=10 pF.

  1. Choose the governing model and isolate the requested quantity.

    fc=1/(2πRC)f_c=1/(2\pi RC)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    fc=1/(2π⋅10−7)f_c=1/(2\pi\cdot10^{-7})

  3. Evaluate the expression; the result uses the units shown.

    Result=1591549 Hz\mathrm{Result}=1591549\ {\rm Hz}

Interpretation. The −3 dB frequency is about 1.59 MHz.

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Example 18. Resistor thermal noise

Definitions & inputs. R=1 kΩ, T=300 K, noise bandwidth B=1 MHz.

  1. Choose the governing model and isolate the requested quantity.

    vn,rms=4kBTRBv_{n,\rm rms}=\sqrt{4k_BTRB}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    4(1.380649×10−23)(300)(1000)(106)\sqrt{4(1.380649\times10^{-23})(300)(1000)(10^6)}

  3. Evaluate the expression; the result uses the units shown.

    Result=4.070355×10−6 V\mathrm{Result}=4.070355\times10^{-6}\ {\rm V}

Interpretation. Noise is an RMS voltage, not a DC offset.

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Example 19. Sampled thermal noise

Definitions & inputs. C=1 pF, T=300 K.

  1. Choose the governing model and isolate the requested quantity.

    σV=kBT/C\sigma_V=\sqrt{k_BT/C}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    (1.380649×10−23)(300)/10−12\sqrt{(1.380649\times10^{-23})(300)/10^{-12}}

  3. Evaluate the expression; the result uses the units shown.

    Result=6.435796×10−5 V\mathrm{Result}=6.435796\times10^{-5}\ {\rm V}

Interpretation. Larger sampling capacitance lowers this thermal-noise scale.

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Example 20. BJT transconductance

Definitions & inputs. IC=1 mA, VT=25.85 mV.

  1. Choose the governing model and isolate the requested quantity.

    gm=IC/VTg_m=I_C/V_T

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    gm=0.001/0.02585g_m=0.001/0.02585

  3. Evaluate the expression; the result uses the units shown.

    Result=0.03868472 S\mathrm{Result}=0.03868472\ {\rm S}

Interpretation. BJT gm is set directly by collector bias current in this model.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.