MOSFET: Mathematical Modeling

Keywords: mosfet equations,mosfet modeling,threshold voltage,drain current,NMOS PMOS,short channel effects,subthreshold,device physics equations

MOSFET: Mathematical Modeling

Metal-Oxide-Semiconductor Field-Effect Transistor (MOSFET) Comprehensive equations, mathematical modeling, and process-parameter relationships

1. Fundamental Device Structure

1.1 MOSFET Components

A MOSFET is a four-terminal semiconductor device consisting of:

• Source (S) : Heavily doped region where carriers originate
• Drain (D) : Heavily doped region where carriers are collected
• Gate (G) : Control electrode separated from channel by dielectric
• Body/Substrate (B) : Semiconductor bulk (p-type for NMOS, n-type for PMOS)

1.2 Operating Principle

The gate voltage modulates channel conductivity through field effect:

$$ \text{Gate Voltage} \rightarrow \text{Electric Field} \rightarrow \text{Channel Formation} \rightarrow \text{Current Flow} $$

1.3 Device Types

2. Core MOSFET Equations

2.1 Threshold Voltage

The threshold voltage $V_{th}$ determines device turn-on and is highly process-dependent:

$$ V_{th} = V_{FB} + 2\phi_F + \frac{\sqrt{2\varepsilon_{Si} \cdot q \cdot N_A \cdot 2\phi_F}}{C_{ox}} $$

Component Equations

• Flat-band voltage :

$$ V_{FB} = \phi_{ms} - \frac{Q_{ox}}{C_{ox}} $$

• Fermi potential :

$$ \phi_F = \frac{kT}{q} \ln\left(\frac{N_A}{n_i}\right) $$

• Oxide capacitance per unit area :

$$ C_{ox} = \frac{\varepsilon_{ox}}{t_{ox}} = \frac{\kappa \cdot \varepsilon_0}{t_{ox}} $$

• Work function difference :

$$ \phi_{ms} = \phi_m - \phi_s = \phi_m - \left(\chi + \frac{E_g}{2q} + \phi_F\right) $$

Parameter Definitions

2.2 Drain Current Equations

2.2.1 Linear (Triode) Region

Condition : $V_{DS} < V_{GS} - V_{th}$ (channel not pinched off)

$$ I_D = \mu_n C_{ox} \frac{W}{L} \left[ (V_{GS} - V_{th}) V_{DS} - \frac{V_{DS}^2}{2} \right] $$

Simplified form (for small $V_{DS}$):

$$ I_D \approx \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th}) V_{DS} $$

Channel resistance :

$$ R_{ch} = \frac{V_{DS}}{I_D} = \frac{L}{\mu_n C_{ox} W (V_{GS} - V_{th})} $$

2.2.2 Saturation Region

Condition : $V_{DS} \geq V_{GS} - V_{th}$ (channel pinched off)

$$ I_D = \frac{1}{2} \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th})^2 (1 + \lambda V_{DS}) $$

Without channel-length modulation ($\lambda = 0$):

$$ I_{D,sat} = \frac{1}{2} \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th})^2 $$

Saturation voltage :

$$ V_{DS,sat} = V_{GS} - V_{th} $$

2.2.3 Channel-Length Modulation

The parameter $\lambda$ captures output resistance degradation:

$$ \lambda = \frac{1}{L \cdot E_{crit}} \approx \frac{1}{V_A} $$

Output resistance :

$$ r_o = \frac{\partial V_{DS}}{\partial I_D} = \frac{1}{\lambda I_D} = \frac{V_A + V_{DS}}{I_D} $$

Where $V_A$ is the Early voltage (typically $5$ to $50$ V/μm × L).

2.3 Subthreshold Conduction

2.3.1 Weak Inversion Current

Condition : $V_{GS} < V_{th}$ (exponential behavior)

$$ I_D = I_0 \exp\left(\frac{V_{GS} - V_{th}}{n \cdot V_T}\right) \left[1 - \exp\left(-\frac{V_{DS}}{V_T}\right)\right] $$

Characteristic current :

$$ I_0 = \mu_n C_{ox} \frac{W}{L} (n-1) V_T^2 $$

Thermal voltage :

$$ V_T = \frac{kT}{q} \approx 26 \text{ mV at } T = 300\text{K} $$

2.3.2 Subthreshold Swing

The subthreshold swing $S$ quantifies turn-off sharpness:

$$ S = \frac{\partial V_{GS}}{\partial (\log_{10} I_D)} = n \cdot V_T \cdot \ln(10) = 2.3 \cdot n \cdot V_T $$

Numerical values :

• Ideal minimum: $S_{min} = 60$ mV/decade (at 300K, $n = 1$)
• Typical range: $S = 70$ to $100$ mV/decade
• $n = 1 + \frac{C_{dep}}{C_{ox}}$ (subthreshold ideality factor)

2.3.3 Depletion Capacitance

$$ C_{dep} = \frac{\varepsilon_{Si}}{W_{dep}} = \sqrt{\frac{q \varepsilon_{Si} N_A}{4 \phi_F}} $$

2.4 Body Effect

When source-to-body voltage $V_{SB} eq 0$:

$$ V_{th}(V_{SB}) = V_{th0} + \gamma \left(\sqrt{2\phi_F + V_{SB}} - \sqrt{2\phi_F}\right) $$

Body effect coefficient :

$$ \gamma = \frac{\sqrt{2 q \varepsilon_{Si} N_A}}{C_{ox}} $$

Typical values : $\gamma = 0.3$ to $1.0$ V$^{1/2}$

2.5 Transconductance and Output Conductance

2.5.1 Transconductance

Saturation region :

$$ g_m = \frac{\partial I_D}{\partial V_{GS}} = \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th}) = \sqrt{2 \mu_n C_{ox} \frac{W}{L} I_D} $$

Alternative form :

$$ g_m = \frac{2 I_D}{V_{GS} - V_{th}} $$

2.5.2 Output Conductance

$$ g_{ds} = \frac{\partial I_D}{\partial V_{DS}} = \lambda I_D = \frac{I_D}{V_A} $$

2.5.3 Intrinsic Gain

$$ A_v = \frac{g_m}{g_{ds}} = \frac{2}{\lambda(V_{GS} - V_{th})} = \frac{2 V_A}{V_{GS} - V_{th}} $$

3. Short-Channel Effects

3.1 Velocity Saturation

At high lateral electric fields ($E > E_{crit} \approx 10^4$ V/cm):

$$ v_d = \frac{\mu_n E}{1 + E/E_{crit}} $$

Saturation velocity :

$$ v_{sat} = \mu_n E_{crit} \approx 10^7 \text{ cm/s (electrons in Si)} $$

3.1.1 Modified Saturation Current

$$ I_{D,sat} = W C_{ox} v_{sat} (V_{GS} - V_{th}) $$

Note: Linear (not quadratic) dependence on gate overdrive.

3.1.2 Critical Length

Velocity saturation dominates when:

$$ L < L_{crit} = \frac{\mu_n (V_{GS} - V_{th})}{2 v_{sat}} $$

3.2 Drain-Induced Barrier Lowering (DIBL)

The drain field reduces the source-side barrier:

$$ V_{th} = V_{th,long} - \eta \cdot V_{DS} $$

DIBL coefficient :

$$ \eta = -\frac{\partial V_{th}}{\partial V_{DS}} $$

Typical values : $\eta = 20$ to $100$ mV/V for short channels

3.2.1 Modified Threshold Equation

$$ V_{th}(V_{DS}, V_{SB}) = V_{th0} + \gamma(\sqrt{2\phi_F + V_{SB}} - \sqrt{2\phi_F}) - \eta V_{DS} $$

3.3 Mobility Degradation

3.3.1 Vertical Field Effect

$$ \mu_{eff} = \frac{\mu_0}{1 + \theta (V_{GS} - V_{th})} $$

Alternative form (surface roughness scattering):

$$ \mu_{eff} = \frac{\mu_0}{1 + (\theta_1 + \theta_2 V_{SB})(V_{GS} - V_{th})} $$

3.3.2 Universal Mobility Model

$$ \mu_{eff} = \frac{\mu_0}{\left[1 + \left(\frac{E_{eff}}{E_0}\right)^ u + \left(\frac{E_{eff}}{E_1}\right)^\beta\right]} $$

Where $E_{eff}$ is the effective vertical field:

$$ E_{eff} = \frac{Q_b + \eta_s Q_i}{\varepsilon_{Si}} $$

3.4 Hot Carrier Effects

3.4.1 Impact Ionization Current

$$ I_{sub} = \frac{I_D}{M - 1} $$

Multiplication factor :

$$ M = \frac{1}{1 - \int_0^{L_{dep}} \alpha(E) dx} $$

3.4.2 Ionization Rate

$$ \alpha = \alpha_\infty \exp\left(-\frac{E_{crit}}{E}\right) $$

3.5 Gate Leakage

3.5.1 Direct Tunneling Current

$$ J_g = A \cdot E_{ox}^2 \exp\left(-\frac{B}{\vert E_{ox} \vert}\right) $$

Where:

$$ A = \frac{q^3}{16\pi^2 \hbar \phi_b} $$

$$ B = \frac{4\sqrt{2m^* \phi_b^3}}{3\hbar q} $$

3.5.2 Gate Oxide Field

$$ E_{ox} = \frac{V_{GS} - V_{FB} - \psi_s}{t_{ox}} $$

4. Parameters

4.1 Gate Oxide Engineering

4.1.1 Oxide Capacitance

$$ C_{ox} = \frac{\varepsilon_0 \cdot \kappa}{t_{ox}} $$

4.1.2 Equivalent Oxide Thickness (EOT)

$$ EOT = t_{high-\kappa} \times \frac{\varepsilon_{SiO_2}}{\varepsilon_{high-\kappa}} = t_{high-\kappa} \times \frac{3.9}{\kappa} $$

4.1.3 Capacitance Equivalent Thickness (CET)

Including quantum effects and poly depletion:

$$ CET = EOT + \Delta t_{QM} + \Delta t_{poly} $$

Where:

• $\Delta t_{QM} \approx 0.3$ to $0.5$ nm (quantum mechanical)
• $\Delta t_{poly} \approx 0.3$ to $0.5$ nm (polysilicon depletion)

4.2 Channel Doping

4.2.1 Doping Profile Impact

$$ V_{th} \propto \sqrt{N_A} $$

$$ \mu \propto \frac{1}{N_A^{0.3}} \text{ (ionized impurity scattering)} $$

4.2.2 Depletion Width

$$ W_{dep} = \sqrt{\frac{2\varepsilon_{Si}(2\phi_F + V_{SB})}{qN_A}} $$

4.2.3 Junction Capacitance

$$ C_j = C_{j0}\left(1 + \frac{V_R}{\phi_{bi}}\right)^{-m} $$

Where:

• $C_{j0}$ = zero-bias capacitance
• $\phi_{bi}$ = built-in potential
• $m = 0.5$ (abrupt junction), $m = 0.33$ (graded junction)

4.3 Gate Material Engineering

4.3.1 Work Function Values

4.3.2 Flat-Band Voltage Engineering

For symmetric CMOS threshold voltages:

$$ V_{FB,NMOS} + V_{FB,PMOS} \approx -E_g/q $$

4.4 Channel Length Scaling

4.4.1 Characteristic Length

$$ \lambda = \sqrt{\frac{\varepsilon_{Si}}{\varepsilon_{ox}} \cdot t_{ox} \cdot x_j} $$

For good short-channel control: $L > 5\lambda$ to $10\lambda$

4.4.2 Scale Length (FinFET/GAA)

$$ \lambda_{GAA} = \sqrt{\frac{\varepsilon_{Si} \cdot t_{Si}^2}{2 \varepsilon_{ox} \cdot t_{ox}}} $$

4.5 Strain Engineering

4.5.1 Mobility Enhancement

$$ \mu_{strained} = \mu_0 (1 + \Pi \cdot \sigma) $$

Where:

• $\Pi$ = piezoresistive coefficient
• $\sigma$ = applied stress

Enhancement factors :

• NMOS (tensile): $+30\%$ to $+70\%$ mobility gain
• PMOS (compressive): $+50\%$ to $+100\%$ mobility gain

4.5.2 Stress Impact on Threshold

$$ \Delta V_{th} = \alpha_{th} \cdot \sigma $$

Where $\alpha_{th} \approx 1$ to $5$ mV/GPa

5. Advanced Compact Models

5.1 BSIM4 Model

5.1.1 Unified Current Equation

$$ I_{DS} = I_{DS0} \cdot \left(1 + \frac{V_{DS} - V_{DS,eff}}{V_A}\right) \cdot \frac{1}{1 + R_S \cdot G_{DS0}} $$

5.1.2 Effective Overdrive

$$ V_{GS,eff} - V_{th} = \frac{2nV_T \cdot \ln\left[1 + \exp\left(\frac{V_{GS} - V_{th}}{2nV_T}\right)\right]}{1 + 2n\sqrt{\delta + \left(\frac{V_{GS}-V_{th}}{2nV_T} - \delta\right)^2}} $$

5.1.3 Effective Saturation Voltage

$$ V_{DS,eff} = V_{DS,sat} - \frac{V_T}{2}\ln\left(\frac{V_{DS,sat} + \sqrt{V_{DS,sat}^2 + 4V_T^2}}{V_{DS} + \sqrt{V_{DS}^2 + 4V_T^2}}\right) $$

5.2 Surface Potential Model (PSP)

5.2.1 Implicit Surface Potential Equation

$$ V_{GB} - V_{FB} = \psi_s + \gamma\sqrt{\psi_s + V_T e^{(\psi_s - 2\phi_F - V_{SB})/V_T} - V_T} $$

5.2.2 Charge-Based Current

$$ I_D = \mu W \frac{Q_i(0) - Q_i(L)}{L} \cdot \frac{V_{DS}}{V_{DS,eff}} $$

Where $Q_i$ is the inversion charge density:

$$ Q_i = -C_{ox}\left[\psi_s - 2\phi_F - V_{ch} + V_T\left(e^{(\psi_s - 2\phi_F - V_{ch})/V_T} - 1\right)\right]^{1/2} $$

5.3 FinFET Equations

5.3.1 Effective Width

$$ W_{eff} = 2H_{fin} + W_{fin} $$

For multiple fins:

$$ W_{total} = N_{fin} \cdot (2H_{fin} + W_{fin}) $$

5.3.2 Multi-Gate Scale Length

Double-gate :

$$ \lambda_{DG} = \sqrt{\frac{\varepsilon_{Si} \cdot t_{Si} \cdot t_{ox}}{2\varepsilon_{ox}}} $$

Gate-all-around (GAA) :

$$ \lambda_{GAA} = \sqrt{\frac{\varepsilon_{Si} \cdot r^2}{4\varepsilon_{ox}} \cdot \ln\left(1 + \frac{t_{ox}}{r}\right)} $$

Where $r$ = nanowire radius

5.3.3 FinFET Threshold Voltage

$$ V_{th} = V_{FB} + 2\phi_F + \frac{qN_A W_{fin}}{2C_{ox}} - \Delta V_{th,SCE} $$

6. Process-Equation Coupling

6.1 Parameter Sensitivity Analysis

6.2 Variability Equations

6.2.1 Random Dopant Fluctuation (RDF)

$$ \sigma_{V_{th}} = \frac{A_{VT}}{\sqrt{W \cdot L}} $$

Where $A_{VT}$ is the Pelgrom coefficient (typically $1$ to $5$ mV·μm).

6.2.2 Line Edge Roughness (LER)

$$ \sigma_{V_{th,LER}} \propto \frac{\sigma_{LER}}{L} $$

6.2.3 Oxide Thickness Variation

$$ \sigma_{V_{th,tox}} = \frac{\partial V_{th}}{\partial t_{ox}} \cdot \sigma_{t_{ox}} = \frac{V_{th} - V_{FB} - 2\phi_F}{t_{ox}} \cdot \sigma_{t_{ox}} $$

6.3 Equations:

6.3.1 Drive Current

$$ I_{on} = \frac{W}{L} \cdot \mu_{eff} \cdot C_{ox} \cdot \frac{(V_{DD} - V_{th})^\alpha}{1 + (V_{DD} - V_{th})/E_{sat}L} $$

Where $\alpha = 2$ (long channel) or $\alpha \rightarrow 1$ (velocity saturated).

6.3.2 Leakage Current

$$ I_{off} = I_0 \cdot \frac{W}{L} \cdot \exp\left(\frac{-V_{th}}{nV_T}\right) \cdot \left(1 - \exp\left(\frac{-V_{DD}}{V_T}\right)\right) $$

6.3.3 CV/I Delay Metric

$$ \tau = \frac{C_L \cdot V_{DD}}{I_{on}} \propto \frac{L^2}{\mu (V_{DD} - V_{th})} $$

Constants:

Equations:

Threshold Voltage

$$ V_{th} = V_{FB} + 2\phi_F + \frac{\sqrt{2\varepsilon_{Si} q N_A (2\phi_F)}}{C_{ox}} $$

Linear Region Current

$$ I_D = \mu C_{ox} \frac{W}{L} \left[(V_{GS} - V_{th})V_{DS} - \frac{V_{DS}^2}{2}\right] $$

Saturation Current

$$ I_D = \frac{1}{2}\mu C_{ox}\frac{W}{L}(V_{GS} - V_{th})^2(1 + \lambda V_{DS}) $$

Subthreshold Current

$$ I_D = I_0 \exp\left(\frac{V_{GS} - V_{th}}{nV_T}\right) $$

Transconductance

$$ g_m = \sqrt{2\mu C_{ox}\frac{W}{L}I_D} $$

Body Effect

$$ V_{th} = V_{th0} + \gamma\left(\sqrt{2\phi_F + V_{SB}} - \sqrt{2\phi_F}\right) $$

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