8.3 Current-voltage characteristics

This topic covers how current flows through MOSFETs under different gate and drain voltage conditions. Understanding MOSFET I-V characteristics is essential for analyzing transistor operation in amplifier and switching circuits, and for connecting device physics to practical circuit design.

A note on this guide: While the sections below also review p-n junction diode I-V behavior (which forms the foundation for understanding MOSFET operation), the primary focus of Unit 8 is the MOSFET. Where possible, connections to MOSFET behavior are highlighted.

Current-voltage characteristics of diodes

The current-voltage (I-V) characteristic describes the relationship between the current flowing through a diode and the voltage applied across it. Diodes exhibit nonlinear I-V curves because of how the p-n junction responds to applied voltage. Grasping this nonlinearity is the starting point for understanding all semiconductor device behavior, including MOSFETs.

Ideal diode vs real diode

An ideal diode has zero resistance in forward bias (short circuit) and infinite resistance in reverse bias (open circuit). It's a useful mental model, but real devices differ in important ways:

• Real silicon diodes require about 0.6–0.7 V of forward bias before significant current flows. This is the turn-on voltage.
• A small reverse saturation current (often in the nanoamp range for silicon) flows even under reverse bias.
• Real diodes also exhibit reverse leakage current and parasitic capacitance effects that matter at high frequencies.

Forward bias vs reverse bias

• Forward bias: The p-type region connects to the positive terminal, the n-type to the negative terminal. This shrinks the depletion region, lowers the potential barrier, and allows current to flow easily once the turn-on voltage is reached.
• Reverse bias: The p-type region connects to the negative terminal, the n-type to the positive terminal. This widens the depletion region and raises the barrier, so only a tiny leakage current flows.

The key idea: forward bias thins the barrier and lets carriers through; reverse bias thickens it and blocks them.

Diode equation

The Shockley diode equation quantifies the I-V relationship:

$I = I_s\left(e^{qV/nkT} - 1\right)$

where:

• $I_s$ = reverse saturation current (device- and temperature-dependent)
• $q$ = elementary charge ($1.6 \times 10^{-19}$ C)
• $V$ = voltage across the diode
• $n$ = ideality factor (1 for an ideal diode, up to ~2 when recombination in the depletion region dominates)
• $k$ = Boltzmann's constant ($1.38 \times 10^{-23}$ J/K)
• $T$ = absolute temperature in Kelvin

The exponential term is what produces the sharp nonlinear turn-on. At room temperature ($T \approx 300$ K), the thermal voltage $kT/q \approx 26$ mV, so even small changes in $V$ cause large changes in current once you're past the turn-on region.

Diode current components

Total diode current has two main components:

• Diffusion current: Caused by majority carriers diffusing across the junction. This dominates in forward bias and is responsible for the exponential rise in current.
• Drift current: Caused by minority carriers swept through the depletion region by the built-in electric field. This dominates in reverse bias and accounts for the small, roughly constant reverse saturation current $I_s$.

Diffusion current vs drift current

• Diffusion current is proportional to the carrier concentration gradient and grows exponentially with forward voltage.
• Drift current is proportional to the electric field in the depletion region and stays nearly constant regardless of reverse voltage (until breakdown).
• In forward bias, diffusion current far exceeds drift current. In reverse bias, drift current is the dominant (though tiny) component. The balance between these two shapes the entire I-V curve.

Temperature dependence of diode current

Temperature affects diode behavior in two competing ways:

• The reverse saturation current $I_s$ increases strongly with temperature, approximately as: $I_s \propto T^3 e^{-E_g/kT}$ where $E_g$ is the semiconductor bandgap energy. As a rough rule, $I_s$ roughly doubles for every 10 °C increase in silicon.
• The forward voltage drop decreases with increasing temperature, at roughly $-2$ mV/°C for silicon. This happens because the rising $I_s$ shifts the I-V curve leftward.

These temperature effects matter in circuit design, especially for biasing and thermal stability.

Breakdown mechanisms in diodes

When reverse bias exceeds a critical threshold, current suddenly surges. This is reverse breakdown. Two distinct physical mechanisms cause it, and knowing which one applies depends on the doping level.

Zener breakdown

Zener breakdown occurs in heavily doped junctions where the depletion region is very narrow (typically < 10 nm).

• The intense electric field across this thin region enables quantum-mechanical tunneling: electrons tunnel directly from the valence band on the p-side to the conduction band on the n-side.
• This produces a sharp, well-defined breakdown knee on the I-V curve.
• Zener breakdown typically dominates for breakdown voltages below about 5 V.

Avalanche breakdown

Avalanche breakdown occurs in lightly doped junctions with wider depletion regions.

1. Under high reverse bias, the strong electric field accelerates carriers to high kinetic energies.
2. These fast-moving carriers collide with lattice atoms and knock out electron-hole pairs through impact ionization.
3. The newly created carriers are also accelerated and cause further ionization, creating a chain reaction (avalanche).

Avalanche breakdown typically dominates for breakdown voltages above about 7 V. Between 5 V and 7 V, both mechanisms can contribute.

Reverse breakdown voltage

The reverse breakdown voltage $V_{BR}$ depends on:

• Doping concentrations: Higher doping → narrower depletion region → lower $V_{BR}$ (Zener mechanism). Lower doping → wider depletion region → higher $V_{BR}$ (avalanche mechanism).
• Junction geometry and semiconductor material: Engineers control $V_{BR}$ by tailoring the doping profile during fabrication.

Diodes designed to operate reliably in breakdown (Zener diodes) are intentionally engineered with a specific $V_{BR}$.

Diode equivalent circuits

Equivalent circuit models simplify diode behavior for circuit analysis. You pick the model based on how much accuracy you need.

Ideal diode model

• Forward bias → short circuit (zero voltage drop, unlimited current).
• Reverse bias → open circuit (zero current).
• Useful for quick, first-pass analysis where the 0.6–0.7 V drop is negligible compared to other voltages in the circuit.

Constant voltage drop model

• Forward bias → the diode holds a fixed voltage $V_D$ (typically 0.7 V for silicon). The rest of the circuit determines the current.
• Reverse bias → open circuit.
• This is the most commonly used model for hand calculations in introductory courses.

Piecewise linear model

• Forward bias → constant voltage drop $V_D$ plus a small forward resistance $r_f$ (so $V = V_D + I \cdot r_f$).
• Reverse bias → modeled as a very high resistance $R_R$, allowing a small leakage current.
• More accurate than the constant voltage drop model, especially at higher currents where the resistive drop matters.

Small-signal model of diodes

When a diode operates at a DC bias point and you apply a small AC signal on top of it, the small-signal model applies. The diode looks like a dynamic resistance:

$r_d = \frac{nkT}{qI_D}$

where $I_D$ is the DC operating current. At room temperature with $n = 1$, this simplifies to $r_d \approx 26 \text{ mV} / I_D$.

For example, at $I_D = 1$ mA, $r_d \approx 26 \;\Omega$. This model is valid only for small perturbations around the bias point and is essential for analyzing diodes in amplifier and filter circuits.

Diode capacitance effects

Diodes store charge, and that stored charge creates capacitance. These capacitance effects determine how fast a diode can switch and how it behaves at high frequencies.

Depletion layer capacitance

The depletion (junction) capacitance $C_j$ comes from the fixed charges in the depletion region, which acts like a parallel-plate capacitor:

$C_j = \frac{C_{j0}}{\left(1 + V_R/V_{bi}\right)^{1/2}}$

• $C_{j0}$ = zero-bias junction capacitance
• $V_R$ = applied reverse bias voltage
• $V_{bi}$ = built-in potential (~0.7 V for silicon)

As reverse bias increases, the depletion region widens and $C_j$ decreases. This voltage-dependent capacitance is exploited in varactor diodes for tuning circuits.

Diffusion capacitance

The diffusion capacitance $C_d$ arises from minority carriers stored in the neutral regions during forward bias:

$C_d = \frac{\tau \, I_D}{nkT/q}$

• $\tau$ = minority carrier lifetime
• $I_D$ = forward bias current

$C_d$ is proportional to the forward current, so it dominates over $C_j$ in forward bias. In reverse bias, $C_d$ is negligible because there's minimal minority carrier injection.

Diode switching characteristics

Capacitance effects directly impact switching speed:

• Turn-on delay: The junction capacitances must charge up before the diode fully conducts. This creates a finite rise time.
• Turn-off (reverse recovery): When switching from forward to reverse bias, the stored minority carriers in the neutral regions must be removed. This causes a brief reverse recovery current that flows until the excess charge is swept out or recombines. The reverse recovery time $t_{rr}$ is a key spec for diodes used in fast-switching applications like power supplies and digital circuits.

MOSFET current-voltage characteristics

Since this is Unit 8, the core focus is the MOSFET. The I-V characteristics of a MOSFET are fundamentally different from a diode because the MOSFET is a three-terminal device where the gate voltage controls the drain current.

MOSFET operating regions

A MOSFET operates in three distinct regions depending on the gate-to-source voltage $V_{GS}$ and drain-to-source voltage $V_{DS}$:

• Cutoff (subthreshold): $V_{GS} < V_T$ (threshold voltage). The channel is not formed, and $I_D \approx 0$ (only a tiny subthreshold leakage current flows).
• Linear (triode) region: $V_{GS} > V_T$ and $V_{DS} < V_{GS} - V_T$. The channel is fully formed, and the MOSFET behaves like a voltage-controlled resistor.
• Saturation region: $V_{GS} > V_T$ and $V_{DS} \geq V_{GS} - V_T$. The channel is pinched off at the drain end, and $I_D$ becomes nearly independent of $V_{DS}$.

MOSFET drain current equations

Linear region:

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

Saturation region:

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

where:

• $\mu_n$ = electron mobility in the channel
• $C_{ox}$ = oxide capacitance per unit area ($\varepsilon_{ox}/t_{ox}$)
• $W/L$ = channel width-to-length ratio (a key design parameter)
• $V_T$ = threshold voltage

The product $\mu_n C_{ox}$ is often written as $k'_n$ (the process transconductance parameter), and $k'_n(W/L)$ is sometimes written as $k_n$.

Output characteristics and channel-length modulation

The ideal saturation equation predicts that $I_D$ is constant for all $V_{DS} \geq V_{GS} - V_T$. In reality, increasing $V_{DS}$ slightly shortens the effective channel length, causing $I_D$ to increase gradually. This is channel-length modulation, modeled by:

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

where $\lambda$ is the channel-length modulation parameter (units: $V^{-1}$). Smaller $\lambda$ means flatter output curves and more ideal current-source behavior in saturation.

Transfer characteristics

The transfer characteristic plots $I_D$ vs. $V_{GS}$ at a fixed $V_{DS}$. In saturation, this is a parabola (square-law relationship). The transconductance $g_m$ describes how effectively the gate voltage controls the drain current:

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

Higher $g_m$ means the transistor is more sensitive to gate voltage changes, which is desirable for amplifier applications.

Applications of diode I-V characteristics

Rectifiers and power supplies

Diodes convert AC to DC through rectification, exploiting the fact that they conduct in only one direction.

• A half-wave rectifier uses one diode and passes only the positive (or negative) half of the AC cycle. The output is a pulsating DC signal.
• A full-wave rectifier uses two or four diodes (bridge configuration) to pass both halves of the AC cycle, producing a smoother DC output with twice the ripple frequency.

Capacitor filters are typically added after rectification to smooth the output further.

Voltage regulators using Zener diodes

A Zener diode operated in reverse breakdown provides a stable reference voltage:

1. Connect the Zener diode in parallel with the load, reverse-biased.
2. Place a series resistor between the supply and the Zener to limit current.
3. As long as the Zener stays in breakdown, the voltage across the load remains approximately equal to the Zener voltage $V_Z$, even if the supply voltage or load current fluctuates.

This is one of the simplest voltage regulation techniques and is widely used for low-power reference circuits.

Diode clipping and clamping circuits

• Clipping circuits use the diode's turn-on voltage to "clip" portions of a waveform that exceed a set level. Series clippers place the diode in series with the signal path; parallel (shunt) clippers place it across the load.
• Clamping circuits shift the entire DC level of an AC signal. A capacitor and diode work together to "clamp" the waveform's peak (or trough) to a specific DC voltage.

Diodes in logic gates and switches

Diodes can implement basic logic functions:

• A diode OR gate connects multiple diode anodes to different inputs and ties the cathodes to a common output through a resistor. If any input is high, the output goes high.
• A diode AND gate connects cathodes to inputs and the anode side to the output. The output is high only if all inputs are high.

These diode-resistor logic (DRL) gates are simple but limited in fan-out and speed. Diode-transistor logic (DTL) adds a transistor stage for signal restoration and better drive capability.