5.3 Current-voltage characteristics

Current-voltage characteristics of semiconductor devices

The current-voltage (I-V) characteristic of a device tells you exactly how much current flows for a given applied voltage. For P-N junctions, this relationship reveals the physics of carrier transport across the junction and determines how the device behaves in a circuit. The Shockley equation captures the ideal case, but real diodes introduce complications like series resistance and breakdown that you need to account for.

Ideal diode current-voltage relationship

An ideal diode acts as a one-way valve for current. Under forward bias (positive voltage applied to the P-side), current increases exponentially once the voltage exceeds a threshold. Under reverse bias, essentially no current flows. This asymmetry is what makes diodes useful.

Shockley diode equation

The Shockley equation describes the I-V relationship of an ideal P-N junction diode:

$I = I_s (e^{V_D / nV_T} - 1)$

where:

• $I$ = diode current
• $I_s$ = reverse saturation current (typically $10^{-12}$ to $10^{-15}$ A for silicon)
• $V_D$ = voltage across the diode
• $n$ = ideality factor
• $V_T$ = thermal voltage

This equation assumes current arises solely from diffusion of carriers across the junction. In forward bias ($V_D \gg V_T$), the $-1$ term becomes negligible and the current grows exponentially. In reverse bias ($V_D$ large and negative), the exponential term vanishes and $I \approx -I_s$.

Diode ideality factor

The ideality factor $n$ captures how much a real diode deviates from pure diffusion-dominated behavior.

• $n = 1$: current is dominated by diffusion (ideal case)
• $n = 2$: recombination-generation current in the depletion region dominates
• Most real diodes fall between 1 and 2, depending on the bias level and device structure

At low forward bias, recombination in the depletion region is more significant, pushing $n$ closer to 2. At moderate forward bias, diffusion dominates and $n$ approaches 1.

Thermal voltage

The thermal voltage represents the voltage equivalent of the average thermal energy of charge carriers:

$V_T = kT / q$

• $k$ = Boltzmann's constant ($1.38 \times 10^{-23}$ J/K)
• $T$ = absolute temperature in Kelvin
• $q$ = elementary charge ($1.6 \times 10^{-19}$ C)

At room temperature ($T = 300$ K), $V_T \approx 26$ mV. This value sets the scale for the exponential behavior: every increase of about $nV_T$ in forward voltage roughly doubles the current (for $n = 1$, a ~60 mV increase gives a 10x current increase).

Real diode current-voltage characteristics

Real diodes deviate from the Shockley equation in several important ways, especially at high forward currents and in reverse bias near breakdown.

Series resistance effects

Every real diode has a finite series resistance ($R_s$) from the bulk semiconductor and metal contacts. The actual terminal voltage is:

$V_{terminal} = V_D + I \cdot R_s$

At low currents, the $IR_s$ drop is negligible and the I-V curve looks exponential. At high currents, the resistive drop dominates and the curve becomes nearly linear. On a semi-log plot, this shows up as the forward characteristic bending away from the ideal straight line at high current levels.

Reverse breakdown mechanisms

When reverse bias exceeds a critical voltage, the diode enters breakdown and current increases sharply. Two distinct mechanisms cause this:

Zener vs avalanche breakdown

The sign of the temperature coefficient is a practical way to distinguish which mechanism dominates in a given diode.

PN junction diode current components

The total current through a P-N junction has three main components, each with a different physical origin.

Diffusion current

Diffusion current arises from the concentration gradient of carriers across the junction. Majority carriers (electrons in the N-region, holes in the P-region) that have enough energy diffuse across the junction into the opposite side. This component dominates in forward bias and is responsible for the exponential I-V relationship described by the Shockley equation.

Drift current

The built-in electric field in the depletion region sweeps minority carriers (electrons in the P-region, holes in the N-region) across the junction. This drift current opposes the diffusion current. In reverse bias, drift current is the primary contributor to the small reverse saturation current $I_s$. In forward bias, it's typically negligible compared to diffusion current.

Recombination-generation current

Within the depletion region itself, electron-hole pairs can recombine or be thermally generated.

• Recombination (forward bias): some carriers recombine before making it across the depletion region, adding an extra current component. This is why $n$ approaches 2 at low forward bias.
• Generation (reverse bias): thermal energy creates electron-hole pairs in the depletion region, which the electric field sweeps out. This contributes to reverse leakage current beyond the ideal $I_s$.

Diode equivalent circuit models

Circuit models simplify the diode's nonlinear I-V curve into something you can use in hand calculations. Each model trades accuracy for simplicity.

Ideal diode model

The simplest model: the diode is a short circuit (zero resistance) when forward biased and an open circuit (infinite resistance) when reverse biased. There's no voltage drop in the on state. This is useful for quick qualitative analysis but ignores the ~0.7 V forward drop of real silicon diodes.

Constant voltage drop model

This model adds a fixed voltage source (typically 0.7 V for silicon, 0.3 V for germanium) in series with an ideal diode. When forward biased, the diode conducts with a constant 0.7 V drop regardless of current. When reverse biased, no current flows. This is the most commonly used model for hand analysis of silicon circuits.

Piecewise linear model

This model approximates the I-V curve using straight-line segments:

• A series combination of a voltage source ($V_{on}$) and a resistance ($r_f$) for the forward region
• A large parallel resistance ($r_r$) for the reverse region

The parameters are chosen to match the actual diode curve at the operating point. This model captures the fact that forward voltage increases with current (due to series resistance) and offers a good balance between accuracy and computational simplicity.

Temperature effects on diode characteristics

Diode behavior is strongly temperature-dependent. The two most important effects work in opposite directions on the forward voltage.

Saturation current temperature dependence

The reverse saturation current increases roughly exponentially with temperature:

$I_s(T) = I_s(T_0) \cdot (T/T_0)^{3/n} \cdot e^{-E_g / nkT}$

• $I_s(T_0)$ = saturation current at reference temperature $T_0$
• $E_g$ = bandgap energy of the semiconductor
• $n$ = ideality factor

As a rough rule, $I_s$ approximately doubles for every 10°C increase in temperature for silicon diodes. This is the dominant temperature effect and causes the forward voltage to decrease by about 2 mV/°C at constant current.

Bandgap voltage temperature dependence

The bandgap energy decreases approximately linearly with temperature:

$V_g(T) = V_g(0) - \alpha T$

• $V_g(0)$ = bandgap voltage extrapolated to 0 K (~1.17 eV for silicon)
• $\alpha$ = temperature coefficient

This reduction in bandgap makes it easier to generate carriers at higher temperatures, which feeds into the increase of $I_s$.

Reverse leakage current temperature dependence

Reverse leakage current grows with temperature because thermal generation of electron-hole pairs in the depletion region accelerates. This follows an Arrhenius-type relationship:

$I_R(T) = I_R(T_0) \cdot e^{-E_a / kT}$

• $E_a$ = activation energy for the generation process (roughly $E_g / 2$ for generation via mid-gap traps)

In practice, reverse leakage roughly doubles for every 10°C rise, which can become a reliability concern in high-temperature applications.

Graphical analysis of diode I-V curves

Plotting the I-V curve on a semi-logarithmic scale (log of current vs. linear voltage) is the standard way to analyze diode characteristics, because the exponential forward region appears as a straight line.

Forward bias region

On a semi-log plot, the forward bias region shows a linear portion whose slope is $q / nkT$ (or equivalently $1 / nV_T$ per decade on a natural log scale). You can extract two key parameters from this region:

• Ideality factor $n$: determined from the slope of the linear region
• Saturation current $I_s$: found by extrapolating the straight-line portion back to $V_D = 0$

At high currents, the curve bends due to series resistance. At very low currents, recombination current (with $n \approx 2$) may produce a second linear region with a shallower slope.

Reverse bias region

In reverse bias, the current stays nearly flat at $-I_s$ for an ideal diode. Real diodes show a gradual increase in leakage with reverse voltage due to generation current. At the breakdown voltage, current rises steeply. The sharpness of the breakdown knee depends on the mechanism (Zener tends to be softer, avalanche sharper).

Diode turn-on voltage

The turn-on voltage ($V_{on}$) is the forward voltage at which the diode begins conducting appreciable current. It's not a sharp threshold but rather the point where the exponential curve becomes significant on a linear scale.

• Silicon diodes: $V_{on} \approx 0.6 - 0.7$ V
• Germanium diodes: $V_{on} \approx 0.2 - 0.3$ V
• GaAs diodes: $V_{on} \approx 1.0 - 1.2$ V

These values reflect the different bandgap energies of each semiconductor material.

Small-signal diode parameters

When a diode operates at a DC bias point and you apply a small AC signal on top of it, you can linearize the I-V curve around that point. The resulting small-signal model consists of a resistance and a capacitance.

Incremental resistance

The incremental (dynamic) resistance is the inverse of the slope of the I-V curve at the operating point:

$r_d = \frac{nV_T}{I_D}$

This comes directly from differentiating the Shockley equation. At a DC bias of $I_D = 1$ mA with $n = 1$, you get $r_d = 26 \, \Omega$ at room temperature. Higher bias current means lower incremental resistance, which makes sense: the I-V curve gets steeper at higher currents.

Diffusion capacitance

Charge stored in the neutral regions near the junction gives rise to the diffusion capacitance:

$C_d = \tau I_D / nV_T$

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

This capacitance is proportional to current, so it becomes significant at high forward bias. It limits how fast the diode can respond to signal changes and is the dominant capacitance in forward bias (as opposed to the junction/depletion capacitance, which dominates in reverse bias).

Diode switching characteristics

When a diode transitions between forward and reverse bias, stored charge must be removed before the diode can block current. The key switching parameters are:

• Forward recovery time: time for the diode to reach its steady-state forward voltage after being switched on
• Reverse recovery time ($t_{rr}$): time for the diode to stop conducting after switching from forward to reverse bias. During this interval, reverse current flows as stored charge is extracted.
• Reverse recovery charge ($Q_{rr}$): total charge that must be removed during reverse recovery

Fast-recovery and Schottky diodes are designed to minimize these parameters for high-frequency switching applications.

Applications of diode I-V characteristics

Rectifier circuits

Rectifiers exploit the diode's unidirectional conduction to convert AC to DC.

• Half-wave rectifier: a single diode passes only positive half-cycles, producing a pulsating DC output. The output has a DC component of $V_p / \pi$, where $V_p$ is the peak input voltage.
• Full-wave rectifier: uses two diodes with a center-tapped transformer, or four diodes in a bridge configuration, to conduct on both half-cycles. This doubles the DC output and reduces ripple.

The forward voltage drop of the diode(s) reduces the output voltage, which matters more at low signal levels.

Voltage regulator circuits

Zener diodes are operated in reverse breakdown to provide a stable reference voltage. The circuit works by placing the Zener in parallel with the load, with a series resistor to limit current. As input voltage or load current varies, the Zener adjusts its current to maintain a nearly constant voltage across the load.

The quality of regulation depends on the Zener's dynamic resistance in the breakdown region: lower dynamic resistance means tighter voltage regulation.

Diode logic gates

Diodes can implement basic AND and OR logic:

• OR gate: diodes with anodes connected to separate inputs and cathodes tied to the output through a pull-down resistor. If any input is HIGH, that diode conducts and pulls the output HIGH.
• AND gate: diodes with cathodes connected to separate inputs and anodes tied to the output through a pull-up resistor. The output is HIGH only when all inputs are HIGH.

Diode logic is simple and fast, but it can't restore signal levels (no gain), so cascading multiple stages degrades noise margins. Transistor-based logic (DTL, TTL, CMOS) solves this problem.