9.3 Current-voltage characteristics

Current-voltage characteristics

Current-voltage (I-V) characteristics map out how current flows through a device as you vary the voltage across it. For BJTs specifically, these curves tell you which region of operation the transistor is in and how it will behave as an amplifier or switch. This topic ties together the device physics you've learned with practical circuit design.

Ideal vs practical devices

Ideal devices follow theoretical I-V equations exactly. They assume abrupt junctions, perfectly uniform doping, and zero parasitic effects. Practical devices always deviate from this because of real-world factors:

• Series resistance in the bulk semiconductor and contacts adds voltage drops
• Leakage currents flow even when the device should be "off"
• High-injection effects alter carrier distributions at large forward bias

The gap between ideal and practical behavior matters most when you're trying to build accurate circuit models. You'll often start with the ideal equation, then layer in corrections.

Linear vs nonlinear behavior

A linear device obeys Ohm's law: double the voltage, double the current. Resistors behave this way. Semiconductor devices like diodes and transistors are nonlinear, meaning the current-voltage relationship follows a more complex function (often exponential).

This nonlinearity is actually what makes semiconductors useful. Without it, you couldn't build rectifiers, amplifiers, or switches. When analyzing nonlinear devices in circuits, you'll often linearize them around a specific operating point (the small-signal approach).

Ohmic vs non-ohmic contacts

• Ohmic contacts have a linear I-V relationship with low resistance. They're what you want at the metal-semiconductor interface so current flows efficiently into and out of the device.
• Non-ohmic contacts show a nonlinear I-V curve and higher resistance. Sometimes this is intentional (Schottky diodes use a metal-semiconductor barrier on purpose), but it can also result from poor fabrication.

Getting contacts right is a practical concern that directly affects device performance and reliability.

Diodes

Diodes are two-terminal devices that conduct current easily in one direction (forward bias) and block it in the other (reverse bias). Their I-V characteristics are the foundation for understanding more complex devices like BJTs.

p-n junction diodes

A p-n junction forms where p-type and n-type semiconductors meet. Under forward bias, the external voltage reduces the built-in potential barrier, allowing majority carriers to be injected across the junction and recombine. Current increases exponentially with voltage.

Under reverse bias, the barrier widens, and only a tiny leakage current flows due to minority carriers and thermal generation. If you push the reverse voltage high enough, breakdown occurs. Applications include rectification, voltage regulation, and signal conditioning.

Schottky diodes

Schottky diodes use a metal-semiconductor junction instead of a p-n junction. The metal creates a potential barrier (the Schottky barrier) at the interface. Because conduction relies on majority carriers only (no minority carrier storage), Schottky diodes have two practical advantages:

• Lower forward voltage drop (typically 0.2–0.3 V vs. 0.6–0.7 V for silicon p-n diodes)
• Faster switching speed (no minority carrier recombination delay)

These properties make them ideal for high-frequency rectification and power conversion.

Zener diodes

Zener diodes are engineered to operate in reverse breakdown at a specific, well-defined voltage. Once breakdown occurs, the voltage across the diode stays nearly constant even as current varies, which makes them excellent for voltage regulation.

The actual breakdown mechanism depends on doping:

• Zener breakdown (tunneling) dominates in heavily doped, narrow junctions (breakdown voltage below ~5 V)
• Avalanche breakdown dominates in lightly doped, wider junctions (breakdown voltage above ~7 V)

Between roughly 5–7 V, both mechanisms can contribute.

Diode I-V equation

The Shockley diode equation describes the ideal diode:

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

where:

• $I_s$ is the reverse saturation current (typically $10^{-12}$ to $10^{-15}$ A for silicon)
• $V_T = kT/q$ is the thermal voltage (about 26 mV at room temperature)
• $n$ is the ideality factor (1 for pure diffusion current, 2 when generation-recombination dominates)

Practical diodes deviate from this equation at high currents (series resistance causes the voltage to increase faster than the exponential predicts) and at very low currents (generation-recombination currents dominate over diffusion).

Diode equivalent circuits

Circuit analysis uses simplified diode models of increasing accuracy:

1. Ideal diode model: Acts as a perfect switch. Short circuit in forward bias, open circuit in reverse bias.
2. Constant voltage drop model: Adds a fixed forward voltage (e.g., 0.7 V for silicon). More realistic for hand calculations.
3. Piecewise-linear model: Includes series resistance $r_s$ along with the voltage drop, giving $V = V_{on} + I \cdot r_s$.
4. Small-signal model: For AC analysis around a DC operating point, the diode looks like a small resistance $r_d = nV_T / I_D$ in parallel with a junction capacitance.

Choose the simplest model that gives you acceptable accuracy for your application.

Bipolar junction transistors (BJTs)

BJTs are three-terminal devices (emitter, base, collector) that use both electrons and holes as charge carriers. The device consists of two back-to-back p-n junctions (emitter-base and base-collector), and the thin base region between them is what makes transistor action possible.

npn vs pnp transistors

• npn: n-type emitter and collector sandwich a thin p-type base. Electrons are the majority carriers doing the work. Conventional current flows from collector to emitter.
• pnp: p-type emitter and collector sandwich a thin n-type base. Holes are the majority carriers. Conventional current flows from emitter to collector.

npn transistors are more common in practice because electron mobility is higher than hole mobility, giving npn devices better speed and gain. The analysis for both types is symmetric; just flip the voltage polarities and current directions.

Common emitter configuration

The common emitter (CE) configuration has the emitter terminal shared between input and output. The input signal drives the base-emitter junction, and the output is taken from the collector-emitter terminals.

• Provides both voltage gain and current gain
• Current gain is $\beta = I_C / I_B$, typically 50–300
• Output signal is inverted (180° phase shift) relative to the input
• Most widely used configuration for general-purpose amplification

Common base configuration

The common base (CB) configuration shares the base terminal. Input is applied at the emitter, output is taken at the collector.

• Low input impedance, high output impedance
• Current gain $\alpha = I_C / I_E$ is slightly less than 1 (typically 0.95–0.99)
• No phase inversion between input and output
• Useful for current buffering, impedance matching, and high-frequency applications (the Miller effect is minimized)

Common collector configuration

The common collector (CC) configuration, also called the emitter follower, shares the collector terminal. Input goes to the base, output is taken from the emitter.

• High input impedance, low output impedance
• Voltage gain is approximately 1 (unity), so the output "follows" the input
• No phase inversion
• Primarily used as a voltage buffer and for impedance transformation between stages

BJT I-V characteristics

BJT I-V characteristics are typically displayed as a family of curves. For the common emitter configuration, you plot $I_C$ vs. $V_{CE}$ for several values of base current $I_B$.

The Ebers-Moll model provides the foundational equations. In the forward active region, the collector current is:

$I_C = I_S \cdot e^{V_{BE}/V_T}$

where $I_S$ is the transport saturation current. The base current is related by $I_B = I_C / \beta$.

Key parameters to know:

• Current gain $\beta$ (or $h_{FE}$): ratio of collector current to base current
• Early voltage $V_A$: accounts for the slight upward slope of $I_C$ vs. $V_{CE}$ curves in the active region (base-width modulation). Higher $V_A$ means flatter curves and a better current source.
• Saturation currents $I_{CS}$ and $I_{ES}$: set the scale of the exponential I-V relationship

BJT regions of operation

Understanding these regions is critical for circuit design:

For amplifier design, you bias the BJT in the active region. For digital switching, you drive it between saturation (ON) and cutoff (OFF).

Field effect transistors (FETs)

FETs are voltage-controlled devices: the gate voltage modulates the conductivity of a semiconductor channel between source and drain. Compared to BJTs, FETs draw almost no gate current, giving them very high input impedance and lower power consumption. They also dominate integrated circuit fabrication because of their simpler structure.

Junction FETs (JFETs)

JFETs control current by varying the width of a reverse-biased p-n junction's depletion region. As you increase the reverse gate-source voltage, the depletion region expands into the channel, narrowing it and reducing current flow.

• n-channel JFET: n-type channel with p-type gate regions. Conducts with $V_{GS} = 0$ (depletion-mode device).
• p-channel JFET: p-type channel with n-type gate regions. Voltage polarities are reversed.

At $V_{GS} = V_P$ (the pinch-off voltage), the channel is fully depleted and current drops to near zero.

Metal-oxide-semiconductor FETs (MOSFETs)

MOSFETs use an insulated gate (a thin oxide layer separates the metal/polysilicon gate from the semiconductor). The gate voltage creates an electric field through the oxide that either attracts or repels carriers in the channel.

• Enhancement-mode (normally off): No channel exists at $V_{GS} = 0$. You must apply a gate voltage exceeding the threshold voltage $V_{th}$ to create a conduction channel. This is the most common type in digital circuits.
• Depletion-mode (normally on): A channel exists at $V_{GS} = 0$. Applying a gate voltage of opposite polarity depletes the channel.

Both n-channel (NMOS) and p-channel (PMOS) variants exist. CMOS technology uses complementary pairs of NMOS and PMOS transistors.

FET I-V characteristics

For an n-channel enhancement MOSFET, the drain current equations are:

Triode region ($V_{DS} < V_{GS} - V_{th}$):

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

Saturation region ($V_{DS} \geq V_{GS} - V_{th}$):

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

Key parameters:

• Threshold voltage $V_{th}$: the minimum gate-source voltage needed to form a conducting channel
• Transconductance $g_m = \partial I_D / \partial V_{GS}$: measures how effectively the gate voltage controls drain current
• Channel length modulation $\lambda$: accounts for the slight increase in $I_D$ with $V_{DS}$ in saturation (analogous to the Early effect in BJTs)

FET regions of operation

Thyristors

Thyristors are four-layer (pnpn) devices with three terminals: anode, cathode, and gate. Their defining feature is bistable switching: once triggered into conduction, they stay on until the current drops below a minimum holding current. This latching behavior makes them well-suited for high-power applications.

SCRs and triacs

• Silicon-controlled rectifiers (SCRs) conduct current in one direction only. A gate pulse triggers the device into conduction, and it remains on as long as anode current exceeds the holding current.
• Triacs are bidirectional, conducting in both directions. They're essentially two SCRs connected in anti-parallel, making them useful for AC power control (e.g., light dimmers).

Both devices require only a brief gate trigger to turn on but cannot be turned off by the gate alone. The current must drop below the holding current for the device to return to its blocking state.

Thyristor I-V characteristics

The thyristor I-V curve has three distinct regions:

1. Forward blocking: The device blocks forward voltage like an open switch. Only a small leakage current flows.
2. Forward conducting: Once triggered (by gate current or by exceeding the breakover voltage), the device snaps to a low-impedance state. The voltage drops to about 1–2 V, and current is limited only by the external circuit.
3. Reverse blocking: The device blocks reverse voltage, similar to a reverse-biased diode, up to its rated reverse voltage.

The transition from blocking to conducting is regenerative: once it starts, it completes very quickly. The device stays conducting until current falls below the holding current $I_H$.

Thyristor triggering methods

• Gate triggering: The standard method. A current pulse at the gate initiates conduction. Most controllable and reliable.
• Voltage triggering: If the forward voltage exceeds the breakover voltage $V_{BO}$, the thyristor turns on without a gate signal. This is generally undesirable in controlled applications.
• dv/dt triggering: A rapid voltage rise across the device can cause unintended turn-on through displacement current charging the internal junction capacitances. Snubber circuits (typically an RC network) are used to limit dv/dt and prevent this.
• Light triggering: Light-activated SCRs (LASCRs) use photons to generate the triggering current, providing electrical isolation between the control signal and the power circuit.

Optoelectronic devices

Optoelectronic devices convert between electrical energy and light. Their I-V characteristics reflect this dual nature and are essential for designing optical communication systems, displays, sensors, and energy harvesting systems.

Photodiodes and phototransistors

Photodiodes are p-n junctions operated in reverse bias. Incident photons generate electron-hole pairs in or near the depletion region, producing a photocurrent that's proportional to light intensity. The I-V curve shifts downward with increasing illumination, similar to a solar cell.

Phototransistors are BJTs with an optically accessible base. Light generates base current, which is then amplified by the transistor's current gain $\beta$. This gives phototransistors much higher sensitivity than photodiodes, but at the cost of slower response time.

Both are used in optical sensing, fiber-optic receivers, and position detection.

Light-emitting diodes (LEDs)

LEDs emit photons when forward-biased carriers recombine across the junction. The wavelength (color) of emitted light is determined by the semiconductor's bandgap energy: $\lambda = hc / E_g$.

The I-V curve of an LED looks similar to a regular diode but with a higher turn-on voltage that corresponds to the photon energy. For example, red LEDs (GaAsP) turn on around 1.8 V, while blue LEDs (GaN) turn on around 3.0 V. Above the turn-on voltage, current increases steeply, so LEDs always need a current-limiting resistor or driver circuit.

Solar cells

Solar cells are p-n junctions that convert sunlight into electricity via the photovoltaic effect. Under illumination, the I-V curve of the diode shifts downward by the photocurrent $I_{ph}$:

$I = I_s (e^{V/nV_T} - 1) - I_{ph}$

Key performance metrics from the I-V curve:

• Open-circuit voltage $V_{OC}$: voltage when $I = 0$
• Short-circuit current $I_{SC}$: current when $V = 0$ (approximately equal to $I_{ph}$)
• Fill factor $FF$: ratio of maximum power to $V_{OC} \times I_{SC}$. Typical values are 0.7–0.85 for good cells.
• Efficiency $\eta = P_{max} / P_{incident}$

Temperature effects

Temperature changes shift I-V curves and alter device parameters. For reliable circuit design, you need to account for how your devices behave across the expected temperature range.

Temperature dependence of I-V curves

Two competing effects occur in most semiconductor devices as temperature rises:

• Forward voltage decreases: The built-in potential of p-n junctions drops by roughly $-2 \text{ mV/°C}$ for silicon. This is because the intrinsic carrier concentration increases with temperature.
• Leakage current increases: Reverse saturation current $I_s$ roughly doubles for every 10°C increase in temperature, since it depends exponentially on $-E_g / kT$.

For BJTs, $\beta$ also increases with temperature, which can shift the operating point if the biasing circuit isn't designed to compensate.

Thermal runaway in devices

Thermal runaway is a destructive positive feedback loop:

1. Device dissipates power, raising its temperature.
2. Higher temperature increases current (due to lower $V_{BE}$ or higher leakage).
3. More current means more power dissipation.
4. Temperature rises further, and the cycle accelerates.

This is a particular concern for BJTs in the active region, where collector current increases with temperature. Prevention strategies include:

• Emitter degeneration resistors that provide negative feedback (as $I_C$ rises, the voltage drop across $R_E$ increases, reducing $V_{BE}$)
• Proper heat sinking to keep junction temperature within safe limits
• Thermally stable bias circuits (e.g., voltage divider bias rather than fixed base bias)

Breakdown mechanisms

Breakdown occurs when a high electric field in the device causes a sudden, large increase in current. Understanding which mechanism dominates helps you select devices with appropriate voltage ratings and design protection circuits.

Avalanche breakdown

In a reverse-biased junction with a wide, lightly doped depletion region, the electric field accelerates carriers to high kinetic energies. When a carrier gains enough energy, it collides with the crystal lattice and knocks out a new electron-hole pair. These new carriers are also accelerated, creating more pairs in a chain reaction (impact ionization).

The result is a multiplicative current increase at the avalanche breakdown voltage. This mechanism dominates in devices with breakdown voltages above roughly 7 V. Avalanche breakdown has a positive temperature coefficient: the breakdown voltage increases slightly with temperature because more frequent lattice collisions reduce the mean free path of carriers.

Zener breakdown

In heavily doped junctions, the depletion region is very narrow. Even at moderate reverse voltages, the electric field becomes strong enough for electrons to tunnel directly from the valence band of the p-side to the conduction band of the n-side (quantum mechanical tunneling).

Zener breakdown dominates for breakdown voltages below about 5 V. It has a negative temperature coefficient: breakdown voltage decreases slightly with temperature. This opposite temperature behavior compared to avalanche breakdown is useful for designing temperature-compensated voltage references (by combining both mechanisms around 5–6 V).

Punch-through breakdown

Punch-through occurs when the depletion region of one junction extends all the way through a thin, lightly doped region and merges with the depletion region of an adjacent junction. Once the regions merge, there's no longer a neutral region to limit current flow, and current increases rapidly.

This is a concern in:

• Short-channel MOSFETs where the source and drain depletion regions can meet
• BJTs with thin base regions under high collector-base reverse bias

Mitigation techniques include halo implants, retrograde well doping profiles, and simply ensuring adequate spacing between junctions.

High-frequency effects

At high frequencies, parasitic capacitances and resistances that you can ignore at DC start to dominate device behavior. Accurate high-frequency models are essential for RF, microwave, and high-speed digital design.

Capacitance in devices

Semiconductor devices have two main types of internal capacitance:

• Junction (depletion) capacitance: Arises from the charge stored in the depletion region. It varies with applied voltage as $C_j = C_{j0} / (1 - V/V_{bi})^m$, where $m$ depends on the doping profile ($m = 1/2$ for abrupt junctions, $m = 1/3$ for linearly graded).
• Diffusion capacitance: Associated with minority carrier charge stored in the neutral regions during forward bias. Proportional to the forward current and the carrier transit time.

Parasitic capacitances from device geometry (overlap capacitance between gate and source/drain in MOSFETs, fringing capacitance) add to these intrinsic capacitances and must be minimized through careful layout.

Frequency limitations of devices

The unity-gain frequency (also called transition frequency) $f_T$ is the frequency at which the small-signal current gain drops to 1. For a BJT:

$f_T = \frac{g_m}{2\pi (C_\pi + C_\mu)}$

where $C_\pi$ is the base-emitter capacitance and $C_\mu$ is the base-collector capacitance.

For a MOSFET:

$f_T = \frac{g_m}{2\pi (C_{gs} + C_{gd})}$

Higher $f_T$ means the device can amplify at higher frequencies. Strategies to improve $f_T$ include reducing device dimensions (shorter channels, thinner bases), optimizing doping profiles, and using advanced structures like heterojunction bipolar transistors (HBTs) or high-electron-mobility transistors (HEMTs).

Measurement techniques

Accurate I-V measurements are the link between device physics theory and real device behavior. They're used for characterization, model parameter extraction, and quality control in fabrication.

I-V curve tracing

To measure an I-V curve:

1. Connect the device under test to a source-measure unit (SMU) or curve tracer.
2. Sweep the voltage across the device terminals in controlled steps.
3. At each voltage step, measure the resulting current.
4. Plot the data as $I$ vs. $V$.

Modern curve tracers and semiconductor parameter analyzers offer pulsed measurement modes (to avoid self-heating), high-voltage capability, and temperature-controlled stages. For multi-terminal devices like BJTs, you sweep one terminal voltage while stepping another (e.g., sweep $V_{CE}$ for several fixed values of $I_B$).

Parameter extraction from I-V curves

Device model parameters are extracted by fitting measured data to the model equations. For example, to extract Shockley diode parameters:

1. Measure the forward I-V curve over several decades of current.
2. Plot $\ln(I)$ vs. $V$. The slope gives $1/nV_T$, from which you find the ideality factor $n$.
3. The y-intercept gives $\ln(I_s)$, from which you find the saturation current.
4. At high currents, deviation from the straight line reveals the series resistance $R_s$.

For BJTs, you'd extract $\beta$, $V_A$, and $I_S$ from families of output and transfer curves. Automated extraction tools use least-squares fitting or optimization algorithms to match models across all operating regions simultaneously.