BJT collector current

In active mode the BJT collector current depends exponentially on the base–emitter voltage, the same way a diode current depends on its forward voltage:

$i_C=I_S{e}^{v_{BE}/V_T}$

• $i_C$ — collector current.
• $v_{BE}$ — base–emitter voltage (the forward voltage across the EBJ).
• $I_S$ — the transistor’s saturation current (a scaling constant, often around $10^{-15}\text{A}$). Set by device geometry and doping. It’s the analogue of the diode’s Reverse saturation current, and itself depends weakly on $v_{CE}$ through the Early effect.
• $V_T$ — the Thermal voltage, $kT/q\approx 25\text{mV}$ at room temperature.

Compare the MOSFET square law $i_D=\frac{1}{2}{k}_n{V}_{OV}^2$. The MOSFET’s drain current grows as the square of the overdrive; the BJT’s collector current grows exponentially with $v_{BE}$. An exponential is far steeper than a square law, so a given fractional change in current needs a far smaller change in control voltage. That steepness is why the BJT has a higher BJT transconductance per unit current than a MOSFET, and why it delivers more current for the same bias. The same exponential underlies the Diode equation; a BJT’s EBJ really is a forward-biased PN junction.

$i_C$ is exponential in $v_{BE}$ (like a diode). At higher temperature the same $i_C$ is reached at a lower $v_{BE}$, roughly $-2\text{mV/°C}$ in $V_{BE}$ at constant current. The right side shows the underlying carrier-diffusion mechanism.

$V_{BE}$ temperature coefficient

Hold $i_C$ fixed and warm the device up: the $v_{BE}$ needed to sustain that same collector current drops by about

${\frac{\partial V_{BE}}{\partial T}|}_{i_C}\approx -2\text{mV/°C}$

Intuition: $I_S$ rises steeply with temperature (it depends on the intrinsic carrier concentration, which climbs fast with $T$), and $V_T=kT/q$ also rises. Both effects mean that at a higher temperature you reach the same $i_C$ at a lower $v_{BE}$. The net result is the $-2\text{mV/°C}$ figure. This is why a fixed-$V_{BE}$ bias is a bad idea: a small temperature rise would run the collector current away. Good bias circuits set the current (via Emitter degeneration / Voltage-divider bias) rather than the voltage. The same temperature coefficient gets exploited deliberately in temperature sensors and bandgap references.

Because the exponential is so steep, $v_{BE}$ moves by only a few tens of millivolts across several decades of $i_C$. That is what justifies the constant-voltage-drop simplification $V_{BE}\approx 0.7\text{V}$ used in the BJT large-signal model and BJT DC analysis. Linearising this exponential around a bias point is exactly how you get the small-signal BJT transconductance $g_m=I_C/V_T$.