Theory

Introduction:
BJT Parameter Extraction from Reverse Gummel Plots

Introduction

The reverse Gummel plot is the counterpart to the forward Gummel plot. It is essential for extracting the parameters that govern the BJT's operation in the reverse-active and saturation regions.

A reverse Gummel plot is generated by plotting the emitter current ($I_E$) and base current ($I_B$) on a logarithmic scale against the base-collector voltage ($V_{BC}$) on a linear scale. During this measurement, the base-emitter voltage is held constant at zero ($V_{BE} = 0$).

This measurement configuration effectively operates the transistor "backwards," with the collector acting as the emitter and the emitter acting as the collector.

The Gummel-Poon Model (Reverse Active)

With $V_{BE} = 0$, the forward-active terms in the Gummel-Poon model become negligible. The model simplifies to describe the reverse-active operation:

$$ I_E \approx \frac{IS}{q_b} \cdot \left( e^{\frac{V_{BC}}{NR \cdot V_T}} - 1 \right) $$

$$ I_B \approx \left[ \frac{IS}{BR} \cdot \left( e^{\frac{V_{BC}}{NR \cdot V_T}} - 1 \right) \right] + \left[ ISC \cdot \left( e^{\frac{V_{BC}}{NC \cdot V_T}} - 1 \right) \right] $$

Where the key reverse-mode parameters are:

• $IS$: Transport Saturation Current (the same as in the forward model, per reciprocity).
• $BR$: Ideal Maximum Reverse Beta.
• $NR$: Reverse Current Emission Coefficient (ideality factor for $I_E$).
• $ISC$: Base-Collector Leakage Saturation Current.
• $NC$: Base-Collector Leakage Emission Coefficient (ideality factor for non-ideal $I_B$).
• $q_b$: Normalized base charge, which includes high-injection effects modeled by IKR.
• $V_T$: Thermal Voltage ($kT/q$).

Graphical Extraction from Plot Regions

Similar to the forward plot, we analyze the different regions of the reverse Gummel plot.

1. Ideal Mid-Current Region

In this region, $I_E$ and the ideal component of $I_B$ are dominated by diffusion. They appear as parallel straight lines.

• IS (Transport Saturation Current): Extrapolate the linear (ideal) portion of the $log(I_E)$ curve back to its intercept at $V_{BC} = 0$. Due to the Ebers-Moll reciprocity principle, this value must be the same IS extracted from the forward Gummel plot's $I_C$ curve. This provides a critical consistency check. $IS = 10^{\text{intercept}}$

• NR (Reverse Emission Coefficient): Find the slope of the linear (ideal) portion of the $log(I_E)$ curve. $\text{Slope}{IE} = \frac{\Delta \log{10}(I_E)}{\Delta V_{BC}} = \frac{1}{NR \cdot V_T \cdot \ln(10)}$ NR is typically very close to 1.0.

• BR (Ideal Maximum Reverse Beta): This is the ideal current gain in the reverse-active mode. It is found from the vertical separation between the ideal $I_E$ line and the (extrapolated) ideal $I_B$ line. $BR = \frac{I_E}{I_{B, \text{ideal}}}$ For standard asymmetric BJTs (where the emitter is much more heavily doped than the collector), BR is very small (e.g., 0.1 to 5). This means the $log(I_E)$ and $log(I_B)$ curves will be very close together in this region.

2. Low-Current Region

At low $V_{BC}$, the non-ideal recombination current in the base-collector space-charge region dominates the base current $I_B$.

• NC (B-C Leakage Emission Coefficient): The $log(I_B)$ curve deviates from the ideal $N=1$ slope and follows a new, shallower slope. Find the slope of this line. $\text{Slope}{IB} = \frac{\Delta \log{10}(I_B)}{\Delta V_{BC}} = \frac{1}{NC \cdot V_T \cdot \ln(10)}$ NC is typically between 1.5 and 2.0.

• ISC (B-C Leakage Saturation Current): Extrapolate this non-ideal (low-current) $log(I_B)$ line back to its intercept at $V_{BC} = 0$. $ISC = 10^{\text{intercept}}$ Since the B-C junction area is typically much larger than the B-E junction area, ISC is usually much larger than ISE.

3. High-Current Region (Roll-off)

At high $V_{BC}$ and high currents, the curves "roll off" due to high-level injection and parasitic resistances.

• IKR (Reverse High-Injection Knee): This is the "knee" in the $log(I_E)$ curve where its slope decreases. It models high-level injection in the collector region (which is acting as the emitter). IKR is the approximate emitter current at this knee. IKR is typically much smaller than its forward counterpart IKF.

• Parasitic Resistances (RB, RC): These resistances cause voltage drops that reduce the internal junction voltages.

  • RB (Base Resistance): The $I_B \cdot RB$ drop causes the $log(I_B)$ curve to roll off.
  • RC (Collector Resistance): The $I_E \cdot RC$ drop (since $I_E \approx I_C$ in this mode) is often the dominant resistive effect, causing both $I_E$ and $I_B$ curves to roll off.

Summary

The reverse Gummel plot is critical for modeling saturation. The parameters extracted (BR, ISC, NC, IKR) are essential for any simulation where the BJT's base-collector junction becomes forward-biased, which is common in digital logic (ECL, TTL) and analog switching applications.