Editing Bipolar junction transistor (section)

==== Ebers–Moll model <span class="anchor" id="Ebers-Moll model"></span><span class="anchor" id="Ebers-Moll"></span><span class="anchor" id="Ebers–Moll"></span> ====
[[File:Ebers-Moll model schematic (NPN).svg|frame|Ebers&ndash;Moll model for an NPN transistor.<ref>
{{cite book |title=Microelectronic Circuits |edition=2nd |first1=Adel S. |last1=Sedra |first2=Kenneth C. |last2=Smith |isbn=978-0-03-007328-1 |date=1987 |page=[https://archive.org/details/microelectronicc0000sedr/page/903 903] |publisher=Holt, Rinehart, and Winston |url=https://archive.org/details/microelectronicc0000sedr/page/903 }}</ref> ''I''<sub>B</sub>, ''I''<sub>C</sub> and ''I''<sub>E</sub> are the base, collector and emitter currents; ''I''<sub>CD</sub> and ''I''<sub>ED</sub> are the collector and emitter diode currents; ''α''<sub>F</sub> and ''α''<sub>R</sub> are the forward and reverse common-base current gains.]]
[[File:Ebers-Moll model schematic (PNP).svg|frame|Ebers&ndash;Moll model for a PNP transistor]]
[[File:Approximated Ebers Moll.svg|frame|Approximated Ebers&ndash;Moll model for an NPN transistor in the forward active mode. The collector diode is reverse-biased so ''I''<sub>CD</sub> is virtually zero. Most of the emitter diode current (''α''<sub>F</sub> is nearly 1) is drawn from the collector, providing the amplification of the base current.]]

The DC emitter and collector currents in active mode are well modeled by an approximation to the Ebers–Moll model:
: <math>\begin{align}
  I_\text{E} &= I_\text{ES} \left(e^\frac{V_\text{BE}}{V_\text{T}} - 1\right) \\
  I_\text{C} &= \alpha_\text{F} I_\text{E} \\
  I_\text{B} &= \left(1 - \alpha_\text{F}\right) I_\text{E}
\end{align}</math>

The base internal current is mainly by diffusion (see [[Fick's law]]) and
: <math>J_{n\,(\text{base})} = \frac{1}{W} q D_n n_{bo} e^{\frac{V_\text{EB}}{V_\text{T}}}</math>
where
* <math>V_{\text{T}}</math> is the [[Boltzmann constant#Thermal voltage|thermal voltage]] <math>kT/q</math> (approximately 26&nbsp;mV at 300&nbsp;K ≈ room temperature).
* <math>I_{\text{E}}</math> is the emitter current
* <math>I_{\text{C}}</math> is the collector current
* <math>\alpha_\text{F}</math> is the common base forward short-circuit current gain (0.98 to 0.998)
* <math>I_{\text{ES}}</math> is the reverse saturation current of the base–emitter diode (on the order of 10<sup>−15</sup> to 10<sup>−12</sup> amperes)
* <math>V_{\text{BE}}</math> is the base–emitter voltage
* <math>D_n</math> is the diffusion constant for electrons in the p-type base
* ''W'' is the base width

The <math>\alpha</math> and forward <math>\beta</math> parameters are as described previously. A reverse <math>\beta</math> is sometimes included in the model.

The unapproximated Ebers&ndash;Moll equations used to describe the three currents in any operating region are given below. These equations are based on the transport model for a bipolar junction transistor.<ref name=Sedra>{{cite book |first1=A.S. |last1=Sedra |first2=K.C. |last2=Smith |title=Microelectronic Circuits |date=2004 |edition=5th |publisher=Oxford |location=New York |isbn=978-0-19-514251-8 |pages=Eqs.&nbsp;4.103&ndash;4.110, p.&nbsp;305 |no-pp=true }}</ref>
: <math>\begin{align}
  i_{\text{C}} &= I_\text{S} \left[                   \left(e^\frac{V_\text{BE}}{V_\text{T}} - e^\frac{V_\text{BC}}{V_\text{T}}\right) - \frac{1}{\beta_\text{R}} \left(e^\frac{V_\text{BC}}{V_\text{T}} - 1\right) \right]\\
  i_{\text{B}} &= I_\text{S} \left[ \frac{1}{\beta_\text{F}} \left(e^\frac{V_\text{BE}}{V_\text{T}} - 1                               \right) + \frac{1}{\beta_\text{R}} \left(e^\frac{V_\text{BC}}{V_\text{T}} - 1\right) \right]\\
  i_{\text{E}} &= I_\text{S} \left[                   \left(e^\frac{V_\text{BE}}{V_\text{T}} - e^\frac{V_\text{BC}}{V_\text{T}}\right) + \frac{1}{\beta_\text{F}} \left(e^\frac{V_\text{BE}}{V_\text{T}} - 1\right) \right]
\end{align}</math>
where
* <math>i_\text{C}</math> is the collector current
* <math>i_\text{B}</math> is the base current
* <math>i_\text{E}</math> is the emitter current
* <math>\beta_\text{F}</math> is the forward common emitter current gain (20 to 500)
* <math>\beta_\text{R}</math> is the reverse common emitter current gain (0 to 20)
* <math>I_\text{S}</math> is the reverse saturation current (on the order of 10<sup>−15</sup> to 10<sup>−12</sup> amperes)
* <math>V_\text{T}</math> is the thermal voltage (approximately 26&nbsp;mV at 300 K ≈ room temperature).
* <math>V_\text{BE}</math> is the base–emitter voltage
* <math>V_\text{BC}</math> is the base–collector voltage

===== Base-width modulation =====
{{Main|Early effect}}
[[File:Early effect (NPN).svg|frame|Top: NPN base width for low collector–base reverse bias; Bottom: narrower NPN base width for large collector–base reverse bias. Hashed regions are [[Depletion width|depleted regions]].]]

As the collector–base voltage (<math>V_\text{CB} = V_\text{CE} - V_\text{BE}</math>) varies, the collector–base depletion region varies in size. An increase in the collector–base voltage, for example, causes a greater reverse bias across the collector–base junction, increasing the collector–base depletion region width, and decreasing the width of the base. This variation in base width often is called the ''[[Early effect]]'' after its discoverer [[James M. Early]].

Narrowing of the base width has two consequences:
* There is a lesser chance for recombination within the "smaller" base region.
* The charge gradient is increased across the base, and consequently, the current of minority carriers injected across the emitter junction increases.

Both factors increase the collector or "output" current of the transistor in response to an increase in the collector–base voltage.

===== Punchthrough =====
When the base–collector voltage reaches a certain (device-specific) value, the base–collector depletion region boundary meets the base–emitter depletion region boundary. When in this state the transistor effectively has no base. The device thus loses all gain when in this state.