Early effect

In Figure 1, the neutral (i.e. active) base is green, and the depleted base regions are hashed light green. The neutral emitter and collector regions are dark blue and the depleted regions hashed light blue. Under increased collector–base reverse bias, the lower panel of Figure 1 shows a widening of the depletion region in the base and the associated narrowing of the neutral base region.

The collector depletion region also increases under reverse bias, more than does that of the base, because the collector is less heavily doped than the base. The principle governing these two widths is charge neutrality. The narrowing of the collector does not have a significant effect as the collector is much longer than the base. The emitter–base junction is unchanged because the emitter–base voltage is the same.

Base-narrowing has two consequences that affect the current:

• 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 collector-base junction increases, which net current is called $I\_\backslash text\{CB0\}$.

Both these factors increase the collector or "output" current of the transistor with an increase in the collector voltage, but only the second is called Early effect. This increased current is shown in Figure 2. Tangents to the characteristics at large voltages extrapolate backward to intercept the voltage axis at a voltage called the Early voltage, often denoted by the symbol V_A.

In the forward active region the Early effect modifies the collector current ($I\_\backslash mathrm\{C\}$) and the forward common-emitter current gain ($\backslash beta\_\backslash mathrm\{F\}$), as typically described by the following equations:{{Cite book| title = Microelectronic Circuit Design | author =R.C. Jaeger and T.N. Blalock | publisher = McGraw-Hill Professional | year = 2004 |page=317| isbn = 0-07-250503-6 | url = https://books.google.com/books?id=u6vH4Gsrlf0C&dq=early-effect+collector+depletion+collector-base&pg=PA317 }}{{Cite book| title = Model and Design of Bipolar and Mos Current-Mode Logic: CML, ECL and SCL Digital Circuits | author = Massimo Alioto and Gaetano Palumbo | publisher = Springer | year = 2005 | isbn = 1-4020-2878-4 | url = https://books.google.com/books?id=rv13_kMvjFEC&dq=early-effect+collector+depletion&pg=PA12 }}

: $\backslash begin\{align\}$

I_\mathrm{C} &= I_\mathrm{S} e^{\frac{V_\mathrm{BE}}{V_\mathrm{T}}} \left(1 + \frac{V_\mathrm{CE}}{V_\mathrm{A}}\right) \\

\beta_\mathrm{F} &= \beta_\mathrm{F0} \left(1 + \frac{V_\mathrm{CE}}{V_\mathrm{A}}\right)

\end{align}

where

• $V\_\backslash mathrm\{CE\}$ is the collector–emitter voltage
• $V\_\backslash mathrm\{BE\}$ is the base–emitter voltage
• $I\_\backslash mathrm\{S\}$ is the reverse saturation current
• $V\_\backslash mathrm\{T\}$ is the thermal voltage $\backslash mathrm\{kT/q\}$; see thermal voltage: role in semiconductor physics
• $V\_\backslash mathrm\{A\}$ is the Early voltage (typically {{val|15|–|150|u=V}}; less for smaller devices)
• $\backslash beta\_\backslash mathrm\{F0\}$ is forward common-emitter current gain at zero bias.

Some models base the collector current correction factor on the collector–base voltage V_CB (as described in base-width modulation) instead of the collector–emitter voltage V_CE.{{Cite book| title = Semiconductor Device Modeling with Spice | author = Paolo Antognetti and Giuseppe Massobrio | publisher = McGraw-Hill Professional | year = 1993 | isbn = 0-07-134955-3 | url = https://books.google.com/books?id=5IBYU9xrGaIC&dq=early-effect+collector+depletion+collector-base&pg=PA59 }} Using V_CB may be more physically plausible, in agreement with the physical origin of the effect, which is a widening of the collector–base depletion layer that depends on V_CB. Computer models such as those used in SPICE use the collector–base voltage V_CB.[https://web.archive.org/web/20160530183304/http://people.clarkson.edu/~tortmeye//ee211/pspice%20files/pspcref.pdf Orcad PSpice Reference Manual named PSpcRef.pdf], p. 209. (archived from [http://people.clarkson.edu/~ortmeyer/ee211/pspice%20files/pspcref.pdf this URL])This manual is included with the free version of Orcad PSpice.

The Early effect can be accounted for in small-signal circuit models (such as the hybrid-pi model) as a resistor defined as{{Cite book| title = Microelectronic Circuit Design |author=R.C. Jaeger and T.N. Blalock |edition=Second |publisher = McGraw-Hill Professional |year = 2004 |isbn =0-07-232099-0 |url = http://worldcat.org/isbn/0072320990 |pages=Eq. 13.31, p. 891}}

: $r\_\backslash text\{O\}\; =\; \backslash frac\{V\_\backslash text\{A\}\; +\; V\_\backslash text\{CE\}\}\{I\_\backslash text\{C\}\}\; \backslash approx\; \backslash frac\{V\_\backslash text\{A\}\}\{I\_\backslash text\{C\}\}$

in parallel with the collector–emitter junction of the transistor. This resistor can thus account for the finite output resistance of a simple current mirror or an actively loaded common-emitter amplifier.

In keeping with the model used in SPICE and as discussed above using $V\_\{CB\}$ the resistance becomes:

: $r\_\backslash text\{O\}\; =\; \backslash frac\{V\_\backslash text\{A\}\; +\; V\_\backslash text\{CB\}\}\{I\_\backslash text\{C\}\}$

which almost agrees with the textbook result. In either formulation, $r\_O$ varies with DC reverse bias $V\_\{CB\}$, as is observed in practice.{{Citation needed|date=November 2007}}

In the MOSFET the output resistance is given in Shichman–Hodges model[http://www.m21medsense.com/media/c4c15e87c7266bdffff81baffffe415.pdf, The Shichman-Hodges Enhancement MOSFET Model and SwitcherCAD III SPICE, Report NDT14-08-2007, NanoDotTek, 12 August 2007]{{Dead link|date=August 2019 |bot=InternetArchiveBot |fix-attempted=yes }} (accurate for very old technology) as:

: $r\_\backslash text\{O\}\; =\; \backslash frac\{1\; +\; \backslash lambda\; V\_\backslash text\{DS\}\}\{\backslash lambda\; I\_\backslash text\{D\}\}\; =\; \backslash frac\{1\}\{I\_\backslash text\{D\}\}\backslash left(\backslash frac\{1\}\{\backslash lambda\}\; +\; V\_\backslash text\{DS\}\backslash right)$

where $V\_\backslash text\{DS\}$ = drain-to-source voltage, $I\_\backslash text\{D\}$ = drain current and $\backslash lambda$ = channel-length modulation parameter, usually taken as inversely proportional to channel length L.

Because of the resemblance to the bipolar result, the terminology "Early effect" often is applied to the MOSFET as well.

The expressions are derived for a PNP transistor. For an NPN transistor, n has to be replaced by p, and p has to be replaced by n in all expressions below.

The following assumptions are involved when deriving ideal current-voltage characteristics of the BJT

{{cite book

|author=R S Muller, Kamins TI & Chan M

|title=Device electronics for integrated circuits

|page=280 ff

|publisher= Wiley

|location=New York

|year=2003

|edition=Third

|isbn=0-471-59398-2

|url=http://worldcat.org/isbn/0-471-59398-2

}}

• Low level injection
• Uniform doping in each region with abrupt junctions
• One-dimensional current
• Negligible recombination-generation in space charge regions
• Negligible electric fields outside of space charge regions.

It is important to characterize the minority diffusion currents induced by injection of carriers.

With regard to pn-junction diode, a key relation is the diffusion equation.

: $\backslash frac\{d^2\; \backslash Delta\; p\_\{\backslash text\{B\}\}\; (x)\}\{dx^2\}\; =\; \backslash frac\{\backslash Delta\; p\_\{\backslash text\{B\}\}\; (x)\}\{L\_\{\backslash text\{B\}\}^2\}$

A solution of this equation is below, and two boundary conditions are used to solve and find $C\_1$ and $C\_2$.

: $\backslash Delta\; p\_\{\backslash text\{B\}\}\; (x)\; =\; C\_1\; e^\{\backslash frac\{x\}\{L\_\{\backslash text\{B\}\}\}\}\; +\; C\_2\; e^\{-\backslash frac\{x\}\{L\_\{\backslash text\{B\}\}\}\}$

The following equations apply to the emitter and collector region, respectively, and the origins $0$, $0\text{'}$, and $0\text{'}\text{'}$ apply to the base, collector, and emitter.

: $\backslash begin\{align\}$

\Delta n_{\text{B}} (x) &= A_1 e^{\frac{x}{L_{\text{B}}}} + A_2 e^{-\frac{x''}{L_{\text{B}}}} \\

\Delta n_{\text{c}} (x') &= B_1 e^{\frac{x' }{L_{\text{B}}}} + B_2 e^{-\frac{x' }{L_{\text{B}}}}

\end{align}

A boundary condition of the emitter is below:

: $\backslash Delta\; n\_\{\backslash text\{E\}\}\; (0\text{'}\text{'})\; =\; n\_\{\backslash text\{E\}O\}\; \backslash left(e^\{\backslash frac\{1\}\{kT\}\; q\; V\_\{\backslash text\{EB\}\}\}\; -\; 1\backslash right)$

The values of the constants $A\_1$ and $B\_1$ are zero due to the following conditions of the emitter and collector regions as $x\text{'}\text{'}\; \backslash rightarrow\; 0$ and $x\text{'}\; \backslash rightarrow\; 0$.

: $\backslash begin\{align\}$

\Delta n_{\text{E}} (x'') &\rightarrow 0 \\

\Delta n_{\text{c}} (x' ) &\rightarrow 0

\end{align}

Because $A\_1\; =\; B\_1\; =\; 0$, the values of $\backslash Delta\; n\_\{\backslash text\{E\}\}\; (0\text{'}\text{'})$ and $\backslash Delta\; n\_\{\backslash text\{c\}\}\; (0\text{'})$ are $A\_2$ and $B\_2$, respectively.

: $\backslash begin\{align\}$

\Delta n_{\text{E}} (x) &= n_{\text{E}0} \left(e^{\frac{q V_{\text{EB}}}{kT}} - 1\right) e^{-\frac{x}{L_{\text{E}}}} \\

\Delta n_{\text{C}} (x' ) &= n_{\text{C}0} \left(e^{\frac{q V_{\text{CB}}}{kT}} - 1\right) e^{-\frac{x' }{L_{\text{C}}}}

\end{align}

Expressions of $I\_\{\backslash text\{E\}n\}$ and $I\_\{\backslash text\{C\}n\}$ can be evaluated.

: $\backslash begin\{align\}$

I_{\text{E}n} &= \left. -q A D_{\text{E}} \frac{d \Delta_{\text{E}} (x)}{dx} \right|_{x=0''} \\

I_{\text{C}n} &= -q A \frac{D_{\text{C}}}{L_{\text{C}}} n_{\text{C}0} \left(e^{\frac{q V_{\text{CB}}}{kT}} - 1\right)

\end{align}

Because insignificant recombination occurs, the second derivative of $\backslash Delta\; p\_\{\backslash text\{B\}\}\; (x)$ is zero. There is therefore a linear relationship between excess hole density and $x$.

: $\backslash Delta\; p\_\{\backslash text\{B\}\}\; (x)\; =\; D\_1\; x\; +\; D\_2$

The following are boundary conditions of $\backslash Delta\; p\_\{\backslash text\{B\}\}$.

: $\backslash begin\{align\}$

\Delta p_{\text{B}}(0) &= D_2 \\

\Delta p_{\text{B}}(W) &= D_1 W + \Delta p_{\text{B}}(0)

\end{align}

with W the base width. Substitute into the above linear relation.

: $\backslash Delta\; p\_\backslash text\{B\}(x)\; =\; -\backslash frac\{1\}\{W\}\backslash left[\backslash Delta\; p\_\backslash text\{B\}(0)\; -\; \backslash Delta\; p\_\backslash text\{B\}(W)\backslash right]x\; +\; \backslash Delta\; p\_\backslash text\{B\}(0)$

With this result, derive value of $I\_\{\backslash text\{E\}p\}$.

: $\backslash begin\{align\}$

I_{\text{E}p} (0) &= \left.-q A D_{\text{B}} \frac{d \Delta p_\text{B}}{dx}\right|_{x=0} \\

I_{\text{E}p} (0) &= \frac{q A D_\text{B}}{W} \left[\Delta p_\text{B}(0) - \Delta p_\text{B}(W)\right]

\end{align}

Use the expressions of $I\_\{\backslash text\{E\}p\}$, $I\_\{\backslash text\{E\}n\}$, $\backslash Delta\; p\_\{\backslash text\{B\}\}(0)$, and $\backslash Delta\; p\_\{\backslash text\{B\}\}(W)$ to develop an expression of the emitter current.

: $\backslash begin\{align\}$

\Delta p_{\text{B}}(W) &= p_{\text{B}0} e^{\frac{q V_\text{CB}}{kT}} \\

\Delta p_{\text{B}}(0) &= p_{\text{B}0} e^{\frac{q V_\text{EB}}{kT}} \\

I_{\text{E}} &= qA \left[ \left(\frac{D_\text{E} n_{\text{E}0}}{L_\text{E}} + \frac{D_\text{B} p_{\text{B}0}}{W}\right)

\left(e^{\frac{q V_\text{EB}}{kT}} - 1 \right) -

\frac{D_{\text{B}}}{W} p_{\text{B}0}\left( e^{\frac{q V_{\text{CB}}}{k T}} - 1 \right)

\right]

\end{align}

Similarly, an expression of the collector current is derived.

: $\backslash begin\{align\}$

I_{\text{C}p}(W) &= I_{\text{E}p}(0) \\

I_{\text{C}} &= I_{\text{C}p}(W) + I_{\text{C}n}(0') \\

I_{\text{C}} &= q A \left[ \frac{D_\text{B}}{W} p_{\text{B}0}\left(e^{\frac{q V_\text{EB}}{kT}} - 1\right) -

\left(\frac{D_\text{C} n_{\text{C}0}}{L_\text{C}} + \frac{D_\text{B} p_{\text{B}0}}{W}\right)

\left(e^{\frac{q V_\text{CB}}{kT}} - 1\right)

\right]

\end{align}

An expression of the base current is found with the previous results.

: $\backslash begin\{align\}$

I_{\text{B}} &= I_{\text{E}} - I_{\text{C}} \\

I_{\text{B}} &= q A \left[ \frac{D_\text{E}}{L_\text{E}} n_{\text{E}0}\left(e^{\frac{q V_\text{EB}}{kT}} - 1\right) + \frac{D_\text{C}}{L_\text{C}} n_{\text{C}0}\left(e^{\frac{q V_\text{CB}}{kT}} - 1\right) \right]

\end{align}

{{reflist}}

• Small-signal model

{{DEFAULTSORT:Early Effect}}

Category:Transistor modeling