Early effect

. . The Early effect, named after its discoverer James M. Early, is the variation in the effective width of the base in a bipolar junction transistor (BJT) due to a variation in the applied base-to-collector voltage. A greater reverse bias across the collector–base junction, for example, increases the collector–base depletion width, thereby decreasing the width of the charge carrier portion of the base.

. . The Early effect, named after its discoverer James M. Early, is the variation in the effective width of the base in a bipolar junction transistor (BJT) due to a variation in the applied base-to-collector voltage. A greater reverse bias across the collector–base junction, for example, increases the collector–base depletion width, thereby decreasing the width of the charge carrier portion of the base. 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_\mathrm{CE} is the collector–emitter voltage V_\mathrm{BE} is the base–emitter voltage I_\mathrm{S} is the reverse saturation current V_\mathrm{T} is the thermal voltage \mathrm{kT/q} , which is approximately 26 mV; see thermal voltage: role in semiconductor physics V_\mathrm{A} is the Early voltage (typically ; less for smaller devices) \beta_\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 . 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 . r_\text{O} = \frac{V_\text{A} + V_\text{CE}}{I_\text{C}} \approx \frac{V_\text{A}}{I_\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_\text{O} = \frac{V_\text{A} + V_\text{CB}}{I_\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. In the MOSFET the output resistance is given in Shichman–Hodges model (accurate for very old technology) as: r_\text{O} = \frac{1 + \lambda V_\text{DS}}{\lambda I_\text{D}} = \frac{1}{I_\text{D}}\left(\frac{1}{\lambda} + V_\text{DS}\right) where V_\text{DS} = drain-to-source voltage, I_\text{D} = drain current and \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. \frac{d^2 \Delta p_{\text{B}} (x)}{dx^2} = \frac{\Delta p_{\text{B}} (x)}{L_{\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 . \Delta p_{\text{B}} (x) = C_1 e^{\frac{x}{L_{\text{B}}}} + C_2 e^{-\frac{x}{L_{\text{B}}}} The following equations apply to the emitter and collector region, respectively, and the origins 0 , 0' , and 0 apply to the base, collector, and emitter. \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: \Delta n_{\text{E}} (0) = n_{\text{E}O} \left(e^{\frac{1}{kT} q V_{\text{EB}}} - 1\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 \rightarrow 0 and x' \rightarrow 0 . \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 \Delta n_{\text{E}} (0) and \Delta n_{\text{c}} (0') are A_2 and B_2 , respectively. \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_{\text{E}n} and I_{\text{C}n} can be evaluated. \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 \Delta p_{\text{B}} (x) is zero. There is therefore a linear relationship between excess hole density and x . \Delta p_{\text{B}} (x) = D_1 x + D_2 The following are boundary conditions of \Delta p_{\text{B}} . \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. \Delta p_\text{B}(x) = -\frac{1}{W}\left[\Delta p_\text{B}(0) - \Delta p_\text{B}(W)\right]x + \Delta p_\text{B}(0) With this result, derive value of I_{\text{E}p} . \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_{\text{E}p} , I_{\text{E}n} , \Delta p_{\text{B}}(0) , and \Delta p_{\text{B}}(W) to develop an expression of the emitter current. \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. \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. \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}

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