PHT.301 Physics of Semiconductor Devices

## Common emitter configuration

In a common emitter configuration the base current controls the collector current.

The Ebers-Moll model for a bipolar transistor is,

$I_E=I_{ES}\left[\exp\left(\frac{eV_{BE}}{k_BT}\right)-1\right]-\alpha_R I_{CS}\left[\exp\left(\frac{eV_{BC}}{k_BT}\right)-1\right],$

$I_C=\alpha_F I_{ES}\left[\exp\left(\frac{eV_{BE}}{k_BT}\right)-1\right]- I_{CS}\left[\exp\left(\frac{eV_{BC}}{k_BT}\right)-1\right],$

$I_B=I_E-I_C.$

The saturation currents are,

$$$I_{ES} = Aen_i^2\left(\frac{D_{pe}}{(x_e-d_e)N_{de}} + \frac{D_{nb}}{(x_{bc}-d_{eb})N_{ab}}\right)\qquad I_{CS} = Aen_i^2\left(\frac{D_{pc}}{(d_c-x_c)N_{dc}} + \frac{D_{nb}}{(x_{bc}-x_{eb})N_{ab}}\right),$$$

and

$$$\alpha_RI_{CS} =\alpha_FI_{ES} = Aen_i^2\left(\frac{D_{nb}}{(x_{bc}-x_{eb})N_{ab}}\right).$$$

Here $A$ is the area of the pn-junctions, $e$ is the elementary charge, $N_{de}$ is the donor concentration of the emitter, $N_{ab}$ is the acceptor concentration of the base, $N_{dc}$ is the donor concentration of the collector, $x_e-d_e$ is the distance from the metal contact of the emitter to the edge of the depletion with in the emitter, $x_{bc}-x_{eb}$ is the neutral base width (between the depletion widths in the base), $d_c-x_c$ is the distance from the metal contact of the colector to the edge of the depletion with in the collector, $D_{pe}$ is the diffusion constant of minority holes in the emitter, $D_{nb}$ is the diffusion constant of minority electrons in the base, and $D_{pc}$ is the diffusion constant of minority holes in the collector. The diffusion constants of the electrons and holes are related to the mobilities by the Einstein relation,

$$$D_n=\frac{\mu_nk_BT}{e}\hspace{1.5cm}D_p=\frac{\mu_pk_BT}{e},$$$

where $\mu_n$ is the mobility of the electrons and $\mu_p$ is the mobility of the holes. The intrinsic carrier concentration is,

$$n_i=\sqrt{N_c\left(\frac{T}{300}\right)^{3/2}N_v\left(\frac{T}{300}\right)^{3/2}}\exp\left(\frac{-E_g}{2k_BT}\right),$$

where $N_c$ is the effective density of states in the conduction band at 300 K, $N_v$ is the effective density of states in the valence band at 300 K, and $E_g$ is the band gap. The following

 n-Emitter $A=$ cm2 Minority $\mu_{pe}=$ cm2/Vs $N_c(300 K)=$ cm-3 $N_{de}=$ cm-3 $N_v(300 K)=$ cm-3 $x_e-d_e=$ μm $E_g=$ eV $T=$ K p-Base Minority $\mu_{nb}=$ cm2/Vs $N_{ab}=$ cm-3 $x_{bc}-x_{eb}=$ μm n-Collector Minority $\mu_{pc}=$ cm2/Vs $N_{dc}=$ cm-3 $d_{c}-x_{c}=$ μm

Once $I_{ES},\,I_{CS},\,\alpha_F$ and $\alpha_R$ are known, the Ebers-Moll model can be solved to plot the collector current $I_C$ as a function of $V_{CE}$ for different base currents $I_{B}$. The Ebers-Moll equations used here do not include the Early effect or breakdown at large $V_{CE}$. Below $I_C$ is plotted against $V_{CE}$ for the values of $I_B$ in the table on the right.

 $I_C$ [mA] $V_{CE}$ [V]
 aF = aR = IES = A ICS = A T = K VCE(max) = V IB[1] = mA IB[2] = mA IB[3] = mA IB[4] = mA IB[5] = mA IB[6] = mA IB[7] = mA IB[8] = mA IB[9] = mA IB[10] = mA