PHT.301 Physics of Semiconductor Devices  

PN diode IV characteristicsThe currentvoltage characteristic of a diode is described by the diode equation, $$ I = I_S\left(\exp\left(\frac{eV}{\eta k_BT}\right)  1\right)\hspace{0.5cm}\text{[A]}.$$Where $I$ is the current, $I_S$ is the saturatuion current, $e$ is the charge of an electron, $V$ is the voltage, $k_B$ is Boltzmann's constant, $\eta$ is the nonideality factor and $T$ is the absolute temperature. For a real diode there is always a resistance $R_S$ in series. This modifies the diode equation to, $$ I = I_S\left(\exp\left(\frac{e(VIR_S)}{\eta k_BT}\right)  1\right)\hspace{0.5cm}\text{[A]}.$$This complicates the calculation since $I$ now appears on both sides of the equation. However, such equations can be solved numerically, for instance by using a binary search. For a long diode, the saturation current can be written as, $$ I_S = Aen_i^2\left(\frac{D_p}{L_pN_d} + \frac{D_n}{L_nN_a}\right).$$Here $A$ is the area of the diode perpendicular to the current flow, $n_i$ is the intrinsic carrier concentration, $N_d$ is the donor concentration, $N_a$ is the acceptor concentration, $D_n$ is the diffusion constant for electrons, $D_p$ is the diffusion constant for holes, $L_n=\sqrt{D_n\tau_n}$ is the diffusion length for electrons, $L_p=\sqrt{D_p\tau_p}$ is the diffusion constant for holes, $\tau_n$ is the minority carrier lifetime for electrons, and $\tau_p$ is the minority carrier lifetime for holes. The derivation of the saturation current assumes that the dominant current mechanism is diffusion as it is in forward bias. While this equation describes forward bias well, the measured diode current in reverse bias often differs significantly from $I_S$. The intrinsic carrier density is a strong function of temperature, $$ n_i=\sqrt{N_c\left(\frac{T}{300}\right)^{3/2}N_v\left(\frac{T}{300}\right)^{3/2}}\exp\left(\frac{E_g}{2k_BT}\right).$$Here $N_c$ 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 temperture dependence of the band gap can be input into the form below. The diffusion constants 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.
