PHT.301 Physics of Semiconductor Devices

## PN diode Voltage-Temperature Characteristics

A pn-diode can be used as a thermometer because the saturation current of a diode is temperature dependent. This is convenient for measuring the temperature of a circuit. The current-voltage characteristic of a diode is described by the diode equation,

$$$\large I = I_S\left(\exp\left(\frac{eV}{k_BT}\right) - 1\right)\hspace{0.5cm}\text{[A]}.$$$

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, and $T$ is the absolute temperature. For a pn-diode, the saturation current can be written as,

$$$\large I_S = Aen_i^2\left(\frac{D_p}{L_pN_d} + \frac{D_n}{L_nN_a}\right).$$$

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 intrinsic carrier density is a strong function of temperature,

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

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,

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

where $\mu_n$ is the mobility of the electrons and $\mu_p$ is the mobility of the holes.

The form below will calculate the current through a pn-diode biased at a current $I$ for temperatures between $T_{start}$ and $T_{stop}$.

 $V$ [V] $T$ [K]
 $A=$ cm2 $N_c(300 K)=$ cm-3 $N_v(300 K)=$ cm-3 $E_g=$ eV $\mu_p=$ cm2/Vs $\tau_p=$ s $N_a=$ cm-3 $\mu_n=$ cm2/Vs $\tau_n=$ s $N_d=$ cm-3 $T_{start}=$ K $T_{stop}=$ K $I=$ A
 $T$ [K] $V$ [V]