PHT.301 Physics of Semiconductor Devices

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Introduction

Electrons in crystals

Intrinsic Semiconductors

Extrinsic Semiconductors

Transport

pn junctions

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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,

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

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,

\[ \begin{equation} \large I_S = Aen_i^2\left(\frac{D_p}{L_pN_d} + \frac{D_n}{L_nN_a}\right). \end{equation} \]

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,

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

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,

\[ \begin{equation} \large D_n=\frac{\mu_nk_BT}{e}\hspace{1.5cm}D_p=\frac{\mu_pk_BT}{e}, \end{equation} \]

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]