In a forward biased pn-junction, electrons are injected into the p-side as minority carriers and holes are injected into the n-side as minority carriers. This establishes a concentration gradient of minority electrons on the p-side and a concentration gradient of minority holes on the n-side. A diffusion current flows because of the concentration gradients. This diffusion current is calculated on the page Long/Short diodes. The short diode limit is when the metal contacts are much closer to the junction than the diffusion lengths so that the minority carriers decay linearly between the edges of the depletion zone and the metal contacts. In this case the minority carrier concentrations are,
$$n_p(x)=n_{p_0} + \frac{n_{p_0}\left(\exp\left(\frac{eV}{k_BT}\right)-1\right)}{x_p(V)-d_p}(x-d_p), \quad \text{for} \quad d_p < x < x_p$$ $$p_n(x)=p_{n_0} + \frac{p_{n_0}\left(\exp\left(\frac{eV}{k_BT}\right)-1\right)}{d_n-x_n(V)}(d_n-x), \quad \text{for} \quad x_n < x < d_n$$Here $d_p$ is the position of the metal-semiconductor interface on the p-side, $x_p$ is the edge of the depletion region on the p-side, $(d_p < x_p < 0)$, $x_n$ is the edge of the depletion region on the n-side, and $d_n$ is the position of the metal-semiconductor interface on the n-side $(0 < x_n < d_n)$. The junction between n and p is located at $x = 0$. To calculate the minority carrier densities at the edges of the depletion zone, substitute $x = x_p$ in the equation for minority electrons and $x= x_n$ for minority holes. The results are the concentrations we expect at the edges of the depletion region,
$$n_p(x_p) = n_{p_0}\exp\left(\frac{eV}{k_BT}\right),\qquad p_n(x_n) = p_{n_0}\exp\left(\frac{eV}{k_BT}\right).$$The minority carrier concentrations are fixed to the equlibrium concentrations at the metal contacts $n_p(d_p) = n_{p_0}$, $p_n(d_n) = p_{n_0}$.
The electron and hole diffusion current densities are,
$$j_{diff,n} = e D_n \frac{dn}{dx} = e D_n\frac{n_{p_0}\left(\exp\left(\frac{eV}{k_BT}\right)-1\right)}{x_p(V)-d_p}, \quad \text{for} \quad d_p < x < x_p$$ $$j_{diff,p} = - e D_p \frac{dp}{dx} = e D_p\frac{p_{n_0}\left(\exp\left(\frac{eV}{k_BT}\right)-1\right)}{d_n-x_n(V)}, \quad \text{for} \quad x_n < x < d_n$$The hole current that passes through the depletion region is the hole diffusion current at $x_n$ and the electron current that passes through the depletion zone is the electron diffusion current at $x_p$ so the total current density is,
$$j = j_{diff,n}(x_p) + j_{diff,p}(x_n)= e \left ( \frac{p_{n_0} D_p}{d_n-x_n(V)} + \frac{n_{p_0} D_n}{x_p(V)-d_p} \right ) \left (\exp \left(\frac{eV}{k_BT}\right) -1 \right).$$By multiplying with the cross-sectional area of the diode $A$, one obtains the total current:
$$I =eA \left ( \frac{p_{n_0} D_p}{d_n-x_n(V)} + \frac{n_{p_0} D_n}{x_p(V)-d_p} \right ) \left (\exp \left(\frac{eV}{k_BT}\right) -1 \right).$$The usual form for the diode formula is obtained by defining a saturation current,
$$I_S(V)=eA \left ( \frac{p_{n_0} D_p}{d_n-x_n(V)} + \frac{n_{p_0} D_n}{x_p(V)-d_p} \right ),$$so that the diode current can be written in the form,
$$I = I_s(V) \left (\exp \left(\frac{eV}{k_BT}\right) -1 \right).$$Notice that the saturation current depends on the voltage because the positions of the edges of the depletion width depend on voltage. However, this calculation assumes that diffusion is the dominant current mechanism as it would be in forward bias. In forward bias $d_n-x_n(V)$ and $x_p(V)-d_p$ will not change very much.
$E_g=$ eV $W=$ μm $x_p=$ μm $x_n=$ μm $V_{bi}=$ V $n_i=$ cm-3
$D_p=$ cm²/s $D_n=$ cm²/s $L_p=$ μm $L_n=$ μm $n_{p_0}=$ cm-3 $p_{n_0}=$ cm-3
$d_n - x_n=$ μm $x_p - d_p=$ μm $j=$ A/cm² $I_s=$ A
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Carrier Densities
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log(Carrier Densities)
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Current densities
$\vec{j}_{n,\text{drift}}= ne\mu_n\vec{E}$, $\vec{j}_{p,\text{drift}}= pe\mu_p\vec{E}$, |
Current-Voltage Characteristics
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