3.2.1.2 Closed Form of the BV and On-Resistance of the Punch-Through Unipolar and Bipolar Devices

A minimum specific on-resistance $R_\mathrm{sp}$ for unipolar power devices (or small voltage drop for bipolar power devices) can be obtained if the drift doping is increased and the drift length is decreased. When a device is designed to breakdown in the punch-through (PT) mode, it is suitable to obtain a lower $R_\mathrm{sp}$ while maintaining the required BV compared to the avalanche breakdown case. Therefore, most power MOSFETs and bipolar devices are designed to have a PT mode.

Figure 3.6: Electric field of a punch-through diode and a normal diode.

The punch-through structure has a lower doping concentration on the lightly doped side with a high concentration region for the contact. The thickness of the lightly doped region is smaller than that of the normal abrupt junction. The electric field profile of the punch-through structure has a rectangular shape compared to the triangular shape of the abrupt junction structure. As shown in Figure 3.6, the electric field has a maximum value at the $n^+$$p^-$-junction and decreases linearly in the $p^-$-region to a value of $E_1$ at the $p^-$$p^+$-interface.

From (3.25) and (3.29) the avalanche breakdown voltage $BV_\mathrm{pp}$ for the abrupt nonpunch through $n^+$$p^-$ case is

$$BV_\mathrm{pp} = \frac{q N_\mathrm{A}\, W^2_\mathrm{pp}}{2 \varepsilon_{si}} = 5.34 \times 10^{13}\,{N_\mathrm{A}}^{-\frac{3}{4}},$$ (3.30)

and the corresponding depletion width $W_\mathrm{pp}$ and the critical electric field $E_c$ are

$$W_\mathrm{pp} = \biggl(\frac{8}{a_\alpha}\biggr)^\frac{1}{8}\,\bi... ...hrm{A}}^{-\frac{7}{8}} = 2.67 \times 10^{10}\, {N_\mathrm{A}}^{-\frac{7}{8}}\,,$$ (3.31)

$$E_c = \frac{q\, N_\mathrm{A}\, W_{pp}}{\varepsilon_{si}}\,,$$ (3.32)

where $N_\mathrm{A}$ is the doping concentration in the $p^-$-region.

Assuming for the nonpunch-through and punch-through junction the same $p^-$-doping concentration $N_\mathrm{A}$, the $BV_\mathrm{PT}$ of a punch-through junction (see Figure 3.6) can be written as

$$BV_\mathrm{PT} = \frac{1}{2}\, (E_c + E_1 ) = E_c\, W_\mathrm{PT} - \frac{q N_\mathrm{A}\, W^2_\mathrm{PT}}{2 \varepsilon_{si}}\,,$$ (3.33)

where $W_\mathrm{PT}$ is the $p^-$ length of the punch-through structure and $E_1$ is

$$E_1 = E_c - \frac{q\, N_\mathrm{A}\, W_\mathrm{PT}}{\varepsilon_{si}}\,.$$ (3.34)

Using (3.25) and (3.33), and considering the normalized depletion width $\lambda$

$$\lambda = \frac{W_\mathrm{PT}}{W_\mathrm{pp}}\,.$$ (3.35)

The $BV_\mathrm{PT}$, $W_\mathrm{PT}$ and $N_\mathrm{A}$ of the punch-through structure can be expressed as follows

$$BV_\mathrm{PT} = (\frac{1}{\lambda} - \frac{1}{2})\, [\frac{a_\al... ...]^{-\frac{1}{4}}\, (\frac{q\,N_\mathrm{A}}{\varepsilon_{si}})^{-\frac{3}{4}}\,,$$ (3.36)

$$W_\mathrm{PT} = \lambda\, W_\mathrm{pp} = \lambda\, (\frac{8}{a_\alpha})^\frac{1}{8}\, (\frac{q\,N_\mathrm{A}}{\varepsilon_{si}})^{-\frac{7}{8}}\,.$$ (3.37)

From (3.36) and (3.37) $BV_\mathrm{PT}$ and $N_\mathrm{A}$ of the punch-through structure can be determined with the proper choice of $W_\mathrm{PT}$ and $\lambda$.

Figure 3.7: Avalanche breakdown voltage for punch-through diodes.

Figure 3.7 shows the breakdown voltages as a function of doping concentration in the lightly doped region calculated for punch-through diodes. With the increased doping concentration and thickness of the lightly doped region, the breakdown voltage becomes equal to that of the avalanche breakdown for the abrupt junction. As shown in the Figure 3.7 the breakdown voltage of the punch-through diode is a week function of the doping concentration in the lightly doped region.

Simultaneously with the desired BV a minimum on-resistance is important for unipolar devices. In addition doping and length of the drift region contribute mainly to the total on-resistance of a high-voltage device. From Figure 3.6 the on-resistance $R_\mathrm{on}$ for unit area is

$$R_\mathrm{on} = \int \frac{dx}{q\,\mu\,N_\mathrm{A}} = \frac{W_\mathrm{PT}}{q\,\mu\,N_\mathrm{A}}\,.$$ (3.38)

By combining (3.37) and (3.38) it is possible to choose the normalized depletion width $\lambda$ which has a minimum $R_\mathrm{on}$.