Abstract
The mathematical model of semiconductor devices, derived in Chap. 19, is applied here to the description of the fundamental bipolar device, the p–n junction. The term bipolar indicates that both electrons and holes contribute to the current. The analysis is carried out using the simple example of a one-dimensional abrupt junction in steady state, with the hypotheses of non-degeneracy and complete ionization, that lend themselves to an analytical treatment. The equilibrium condition is considered first, and the solution of Poisson’s equation is tackled, showing that the structure can be partitioned into space-charge and quasi-neutral regions. Then, the Shockley theory is illustrated, leading to the derivation of the ideal I(V) characteristic. The semiconductor model is then applied to illustrating two features of the reverse-bias condition, namely, the depletion capacitance and the avalanche due to impact ionization. The complements justify the simplification of considering only the diffusive transport for the minority carriers in a quasi-neutral region, and provide the derivation of the Shockley boundary conditions. Finally, the expression of the depletion capacitance is worked out for the case of an arbitrary charge-density profile.
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Notes
- 1.
The metallurgical junction is often indicated with the same term used for the whole device, namely, p–n junction or simply junction.
- 2.
The use of asymptotic conditions is not applicable to shallow junctions like, e.g., those used for the fabrication of solar cells. In this case, the theory is slightly more involved.
- 3.
The same quantity is also called barrier potential and is sometimes indicated with ψ B .
- 4.
The numerical procedure is outlined in the note of Sect. 22.2.1.
- 5.
The electric potential can not be exactly constant, because the solution of ({21.8}) is an analytical function; as a consequence, if ϕ were constant in a finite interval, it would be constant everywhere.
- 6.
The inverse reasoning would not be correct: in fact, \(\rho = 0\) may yield \({\rm d}\varphi/{\rm d} x = \hbox{const} \ne 0\), which makes ϕ a linear function of x; deducing \(\varphi = \hbox{const}\) from \(\rho = 0\) is correct only if the additional condition of spatial uniformity holds.
- 7.
If it is not so, one must add to the left hand side of ({21.14}) the barrier between the two materials.
- 8.
The width of the space-charge region decreases as well (Fig. 21.8); such a decrease, however, is small, and does not compensate for the decrease in the potential drop.
- 9.
- 10.
It is worth reminding that the result holds only in the reverse-bias condition. In the forward-bias condition the injection of carriers from the quasi-neutral regions into the space-charge region prevents one from neglecting the contribution of the carrier concentrations to the charge density and makes the use of ({21.44}) erroneous.
- 11.
Varactors are also manufactured using technologies other than the bipolar one; e.g., with MOS capacitors or metal-semiconductor junctions.
- 12.
If the breakdown is accompanied by current crowding, the junction may be destroyed due to excessive heating. Special p–n junctions, called avalanche diodes, are designed to have breakdown uniformly spread over the surface of the metallurgical junction, to avoid current crowding. Such devices are able to indefinitely sustain the breakdown condition; they are used as voltage reference and for protecting electronic circuits against excessively-high voltages.
- 13.
An example of model for k n , k p is that proposed by Chynoweth [14]: \(k_n = k_{ns} \, {\rm exp} (- \vert E_{cn}/E \vert^{\beta_n})\), \(k_p = k_{ps} \, {\rm exp} (- \vert E_{cp}/E \vert^{\beta_p})\), where the parameters depend on temperature [83, 111].
- 14.
This happens, for instance, if a value of \(\vert V \vert\) larger than the breakdown voltage is used in ({21.51)}.
- 15.
After the change in the boundaries’ positions, x in ({21.69}) is still internal to the space-charge region.
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Problems
Problems
-
21.1
Evaluate the built-in potential at room temperature in an abrupt p–n junction with \(N_A = 10^{16}\) \(\mathrm{cm}^{-3}\) and \(N_D = 10^{15}\) \(\mathrm{cm}^{-3}\).
-
21.2
Show that avalanche due to impact ionization is possible only if both coefficients k n and k p are different from zero.
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Rudan, M. (2015). Bipolar Devices. In: Physics of Semiconductor Devices. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1151-6_21
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DOI: https://doi.org/10.1007/978-1-4939-1151-6_21
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