PN Junctions

Abstract

The pn junction is the fundamental building block of most silicon devices. The junction shape, doping profile, and characteristics have a direct impact on device and circuit performance. The chapter begins with a basic description of junction types and their thermal-equilibrium characteristics. This is followed by a review of junction forward-bias characteristics under low-level and high-level injection, and reverse-bias characteristics under low- and high-field conditions. The junction switching behavior and reverse recovery time are then described, followed by examples of stand-alone junction applications.

Problems

• 1. In a step junction, N _D = 10¹⁸ cm⁻³ and N _A = 10¹⁶ cm⁻³. Find for thermal equilibrium at 25 and 85 °C:

  • (a) The built-in voltage

  • (b) The depletion widths x _dp and x _dn

  • (c) The total positive charge per cm²

  • (d) The peak field

  • (e) The planar capacitance per cm².

• 2. One special case of a non-uniform profile is shown in the figure below. It is characterized by a concentration gradient a where the impurity concentration changes linearly as N _D − N _A = ax. Assume the depletion approximation, a = 5 × 10²¹ cm⁻⁴, and thermal equilibrium at 25 °C.

  • (a) Plot the electrical field as a function of depleted width.

  • (b) Plot the electrostatic potential as a function of depleted width.

    Hint: Use Eq. 3.5.

  • (c) The built-in voltage cannot be extracted analytically. Instead, the following equation can be solved iteratively

    $$V_{\text{b}} = \frac{2kT}{3q}\,\ln \frac{{3\varepsilon_{0} \varepsilon_{\text{Si}} a^{2} V_{\text{b}} }}{{2qn_{i}^{3} }}$$

  • (d) Find the total thermal equilibrium depletion width (Fig. P2).

    Fig. P2

    

    Idealized linearly graded approximation

• 3. Consider a pn junction with uniform N _D on the N-side and N _A on the P-side, and narrow widths W _n, W _p as shown in Fig. 3.14. The ratio of thermal equilibrium electron concentration $\overline{n}_{\text{p}}$ at the boundary of the depletion layer on the P-side to the electron concentration $\overline{n}_{\text{n}}$ at the depletion boundary of the N-side and the corresponding relation for holes are given by (3.7).

  • (a) Assume that when a forward voltage V _F is applied to the junction, the barrier is reduced by V _F and the relations for electrons follow Boltzmann’s distribution law as

    $$n_{\text{p}} = n_{\text{n}} {\text{e}}^{{ - q(V_{\text{bi}} - V_{\text{F}} )/kT}} ;\quad p_{\text{n}} = p_{\text{p}} {\text{e}}^{{ - q(V_{\text{bi}} - V_{\text{F}} )/kT}}$$

    Use the results to show that

    $$\Delta n_{\text{p}} = \overline{n}_{\text{p}} ({\text{e}}^{{qV_{\text{F}} /kT}} - 1);\quad\Delta p_{\text{n}} = \overline{p}_{\text{n}} ({\text{e}}^{{qV_{\text{F}} /kT}} - 1)$$

  • (b) Assume that the injected excess minority-carrier concentration drops linearly from the depletion edge to the contacts and derive (3.25).

• 4. A pn junction is made in a 2 Ω-cm resistivity silicon P-type wafer by implanting and diffusing arsenic at the surface such that the N-type region has a concentration of 10²⁰ cm⁻³. The area of the junction is 100 × 100 μm². The thickness of the wafer is 750 μm. The trap density in the P-type wafer is N _t = 10¹² cm⁻³. Ohmic contacts are made to the N-region and bottom of the wafer. At 25 °C, determine

  • (a) The thermal equilibrium built-in voltage.

  • (b) The reverse current for a reverse voltage V _R = 5 V.

  • (c) The forward current for a forward voltage V _F = 0.5 V.

  • (d) The junction capacitance at a reverse voltage V _R = 5 V.

  • (e) The reverse voltage necessary to spread the depletion region 25 mm in the P-region.

  • (f) The junction breakdown voltage.

• 5. The concentrations in an abrupt pn junction are N _A = N _D = 5 × 10¹⁸ cm⁻³. At what reverse voltage will this junction breakdown. Assume that Zener breakdown occurs when the peak field reaches 10⁶ V/cm.

• 6. Consider the N⁺P junction in the figure below. The N⁺-layer is externally shorted to the P⁺ substrate contact. Impact ionization in an adjacent circuit generates a hole current that passes under the N⁺-region and is collected at the substrate contact. The resistance between point A in the substrate immediately under the N⁺-region and the P⁺-contact is about 4 kΩ. The average trap density in the substrate is 10¹² cm⁻³, and the wafer temperature is 25 °C.

  

  • (a) At what substrate current I _sub will the N⁺P junction develop a forward bias of 0.25 V?

  • (b) For a forward bias of 0.25 V, what is the excess electron concentration at the depletion boundary in the substrate?

  • (c) Estimate the diffusion length minority electrons in the substrate.

• 7. Consider a one-sided N⁺P step junction having a junction depth of 0.3 μm and a uniform background concentration N _A = 10¹⁷ cm⁻³. The effective density of generation-recombination sites in the P-region is 10¹⁰ cm⁻³. An N⁺-region is placed 0.8 μm below the silicon surface and reverse-biased at 5 V.

  • (a) Calculate the electron current density for a forward bias voltage of 0.8 V at 25 and 100 °C.

  • (b) Punch-through occurs when the depletion regions of the top and bottom junction merge in the P-region. The reverse voltage is increased until a current of 1 μA/μm² is measured. Is the main mechanism for this current impact ionization, punch-through, or thermal generation?

• 8. A one-sided N⁺P junction is formed by diffusing a heavily doped N-region to a depth of 0.5 μm into a 10 Ω-cm P-substrate of thickness 725 μm. The density of recombination–generation sites in the P-region is 5 × 10¹¹ cm⁻³. A forward bias V _F = 0.7 V is applied to the junction at 25 °C. Will the minority-carrier electrons reach the backside of the substrate?

• 9. An abrupt N⁺P junction is formed by implanting and diffusing arsenic through a 2 × 2 μm² mask opening into a P-type substrate. The metallurgical junction is 0.25 μm deep, and the substrate is uniformly doped with boron at a concentration of 10¹⁷ cm⁻³. Assume cylindrical junction edges and compare the breakdown voltage at the edge to that in the planar part of the junction at 25 °C.

• 10. The intrinsic region in a PIN junction is 1 μm thick. Assume that the N-region and P-region are heavily doped. For a reverse voltage, V _R = 5 V and 25 °C calculate

  • (a) The junction capacitance.

  • (b) The electric field in the intrinsic region.

  • (c) The transit time for an electron–hole pair generated by a photon at the center of the intrinsic region.