Review of Single-Crystal Silicon Properties

Abstract

A review of single-crystal silicon properties is essential to understanding silicon components. The objective of this chapter is to highlight only those semiconductor properties that are most important to analog (and digital) silicon device operation and characteristics discussed in the following chapters. The chapter covers carrier concentrations and thermal-equilibrium statistics, carrier transport under low- and high-field conditions, and minority-carrier lifetime and diffusion length.

Problems

• 1. A region in silicon is uniformly doped with 10¹⁸ boron atoms/cm³. Assume the region to have a length of 50 nm, a width of 1 μm, and a thickness of 100 nm, and estimate the number of boron atoms in the region. What is the average distance between two boron atoms?

• 2. What is the percentage of covalent bonds broken in pure silicon at 100 °C?

• 3. Silicon is doped with 10¹⁶ cm⁻³ phosphorus atoms/cm³. At what temperature would the hole concentration be equal to 10% of the ionized impurity concentration?

• 4. Silicon is doped with 10¹⁵ cm⁻³ phosphorus atoms/cm³. At what temperature is n = 0.9 N_D? (Use the Fermi–Dirac distribution function.)

• 5. Calculate the conductivity of pure silicon at 25 °C and for silicon doped with 10¹⁶ boron atoms/cm³ plus 10¹⁶ arsenic atoms/cm³.

• 6. The sheet resistance R_S of a film is defined as the resistance measured between two opposite sides of a square of the film. Show that for a film thickness t, the sheet resistance along the surface is

  $$ {R}_{\mathrm{S}}=\frac{\overline{\rho}}{t} $$

  where $\overline{\rho}$ is the average film resistivity.

• 7. The boron profile in a 0.1-μm-deep transistor region in silicon can be approximated by an exponential function of the form

  $$ {N}_{\mathrm{A}}(x)=5\times {10}^{18}{\mathrm{e}}^{-90x} $$

  where x is the depth in μm from the surface.

  • (a) Calculate the average sheet resistance at 25 °C.

  • (b) Show that the built-in field is constant in the region.

• 8. An N-type layer in silicon is formed by phosphorus ion-implantation into a 10 Ω-cm substrate. The phosphorus profile after activation can be approximated by a Gaussian distribution as

  $$ {N}_{\mathrm{D}}(x)=\frac{\phi }{\Delta {R}_{\mathrm{p}}\sqrt{2\pi }}{\mathrm{e}}^{-{\left(x-{R}_{\mathrm{p}}\right)}^2/2\Delta {R}_{\mathrm{p}}^2} $$

  where ϕ is the implanted dose in atoms/cm², R_p is the projected range, and ΔR_p is the straggle. Assume full ionization, and rind the sheet resistance of the layer at 25 °C for ϕ = 5 × 10¹² cm⁻², R_p = 0.13 μm, and ΔR_p = 0.05 μm.

• 9. Consider a homogeneous conductor of conductivity σ and dielectric permittivity ε. Assume a mobile charge concentration

  $$ \rho \left(x,y,z,t=0\right) $$

  in space at a time t = 0.

  The following relations stem from electromagnetism:

  $$ \nabla \cdot D=\rho; \kern1.5em D=\varepsilon E;\kern1.5em J=\sigma E;\kern1.5em \nabla \cdot J=-\frac{\mathrm{d}\rho }{\mathrm{d}t} $$

  Show from these facts that

  $$ \rho \left(x,y,z,t\right)=\rho \left(x,y,z,t=0\right){\mathrm{e}}^{-t/\left(\varepsilon /\sigma \right)} $$

  Interpret this result to show that mobile charge cannot remain in the bulk of uniform conducting material but must accumulate at surfaces of discontinuity or other places of nonuniformity. Find the value of the “dielectric relaxation time” ε/σ for a typical metal and for P-type silicon with N_A = 10¹⁵ cm⁻³ and N-type silicon with N_D = 10¹⁷ cm⁻³ [3].

• 10. An N-type region in silicon is uniformly doped with arsenic at a concentration N_D = 10²⁰ cm⁻³. Assume full ionization and an effective density of recombination centers N_T = 10¹² cm⁻³, and estimate the hole diffusion length in the region at 25 °C.