Abstract
A review of silicon single crystal 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.
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Notes
- 1.
One electron volt (eV) is the energy dissipated or acquired by one electron that goes through a potential difference of 1 V. Since the charge of one electron is 1.6 × 10−19 Coulomb, 1 eV = 1.6 × 10−19 J. In this book, eV and cm are frequently used in place of J and m, as a convenient departure from SI units.
- 2.
Although the electron is free to move, it is still bound to the crystal. Electrons in the conduction band are sometimes described as quasi-free, i.e., behaving as if they were free. For simplicity, the term “free electron” will be used in this book to describe an electron in the conduction band.
- 3.
The unit Siemens (S) is used for 1/Ω.
- 4.
Bold E is used for electric field to distinguish it from energy E.
References
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Problems
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 mm, 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?
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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).
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5.
Calculate the conductivity of pure silicon at 25 °C, and for silicon doped with 1016 boron atoms/cm3 plus 1016 arsenic atoms/cm3.
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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
$$R_{\text{S}} = \frac{{\overline{\rho }}}{t}$$where \(\bar{\rho }\) 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
$$N_{\text{A}} (x) = 5 \times 10^{18} {\text{e}}^{ - 90x}$$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.
-
(a)
-
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
$$N_{\text{D}} (x) = \frac{\phi }{{{\Delta} R_{\text{p}} \sqrt {2\pi } }}{\text{e}}^{{ - (x - R_{\text{p}} )^{2} /2\Delta R_{\text{p}}^{2} }}$$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
$$\rho \,(x,y,z,t = 0)$$in space at a time t = 0.
The following relations stem from electromagnetism:
$$\nabla \cdot D = \rho ;\quad D = \varepsilon E;\quad J = \sigma E;\quad \nabla \cdot J = - \frac{{{\text{d}}\rho }}{{{\text{d}}t}}$$Show from these facts that
$$\rho (x,y,z,t) = \rho (x,y,z,t = 0){\text{e}}^{ - t/(\varepsilon /\sigma )}$$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 non-uniformity. 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.
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El-Kareh, B., Hutter, L.N. (2015). Review of Single-Crystal Silicon Properties. In: Silicon Analog Components. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2751-7_2
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