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Some impurities may be present in the ingot that are undesirable. Zone refining is a common technique for purifying material. A high-temperature coil, or r-f induction coil, is slowly passed along the length of the boulc. The temperature induced by the coil is high enough so that a thin layer of liquid is formed. At the solid-liquid interface, there is a distribution of impurities between the two phases. The parameter that describes this distribution is called the segregation coefficient: the ratio of the concentration of impurities in the solid to the concentration in the liquid. If the segregation coefficient is 0.1, for example, the concentration of impurities in the liquid is a factor of 10 greater than that in the solid. As the liquid zone moves through the material, the impurities are driven along with the liquid. After several passes of the r-f coil, most impurities are at the end of the bar, which can then be cut off. The moving molten zone, or the zone-refining technique, can result in considerable purification.

After the semiconductor is grown, the boule is mechanically trimmed to the proper diameter and a flat is ground over the entire length of the boule to denote the crystal orientation. The flat is peфendicular to the 111 OJ direction or indicates the (110) plane. (See Figure 1.20b.) This then allows the individual chips to be fabricated along given crystal planes so that the chips can be sawed apart more easily. The boule is then sliced into wafers. The wafer must be thick enough to mechanically support itself. A mechanical two-sided lapping operation produces a flat wafer of uniform thickness. Since the lapping procedure can leave a surface damaged and contaminated by the mechanical operation, the surface must be removed by chemical etching. The final step is polishing. This provides a smooth surtace on which devices may be fabricated or further growth processes may be carried out. This final semiconductor wafer is called the substrate material.

1.6.2 Epitaxial Growth

A common and versatile growth technique that is used extensively in device and integrated circuit fabrication is epitaxial growth. Epitaxial growth is a process whereby a thin, single-crystal layer of material is grown on the surface of a single-crystal substrate. In the epitaxial process, the single-crystal substrate acts as the seed, although the process takes place far below the melting temperature. When an epitaxial layer is grown on a substrate of the same material, the process is termed homoepitaxy. Growing silicon on a silicon substrate is one example of a homoepitaxy process. At present, a great deal of work is being done with heteroepitaxy. In a heteroepitaxy process, although the substrate and epitaxial materials are not the same, the two crystal structures should be very similar if single-crystal growth is to be obtained and if a large number of defects are to be avoided at the epitaxial-substrate interface. Growing epitaxial layers of the ternary alloy AI GaAs on a GaAs substrate is one example of a heteroepitaxy process.

One epitaxial growth technique that has been used extensively is called chemical vapor-phase deposition (CVD). Silicon epitaxial layers, for example, are grown on silicon substrates by the controlled deposition of silicon atoms onto the surtace from a chemical vapor containing silicon. In one method, silicon tetrachloride reacts with hydrogen at the surface of a heated substrate. The silicon atoms are released in



1 - 7 Summary 19

the reaction and can be deposited onto the substrate, while the other chemical reac-tant, HCl, is in gaseous form and is swept out of the reactor. A shaф demarcation between the impurity doping in the substrate and in the epitaxial layer can be achieved using the CVD process. This technique allows great flexibility in the fabrication of semiconductor devices.

Liquid-phase epitaxy is another epitaxial growth technique. A compound of the semiconductor with another element may have a melting temperature lower than that of the semiconductor itself. The semiconductor substrate is held in the liquid compound and, since the temperature of the melt is lower than the melting temperature of the substrate, the substrate does not melt. As the solution is slowly cooled, a single-crystal semiconductor layer grows on the seed crystal. This technique, which occurs at a lower temperature than the Czochralski method, is useful in growing group ill-V compound semiconductors.

A versatile technique for growing epitaxial layers is the molecular beam epitaxy (MBE) process. A substrate is held in vacuum at a temperature normally in the range of 400 to 800С, a relatively low temperature compared with many semiconductor-processing steps. Semiconductor and dopant atoms are then evaporated onto the sur-face of the substrate. In this technique, the doping can be precisely controlled resulting in very complex doping profiles. Complex ternary compounds, such as AlGaAs, can be grown on substrates, such as GaAs, where abrupt changes in the crystal com-posifion are desired. Many layers of various types of epitaxial compositions can be grown on a substrate in this manner. These structures are extremely beneficial in optical devices such as laser diodes.

1.7 I SUMMARY

A few of the most common semiconductor materials were listed, Silicon is the most common semiconductor material.

The properties of semiconductors and other materials are determined to a large extent by the single-crystal laUice structure. The unit cell is a small volume of the crystal that is used to reproduce the entire crystal. Three basic unit cells are the simple cubic, body-centered cubic, and face-centered cubic.

Silicon has the diamond crystal structure. Atoms are formed in a tetrahedral configuration with four nearest neighbor atoins. The binary semiconductors have a zincblende lattice, that is basically the same as the diainond lattice.

Miller indices are used to describe planes in a crystal lattice. These planes may be used to describe the surface of a semiconductor material. The Miller indices are also used to describe directions in a crystal.

Imperfections do exist in semiconductor materials. A few of these imperfections are vacancies, substitutional impurities, and interstitial impurities. Small amounts of controlled substitutional impurities can favorably alter semiconductor properties as we will see in later chapters.

A brief description of semiconductor growth methods was given. Bulk growth produces the starting semiconductor material or substrate. Epitaxial growth can be used to control the surface properties of a semiconductor. Most semiconductor devices are fabricated

in the epitaxial layer.



CHECKPOINT

After studying this chapter, the reader should have the ability to:

Determine the volume density of atoms for various lattice structures. Determine the Miller indices of a crystal-lattice plane. Sketch a lattice plane given the Miller indices.

Determine the surface density of atoms on a given crystaMattice plane. Understand and describe various defects in a single-crystal lattice.

REVIEW QUESTIONS

1. List two elemental semiconductor materials and two compound semiconductor materials.

2. Sketch three lattice structures: (a) simple cubic, (h) body-centered cubic, and (c) face-centered cubic.

3. Describe the procedure for finding the volume density of atoms in a crystal.

4. Describe the procedure for obtaining the Miller indices that describe a plane in a crystal.

5. What is meant by a substitutional impurity in a crystal? What is meant by an interstitial impurity?

GLOSSARY OF IMPORTANT TERMS

binary semiconductor A iwo-element compound semiconductor, such as galhum arsenide (GaAs).

covalent bonding The bonding between atoms in which valence electrons are shared,

diamond lattice The atomic crystal structure of silicon, for example, in which each atom has four nearest neighbors in a tetrahedral configuradon.

doping The process of adding specific types of atoms lo a semiconductor to favorably alter the electrical characteristics.

elemental semiconductor A semiconductor composed of a single species of atom, such as silicon or germanium.

epitaxial layer A thin, single-crystal layer of material formed on the surface of a substrate,

ion implantation One particular process of doping a semiconductor.

lattice The periodic arrangement of atoms in a crystal.

Miller indices The set of integers used to describe a crystal plane.

primitive cell The smallest unit cell that can be repeated to form a lattice.

substrate A semiconductor wafer or other material used as the starting material for further semiconductor processing, such as epitaxial growth or diffusion.

ternary semiconductor A three-element compound semiconductor, such as aluminum gallium arsenide (AlGaAs).

unit cell A small volume of a crystal that can be used to reproduce the entire crystal.

zincblende lattice A lattice structure identical to the diamond laUice except that there arc two types of atoms instead of one.



Problems 21

PROBLEMS

Section 1.3 Space Lattices

1.1 Determine the number of atoms per unit cell in a (a) face-centered cubic,

(b) body-centered cubic, and (c) diamond lattice.

1.2 {a) The lattice constant of GaAs is 5.65 A, Determine the number of Ga atoms and As atoms per cm. (b) Determine the volume density of germanium atoms in a germanium semiconductor. The lattice constant of germanium is 5.65 A.

1.3 Assume that each atom is a hard sphere with the surface of each atom in contact with the surface of its nearest neighbor. Determine the percentage of total unit cell volume that is occupied in (o) a simple cubic lattice, (b) a face-centered cubic lattice,

(c) a body-centered cubic lattice, and (d) a diamond lattice.

1.4 A material, with a volutne of 1 cm , is composed of an fee lattice with a lattice constRntof2.5 mm. The atoms in this material are actuaUy coffee beans. Assume the coffee beans are hard spheres with each bean touching its nearest neighbor. Determine the volume of coffee after the coffee beans have been ground. (Assume 100 percent packing density of the ground coffee.)

1.5 If the lattice constant of silicon is 5.43 A, calculate ia) the distance from the center of one silicon atom to the center of its nearest neighbor, (h) the number density of silicon atoms (# per cm), and (c) the mass density (grams per cnv) of silicon.

1.6 A crystal is composed of two elements, A and B. The basic crystal structure is a body-centered cubic with elements A at each of the corners and element В in the center. The effective radius of element A is 1.02 A. Assume the elements are hard spheres with the surface of each A-type atom in contact with the surface of its nearest A-type neighbor. Calculate (a) the maximum radius of the В-type atom that will fit into this structure, and (b) the volume density (#/cm) of both the A-type atoms and the В-type atoms.

1.7 The crystal structure of sodium chloride (NaCl) is a simple cubic with the Na and CI atoms alternating positions. Each Na atom is then surrounded by six CI atoms and likewise each CI atom is surrounded by six Na atoms, (a) Sketch the atoms in a (100) plane, (b) Assume the atoms arc hard spheres with nearest neighbors touching. The effective radius of Na is 1.0 A and the effective radius of CI is 1.8 A. Determine the lattice constant, (c) Calculate the volume density of Na and CI atoms, (d) Calculate the mass density of NaCl.

1.8 (a) A material is composed of two types of atoms. Atom A has an effective radius of 2.2 A and atom В has an effective radius of 1.8 A. The lattice is a bcc with atoms A at the comers and atom В in the center. Determine the lattice constant and the volume densities of A atoms and В atoms, (b) Repeat part (a) with atoms В at the corners and atom A in the center, (c) What comparison can be made of the materials in parts {a) and {b)l

1.9 Consider the materials described in Problem 1.8 in parts {a) and {b). For each case, calculate the surface density of A atoms and В atoms in the (110) plane. What comparison can be made of the two materials?

1.10 (я) The crystal structure of a particular material consists of a single atom in the center of a cube. The lattice constant is ao and the diameter of the atom is q. Determine the volume density of atoms and the surface density of atoms in the (110) plane.

{b) Compare the results of part {a) to the results for the case of the simple cubic structure shown in Figure 1.5a with the same lattice constant.




Figure 1.21 I Figure for Problem 1Л 2.


1.11

1.16

Consider a three-dimensional eubic lattice with a lattice constant equal to a. {a) Sketch the following planes: (0 (100), (Я) (110), (ш) (310), and (jv) (230). (b) Sketch the following directions: (0 [100], ( ) [110], (ш) [310], and (/v) 1230].

For a simple cubic lattice, determine the Miller indices for the planes shown in Figure 1.21.

The lattice constant of a simple cubic cell is 5.63 A. Calculate the distance between the nearest parallel {a) (100), Kb){\\0), and (c) (111) planes.

The laUice constant of a single crystal is 4.50 A. Calculate the surface density of atoms (# per cm ) on the following planes: (/) (100), ( ) (110), iiii) (111) for each of the following lattice structures: (a) simple cubic, (b) body-centered cubic, and {c) face-centered cubic.

Determine the surface density of atoms for silicon on the {a) (100) plane, (/?) (110) plane, and (c) (111) plane.

Consider a face-centered cubic lattice. Assume the aloms are hard spheres with the surfaces of the nearest neighbors touching. Assume the radius of the atom is 2.25 A. {d) Calculate the volume density of atoms in the crystal. (/?) Calculate the distance between nearest (110) planes, (c) Calculate the surface density of atoms on the (110) plane.

Section \A Atomic Bonding

1.17 Calculate the density of valence electrons in silicon.

о

1Л8 The structure of GaAs is the zincblende lattice. The lattice constant is 5.65 A. Calculate the density of valence electrons in GaAs.



Reading Ust 23

Section L5 Imperfections and Impurities in Solids

1.19 (o) If 2 X 10* boron atoms per cm * are added to silicon as a substitutional impurity, determine what percentage of the silicon atoms are displaced in the single crystal lattice, (b) Repeat part (ci) for 10 boron atoms per cm *.

1.20 (й) Phosphorus atoms, at a concentration of 5 x 10 cm, are added to a pure sample of silicon. Assume the phosphorus atoms are distributed homogeneously throughout the silicon. What is the fraction by weight of phosphorus? (b) If boron atoms, at a concentration of 10 cm, are added to the material in pan

(a), determine the fraction by weight of boron.

1.21 If 2 X 10- gold atoms per cm are added to silicon as a substitutional impurity and are distributed uniformly throughout the semiconductor, determine the distance between gold atoms in terms of the silicon lattice constant. (Assume the gold atoms are distributed in a rectangular or cubic array.)

READING LIST

1. Azaroff, L.V., and J. J. Brophy, Electronic Processes in Materials. New York: McGraw-Hill, 1963.

2. Campbell, S. A. The Science and Engineering of Microelectronic Fabrication. New York: Oxford University Press, 1996.

3. Kittel, C. Introduction to Solid State Physics, 7th ed. Berlin: Springer-Verlag, 1993. *4. Li, S. S. Semiconductor Physical Electronics. New York: Plenum Press, 1993.

5. McKelvey, J, P. Solid State Physics for Engineering and Materials Science. Malabar, FL: Krieger, 1993.

6. Pierret, R. F Semiconductor Device Fundamentals, Reading, MA: Addison-Wesley, 1996.

7. Runyan, W. R., and K, E. Bean. Semiconductor Integrated Circuit Processing and Technology. Reading, MA: Addison-Wesley, 1990.

8. Singh, J. Semiconductor Devices: Basic Principles, New York: John Wiley and Sons, 2001.

9. Streetman, B.C., and S. Banerjee. Solid State Electronic Devices, 5 th ed. Upper Saddle River, NJ: Prentice Hall, 2000.

10. Sze, S. M. VLSI Technology New York: McGraw-Hill, 1983.

*1L Wolfe, C. M., N. Holonyak, Jr., and G. E. Stillman. Physical Properties of Semiconductors, Englewood Cliffs, NJ: Prentice Hall, 1989.

♦Indicates references that are at an advanced level compared to this text.



с Н А Р V Е R


Introduction to Quantum

Mechanics

PREVIEW

The goal of this text is to help readers understand the operation and characteristics of semiconductor devices. Ideally, we would like to begin discussing these devices immediately. However, in order to understand the current-voltage characteristics, we need some knowledge of the electron behavior in a crystal when the electron is subjected to various potential functions.

The motion of large objects, such as planets and satellites, can be predicted to a high degree of accuracy using classical theoretical physics based on Newtons laws of motion. But certain experimental results, involving electrons and high-frequency electromagnetic waves, appear to be inconsistent with classical physics. However, these experimental results can be predicted by the principles of quantum mechanics. The quantum mechanical wave theory is the basis for the theory of semiconductor physics.

We are ultimately interested in semiconductor materials whose electrical properties arc directly related to the behavior of electrons in the crystal lattice. The behavior and characteristics of these electrons can be described by the formulation of quantum mechanics called wave mechanics. The essential elements of this wave mechanics, using Schrodingers wave equation, are presented in this chapter

The goal of this chapter is to provide a brief introduction to quantum mechanics so that readers gain an understanding of and become comfortable with the analysis techniques. This introductory material forms the basis of semiconductor physics.



2.11 PRINCIPLES OF QUANTUM MECHANICS

Before we delve into the mathematics of quantum mechanics, there are three principles we need to consider: the principle of energy quanta, the wave-particle duality principle, and the uncertainty principle.

2ЛЛ Energy Quanta

One experiment that demonstrates an inconsistency between experimental results and the classical theory of light is called the photoelectric effect. If monochromatic light is incident on a clean surface of a material, then under certain conditions, electrons (photoelectrons) are emitted from the surface. According to classical physics, if the intensity of the light is large enough, the work function of the material will be overcome and an electron will be emitted from the surface independent of the incident frequency. This result is not observed. The observed effect is that, at a constant incident intensity, the maximum kinetic energy of the photoelectron varies linearly with frequency with a limiting frequency v vo, below which no photoelectron is produced. This result is shown in Figure 2.1. If the incident intensity varies at a constant frequency, the rate of photoelectron emission changes, but the maximum kinetic energy remains the same.

Planck postulated in 1900 that thermal radiation is emitted from a heated surface in discrete packets of energy called quanta. The energy of these quanta is given by £ - hv, where v is the frequency of the radiation and h is a constant now known as Plancks constant (h = 6.625 x iO - * J-s). Then in 1905, Einstein interpreted the photoelectric results by suggesting that the energy in a light wave is also contained in discrete packets or bundles. The particle-like packet of energy is called a photon, whose energy is also given by E = hv. A photon with sufficient energy, then, can knock an electron from the surface of the material. The minimum energy required to remove an electron is called the work function of the material

Incident monochromatic light

Photoelectron

kinetic energy = T


Material


Frequency, v

Figure 2.11 (a) The photoelectric effect and (b) the maximum kinetic energy of the photoelectron as a function of incident frequency.



and any excess photon energy goes into the kinetic energy of the photoelectron. This result was confirmed experimentally as demonstrated in Figure 2.1. The photoelectric effect shows the discrete nature of the photon and demonstrates the particle-like behavior of the photon.

The maximum kinetic energy of the photoelectron can be written as

Tliiax = =hv- hvo (v > vo) (2Л)

where hv is the incident photon energy and hvQ \s the minimum energy, or work function, required to remove an electron from the surface.

EXAMPLE 2Л Objective

To calculate the photon energy corresponding to a particular wavelength. Consider an x-ray with a wavelength of a = 0.708 x lO * cm.

Solution

The energy is

F , (6.625 X 10-)(3 X 10 )

E = hv = - =----z- =2.81 X 10 - J

X 0.708 X 10-

This value of energy may be given in the more common unit of electron-volt (see Appendix F). We have

2.81 X IQ-i-

E = -:--- = 1.75 X IC* eV

1.6 X 10-

Comment

The reciprocal relation between photon energy and wavelength is demonstrated: A large energy corresponds to a short wavelength.

2.1.2 Wave-Particle Duality

We have seen in the last section that light waves, in the photoelectric effect, behave as if they are particles. The particle-like behavior of electromagnetic waves was also instrumental in the explanation of the Compton effect. In this experiment, an x-ray beam was incident on a solid. A portion of the x-ray beam was deflected and the frequency of the deflected wave had shifted compared to the incident wave. The observed change in frequency and the deflected angle corresponded exactly to the expected results of a billiard ball collision between an x-ray quanta, or photon, and an electron in which both energy and momentum are conserved.

In 1924, de Broglie postulated the existence of matter waves. He suggested that since waves exhibit particle-like behavior, then particles should be expected to show wave-like properties. The hypothesis of de Broglie was the existence of a



wave-particle duality principle. The momentum of a photon is given by

h к

(2.2)

where X is the wavelength of the light wave. Then, de Broglie hypothesized that the wavelength of a particle can be expressed as

(2.3)

where p is the momentum of the particle and X is known as the de Broglie wavelength of the matter wave.

The wave nature of electrons has been tested in several ways. In one experiment by Davisson and Germer in 1927, electrons from a heated filament were accelerated at normal incidence onto a single crystal of nickel. A detector measured the scattered electrons as a function of angle. Figure 2.2 shows the experiinental setup and Figure 2.3 shows the results. The existence of a peak in the density of scattered electrons can be explained as a constructive interference of waves scattered by the periodic atoms in the planes of the nickel crystal. The angular distribution is very similar to an interference pattern produced by light diffracted from a grating.

In order to gain some appreciation of the frequencies and wavelengths involved in the wave-particle duaVity principle. Figure 2.4 shows the electromagnetic

о

frequency spectrum. We see that a wavelength of 72.7 A obtained in the next example is in the ultraviolet range. Typically, we will be considering wavelengths in the

Sample


Scattered electrons

Galvanometer

Figure 2.2 I Experimental arrangement of the Davisson-Germer experiment.

Sample

Incident electron beam


в = 45

Figure 2.31 Scattered electron flux as a function of scattering angle for the Davisson-Germer experiment.




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