System of Units, Conversion Factors, and General Constants

*The cm is the common unit of length and the electron-voit is the common unit of energy (see Appendix F) used in the study of semiconductors. However, the jouie and in some cases the meter should be used in most formulas.

Table B.2 t Conversion factors

Prefixes

Table B.3 I Physical constants

Avogadros number

Boltzmann *s constant

Electronic charge (magnitude)

Free electron rest mass

Permeability of free space

Permitdvity of free space

Plancks constant

Proton rest mass

Speed of light in vacuum

Thermal voltage (Г = 300 K)

УУл = 6.02 X 10+ - atoms per gram molecular weight

A: = L38 x 10- J/K = 8.62 X 10-* eV/K

€ = 1.60 X 10* с

го = 9.11 X 10 kg До = 4jr X 10-- H/m €() = 8.85 X 10 F/cm

= 8-85 X 10 F/m ft =6.625 X 10-3 J-s

= 4.135 X 10- eV-s

= n= 1.054 X 10-3 J-s

M = 1.67 X 10- kg с = 2.998 X 10 cm/s kT

0.0259 volt

kT = 0.0259 eV

Table B.4 I Silicon, gallium arsenide, and germanium properties {T= 300 K)

Effective mass Electrons

Holes

Effective mass (density of states) Electrons ()

m* = 0.98 тГ =0.19

ih *

mr, =0.16

0.49

Holes

1.08

0.56

0.067

0.082 0.45

0.067 0.48

1.64 0.082 0.044

0.28

0.55 0.37

Table B.5 I Other semiconductor parameters

Property SiO; Si3N4

Table B.6 \ Properties of SiO. and [T = 300 K)

А Р

The Periodic Table

Rare earths

VI 57-71

57 La 138.92

58 Ce 140.13

59 Pr

140.92

60 Nd 144.27

61 Pm

62 Sm 150,43

63 Eu 152.0

64 Gd 156.9

65 ТЪ 159.2

66 Dy 162.46

67 Ho 164.90

68 Er 167.2

69 Tm 169.4

70 Yb 173.04

71 Lu 174.9

The numbers in front of the symbols of the elements denote the atomic numbers; the numbers underneath are the atomic weights.

APPENDIX

The Error Function

erf(z)

V Jo

erf(0) = 0 erf(oo) = 1 erfc(z) = 1 - erf(z)

Derivation of Schrodingers

Wave Equation

Schrodingers wave equation was stated in Equation (2.6). The time-independent form of Schrodingers wave equation was then developed and given by Equation (2.13). The time-independent Schrodingers wave equation can also be developed from the classical wave equation. We may think of this development more in terms of a justification of the Schrodingers time-independent wave equation rather than a strict derivation.

The time-independent classical wave equauon, in terms of voltage, is given as

V(x)0 (ЕЛ)

where o) is the radian frequency and Vp is the phase velocity.

If we make a change of variable and let ij/ix) - V{x), then we have

Bxlfix) /a)\

We can write that

where v and X are the wave frequency and wavelength, respecrively.

From the wave-particle duality principle, we can relate the wavelength and momentum as

X = - (E.4)

Then

(E-5)

720 APPENDIX E Derivation of Schrodingers Wave Equation

and since h- -, we can write

(Tj=(fM(£)

- = Г = £ - У (ЕЛ)

where T, E, and Vare the kinetic energy, total energy, and potential energy terms, respectively.

We can then write

2 /2jrV 2m 2m

- (t) = [i,) = F -

Subsrituting Equation (E.8) into Equarion (E.2), we have

+ (E-VWx) = 0 (E.9)

which is the one-dimensional, time-independent Schrodingers wave equation.

А Р

D I X

Unit of Energy-The Electron-Volt

he electron-volt (eV) is a unit of energy that is used constantly in the study of A semiconductor physics and devices. This short discussion may help in getting a feel for the electron-volt.

Consider a parallel plate capacitor with an applied voltage as shown in Figure F. 1. Assume that an electron is released at .v - 0 at time r = 0. We may write

F = ma mo -- - (F. 1)

where e is the magnitude of the electronic charge and E is the magnitude of the electric field as shown. Upon integrating, the velocity and distance versus time are given by

eEt mo

(F.2)

Ю- V +0-

E-held

;c = 0

x = d

Figure F,l I Parallel plate capacitor.

APPENDIXF Unit of Energy-The Efectron-Volt

where we have assumed that v = 0 ai t - 0.

Assume that ait - to the electron reaches the positive plate of the capacitor so that jc - J. Then

eEt}

=T (F-4a)

, 2m 0

r = V (F.4b)

The velocity of the electron when it reaches the positive plate of the capacitor is

eEt(] 2eEd mo V 0

The kinetic energy of the electron at this time is

T = -movitof - -mo - = eEd (E6)

2 2 V mo /

The electric field is

E = - (F.7)

so that the energy is

e V (R8)

If an electron is accelerated through a potential of 1 volt, then the energy is T = e\ = (1.6 X 10-)(1) = 1.6 X 10- joule (F.9)

The electron-volt (eV) unit of energy is defined as

joule

Electron-volt = ~- (ElO)

Then, the electron that is accelerated through a potential of 1 volt will have an energy of

,g 1.6x10

or 1 eV.

We may note that the magnitude of the potential (1 volt) and the magnitude of the electron energy (1 eV) are the same. However, it is important to keep in mind that the unit associated with each number is different.