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513.160 Microelectronics and Micromechanics | |
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SiliconSilicon is the second most common element in the earth's crust and an important semiconducting material. Structural propertiesCrystal structure: DiamondBravais lattice: face centered cubic Space group: 227 (F d -3 m), Strukturbericht: A4, Pearson symbol: cF8 Point group: m3m (Oh) six 2-fold rotations, four 3-fold rotations, three 4-fold rotations, nine mirror planes, inversion Lattice constant: $a=0.543$ nm Atomic weight $28.09$ Atomic density $n_{atoms}=4.995 \times 10^{22}$ 1/cm³ Density $\rho= 2.33$ g/cm³ Density of surface atoms (100) $6.78 \times 10^{14}$ 1/cm² (110) $9.59 \times 10^{14}$ 1/cm² (111) $7.83 \times 10^{14}$ 1/cm² The primitive lattice vectors are, $\vec{a}_1=\frac{a}{2}\hat{x}+\frac{a}{2}\hat{y}$, $\vec{a}_2=\frac{a}{2}\hat{x}+\frac{a}{2}\hat{z}$, $\vec{a}_3=\frac{a}{2}\hat{y}+\frac{a}{2}\hat{z}$. There are two atoms in the basis. In fractional coordinates of the conventional unit, the positions of the two atoms are, $\vec{B}_1=(0,0,0)$, $\vec{B}_2=(0.25,0.25,0.25)$. The conventional unit cell is: \[ \begin{equation} a = 0.543\, \text{nm}, \, b = 0.543 \,\text{nm}, \, c = 0.543 \,\text{nm}, \, \alpha = 90^{\circ} , \,\beta = 90^{\circ} ,\,\gamma = 90^{\circ} . \end{equation} \]Space group: 227 (F d -3 m) 1 x,y,z identity 2 z+1/4,y+1/4,-x+1/4 4-fold screw axis|translation: 0 1/4 0 3 y+1/4,x+1/4,-z+1/4 2-fold screw axis|translation: 1/4 1/4 0 4 x+1/4,z+1/4,-y+1/4 4-fold screw axis|translation: 1/4 0 0 5 z+1/4,x+1/4,-y+1/4 3-bar axis|inversion center at 1/12 1/3 -1/12 6 y+1/4,z+1/4,-x+1/4 3-bar axis|inversion center at 1/3 1/12 -1/12 7 x+1/4,y+1/4,-z+1/4 n-glide plane|translation: 1/4 1/4 0 8 z+1/4,-y+1/4,x+1/4 2-fold screw axis|translation: 1/4 0 1/4 9 y+1/4,-x+1/4,z+1/4 4-fold screw axis|translation: 0 0 1/4 10 x+1/4,-z+1/4,y+1/4 4-fold screw axis|translation: 1/4 0 0 11 z+1/4,-x+1/4,y+1/4 3-bar axis|inversion center at 1/3 -1/12 1/12 12 y+1/4,-z+1/4,x+1/4 3-bar axis|inversion center at 1/12 -1/12 1/3 13 x+1/4,-y+1/4,z+1/4 n-glide plane|translation: 1/4 0 1/4 14 -z+1/4,y+1/4,x+1/4 4-fold screw axis|translation: 0 1/4 0 15 -y+1/4,x+1/4,z+1/4 4-fold screw axis|translation: 0 0 1/4 16 -x+1/4,z+1/4,y+1/4 2-fold screw axis|translation: 0 1/4 1/4 17 -z+1/4,x+1/4,y+1/4 3-bar axis|inversion center at -1/12 1/12 1/3 18 -y+1/4,z+1/4,x+1/4 3-bar axis|inversion center at -1/12 1/3 1/12 19 -x+1/4,y+1/4,z+1/4 n-glide plane|translation: 0 1/4 1/4 20 -z+1/4,-y+1/4,-x+1/4 C2 axis 21 -y+1/4,-x+1/4,-z+1/4 C2 axis 22 -x+1/4,-z+1/4,-y+1/4 C2 axis 23 -z+1/4,-x+1/4,-y+1/4 3-bar axis|inversion center at 1/12 1/12 1/12 24 -y+1/4,-z+1/4,-x+1/4 3-bar axis|inversion center at 1/12 1/12 1/12 25 -x+1/4,-y+1/4,-z+1/4 Ci: 1/12 1/12 1/12 26 -z,-y,x 4-bar axis|inversion center at 0 0 0 27 -y,-x,z mirror plane 28 -x,-z,y 4-bar axis|inversion center at 0 0 0 29 -z,-x,y C3 axis 30 -y,-z,x C3 axis 31 -x,-y,z C2 axis 32 -z,y,-x mirror plane 33 -y,x,-z 4-bar axis|inversion center at 0 0 0 34 -x,z,-y 4-bar axis|inversion center at 0 0 0 35 -z,x,-y C3 axis 36 -y,z,-x C3 axis 37 -x,y,-z C2 axis 38 z,-y,-x 4-bar axis|inversion center at 0 0 0 39 y,-x,-z 4-bar axis|inversion center at 0 0 0 40 x,-z,-y mirror plane 41 z,-x,-y C3 axis 42 y,-z,-x C3 axis 43 x,-y,-z C2 axis 44 z,y,x mirror plane 45 y,x,z mirror plane 46 x,z,y mirror plane 47 z,x,y C3 axis 48 y,z,x C3 axis 49 z+1/4,y-1/4,-x-1/4 4-fold screw axis|translation: 0 -1/4 0 50 y+1/4,x-1/4,-z-1/4 C2 axis 51 x+1/4,z-1/4,-y-1/4 4-fold screw axis|translation: 1/4 0 0 52 z+1/4,x-1/4,-y-1/4 3-bar axis|inversion center at 1/12 -1/12 -1/12 53 y+1/4,z-1/4,-x-1/4 3-bar axis|inversion center at -1/12 -1/3 -1/12 54 x+1/4,y-1/4,-z-1/4 n-glide plane|translation: 1/4 -1/4 0 55 z+1/4,-y-1/4,x-1/4 C2 axis 56 y+1/4,-x-1/4,z-1/4 4-fold screw axis|translation: 0 0 -1/4 57 x+1/4,-z-1/4,y-1/4 4-fold screw axis|translation: 1/4 0 0 58 z+1/4,-x-1/4,y-1/4 3-bar axis|inversion center at -1/12 -1/12 -1/3 59 y+1/4,-z-1/4,x-1/4 3-bar axis|inversion center at 1/12 -1/12 -1/12 60 x+1/4,-y-1/4,z-1/4 n-glide plane|translation: 1/4 0 -1/4 61 -z+1/4,y-1/4,x-1/4 4-fold screw axis|translation: 0 -1/4 0 62 -y+1/4,x-1/4,z-1/4 4-fold screw axis|translation: 0 0 -1/4 63 -x+1/4,z-1/4,y-1/4 2-fold screw axis|translation: 0 -1/4 -1/4 64 -z+1/4,x-1/4,y-1/4 3-bar axis|inversion center at 1/3 1/12 -1/12 65 -y+1/4,z-1/4,x-1/4 3-bar axis|inversion center at 1/3 -1/12 1/12 66 -x+1/4,y-1/4,z-1/4 n-glide plane|translation: 0 -1/4 -1/4 67 -z+1/4,-y-1/4,-x-1/4 2-fold screw axis|translation: 1/4 0 -1/4 68 -y+1/4,-x-1/4,-z-1/4 2-fold screw axis|translation: 1/4 -1/4 0 69 -x+1/4,-z-1/4,-y-1/4 C2 axis 70 -z+1/4,-x-1/4,-y-1/4 3-bar axis|inversion center at 1/12 -1/3 1/12 71 -y+1/4,-z-1/4,-x-1/4 3-bar axis|inversion center at 1/12 1/12 -1/3 72 -x+1/4,-y-1/4,-z-1/4 Ci: 1/12 -1/12 -1/12 73 -z,-y+1/2,x+1/2 4-bar axis|inversion center at -1/4 1/4 1/4 74 -y,-x+1/2,z+1/2 g-glide plane|translation: -1/4 1/4 1/2 75 -x,-z+1/2,y+1/2 4-bar axis|inversion center at 0 0 1/2 76 -z,-x+1/2,y+1/2 3-fold screw axis|translation: -1/3 1/3 1/3 77 -y,-z+1/2,x+1/2 C3 axis 78 -x,-y+1/2,z+1/2 2-fold screw axis|translation: 0 0 1/2 79 -z,y+1/2,-x+1/2 g-glide plane|translation: -1/4 1/2 1/4 80 -y,x+1/2,-z+1/2 4-bar axis|inversion center at -1/4 1/4 1/4 81 -x,z+1/2,-y+1/2 4-bar axis|inversion center at 0 1/2 0 82 -z,x+1/2,-y+1/2 C3 axis 83 -y,z+1/2,-x+1/2 3-fold screw axis|translation: -1/3 1/3 1/3 84 -x,y+1/2,-z+1/2 2-fold screw axis|translation: 0 1/2 0 85 z,-y+1/2,-x+1/2 4-bar axis|inversion center at 1/4 1/4 1/4 86 y,-x+1/2,-z+1/2 4-bar axis|inversion center at 1/4 1/4 1/4 87 x,-z+1/2,-y+1/2 mirror plane 88 z,-x+1/2,-y+1/2 C3 axis 89 y,-z+1/2,-x+1/2 C3 axis 90 x,-y+1/2,-z+1/2 C2 axis 91 z,y+1/2,x+1/2 g-glide plane|translation: 1/4 1/2 1/4 92 y,x+1/2,z+1/2 g-glide plane|translation: 1/4 1/4 1/2 93 x,z+1/2,y+1/2 n-glide plane|translation: 0 1/2 1/2 94 z,x+1/2,y+1/2 3-fold screw axis|translation: 1/3 1/3 1/3 95 y,z+1/2,x+1/2 3-fold screw axis|translation: 1/3 1/3 1/3 96 x,y+1/2,z+1/2 translation: 0 1/2 1/2 97 z-1/4,y+1/4,-x-1/4 4-fold screw axis|translation: 0 1/4 0 98 y-1/4,x+1/4,-z-1/4 C2 axis 99 x-1/4,z+1/4,-y-1/4 4-fold screw axis|translation: -1/4 0 0 100 z-1/4,x+1/4,-y-1/4 3-bar axis|inversion center at -1/3 -1/12 -1/12 101 y-1/4,z+1/4,-x-1/4 3-bar axis|inversion center at -1/12 1/12 -1/12 102 x-1/4,y+1/4,-z-1/4 n-glide plane|translation: -1/4 1/4 0 103 z-1/4,-y+1/4,x-1/4 2-fold screw axis|translation: -1/4 0 -1/4 104 y-1/4,-x+1/4,z-1/4 4-fold screw axis|translation: 0 0 -1/4 105 x-1/4,-z+1/4,y-1/4 4-fold screw axis|translation: -1/4 0 0 106 z-1/4,-x+1/4,y-1/4 3-bar axis|inversion center at -1/12 1/3 1/12 107 y-1/4,-z+1/4,x-1/4 3-bar axis|inversion center at 1/12 1/3 -1/12 108 x-1/4,-y+1/4,z-1/4 n-glide plane|translation: -1/4 0 -1/4 109 -z-1/4,y+1/4,x-1/4 4-fold screw axis|translation: 0 1/4 0 110 -y-1/4,x+1/4,z-1/4 4-fold screw axis|translation: 0 0 -1/4 111 -x-1/4,z+1/4,y-1/4 C2 axis 112 -z-1/4,x+1/4,y-1/4 3-bar axis|inversion center at -1/12 1/12 -1/12 113 -y-1/4,z+1/4,x-1/4 3-bar axis|inversion center at -1/12 -1/12 -1/3 114 -x-1/4,y+1/4,z-1/4 n-glide plane|translation: 0 1/4 -1/4 115 -z-1/4,-y+1/4,-x-1/4 C2 axis 116 -y-1/4,-x+1/4,-z-1/4 2-fold screw axis|translation: -1/4 1/4 0 117 -x-1/4,-z+1/4,-y-1/4 2-fold screw axis|translation: 0 1/4 -1/4 118 -z-1/4,-x+1/4,-y-1/4 3-bar axis|inversion center at 1/12 1/12 -1/3 119 -y-1/4,-z+1/4,-x-1/4 3-bar axis|inversion center at -1/3 1/12 1/12 120 -x-1/4,-y+1/4,-z-1/4 Ci: -1/12 1/12 -1/12 121 -z+1/2,-y,x+1/2 4-bar axis|inversion center at 0 0 1/2 122 -y+1/2,-x,z+1/2 g-glide plane|translation: 1/4 -1/4 1/2 123 -x+1/2,-z,y+1/2 4-bar axis|inversion center at 1/4 -1/4 1/4 124 -z+1/2,-x,y+1/2 C3 axis 125 -y+1/2,-z,x+1/2 3-fold screw axis|translation: 1/3 -1/3 1/3 126 -x+1/2,-y,z+1/2 2-fold screw axis|translation: 0 0 1/2 127 -z+1/2,y,-x+1/2 mirror plane 128 -y+1/2,x,-z+1/2 4-bar axis|inversion center at 1/4 1/4 1/4 129 -x+1/2,z,-y+1/2 4-bar axis|inversion center at 1/4 1/4 1/4 130 -z+1/2,x,-y+1/2 C3 axis 131 -y+1/2,z,-x+1/2 C3 axis 132 -x+1/2,y,-z+1/2 C2 axis 133 z+1/2,-y,-x+1/2 4-bar axis|inversion center at 1/2 0 0 134 y+1/2,-x,-z+1/2 4-bar axis|inversion center at 1/4 -1/4 1/4 135 x+1/2,-z,-y+1/2 g-glide plane|translation: 1/2 -1/4 1/4 136 z+1/2,-x,-y+1/2 3-fold screw axis|translation: 1/3 -1/3 1/3 137 y+1/2,-z,-x+1/2 C3 axis 138 x+1/2,-y,-z+1/2 2-fold screw axis|translation: 1/2 0 0 139 z+1/2,y,x+1/2 n-glide plane|translation: 1/2 0 1/2 140 y+1/2,x,z+1/2 g-glide plane|translation: 1/4 1/4 1/2 141 x+1/2,z,y+1/2 g-glide plane|translation: 1/2 1/4 1/4 142 z+1/2,x,y+1/2 3-fold screw axis|translation: 1/3 1/3 1/3 143 y+1/2,z,x+1/2 3-fold screw axis|translation: 1/3 1/3 1/3 144 x+1/2,y,z+1/2 translation: 1/2 0 1/2 145 z-1/4,y-1/4,-x+1/4 4-fold screw axis|translation: 0 -1/4 0 146 y-1/4,x-1/4,-z+1/4 2-fold screw axis|translation: -1/4 -1/4 0 147 x-1/4,z-1/4,-y+1/4 4-fold screw axis|translation: -1/4 0 0 148 z-1/4,x-1/4,-y+1/4 3-bar axis|inversion center at 1/12 -1/12 1/3 149 y-1/4,z-1/4,-x+1/4 3-bar axis|inversion center at -1/12 1/12 1/3 150 x-1/4,y-1/4,-z+1/4 n-glide plane|translation: -1/4 -1/4 0 151 z-1/4,-y-1/4,x+1/4 C2 axis 152 y-1/4,-x-1/4,z+1/4 4-fold screw axis|translation: 0 0 1/4 153 x-1/4,-z-1/4,y+1/4 4-fold screw axis|translation: -1/4 0 0 154 z-1/4,-x-1/4,y+1/4 3-bar axis|inversion center at -1/12 -1/12 1/12 155 y-1/4,-z-1/4,x+1/4 3-bar axis|inversion center at -1/3 -1/12 -1/12 156 x-1/4,-y-1/4,z+1/4 n-glide plane|translation: -1/4 0 1/4 157 -z-1/4,y-1/4,x+1/4 4-fold screw axis|translation: 0 -1/4 0 158 -y-1/4,x-1/4,z+1/4 4-fold screw axis|translation: 0 0 1/4 159 -x-1/4,z-1/4,y+1/4 C2 axis 160 -z-1/4,x-1/4,y+1/4 3-bar axis|inversion center at -1/12 -1/3 -1/12 161 -y-1/4,z-1/4,x+1/4 3-bar axis|inversion center at -1/12 -1/12 1/12 162 -x-1/4,y-1/4,z+1/4 n-glide plane|translation: 0 -1/4 1/4 163 -z-1/4,-y-1/4,-x+1/4 2-fold screw axis|translation: -1/4 0 1/4 164 -y-1/4,-x-1/4,-z+1/4 C2 axis 165 -x-1/4,-z-1/4,-y+1/4 2-fold screw axis|translation: 0 -1/4 1/4 166 -z-1/4,-x-1/4,-y+1/4 3-bar axis|inversion center at -1/3 1/12 1/12 167 -y-1/4,-z-1/4,-x+1/4 3-bar axis|inversion center at 1/12 -1/3 1/12 168 -x-1/4,-y-1/4,-z+1/4 Ci: -1/12 -1/12 1/12 169 -z+1/2,-y+1/2,x 4-bar axis|inversion center at 1/4 1/4 1/4 170 -y+1/2,-x+1/2,z mirror plane 171 -x+1/2,-z+1/2,y 4-bar axis|inversion center at 1/4 1/4 1/4 172 -z+1/2,-x+1/2,y C3 axis 173 -y+1/2,-z+1/2,x C3 axis 174 -x+1/2,-y+1/2,z C2 axis 175 -z+1/2,y+1/2,-x g-glide plane|translation: 1/4 1/2 -1/4 176 -y+1/2,x+1/2,-z 4-bar axis|inversion center at 0 1/2 0 177 -x+1/2,z+1/2,-y 4-bar axis|inversion center at 1/4 1/4 -1/4 178 -z+1/2,x+1/2,-y 3-fold screw axis|translation: 1/3 1/3 -1/3 179 -y+1/2,z+1/2,-x C3 axis 180 -x+1/2,y+1/2,-z 2-fold screw axis|translation: 0 1/2 0 181 z+1/2,-y+1/2,-x 4-bar axis|inversion center at 1/4 1/4 -1/4 182 y+1/2,-x+1/2,-z 4-bar axis|inversion center at 1/2 0 0 183 x+1/2,-z+1/2,-y g-glide plane|translation: 1/2 1/4 -1/4 184 z+1/2,-x+1/2,-y C3 axis 185 y+1/2,-z+1/2,-x 3-fold screw axis|translation: 1/3 1/3 -1/3 186 x+1/2,-y+1/2,-z 2-fold screw axis|translation: 1/2 0 0 187 z+1/2,y+1/2,x g-glide plane|translation: 1/4 1/2 1/4 188 y+1/2,x+1/2,z n-glide plane|translation: 1/2 1/2 0 189 x+1/2,z+1/2,y g-glide plane|translation: 1/2 1/4 1/4 190 z+1/2,x+1/2,y 3-fold screw axis|translation: 1/3 1/3 1/3 191 y+1/2,z+1/2,x 3-fold screw axis|translation: 1/3 1/3 1/3 192 x+1/2,y+1/2,z translation: 1/2 1/2 0 PhononsThe phonon dispersion relation for silicon is shown below. There are two atoms in the basis so there are three acoustic branches and three optical branches. The phonon density of states is plotted in the right panel.
The phonon density of states of silicon is,
The density of phonon modes is three times the density of atoms. Integrating over the phonon density of states should be the number of degrees of freedom in a cubic meter of Si. The integral of the data above is 1.498 × 1029. The density of silicon is 2.33 g/cm³ = 2330 kg/m³. The mass of a silicon atom is 4.664 × 10-26 kg. The number of silicon atoms in a cubic meter is 4.99 × 1028. Multipying this by 3 to get the number of degrees of freedom yields 1.498 × 1029 so this density of states is consistent with the known number of vibrational modes in silicon. The energy spectral density specifies which phonon modes are occupied for a specific temperature. This is analogous to the Planck radiation curves for blackbody radiation. The energy spectral density is plotted for temperatures $T=60i\quad\text{[K]}\quad i=1,2,\cdots,10$.
The integral of the energy spectral density over all frequencies is the phonon contribution to the internal energy density.
The derivative of the internal energy with respect to temperature is the phonon contribution to the specific heat, $c_v=\frac{du}{dT}$.
The phonon contribution to the entropy density can be calculated from the specific heat $s=\int\frac{c_v}{T}dT$.
A crystal held at constant temperature will go to a minimum of the Helmholtz free energy, $f=u-Ts$.
Data for the Gibbs free energy, $G=U+pV-TS$, for silicon is available from the SGTE pure element database. The Gibbs energies of different phases of silicon are plotted below.
Electrical Band StructureSilicon is an indirect bandgap semiconductor.
The six 6 conduction band minima can be approximated by the paraboloids, \[ \begin{equation} E_{c100}=E_g+\frac{\hbar^2}{2m_l}\left(k_x-\frac{1.7\pi}{a}\right)^2+\frac{\hbar^2}{2m_t}k_y^2+\frac{\hbar^2}{2m_t}k_z^2, \end{equation} \] \[ \begin{equation} E_{c\overline{1}00}=E_g+\frac{\hbar^2}{2m_l}\left(k_x+\frac{1.7\pi}{a}\right)^2+\frac{\hbar^2}{2m_t}k_y^2+\frac{\hbar^2}{2m_t}k_z^2, \end{equation} \] \[ \begin{equation} E_{c010}=E_g+\frac{\hbar^2}{2m_t}k_x^2+\frac{\hbar^2}{2m_l}\left(k_y-\frac{1.7\pi}{a}\right)^2+\frac{\hbar^2}{2m_t}k_z^2, \end{equation} \] \[ \begin{equation} E_{c0\overline{1}0}=E_g+\frac{\hbar^2}{2m_t}k_x^2+\frac{\hbar^2}{2m_l}\left(k_y+\frac{1.7\pi}{a}\right)^2+\frac{\hbar^2}{2m_t}k_z^2, \end{equation} \] \[ \begin{equation} E_{c001}=E_g+\frac{\hbar^2}{2m_t}k_x^2+\frac{\hbar^2}{2m_t}k_y^2+\frac{\hbar^2}{2m_l}\left(k_z-\frac{1.7\pi}{a}\right)^2, \end{equation} \] \[ \begin{equation} E_{c00\overline{1}}=E_g+\frac{\hbar^2}{2m_t}k_x^2+\frac{\hbar^2}{2m_t}k_y^2+\frac{\hbar^2}{2m_l}\left(k_z+\frac{1.7\pi}{a}\right)^2. \end{equation} \]Here $a = 0.543$ nm is the lattice constant, $m_t = 0.19m_e$ is the transverse electron effective mass, $m_l = 0.98m_e$ is the longitudinal electron effective mass, and $m_e = 9.11\times 10^{-31}$ kg is the mass of an electron. The valence bands consist of a light hole band, a heavy hole band, and a split-off band. All three bands have a maximum at $k=0$. The light hole band and the heavy hole band are degenerate at $k=0$ while the energy of the split-off band is $E_{so}$ lower at $k=0$. The dispersion for the light holes and the heavy holes are given approximately by, [Askerov] \[ \begin{equation} E_{v,lh}=-\frac{\hbar^2}{2m_{e}}\left(4.1k^2-\sqrt{1.21k^4+4.1(k_x^2k_y^2+k_x^2k_z^2+k_y^2k_z^2)}\right), \end{equation} \]and \[ \begin{equation} E_{v,hh}=-\frac{\hbar^2}{2m_{e}}\left(4.1k^2+\sqrt{1.21k^4+4.1(k_x^2k_y^2+k_x^2k_z^2+k_y^2k_z^2)}\right), \end{equation} \]These bands are quadratic but anisotropic. They are sometimes approximated by isotropic bands described by effective masses, \[ \begin{equation} E_{v,lh}\approx -\frac{\hbar^2k^2}{2m_{lh}}, \end{equation} \]and \[ \begin{equation} E_{v,hh}\approx -\frac{\hbar^2k^2}{2m_{hh}}. \end{equation} \]The split-off band is isotropic, \[ \begin{equation} E_{v,so}=-E_{so}-\frac{\hbar^2k^2}{2m_{so}}. \end{equation} \]For silicon, $m_{lh}= 0.16m_e$, $m_{hh}= 0.49m_e$, $m_{so}= 0.24m_e$, and $E_{so} = 0.035\,\text{eV}$. The electron density of states for silicon is shown below. At temperature $T=0$ the valence band is occupied and the conduction band is empty. At room temperature, a few electrons are thermally excited into the conduction band and they leave and equal number of holes behind in the valence band.
Near the top of the valence band, the density of states have the form:
The increase of the density of states at -0.035 eV is due to the split-off band. A simple model of the density of states near the top of the valence band is $D(E)=D_v\sqrt{E_v-E}$. By fitting a straight line to a plot of $D^2$ vs. $E$, the constant $D_v$ can be determined: $D_v = 10.17\times 10^{26}$ eV-3/2m-3.
Mechanical propertiesThe stress-strain relation for silicon is described by the tensor equation, \[ \begin{equation} \sigma_{ij}=c_{ijkl}\epsilon_{kl}, \end{equation} \]where the elements of the compliance tensor are,
$c_{11}=165.7$ GPa $=c_{xxxx}=c_{yyyy}=c_{zzzz}$, For isotropic materials like silicon, the compliance tensor elements are related to Young's modulus $E$ and Poisson's ratio $\nu$ by $$c_{11} = \frac{E(1-\nu)}{(1+\nu)(1-2\nu)},$$ $$c_{12} = \frac{E\nu}{(1+\nu)(1-2\nu)},$$ $$c_{44} = \frac{E}{2(1+\nu)}.$$Yield strength: 7 GPa When an electric field is applied to a silicon crystal, the resulting strain can be expressed as a Taylor series, \[ \begin{equation} \epsilon_{ij}=d_{ijk}E_k+Q_{ijkl}E_kE_l+\cdots, \end{equation} \]where $\epsilon_{ij}$ is the strain, $E_k$ is the electric field, $d_{ijk}$ is the reciprocal piezoelectric tensor, and $Q_{ijkl}$ is the electrostriction tensor. Silicon has inversion symmetry in its point group so the reciprocal piezoelectric tensor is zero by symmetry and the leading order term is electrostriction. Optical propertiesDielectric function
There are two conventions for dielectric function. Either it is assumed that the time dependence of $\vec{D}$, $\vec{P}$, and $\vec{E}$ is $\exp (-i\omega t)$ and the plot of the dielectric function looks as it is shown above, or it is assumed that the time dependence of $\vec{D}$, $\vec{P}$, and $\vec{E}$ is $\exp (i\omega t)$ and the imaginary part of the has the opposite sign as in the plot above. Here we will assume a time dependence of $\exp (-i\omega t)$. The data show here was digitized from D. E. Aspnes and A. A. Studna, Phys. Rev. B 27, 985-1009 (1983). Electric susceptibility
Complex conductivity \[ \begin{equation} \sigma\left(\omega \right)=-i\omega \epsilon_0 \chi_E(\omega). \end{equation} \]
The dielectric function, the electric susceptibility, and the complex conductivity are three different ways to present the same information. Usually for frequencies below about 1THz, we use the complex conductivity and for higher frequencies we use the dielectric function or the electric susceptibility. The index of refraction $n$ and the extinction coefficient $K$ \[ \begin{equation} \sqrt{\epsilon_r}= n+iK \end{equation} \] When waves travel from vacuum into some material, the frequency remains constant. A plane wave moving to the right in vaccuum has the form $\exp\left(i\left(\omega x/c -\omega t\right)\right)$ where $c$ is the speed of light in vacuum. When this wave enters some material, $c \rightarrow c/ \left(n+iK\right)$. The speed of the electromagnetic waves is smaller than the speed of light in vacuum by a factor of $n$. The extinction coeffcient describes the exponential decay of the amplitude of the electromagnetic waves. For waves propagating in the $x$-direction, the amplitude decays like $\exp\left(-\omega Kx/c \right)$.
Absorption coefficient $\alpha$ \[ \begin{equation} \alpha =\frac{2\omega K}{c} \end{equation} \]
Reflection \[ \begin{equation} R=\frac{\left(n-1\right)^2+K^2}{\left(n+1\right)^2+K^2} \end{equation} \]
[Askerov] B. M. Askerov, Electron Transport Phenomena in Semiconductors, World Scientific (1994). |
