Boltzmann approximation

The table below gives the contribution of electrons in intrinsic semiconductors and insulators to some thermodynamic quantities. These results where calculated in the Boltzmann approximation where it is assumed that the chemical potential lies in the band gap more than 3k_BT from the band edge. The electronic contribution to the thermodynamic quantities are usually much smaller than the contribution of the phonons and thus the electronic components are often simply ignored.

	1-d	2-d	3-d
Density of states	$D(E) = \begin{cases} \frac{D_v}{\sqrt{E_v-E}}, & \mbox{for } E\lt E_v \\ 0, & \mbox{for } E_v\lt E\lt E_c \\ \frac{D_c}{\sqrt{E-E_c}}, & \mbox{for } E_c \lt E \end{cases}$	$D(E) = \begin{cases} D_v, & \mbox{for } E\lt E_v \\ 0, & \mbox{for } E_v\lt E\lt E_c \\ D_c, & \mbox{for } E_c \lt E \end{cases}$	$D(E) = \begin{cases} D_v\sqrt{E_v-E}, & \mbox{for } E\lt E_v \\ 0, & \mbox{for } E_v\lt E\lt E_c \\ D_c\sqrt{E-E_c}, & \mbox{for } E_c \lt E \end{cases}$
Density of states $m_e^$ and $m_h^$ are 'density of states' effective masses	$D(E) = \begin{cases} \frac{1}{\hbar\pi}\sqrt{\frac{2m^_h}{E_v-E}}, & \mbox{for } E\lt E_v \\ 0, & \mbox{for } E_v\lt E\lt E_c \\ \frac{1}{\hbar\pi}\sqrt{\frac{2m^_e}{E-E_c}}, & \mbox{for } E_c \lt E \end{cases}$	$D(E) = \begin{cases} \frac{m_h^}{\hbar^2\pi}, & \mbox{for } E\lt E_v \\ 0, & \mbox{for } E_v\lt E\lt E_c \\ \frac{m_e^}{\hbar^2\pi}, & \mbox{for } E_c \lt E \end{cases}$	$D(E) = \begin{cases} \frac{(2m_h^)^{\frac{3}{2}}}{2\pi^2\hbar^3}\sqrt{E_v-E}, & \mbox{for } E\lt E_v \\ 0, & \mbox{for } E_v\lt E\lt E_c \\ \frac{(2m_e^)^{\frac{3}{2}}}{2\pi^2\hbar^3}\sqrt{E-E_c}, & \mbox{for } E_c \lt E \end{cases}$
Density of states $N_v$ and $N_c$ are effective densities of states	$D(E) = \begin{cases} N_v(300)\sqrt{\frac{2}{300\pi k_B(E_v-E)}}, & \mbox{for } E\lt E_v \\ 0, & \mbox{for } E_v\lt E\lt E_c \\ N_c(300)\sqrt{\frac{2}{300\pi k_B(E-E_c)}}, & \mbox{for } E_c \lt E \end{cases}$	$D(E) = \begin{cases} \frac{N_v(300)}{300k_B}H(E_v-E), & \mbox{for } E\lt E_v \\ 0, & \mbox{for } E_v\lt E\lt E_c \\ \frac{N_c(300)}{300k_B}H(E-E_c), & \mbox{for } E_c \lt E \end{cases}$	$D(E) = \begin{cases} \frac{2N_v(300)}{\sqrt{\pi}}\left(\frac{1}{300k_B}\right)^{\frac{3}{2}}\sqrt{E_v-E}, & \mbox{for } E\lt E_v \\ 0, & \mbox{for } E_v\lt E\lt E_c \\ \frac{2N_c(300)}{\sqrt{\pi}}\left(\frac{1}{300k_B}\right)^{\frac{3}{2}}\sqrt{E-E_c}, & \mbox{for } E_c \lt E \end{cases}$
Density of electrons in the conduction band $n=\int\limits_{E_c}^{\infty}D(E)f(E)dE$	$n=\sqrt{\frac{m_e^*k_BT}{\hbar^2\pi}}\exp\left(\frac{\mu-E_c}{k_BT}\right)\quad\text{m}^{-1}$ $=N_c(T)\exp\left(\frac{\mu-E_c}{k_BT}\right)$
Density of electrons in the conduction band $n=\int\limits_{E_c}^{\infty}D(E)f(E)dE$
Density of holes in the valence band
Law of mass action
Intrisic carrier density
Chemical potential Set n = p, solve for μ
Internal energy density
Helmholtz free energy f = u - Ts
Specific heat
Entropy

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<table align='center'><tr><td><ul><li>abs(x) - absolute value</li><li>acos(x) - inverse cosine</li><li>acosh(x) - inverse hyperbolic cosine</li>
<li>asin(x) - inverse sine</li><li>asinh(x) - inverse hyperbolic sine</li><li>atan(x) - inverse tangent</li><li>atanh(x) - inverse hyperbolic tangent</li><li>cos(x) - cosine</li><li>cosh(x) - hyperbolic cosine</li>
<li>exp(x) - et</li></ul></td>
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<table align='center'><tr><td><ul><li>cross(A,B) - cross product</li><li>dot(A,B) - scalar product of two vectors </li><li>dot(a,B) - scalar a times vector B</li><li>length(A) - length of a vector</li></ul></td><td valign="top"><ul><li>unit(A) - unit vector in direction of A</li><li>vadd(A,B) - vector addition A + B</li><li>vsub(A,B) - vector subtraction A -B</li></ul></td></tr></table>

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//For Silicon kB = 1.380649E-23; // [J/K] Boltzmann's constant hbar = 1.054571817E-34; // [J s] reduced Planck's constant e = 1.602176634E-19; // [C] elementary charge pi = 3.141592653589793; Dc = 6.874E21; // [eV^(-3/2) cm^(-3)] Constant describing the density of states at the conduction band edge Dv = 3.8775E21; // [eV^(-3/2) cm^(-3)] Constant describing the density of states at the valence band edge DcSI = Dcpow(e,-3/2)1E6; //[J^(-1) m^(-3)] DvSI = Dvpow(e,-3/2)1E6; //[J^(-1) m^(-3)] mh = 0.5pow(2pipihbarhbarhbarDvSI,2/3); // [kg] density of states effective hole mass me = 0.5pow(2pipihbarhbarhbarDcSI,2/3); // [kg] density of states effective electron mass Nc300 = (2Dc/sqrt(pi))pow(kB300/e,3/2); // Effective density of states of the conduction band at 300 K Nv300 = (2Dv/sqrt(pi))pow(kB300/e,3/2); // Effective density of states of the valence band at 300 K T = 300; // temperature [K] Eg = 1.166-4.73E-4TT/(T+636); //[eV] mu = Eg/2 + kBTlog(Dv/Dc)/e; // [eV] chemical potential ni = sqrt(0.25piDcDvpow(kBT/e,3)exp(-eEg/(kBT))); // [1/cm^3] intrinsic carrier concentration ni=n=p u = (sqrt(2pi)/(2pipihbarhbarhbar))pow(memh,3/4)exp(-eEg/(2kBT))pow(kBT,3/2)(eEg + 3kBT); // [J/m^3] internal energy density f = -(sqrt(2pi)/(pipihbarhbarhbar))pow(memh,3/4)exp(-eEg/(2kBT))pow(kB*T,5/2); // [J/m^3] Helmholtz free energy density
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