3 Introduction to the Quantum Theory of Solids

3.1 Allowed and Forbidden Energy Bands

3.1.3 The $k$-Space Diagram

the k comes from that all one-electron wave functions, for problems involving periodically varying potential energy functions, must be of the form $\psi(x)=u(x)e^{jkx}$

\[ \begin{gather*} P'\frac{\sin\alpha a}{\alpha a}+\cos \alpha a=\cos ka\\ P'=\frac{mV_0ba}{\hbar^2} \end{gather*} \]

Equation gives the relation between the parameter $k$ total energy $E$ (through the parameter $\alpha$), and the potential barrier $bV_0$.

We assume that $V_0=0$, then we acquire

\[ \begin{gather*} \cos \alpha a=\cos ka\\ or\quad \alpha =k \end{gather*} \]

$\alpha$ is given by$\sqrt{\frac{2mE}{\hbar^2}}=\frac{p}{\hbar}=k$, so we obtain the relationship between $k$ and $E$

\[ E=\frac{k^2\hbar^2}{2m} \]

Now, we consider the relation between $E$ and $k$ form Equation for the particle in the single-crystal lattice.

define the left side of the Equation as

$$ f(\alpha a)=P'\frac{\sin\alpha a}{\alpha a}+\cos\alpha a $$ to the right side, we also have,

$$ f(\alpha a)=\cos ka $$

the right side can also be written as $\cos ka=\cos(ka+2n\pi)=\cos(ka-2n\pi)$

then we gain

3.2 Electrical Conduction In Solids

3.2.1 The Energy Band and the Bond Model

3.2.2 Drift Current

the drift current density

\[ \begin{gather*} J=qNv_d\qquad A/cm^2\\ or\quad J=q\sum^N_{i=1}v_i \end{gather*} \]

if a force is applied to a particle and the particle moves.

\[ dE=Fdx=Fv\cdot dt \]

3.2.3 Electron Effective Mass

\[ F_{\text{total}}=F_{\text{ext}}+F_{\text{int}}=ma \]

and we add $m^*$ as effective mass

\[ F_{ext}=m^*a \]

then back to the Equation, we obtain,

\[ \begin{gather*} \frac{dE}{dk}=\frac{\hbar^2k}{m}=\frac{\hbar p}{m}\\ \frac1{\hbar}\frac{dE}{dk}=\frac{p}m=v \end{gather*} \]

and, if we take the second derivative of $E$ with respect to $k$

\[ \frac1{\hbar^2}\frac{d^2E}{dk^2}=\frac1{m} \]

apply the Newton's classical equation of motion:

\[ \begin{align*} F&=ma=-eE\\ a&=\frac{-eE}{m} \end{align*} \]

对$k$和$E$的关系进行近似，即

\[ E-E_c=C_1(k)^2 \]

since that,

\[ \begin{gather*} \frac{d^2E}{dk^2}=2C_1\\ \frac1{\hbar^2}\frac{d^2E}{dk^2}=\frac{2C_1}{\hbar^2}=\frac1{m^*} \end{gather*} \]

then the acceleration,

\[ a=\frac{-eE}{m^*_n} \]

$m^*_N$ denotes the effective mass of the electron

3.2.4 Concept of the Hole

\[ \begin{gather*} J=-e\sum_{i(filled)}v_i\\ J=-e\sum_{i(total)}v_i+e\sum_{i(empty)}v_i\\ \text{(take into account a very large number of states)} \end{gather*} \]