checkin for sync to d81

[lectures/latex.git] / solid_state_physics / tutorial / 2n_01s.tex

diff --git a/solid_state_physics/tutorial/2n_01s.tex b/solid_state_physics/tutorial/2n_01s.tex
index b16cf30..38566a8 100644 (file)
--- a/solid_state_physics/tutorial/2n_01s.tex
+++ b/solid_state_physics/tutorial/2n_01s.tex
@@ -92,7 +92,7 @@ Use: $x=(\epsilon - E_{\text{c}})/k_{\text{B}}T$
 $\Rightarrow$
 $n=\frac{1}{2\pi^2}(\frac{2m_nk_{\text{B}}T}{\hbar^2})^{3/2}
  \exp(-\frac{E_{\text{c}}-\mu_{\text{F}}}{k_{\text{B}}T})
- \underbrace{\int_0^{\infty}x^{1/2}e^{-x}dx}_{=1/2\sqrt{\pi}}=
+ \underbrace{\int_0^{\infty}x^{1/2}e^{-x}dx}_{=\frac{\sqrt{\pi}}{2}}=
  \underbrace{2(\frac{m_nk_{\text{B}}T}{2\pi\hbar^2})^{3/2}}_{=N_{\text{c}}}
  \exp(-\frac{E_{\text{c}}-\mu_{\text{F}}}{k_{\text{B}}T})=
  N_{\text{c}}\exp(-\frac{E_{\text{c}}-\mu_{\text{F}}}{k_{\text{B}}T})$
@@ -101,7 +101,7 @@ $\forall \epsilon$ of states within conduction band:
 $\mu_{\text{F}}-\epsilon >> k_{\text{B}}T$
 $\Rightarrow$
 $1-f(\epsilon,T)=
- 1-\frac{1}{\exp(\frac{\mu_{\text{F}}-\epsilon}{k_{\text{B}}T})+1}\approx
+ 1-\frac{1}{\exp(\frac{\epsilon-\mu_{\text{F}}}{k_{\text{B}}T})+1}\approx
  \exp(\frac{\epsilon-\mu_{\text{F}}}{k_{\text{B}}T})$\\
 Parabolic approximation:
 $D_v(\epsilon)=\frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2}(E_v-\epsilon)^{1/2}$
@@ -110,14 +110,14 @@ $p=\int_{-\infty}^{E_{\text{v}}}D_v(\epsilon)(1-f(\epsilon,T))d\epsilon\approx
  \frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2}
  \exp(-\frac{\mu_{\text{F}}}{k_{\text{B}}T})
  \int_{-\infty}^{E_{\text{v}}}(E_v-\epsilon)^{1/2}
- \exp(-\frac{\epsilon}{k_{\text{B}}T})d\epsilon$\\
+ \exp(\frac{\epsilon}{k_{\text{B}}T})d\epsilon$\\
 Use: $x=(E_{\text{v}}-\epsilon)/k_{\text{B}}T$
      $\Rightarrow\epsilon=E_{\text{v}}-xk_{\text{B}}T$ and
-     $d\epsilon=-k_{\text{B}}Tdx$\\
+     $d\epsilon={\color{red}-}k_{\text{B}}Tdx$\\
 $\Rightarrow$
 $p=\frac{1}{2\pi^2}(\frac{2m_pk_{\text{B}}T}{\hbar^2})^{3/2}
  \exp(\frac{E_{\text{v}}-\mu_{\text{F}}}{k_{\text{B}}T})
- \underbrace{\int_0^{\infty}x^{1/2}e^{-x}dx}_{=1/2\sqrt{\pi}}=
+ \underbrace{\int_{{\color{red}0}}^{{\color{red}\infty}}x^{1/2}e^{-x}dx}_{=\frac{\sqrt{\pi}}{2}}=
  \underbrace{2(\frac{m_pk_{\text{B}}T}{2\pi\hbar^2})^{3/2}}_{=N_{\text{v}}}
  \exp(\frac{E_{\text{v}}-\mu_{\text{F}}}{k_{\text{B}}T})=
  N_{\text{v}}\exp(\frac{E_{\text{v}}-\mu_{\text{F}}}{k_{\text{B}}T})$