From 2d30c1f35b4c629fe2c34ce33b8f5bb5e50ba1dd Mon Sep 17 00:00:00 2001
From: Anton Akhmerov <anton.akhmerov@gmail.com>
Date: Sat, 14 Apr 2018 19:14:50 +0200
Subject: [PATCH] use collapsed questions, provide a couple of answers
---
docs/lecture_6.md | 9 +++++++--
docs/lecture_7.md | 8 ++++----
2 files changed, 11 insertions(+), 6 deletions(-)
diff --git a/docs/lecture_6.md b/docs/lecture_6.md
index 26a065a9..7823068a 100644
--- a/docs/lecture_6.md
+++ b/docs/lecture_6.md
@@ -64,7 +64,9 @@ $$H\begin{pmatrix}\alpha \\ \beta \end{pmatrix} =
$$
Here we used $\delta p = \hbar \delta k$, and we expanded the quadratic function into a linear plus a small correction.
-**Question: can you calculate $E_0$ and the velocity $v$?**
+
+??? question "calculate $E_0$ and the velocity $v$"
+ The edge of the Brilloin zone has $k = \pi/a$. Substituting this in the free electron dispersion $E = \hbar^2 k^2/2m$ we get $E_0 = \hbar^2 \pi^2/2 m a^2$, and $v=\hbar p/m=\hbar \pi/ma$.
Without $V(x)$ the two wave functions $\psi_+$ and $\psi_-$ are independent since they have a different momentum. When $V(x)$ is present, it may couple these two states.
@@ -78,7 +80,10 @@ $$
Here the coupling strength $W = \langle \psi_+ | V(x) | \psi_- \rangle$ is the matrix element of the potential between two states. *(This where we need to apply the perturbation theory, and this is very similar to the LCAO Hamiltonian)*.
-**Question: how does our solution satisfy the Bloch theorem? What is $u(x)$ in this case?**
+??? question "how does our solution satisfy the Bloch theorem? What is $u(x)$ in this case?"
+ The wave function has a form $\psi(x) = \alpha \exp[ikx] + \beta \exp[i(k - 2\pi/a)x]$
+ (here $k = \pi/a + \delta k$). Choosing $u(x) = \alpha + \beta \exp(2\pi i x/a)$ we see
+ that $\psi(x) = u(x) \exp(ikx)$.
#### Dispersion relation near the avoided level crossing
diff --git a/docs/lecture_7.md b/docs/lecture_7.md
index a4c06fab..f4ab67aa 100644
--- a/docs/lecture_7.md
+++ b/docs/lecture_7.md
@@ -231,9 +231,8 @@ $$E_F = E_G - kT\log[N_C/(N_D-N_A)], \textrm{ for } N_D > N_A$$
and
$$E_F = kT\log[N_V/(N_A-N_D)], \textrm{ for } N_A > N_D$$
-**Question:** When is a semiconductor intrinsic, and when it is extrinsic?
-
-**Answer:** The semiconductor is intrinsic when $|N_D-N_A| \ll n_i$, so $kT \gtrsim E_G/\log[N_C N_V/(N_D-N_A)^2]$.
+??? question "When is a semiconductor intrinsic, and when it is extrinsic?"
+ By definition the semiconductor is intrinsic when $|N_D-N_A| \ll n_i$, so $kT \gtrsim E_G/\log[N_C N_V/(N_D-N_A)^2]$.
## Temperature dependence of the carrier density and Fermi level
@@ -247,7 +246,8 @@ Several noteworthy features:
* Once the temperature is sufficiently low, we expect the electrons to "freeze away" from the conduction band to the donor band, so that the donor band starts playing a role of the new valence band at $kT \ll E_G - E_D$.
* At zero temperature $E_F$ should match the donor band since it has partially occupied states. If there are no acceptors, $E_F$ would be halfway between $E_D$ and $E_G$, and if there was no doping at all it would be at $E_G/2$.
-**Exercise:** check that you can reproduce all the relevant limits in a calculation.
+!!! check "Exercise"
+ check that you can reproduce all the relevant limits in a calculation.
## Measuring band gaps
--
GitLab