This is Eric's logbook for Nano Japan project from 2012.6.3-2012.7.27.

updated 2012.6.1 by Nugraha

Daily schedule

Goal of the project.

Schedule for discussion

Time and DayMonTueWedThuFri
09:00-10:00SatoNugrahaTapsanitTatsumiNugraha
13:30-14:30SaitoSimonHasdeoSimonTapsanit

Questions and Answers

Q: About Brillouin Zone

Hey guys! So I believe this is where I should post questions. I don't have any specific questions today, but was just trying to understand Brillouin zones and Bloch's theorem, which appear a lot in Saito-sensei's book on Carbon Nanotubes.

From google I have an ok idea of what Brillouin zones are, but am still trying to figure out how they relate to the energy gap of the cell such as on page 28 of the CN book. I was also wondering how on page 47 the Brillouin zone can be a long line segment instead of a polygon.

If anybody has any quick words of advice for how to understand Bloch's theorem that would be great as well! It's a little bit intimidating and I'm not sure if there's a simple way to begin to understand it.

Thanks so much and see you guys in a couple of weeks!

Eric

Answer (by Tapsanit)

Please consider my answers one by one in order to understand the Brillouin zone.

Q: About copy machine

Can I use a copy machine or printer?

A: (not solved)

Q: About library

I would like to know where the library is and how to use it.

A: (not solved)

Report

This part is basically written by Eric-san. Any other people can add this. Here the information should be from new to old so that we do not need to scroll.

Note: Format didn't carry over from powerpoint. (Eric)

Task from Tapsanit (Bloch's theorem and reciprocal lattice)

  1. Check that the &tex(): Error! The expression contains invalid characters.; satisfies the Bloch's theorem.
    • Answer: Bloch's theorem is: \[ T_{\vec a_i}\Psi = e^{(i\vec{k}\cdot\vec{a_i})}\Psi \] Here &tex(): Error! The expression contains invalid characters.; is a translation operator: &tex(): Error! The expression contains invalid characters.;. Then, also note that &tex(): Error! The expression contains invalid characters.; is a periodic function. We have &tex(): Error! The expression contains invalid characters.; because &tex(): Error! The expression contains invalid characters.; is a lattice vector. Thus: \[ T_{\vec{a}_i} \Psi(\vec{r}) = \Psi(\vec{r} + \vec{a}_i) \] \[ = e^{(i\vec{k} \cdot (\vec{r} + \vec{a}_i))} u(\vec{r} + \vec{a}_i) \] \[ = e^{i\vec{k} \cdot \vec{a}_i} e^{i\vec{k}\cdot \vec{r}} u(\vec{r} + \vec{a}_i) \] \[ =e^{i\vec{k} \cdot \vec{a}_i} [e^{i\vec{k} \cdot \vec{r}} u(\vec{r})] \] The Bloch theorem is satisfied.
  2. Derive the unit vectors of reciprocal lattice, &tex(): Error! The expression contains invalid characters.; and &tex(): Error! The expression contains invalid characters.;, of graphene in Eq. (2.23) of CN book using &tex(): Error! The expression contains invalid characters.; and &tex(): Error! The expression contains invalid characters.; in Eq. (2.22).
    • Answer: Here we use \[ \vec{a}_i = {\rm unit~vector~of~real~lattice} \] \[ \vec{b}_i = {\rm unit~vector~of~reciprocal~lattice} \] \[ \vec{a}_i \vec{b}_i = 2\pi \] \[ \vec{a}_i \vec{b}_j = 0 \quad {\rm if} \quad i \neq j \] From Eq. (2.22) of CN book: \[
      \vec {a}_1 = \left(\frac{\sqrt{3}}{2}a,\frac{a}{2}\right), \quad
      \vec {a}_2 = \left(\frac{\sqrt{3}}{2}a,-\frac{a}{2}\right)
      \]
      	a ?_1=(?3/2 a,a/2), a ?_2=(?3/2 a,(-a)/2)
      	a ?_1?b ?_1=a_1x b_1x+a_1y b_1y=b_1x  (a?3)/2+b_1y   a/2=2?
      	a ?_2?b ?_1=	a_2x b_1x+a_2y b_1y=b_1x  (a?3)/2-b_1y   a/2=0
      	?b_1y   a/2  = b_1x  (a?3)/2
      	?b_1x  (a?3)/2+b_1x  (a?3)/2=2?
      	?b_1x=2?/(a?3)  ? ?-b_1y   a/2=0 ? b_1y=2?/a 
                         b_1=(2?/(a?3),2?/a)
      Same process gives: b_2=(2?/(a?3),(-2?)/a)

June 5 (Hasdeo teaches XX and YY. from 9:00-9:30) etc

June 4


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