Assuming the following potential:
The 2V_{02} is the coupling between the zeroth and second energy band, is this also the equal to the hopping parameter between any two next-nearest points in the y-direction?
Meaning that a potential lattice could look like this:
Yeah i got the same solution, however you switch V01 en V02 in your drawing (I think). I am assuming you are doing the 2020 tenta, in this case it’s a little different as The question explicity asks for the “the reciprocal lattice”.
PS:
This is just the solution we came up with so I am not 100% certain of answer
Yup, that’s right. If we use the potential and the notation above, then V_{mn} couples wavevectors that are m \overrightarrow{b_1} + n \overrightarrow{b_2} reciprocal lattice vectors apart.
Ah that clears it up, thank you both!
Shouldn’t the spacing be \frac{2 \pi}{a} and \frac{2 \pi}{b}? Or have you drawn the real space lattice?
I thought you could write V = \sum_\vec{G} V_{\vec{G}} e^{i \vec{G} \cdot \vec{r}} , with \vec{G} = m_1 \vec{b_1} + m_2\vec{b_2}. This would give \vec{G_1} = \frac{2 \pi}{a} \hat{x}, \vec{G_2} = \frac{2 \pi}{b} \hat{y} and \vec{G_3} = \frac{4 \pi}{b} \hat{y}. Reading this off I would think \vec{b_1} = \frac{2 \pi}{a} \hat{x} and \vec{b_2} = \frac{2 \pi}{b} \hat{y}, but in that case I do not see how it would change if V_{02} = 0.
b here I assumed to be the reciprocal lattice vectors
