Lecture 8 (EM21) -- Calculation examples of periodic structures

[@sphericalcow3143]

You are a fantastic teacher! I am in school and am taking a graduate course on condensed matter physics. This short lesson has helped me clear up a lot of confusion that has lingered the entire semester. I plan to watch more of your lectures a fantastic supplement to my class. Thank you.

[@empossible1577]

+SphericalCow Wow! You made my day! Thank you!!

[@hairuo]

Thanks for sharing this lecture, this is the most clearly explain on relationship between Brillouin zone and band diagrams I had ever seen.

[@Bhaumikpk]

Thanks for the nice lecture.

[@empossible1577]

Thank you!

[@橋詰仁人]

I'm trying to calculate band diagram of hexagonal lattice. but I can't calculate dielectric constant distribution in Fourier space in way of using FFT. please tell me how to do that!!

[@empossible1577]

You have to setup an oblique meshgrid to do this and then build your unit cell on that oblique meshgrid. This is done to be consistent with the lattice vectors. You may also be interested in the course website for Computational Electromagnetics where there are more learning resources for you. empossible.net/emp5337/ We have a tutorial on calculating convolution matrices on oblique grids, but currently that is only available in one of our paid courses. Here is a link to our least expensive course with this tutorial: empossible.thinkific.com/courses/pwem2dbands

[@wabidemeke3067]

In the lecture presented here, the unit cell of periodic structure is contains circular hole what if it is not circular(hexagonal) or it has complex shape but let's say it has the same direct lattice vector as the unit cell of periodic structure that contains circular hole, is it appropriate to sample (kx ky) only along the boundary of irreversible Brillouin zone?

[@empossible1577]

The lattice vectors only describe the symmetry of the unit cell, not the pattern within the unit cell. Therefore, no matter what, all unit cells of the same symmetry will have the same Brillouin zone. Now, the IBZ can be different depending on the pattern within the BZ. If the unit cell has left/right symmetry, like a heart, then you can cut your BZ in half. A heart does not not have up/down symmetry or any additional symmetry so the IBZ would be half the BZ. A circle, as you know allows the IBZ to be a little triangle within the BZ. The IBZ does depend on the pattern within the unit cell. Marching around the perimeter of the IBZ can be said not matter what the pattern is within the unit cells, but the shape of the IBZ will change. Did I make sense?

[@wabidemeke3067]

@@empossible1577 Thank you for your nice response! Just to make sure I got this right, so for example you mentioned (heart like shape) the corresponding IBZ is half of BZ, therefore, sampling (kx ky) around boundary (perimeter) of half of BZ would be enough? or we have to sample meshgird (kx ky) of over half BZ(IBZ)? Since I am planning to run multiple simulations for different shapes, I have to reduce my simulation time by sampling as small as possible(kx ky).

[@empossible1577]

@@wabidemeke3067 Each shape can have a different IBZ. You may need to just always do the full BZ. I suppose you could have your code check for symmetries and then calculate different IBZs. Whenever I have done this, I just calculated throughout the entire BZ. Yes, that wasted some computation time. What numerical method are you using for calculating the bands?

[@wabidemeke3067]

@@empossible1577 I am using Comsol Multiphysics and I am actually working on phononic crystals not photonic crystals. I have attached a link below on band gap calculation by Dr Nagi Elabbasi using Comsol Multiphysics. www.comsol.com/blogs/modeling-phononic-band-gap-materials-and-structures/

[@empossible1577]

@@wabidemeke3067 Very good. I noticed you are watching an older video. In fact, I have actually greatly improved this section of material. Checkout "Solid State Electromagnetics" under Topic 4 at the following site: empossible.net/academics/emp6303/

[@noahnguyen6924]

at 21:46, i don't get the formula you use to find reciprocal lattices T1 and T2. what is [t2,y -t2,x]??

[@empossible1577]

Noah Nguyen The lower case letters (t1 and t2) are the primitive translation vectors of the direct lattice. It is from these that you calculate the primitive translation vectors (T1 and T2) of the reciprocal lattice.

[@taolin4569]

Could you explain how to resolve the band crossing problem as you mentioned at 12:35 ?

[@empossible1577]

This is something I have spent some time studying. I don't have an answer other than I think it may be ambiguous if the bands cross or not.

[@taolin4569]

@@empossible1577 Thank you for quick answer. I guess this have something to do with the mode shapes (eigenvectors). It's natural to think that all the eigenvectors should have the same shape if they belong to the same band.

[@empossible1577]

@@taolin4569 That was my thought as well so I created an animation where I visualized the Bloch mode of two bands as they approached and passed where they cross. The Bloch modes abruptly change where the bands touch making it impossible for me to determine continuity.

[@enstien100]

Dear, Can you share matlab code ?

[@empossible1577]

Let me point you to some resources where I teach all of these different numerical methods and even provide MATLAB codes. A great place to get started is my Computational Methods course. Specifically, the finite-difference subjects taught in Topics 6 and 7. empossible.net/academics/emp4301_5301/ After this, you can move into learning either finite-difference frequency-domain (FDFD) or finite-difference time-domain (FDTD). I just published a book with MATLAB codes that teaches FDFD for beginners. The book shows how to simulate a very wide variety of devices. Here is a link to the book website: empossible.net/fdfdbook/ If you wish to learn FDTD, I have created some excellent online courses for that. Here is a link to a video showing the contents of these courses. kzbin.info/www/bejne/q3PMoaV_g52anLs Here is a link to get the courses. The entire first half of the first course is completely free to see if you like it. empossible.thinkific.com/collections?category=FDTD-in-MATLAB As you get more advanced, here is a link to my Computational Electromagnetics course: empossible.net/emp5337/ Hope this helps!

[@somayehnodehi3657]

Nice lecture. To obtain bandstructure using FDTD method, we have to locate the dipoles far enough from the pml boundary. I say it because they cause some noise in a bandstructure result but I dont know why? Do you have any justification for it? Thanks

[@empossible1577]

Unless you are doing something special, you should not need a PML in FDTD to calculate the band structure. Instead, you should use periodic boundary conditions on all edges. Calculating bands with FDTD is a bit tricky. I talk about that in Lecture 10 here: emlab.utep.edu/ee5390fdtd.htm You may want to consider a different numerical technique for calculating band diagrams. Take a look at finite-difference frequency-domain and the plane wave expansion method here: emlab.utep.edu/ee5390cem.htm

[@dalilarobledodebasabe191]

What program did you use to get the band diagrams? And thanks for the lectures, they're super clear and helpful.

[@empossible1577]

Everything here was done in MATLAB. The band diagrams are usually calculated using the plane wave expansion method (PWEM). This technique is covered in course prior to this one, Computational Electromagnetics. Here is the course website for CEM: emlab.utep.edu/ee5390cem.htm

[@dj2544]

Could anybody write me the periodic boundary condition for hexagonal lattice, like w(x,y)=w(x+a,y)e^ik_xa,w(x,y)=w(x,y+a)e^ik_ya for square lattice

[@empossible1577]

The lectures for the plane wave expansion method and Fourier space methods have been completely revised in order to handle any symmetry. Here is a link to the official course website: empossible.net/academics/emp5337/

[@alaaal-jobory6994]

Thanks for the nice lecture could you explain why K = 1/3T1+1/3T2 in 25

[@empossible1577]

This just comes down to geometry. If you draw the Brillouin zone and work through the geometry to calculate the positions of the key points, this equation falls out. There really is nothing more to it. BTW, this section of the course has been considerably revised. I recommend accessing the material through the course website: empossible.net/academics/emp6303/ The revised version of this video is now in Topic 4. Hope this helps!

[@alaaal-jobory6994]

@@empossible1577 Thank you so much

[@mohibmohib6442]

I always noticed that the points (111) became (kx=ky=kz=pi/a) and X(011) becam (Kx=Ky=pi/a , Kz=0) Is there a rule for transferring them..? please

[@empossible1577]

I think you are referring to the Miller indices? A reciprocal lattice vector is an integer multiple of the primitive reciprocal lattice vectors. The set of integers for that lattice vector are the Miller indices of the plane described by the lattice vector. This may be the connection you are looking for. kx, ky and kz are essentially the components of the lattice vector.

[@VinodKumariist]

Is this Brillouin Zone calculation is valid for non symmetric lattice also??

[@empossible1577]

+Vinod Kumar That depends on what you mean by non symmetric. There are pseudo-periodic lattices that have some randomness to their geometry, but otherwise appear to repeat. Some examples are penrose tilings and such. Those do have a BZ although it is derived statistically due to the randomness. If it does not repeat at all, like in a functionally graded lattice, the lattice does not have a BZ. It still may help to think about localized portions of the lattice having a local BZ although it would only be an approximation.

[@BinhNguyen-if5yd]

Why we use band here instead of curves?

[@empossible1577]

For ordinary homogeneous materials, there is just a single dispersion curve, or "light line." However in periodic structures, there arise multiple solutions that have multiple lines. Each of these is a band.

[@BinhNguyen-if5yd]

why each line form a band? If we see graphic at 10:51 we only see lines but if we see each of them over the k-axis so it can be said each line form a band ?

[@empossible1577]

This diagram shows a bunch of discrete solutions plotted as a function of the Bloch wave vector. If done at high enough resolution, the discrete solutions fall along lines that we call bands. This comes from solid state theory and is why we talk about bands in semiconductors. If the discrete solutions jumped around more randomly, that would imply that small changes in frequency produce wild changes in the Bloch modes and that would not make much sense. There are multiple bands because there are multiple solutions to the eigen-value problem. This is the same reasoning behind why there are multiple solutions for modes in a waveguide. In a photonic crystal, we can keep calculating more and more bands simply by moving to a higher frequency.

[@GBabuu]

Greetings Dr. Rumpf. THIS IS THE BEST LECTURE I HAVE EVER COME ACROSS.... The material were cogently presented, well illustrated and with so much ease... I can't thank you enough for this video sir. I have two questions regarding unit cells? 1.) If i use Finite Element approach in stead of PWEM. I will need to mesh the unit cell (that has the matrix and inclusion). My question is; should i mesh the physical unit cell, or the unit cell in a reciprocal space? If I have to mesh unit cell in a reciprocal space, how would i provide the dimensions for the inclusion, supposedly its a circular inclusion like the one you used to illustrate in this lecture 2.) My second question is, for 1D periodicity, would the two options for the choice of unit cells give exact same band structures? Thank you in advance sir! And again, thank you very very much for this video lecture!

[@empossible1577]

Thank you very much! I am so glad the videos are helping you. 1) For FEM, you will mesh the unit cell in real-space (i.e. xyz). There are methods that mesh in reciprocal space, but I do not think that is what you are talking about or would want to do. 2. What two options do you have for the unit cell? A unit cell is the smallest block of something that can be stacked to recreate that something without any overlaps or voids. Given the boundaries of the unit cell, you can slide that through your lattice and grab whatever pattern is enclosed within those boundaries without any problem or change to the band structure. Maybe this is what you mean?

[@GBabuu]

Thank you for your timely response Dr. Rumpf 1) Yes sir, my project is to use FEM to solve the Linear Eigenvalue problem for the "omega given k" approach as well as solving Quadratic Eigenvalue problem for the "k given omega" approach whereas the later gives the real and imaginary k for the complex band structure.... Now since, the traditional (or rather conventional) way of computing the dispersion relations (band structures) uses First brillouin zone, which conveniently, gives the band structures that runs from through high symmetry points... I was just wondering how would i be able to replicate these points if my unit cell is meshed in physical space (xyz) 2.) Yes sir, by choice of unit cell, I meant something like 2:30 in this video lecture on the top. if you use either of the unit cells, weather the band structures are going to be identical I could subscribe two times if i could in your channel, what a fantastic lecturer and a person you are. From the bottom of my heart, Thank you sir

[@empossible1577]

@@GBabuu Thank you again!! 1. Build your mesh in real-space and then build your matrix. The information from your Bloch wave vector goes into the periodic boundary conditions to enforce whatever phase condition is necessary at the edges. I cover band diagrams with finite-difference frequency-domain (another real-space method) in Lecture 15 here: emlab.utep.edu/ee5390cem.htm 2. Both unit cell descriptions describe the same lattice so both will lead to the same band structure. In fact, this is a great way to test your code. If you get a different band structure, something it wrong. Most likely you have not built the lattice correctly in one of the descriptions. If you calculate the eigen-vectors, you may notice the raw numbers are different only because the fields are shifted along with the unit cell. They should look the same if plotted over several unit cells. Keep at it!

[@GBabuu]

@@empossible1577 Dear Dr. Rumpf. I'm reading your response with widest smile on my face. How ironic is it, you mentioned that, the way to know my code is correct, is by using two unit cell descriptions of the same lattice with anticipation to get same exact solutions....because that's what my project advisor insisted as well. And its also amazing that you mentioned that the eigenvalues and modes, will not be of same exact numbering because of the phase shift. THIS IS SUCH AN IMPORTANT INFORMATION because I wondered about this too. I'm now watching lecture 15. YOU SIR, ARE SUCH A JOY... GOD BLESS YOU!!!!!!

[@empossible1577]

@@GBabuu Ha ha. You have giving me a big head! Thx!

[@jaimetpearl]

Dear Dr. Rumpf, I've been trying to simulate the band structure of a 2D hexagonal lattice using the PWEM as described in your lectures. I'm sure my reciprocal lattice vectors and symmetry points are correct; however, when defining the device, I ran into some problems since the unit cell is not Cartesian but an hexagon. In order to work around this issue I defined my unit cell as a body centered structure as follows: a = 1; b = sqrt(3) * a; Nx = 500; Ny = sqrt(3) * Nx; P = 5; Q = round( P * sqrt( 3 ) ); ER0 = ( X.^2 + Y.^2 ) > r^2; ER1 = ( (X-a/2).^2 + (Y-b/2).^2 ) > r^2; ER2 = ( (X+a/2).^2 + (Y-b/2).^2 ) > r^2; ER3 = ( (X-a/2).^2 + (Y+b/2).^2 ) > r^2; ER4 = ( (X+a/2).^2 + (Y+b/2).^2 ) > r^2; ER = ER0 + ER1 + ER2 + ER3 + ER4 - 4; ER = 1.0 + ( er - 1.0 ) * ER; However, once I run the simulation the band structure doesn't look right. Could you comment whether the way I'm defining the device is correct or if I should use a different unit cell?. Thank you very much and I really appreciate all the effort you have put into these fantastic lectures!.

[@empossible1577]

Your code did not have everything I needed to verify what you are doing exactly. I calculated X, Y, er, and r based on what I think you are doing. Overall, the approach is correct. The simulate a hexagonal lattice the trick will be how you calculate the Fourier coefficients. There are two approaches. First, you can perform a coordinate transform and then use a standard FFT. For this approach, most people cite [Ward, A. J., and John B. Pendry. "Refraction and geometry in Maxwell's equations." Journal of modern optics 43.4 (1996): 773-793.]. The second approach is to perform the numerical integration of the equations on slide 8 of Lecture 18 here: emlab.utep.edu/ee5390cem.htm

[@jaimetpearl]

Thank you very much for your answer!. My apologies for not including the whole code, but I decided not to do so, since it is basically based on your code for the square lattice. The only modifications I made was the way to construct the device, the reciprocal vectors definitions (T1 and T2), and the symmetry points (G, M and K). I will double check the Fourier coefficients since I only changed Bx and By (through the definition of BETA). Thank you also for the reference you suggest in your reply!

[@elmaftahsofia9564]

@@empossible1577 can you explain me how to calculate the frequency of fonctionnement for 3D structure

[@empossible1577]

@@elmaftahsofia9564 fonctionnement? Do you mean the operating frequency? That is up to you as the engineer to choose the operating frequency. Once you have that, you use the normalized frequency from your band diagram to scale the lattice so that it operates at your design frequency. For example, suppose you have a band gap centered at wn = 0.5. Now suppose you want at lattice that has its bandgap at a frequency of 20 GHz. First, the free space wavelength of 20 GHz is 1.5 cm. The lattice spacing should be a = 0.5(1.5 cm) = 0.75 cm.

[@elmaftahsofia9564]

@@empossible1577 thank you

[@amritaarun5625]

Can you point me to lecture on the calculation of band-gap eigen-values.

[@empossible1577]

Here is a link the official course website: empossible.net/academics/emp5337/ The section on the plane wave expansion method is what you are looking for.