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Using lattice symmetry to reduce the number of simulations

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For OLEDs with patterning that have square or hexagonal symmetry, one can take advantage of this lattice symmetry and reduce the number simulations that are required.

PC with square symmetry

If we consider a rectangular lattice structure, we might want to consider 4x4 dipole locations across the unit cell (more may be required, but we have found this to work quite well). For each dipole location, we need 3 simulations for the 3 dipole orientations (x/y/z). In principle, this means that we need 4x4x3=48 simulations. However, in this case, we can follow the steps below to apply symmetry and reduce the total number of simulations to 7 for the patterned device, and 2 for the unpatterned device.

 

Step 1: Starting with the unit cell, we have the 4x4 dipole locations that may be necessary to fully characterize the OLED structure.

 

oled_setup1_zoom25

 

Step 2: We recognize that only 1/4 of the unit cell needs to be simulated because the structure has symmetry across the x and y planes, the remaining simulations can be reproduced using symmetry operations. Note that if your unit cell is rectangular, or if it does not have 45 degree rotational symmetry, one will not be able to reduce the 1/4 unit cell any further (ie. this leads to 4 dipoles locations x 3 dipole orientations = 12 simulations).

oled_setup2_zoom25

 

Step 3: Since the unit cell is square and has 45 degree rotational symmetry, we can further reduce the number of simulations to 1/8 of the unit cell so that only 3 dipole locations are required.

 

oled_setup3_zoom25

 

Step 4: Lastly, we recognize that for the 2 locations along the diagonal, the dipole oriented along the y axis can be determined by the results from the dipole oriented along the x axis. This means that we can avoid simulating both x and y oriented dipoles.

 

In summary, we now have 3 dipole locations to simulate, with 2 simulations for the two dipole along the diagonal (x/z orientations) and 3 simulations for the third dipole (x/y/z orientations). This means that we have reduced the total number of simulations from 48 to 7 for the patterned device.

 

PC with hexagonal symmetry

As the above discussion for the PC lattice with square lattice, we can also use a similar approach for an OLED with hexagonal PC symmetry, and reduce the total number of simulations from 48 to 6. The steps are outlined below:

 

Step 1: In the case of a hexagonal lattice, the unit cell is given by a parallelogram. In order to understand how to apply symmetry, we divide the unit cell into 8 triangular regions as shown below:

 

oled_hex_setup1_zoom113

 

Step 2: Using the fact that the lattice has 60 degree rotational symmetry, this can be reduced to regions A and 1 (shaded regions above).

 

Step 3: We further recognize the symmetry along the 30 degree line, and further reduce regions A and 1 to A' and 1'. This means that the result of a dipole place in an arbitrary location in 1'' can be determined from the results for a dipole in the same relative position in 1'.

oled_hex_setup2_zoom65

 

Step 4: Using the results from steps 2 and 3, this means that the result of a dipole at position X in 1' can be used to determine the result of a dipole in the positions "O":

 

oled_hex_setup3_zoom50

 

And the result of a dipole at position X in A' can be used to determine the result of a dipole in the positions "O":

 

oled_hex_setup4_zoom114

 

We have, therefore, reduced the number of dipole locations from 16 to 2, which means that we have reduced the total number of simulations from 48 to 6.

 

Step 5 (optional): Here, we must recognize that the unit cell may also be given by a parallelogram shown below

 

oled_hex_setup5_zoom48

 

In order to take the 2 types of unit cells into account, one can average the far-fields from these 2 unit cells. No extra simulations are required to do this since the results for one type of unit cell can be completely determined by the results from the other. However, if enough points are simulated within one orientation of the unit cell, this step over averaging over different orientations is not necessary and may not improve the results.

 

Note: even though we were able to reduce the number of simulations significantly, it is likely that 2 dipole locations (in 1'+A') will not be enough for an accurate simulation, and one should run a convergence test to make sure of this.

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