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## Piece-Wise Constant Potential Barriers Tool

Transmission and the reflection coefficient of a five, seven, nine, eleven and 2n-segment piece-wise constant potential energy profile

Version **1.2.8** - published on 21 Jul 2014

doi:10.4231/D3R20RX39 cite this

This tool is closed source.

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#### Abstract

Detailed description of the physics that needs to be understood to correctly use this tool and interpret the results obtained, is provided in the reading materials listed below:

Open Systems
Double-Barrier Case Explained

Exercises that illustrate the importance of quantum-mechanical reflections in state of the art devices and the resonance width dependence upon the geometry in the double-barrier structure that is integral part of resonant tunneling diodes are given below:

Quantum-Mechanical Reflections
Quantum-Mechanical Reflections in Nanodevices
Double-Barrier Structure

The formation of bands in periodic potentials and how the width and the number of the energy bands varies by varying the geometry of the n-well potential is illustrated via the following homework assignments:

From one well, to two wells, to five wells, to periodic potentials
Bands as a function of the geometry of the n-well potential

One can also use this tool to calculate the transmission coefficient through barriers that are approximated with piece-wise constant segments.

Tunneling through triangual barrier encountered in Schottky contacts

One can also use this tool to test the validity of first-order and second order stationary perturbation theory.

Application of stationary perturbation theory example

Improvements / modifications in subsequent releases:

- 1.2 – the energy and transmission coefficent axis are exchanged, so the resonance peaks now line up with the spatial resonances in the barrier structure.

- 1.2 – bug-fix: transmission through a single barrier can be simulated now in the “n” barrier case. The code no longer provides an empty output.

- 1.2 – the adaptive energy refinement was improved through a different algorithm. The tool no longer utilizes the Matlab built-in adaptive integration routine but an adaptive resonance finding and grid refinement technique as used in the NEMO1D tool or the Resonant Tunneling Diode Tool.

- 1.2 – The single barrier case has been corrected and should be functional.

- 1.2 – The tool now has a progress update for the adaptive resonance finding.

- 1.4- The tool now has the tight-binding Green’s function based formalism built into it. This will enable the user to make a comparison between the Transfer matrix method and the single band tight-binding calculation.
- 1.5 – spatially varying effective masses are introduced.
- 1.6 – Fixed a plotting problem for very small numbers that cannot be properly represented in Rappture.
- 1.7 – the effective mass treatment in the tight binding approach has been corrected. The Transfer matrix approach still appears to have some problems when masses are varied across the device.
- 1.7 – the natural lattice constant is set to 0.5 for the tight binding calculation such that there is no rounding of the barrier and well heights which will result in deviations from the Transfer matrix method. Now if there is no effective mass variation the two methods give virtually the same results for the default structures.
- 1.8 – the effective mass variation in the Transfer Matrix approach is now properly implemented. Tight Binding and Transfer Matrix method now deliver virtually the same result.
- 1.8 – proper tool label in the effective mass assignment window.
- 1.9 – A problem pertaining to the display of the geometry adjustment in the text log when a tight-binding calculation is called, has been corrected. Rappture plots remain unaffected.
- 2- Composite plots have now been introduced. Potential profile, transmission data and band-structure are shown in combination on the same plot.
- 2.1-Composite plots now have resonances. X axis is normalized to improve transition from one slider element to another.
- 2.1- Transmission and reflection data zoom into not more than seven orders below one.
- 2.1- Peaks in transmission are now reported as resonances above the highest barrier also.
- 2.1- Outputs rearranged to make them easier to locate.
- 2.1- The transmission co-efficient at zero energy is corrected for the case of a flat potential profile. It goes to one for transfer matrices, which can also be seen analytically.
- 2.2- Transmission defined as a real quantity to avoid small imaginary values which might cause errors in plotting.
- 2.2- An error in the tight-binding calculation pertaining to the potential profiles affect on the BCs has been corrected.
- 2.2- Bulk bandstructure now displayed as an output.
- 2.2- The adaptive mesh algorithm was refined to significantly reduce the number of energies needed in cases of very sharp resonances. For example in the

case of 30 barriers the number of energy nodes was reduced by over half from around 12,000 points to just over 5,000 points. The computation time was reduced by around 50% in those cases. In the case of small number of barriers the reduction of number of nodes is slightly less and the mesh refinement algorithm itself requires more time. The code slows down by about 10-20%. Those compute times are within 2-3 seconds, so the slow-down is not that significant.

- 2.3- Introduced Local Density of States plot for the tight-binding model.
- 2.3- Wave-functions are now calculated for resonances for the tight-binding model.
- 2.3- Bulk Bandstructure plots are generated for each region specified in the geometry.
- 2.3- A problem pertaining to slow rappture plotting, because of a large number of wave-functions(each having a separate curve for each peak), has been avoided. The user now sees one scatter plot when the device is long or if many regions are specified.
- 2.4- Rappture plotting is now faster. Plots that can be plotted as scatter plots are being plotted in that manner rather than independent curves for each point.
- 2.4- 1 barrier(3 segments) case included.
- 2.4- Composite plots are now explicitly labeled as having an arbitrary x-axis.
- 2.4- Resonance finder input options have been moved to a separate tab.
- 2.5- Reverted back to the 4-point adaptive grid scheme.
- 2.5- Plot comparing the analytic bulk band-structure with the calculated dispersion has been added.
- 2.6- Bulk band-structure plot goes to the maximum in the energy range of simulation. The effective lengths in the particle in a box are only shown as labels.

#### Credits

Vasileska | ... core transfer matrix engine in matlab |

Luisier, Agarwal | ... core effective mass engine in matlab |

Wang,Agarwal | ... GUI implementation |

Klimeck | ... GUI and output design |

Sellier | ... adaptive mesh refinement |

#### Sponsored by

NSF

#### References

Dragica Vasileska lecture notes on Quantum mechanics http://www.eas.asu.edu/~vasilesk .

#### Cite this work

Researchers should cite this work as follows: