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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

Launch Tool

**Archive** Version **1.2**

Published on 17 Apr 2009, unpublished on 05 May 2009

Latest version: 1.2.9. All versions

doi:10.4231/D3J96088X cite this

This tool is closed source.

#### Category

#### Published on

#### 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.

#### Sponsored by

NSF

#### References

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

#### Cite this work

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