nnp:cbr_1d_potential
Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision | |||
|
nnp:cbr_1d_potential [2024/01/03 16:50] stefan.birner removed |
— (current) | ||
|---|---|---|---|
| Line 1: | Line 1: | ||
| - | ===== 1D Transmission (CBR) ===== | ||
| - | Author: Takuma Sato, nextnano GmbH | ||
| - | In this tutorial we calculate the transmission coefficient $T(E)$ as a function of energy $E$. We consider the following pedagogical examples we learn in undergraduate quantum mechanics courses. | ||
| - | * Single potential barrier | ||
| - | * Step potential | ||
| - | * Quantum well | ||
| - | * Double potential barrier | ||
| - | |||
| - | To calculate transmission spectra with nextnano++, we use **C**ontact **B**lock **R**eduction (CBR) method (see also [[https://www.nextnano.com/nextnanoplus/software_documentation/input_file/cbr.htm|documentation]]). This tutorial is an analog of [[https://www.nextnano.de/nextnano3/tutorial/1Dtutorial_Transmission_NEGF.htm|nextnano3 tutorial]]. | ||
| - | |||
| - | === Reference === | ||
| - | * Ballistic Quantum Transport using the Contact Block Reduction (CBR) Method - An introduction, S. Birner, C. Schindler, P. Greck, M. Sabathil, P. Vogl, Journal of Computational Electronics (2009) | ||
| - | |||
| - | The corresponding input files are | ||
| - | * Transmission_CBRtutorial_1Dsinglebarrier_nnp.in | ||
| - | * Transmission_CBRtutorial_1Dstep_nnp.in | ||
| - | * Transmission_CBRtutorial_1Dwell_nnp.in | ||
| - | * Transmission_CBRtutorial_1Ddoublebarrier_nnp.in | ||
| - | * Transmission_CBRpaper_1Ddoublebarrier_nnp.in | ||
| - | ==== Single Potential Barrier ==== | ||
| - | We first consider transmission through a finite quantum barrier. 10 nm barrier is located in a 50 nm sample. After running the input file, we obtain the following bandedge profile. The barrier height is set to $E_{\mathrm{barrier}}=0.3$ eV. | ||
| - | |||
| - | <figure single_barrier_gamma> | ||
| - | {{:nnp::transmission_1dsinglebarrier_gamma.png}} | ||
| - | <caption>The conduction bandedge profile (''bandedge_Gamma.dat'').</caption> | ||
| - | </figure> | ||
| - | |||
| - | With nextnano++, one can calculate the transmission spectrum using the CBR method (cf. [[https://www.nextnano.com/nextnanoplus/software_documentation/input_file/cbr.htm|here]] for details). The sample input file is generalised so that you can change the barrier width and alloy content (which determines the barrier height). | ||
| - | |||
| - | Here we look into the barrier width dependence. In nextnanomat, go to **'Template'** tab and select the input file. Then at the bottom you can select how to sweep the value (here ''List of values'') and variable ''Barrier_Width''. The list of values shows up automatically, as it is specified in the input file with the tag ''ListOfValues''. Clicking the button ''Create input files'' generates multiple input files by sweeping variables. Please go to **'Simulation'** tab and run the simulation. | ||
| - | |||
| - | {{:nnp::transmission_1dsinglebarrier_nnm.png}} | ||
| - | |||
| - | The result is written in ''transmission_cbr_Gamma1.dat''. The barrier width $w$ affects the transmission coefficient as shown in Figure {{ref>single_barrier}}. | ||
| - | |||
| - | <figure single_barrier> | ||
| - | {{:nnp::transmission_1dsinglebarrier.png}} | ||
| - | <caption>Transmission coefficient as a function of energy for different barrier width $w$ [nm]. The dashed line marks $E_{\mathrm{barrier}}$.</caption> | ||
| - | </figure> | ||
| - | |||
| - | Classical mechanics argues that the transmission is $0$ below $E_{\mathrm{barrier}}$ and abruptly increases to $1$ at $E_{\mathrm{barrier}}$. However, quantum mechanics allows electrons with energy below $E_{\mathrm{barrier}}$ to go through the barrier. This effect becomes even larger when the barrier is thin ($w=5$ nm in this example). Quantum mechanics also predicts a oscillatory behaviour above $E_{\mathrm{barrier}}$. | ||
| - | ==== Step Potential ==== | ||
| - | For a step potential structure as shown in Figure {{ref>step_gamma}}, the transmission of electrons with energy below $E_{\mathrm{barrier}}$ is prohibited because the barrier is infinitely thick. | ||
| - | |||
| - | <figure step_gamma> | ||
| - | {{:nnp::transmission_1dstep_gamma.png}} | ||
| - | </figure> | ||
| - | |||
| - | <figure step> | ||
| - | {{:nnp::transmission_1dstep.png}} | ||
| - | <caption>Transmission spectrum for a step potential. Transmission is only allowed above the step.</caption> | ||
| - | </figure> | ||
| - | ==== Quantum Well ==== | ||
| - | Similarly a quantum well structure can be simulated. The well width is w=10nm here. | ||
| - | |||
| - | <figure well gamma> | ||
| - | {{:nnp::transmission_1dwell_gamma.png}} | ||
| - | </figure> | ||
| - | |||
| - | Again the transmission of electron within the barriers is impossible because the barrier is infinitely thick. Above 0eV, the spectrum shows an oscillatory behaviour. | ||
| - | |||
| - | <figure well> | ||
| - | {{:nnp::transmission_1dwell.png}} | ||
| - | <caption>Transmission spectrum for the quantum well structure. The dashed line marks the top of the barrier.</caption> | ||
| - | </figure> | ||
| - | ==== Double Potential Barrier ==== | ||
| - | Finally we consider a double barrier structure with wall width 10 nm. The barrier interval is 10 nm. | ||
| - | <figure double barrier gamma> | ||
| - | {{:nnp::transmission_1ddoublebarrier_gamma.png}} | ||
| - | </figure> | ||
| - | |||
| - | This system has two resonant modes localized between the barriers. The bandstructure and wavefunctions are written in ''bandedge_Gamma.dat'' and ''\Quantum\wf_probabilities_shift_cbr_Gamma_0000.dat'', respectively. | ||
| - | |||
| - | <figure resonant> | ||
| - | {{:nnp::transmission_1ddoublebarrier_resonant.png}} | ||
| - | <caption>Probability distribution $|\psi(x)|^2$ of the two resonant modes.</caption> | ||
| - | </figure> | ||
| - | |||
| - | In the transmission spectrum, one can clearly see the sharp transmission at the energies of the resonant states in the quantum well. Please note that the vertical axis is logarithmic scale. | ||
| - | <figure double_barrier> | ||
| - | {{:nnp::transmission_1ddoublebarrier.png}} | ||
| - | <caption>Transmission coefficient of the double barrier structure. The spectrum has two sharp peaks below the barrier height 3.084 eV, which corresponds to the resonant mode within the barriers.</caption> | ||
| - | </figure> | ||
| - | |||
| - | A [[https://www.nextnano.de/nextnano3/tutorial/1Dtutorial_RTD_green.htm|resonant tunneling diode (RTD)]] is an example of a device that exploits this $\delta$-function-like behaviour of transmission coefficient $T(E)$. | ||
| - | |||
| - | |||
| - | |||
| - | ==== CBR efficiency assessment ==== | ||
| - | * Input file: Transmission_CBRpaper_1Ddoublebarrier_nnp.in | ||
| - | |||
| - | Figure 4 in [Birner2009] compares the transmission coefficient of a double barrier structure for different number of eigenstates considered in the CBR method. The following figure shows the result reproduced by nextnano++ and demonstrates that the first resonant peak is accurately reproduced using incomplete set of eigenstates. The spectrum is not identical to the previous result because the barrier width here is 2 nm. | ||
| - | |||
| - | <figure assessment> | ||
| - | {{:nnp::transmission_1ddoublebarrier_paper.png?direct&600}} | ||
| - | <caption>Transmission coefficient for three different CBR parameters. The red curve is the result considering complete set of device eigenstates, while green and blue curves take into account only 40% and 10% of them, respectively.</caption> | ||
| - | </figure> | ||
| - | |||
| - | See also [[https://www.nextnano.com/dokuwiki/doku.php?id=nnp:cbr_3d_nanowire|3D case]] for another CBR efficiency assessment. | ||
| - | |||
| - | * Please help us to improve our tutorial. Should you have any questions or comments, feel free to send to: support [at] nextnano.com | ||
Page Tools
