nnp:cbr_1d_potential
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nnp:cbr_1d_potential [2019/10/21 14:30] takuma.sato [Step Potential] |
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| - | ===== 1D Transmission (CBR) ===== | ||
| - | 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 Al<sub>0.3</sub>Ga<sub>0.7</sub>As barrier is located in a 50 nm GaAs sample. For this compound the top of the barrier is at $E_{\mathrm{barrier}}=3.243$ eV. | ||
| - | |||
| - | <figure single barrier gamma> | ||
| - | {{:nnp::transmission_1dsinglebarrier_gamma.png?direct&600}} | ||
| - | <caption>The conduction bandedge profile.</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). The result is written in ''~\bias_000_000\transmission_cbr_Gamma1.dat''. | ||
| - | <figure single barrier> | ||
| - | {{:nnp::transmission_1dsinglebarrier.png?direct&600}} | ||
| - | <caption>Transmission coefficient as a function of energy. 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 transmit the barrier, and 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 it can be seen as an infinitely thick barrier. | ||
| - | |||
| - | <figure step gamma> | ||
| - | {{:nnp::transmission_1dstep_gamma.png?direct&600}} | ||
| - | </figure> | ||
| - | |||
| - | <figure step> | ||
| - | {{:nnp::transmission_1dstep.png?direct&600}} | ||
| - | </figure> | ||
| - | ==== Quantum Well ==== | ||
| - | <figure well gamma> | ||
| - | {{:nnp::transmission_1dwell_gamma.png?direct&600}} | ||
| - | <caption></caption> | ||
| - | </figure> | ||
| - | |||
| - | <figure well> | ||
| - | {{:nnp::transmission_1dwell.png?direct&600}} | ||
| - | <caption></caption> | ||
| - | </figure> | ||
| - | ==== Double Potential Barrier ==== | ||
| - | Finally we consider a double barrier structure with wall width 10 nm. The distance between the barriers is 10 nm. | ||
| - | <figure double barrier gamma> | ||
| - | {{:nnp::transmission_1ddoublebarrier_gamma.png?direct&600}} | ||
| - | </figure> | ||
| - | |||
| - | This system has two resonant modes localized between the barriers. The bandstructure and wavefunctions are written in ''~\bias_000_000\bandedge_Gamma.dat'' and ''~\Quantum\wf_probabilities_shift_cbr_Gamma_0000.dat'', respectively. | ||
| - | |||
| - | <figure resonant> | ||
| - | {{:nnp::transmission_1ddoublebarrier_resonant.png?direct&600}} | ||
| - | <caption>Probability distribution $|\psi(x)|^2$ of the resonant modes.</caption> | ||
| - | </figure> | ||
| - | |||
| - | In the transmission spectrum, one can clearly see the 100% transmission at the energies of the resonant states in the quantum well. | ||
| - | <figure double barrier> | ||
| - | {{:nnp::transmission_1ddoublebarrier.png?direct&600}} | ||
| - | <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> | ||
| - | |||
| - | ==== 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. | ||
nnp/cbr_1d_potential.1571661024.txt.gz ยท Last modified: 2019/10/21 14:30 by takuma.sato
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