nextnano - Software for
semiconductor nanodevices

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 Contact Block Reduction (CBR) method (see also documentation). This tutorial is an analog of nextnano3 tutorial.

• 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

We first consider transmission through a finite quantum barrier. 10 nm Al_0.3Ga_0.7As 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> <caption>The conduction bandedge profile.</caption> </figure>

With nextnano++, one can calculate the transmission spectrum using the CBR method (cf. 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> <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 $E_{\mathrm{barrier}}$.

For a step potential structure as shown in Figure search?q=step%20gamma&amp;btnI=lucky, 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> </figure>

<figure step> </figure>

<figure well gamma> <caption></caption> </figure>

<figure well> <caption></caption> </figure>

Finally we consider a double barrier structure with wall width 10 nm. The distance between the barriers is 10 nm. <figure double barrier gamma> </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> <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> <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>

• 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> <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 3D case for another CBR efficiency assessment.