nextnano - Software for
semiconductor nanodevices

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 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 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> <caption>The conduction bandedge profile (bandedge_Gamma.dat).</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).

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.

The result is written in transmission_cbr_Gamma1.dat. The barrier width $w$ affects the transmission coefficient as shown in Figure search?q=single_barrier&amp;btnI=lucky.

<figure single_barrier> <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 $E_{\mathrm{barrier}}$.

For a step potential structure as shown in Figure search?q=step_gamma&amp;btnI=lucky, the transmission of electrons with energy below $E_{\mathrm{barrier}}$ is prohibited because the barrier is infinitely thick.

<figure step_gamma> </figure>

<figure step> </figure>

Similarly a quantum well structure can be simulated. The well width is w=10nm here.

<figure well gamma> </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> <caption>Transmission spectrum for the quantum well structure. The dashed line marks the top of the barrier.</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 bandedge_Gamma.dat and \Quantum\wf_probabilities_shift_cbr_Gamma_0000.dat, respectively.

<figure resonant> <caption>Probability distribution $|\psi(x)|^2$ of the two 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. Please note that the vertical axis is logarithmic scale. <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>

A resonant tunneling diode (RTD) is an example of a device that exploits this $\delta$-function-like behaviour of transmission coefficient $T(E)$.

• 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.

• Please help us to improve our tutorial. Should you have any questions or comments, feel free to send to: support [at] nextnano.com