Exercise: Electron Tunneling in Gallium Arsenide

The applet simulates electron transport in a semiconductor device constructed as a three-layer gallium arsenide–gallium aluminum arsenide (GaAs–Ga_1–xAl_xAs) sandwich. The GaAs layer constitutes a potential well between two confining barriers that are formed from the GaAs matrix by doping with about 30% Al content. The device is modeled using the double-barrier potential energy V(x) shown in the applet. The barriers are 0.25 eV high and 5.0 nm wide, with a gap of equal width separating them; the effective mass for electrons in GaAs is 34.24 keV/c² [see Am. J. Phys. 62(2), 143 (1994)]. These values appear on the Math tab of the applet (the barrier/gap width is a – b). On the Graphics: [x] tab the double-barrier potential energy V(x) is plotted over the interval [–20 nm, +20 nm]. The listing to the right of the graph includes a placeholder for a single scattering (stationary) state of the electron in this environment; the energy of this state is adjustable. Unusually large transmission (resonant tunneling) through the device occurs when the electron energy coincides with the energy of a bound state in the well formed by the two barriers. In this exercise, we will determine the lowest electron energy that results in peak transmission, and investigate the width of the resonance. [In practice, the electron energy is fixed (at the Fermi energy of GaAs) and the device is 'tuned' to resonance by applying a suitable bias voltage that alters the bound state energies in the central well.]

1. Show the scattering waveform ψ₀ by right-clicking its placeholder in the list, clicking on the visibility icon in the Display Options field, then choosing the OK button. This is a stationary state of the electron in this environment, constructed as a purely transmitted wave to the right of the double barrier to model the case where electrons are incident from the left. The electron energy defaults to 0.03 eV. For this complex-valued waveform, the plotting style defaults to a color-for-phase scheme (see Technical Notes below). Note the uniform amplitude of the transmitted wave. In contrast, the wave to the left of the barrier is a mixture of incident and reflected components.
2. Display the reflected component of ψ₀ by right-clicking on the waveform and choosing "Show Reflected Wave" from the popup menu. Note the uniform amplitude of the reflected component on the incident side of the barrier.
3. The reflected wave amplitude is zero at a transmission resonance. To find the lowest such energy, go to the Math tab, where the energy of this state is recorded as E₀ = 0.03. Right-click anywhere in the value field for E₀ and choose "Edit Parameter..." from the popup menu to bring up the Energy Editor. Return to the Graphics: [x] tab and re-position the editor so as to afford an unobstructed view of the waveform. Now use the slider to manipulate the highlighted digits in the energy field (labeled E) while observing the waveform. Increase the energy upward from the initial value until no reflected amplitude is visible on the graph. Fine tune the search by reducing the number of highlighted digits in the text field; the first digit highlighted can be moved left or right by clicking on the corresponding arrow to the right of this field. The energy yielding no visible reflected wave is an approximation to the resonance value.
   
   
   Extra digits can be added before or after the decimal by typing directly in the editor energy field, then pressing the Enter key. The sum of leading and trailing digits is limited to 9.
   
   
4. Now let's fine tune even more the energy found in the previous instruction. From the Math tab, observe that E₀ is the rightmost argument of the refl(..) function, and that the function value changes interactively as we adjust the energy. The refl(..) function calculates directly the reflection probability at the given energy. Continue adjusting E₀ until you find a value that gives near-zero reflection; this is our most accurate result for the resonance energy. How is the wave for this case unique? Finish by selecting OK to end the edit session with the current settings.
5. Analogous to the refl(..) function, the trns(..) function calculates directly the transmission probability for the given energy. Note the value of the trns(..) function for the resonance energy just found. Try energies below and above resonance to find values where the transmission probability drops to half of its peak value. The difference between these bounding values is commonly termed the 'width' of the resonance. The very narrow width, even compared to the resonance energy, is evidence that this resonance is quite 'sharp'.

Technical Notes

• The Numerov method is used to construct the scattering wavefunctions for a given energy. The correct asymptotic boundary conditions at the right endpoint of the interval are derived from the requirement that the wave be a purely transmitted one in this region. The Schrödinger equation is integrated from there to the left endpoint, first for the real part and then again for the imaginary part of the scattering waveform. The resulting wave near the left endpoint of the interval is a mixture of an incident and a reflected wave. The coefficients of each are extracted from the expected asymptotic behavior of the full waveform. These coefficients furnish the needed boundary conditions at the left endpoint to integrate the Schrödinger equation (again using the Numerov method) from there to the right endpoint, thereby generating the incident and/or reflected waves as desired. Further details are reported in Computing in Science & Eng. 8(4), 32 (2006).
• A complex-valued function can be rendered as a single color-for-phase plot rather than two separate plots for the real and imaginary parts. In this display style, the phase of a complex number (the function value) is represented by color which, like phase angle, repeats with a definite period. [The 'colorwheel' follows the rainbow from red to green to violet, then back to red again through magenta.] In computer jargon, the phase is mapped to the hue (the amount of red, green, or blue) component of an hue-saturation-brightness color model.