nanoHUB IGNITE 2021
Artificial Atoms Challenge
Instructions
Quantum Dot Design for optical absorption at specific wavelengths
Use nanoHUB’s Quantum Dot Lab (https://nanohub.org/tools/qdot) to design a system that absorbs light at about 340meV when illuminated in the x-direction and about 490meV when illuminated from the y-direction. Minimize absorption at other wavelengths when illuminated from the top (z-direction). Document the approach and all the final settings (or range of settings) in Quantum Dot Lab.
Brief background information
Quantum dots are like artificial atoms which can absorb and emit light at colors / frequencies / energies that can be controlled by design. There are many ways to create quantum dots. Here we consider quantum dots that can be grown on semiconductor substrates like GaAs. These dots typically have dome-like or pyramidal shape. Different shapes and sizes determine the energies in the quantum dot. The energy differences in that system can lead to quantum transitions that can be detected by light absorption or light emission.
The system considered here just elucidates the fundamental principles of light absorption in the conduction band (electrons only) in the most simple material model. The simulations run rather fast but do not include all the necessary physics that are present in real systems. So the outcome is not going to be a new, realistic quantum dot design, but a much clearer understanding of light absorption in 3D quantum dot systems. With that knowledge, you can begin to explore realistic quantum dots.
Additional constraints are provided to make the exploration and analysis easier:
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Consider only a simple quantum dot and include 20 eigenstates in the calculation.
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Consider a lattice constant of 0.3nm or 0.5nm (code will run faster).
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Consider a single effective mass of GaAs or 0.067.
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Consider the fundamental energy gap 1.43eV.
Some tips and explanation of the tool settings
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The problem should be as simple as possible. Let’s imagine that the lowest quantum dot state (ground state) is occupied with electrons. Incoming light can then cause transitions to excited states and therefore light that travels through the system would be absorbed. Various excited states will have different energies compared to the ground state energy and therefore different absorption “lines” will occur.
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The Fermi level determines the occupation of states in equilibrium. The default setting in the tool is such that the electron ground state is occupied. Do NOT change the setting that determines the Fermi level.
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State Broadening refers to converting discrete Delta functions of eigen energies to a Lorentzian Broadening that resembles non-idealities such as scattering and other effects. There is no need to change this number.
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The temperature refers to the thermal distribution around the Fermi level. Unless the first excited state is within a few kBT within the ground state, temperature should not matter…. This is an invitation to play :-) - see additional challenges below.
Coordinate systems
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Here we consider the vertical z-axis the direction in which the dots are grown.
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The growth plane is considered to be in the x- and y-direction:
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The light injection is measured against a spherical coordinate system, where by typical convention the angle against the polar coordinate (vertical z axis) is measured as Theta (θ) and the angle from the x-axis is measured in Phi.
Sweeping over light injection angles. The exploration challenge is to determine a set of quantum dots that have specific absorption characters for light injection from different angles. You may find it useful to sweep over the angles Phi and Theta to understand the angle dependence in the first place and then to optimize your design.
Selecting and viewing output
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Once the simulations finish, there will be a result visible on the screen. By default, you will see the spherical ground state. By clicking on the bar underneath you can also view additional states.
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In the “Results” bar across you can select different outputs, for example “Absorption (phi=0, theta=45)”:
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Selecting “Absorption sweep of angle theta” shows three default curves for theta = 0, 45, 90 degrees. The theta = 90 degrees shows an extremely low absorption by a factor of 1e20 smaller than the other angles…. You will begin to understand this after some exploration.
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You can download the results as raw data or as well-formatted images by clicking on the green download button on the top right (). You can also put the cursor over individual line points and read off individual values.
Suggestions for a solution strategy
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The whole problem can, in principle, be designed completely empirically by a design of experiments. However, understanding some of the underlying physics might speed that process up a little bit.
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It is helpful to understand a simple particle in a box problem which can also be solved analytically - see notes below
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You need to understand (or discover through experimentation) the concept of allowed and forbidden transitions in a quantum system.
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Understand angular dependence of absorption in a simple cubic system:
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To calibrate yourself start from a perfectly symmetric cube (say 10nm x 10nm x 10nm) and run an absorption sweep versus phi and another simulation sweep versus theta for say 7 angles. Convince yourself numerically that the principal absorption peak is virtually unchanged as a function of sweep angles. The wavefunctions do not look “clean” - you can leave an interpretation as to why that might be for later. (see additional challenges)
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Repeat the experiment with a cuboid that is asymmetric in one dimension (say 10nm x 10nm x 9.5nm). Observe the absorption changes a function angles phi and theta. Some new peaks emerged or split. Some peaks have a strong angle dependence.
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Repeat the experiment above with an asymmetric cuboid say 10nm x 9.5nm x 9nm).
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Repeat additional experiments where you reduce the quantum dot height systematically.
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Realistic quantum dots in these material systems are typically flatter in the z-direction (5-10nm) compared to the x and y directions (typically 10-50nm). They typically grow by self-assembly into dome-like or pyramidal-like shapes - you can now begin your exploration for the desired absorption lines.
Additional challenge questions
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How stable is the solution against variations of the physical structure?
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Do your results depend on the lattice constant you are setting?
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Can you interpret the “not-so-clean” wavefunctions in the perfect cubic system?
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What happens when the Fermi level or the temperature are changed?
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Discuss the validity of the absorption spectra results for high energies - consider the fundamental bandgap
Additional background information that may or may not be helpful, depending on your background
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Introduction to quantum dot lab. https://nanohub.org/resources/4194
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A complementary view of a particle in a box and selection rules: http://ritchie.chem.ox.ac.uk/Grant%20Teaching/2010/Lecture%204%202010.pdf
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Wikipedia particle in a box https://en.wikipedia.org/wiki/Particle_in_a_box
nanoHUB Tools to use
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Quantum Dot Lab - https://nanohub.org/tools/qdot