Prime, Compress, Author

When I was in graduate school studying for my qualifying/comprehensive exams, I had to devise a way I could learn as much information as I could. For one, I'm a terrible timed-test taker, I do not compete well in this setting. Remove the time constraint and I'm much more competitive; probably the case with others as well. However, that is a luxury that is rarely afforded to students since the goal of timed examination is to determine how well an individual can recall and utilize concepts in comparison to their peers. Because of my poor test taking abilities I was obsessed with preparing the best I could and formulated a technique that works for me. I'm positive elements of this approach exist in more well known standard methods.

My reason for writing about this approach now rather than 10 years ago is that I've been spending a lot of my time learning new things in the fields of AI, neuroscience, and thermodynamics computing. Because I have no formal background in these topics, I needed to leverage my previous approaches that worked well for me, hence my method used for my comprehensive exams.

Overview

My method is primarily designed for learning from books, but could potentially be adapted for audio or video content. It involves three key steps:

1. Prime: Prepare your mind by uninterrupted reading.
2. Compress: Annotate key ideas in your own concise style.
3. Author: Write as if you were the author of a textbook.

This is what I refer to as the PCA method, which is somewhat ironic given that principal component analysis is a method of compressing data, yet retains a richness of information.

🧨 Prime

The goal here is to familiarize yourself with the concepts and ideas in the learning material without interruptions. During this initial exposure, do not take notes; focus solely on understanding the content. Additionally, avoid consulting supplementary materials, even if parts of the text are unclear. This maintains the focus and efficiency of the session.

For effective priming, shorter segments are generally more manageable than entire chapters, as chapters often cover broad topics with complex details. Engaging with smaller content blocks allows for a more focused approach.

At the end of reading you should feel like you have a general sense of the content and what is important.

📦 Compress

I find this step crucial as it solidifies the understanding of the material. Re-read the text from the Prime step and take notes using paper and pen in real-time. The format of your notes should reflect your personal preferences, whether as text chunks or bullet points, etc. Include diagrams and summaries of important figures in your own style. Add footnotes for references or important details to revisit later.

The aim is to distill the content into a personalized, dense, and informative resource. These notes are meant for your use, tailored to your understanding. Don't write them thinking you'll share them with others, so "being messy" is okay!

📚 Author

Now, adopt the mindset of an "Author" or "Teacher." Use the notes from the Compress step as a basis to expand into a clear, detailed, and presentable format. This step involves refining your notes into a resource useful to others. Aim for thoroughness and professionalism.

Initially, try to rely on your memory and notes to guide your writing, ensuring the content is conveyed effectively. However, feel free to refer back to the original material and your notes, research further as needed, and enhance your explanations with additional resources.

Sometimes I call this step the "Being Feynman" step, in reference to the Richard Feynman technique. You could say PCA is a form of the Feynman technique, but as Dirac's self-appointed student it would be blasphemous to do so, 😉.

Re-read & Polish (Optional)

If time allows, revisiting the original text can enhance the accuracy of your work. Review your notes and the final document, possibly consulting with peers to identify and correct misunderstandings. Recognizing errors independently can be challenging, making peer feedback invaluable.

Tools You May Want to Use

1. Prime: Utilize the primary text, audio, or video only.
2. Compress: Choose quality writing tools and your preferred type of paper; I favor plain white printer paper.
3. Authoring: Consider various tools based on your needs:

App	Scope
$\LaTeX$	Ideal for non-computational writing, with Overleaf providing a quick start.
Obsidian	Excellent for non-computational, markdown-based documentation.
Jupyter Notebooks	Best for including computations and visualizations.
Quarto	Similar to Jupyter Notebooks but with expanded document format options.

Footnotes

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1. Dividing the Prime step into smaller segments simplifies the Compress step by reducing the volume of notes and easing cross-reference with previous annotations. ↩

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Why does Fermi have a level?

Note

This is an old post that I started to write back in 2021, but never got around to extending and publishing it, hence why it may seem a little out of place with my more recent posts. I'm publishing in the current state with hopes to provide some more derivations in the future.

When you take your first solid-state physics¹ or electronic materials class, you will be regularly confronted with band structure diagrams. At first, you'll be like "what in the hell" as there is so much information packed into a single plot. You have the x-axis, which contains Greek letters that indicate high-symmetry points, and then the paths between them indicate directions. Moreover, you have to remember this is in reciprocal space of the crystal structure, so you're dealing with wave-vectors ². The y-axis isn't as frustrating, for each k-point represents the allowable energies.

Now you may ask, "How do I use this kind of plot to extract information?" Yeah, that's not trivial, and understanding it is an objective of a solid-state physics course. However, in brief, you have things like phase and group velocity, which correspond to derivatives, i.e., $\frac{\partial E(\mathbf{k})}{\partial \mathbf{k}}$, and tell you about how electrons conduct.

One particular aspect of band structure I want to cover is the Fermi energy level. The Fermi level plays a crucial role as it helps in understanding the behavior of electrons in a material. We typically use it to denote whether a material is a conductor, semiconductor, or insulator.

Electronic Band Structure

An electronic band structure is a representation of the allowed energy levels of electrons in a solid material. It is obtained by solving the quantum mechanical problem for electrons moving in a periodic potential, which represents the regular array of atoms in a crystal. The solutions to the Schrödinger equation for this periodic potential yield the energy bands, which are key features in the electronic band structure. In the band structure, the energy of each state (or eigen-energy) is plotted against the wavevector (or k-point), which is a measure of the wavelength or crystal momentum of the state³.

Example of band structure and density of states forSiC (mp-8062)[1]

Fermi Level as a Reference

To compare the electronic energy levels, we need a reference, and this is where the Fermi level comes in. The Fermi level is taken as the chemical potential of electrons in a periodic box at absolute zero temperature (Definition 4.1 in ref. [2]), representing the highest energy level occupied by electrons.

Warning

Its more correct to say the Fermi level is the energy that is half-way between the lowest and highest occupied electron state.

For brevity, the chemical potential is defined as:

$$ \begin{equation} \mu = \left(\frac{\partial U}{\partial N}\right)_{V, S} \end{equation} $$

where $\mu$ is the chemical potential (or Fermi level), $U$ is the internal energy, $V$ is the volume, $S$ is the entropy, and $N$ is the number of particles.

Fermi Level and Conductivity

In metals, the Fermi level often lies within an energy band, leading to partially filled bands. This allows for easy movement of electrons within the band and thus good electrical conductivity. In semiconductors and insulators, the Fermi level lies in a band gap, with all states below it fully occupied (forming the valence band) and all states above it empty (forming the conduction band).

Calculating the Fermi Level

The Fermi energy is calculated based on the number of electron states and the number of electrons in the system. It can be obtained from the following equation:

$$ \begin{equation} N = \int_{-\infty}^{E_F} g(E) \, dE \end{equation} $$

where $N$ is the total number of electrons, $E_F$ is the Fermi energy, and $g(E)$ is the density of states⁵. The Fermi energy $E_F$ is the energy that satisfies this equation.

Utility

The Fermi level serves as a crucial reference point, dividing occupied from unoccupied electronic states at zero temperature. It significantly influences the electronic, magnetic, and optical properties of materials. Understanding the band structure helps reveal how crystal symmetry affects electronic properties, while the density of states focuses on energy level distribution.

The next step would be to classify a material as metals, insulators, semiconductors, etc. based on the bandstructure. I'm not going to cover this in detail here, but there are many many good resources on this and a good start is Ashcroft and Mermin[3] or Simon[2].

Footnotes

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1. Its probably "old-fashion" to use solid-state physics as this is considered a specific focus within condensed matter physics. ↩

2. You can think of the wavevector as a type of momentum representation of the electronic wavefunction. This is sometimes called the crystal momentum as well. It provides information about how the electronic state is moving in space. ↩

3. The wavevector is also a measure of the phase change of the electron wavefunction as it moves across the crystal. ↩

4. At non-zero temperatures, the Fermi-Dirac distribution comes into play, affecting the occupation of states. ↩

5. This equation is often solved numerically, especially for complex materials. ↩

References

[1] Materials Data on SiC by Materials Project. LBNL Materials Project; Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States), 2020. https://doi.org/10.17188/1282015.

[2] Simon, Steven H.. The Oxford Solid State Basics. United Kingdom, OUP Oxford, 2013.

[3] Ashcroft, Neil W., and Mermin, N. David. Solid State Physics. United States, Cengage Learning, 1976.

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