Density of States Derivation Part 1

/ edmundsj
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In this video, I derive the equation for the density of states, and I show how finding the most convoluted way possible to count apples helps us do this.
This is part of my series on semiconductor physics (often called Electronics 1 at university). This is based on the book Semiconductor Physics and Devices by Donald Neamen, as well as the EECS 170A/174 courses taught at UC Irvine.
Hope you found this video helpful, please post in the comments below anything I can do to improve future videos, or suggestions you have for future videos.

[@aselim20.]

I watched it twice and I took notes.

[@christosvasiliou219]

i am a MSc studend in adv. materials and nano, and i have a module of semiconductors. Your video series helped a lot with my midterm and realising to the full everything i have been told to the lectures. Godd job and thank you in advnace

[@lowerlowerhk]

The apple counting part helps me a lot. For this concept of density of state, I do need somebody explaining it to me like I am a 5 year old.

[@priyanksharma4178]

So beautifully explained.

[@shrutijain3865]

Excellent explanations sir!! Love n support from India!

[@jozokukavica9814]

Oh thank you so much for this. A year and a half ago i was googling and youtubeing to find a good explanation on this and couldn't find it. Embarking upon this is such a delight, thank you. Finally a good enough explanation. Thank you!

[@JordanEdmundsEECS]

My pleasure :D

[@seatonehoo3222]

Awesome lecture!

[@abebe7017]

I think number of state should be volume*volume density. I mean N=v*(n /a^3), not N=v/(n/z^3). I think the equation written at 4:37 is not right: even the unit does not give us the number of apples. Other than that your videos are life savers!!!

[@JordanEdmundsEECS]

Yes, you are correct, the number of states will be volume * volume density xD. Thank you so much for pointing that out!

[@dqlin3609]

oh that what I'm confused. Thanks

[@tim40gabby25]

@Kyle Rylan you seem to be sharing this um.. apparently useful information on a lot of channels. Probably best if you leave the class, quietly?. if you wish to troll me, at least spellcheck, good lad :)

[@sl2357]

The volume of a spherical shell: recall volume of a sphere V = 4/3.pi.r^3, so dV = 4.pi.r^2. dr//

[@jorgegoizueta4911]

So useful thank you!!!

[@lorenzo0008]

At minute 2:26 n is not the number of electrons but the concentration of electrons per unit of volume. Thanks for the wonderful lesson!

[@haya4895]

thank you so!!

[@willyassenga2943]

awesome,, i like it

[@rsbpg]

Congrats on your channel! Clear and easy to understand. One question... if we use finite quantum wells instead of infinite ones, would the final g(E) change much? I mean, do you think g(E) (as in the video) would be accurate for a metal like copper?

[@JordanEdmundsEECS]

Great question. So the DOS will be accurate as long as you can approximate a finite number of states with a continuous integral. In practice, the number of states you are summing over is so huge even for something like a quantum dot (tens of nm on a side corresponds to about 10^6 atoms) that the core approximation is usually pretty fantastic. g(E) is definitely going to be accurate for metals as well as semiconductors. The underlying model common between all of them is the "free electron gas" or the "Fermi gas".

[@shashankravibhagwat1]

Great!

[@SampleroftheMultiverse]

For those That prefer a mechanical analog you can look at harmonics of a guitar string and such. The video I present is another mechanical method of quantizing a system. It is one of two methods where structures can actually be produced. kzbin.info/www/bejne/raOlpKSfepWpfZYsi=waT8lY2iX-wJdjO3 Area under a curve is often equivalent to energy. Buckling of an otherwise flat field shows a very rapid growth of this area. If my model applies, it may show how the universe’s energy naturally developed from the inherent behavior of fields. Under the right conditions, the quantization of a field is easily produced. The ground state energy is induced via Euler’s contain column analysis. Containing the column must come in to play before over buckling, or the effect will not work. The sheet of elastic material “system” response in a quantized manor when force is applied in the perpendicular direction. Bonding at the points of highest probabilities and maximum duration( ie peeks and troughs) of the fields “sheet” produced a stable structure when the undulations are bonded to a flat sheet that is placed above and below the core material.

[@nellvincervantes6233]

The confusing part of this is there is stil a probability of occupancy of electrons on forbidden band based on Fermi Dirac statistics. But when you get the density of state (I think it is the average = integral g(E)P(E)dE) on forbidden gap, it will be zero.

[@JordanEdmundsEECS]

Yes, exactly! This is the part I myself was confused on when first learning this. Even if the probability of finding an electron is 50%, because there aren't any states in the first place, it's 50% of zero, which is zero.

[@suruchiverma48]

At 4.34 you say that we take the volume and divide by volume density. But if we do volume divided by (number per volume) it doesn't work. So either it should be volume divided by (volume per number) or else volume multiplied by (number per volume). Am i right?

[@sathwikmaganti9042]

Nice

[@ano255t3]

Did we choose the shell because different radial distances corresponds to different energy levels? so that we can find the quantity of different energy levels.

[@JordanEdmundsEECS]

Yes, everywhere on the shell has the same magnitude of k, and hence the same energy. I actually hadn’t thought of that until you mentioned it :) that’s actually really important to do the math!

[@user-ge8hj9br6w]

discrete sum can be found from integration

[@kirthi099]

Hi, I'm confused what you mean by a "state" of an electron. You said that each "state" can have 2 electrons, that means a state corresponds to a given (n,m,l) value. But here, you considered each state to correspond to a single "n" value only. For a given "n", wouldnt there be multiple states for different m and l values? I think I'm missing something.. pls let me know if you can clarify! Thanks!

[@JordanEdmundsEECS]

That is a deep and excellent question. The problem is that you are confusing electrons confined to an *atom* (orbital electrons, or the states of a hydrogen atom) with electrons confined to a *box*. In the hydrogen atom, the "states" are labelled by 4 numbers: n,l,m, and s. However, in the particle-in-a-box model, there are only 2 quantum numbers: n (the energy), and s (spin). It's a completely different model (different potential that you plug into Schrodinger's equation), and so leads to a different solution.

[@user-si1zn3ir7x]

Hello! I have a question.. in 2:28 (the number of charge carriers in unit voulme) what you did is you multiplied #of state with probability of that state occupied. However, shouldn't it be #of particles for that state instead of probability of occupancy? I'm quite confused about it because if was probability, then it should give you 1 if properly integrated(without density of state multiplied)... By the way, it was a really good lecture thank you!

[@suruchiverma48]

No, because probability multiplied possible states will give the number that actually will be there.

[@AndreKuhlmannDesign]

Thanks for the video Jordan. There is one thing though I cannot wrap my mind around "k-space". Can you give me a hint on what "k-space" is?

[@JordanEdmundsEECS]

Yeah, so “k” is just another way of saying the momentum of the electron (which is hbar * k). Just as we can specify particles’ positions (which is the ‘real space’ you are familiar with), its completely reasonable to specify each particles momentum (or velocity if you prefer). In classical mechanics these are two different things, and you need both, but in quantum mechanics they are coupled - they represent the same underlying information about the particle. So “k-space” is just a fancy way of saying all the possibilities for momentum of an electron. If this were the classical world this space would be continuous, but because quantum mechanics it’s made up of a bunch of points.

[@ravuruvasudevareddy3347]

@@JordanEdmundsEECS what it means "they represent the same underlying information about the particle" can u elaborate please..

[@xandersafrunek2151]

I followed this video and the next and everything is understandable, EXCEPT shouldn't the distance from the center to the edge of the sphere in "k-space" just be k and not k^2? This makes sense to me because of the way you plug in k as if it were equal to r as explained previously. Also, if possible can you better explain this transition from euclidean space to "k-space"? Is this just a geometric interpretation or does it have physical meaning?

[@theleviathan3902]

He uses k^2 because he's using 3d Pythagorean theorum to find the distance from the center

[@aneelahassan3425]

Asslam o alikum sir.. I have a question that this particular derivation of density of state (you have done) and derivations of density of states in 3D for nanomaterials is same or different?

[@aneelahassan3425]

Can you plz send me this derivation in soft form for nanomaterials?

[@seandafny]

Seems kind of backwards to me to use the volume per states to find the state per volume but I guess the difference come in because one of the volumes uses real space and one of them uses only k space

[@JordanEdmundsEECS]

Yeah it is kinda backwards, I find it more natural to ask the question “how much space does something take up” than “how many things are there per unit space”

[@ArsenedeBienne]

10:50 the radius is not k^2, but k

[@pawellisowski6379]

Hi, thanks for the great video! I just have a question: a document I have states that k can take values of (2πn/L), whereas your video says it can take (πn/L), do you know why this may be?

[@ElPrestigo]

This depends on the boundary conditions. In this video fixed boundary conditions are used which lead to a grid spacing of pi/L, where kx, ky and kz can only assume positive values leading to the octant in k-space. If on the other hand periodic boundary conditions were used, then the grid spacing would be 2pi/L with kx, ky and kz taking on both positive and negative values leading to the full k-space.

[@jimitsoni18]

Wait, why is k the length of radius? Shouldn't it be L? I mean isn't it the wave number?

[@kaank6605]

Why is the globe volume 4*pi*r^2*deltar? It was supposed to be 4*pi*r^2*deltar/3?

[@JordanEdmundsEECS]

Kaan Karaköse It’s not the volume of the entire sphere, just the (approximate) volume of a spherical shell, which is the area of that shell (4*pi*r^2) multiplied by the thickness (delta-r).

[@gabrieleeraquelmotta174]

I think you forgot the 1/8 factor in the last bit

[@correa95100]

why the length between atoms is Pi/L?

[@JordanEdmundsEECS]

The distance between the *states* is pi/L. This is from the particle in a box model, the wavevector is an integer multiple of pi/L.

[@stormwatcheagle5448]

Think Tim Cook would disagree about that apple bit.

[@zubairkabir2293]

What exactly is a state?

[@jimitsoni18]

See, if k can only take those integral multiple values, then the solution for psi will always be zero which means the probability will always be zero... Help me get through...

[@JordanEdmundsEECS]

k can take integer multiples of pi/L but x can be any number between 0 and L (it’s just a sine wave that fits so that it goes to 0 and 0 and L). You can plot sin(kx) from 0 < x < L and check :)

[@jimitsoni18]

@@JordanEdmundsEECS Thank you so much for that!

[@jimitsoni18]

@@JordanEdmundsEECS btw why are we taking the k space... Because the electrons lie in the L space and I'm not able to see how k space exists...

[@JordanEdmundsEECS]

It helps if you don't think about the electrons as "existing" in k-space: it's just that each electron has a well-defined momentum (hbar * k), just like I have a well-defined momentum right now sitting on the couch (zero). It just turns out that for the underlying system we are dealing with (the particle in a box), a well-defined momentum corresponds to a well-defined energy, and hence a *state*.

[@farazaliahmad5226]

What is state here???

[@manriqueorellanarios6986]

Could someone here explain what is a state?? I mean, is it related to the energy levels we learned about in school??? Wooow I am just about to cry :(

[@JordanEdmundsEECS]

A ‘state’ is a solution to the time-independent Schrodinger equation (it comes from quantum mechanics). In this case with the density of states, we are solving the Schrödinger equation for a particle in a 3D box, and the solutions we get are called ‘states’. These states do have discrete energy levels as you learned about in school. They also have discrete momenta (just hbar * k).

[@manriqueorellanarios6986]

@@JordanEdmundsEECS I really appreciate the time you take to share the knowledge!! I am studying solar cells physics and it brought me here, now I understand I need some quantum mechanics background.

[@SiddharthRanjan6197]

Why did you take a sphere after all?

[@JordanEdmundsEECS]

Also a great question, and a subtle answer. The reason is because the energy is related to the *magnitude* squared of the k-vector. Our goal is ultimately to figure out an expression in terms of energy, and so we want to perform our integral using the magnitude of k. The surface of constant k-magnitude (and hence constant energy) is a spherical shell, so that’s why the sphere.

[@DaytonaStation]

too many gaps in this. For example going to k space or finding the volume of a unit region.

[@varunshrivastav8876]

You make so Many mistakes in your videos

[@alis5893]

This lecture was not explained properly along with some wrong calculations

[@tim40gabby25]

More detail, please.