Density of States Derivation Part 1

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  • เผยแพร่เมื่อ 4 ธ.ค. 2024

ความคิดเห็น • 69

  • @aselim20.
    @aselim20. 8 หลายเดือนก่อน +3

    I watched it twice and I took notes.

  • @christosvasiliou219
    @christosvasiliou219 ปีที่แล้ว +4

    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
    @lowerlowerhk ปีที่แล้ว +4

    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.

  • @abebe7017
    @abebe7017 5 ปีที่แล้ว +40

    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
      @JordanEdmundsEECS  5 ปีที่แล้ว +14

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

    • @dqlin3609
      @dqlin3609 4 ปีที่แล้ว +4

      oh that what I'm confused. Thanks

    • @tim40gabby25
      @tim40gabby25 3 ปีที่แล้ว

      @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 :)

  • @lorenzo0008
    @lorenzo0008 2 ปีที่แล้ว

    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!

  • @priyanksharma4178
    @priyanksharma4178 6 ปีที่แล้ว +6

    So beautifully explained.

  • @jozokukavica9814
    @jozokukavica9814 5 ปีที่แล้ว +8

    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!

  • @sl2357
    @sl2357 2 ปีที่แล้ว +2

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

  • @shrutijain3865
    @shrutijain3865 4 ปีที่แล้ว

    Excellent explanations sir!!
    Love n support from India!

  • @ano255t3
    @ano255t3 5 ปีที่แล้ว +1

    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
      @JordanEdmundsEECS  5 ปีที่แล้ว +1

      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!

  • @rsbpg
    @rsbpg 4 ปีที่แล้ว +3

    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
      @JordanEdmundsEECS  4 ปีที่แล้ว +2

      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".

  • @ArsenedeBienne
    @ArsenedeBienne 8 หลายเดือนก่อน

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

  • @seatonehoo3222
    @seatonehoo3222 5 ปีที่แล้ว +1

    Awesome lecture!

  • @user-ge8hj9br6w
    @user-ge8hj9br6w 4 ปีที่แล้ว

    discrete sum can be found from integration

  • @AndreKuhlmannDesign
    @AndreKuhlmannDesign 4 ปีที่แล้ว +1

    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
      @JordanEdmundsEECS  4 ปีที่แล้ว +19

      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
      @ravuruvasudevareddy3347 3 ปีที่แล้ว +1

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

  • @seandafny
    @seandafny 3 ปีที่แล้ว +1

    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
      @JordanEdmundsEECS  2 ปีที่แล้ว

      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”

  • @kirthi099
    @kirthi099 4 ปีที่แล้ว +1

    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
      @JordanEdmundsEECS  4 ปีที่แล้ว +4

      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.

  • @RohitSharma-mi8gt
    @RohitSharma-mi8gt 24 วันที่ผ่านมา

    for 2d materials, would the vol be replaced by area?

  • @xandersafrunek2151
    @xandersafrunek2151 3 ปีที่แล้ว +1

    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
      @theleviathan3902 ปีที่แล้ว

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

  • @한두혁
    @한두혁 3 ปีที่แล้ว

    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
      @suruchiverma48 ปีที่แล้ว

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

  • @suruchiverma48
    @suruchiverma48 3 ปีที่แล้ว

    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?

  • @nellvincervantes6233
    @nellvincervantes6233 3 ปีที่แล้ว

    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
      @JordanEdmundsEECS  2 ปีที่แล้ว

      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.

  • @aneelahassan3425
    @aneelahassan3425 2 ปีที่แล้ว

    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
      @aneelahassan3425 2 ปีที่แล้ว

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

  • @jorgegoizueta4911
    @jorgegoizueta4911 3 ปีที่แล้ว

    So useful thank you!!!

  • @SampleroftheMultiverse
    @SampleroftheMultiverse 5 หลายเดือนก่อน

    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.
    th-cam.com/video/wrBsqiE0vG4/w-d-xo.htmlsi=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.

  • @haya4895
    @haya4895 4 ปีที่แล้ว

    thank you so!!

  • @jimitsoni18
    @jimitsoni18 4 ปีที่แล้ว

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

  • @jimitsoni18
    @jimitsoni18 4 ปีที่แล้ว +1

    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
      @JordanEdmundsEECS  4 ปีที่แล้ว

      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
      @jimitsoni18 4 ปีที่แล้ว

      @@JordanEdmundsEECS Thank you so much for that!

    • @jimitsoni18
      @jimitsoni18 4 ปีที่แล้ว

      @@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
      @JordanEdmundsEECS  4 ปีที่แล้ว +1

      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*.

  • @correa95100
    @correa95100 4 ปีที่แล้ว

    why the length between atoms is Pi/L?

    • @JordanEdmundsEECS
      @JordanEdmundsEECS  4 ปีที่แล้ว

      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.

  • @willyassenga2943
    @willyassenga2943 5 ปีที่แล้ว +2

    awesome,, i like it

  • @zubairkabir2293
    @zubairkabir2293 ปีที่แล้ว

    What exactly is a state?

  • @gabrieleeraquelmotta174
    @gabrieleeraquelmotta174 5 ปีที่แล้ว +1

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

  • @pawellisowski6379
    @pawellisowski6379 4 ปีที่แล้ว +1

    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
      @ElPrestigo 3 ปีที่แล้ว +1

      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.

  • @kaank6605
    @kaank6605 6 ปีที่แล้ว

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

    • @JordanEdmundsEECS
      @JordanEdmundsEECS  6 ปีที่แล้ว +2

      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).

  • @farazaliahmad5226
    @farazaliahmad5226 4 ปีที่แล้ว

    What is state here???

  • @stormwatcheagle5448
    @stormwatcheagle5448 3 ปีที่แล้ว

    Think Tim Cook would disagree about that apple bit.

  • @shashankravibhagwat1
    @shashankravibhagwat1 3 ปีที่แล้ว

    Great!

  • @manriqueorellanarios6986
    @manriqueorellanarios6986 4 ปีที่แล้ว +1

    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
      @JordanEdmundsEECS  4 ปีที่แล้ว +3

      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
      @manriqueorellanarios6986 4 ปีที่แล้ว +1

      @@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.

  • @sathwikmaganti9042
    @sathwikmaganti9042 4 ปีที่แล้ว

    Nice

  • @KarensheilahWafula
    @KarensheilahWafula หลายเดือนก่อน

    😊

  • @DaytonaStation
    @DaytonaStation 5 ปีที่แล้ว

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

  • @SiddharthRanjan6197
    @SiddharthRanjan6197 5 ปีที่แล้ว

    Why did you take a sphere after all?

    • @JordanEdmundsEECS
      @JordanEdmundsEECS  5 ปีที่แล้ว +2

      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.

  • @varunshrivastav8876
    @varunshrivastav8876 3 ปีที่แล้ว

    You make so Many mistakes in your videos

  • @alis5893
    @alis5893 5 ปีที่แล้ว

    This lecture was not explained properly along with some wrong calculations

    • @tim40gabby25
      @tim40gabby25 3 ปีที่แล้ว

      More detail, please.