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Density of States

Discussion in 'Electronic Basics' started by Sunil, Feb 5, 2007.

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  1. Sunil

    Sunil Guest

    I was reading up on the density of states and i hit on a snag...

    I dint understand how they take the "volume" of the density of
    states...i mean density of states corresponds to the density of energy
    level can u consider the "volume" of energy level??

    Moreover....can you also give me the qualitative reason for the
    density of states function??

    thank you
  2. redbelly

    redbelly Guest

    Um, this really isn't a "basic electronics" question.


  3. Bob Pownall

    Bob Pownall Guest

    I'd have to agree.

    I'd suggest that the OP try posting to sci.engr.semiconductors instead.

    Bob Pownall
  4. Its exactly what it says. Given a specific energy level E we can find the
    density of the states at that level per unit volume per unit energy(we have
    to normalize so its more useful).

    Now any time your talking about density you must count something and then
    divide by some related factor that in some sense normalizes it. Mass
    density is the the mass(which is effectively the "count" of the number of
    atoms) divided by the volume that contains that mass. Ofcourse this is a
    little easier to do as we don't have to know the mass of the atoms nor how
    many because we have newtons laws to help us out. (we just weigh the thing
    and measure its volume through whatever means we can).

    So in this case we have to somehow count the states of the electrons(or
    whatever were describing). Think of a material that has its electrons all
    in different states. We sorta want to have a means of knowing the density of
    those states in the material. Basically a proportion of the states that
    contribute to the total energy in the volume.

    That is, we need to find all the states that contribute to energy E and then
    divide by the volume we looked at and divide by the total energy. Think of
    an atom with its electrons in different states. These states correspond to
    definite energy levels but what proportions correspond to what energy

    For a free electron we have E = (hbar*k/2/m)^2. This gives us the energy in
    terms of phase space.

    The key here is that phase space is a space of states. Each unit in the
    phase space corresponds to a specific state. That means now its really easy
    to add up the states because we just have to integerate over a volume in it.
    But what volume? Well we know that its spheres because constant energy
    levels result in sphere in k space. Remember, if you solve shrodingers wave
    equation you get that psi_k is dependent on k. Think of k as an independent
    variable you then just have to add up the right number of psi_k's that
    depend on E because thats what were after. (states in terms of energy)

    So take our free particle,

    E = (hbar*k)^2/2/m

    and then integrate over a sphere with radius k = sqrt(2mE/hbar^2). Remember,
    essentially we want to count all the states within our constant energy
    surface. In k space we just have to integrate. Since we are dealing with
    spheres in k space and volume tells us the number of k spaces(if say, we
    were dealing with a 1x1x1 = 1 cube then we would have exactly one state, if
    a 2x3x1 cube we would have 6 states). Therefor the number of states that
    give us an energy less or equal to E is

    2*4/3*pi* sqrt(2mE/hbar^2)^3

    which is simply the volume of a sphere in k space with radius
    sqrt(2mE/hbar^2). The extra factor of 2 comes in because each state is
    actually 2 states(spin up and spin down for electron).

    So now we are counting the number of states that have at most energy E. But
    here we assumed that the unit length was a state when in fact its L/2/pi. (I
    didn't scale my axis because I forgot). This means that my volume is off by
    a factor of (L/2/pi)^3. (instead of counting unit cubes I have to count
    different sized cubes and that means I just have to multiply by the scaling

    So what I really have is

    (L/2/pi)^3*2*4/3*pi* sqrt(2mE/hbar^2)^3
    = L^3/3/pi^2/hbar^3*(2mE)^(3/2)

    This counts the states who has energy <= E. Its not hard but you have to
    just not think about it as being to hard. Just count the states. To do that
    you just have to know 3 facts:

    Each cube with edge L/2/pi is a state in k space.
    A sphere of radius E represents a constant energy.
    Each state needs to be counted twice for an valence electron.

    So now we have counted the total number of states that give an energy <= E
    in a volume of L^3(were talking about a free electron in a cube with edge

    Now we just have divide out the total volume and differentiate it w.r.t E to
    get number of states that contribute to energy E(not <= E). Actually with
    would be E*dE.

    Dividing out the volume(effectively normalizing it w.r.t volume) gives


    and differentiating will give you the density of states of E. Now I'm not
    going to differentiate it but approximate it. We are trying to find the
    number of states that are between two energy levels that are close together,
    say E + dE and E.

    [1/3/pi^2/hbar^3*(2mE + dE)^(3/2) - 1/3/pi^2/hbar^3*(2mE)^(3/2)]/dE

    but as you can see this really is just the derivative. We had to divide by
    dE so the limit would not be zero but technically its the
    derivative*dE(which you might not see the factor of dE there in your

    The resulting equation for the density of states is

    d/dE[1/3/pi^2/hbar^3*(2mE)^(3/2)] = [1/2/pi^2/hbar^3*(2m)^(3/2)*E^(1/2)]dE

    Hopefully that made sense. Essentially we go into phase space where its much
    easier to count states, add up all the states that give us an energy between
    E and E + dE(which is easier if we first just count those that give us an
    energy <= E), this evolves finding volumes over spheres, then substitute k =
    K(E) into the equation to get it in terms of E and divide by the volume to
    normalize it w.r.t to volume.

    I'm not sure if I did a decent explination as it was off the top of my head
    but I'm sure there are plenty of resources on the net. Its not necessarily
    a hard thing to understand but you need not get confused on the details. Its
    just a counting problem. (and if I didn't say it I was doing it for a free
    particle in a box because thats easier)

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