Support

Support Options

Submit a Support Ticket

 
Home › Questions and Answers › defining the 3D electron density

Questions and Answers

Status: Closed

0 Like

George Pau

defining the 3D electron density

Hi,

In the reference cited, Nanowire uses the NEGF formalism and the 3D density is expressed in terms of that formalism. Could someone tell me what the equation will be like in terms of wavefunctions? I do not think it is very different from the reference but I would like to see what the exact form is like if one decide not to use NEGF formalism.

Thanks, George

Tags:

Report abuse

Chosen Answer

  1. 0 Dislike

    Saumitra Raj Mehrotra

    Hello,

    I could not understand the reference you refer to. However to answer about calculating 3D electron density.Nanowire mode space approach breaks the 3D problem into 2D Schodinger and 1D NEGF transport problem. On solving 1D NEGF we get 1D electron density (Ne1D).Electron density in terms of wavefunction can now be written as,

    Ne3D=Sum over mode m(Ne1D.{psi_m.psi*_m)

    A more correct representation should be (code being updated), Ne3D=Sum over mode m over mode n

    Refer:http://www.iis.ee.ethz.ch/~schenk/JApplPhys_100_043713.pdf

    I have written nth mode here which can be understood as nth subband.

    I hope it helps

    thank, Saumitra

    Report abuse

    1. George Pau

      Let’s suppose I have a lot of computing resources and I could solve the open Schrodinger equation in 3D. So, for each electron of energy E coming in from source or drain, I can compute an associated wavefunction, say \phi(E). I can then determine the density directly based on \phi(E). The question then is what is the resulting formula? Is the density given by \int_k |\phi(E(k))|^2 f(E(k)) where k is the wavevector associated with E based on some E-k relation and f is the Fermi-Dirac function? Or am I missing some constants?

      Thanks, George

      Report abuse

      1. Mathieu Luisier

        Your equation is basically correct. You must not forget to sum over all the contacts that you use to inject states into your device. Furthermore, the sum over k should be transformed into an integral, then a variable transformation should be applied (k->E) so that a term |dE/dk|^ appears in the equation. You can take a look at the documentation and the references of OMENnanowire. This tool uses a wave function approach instead of NEGF.

        Mathieu

        Report abuse

Responses (1)

  1. 0 Like 1 Dislike

    Daniel Lee Whitenack

    Hello George,

    If you have the wavefunction (call it phi) for one particle in one dimension, the probability density is defined as the square modulus of phi:

    n(x) = |phi(x)|^2

    Where this should satisfy the condition that if you integrate over all space you just get 1. (phi should be normalized) In 3D, say you have the one particle wavefunction phi(x,y,z). Again the density is interpreted as just |phi(x,y,z)|^2.

    Now for many particles you are going to have the wavefuntion as a function of all of the particle coordinates:

    phi(r1,r2,r3,…)

    Where r1, r2, r3 specify the location of particle 1, 2, 3, etc. However, the density is always going to be a funtion of just three spatial coordinates. So now the density is defined as

    n® = N int |phi(r,r2,r3,…)|^2 dr2 dr3 …

    Where N is the number of particles. Therefore if your wavefunction is normalized, the density integrated over all space is N.

    Note I have ignored spin. If you want to include spin, you have to include a sum over all spin states. (these are discrete as opposed to the continuous position)

    Hope that helps at least a little, Daniel

    Report abuse

Get Help

Get Involved

Legal

nanoHUB.org, a resource for nanoscience and nanotechnology, is supported by the National Science Foundation and other funding agencies.