- #1
mhsd91
- 22
- 4
Considering the Nearly Free Electron model of solids, where we assume the valence electrons of some one dimensional(!) solid to move in a weak, periodic (with respect to the solids lattice constant) potensial.
We may derive (which I assume you are familiare with, and will not do here) the central equation as an algebraic reformulation of the time independent Schrödinger eq. corresponding to the model/potential at hand,
<br /> (\lambda_k - \epsilon)C_k + \Sigma_G U_G C_{k-G} = 0<br />
where \lambda_k = (\hbar^2 k^2) / (2m_e), G is the set of possible reciprocal lattice vectors and C_k is det Fourier coefficients corresponding to the solution of the Schrödinger eq.:
<br /> \psi_k = \Sigma_k C_k e^{ikx}<br />.
My problem is that I do not understand what exactly we do find if we solve the central equation.
Say for instance I solve it and find the energy \epsilon_\pm = \lambda_k \pm U_0 for some k. Then I am told the energy gap, \epsilon_{gap} = \epsilon_+ - \epsilon_-, between two energy bands for this k at hand. Please (dis)confirm!?
... and then WHICH two bands are this gap between? (If that makes sense). And is it possible to find values for C_k, how? .. Assuming we know the periodicity of the potential and k.
We may derive (which I assume you are familiare with, and will not do here) the central equation as an algebraic reformulation of the time independent Schrödinger eq. corresponding to the model/potential at hand,
<br /> (\lambda_k - \epsilon)C_k + \Sigma_G U_G C_{k-G} = 0<br />
where \lambda_k = (\hbar^2 k^2) / (2m_e), G is the set of possible reciprocal lattice vectors and C_k is det Fourier coefficients corresponding to the solution of the Schrödinger eq.:
<br /> \psi_k = \Sigma_k C_k e^{ikx}<br />.
My problem is that I do not understand what exactly we do find if we solve the central equation.
Say for instance I solve it and find the energy \epsilon_\pm = \lambda_k \pm U_0 for some k. Then I am told the energy gap, \epsilon_{gap} = \epsilon_+ - \epsilon_-, between two energy bands for this k at hand. Please (dis)confirm!?
... and then WHICH two bands are this gap between? (If that makes sense). And is it possible to find values for C_k, how? .. Assuming we know the periodicity of the potential and k.
