Meaning of soulution of Central Equation: Nearly free electron model

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Main Question or Discussion Point

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,

[itex]
(\lambda_k - \epsilon)C_k + \Sigma_G U_G C_{k-G} = 0
[/itex]



where [itex] \lambda_k = (\hbar^2 k^2) / (2m_e) [/itex], [itex] G [/itex] is the set of possible reciprocal lattice vectors and [itex] C_k [/itex] is det fourier coefficients corresponding to the solution of the Schrödinger eq.:

[itex]
\psi_k = \Sigma_k C_k e^{ikx}
[/itex].


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 [itex] \epsilon_\pm = \lambda_k \pm U_0 [/itex] for some [itex]k[/itex]. Then I am told the energy gap, [itex] \epsilon_{gap} = \epsilon_+ - \epsilon_- [/itex], between two energy bands for this [itex]k[/itex] 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 [itex] C_k [/itex], how? .. Assuming we know the periodicity of the potential and [itex] k [/itex].
 

Answers and Replies

  • #2
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,

[itex]
(\lambda_k - \epsilon)C_k + \Sigma_G U_G C_{k-G} = 0
[/itex]



where [itex] \lambda_k = (\hbar^2 k^2) / (2m_e) [/itex], [itex] G [/itex] is the set of possible reciprocal lattice vectors and [itex] C_k [/itex] is det fourier coefficients corresponding to the solution of the Schrödinger eq.:

[itex]
\psi_k = \Sigma_k C_k e^{ikx}
[/itex].


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 [itex] \epsilon_\pm = \lambda_k \pm U_0 [/itex] for some [itex]k[/itex]. Then I am told the energy gap, [itex] \epsilon_{gap} = \epsilon_+ - \epsilon_- [/itex], between two energy bands for this [itex]k[/itex] 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 [itex] C_k [/itex], how? .. Assuming we know the periodicity of the potential and [itex] k [/itex].
yes we can;
in this case C is equal to: +_sgn(U)C
u can find the exact equation in,Solid State Physics By Ashcroft&Mermin.chapter9,equation (9.29)-
 

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