I am looking into the $k\cdot p$ Hamiltonian approach to describe a semiconductor system. The simplest system appears to be the 2x2 system which can be visualised as: \begin{pmatrix} \epsilon(k) && \frac{\hbar kP}{m_0} \\ \frac{\hbar kP}{m_0} && \epsilon(k)-E_0 \end{pmatrix} this is for a two-band problem. Which I assume describes the upmost valence band and the lowest conduction band. This Hamiltonian can be diagonalized to give eigenvalues and eigenfunctions of the above Hamiltonian. In the literature, the eigenfunctions appear to be written as the coefficients of the eigenfunctions, i.e. $C_{v,k}$ and $C_{c,k}$. For example in [1] we have:

$$E(k)=\epsilon(k)-\frac{E_0}{2} \pm\left[\frac{E_0^2}{4}+\frac{\hbar^2}{m_0^2} P^2 k^2\right]^{1 / 2}$$ and $$ P_{c v}(k)=\left[C_{c k}^*(s) C_{v k}(i x)+C_{c k}^*(i x) C_{v k}(s)\right] P$$

I am trying to understand what are the $C_{v k}$ and $C_{c k}$ here. Are they simply the result of the eigenfunctions i.e. is $C_{v k}$ the two-component column vector that is obtained via the diagonalisation or is it a specific number inside the column vector?

In addition to this, for a more complicated system, one may need to go beyond the simple two-band model above. Therefore, say one describes a system in which we have one conduction band (lowest) and the three highest valence bands. Is the extension to the above as simple as extending the Hamiltonian to a 4x4 matrix which results in a more complicated,$E(k)$, and a four-component eigenfunction for each of the bands, which indicates how each band is affected by one another.

Any help would be greatly appreciated.

[1] - Lew Yan Voon LC, & Ram-Mohan, L. R. (1993). Tight-binding representation of the optical matrix elements: Theory and applications. Physical review. B, Condensed matter, 47(23), 15500–15508. https://doi.org/10.1103/physrevb.47.15500


1 Answer 1


I am not sure what $P_{cv}(k)$ and $C(s)$ are without some more context, but we can work through the $k\cdot p$ model from the beginning and continue the dialogue if you're still unsure. $k\cdot p$ literature is rife with different notations which can get very confusing. The 1D Schrodinger equation and Bloch's theorem:

$\big(p^2/(2m) + V(r)\big)\psi(x) = E\psi(x)$ and $\psi(x) = u_{n,k}(x)e^{ikx}$

The $k\cdot p$ model comes from substituting in the Bloch wavefunction into the Schrodinger equation, where get with a new equation just in terms of the Bloch functions $u$:

$\big( \frac{p^2}{2m} + \frac{\hbar k p}{m} + \frac{\hbar^2k^2}{2m} + V(x)\big)u_{n,k}(x) = Eu_{n,k}(x)$

Let's divide up the $k\cdot p$ Hamiltonian: $H_0 = p^2/(2m) + V(x)$ and $H_1 = \hbar kp/m + \hbar^2k^2/2m$

There are two ways we can approach this problem. We can treat the k-dependent terms with perturbation theory, or we can try to diagonalize the matrix directly in some minimal, tractable basis. The latter is called the Kane model and is what you are trying to do in your example.

We want to evaluate $H_0 + H_1$ in the basis of Bloch functions at k = 0. For two Bloch functions $|u_{c,0}\rangle$ and $|u_{v,0}\rangle$, the matrix elements of $H_0$ are the band extrema on the diagonal and zero on the off-diagonal. This is 0 and $-E_0$ in your notation. The matrix elements of $H_1$ are $\hbar^2k^2/2m = \epsilon(k)$ on the diagonal and $\frac{\hbar k}{m}\langle u_{c,0}|p|u_{v,0}\rangle \equiv \frac{\hbar k}{m}P$ on the off-diagonal. So, $P = -i\langle u_{c,0}|p|u_{v,0}\rangle$ comes from the matrix element of the momentum operator $p$ between the zone-center Bloch functions.

To get the k-dependent wavefunctions, we just solve the Schrodinger equation in matrix form using the energies obtained from the determinant.

The two-band Kane model describes the electron and the light hole in II-VI and III-V crystals (not the ``upmost" valence band, which is the heavy hole). To include more bands, you follow a completely equivalent procedure but it gets more complicated because (a) it's hard to analytically solve bigger matrices, and (b) you have to make bigger assumptions about your zone center Bloch functions to evaluate the matrix elements.

Although you will find 15-band Kane models used routinely for small-gap semiconductors like HgCdTe and sometimes even 30-band models, at that point much of the physical intuition and analytical elegance of $k\cdot p$ is gone and it is often better to move to tight-binding, DFT, etc. since they are more accurate (but computationally much more expensive).

A resource that's been floating around the internet is this PhD thesis which looks like it has decent intro to $k\cdot p$ with references in the appendix.

Hope this helps


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