k-p Method and Effective Mass Theory

August 18, 2020

Tags: [notes] [math] [quantum-mechanics]

\(\vec{k} \cdot \vec{p}\) method

Purpose: derive analytical expressions for the band dispersion and the effective masses in the immediate vicinity of a \(k-\)point at which all the single-electron states are known.

In other words, \(\vec{k} \cdot \vec{p}\) method \(\rightarrow\) extrapolation method applied to compute valence bands of SEMICONDUCTOR!

From Bloch’s theorem, the wavefumction takes a form of

\[\psi_{n \vec{k}}=e^{i \vec{k} \cdot \vec{r}} u_{n \vec{k}}(\vec{r})\tag{1.1}\]

with \(n\) being energy band index. Eq. (1.1) is a solution to single-electron Schrodinger eqn.

\[\left(\frac{\hat{p}^{2}}{2 m}+U(\vec{r})\right) \psi_{n \vec{k}}=E_{n \vec{k}} \vec{\psi}_{n \vec{k}}\tag{1.2}\]

Using (1.1) in (1.2) to give,

\[\begin{aligned}\left[\frac{\hat{p}^{2}}{2 m}+U(\vec{r})\right] e^{i \vec{k} \cdot \vec{r}} u_{n \vec{k}}(\vec{r})&=E_{n \vec{k}} \psi_{n k}\\ \left[\frac{\hat{p}^{2}}{2 m}+\frac{\hbar^{2} k^{2}}{2 m}+\frac{\hbar k \cdot \hat{p}}{m}+U(\vec{r})\right]u_{n\vec{k}}&=E_{n\vec{k}}u_{n\vec{k}}\\ \left[\frac{\hat{p}^{2}}{2 m}+\frac{\hbar k \cdot \hat{p}}{m}+U(\vec{r})\right]u_{n\vec{k}}&=\left(E_{n\vec{k}}-\frac{\hbar^{2} k^{2}}{2 m}\right)u_{n\vec{k}}. \end{aligned}\tag{1.3}\]

Now we use the information above to derive a secular relation for calculating \(E_{nk}\). Assume that we know \(u_{n k}(r)\) and \(E_{nk}\) for all “\(n\)” at \(\vec{k_0}\), then we can expand \(u_{n\vec{k}}\) at \(\vec{k}\) in terms of \(u_{n\vec{k_0}}\) as

\[u_{n \vec{k}}(\vec{r})=\sum_{n^{\prime}} C_{n^{\prime}}\left(\vec{k}, \vec{k}_{0}\right) u_{n^{\prime}\vec{k}_{0}}(\vec{r}) \tag{1.4},\]

and

\[\left[\frac{\hat{p}^{2}}{2 m}+\frac{\hbar \overrightarrow{k_{0}} \cdot \hat{p}}{m}+U(\vec{r})\right] u_{n \vec{k}_0}(\vec{r})=\left(E_{n\vec{k}_0}-\frac{\hbar^{2} \vec{k}^2_{0}}{2 m}\right) u_{n \vec{k}_{0}}(\vec{r})\tag{1.5}.\]

Use (1.4) and (1.5) in (1.3) to give:

\[\begin{aligned} &\sum_{n^{\prime}} C_{n^{\prime}}(\vec{k}, \vec{k}_0)\left[\left(E_{n^{\prime}\vec{k}_0} -\frac{\hbar \overrightarrow{k_{0}}^{2}}{2 m}\right) u_{n^{\prime} \vec{k_{0}}}(\vec{r})+\frac{\hbar}{i m}\left(\vec{k}-\vec{k}_{0}\right) \cdot \nabla u_{n^{\prime}\vec{k}_{0}}(\vec{r})\right]\\ &=\sum_{n^{\prime}} C_{n^{\prime}}(\vec{k}, \vec{k}_0)\left(E_{n \vec{k}}-\frac{\hbar^{2} k^{2}}{2 m}\right) u_{n^{\prime}\overrightarrow{k_{0}}}(\vec{r}). \end{aligned}\]

Mutiply both sides by \(u^*_{nk_0}(\vec{r})\) and integrating over \(\vec{r}\) yields:

\[\begin{aligned}&\sum_{n^{\prime}} C_{n^{\prime}}\left(\vec{k}, \vec{k}_{0}\right)\left[\left(E_{n^{\prime}\vec{k}_{0}} -\frac{\hbar^{2} \vec{k}_{0}^{2}}{2 m}\right) \delta_{n n^{\prime}}+\frac{\hbar}{m}\left(\vec{k}-\overrightarrow{k_{0}}\right) \cdot \vec{P}_{n n^{\prime}}\left(k_{0}\right)\right]\\ &=\sum_{n^{\prime}} C_{n^{\prime}}\left(\vec{k}, \overrightarrow{k_{0}}\right)\left(E_{n \vec{k}}-\frac{\hbar^{2} \vec{k}^{2}}{2 m}\right) \delta_{n n^{\prime}}. \end{aligned}\tag{1.6}\]

where

\[\int d \vec{r} u_{n\vec{k}_0}^{*} u_{n^{\prime} k_{0}}=\delta_{n n^{\prime}}\]

and the momentum matrix element is

\[-i \int d \vec{r} u_{n\vec{k}} \nabla u_{n^{\prime}\vec{k}_{0}}=\vec{P}_{n n^{\prime}}\left(k_{0}\right).\]

Move the RHS of (1.6) to LHS to give:

\[\begin{aligned}\sum_{n^{\prime}}&\left\{\frac{\hbar}{m}\left(\vec{k}-\vec{k}_{0}\right) \cdot \vec{P}_{n n'}\left(\vec{k}_{0}\right)+\right.\\ &\left.\left[E_{n \vec{k}_{0}}+\frac{\hbar^{2}}{2 m}\left(\vec{k}^{2}-\vec{k}_{0}^{2}\right)-E_{n \vec{k}}\right] \delta_{n n'}\right\}C_{n'}(\vec{k},\vec{k}_0)=0.\end{aligned}\tag{1.7}\]

The “secular equation” for \(E_{nk}\) follows from the condition for the existence of nontrivial solutions to this set of homogeneous linear equations,

\[det\left|\frac{\hbar}{m}\left(\vec{k}-\vec{k}_{0}\right) \cdot \vec{P}_{n n'}\left(\vec{k}_{0}\right)+\left[E_{n \vec{k}_{0}}+\frac{\hbar^{2}}{2 m}\left(\vec{k}^{2}-\vec{k}_{0}^{2}\right)-E_{n \vec{k}}\right] \delta_{n n'}\right|=0.\tag{1.8}\]

Special things about the derivation of (1.8)

No restriction on \(\vec{k}\);
If only an under-complete set of \(u_{nk_0}\) is used, the k-p method is accurate only for \(\vec{k}\) near \(\vec{k}_0\).
When sufficiently large set of \(u_{nk_0}\) is used, the band structure in FBZ can be approximated.

Effective mass tensor theory

Purpose: The k-p method above deals with unperturbed single-electron Schrodinger equation. Using effective mass tensor theory, we are able to take into account non-periodic perturbation in the Hamiltonian operator, and study the motion of electron or holes in a perturbed potential.

When \(\vec{k}\) approaches an extreme point of an energy band \(\vec{k_0}\), we can expand \(E_{n\vec{k}}\) as a Taylor series about \(\vec{k_0}\):

\[\begin{aligned} E_{n \vec{k}} & \approx E_{n \vec{k}_{0}}+\frac{1}{2} \sum_{\alpha \beta}\left(\frac{\partial^{2} E_{n\vec{k}} }{\partial k_{\alpha} \partial k_{\beta}}\right)\left(\vec{k}-\overrightarrow{k_{0}}\right)_{\alpha}\left(\vec{k}-\vec{k}_{0}\right)_{\beta} \\ &=E_{n k_{0}}+\left.\sum_{\alpha \beta}\left(\frac{\hbar^{2}}{2 m^{*}}\right)_{\alpha \beta}\right|_{\vec{k}_{0}}\left(\vec{k}-\vec{k}_{0}\right)_{\alpha}\left(\vec{k}-\vec{k}_{0}\right) \beta \end{aligned}\]

Where the inverse effective mass tensor element is defined as

\[\quad\left(\frac{1}{m^{*}}\right)_{\alpha \beta}=\frac{1}{\hbar^{2}} \frac{d^{2} E_{n \vec{k}}}{d k_{\alpha} d k_{\beta}}\tag{2.1}.\]

If the solid possesses the cubic symmetry then the effective mass tersor is a diagonal matrix.

Consider the ettective mass theory for a single nondegenerate energy band and let the single-electron Hamiltonian to be

\[\widehat{H}=\widehat{H}_0+V(\vec{r})\]

where

\[\widehat{H}_{0}|n \vec{k}\rangle=E_{n \vec{k}}|n \vec{k}\rangle\]

and

\[|\psi\rangle=\sum_{n\vec{k}}^{\prime} a_{n\vec{k}} |n \vec{k}\rangle\]

with the summation over FBZ only. Using the summation we have

\[\sum_{n_{k}}^{\prime} a_{n \vec{k}}\left(E_{n k}+V\right)|n \vec{k}\rangle=E\sum_{n\vec{k}}^{\prime} a_{n\vec{k}} |n \vec{k}\rangle.\]

Multiply \(\|n'\vec{k'}\rangle\) on both sides to give

\[\left\langle n^{\prime} \vec{k}^{\prime}\left|\sum_{n \vec{k}}^{\prime} a_{n \vec{k}}\left(E_{n \vec{k}}+v\right)\right| n \vec{k}\right\rangle=\left\langle n^{\prime} \vec{k}^{\prime}\right| E \sum_{nk}^{\prime} a_{n\vec{k}} \left|n\vec{k}\right\rangle\tag{2.2}.\]

From

\[\left\langle n' \vec{k}^{\prime} \mid n \vec{k}\right\rangle=\delta_{n n} \cdot \delta_{k k^{\prime}},\]

we have from (2.2) that

\[\left(E_{n k}-E\right) a_{n k}+\sum_{n=k^{\prime}}^{\prime} V_{n \vec{k} n^{\prime} \vec{k}^{\prime}} a_{n^{\prime}\vec{k^{\prime}} }=0\]

where

\[V_{n\vec{k}n'\vec{k}'}=\left\langle n \vec{k}|V| n^{\prime} \vec{k}^{\prime}\right\rangle=\int d \vec{r} e^{-i (\vec{k}-\vec{k}^{\prime}) \cdot \vec{r}} u_{n \vec{k}}^{*}(\vec{r}) V u_{n^{\prime}\vec{k}'}\tag{2.3}.\]

If we express \(u_{nk}^*u_{n'k'}\) as a Fourier series w.r.t the reciprocal lattice vector, \(\mathbf{K}\),

\[u_{n \vec{k}}^{*} u_{n^{\prime}\vec{k'}}=\sum_{\mathbf{K}} C_{\mathbf{K}}\left(n \vec{k}, n^{\prime} \vec{k}^{\prime}\right) e^{i \mathbf{K} \cdot \vec{r}}\]

and

\[C_{\mathbf{K}}\left(n \vec{k}, n^{\prime} \vec{k}^{\prime}\right)=\frac{1}{V} \int_{V} d \vec{r} e^{-i \mathbf{K} \cdot \vec{r}} u_{n \vec{k}}^{*} u_{n^{\prime}\vec{k'}}\]

Then

\[\begin{aligned} V_{n \vec{k}, n^{\prime} \vec{k}^{\prime}} &=\sum_{k} C_{\mathbf{K}}\left(n \vec{k}, n^{\prime} \vec{k}^{\prime}\right) \int d r e^{-i\left(\vec{k}-\vec{k}^{\prime}-\mathbf{K}\right) \cdot \vec{r}} V(r) \\ &=\sum_{\mathbf{K}} C_{\mathbf{K}}\left(n \vec{k}, n^{\prime} \vec{k}^{\prime}\right) V_{\vec{k}-\vec{k}^{\prime}-\mathbf{K}} \end{aligned}\]

Note that \(V\) is usually small, so it doesn’t cause band mixing. Thus we can ignore the matrix element where \(n\neq n'\) to give

\[\begin{aligned}\sum^{\prime}_{n'\vec{k'}}V_{n\vec{k},n'\vec{k}'}a_{n'\vec{k}'}&\approx\sum^{\prime}_{\vec{k'}}V_{n\vec{k},n\vec{k}'}a_{n\vec{k}'}\\ &=\sum_{\mathbf{K}}\sum_{\vec{k'}}C_{\mathbf{K}}(n\vec{k},n\vec{k}')V_{\vec{k}-\vec{k'}-\mathbf{K}}a_{n\vec{k'}}.\end{aligned}\tag{2.4}\]

Change summation variable \(\vec{k}^{\prime}\rightarrow\vec{k}^{\prime}+\mathbf{K}\), we have from (2.4)

\[\sum^{\prime}_{n'\vec{k'}}V_{n\vec{k},n'\vec{k}'}a_{n'\vec{k}'}\approx\sum_{\vec{k}^{\prime}} \sum_{\mathbf{K}} C_{\mathbf{K}}\left(n \vec{k}, n \vec{k}^{\prime}\right) V_{\vec{k}-\vec{k'}} a_{n \vec{k}^{\prime}} \tag{2.5},\]

where we applied the periodicity of \(a_{n\vec{k}}\) and \(C_{K}\) on reciprocal Latrice. Now we ignore the terms with \(C_{K\neq0}\) to give

\[\begin{aligned}\sum_{n=1} V_{n \vec{k}n^{\prime} \vec{k}^{\prime}} a_{n\vec{k}^{\prime}} &\approx \sum_{k^{\prime}} C_{0}\left(n \vec{k}, n \vec{k}^{\prime}\right) V_{k-k^{\prime}} a_{n \vec{k}^{\prime}}\\ &=\frac{1}{V} \sum_{\vec{k}^{\prime}} V_{\vec{k}-\vec{k}^{\prime}} a_{n \vec{k}^{\prime}} \end{aligned}\tag{2.6}.\]

According to the definition of \(C_{K}\) we have

\[C_{0}(n \vec{k}, n \vec{k})=\frac{1}{N v_{c}}\]

with \(v_c\) being unit cell volumn. For valence electron \(u_{n\vec{k}}\) varies slowly with \(\vec{k}\), so we have

\[C_{0}\left(n \vec{k}, n \vec{k}^{\prime}\right) \approx C_{0}\left(n\vec{k}, n \vec{k}\right)=\frac{1}{N v_{c}}\]

Therefore

\[(E_{n \vec{k}}-E) a_{n \vec{k}}+\frac{1}{N v_{c}} \sum_{\vec{k}^{\prime}} V_{\vec{k}-\vec{k}^{\prime}} a_{n \vec{k}^{\prime}}=0 \tag{2.7}\]

To transtiom (2.7) into a form for real space, we multiply both sides With \(e^{i\vec{k}\cdot r}\) and sum over all \(\vec{k}\) in reciprocal space to give

\[\sum_{\vec{k}}\left[\left(E_{n k}-E\right) a_{n \vec{k}} e^{i \vec{k} \cdot r}+\frac{1}{Nv_c}\sum_{\vec{k}'}V_{\vec{k}-\vec{k}'}a_{n\vec{k}'}e^{i\vec{k}\cdot\vec{r}}\right]=0\tag{2.8}.\]

where the envelop function is defined as

\[F_{n}(\vec{r})=\sum_{\vec{k}} e^{i \vec{k} \cdot \vec{r}} a_{n \vec{k}}.\]

In (2.8) we also rewrite \(E_{n\vec{k}}\) as \(E_n(\vec{k})\) with the operator being \(E_{n}(-i\nabla)\). The second term in (2.8) is then

\[\frac{1}{N v_{L}} \int d \vec{r'} \sum_{\vec{k} \vec{k}^{\prime}} e^{i \vec{k} \cdot(\vec{r}-\vec{r'})} e^{i \vec{k'} \cdot \vec{r}^{\prime}} V\left(\vec{r}^{\prime}\right) a_{n \vec{k'}}=V(\vec{r}) F_{n}(\vec{r})\]

Thus,

\[\left[\widehat{E}_{n}(-i \nabla)+V(\vec{r})\right] F_{n}(\vec{r})=E F_{n}(\vec{r}) \tag{2.9}.\]

To solve the original perturbed Schrodinger equation, we can first solve (2.9) to get envelop function \(F_n\), and then use it to construct the solution to the Schrodinger equation as

\[\left\langle\vec{r} \mid \vec{\psi}_{n}\right\rangle=\vec{\psi}_{n}(\vec{r})=\sum_{\vec{k}} a_{n\vec{k}} \psi_{n \vec{k}}(\vec{r}) \approx\sum_{n\vec{k}} a_{n \vec{k}} e^{i \vec{k} \cdot \vec{r}} u_{n \vec{k}_{0}}(\vec{r})=F_nu_{n\vec{k}_0}.\]

To get the format of \(\hat{E}_n\) in (2.9), we first expand \(E_n\) about \(\vec{k}=0\) to the second order

\[E_{n}(\vec{k})=E_{n}(0)+\sum_{\alpha \beta}\left(\frac{\hbar^{2}}{2 m^{*}}\right)_{\alpha \beta} k_{\alpha} k_{\beta} \tag{2.10}\]

\[\left[\sum_{\alpha_{\beta}}\left(\frac{\hbar^{2}}{2 m^{*}}\right)_{\alpha{\beta}}\left(-i \frac{\partial}{\partial k_{\alpha}}\right)\left(-i \frac{\partial}{\partial k_{\beta}}\right)+V(\vec{r})\right] F_{n}=(E-E_n(0))F_n.\]

Summary

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