Physics:Luttinger–Kohn model

A flavor of the k·p perturbation theory used for calculating the structure of multiple, degenerate electronic bands in bulk and quantum well semiconductors. The method is a generalization of the single band k·p theory. In this model the influence of all other bands is taken into account by using Löwdin's perturbation method.

Background

All bands can be subdivided into two classes:

• Class A: six valence bands (heavy hole, light hole, split off band and their spin counterparts) and two conduction bands.
• Class B: all other bands.

The method concentrates on the bands in Class A, and takes into account Class B bands perturbatively.

We can write the perturbed solution [math]\displaystyle{ \phi^{}_{} }[/math] as a linear combination of the unperturbed eigenstates [math]\displaystyle{ \phi^{(0)}_{i} }[/math]:

[math]\displaystyle{ \phi = \sum^{A,B}_{n} a_{n} \phi^{(0)}_{n} }[/math]

Assuming the unperturbed eigenstates are orthonormalized, the eigenequation are:

[math]\displaystyle{ (E-H_{mm})a_m = \sum^{A}_{n\neq m}H_{mn}a_{n} + \sum^{B}_{\alpha \neq m}H_{m\alpha}a_{\alpha} }[/math],

where

[math]\displaystyle{ H_{mn} = \int \phi^{(0)\dagger}_{m} H \phi^{(0)}_{n}d^3 \mathbf{r} = E^{(0)}_{n}\delta_{mn}+H^{'}_{mn} }[/math].

From this expression we can write:

[math]\displaystyle{ a_{m} = \sum^{A}_{n\neq m} \frac{H_{mn}}{E-H_{mm}} a_{n} + \sum^{B}_{\alpha\neq m} \frac{H_{m\alpha}}{E-H_{mm}} a_{\alpha} }[/math],

where the first sum on the right-hand side is over the states in class A only, while the second sum is over the states on class B. Since we are interested in the coefficients [math]\displaystyle{ a_{m} }[/math] for m in class A, we may eliminate those in class B by an iteration procedure to obtain:

[math]\displaystyle{ a_{m} = \sum^{A}_{n} \frac{U^{A}_{mn} - \delta_{mn} H_{mn}}{E-H_{mm}} a_{n} }[/math],

[math]\displaystyle{ U^{A}_{mn} = H_{mn} + \sum^{B}_{\alpha\neq m} \frac{H_{m\alpha}H_{\alpha n}}{E-H_{\alpha\alpha}} + \sum_{\alpha,\beta\neq m,n;\alpha\neq\beta} \frac{H_{m \alpha} H_{\alpha\beta}H_{\beta n} }{(E-H_{\alpha\alpha})(E-H_{\beta\beta})} + \ldots }[/math]

Equivalently, for [math]\displaystyle{ a_{n} }[/math] ([math]\displaystyle{ n \in A }[/math]):

[math]\displaystyle{ a_{n} = \sum^{A}_{n} (U^{A}_{mn} - E\delta_{mn})a_{n} = 0, m \in A }[/math]

and

[math]\displaystyle{ a_{\gamma} = \sum^{A}_{n} \frac{ U^{A}_{\gamma n} - H_{\gamma n}\delta_{\gamma n}}{E-H_{\gamma\gamma}} a_{n} = 0, \gamma \in B }[/math].

When the coefficients [math]\displaystyle{ a_{n} }[/math] belonging to Class A are determined so are [math]\displaystyle{ a_{\gamma} }[/math].

Schrödinger equation and basis functions

The Hamiltonian including the spin-orbit interaction can be written as:

[math]\displaystyle{ H = H_0 + \frac{\hbar}{4m_{0}^{2}c^{2}}\bar{\sigma}\cdot\nabla V \times \mathbf{p} }[/math],

where [math]\displaystyle{ \bar{\sigma} }[/math] is the Pauli spin matrix vector. Substituting into the Schrödinger equation in Bloch approximation we obtain

[math]\displaystyle{ H u_{n\mathbf{k}}(\mathbf{r}) = \left( H_0 + \frac{\hbar}{m_{0}}\mathbf{k}\cdot\mathbf{\Pi} + \frac{\hbar^2 k^2}{4m_{0}^{2}c^{2}} \nabla V \times \mathbf{p} \cdot \bar{\sigma} \right) u_{n\mathbf{k}}(\mathbf{r}) = E_{n}(\mathbf{k}) u_{n\mathbf{k}}(\mathbf{r}) }[/math],

where

[math]\displaystyle{ \mathbf{\Pi} = \mathbf{p} + \frac{\hbar}{4m_{0}^{2}c^{2}}\bar{\sigma} \times \nabla V }[/math]

and the perturbation Hamiltonian can be defined as

[math]\displaystyle{ H' = \frac{\hbar}{m_0}\mathbf{k}\cdot\mathbf{\Pi}. }[/math]

The unperturbed Hamiltonian refers to the band-edge spin-orbit system (for k=0). At the band edge, conduction band Bloch waves exhibit s-like symmetry, while valence band states are p-like (3-fold degenerate without spin). Let us denote these states as [math]\displaystyle{ |S \rangle }[/math], and [math]\displaystyle{ |X \rangle }[/math], [math]\displaystyle{ |Y \rangle }[/math] and [math]\displaystyle{ |Z \rangle }[/math] respectively. These Bloch functions can be pictured as periodic repetition of atomic orbitals, repeated at intervals corresponding to the lattice spacing. The Bloch function can be expanded in the following manner:

[math]\displaystyle{ u_{n \mathbf{k}} (\mathbf{r}) = \sum^{A}_{j'} a_{j'}(\mathbf{k}) u_{j'0}(\mathbf{r}) + \sum^{B}_{\gamma} a_{\gamma}(\mathbf{k}) u_{\gamma 0}(\mathbf{r}) }[/math],

where j' is in Class A and [math]\displaystyle{ \gamma }[/math] is in Class B. The basis functions can be chosen to be

[math]\displaystyle{ u_{10}(\mathbf{r}) = u_{el}(\mathbf{r}) = \left | S\frac{1}{2},\frac{1}{2} \right \rangle = \left|S\uparrow\right\rangle }[/math]

[math]\displaystyle{ u_{20}(\mathbf{r}) = u_{SO}(\mathbf{r}) = \left | \frac{1}{2},\frac{1}{2} \right \rangle = \frac{1}{\sqrt 3} |(X+iY)\downarrow\rangle + \frac{1}{\sqrt 3} |Z\uparrow\rangle }[/math]

[math]\displaystyle{ u_{30}(\mathbf{r}) = u_{lh}(\mathbf{r}) = \left | \frac{3}{2},\frac{1}{2} \right \rangle = -\frac{1}{\sqrt 6} |(X+iY)\downarrow\rangle + \sqrt{\frac{2}{3}} |Z\uparrow\rangle }[/math]

[math]\displaystyle{ u_{40}(\mathbf{r}) = u_{hh}(\mathbf{r}) = \left | \frac{3}{2},\frac{3}{2} \right \rangle = -\frac{1}{\sqrt 2}|(X+iY)\uparrow\rangle }[/math]

[math]\displaystyle{ u_{50}(\mathbf{r}) = \bar{u}_{el}(\mathbf{r}) = \left | S\frac{1}{2},-\frac{1}{2} \right \rangle = -|S\downarrow\rangle }[/math]

[math]\displaystyle{ u_{60}(\mathbf{r}) = \bar{u}_{SO}(\mathbf{r}) = \left | \frac{1}{2},-\frac{1}{2} \right \rangle = \frac{1}{\sqrt 3} |(X-iY)\uparrow\rangle - \frac{1}{\sqrt 3} |Z\downarrow\rangle }[/math]

[math]\displaystyle{ u_{70}(\mathbf{r}) = \bar{u}_{lh}(\mathbf{r}) = \left | \frac{3}{2},-\frac{1}{2} \right \rangle = \frac{1}{\sqrt 6} |(X-iY)\uparrow\rangle + \sqrt{\frac{2}{3}} |Z\downarrow\rangle }[/math]

[math]\displaystyle{ u_{80}(\mathbf{r}) = \bar{u}_{hh}(\mathbf{r}) = \left | \frac{3}{2},-\frac{3}{2} \right \rangle = -\frac{1}{\sqrt 2}|(X-iY)\downarrow\rangle }[/math].

Using Löwdin's method, only the following eigenvalue problem needs to be solved

[math]\displaystyle{ \sum^{A}_{j'} (U^{A}_{jj'}-E\delta_{jj'})a_{j'}(\mathbf{k}) = 0, }[/math]

where

[math]\displaystyle{ U^{A}_{jj'} = H_{jj'} + \sum^{B}_{\gamma \neq j,j'} \frac{H_{j\gamma}H_{\gamma j'}}{E_0-E_{\gamma}} = H_{jj'} + \sum^{B}_{\gamma \neq j,j'} \frac{H^{'}_{j\gamma}H^{'}_{\gamma j'}}{E_0-E_{\gamma}} }[/math],

[math]\displaystyle{ H^{'}_{j\gamma} = \left \langle u_{j0} \right | \frac{\hbar}{m_0} \mathbf{k} \cdot \left ( \mathbf{p} + \frac{\hbar}{4 m_0 c^2} \bar{\sigma} \times \nabla V \right ) \left | u_{\gamma 0} \right \rangle \approx \sum_{\alpha} \frac{\hbar k_{\alpha}}{m_0}p^{\alpha}_{j \gamma}. }[/math]

The second term of [math]\displaystyle{ \Pi }[/math] can be neglected compared to the similar term with p instead of k. Similarly to the single band case, we can write for [math]\displaystyle{ U^{A}_{jj'} }[/math]

[math]\displaystyle{ D_{jj'} \equiv U^{A}_{jj'} = E_{j}(0)\delta_{jj'} + \sum_{\alpha\beta} D^{\alpha\beta}_{jj'}k_{\alpha}k_{\beta}, }[/math]

[math]\displaystyle{ D^{\alpha\beta}_{jj'} = \frac{\hbar^2}{2 m_0} \left [ \delta_{jj'}\delta_{\alpha\beta} + \sum^{B}_{\gamma} \frac{ p^{\alpha}_{j\gamma}p^{\beta}_{\gamma j'} + p^{\beta}_{j\gamma}p^{\alpha}_{\gamma j'} }{ m_0 (E_0-E_{\gamma}) } \right ]. }[/math]

We now define the following parameters

[math]\displaystyle{ A_0 = \frac{\hbar^2}{2 m_0} + \frac{\hbar^2}{m_0^2} \sum^{B}_{\gamma} \frac{ p^{x}_{x\gamma}p^{x}_{\gamma x} }{ E_0-E_{\gamma} }, }[/math]

[math]\displaystyle{ B_0 = \frac{\hbar^2}{2 m_0} + \frac{\hbar^2}{m_0^2} \sum^{B}_{\gamma} \frac{ p^{y}_{x\gamma}p^{y}_{\gamma x} }{ E_0-E_{\gamma} }, }[/math]

[math]\displaystyle{ C_0 = \frac{\hbar^2}{m_0^2} \sum^{B}_{\gamma} \frac{ p^{x}_{x\gamma}p^{y}_{\gamma y} + p^{y}_{x\gamma}p^{x}_{\gamma y} }{ E_0-E_{\gamma} }, }[/math]

and the band structure parameters (or the Luttinger parameters) can be defined to be

[math]\displaystyle{ \gamma_1 = - \frac{1}{3} \frac{2 m_0}{\hbar^2} (A_0 + 2B_0), }[/math]

[math]\displaystyle{ \gamma_2 = - \frac{1}{6} \frac{2 m_0}{\hbar^2} (A_0 - B_0), }[/math]

[math]\displaystyle{ \gamma_3 = - \frac{1}{6} \frac{2 m_0}{\hbar^2} C_0, }[/math]

These parameters are very closely related to the effective masses of the holes in various valence bands. [math]\displaystyle{ \gamma_1 }[/math] and [math]\displaystyle{ \gamma_2 }[/math] describe the coupling of the [math]\displaystyle{ |X \rangle }[/math], [math]\displaystyle{ |Y \rangle }[/math] and [math]\displaystyle{ |Z \rangle }[/math] states to the other states. The third parameter [math]\displaystyle{ \gamma_3 }[/math] relates to the anisotropy of the energy band structure around the [math]\displaystyle{ \Gamma }[/math] point when [math]\displaystyle{ \gamma_2 \neq \gamma_3 }[/math].

Explicit Hamiltonian matrix

The Luttinger-Kohn Hamiltonian [math]\displaystyle{ \mathbf{D_{jj'}} }[/math] can be written explicitly as a 8X8 matrix (taking into account 8 bands - 2 conduction, 2 heavy-holes, 2 light-holes and 2 split-off)

[math]\displaystyle{ \mathbf{H} = \left( \begin{array}{cccccccc} E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\ P_z^{\dagger} & P+\Delta & \sqrt{2}Q^{\dagger} & -S^{\dagger}/\sqrt{2} & -\sqrt{2}P_{+}^{\dagger} & 0 & -\sqrt{3/2}S & -\sqrt{2}R \\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\ \end{array} \right) }[/math]

References

2. Luttinger, J. M. Kohn, W., "Motion of Electrons and Holes in Perturbed Periodic Fields", Phys. Rev. 97,4. pp. 869-883, (1955). https://journals.aps.org/pr/abstract/10.1103/PhysRev.97.869