Luttinger–Kohn model

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, $\backslash phi^\{\}\_\{\}$, as a linear combination of the unperturbed eigenstates $\backslash phi^\{(0)\}\_\{i\}$:

:$\backslash phi\; =\; \backslash sum^\{A,B\}\_\{n\}\; a\_\{n\}\; \backslash phi^\{(0)\}\_\{n\}$

Assuming the unperturbed eigenstates are orthonormalized, the eigenequations are:

:$(E-H\_\{mm\})a\_m\; =\; \backslash sum^\{A\}\_\{n\backslash neq\; m\}H\_\{mn\}a\_\{n\}\; +\; \backslash sum^\{B\}\_\{\backslash alpha\; \backslash neq\; m\}H\_\{m\backslash alpha\}a\_\{\backslash alpha\}$,

where

:$H\_\{mn\}\; =\; \backslash int\; \backslash phi^\{(0)\backslash dagger\}\_\{m\}\; H\; \backslash phi^\{(0)\}\_\{n\}d^3\; \backslash mathbf\{r\}\; =\; E^\{(0)\}\_\{n\}\backslash delta\_\{mn\}+H^\{\text{'}\}\_\{mn\}$.

From this expression, we can write:

:$a\_\{m\}\; =\; \backslash sum^\{A\}\_\{n\backslash neq\; m\}\; \backslash frac\{H\_\{mn\}\}\{E-H\_\{mm\}\}\; a\_\{n\}\; +\; \backslash sum^\{B\}\_\{\backslash alpha\backslash neq\; m\}\; \backslash frac\{H\_\{m\backslash alpha\}\}\{E-H\_\{mm\}\}\; a\_\{\backslash alpha\}$,

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 $a\_\{m\}$ for m in class A, we may eliminate those in class B by an iteration procedure to obtain:

:$a\_\{m\}\; =\; \backslash sum^\{A\}\_\{n\}\; \backslash frac\{U^\{A\}\_\{mn\}\; -\; \backslash delta\_\{mn\}\; H\_\{mn\}\}\{E-H\_\{mm\}\}\; a\_\{n\}$,

:$U^\{A\}\_\{mn\}\; =\; H\_\{mn\}\; +\; \backslash sum^\{B\}\_\{\backslash alpha\backslash neq\; m\}\; \backslash frac\{H\_\{m\backslash alpha\}H\_\{\backslash alpha\; n\}\}\{E-H\_\{\backslash alpha\backslash alpha\}\}\; +\; \backslash sum\_\{\backslash alpha,\backslash beta\backslash neq\; m,n;\backslash alpha\backslash neq\backslash beta\}\; \backslash frac\{H\_\{m\; \backslash alpha\}\; H\_\{\backslash alpha\backslash beta\}H\_\{\backslash beta\; n\}\; \}\{(E-H\_\{\backslash alpha\backslash alpha\})(E-H\_\{\backslash beta\backslash beta\})\}\; +\; \backslash ldots$

Equivalently, for $a\_\{n\}$ ($n\; \backslash in\; A$):

:$a\_\{n\}\; =\; \backslash sum^\{A\}\_\{n\}\; (U^\{A\}\_\{mn\}\; -\; E\backslash delta\_\{mn\})a\_\{n\}\; =\; 0,\; m\; \backslash in\; A$

and

:$a\_\{\backslash gamma\}\; =\; \backslash sum^\{A\}\_\{n\}\; \backslash frac\{\; U^\{A\}\_\{\backslash gamma\; n\}\; -\; H\_\{\backslash gamma\; n\}\backslash delta\_\{\backslash gamma\; n\}\}\{E-H\_\{\backslash gamma\backslash gamma\}\}\; a\_\{n\}\; =\; 0,\; \backslash gamma\; \backslash in\; B$.

When the coefficients $a\_\{n\}$ belonging to Class A are determined, so are $a\_\{\backslash gamma\}$.

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

:$H\; =\; H\_0\; +\; \backslash frac\{\backslash hbar\}\{4m\_\{0\}^\{2\}c^\{2\}\}\backslash bar\{\backslash sigma\}\backslash cdot\backslash nabla\; V\; \backslash times\; \backslash mathbf\{p\}$,

where $\backslash bar\{\backslash sigma\}$ is the Pauli spin matrix vector. Substituting into the Schrödinger equation in Bloch approximation we obtain

:$H\; u\_\{n\backslash mathbf\{k\}\}(\backslash mathbf\{r\})\; =\; \backslash left(\; H\_0\; +\; \backslash frac\{\backslash hbar\}\{m\_\{0\}\}\backslash mathbf\{k\}\backslash cdot\backslash mathbf\{\backslash Pi\}\; +\; \backslash frac\{\backslash hbar^2\; k^2\}\{4m\_\{0\}^\{2\}c^\{2\}\}\; \backslash nabla\; V\; \backslash times\; \backslash mathbf\{p\}\; \backslash cdot\; \backslash bar\{\backslash sigma\}\; \backslash right)\; u\_\{n\backslash mathbf\{k\}\}(\backslash mathbf\{r\})\; =\; E\_\{n\}(\backslash mathbf\{k\})\; u\_\{n\backslash mathbf\{k\}\}(\backslash mathbf\{r\})$,

where

:$\backslash mathbf\{\backslash Pi\}\; =\; \backslash mathbf\{p\}\; +\; \backslash frac\{\backslash hbar\}\{4m\_\{0\}^\{2\}c^\{2\}\}\backslash bar\{\backslash sigma\}\; \backslash times\; \backslash nabla\; V$

and the perturbation Hamiltonian can be defined as

:$H\text{'}\; =\; \backslash frac\{\backslash hbar\}\{m\_0\}\backslash mathbf\{k\}\backslash cdot\backslash mathbf\{\backslash Pi\}.$

The unperturbed Hamiltonian refers to the band-edge spin-orbit system (for k=0). At the band edge, the conduction band Bloch waves exhibits s-like symmetry, while the valence band states are p-like (3-fold degenerate without spin). Let us denote these states as $|S\; \backslash rangle$, and $|X\; \backslash rangle$, $|Y\; \backslash rangle$ and $|Z\; \backslash rangle$ 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:

:$u\_\{n\; \backslash mathbf\{k\}\}\; (\backslash mathbf\{r\})\; =\; \backslash sum^\{A\}\_\{j\text{'}\}\; a\_\{j\text{'}\}(\backslash mathbf\{k\})\; u\_\{j\text{'}0\}(\backslash mathbf\{r\})\; +\; \backslash sum^\{B\}\_\{\backslash gamma\}\; a\_\{\backslash gamma\}(\backslash mathbf\{k\})\; u\_\{\backslash gamma\; 0\}(\backslash mathbf\{r\})$,

where j' is in Class A and $\backslash gamma$ is in Class B. The basis functions can be chosen to be

:$u\_\{10\}(\backslash mathbf\{r\})\; =\; u\_\{el\}(\backslash mathbf\{r\})\; =\; \backslash left\; |\; S\backslash frac\{1\}\{2\},\backslash frac\{1\}\{2\}\; \backslash right\; \backslash rangle\; =\; \backslash left|S\backslash uparrow\backslash right\backslash rangle$

:$u\_\{20\}(\backslash mathbf\{r\})\; =\; u\_\{SO\}(\backslash mathbf\{r\})\; =\; \backslash left\; |\; \backslash frac\{1\}\{2\},\backslash frac\{1\}\{2\}\; \backslash right\; \backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\; 3\}\; |(X+iY)\backslash downarrow\backslash rangle\; +\; \backslash frac\{1\}\{\backslash sqrt\; 3\}\; |Z\backslash uparrow\backslash rangle$

:$u\_\{30\}(\backslash mathbf\{r\})\; =\; u\_\{lh\}(\backslash mathbf\{r\})\; =\; \backslash left\; |\; \backslash frac\{3\}\{2\},\backslash frac\{1\}\{2\}\; \backslash right\; \backslash rangle\; =\; -\backslash frac\{1\}\{\backslash sqrt\; 6\}\; |(X+iY)\backslash downarrow\backslash rangle\; +\; \backslash sqrt\{\backslash frac\{2\}\{3\}\}\; |Z\backslash uparrow\backslash rangle$

:$u\_\{40\}(\backslash mathbf\{r\})\; =\; u\_\{hh\}(\backslash mathbf\{r\})\; =\; \backslash left\; |\; \backslash frac\{3\}\{2\},\backslash frac\{3\}\{2\}\; \backslash right\; \backslash rangle\; =\; -\backslash frac\{1\}\{\backslash sqrt\; 2\}|(X+iY)\backslash uparrow\backslash rangle$

:$u\_\{50\}(\backslash mathbf\{r\})\; =\; \backslash bar\{u\}\_\{el\}(\backslash mathbf\{r\})\; =\; \backslash left\; |\; S\backslash frac\{1\}\{2\},-\backslash frac\{1\}\{2\}\; \backslash right\; \backslash rangle\; =\; -|S\backslash downarrow\backslash rangle$

:$u\_\{60\}(\backslash mathbf\{r\})\; =\; \backslash bar\{u\}\_\{SO\}(\backslash mathbf\{r\})\; =\; \backslash left\; |\; \backslash frac\{1\}\{2\},-\backslash frac\{1\}\{2\}\; \backslash right\; \backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\; 3\}\; |(X-iY)\backslash uparrow\backslash rangle\; -\; \backslash frac\{1\}\{\backslash sqrt\; 3\}\; |Z\backslash downarrow\backslash rangle$

:$u\_\{70\}(\backslash mathbf\{r\})\; =\; \backslash bar\{u\}\_\{lh\}(\backslash mathbf\{r\})\; =\; \backslash left\; |\; \backslash frac\{3\}\{2\},-\backslash frac\{1\}\{2\}\; \backslash right\; \backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\; 6\}\; |(X-iY)\backslash uparrow\backslash rangle\; +\; \backslash sqrt\{\backslash frac\{2\}\{3\}\}\; |Z\backslash downarrow\backslash rangle$

:$u\_\{80\}(\backslash mathbf\{r\})\; =\; \backslash bar\{u\}\_\{hh\}(\backslash mathbf\{r\})\; =\; \backslash left\; |\; \backslash frac\{3\}\{2\},-\backslash frac\{3\}\{2\}\; \backslash right\; \backslash rangle\; =\; -\backslash frac\{1\}\{\backslash sqrt\; 2\}|(X-iY)\backslash downarrow\backslash rangle$.

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

:$\backslash sum^\{A\}\_\{j\text{'}\}\; (U^\{A\}\_\{jj\text{'}\}-E\backslash delta\_\{jj\text{'}\})a\_\{j\text{'}\}(\backslash mathbf\{k\})\; =\; 0,$

where

:$U^\{A\}\_\{jj\text{'}\}\; =\; H\_\{jj\text{'}\}\; +\; \backslash sum^\{B\}\_\{\backslash gamma\; \backslash neq\; j,j\text{'}\}\; \backslash frac\{H\_\{j\backslash gamma\}H\_\{\backslash gamma\; j\text{'}\}\}\{E\_0-E\_\{\backslash gamma\}\}\; =\; H\_\{jj\text{'}\}\; +\; \backslash sum^\{B\}\_\{\backslash gamma\; \backslash neq\; j,j\text{'}\}\; \backslash frac\{H^\{\text{'}\}\_\{j\backslash gamma\}H^\{\text{'}\}\_\{\backslash gamma\; j\text{'}\}\}\{E\_0-E\_\{\backslash gamma\}\}$,

:$H^\{\text{'}\}\_\{j\backslash gamma\}\; =\; \backslash left\; \backslash langle\; u\_\{j0\}\; \backslash right\; |\; \backslash frac\{\backslash hbar\}\{m\_0\}\; \backslash mathbf\{k\}\; \backslash cdot\; \backslash left\; (\; \backslash mathbf\{p\}\; +\; \backslash frac\{\backslash hbar\}\{4\; m\_0\; c^2\}\; \backslash bar\{\backslash sigma\}\; \backslash times\; \backslash nabla\; V\; \backslash right\; )\; \backslash left\; |\; u\_\{\backslash gamma\; 0\}\; \backslash right\; \backslash rangle\; \backslash approx\; \backslash sum\_\{\backslash alpha\}\; \backslash frac\{\backslash hbar\; k\_\{\backslash alpha\}\}\{m\_0\}p^\{\backslash alpha\}\_\{j\; \backslash gamma\}.$

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

:$D\_\{jj\text{'}\}\; \backslash equiv\; U^\{A\}\_\{jj\text{'}\}\; =\; E\_\{j\}(0)\backslash delta\_\{jj\text{'}\}\; +\; \backslash sum\_\{\backslash alpha\backslash beta\}\; D^\{\backslash alpha\backslash beta\}\_\{jj\text{'}\}k\_\{\backslash alpha\}k\_\{\backslash beta\},$

:$D^\{\backslash alpha\backslash beta\}\_\{jj\text{'}\}\; =\; \backslash frac\{\backslash hbar^2\}\{2\; m\_0\}\; \backslash left\; [\; \backslash delta\_\{jj\text{'}\}\backslash delta\_\{\backslash alpha\backslash beta\}\; +\; \backslash sum^\{B\}\_\{\backslash gamma\}\; \backslash frac\{\; p^\{\backslash alpha\}\_\{j\backslash gamma\}p^\{\backslash beta\}\_\{\backslash gamma\; j\text{'}\}\; +\; p^\{\backslash beta\}\_\{j\backslash gamma\}p^\{\backslash alpha\}\_\{\backslash gamma\; j\text{'}\}\; \}\{\; m\_0\; (E\_0-E\_\{\backslash gamma\})\; \}\; \backslash right\; ].$

We now define the following parameters

:$A\_0\; =\; \backslash frac\{\backslash hbar^2\}\{2\; m\_0\}\; +\; \backslash frac\{\backslash hbar^2\}\{m\_0^2\}\; \backslash sum^\{B\}\_\{\backslash gamma\}\; \backslash frac\{\; p^\{x\}\_\{x\backslash gamma\}p^\{x\}\_\{\backslash gamma\; x\}\; \}\{\; E\_0-E\_\{\backslash gamma\}\; \},$

:$B\_0\; =\; \backslash frac\{\backslash hbar^2\}\{2\; m\_0\}\; +\; \backslash frac\{\backslash hbar^2\}\{m\_0^2\}\; \backslash sum^\{B\}\_\{\backslash gamma\}\; \backslash frac\{\; p^\{y\}\_\{x\backslash gamma\}p^\{y\}\_\{\backslash gamma\; x\}\; \}\{\; E\_0-E\_\{\backslash gamma\}\; \},$

:$C\_0\; =\; \backslash frac\{\backslash hbar^2\}\{m\_0^2\}\; \backslash sum^\{B\}\_\{\backslash gamma\}\; \backslash frac\{\; p^\{x\}\_\{x\backslash gamma\}p^\{y\}\_\{\backslash gamma\; y\}\; +\; p^\{y\}\_\{x\backslash gamma\}p^\{x\}\_\{\backslash gamma\; y\}\; \}\{\; E\_0-E\_\{\backslash gamma\}\; \},$

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

:$\backslash gamma\_1\; =\; -\; \backslash frac\{1\}\{3\}\; \backslash frac\{2\; m\_0\}\{\backslash hbar^2\}\; (A\_0\; +\; 2B\_0),$

:$\backslash gamma\_2\; =\; -\; \backslash frac\{1\}\{6\}\; \backslash frac\{2\; m\_0\}\{\backslash hbar^2\}\; (A\_0\; -\; B\_0),$

:$\backslash gamma\_3\; =\; -\; \backslash frac\{1\}\{6\}\; \backslash frac\{2\; m\_0\}\{\backslash hbar^2\}\; C\_0,$

These parameters are very closely related to the effective masses of the holes in various valence bands. $\backslash gamma\_1$ and $\backslash gamma\_2$ describe the coupling of the $|X\; \backslash rangle$, $|Y\; \backslash rangle$ and $|Z\; \backslash rangle$ states to the other states. The third parameter $\backslash gamma\_3$ relates to the anisotropy of the energy band structure around the $\backslash Gamma$ point when $\backslash gamma\_2\; \backslash neq\; \backslash gamma\_3$.

The Luttinger-Kohn Hamiltonian $\backslash mathbf\{D\_\{jj\text{'}\}\}$ 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)

:$\backslash mathbf\{H\}\; =\; \backslash left(\; \backslash begin\{array\}\{cccccccc\}$

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)

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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

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Category:Condensed matter physics