The 10-Band Zincblende k·p Model: Dilute Nitrides

Why Do We Need a Ten-Band Model?

In the eight-band zincblende model, we describe a semiconductor using an s-like conduction orbital and three p-like valence orbitals, with two spin states for each. Coupling these states allows us to calculate the conduction and valence bands, including effects that the simpler parabolic-band model cannot reproduce.

For certain alloys, however, even this picture is incomplete. A particularly interesting example is a dilute nitride, such as GaAsN or GaInNAs: a material in which a small fraction of the group-V atoms is replaced by nitrogen. Remarkably, introducing a relatively small amount of nitrogen can reduce the band gap much more strongly than we would expect from simply interpolating the properties of the constituent materials. The electron effective mass can also change substantially.

The reason is that nitrogen does more than slightly alter the existing crystal bands. It introduces a resonant electronic level which interacts with the host semiconductor's conduction band. To describe that interaction, we need to add an additional nitrogen-derived state to the ordinary k·p picture. Accounting for both spin orientations takes us from eight basis states to ten.

What Does Nitrogen Do to the Conduction Band?

Consider the conduction band of the original host semiconductor before nitrogen is added. Close to its minimum we can describe it, to a first approximation, by a parabola:

\[ E_c(k)=E_{c,0}+\frac{\hbar^2 k^2}{2m_c^{*}}. \]

Here, \(E_{c,0}\) is the original conduction-band edge, while \(k\) describes the electron's crystal wavevector. In the dilute-nitride model, nitrogen also introduces a more localised electronic state at an energy \(E_N\). Unlike the extended conduction-band states, this nitrogen-derived level has only weak dispersion in the simplest model; on an energy-versus- wavevector diagram, we can think of it as an approximately horizontal line.

If the two states did not interact, their energies could simply cross. But they do interact. As the host conduction band approaches the nitrogen level in energy, the two states mix, pushing their energies apart. This effect is called band anticrossing (BAC).

Understanding the Anticrossing

We can understand the essential physics without starting with a full ten-band Hamiltonian. Imagine that we have only two electronic states: one belonging to the original host conduction band, with energy \(E_c\), and one belonging to nitrogen, with energy \(E_N\). Their interaction is described by a coupling energy \(V_N\).

We can write this small problem as a matrix:

\[ \boxed{ H_{\mathrm{BAC}}= \begin{pmatrix} E_c & V_N \\ V_N & E_N \end{pmatrix} } \]

The diagonal entries are the energies the two states would have on their own. The off-diagonal entries describe their interaction. When \(V_N=0\), the states are independent. When the coupling is non-zero, the permitted energies are obtained by finding the eigenvalues of this matrix.

Writing the eigenvalue equation as \(\det(H_{\mathrm{BAC}}-EI)=0\) gives:

\[ (E_c-E)(E_N-E)-V_N^2=0. \]

This is a quadratic equation in energy. Solving it gives two answers, corresponding to the two new, mixed electronic states:

\[ \boxed{ E_{\pm}=\frac{E_c+E_N}{2} \pm\frac{1}{2} \sqrt{(E_c-E_N)^2+4V_N^2} } \]

The square-root term is what creates the anticrossing. Even if the original levels have exactly the same energy, the two new states do not cross: they remain separated by \(2|V_N|\). We can see this directly by setting \(E_c=E_N\) in the expression above.

OghmaNano discretises the growth direction into spatial grid points and uses a sparse eigensolver to find the states of interest. If the grid contains \(N_z\) points, the ten basis components give a matrix dimension of \(10N_z\). Here, \(N_z\) is the number of spatial grid points; it should not be confused with the symbol \(N\) used to identify the nitrogen-derived state. A complete GaInNAs/GaAs calculation using this model is demonstrated in the ten-band dilute-nitride k·p quantum-well tutorial.

Why Does the Band Gap Become Smaller?

Suppose the nitrogen level lies above the original conduction-band minimum, as in the standard dilute-nitride band-anticrossing picture. Turning on the interaction moves the lower mixed state below that original minimum. The valence-band edge does not undergo the same anticrossing in this model, so the energy separating the lowest conduction state from the valence band becomes smaller.

The shift of the conduction edge is not simply proportional to the small nitrogen concentration. It depends on both the coupling strength and how close the nitrogen level is to the host band. To illustrate the effect with arbitrary example energies, imagine:

\[ E_c=1.40\,\mathrm{eV},\qquad E_N=1.60\,\mathrm{eV},\qquad V_N=0.10\,\mathrm{eV}. \]

Substituting these values into the anticrossing expression gives a lower branch of approximately \(1.359\,\mathrm{eV}\), rather than the original \(1.40\,\mathrm{eV}\). In this illustrative example, interaction with the nitrogen state has lowered the conduction edge by about \(41\,\mathrm{meV}\). The numbers are chosen to demonstrate the mechanism, not to specify parameters for a particular alloy.

Importantly, the conduction band also changes shape. In our two-level picture, \(E_c\) is itself a function of \(k\), so both new solutions are dispersions \(E_-(k)\) and \(E_+(k)\). The lower branch can become flatter near its minimum, corresponding to a larger electron effective mass. Thus the same interaction can explain both the reduced band gap and the change in band curvature.

How Does Nitrogen Concentration Enter the Model?

The amount of nitrogen matters because there are more nitrogen-derived states available to interact with the extended conduction band as the nitrogen fraction increases. In the standard band-anticrossing model, the effective coupling is written as:

\[ \boxed{V_N=\beta_N\sqrt{x_N}} \]

Here, \(x_N\) is the nitrogen mole fraction (so, for example, one percent nitrogen is entered as \(x_N=0.01\)), and \(\beta_N\) is the coupling parameter for the material system. The square-root dependence reflects the collective coupling of the host conduction state to the dilute population of nitrogen sites. It is not an arbitrary linear shift of the conduction-band edge.

With no nitrogen, \(x_N=0\), the coupling disappears and we recover the uncoupled host-band description. As nitrogen is added, the interaction becomes stronger and the two mixed branches move further apart. The actual behaviour also depends on the location of \(E_N\) relative to the host conduction band, so both \(E_N\) and \(\beta_N\) are important material parameters.

From Two States to the Full Ten-Band Model

We have now understood the anticrossing using just two states. How does this fit into the eight-band zincblende Hamiltonian introduced on the previous page?

The eight-band model already contains the s-like conduction orbital and three p-like valence orbitals, each with two spin orientations. The dilute-nitride model retains these original eight states, then introduces two additional nitrogen-derived basis states, one for each spin:

\[ \begin{aligned} \{&|S\uparrow\rangle,\ |S\downarrow\rangle,\\ &|X\uparrow\rangle,\ |Y\uparrow\rangle,\ |Z\uparrow\rangle,\\ &|X\downarrow\rangle,\ |Y\downarrow\rangle,\ |Z\downarrow\rangle,\\ &|N\uparrow\rangle,\ |N\downarrow\rangle\}. \end{aligned} \]

These nitrogen-derived states are an effective resonant-level representation of the alloy. They should not be interpreted as an attempt to track the wavefunction around every individual nitrogen atom in a disordered crystal.

In the standard ten-band construction, nitrogen couples to the s-like conduction state with the same spin. Its direct coupling to the p-like valence states is neglected in this model. We can represent the full matrix schematically as:

\[ H_{10}= \begin{pmatrix} H_{8} & C_N\\ C_N^{\dagger} & E_N I_2 \end{pmatrix}, \]

where \(H_8\) is the original eight-band zincblende Hamiltonian, \(I_2\) is a two-state identity matrix, and \(C_N\) contains the nitrogen-to-conduction coupling. In the basis order shown above, the only non-zero entries of \(C_N\) connect \(|S\uparrow\rangle\) to \(|N\uparrow\rangle\), and \(|S\downarrow\rangle\) to \(|N\downarrow\rangle\), each with coupling strength \(V_N\).

This is why the model is called ten-band: it is the original eight-band system plus two spin-resolved nitrogen states, not ten separate parabolas added together.

What Changes When We Put the Material in a Quantum Well?

So far, our discussion has described the energies of an alloy as a function of crystal wavevector. In a quantum well, we also need to account for the semiconductor layers and the confinement potential perpendicular to them.

As in the parabolic-band tutorial, we define \(z\) as the direction across the layers, while the electron can still move parallel to them. In a multiband calculation the confined state is no longer described by just one scalar wavefunction. Instead, it has an envelope component for every state in the basis, including the nitrogen-derived components.

At each position, the local material determines the band edges, effective couplings, nitrogen level \(E_N\), and nitrogen fraction \(x_N\). The ten-band Hamiltonian describes how all the components interact. Solving the resulting quantum-well eigenvalue problem produces the permitted confined energies and their mixed electron and hole wavefunctions.

OghmaNano discretises the growth direction into spatial grid points and uses a sparse eigensolver to find the states of interest. If the grid contains \(N_z\) points, the ten basis components give a matrix dimension of \(10N_z\). Here, \(N_z\) is the number of spatial grid points; it should not be confused with the symbol \(N\) used to identify the nitrogen-derived state.

One practical consequence of anticrossing is that the bottom of the conduction band is no longer at the original host value \(E_c\). The lower branch \(E_-\) provides a useful estimate of the shifted conduction edge when targeting the confined states numerically.

What Does This Mean for Optical Transitions?

The band gap is not the only quantity that changes. Since the eigenstates themselves are mixtures of conduction-band and nitrogen-derived character, their interaction with light can also change. A state that is largely host-conduction-band-like need not have the same optical coupling as one with a substantial nitrogen-derived contribution.

The lower anticrossing branch is particularly important because it can become the lowest available conduction state, changing the transition energy between the conduction and valence bands. Its altered dispersion also affects the energies and number of states available for optical transitions. OghmaNano can use the resulting multicomponent wavefunctions in the subsequent optical-transition, spontaneous-emission and gain calculations.

The ten-band BAC model captures the influence of the nitrogen resonant state on the conduction-band structure, but it is still an approximation. It does not resolve the individual atomic positions of nitrogen impurities or, by itself, describe every disorder-related optical and transport effect found in real dilute-nitride alloys.