Band structure

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• Localised nitrogen states
• Band gap bowing
• The hybridization picture
• The band-anticrossing model
• The linear combination of isolated nitrogen states model

Localised nitrogen states

Nitrogen impurities introduced into GaAs are known to give rise to discrete energy levels in the band structure [1,2]. The most prominent of these levels is due to an isolated nitrogen atom and in GaAs is resonant with the conduction band. Deeper lines in the energy gap have been associated with nitrogen pairs.

Early theoretical work using tight-binding (TB) calculations [3] had predicted the existence of deep trap-like states in GaAsP alloys associated with isolated nitrogen. Since then, more intensive supercell calculations incorporating lattice relaxation have been undertaken using the empirical pseudopotential method (EPM) [4] and TB calculations [5] to determine the bandstructure. Both of these approaches confirm that the formation of highly localised electronic states around the nitrogen sites.

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Band gap bowing

As the concentration of nitrogen is increased, the band gap of the material undergoes a dramatic redshift (i.e. narrowing). Such bowing away from a linear interpolation of energy gaps of the binaries is common in semiconductor alloys. However, in GaNAs the degree of bowing (specified by the 'optical bowing coefficient') is an order of magnitude greater than in other III-V alloys. Different interpretations of the perturbation of the conduction band have been given. Zunger and co-workers [4] argue that the conduction bandedge becomes highly localised due to a mixing of the G, L and X valleys via the nitrogen site potential. An immediate physical justification of this is that any deep level impurity giving rise to a localised state must be formed from states from all bands. A more phenomenological explanation is given in terms of the hybridization of the extended conduction band states with the localised nitrogen states. This interpretation has been corroborated via tight-binding calculations by O'Reilly and co-workers [5] and is qualitatively consistent with probing of the dispersion relations via magneto-tunnelling measurements by Patanč et al [6].

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The hybridization picture

The band-anticrossing (BAC) model

The simplest formulation of the hybridization picture is the band-anticrossing (BAC) model due to Shan et al [7]. According to this model, the conduction band states are a superposition of the host semiconductor states and localised nitrogen states. The energy eigenvalues of the system are then found from the Hamiltonian

where E_M and E_N are the energies of the host semiconductor and isolated nitrogen respectively, V is a measure of the hybridization strength and x is the nitrogen concentration. The BAC model predicts a splitting of the conduction band into lower and upper bands (usually denoted E_- and E₊). The resulting downward shift of E_- with nitrogen content gives rise to a red-shift of the energy gap, giving good agreement with the pressure and temperature dependence. If we allow E_M to vary with wavevector k, we can extrapolate the dispersion relations from the model. An important feature of this is that the effective mass is predicted to increase, despite the reduction in band gap.

To see a demonstration applet of the BAC model, click here.

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The linear combination of isolated nitrogen states (LCINS) model

A more general model has been abstracted from tight-binding calculations. Since the nitrogen forms localised states, it is argued that the band-structure can be reproduced from a linear combination of these states with the host semiconductor states. A remarkable success of the LCINS model [8] is its ability to reproduce the non-monotonic variation of effective mass with nitrogen concentration [9].

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References