Optical absorption of dilute nitride alloys using self-consistent Green’s function method
- Masoud Seifikar^{1, 2}Email author,
- Eoin P O’Reilly^{1, 2} and
- Stephen Fahy^{1, 2}
https://doi.org/10.1186/1556-276X-9-51
© Seifikar et al.; licensee Springer. 2014
Received: 20 November 2013
Accepted: 14 January 2014
Published: 29 January 2014
Abstract
Abstract
We have calculated the optical absorption for InGaNAs and GaNSb using the band anticrossing (BAC) model and a self-consistent Green’s function (SCGF) method. In the BAC model, we include the interaction of isolated and pair N levels with the host matrix conduction and valence bands. In the SCGF approach, we include a full distribution of N states, with non-parabolic conduction and light-hole bands, and parabolic heavy-hole and spin-split-off bands. The comparison with experiments shows that the first model accounts for many features of the absorption spectrum in InGaNAs; including the full distribution of N states improves this agreement. Our calculated absorption spectra for GaNSb alloys predict the band edges correctly but show more features than are seen experimentally. This suggests the presence of more disorder in GaNSb alloys in comparison with InGaNAs.
Keywords
Background
The substitution of a small fraction x of nitrogen atoms, for group V elements in conventional III-V semiconductors such as GaAs and GaSb, strongly perturbs the conduction band (CB) of the host semiconductor. The band structure of dilute nitride alloys has been widely investigated [1]. We have recently developed [2] a SCGF approach to calculate the density of states (DOS) near the conduction band edge (CBE) in these alloys.
One way to test the accuracy of this model is to look at optical absorption spectra for dilute nitride samples, where we expect to see features related to the N states present in the samples. The absorption spectrum arises from transitions between valence and conduction band states. It provides knowledge of the energy gap in semiconductors, and also gives significant information about the band structure of materials. Experimental measurements of absorption spectra can be used to benchmark band structure calculations. In this paper, we investigate two different materials: In_{ y }Ga_{1-y}N_{ x }As_{1-x}, for which the band structure has been widely studied and many of the features are well established and GaN _{ x }Sb_{1-x}, for which much less information has been reported in the literature.
We consider two different models for the band structure of dilute nitride alloys, firstly a five-level band anticrossing (BAC) model, including the host semiconductor CB and valence bands, isolated N and pair N-N states and, secondly, the linear combination of isolated nitrogen states (LCINS) model [3, 4], which allows for interaction between N states on nearby sites as well as inhomogeneous broadening and produces a distribution of N state energies. In the LCINS model, the band structure of the alloys is calculated using a SCGF approach [2].
For In_{ y }Ga_{1-y}N_{ x }As_{1-x} alloys, we find that the BAC model reproduces the main features in the absorption spectrum, in agreement with previous work [5, 6]. However this model shows some additional features which are related to the N and N-N state energies, reflecting that in the BAC model, we have ignored the actual distribution of localised states. Including the LCINS distribution of N states in In_{ y }Ga_{1-y}N_{ x }As_{1-x} using the SCGF approach [2] removes the additional features found in the BAC calculations and gives absorption spectra that are in very good agreement with experimental data.
We then apply our methods to GaN _{ x }Sb_{1-x}, where much less information is known theoretically and experimentally. The overall width of the optical spectrum can be well fitted by our models for the absorption spectrum. Both the BAC and LCINS models account for the absorption edge of GaNSb alloys, supporting the presence of a band anti-crossing interaction in these alloys. However, the five-level BAC model gives more features than are seen experimentally in the absorption spectrum. Including a distribution of localised state energies, obtained by modifying those calculated for GaNAs, makes the calculated absorption spectra smoother and gives better agreement with experimental data but still shows some discrepancies around the localised state peak energies. These results suggest the presence of more disorder in GaNSb samples than in InGaNAs. This disorder may be due to sample inhomogeneities or due to an intrinsically broader distribution of N states in GaNSb than in InGaNAs.
The remainder of this paper is organised as follows. In the ‘Methods’ Section we first provide an overview of optical absorption calculation, followed by a description of the band structure models used for dilute nitride alloys. The theoretical results for InGaNAs and GaNSb are presented and compared with experiment in the ‘Results and discussion’ Section. Finally, we summarise our conclusions in the last section.
Methods
where E=E_{ ck }-E_{ vk } is the transition energy between conduction (E_{ ck }) and valence (E_{ vk }) states with wavevector k, J_{ cv }(E) is the joint density of states, and we ignore for now the energy dependence of M_{ b }.
where κ is the static dielectric constant, h is the Planck constant, $\mu =1/({m}_{c}^{-1}+{m}_{v}^{-1})$ is the reduced mass, and ${m}_{v}={({m}_{h}^{3/2}+{m}_{l}^{3/2})}^{2/3}$, where m_{ h } and m_{ l } are the heavy-hole and light-hole effective masses.
where α_{ lh }, α_{ hh } and α_{ so } are the absorption spectra for transitions from the light-hole (LH), heavy-hole (HH) and spin-orbit split-off (SO) bands to the conduction band, respectively.
The effect of the incorporation of N in (In)GaNAs alloys can be described in different ways. We investigate here how the model chosen influences the calculated alloy absorption spectrum. We first present a simple model including isolated and pair N states using the BAC model. This model includes the nonparabolicity of the conduction and light-hole and split-off bands and the interaction between the split-off and conduction bands. In the second model, we then include the full LCINS distribution of localised states using the SCGF model.
Optical absorption of dilute nitride alloys in five-level BAC model
BAC model parameters for In _{ y } Ga _{ 1- y } N _{ x } As _{ 1- x }
Parameter | Symbol | Values |
---|---|---|
N energy | E _{N 0} | 1.706(1-y)+1.44y-0.38y(1-y) (eV) |
N-N energy | E _{N N 0} | 1.486(1-y)+1.44y-0.38y(1-y) (eV) |
dE_{ N }/dT | a _{ N } | -2.5×10^{-4} (eV/K) |
dE_{ NN }/dT | a _{ NN } | -2.5×10^{-4} (eV/K) |
dE_{ N }/dx | γ _{ N } | -0.22 (eV) |
dE_{ NN }/dx | γ _{ NN } | -0.22 (eV) |
dE_{ c }/dx | γ _{ x } | -2.1 eV |
N interaction | β _{ N } | 1.97(1-y)+2y-3.5y(1-y) (eV) |
N-N interaction | β _{ NN } | 2.69(1-y)+2y-3.5y(1-y) (eV) |
Energy gap | E _{c 0} | E_{g,GaAs}-1.33y+0.27y^{2} (eV) |
GaN _{ x } Sb _{ 1- x } parameters at room temperature
Parameter | Symbol | Values |
---|---|---|
Lattice constant [19] | a _{0} | 6.09593 (Å) |
Electron effective mass [19] | ||
Conduction | ${m}_{c}^{\ast}$ | 0.039(m_{0}) |
Light hole | m _{ l } | 0.0439(m_{0}) |
Heavy hole | m _{ h } | 0.25(m_{0}) |
Split-off | m _{ so } | 0.12(m_{0}) |
SO splitting energy | Δ _{ so } | 0.76 (eV) |
Energy gap [20] | E _{ g } | 0.725 (eV) |
Refractive index [19] | n _{ r } | 3.8 |
N energy [17] | E _{N 0} | 0.82-2.3 x (eV) |
N-N energy [17] | E _{N N 0} | 0.48-2.3 x (eV) |
N interaction [17] | β _{ N } | 2.4 (eV) |
N-N interaction [17] | β _{ NN } | 3.39 (eV) |
Here, for each conduction sub-band, we use the appropriate energy E, as shown in Figure 1 by E_{ l }, E_{ m } and E_{ u }.
where α_{SO-l}, α_{SO-m} and α_{SO-u} are the absorption spectra from split-off band to lower, middle and upper sub-bands, respectively, and with a similar notation used for transitions from the HH and LH bands.
Optical absorption of dilute nitride alloys in the LCINS model
where E_{ vi } is the energy of the valence band vi which, as in the previous section, can be the LH, HH, or split-off (SO) band. We take into account the nonparabolicity of the LH band given by Equation 13, but assume parabolic heavy-hole and split-off bands, as we find that the parabolic split-off band dispersion in the relevant energy range is very close to that obtained when nonparabolicity effects are also included.
Results and discussion
Here, we present the absorption spectra calculated using the five-level BAC and LCINS model and compare them with experiments. Perlin et al. [5, 6, 22–24] measured the optical absorption spectra for In_{0.04}Ga_{0.96}N_{0.01}As_{0.99} and In_{0.08}Ga_{0.92}N_{0.015}As_{0.985} and compared them with GaAs absorption data. Turcotte et al. [16, 25] recently measured the optical absorption spectrum of GaN_{ x }As_{1-x} and In_{ y }Ga_{1-y}N_{ x }As_{1-x} for several values of x and y. Here, we calculate the absorption spectra for In_{0.04}Ga_{0.96}N_{0.01}As_{0.99} and compare them with Skierbiszewski measurements at different temperatures.
Five-level model for In_{ y }Ga_{1-y}N_{ x }As_{1-x}
Electrical and optical parameters in GaAs
Parameter | Symbol | Values |
---|---|---|
Lattice constant [19] | a _{0} | 5.65+3.88 ×10^{-5} (T_{ e }-300) (Å) |
Electron effective mass [13] | 300 K/10 K | |
Conduction | ${m}_{c}^{\ast}$ | 0.063/ 0.067(m_{0}) |
Light hole | m _{ l } | 0.076/ 0.082(m_{0}) |
Heavy hole | m _{ h } | 0.50/ 0.51(m_{0}) |
Split-off | m _{ so } | 0.145/ 0.154(m_{0}) |
Energy gap (T=0) | E _{g 0} | 1.519 eV |
Varshni parameters [19] | ||
$\left({E}_{g}={E}_{g0}-\frac{{\alpha}_{T}{T}_{e}^{2}}{\left({\beta}_{T}+{T}_{e}\right)}\right)$ | α _{ T } | 5.408×10^{-4} (eV/K) |
β _{ T } | 204 | |
SO splitting energy [19] | Δ _{ so } | 0.34 eV |
Refractive index [13] | n _{ r } | 3.255(1+4.5×10^{-5}T_{ e }) |
Static dielectric constant [26] | κ | 12.4(1+1.2×10^{-4}T_{ e }) |
The usual BAC model predicts a gap in the DOS [2, 27] of (In)GaNAs alloys. However, it is clear from Figure 3 that the joint DOS for different transitions overlaps and fills this gap. Therefore, no gap is found in the absorption spectrum in (In)GaNAs alloys when using the BAC model.
LCINS approach for In_{ y }Ga_{1-y}N_{ x }As_{1-x}
We then solve Equations 20 and 24 self-consistently [2]. Figure 5 shows the CB DOS calculated by Equation 21 for In _{0.04}Ga_{0.96}N_{0.01}As_{0.99} alloy. We observe that use of the LCINS distribution of states inhibits the gap predicted by the BAC model in the DOS of (In)GaNAs alloys.
The blue squares in Figure 4 show the calculated absorption spectrum at T=10 K including the full LCINS distribution of N states, which is compared with the absorption spectrum measured by Skierbiszewski [5]. Clearly, the sharp steps that we saw in the five-level BAC model disappear due to the inclusion of the distribution of localised states. This gives a better overall agreement with the experimental data. The remaining discrepancies between the calculated and experimental data may be partly due to the fact that we have approximated the N distribution by the one that was previously calculated for GaN _{0.012}As_{0.988}.
The absorption spectrum for GaN _{ x }Sb_{1-x}
We can apply our method to calculate the band structure and absorption spectrum of other dilute nitride alloys. Here, we extend our calculations to investigate the absorption spectrum of GaNSb. The room temperature band gap of GaSb is about 725 meV, around half that of GaAs. Lindsay et al. [17, 28] have reported that N-related defect levels lie close to the CBE in GaNSb and therefore strongly perturb the lowest conduction states in this alloy. The band gap and optical properties in GaN _{ x }Sb_{1-x} have been shown to be strongly affected and highly sensitive to the distribution of the nitrogen atoms. Lindsay et al. [28] found that there is a wide distribution of N levels lying close to and below the CBE. The higher-lying N states push the CBE down in energy, as in GaAs, but the large number of lower energy N states are calculated to mix in strongly with the conduction band edge states, severely disrupting the band edge dispersion in GaNSb.
Here, we first investigate the band structure and optical absorption spectra of GaN _{ x }Sb_{1-x} in the five-level BAC model and compare the results with the absorption spectra measured by Veal et al. [20] and Jefferson et al. [29]. We then apply the SCGF method to GaN _{ x }Sb_{1-x} in the Section ‘LCINS model for GaN _{ x }Sb_{1-x}’. As the LCINS distributions have not yet been calculated for these alloys, we modify those calculated for GaNAs alloys and use them in our calculations.
Five-level BAC model for GaN _{ x }Sb_{1-x}
When a single Sb atom is replaced by N in GaSb, the N atom introduces a localised state with energy E_{ N }. However, a GaNSb alloy can also contain clusters of N atoms, such as N-N nearest neighbour pairs as well as larger clusters that introduce states in the band gap of GaSb. Table 2 contains the band parameters that we use for GaN _{ x }Sb_{1-x}, including the isolated N state energies, E_{ N } and N pair state energies, E_{ NN } relative to the valence band maximum energy and the BAC interaction parameters β_{ N } and β_{ NN }. As shown in this table, isolated N states are calculated to be less than 0.1 eV above the conduction band minimum, while the N pair states have energies that lie in the GaSb band gap. The calculated energy gap of GaN _{ x }Sb_{1-x} depends strongly on the assumed N distribution, reflecting that N cluster states introduce a series of defect levels close to the CBE in this alloy. In addition, the interaction parameters (β_{ N } and β_{ NN }) in GaN _{ x }Sb_{1-x} are calculated to be about 20% larger than for GaNAs alloys.
The middle sub-band (E_{ m }) lies between 0.55 and 0.78 eV, and the upper sub-band (E_{ u }) minimum is close to 1.0 eV. The blue dashed line in Figure 7 shows the non-parabolic spin-orbit split-off band (E_{SO}), calculated by the lowest eigenvalue of Equation 9. Also, the non-parabolicity of the light-hole band (E_{LH}) has been taken into account using Equation 13, while we assumed that the heavy-hole (E_{HH}) band has a parabolic dispersion.
We note however that there are two sharp features in the calculated results that are not observed in experiments. This could be due to the fact that we included only isolated and pair N states in this model and ignored the distribution of N states and their inhomogeneous broadening. The results for the calculated absorption spectrum using a five-level BAC model suggest that we need to include the full distribution of N states in optical absorption calculations.
LCINS model for GaN _{ x }Sb_{1-x}
It has been shown that the calculated electronic structure of GaN _{ x }Sb_{1-x} strongly depends on the assumed distribution of N atoms [28]. Therefore, in order to calculate an accurate band dispersion for this alloy, we need to have the distribution of localised states. Unfortunately, such a distribution has not been calculated for GaN _{ x }Sb_{1-x}. However, we expect that the distribution of N states in GaN _{ x }Sb_{1-x} should have a similar general form to the LCINS distribution that Lindsay et al. [3] have calculated for GaN _{ x }As_{1-x} and for GaN _{ x }P_{1-x} alloys [17]. Therefore, here, we consider the LCINS distribution of N states in GaN _{ x }As_{1-x} and, with some small modifications, use that for GaN _{ x }Sb_{1-x} alloys.
Having the distribution of N states, we are able to calculate the Green’s function for GaN _{0.012}Sb_{0.988} using Equations 19 and 20, self-consistently. Also, the density of CB states can be calculated using Equation 21. Figure 9 shows the DOS of GaN _{0.012}Sb_{0.988} calculated by the SCGF method and including the distribution of localised states shown by the solid blue line in the inset of Figure 9. The gaps corresponding to isolated and pair N states are clearly observed in this plot. Also, at energies around 0.65 eV, the DOS has a small gap that is related to the higher cluster of N states.
We can also calculate the absorption spectrum using the SCGF model. The blue line with diamonds in Figure 8 shows the calculated absorption coefficient using this method. As expected, this method shows a better agreement with experiments than the result of the five-level BAC model (shown by red circles in this plot).
For the considered N distribution, this calculation suggests more gaps in the DOS of GaN _{0.012}Sb_{0.988} compared to GaN _{0.012}As_{0.988}[2]. However, experimental data indicate that there are fewer features in the GaNSb absorption spectra than in the GaNAs ones. This could be due to inhomogeneities in the samples investigated experimentally, either due to fluctuations in the N composition in the experimental samples or because of intrinsic differences between the short-range N ordering in GaNSb and in InGaNAs samples.
Recent work by Mudd et al. [31] has shown that the composition dependence of the energy gap in GaN _{ x }Sb_{1-x} is well described using a three-level model including interactions between the host matrix band edge and the N isolated states and N-N pair states. The energy gap calculated using the LCINS model is also determined primarily by these interactions. The energy gap calculated here using the SCGF and LCINS method is consistent with experiment for the N composition x=1.2% which we consider and should closely follow the theoretical energy gap results presented in [31] as a function of N composition x.
Conclusions
In this paper, we presented an analysis of the optical absorption spectra of dilute nitride alloys, calculated using the band structure model presented in our earlier work [2]. We have considered two different models to calculate the absorption spectra in InGaNAs and GaNSb alloys and compared our results with experimental measurements. We note however that there are some discrepancies between experimental data in similar samples that make quantitative comparison difficult.
Two models have been considered to calculate the absorption spectrum in these materials: a five-level BAC model and a LCINS-based model. The five-level BAC model included isolated and pair N states and their interactions with the host semiconductor valence and conduction bands. The results of this model for InGaNAs alloys give an overall good agreement with experiments, and predict accurate absorption edge for these alloys. However, the results of the five-level BAC model include several additional features not seen experimentally, supporting the need to consider a full distribution of N state energies in the electronic structure calculations.
We therefore extended our calculations to include the LCINS distribution using the SCGF approach presented in [2]. The calculated absorption spectra using this approach for InGaNAs provide very good agreement with experiments, supporting the validity of the LCINS approach to describe dilute nitride conduction band structure.
Our calculated absorption spectra for GaNSb alloys fit well with experiments at the absorption edge [31], and predict the correct band gap in these alloys. However, the absorption spectrum calculated in the BAC model contains features associated with individual transitions to lower and upper sub-bands in the model that are not seen in the measured absorption spectra. Taking the distribution of localised states into account reduces the impact of these features and gives results more similar to experimental absorption. But we still see some dips in our calculated spectra that are not seen in any experiment. We conclude that the distribution of N states in the GaNSb alloys studied are different from that for InGaNAs samples. We conclude that further work is required to address and resolve why more structure is found in the calculated absorption spectra compared to what is observed in the experimentally measured spectra.
Declarations
Acknowledgements
This work was supported by the Science Foundation Ireland (06/IN.1/I90; 10/IN.1/I2994; 07/IN.1/I1810). The authors thank Tim Veal for providing measured value of absorption spectrum data for GaNSb samples.
Authors’ Affiliations
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