Theoretical study of nitride short period superlattices

First, it will be shown how the mInN/nGaN and mGaN/nAlN SLs band gap values differ from the band gaps of the corresponding InGaN and GaAlN alloys (with the same effective In/Ga and Ga/Al composition). In figure 4, the calculated band gaps versus cation (In or Ga) content, x  =  m/(m  +  n), for sets of mInN/nGaN (figure 4(a)) and mGaN/nAlN (figure 4(b)) SLs, are presented. The dashed lines correspond to the calculated band gaps of In_xGa_1−xN and Ga_xAl_1−xN random alloys with 'uniform' cation distribution [64].

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The first observation is that only for the sets of m/1 SLs the band gaps are similar to E_g's of random alloys with the same effective composition x. Generally, the band gaps of SLs differ substantially from the gaps of the alloys. The concept of SLs enables to go far beyond the limitation of E_g evolution realized in ternary alloys. It is especially visible in the case of mInN/nGaN SLs, where calculations give E_g smaller than E_g of binary InN (0.65 eV). It takes place in several cases for m  ⩾  3 and the resulting metallization (closing of the effective band gap) occurs for m  =  n  ⩾  5. The latter effect was also demonstrated by Miao et al [70] where the possibility of the topological insulator formation was predicted.

Band gaps of mGaN/nAlN SLs in accordance with expectations, demonstrate much larger magnitude than E_g's of mInN/nGaN SLs. Their deviation from band gaps of Ga_xAl_1−xN random alloy are much smaller than in case of In-containing SLs. For the 1/n, 2/n, and 3/n SLs E_g increases with increasing barrier thickness. For the set 5/n the gap is almost independent of n, while for the set 7/n there is a slightly decreasing dependence on n. Only for the 7/n type SLs band gaps smaller than the gap of pure GaN (3.6 eV) appear.

The calculated band gaps of mGaN/nAlN SLs (see figure 4(b)) are compared with the PL emission energies measured for 3 samples of SLs grown by MOVPE [4]. The declared layer thicknesses, m/n for these samples are: 0.9/7.2, 1.8/7.3, and 2.5/7.1 (non-integer m and n values reflect the fluctuations of the QW and QB layer thicknesses). The samples were grown pseudomorphically on an AlN substrate and the same pseudomorphic growth conditions were assumed in the calculations. As shown in figure 4(b), a very good agreement is obtained between the calculated E_g and the emission energies measured on the set of samples.

The results for InN/GaN cannot be compared to experimental data. Earlier published experimental data for samples of nominal 1InN/nGaN SLs [71] likely represent SLs with InGaN alloy in a QW. It was found [11] that their real composition was 1In_0.33Ga_0.67N/nGaN. Indium incorporation lower than 100% in the QW may be a general property for all 'InN ML' samples presented so far. The results for the In_xGa_1−xN/nGaN SLs will be presented and discussed in section 6.

To discuss further the band gap engineering in nitride SLs, an example of mInN/nGaN SLs will be used. Instead of using rather complicated dependence of E_g on the effective In/Ga content (which was useful for comparison of SLs and alloys), we plot the band gaps as functions of barrier and well thicknesses, i.e. n and m. Such a choice corresponds to the growth related approach.

Analysing the E_g dependence on QB and QW thickness, presented in figures 5(a) and (b), respectively, one observes that the band gaps are more sensitive to the well thickness than to the barrier width. Regarding the dependence on barrier thickness (figure 5(a)) the E_g at first (for m  =  1) increases rapidly up to n  =  5, then saturates. In contrast, for SLs with more than one InN ML the band gap decreases with increasing well and barrier thickness. As it is shown in figure 5(b), E_g decreases strongly with increasing well thickness and this dependence is more pronounced for larger n values. The crossover from increasing to decreasing trend occurring already for well width m  =  2 (figure 5(a)) is clearly seen in figure 5(b).

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To study the influence of the internal strain on E_g, we can simulate different strain conditions. The SL can be grown pseudomorphically on a substrate, thus having fixed in-plane lattice constant equal to the substrate lattice parameter. In this mode, the relaxation of the SL geometry is performed along the growth direction. The other strain mode corresponds to free-standing structure and involves a full relaxation of the atomic positions and lattice parameters.

In the case of the growth of In(Ga)N/GaN quantum structures, the QWs are strongly strained due to 10.6% of lattice mismatch between InN and GaN. Depending on the layer thicknesses and growth conditions, a relaxation of In(Ga)N can take place. It occurs when a thickness of indium containing layers exceeds so called critical thickness. To answer the question how the internal strain in InN wells influences the E_g values, two cases of growth conditions are compared: the pseudomorphic one, in which a-lattice constants of InN match to GaN and the free-standing one (a-lattice constant of the SL is an average of InN and GaN).

The InN/GaN band gaps presented in figures 4(a) and 5 were obtained assuming the free-standing, F-S, conditions, but some of the calculations for InN/GaN SLs were performed also for the case in which the SL is grown pseudomorphically on GaN substrate, which was motivated by the fact that In(Ga)N/GaN structures are often grown on bulk GaN substrate.

Figure 6 shows the calculated band gaps versus layer thicknesses for sets of mInN/nGaN SLs in the F-S mode and in the pseudomorphic strain mode. One can observe that all trends in the gap evolution are the same. However, the E_g values are generally smaller in the pseudomorphic mode, where closing of the gap occurs already for m  =  n  ⩾  4. In the case of 1/n SLs the difference between both strain modes is very small, but becoming larger for thicker wells, and it is quite pronounced for 5/n SLs (up to 0.5 eV). It reflects the influence of strain coming from the InN–GaN lattice mismatch on the InN layers, which causes the increasing degree of atomic relaxation along the growth direction.

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To get a deeper insight into the problem of internal strain, the calculated atomic positions in three different strain modes are shown in figure 7 for 2InN/2GaN SL.

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In the case of pseudomorphic growth to match the GaN barrier, called 'ps-barrier', InN layer is compressed to match the lattice a-constant of GaN. Consequently, the InN layer is thicker (c parameter larger). Opposite, in the case of pseudomorphic growth on InN, 'ps-well', the GaN layer is expanded to match the lattice constant of InN, which is larger and it results in thinner GaN layer (smaller c parameter). In the free-standing, 'F-S' case, the InN layer is somewhat compressed and GaN expanded, leading to an intermediate c value. The calculated bond lengths and lattice parameters are compared to the atomic positions and bond lengths in the corresponding InN and GaN bulk materials.

We can observe similar bond lengths in all the strain modes, but different angles between them. The next observation is that the effect of strain is larger in the InN well (layer thicknesses are changing from 5.70 Å to 5.87 Å) than in the GaN barrier (layer thicknesses change from 5.09 Å to 5.17 Å). It could be explained by 'softer' InN bonds. The band gaps of 2InN/2GaN SLs being affected mainly by the InN well geometry, change from 1.09 eV in the 'ps-barrier' strain mode to 1.27 eV in the 'F-S' mode and to 1.36 eV in the 'ps-well' growth case.

Concluding, the effects of lattice relaxation and internal strain are quite significant in InN/GaN SLs, due to large lattice mismatch (calculated values of the a lattice parameter of GaN and InN are 3.155 Å and 3.502 Å, respectively). We can compare this result for InN/GaN SLs with the case of GaN/AlN SLs, where the E_g variations with the different strain conditions are almost negligible (not exceeding 0.05 eV) [72] because of small lattice mismatch (calculated values of the a lattice constant are: 3.101 Å in AlN and 3.155 Å in GaN).

The fact that all the trends in evolution of the band gaps with barrier thickness (see figure 6) are the same, independent of the built-in strain, enables us to eliminate the strain effect from further discussion. Now, for a given strain mode, the band gap evolution may be analysed in terms of two counteracting effects: (i) the hybridization of QW and QB wave functions and (ii) the built-in electric fields. The overall SL band gap corresponds to the local E_g of the InN ML. The band gaps in SLs with thin QWs are dominated by the hybridization effect, which leads to a larger gap, due to the influence of the QB wave functions on the states in the QW. In 1/n SLs, contributions to the QW wave functions coming from GaN barrier cause an increase of the E_g from 0.65 eV (bulk InN) to about 2.1 eV in the InN well. Strong influence of the wave functions of GaN QB on the states in InN QW is observed up to n  =  5, then E_g is almost constant, increasing very slowly. On the other hand, in SLs with wider QWs, the effect of the internal field dominates, leading to the reduction of the E_g values.

Wurtzite InN/GaN quantum structures grown along polar [0 0 0 1] axis are characterized by large polarization fields due to the spontaneous part and the piezoelectric polarization, the latter caused by lattice mismatch between QW and QB layers. The total polarization induces an internal electric field whose influence on the band profiles leads to the reduction of the electron–hole wave functions overlap. It results in a strong lowering of the radiative recombination rates and, consequently, the efficiency of optoelectronic devices, both laser diodes LDs and light emitting diodes LEDs drop down. A red-shift of the emitted light is observed. All the above mentioned effects constitute the so-called quantum confined Stark effect.

There are different theoretical methods to estimate the strength of the built-in electric field in SLs. Three of them will be presented below: (i) electric field, E, simply estimated from the slope of the band profile, (ii) E calculated by using the N-1s core-states as reference energies, and (iii) E estimated from polarization properties of bulk constituents. Below we will discuss the results for InN/GaN and GaN/AlN SLs obtained by these methods.

4.3.1. Electric field estimated from the band profiles.

Analyzing the band profiles one can estimate the strengths of the internal electric fields from the slopes of the profile curves. As an example, figure 8 shows band profiles with internal electric field directed from the type A interface towards the type B interface in both well and barrier regions in 5InN/5GaN SL. The different characters of the A and B interfaces and strong effect of the internal electric field are clearly seen. The strength of the electric field can be estimated from a linear approximation to the band profiles inside the QW and QB. The obtained values are: E_el  =  −6.6 MV cm⁻¹ in the InN layers and E_el  =  7.7 MV cm⁻¹ in the GaN layers. The gap is indirect in real space and equal to E_g  =  0.25 eV. The valence band offset, VBO, equal to difference between the valence band maxima in the central layers of InN well and GaN barrier, is estimated to be about 0.5 eV.

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4.3.2. Electric field calculated using the N-1s core-states as reference.

Values of the internal electric fields can be found from the ab initio calculations by using the N-1s core-states as reference energies. In figure 9, the energies of N-1s core-states, E_1s, within layers of the 5/5 SL, are plotted along the [0 0 0 1] growth direction (polar c-axis). The N-1s energy values are different in successive MLs of the SL due to the presence of the induced electric fields. The magnitude of the internal electric field can be found from the slope of the best straight-line fit. The obtained values are: E_el  =  −7.0 MV cm⁻¹ in InN well and E_el  =  8.2 MV cm⁻¹ in GaN barrier. These values are slightly higher than the values estimated from the band profiles presented in figure 8.

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The VBO can be determined by the procedure described in [72], following the method used by Picozzi et al [73]. The VBO in this method is defined as: E_VBO  =  ▵E_I  +  ▵E_B, where ▵E_I is so called 'interface term' being the difference between the N-1s core-states energies in the QW and QB layers, and ▵E_B is the 'bulk term' being the difference between N-1s state energies with respect to the valence band maxima in bulk QW and QB materials. ▵E_I  =  0.46 eV is found by extrapolation, as illustrated in figure 9. The calculated ▵E_B term is: ▵E_B  =  0.10 eV. Thus our calculated VBO for 5InN/5GaN SL is equal: E_VBO  =  0.56 eV. This value is close to the one obtained by Moses et al [74] (0.62 eV) and it is in a good agreement with the experimental data (0.58 eV [75], 0.78 eV [76] and 0.8  ±  0.1 eV [77]).

The electric fields in wells and barriers of mInN/nGaN SLs are illustrated on figures 10(a) and (b), respectively. The internal E_el is ill defined in QW or QB thinner than 3 MLs. The first observation is that the electric field, both in InN well and inside the GaN barrier, depends strongly and almost linearly on the effective In composition. In InN well, the absolute values of the internal electric field decrease with increasing In content from 13 MV cm⁻¹ for 3/7 SL down to 0.2 MV cm⁻¹ for 7/1 SL. In GaN barrier, it increases with increasing In content from 0.9 MV cm⁻¹ for 1/13 SL up to 14.8 MV cm⁻¹ for 7/3 SL. The second observation is that the electric field values obtained for the given effective In composition (x  =  m/(m  +  n)) are slightly different for different layer thicknesses. As an example, we can consider the sequence of 3/3, 4/4, 5/5 SLs (x  =  0.5). Comparing the electric field values for this set, we observe the general trend that internal electric fields, both in QWs and QBs, decrease with increasing number of MLs. Due to this trend we expect in 6/6 SL lower values of the electric field than in 5/5 SL. However, the influence of the electric field on the E_g values depends also on QW thickness, and despite the weaker electric field in the 6/6 SL, the overall band gap is even smaller than in the 5/5 SL due to the thicker QW and we observe closing of the gap (metallization).

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4.3.3. Electric field estimated using a semi-macroscopic model.

A semi-macroscopic model where the built-in electric fields in the QWs and QBs of the SL can be obtained from the spontaneous polarization and piezoelectric constants of the corresponding bulk materials, enables to analyze easily the electric fields in SLs for all the compositions and layer widths. The basic equations describing the electric fields: E_w in a QW and E_b in a QB are [78]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{E}_{{\rm w}}}=\frac{{{L}_{{\rm b}}}~({{P}_{{\rm b}}}-{{P}_{{\rm w}}}~)}{({{L}_{{\rm w}}}{{\lambda }_{{\rm b}}}+{{L}_{{\rm b}}}{{\lambda }_{{\rm w}}})};\nonumber \end{align} \tag{ 3 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{E}_{{\rm b}}}=~\frac{-{{L}_{{\rm w}}}~{{E}_{{\rm w}}}}{{{L}_{{\rm b}}}},\nonumber \end{align} \tag{ 4 }$$

where P_w and, P_b denote the polarizations in QW and QB, L_w and L_b are the QW and the QB thicknesses, λ_w and λ_b are the static dielectric constants of the well and barrier bulk materials.

The polarization contains spontaneous and piezoelectric parts (the (w,b) subscripts are omitted):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle P={{P}_{{\rm sp}}}+{{P}_{{\rm pz}}},\nonumber \end{align} \tag{ 5 }$$

where P_pz originates from the distortions due to the lattice mismatch between quantum layers. P_sp values for bulk materials are given in table 2, whereas P_pz is expressed by [79]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{P}_{{\rm pz}}}=2{{e}_{31}}{{\varepsilon }_{xx}}+{{e}_{33}}{{\varepsilon }_{zz}}\nonumber \end{align} \tag{ 6 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\varepsilon }_{xx}}=({{a}_{{\rm s}}}-a)/a\nonumber \end{align} \tag{ 7 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\varepsilon }_{zz}}=-2{{c}_{13}}{{\varepsilon }_{xx}}/{{c}_{33}},\nonumber \end{align} \tag{ 8 }$$

where a and a_s are the lattice constants of QW or QB bulk material and the substrate, respectively.

Table 2. The spontaneous polarization, P_sp (in C m⁻²), piezoelectric, e_ij, elastic, c_ij, and dielectric, λ, constants used in the model calculations.

	InN	AlN	GaN
a (Å)	3.533	3.112	3.186
Psp	−0.035 [79]	−0.095 [79]	−0.027 [79]
λ	15.3 [80]	8.5 [80]	10.4 [80]
e31	−0.59 [8]	−0.67 [8]	−0.44 [8]
e33	1.14 [8]	1.67 [8]	0.75 [8]
c13	95 [81]	115 [81]	117 [81]
c33	235 [81]	372 [81]	400 [81]

We use the described model to get the electric field strengths for InN/GaN and GaN/AlN SLs. The values of parameters used in the calculations are presented in table 2.

Figure 11 presents the calculated buit-in electric fields in QWs and QBs of mInN/nGaN (a) and mGaN/nAlN (b) SLs, as functions of the effective composition, x. We observe that, the electric fields are larger in mInN/nGaN than in mGaN/nAlN SLs, reaching 18.3 MV cm⁻¹ in the barrier of 7InN/1GaN and 12.5 MV cm⁻¹ in the barrier of 7GaN/1AlN SL. Larger values of built-in electric fields in mInN/nGaN than in mGaN/nAlN SLs are correlated with the larger lattice mismatch between InN and GaN than between GaN and AlN.

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The described model neglects the atomistic structure of the SL structure; the resulting values of electric fields depend on the effective chemical composition (m/n ratio), as follows from equations (1) and (2)), but not on the separate values of m and n (like in ab initio calculations). This originates from using parameters of the bulk materials constituting the SL and neglecting specific SL features. Neverthless, the agreement between 'model' and 'ab initio' calculated values of E_w and E_b is quite good, as we can see from the comparison presented in figure 11. An advantage of the model calculations is that it easily provides the electric field strengths for any values of the QW and QB thicknesses, whereas, by ab initio calculations, we cannot obtain the internal electric field values for very thin layers (<3 MLs) and also for rather thick layers (>10 MLs) due to computational restrictions.

The dependence of the band gaps on both, wave function hybridization and built-in electric fields will be discussed in the next sections.

The selected structure for analyzing the contribution of the wave functions hybridization to the evolution of the band gap will be 1InN/5GaN SL.

In figure 12 the total density of states (DOS), the valence and conduction band profiles, and the valence charge density through the MLs of 1InN/5GaN SL are shown. The band-edge profiles are characteristic for the presence of the internal electric field. Two kinds of interfaces are observed: A—interface (between the last Ga layer in the GaN2 barrier and the N layer in InN1 well), and B—interface (between the In layer in the InN1 well and the adjacent N layer in the GaN6 barrier). At the interface A the valence band maximum,VBM, is in the highest energetic position, whereas at the interface B the conduction band minimum, CBM, is in the lowest position. Spatial separation of the CBM and VBM results in the band gap indirect in real space and with reduced magnitude. It is interesting to note that despite the band gap lowering due to the effect of internal electric field, the value of E_g, about 1.95 eV is significantly larger than that of bulk InN, i.e. 0.7 eV. This effect is caused by hybridization of the InN well and adjacent GaN barrier wave functions.

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Separation of the VBM and CBM is reflected in an asymmetry of the charge distribution. We can see that charge is mainly localized on N sites, and interfaces A and B are not equivalent. The charge at the VBM is mainly accumulated at the interface A, in contrast, the CBM charge density is accumulated mainly at the interface B.

We observe also that the electron charge is more delocalized over the entire supercell, while localization near the InN ML mainly occurs for the hole states. Partly, this stems from the much smaller effective mass of the electrons compared to the holes. The calculations reveal an isotropically averaged effective masses for electrons of m_e  =  0.171 and m_hh  =  1.118 for the heavy holes.

As was already mentioned, the E_g behavior in mInN/nGaN and mGaN/nAlN SLs, shown in figures 4–6 may be understood mainly in terms of the wave functions hybridization and the influence of built-in electric field. The SL band gap can be decomposed as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{E}_{{\rm g}}}\left({\rm SL} \right)={{E}_{{\rm g}}}\left({\rm well} \right)+ \Delta {{E}_{{\rm g1}}}+ \Delta {{E}_{{\rm g2}}},\nonumber \end{align} \tag{ 9 }$$

where E_g(well) denotes the E_g of the bulk semiconductor forming the QW, ΔE_g1 is the change of the E_g due to the internal electric field, and ΔE_g2 includes the effects of hybridization of QW and QB wave functions, and also local atomic relaxations and strains from the substrate matching. ΔE_g1 depends on the strength of the electric field and on the QW thickness:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta {{E}_{{\rm g1}}}=eE{{d}_{{\rm w}}}.\nonumber \end{align} \tag{ 10 }$$

In figure 13, the calculated electric fields (a) and ΔE_g1 (b) as functions of barrier thickness are shown for mInN/nGaN SLs. We observe that the lowering of the SL band gaps for wider wells can be explained by the influence of the electric fields.

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Having determined ΔE_g1 allows us to compare the mInN/nGaN and mGaN/nAlN band gaps (figure 4) with the gaps, E_g(hyp) for the hypothetical case corresponding to the internal electric field 'switched off', i.e.:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle ~{{E}_{{\rm g}}}\left({\rm hyp} \right)={{E}_{{\rm g}}}\left({\rm SL} \right)- \Delta {{E}_{{\rm g1}}}.\nonumber \end{align} \tag{ 11 }$$

Figures 14(a) and (b) illustrate the band gaps E_g(hyp) for the two sets of 3/n and 5/n SLs in mInN/nGaN and mGaN/nAlN SLs, respectively. We observe that the E_g(hyp) are lying closer to the gaps of the corresponding random alloys than E_g(SL) and show an increasing trend as a function of the number n of barrier layers, in both mInN/nGaN and mGaN/nAlN SLs. Hence, we may conclude that the difference in E_g(SL) trends between these two systems (see also figures 4 and 5) is mainly due to the stronger internal fields in the QWs of mInN/nGaN SLs.

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Almost constant values of E_g(SL) for both series 3/n and 5/n in mGaN/nAlN SLs illustrates the compensation between the effects of the internal field and the hybridization as the barrier width n is varied. As a numerical example, we can consider the 5GaN/7AlN SL, where the estimated reduction of the gap due to the electric field is ΔE_g1  =  −0.8 eV. Thus, E_g(SL)  =  3.85 eV is decomposed into E_g(GaN)  =  3.50 eV, ΔE_g1  =  −0.8 eV and ΔE_g2  =  +1.15 eV, i.e. ΔE_g2 is larger than ΔE_g1 originating from the internal electric field. On the other hand, for 5InN/7GaN SL we have E_g(SL)  =  0.05 eV, E_g(InN)  =  0.7 eV, ΔE_g1  =  −1.2 eV, and hence ΔE_g2  =  +0.55 eV. In this case, the contribution from the electric field dominates over that from the strain and hybridization effect.

Generally we observe that the gap evolution in thin well SLs is dominated by the hybridization effect, i.e. a wider barrier (larger n) leads to a larger value of E_g. In contrast, in SLs with wider wells the effect of built-in electric fields dominates influencing the band profiles and causing the band gaps to be indirect in real space, reduced in size, and eventually to vanish. This effect is larger in InN/GaN SLs than in GaN/AlN SLs, due to the stronger internal electric fields due to the larger piezoelectric part of polarization. More details can be found in [69].

4.5.1. Oscillator strength.

Both effects (i) overlap reduction between hole and electron states in the valence and conduction bands caused by the presence of electric field and (ii) weakening of the wave functions hybridization with increasing barrier thickness, result in a decrease not only the E_g values, i.e. PL energy emission, but also the PL intensity. We can connect the PL intensity with the overlap integral between the electron and hole wave functions, the square of which reflects the oscillator strength (OS) of a band-to-band transition. Experimentally, it is related to the intensity of both a light absorption and PL.

Figure 15 shows the ratio of transition matrix elements of edge transitions for SL and bulk GaN. The OS values for different structures were obtained from an implementation of the projector augmented wave (PAW) method [83] in an existing plane-wave code supporting non norm-conserving Vanderbilt-type ultra-soft pseudopotentials [84], i.e. the Vienna ab initio simulation package VASP [17]. Based on the corresponding PAW-derived all electron wave functions, an implementation of the optical matrix elements in the VASP package is developed. The optical transition matrix elements are given by:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{f}_{ij}}=\frac{2{{m}_{{\rm e}}}}{3\hbar }({{\varepsilon }_{j}}-{{\varepsilon }_{i}}){{\sum }_{\alpha =x,y,z}}{{\left| \left\langle {{{\tilde{\Psi }}}_{i}}\left| {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{R}}}_{\alpha }}\left|{{{\tilde{\Psi }}}_{j}}\right.\right.\right\rangle\right|}^{2}},\nonumber \end{align} \tag{ 12 }$$

where ε_i, ε_j are the single-particle energies, m_e is the mass of an electron, $\hbar$ is the reduced Planck constant, ${{\tilde{\Psi }}_{i}}$, ${{\tilde{\Psi }}_{j}}$ are the conduction and the valence wave functions, respectively, and ${{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{R}}_{\alpha }}$ is the position operator. In this formulation, excitonic effects are neglected. The details of this model can be found in [85].

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As one can see from figure 15, the highest electron–hole transition probability, i.e. OS, is found for the set of SLs with the shortest barrier, n  =  1. OS is almost the same for very thin wells, but from m  =  5, it starts to decrease. For thin wells, it can be explained by a weak effect of the electric field and strong well-barrier wave functions hybridization, whereas, for thicker wells, influence of the electric field is dominant reducing the OS. On the other hand, considering SLs with the single ML in the well (m  =  1), a strong reduction of the wave functions hybridization occurs with increasing barrier thickness. It causes rapid reduction of the OS at the beginning, and then for thicker barriers, starting from around n  =  5, the transition rates show tendency to saturate (around n  =  15 the oscillator strength is around 20% of the bulk GaN value). This saturation results from the finite penetration lengths of electron and hole states into the SL barrier (an area in the GaN barrier begins to emerge wherein overlap between hole and electron states is approximately zero).

Concluding, we demonstrated that the effect of wave functions hybridization in SLs is dominant for narrow wells, whereas for wider wells, the effect of internal electric field is more important. We predicted theoretically that the PL emission intensity should drop with the increasing widths of SL layers, especially with QB thickness, as was shown for GaN/AlN SLs [86]. Unfortunately, experimental confirmation for InN/GaN SLs is not possible, due to the lack of structures with binary InN QWs. It seems that large lattice mismatch between InN and GaN as well as low structural stability of InN requiring significantly lower growth temperature make the task of obtaining InN/GaN SL very challenging.

The band structures of nonpolar mInN/nGaN SLs were investigated for two possible growth directions, along the a $\left\langle {\rm 1}\,{\rm 1} - {\rm 2}\,0 \right\rangle$ and m $\left\langle {10} - {10} \right\rangle$ axes. Arrangements of atomic MLs along these directions are illustrated in figure 17 for the example of 2InN/2GaN SL. In the case of SL grown in the a-direction, all the MLs are separated by the same distance a/2, where a is the lattice constant perpendicular to the c-plane, whereas MLs of SL grown in the m-direction can be separated by about 0.29a or 0.58a, and consequently two non-equivalent arrangements of MLs are possible. We call them m1 and m2.

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Figure 18 shows the relaxed atomic positions in 2InN/2GaN SLs for the three cases: a, m1, and m2. We observe that the relaxation of the atomic coordinates is different in the m1 and m2 case. The bond lengths are shorter in the m2 case inside the GaN layers and slightly longer inside the InN layers than in the m1 case.

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The calculated variations of the band gaps with effective In composition are presented in figure 19 for mInN/nGaN SLs grown in the a-direction (a) and the m-direction (b). Comparison with the band gaps of polar SLs (grown in the c-direction) is shown in figure 19(a). The dashed lines in figure 19 represent the calculations for In_xGa_1−xN alloys with a quasi-random ('uniform') cation distribution [64–66].

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Figure 19 shows that all the observed trends are alike for the a- and m- growth directions: E_g increases with increasing barrier thickness (n) and decreases with increasing well thickness (m) reflecting (as in the case of alloys) the dependence of E_g on composition. This tendency is in the case of polar SLs valid only for m  =  1, whereas, for higher values of m, E_g decreases also with increasing barrier thickness (n). All the gaps of the nonpolar SLs are lying rather close to the curve for random In_xGa_1−xN alloys (dashed lines),whereas they are much smaller in the polar case. We observe also that the gaps of SLs grown in the a direction (figure 19(a)) tend to be slightly larger than those of the SLs grown in the m-direction (figure 19(b)), illustrating the dependence of E_g on SL geometry, but the band gaps depend not only on the growth direction (a or m) of SL, but also on the atomic arrangement (m1 or m2) for the m growth direction. Two possible arrangements of atomic layers for even/even SLs grown in the m direction lead to different values of the band gaps (represented by solid and dotted lines).

As already discussed, the low values of the band gaps in polar SLs originate from the strong internal electric fields present in these structures, which determines the effective E_g. The electric field leads to a spatial separation of the VBM and CBM density of states. Consequently, the 'indirect in real space' band gap becomes very small for thick layers. In contrast, in nonpolar SLs, grown along the a- or m-axis of the wurtzite structure, there is no macroscopic polarization and no internal electric field. Polar and nonpolar SPLs with the same composition show difference in the band gap values above 1 eV. A similar energy difference arises when we 'switch off' the electric field in polar SLs (see figure 14(a)).

In figure 20, band gaps of 3InN/nGaN SLs are presented for 3 growth directions, a, m and c, and compared to hypotetical band gaps of c-SLs with 'switched off' electric field. We observe that indeed differences in the band gaps between polar and nonpolar SLs come mainly from the existence of electric field in polar structures, but the effect of geometry itself has to be taken into account also.

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The relaxed atomic geometries, the valence and conduction band profiles, and the probability densities for the VBM and CBM states through the 1InN/5GaN MLs are illustrated in figures 21(a) and (b), for the a-, and m-growth orientations, respectively. Figure 21 shows that the charge distribution along the growth direction is significantly different for nonpolar SLs and polar (see figure 12) SLs. In the absence of an internal electric field, the band profiles are flat, the band gap is 'direct in real space' and significantly larger (3.05 eV—a direction, 2.75 eV—m direction) than in the polar case (1.95 eV). Also, there is no noticeable charge transfer between the layers. The relaxation of atomic positions leads to In–N bond lengths slightly shorter than in pure InN, but different for a and m cases due to different geometry of the supercell.

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In conclusion, comparing the E_g behavior for three differently oriented SLs, two aspects should have been in focus: the influence of internal electric fields (comparison of polar and nonpolar SLs) and the dependence on lattice geometry (comparison of a-and m-oriented nonpolar SLs). It is interesting to note that the band gaps depend not only on the growth direction and the widths of the layers, but also on the arrangements of layers, and atomic positions inside the layers. This was illustrated by the band gaps in two different types of InN/GaN SLs grown in the m direction.

As has been discussed for a variety of semiconductors, the pressure coeficient of the band gap dE_g/dp supplies an important information about mechanisms controlling a behavior of the band gap [95]. In particular, in the case of polar nitride quantum structures dE_g/dp exhibits a significant reduction with respect to the alloys with equivalent composition [96]. The internal electric field present in quantum structures is responsible for this effect. In this context it is interesting to study the dependence of dE_g/dp on the growth directions. The calculated dE_g/dp for selected polar and nonpolar SLs is shown in figure 22. One can see that, in agreement with expectations, the band gap pressure coefficients in nonpolar SLs are lying very close to the curve presenting dE_g/dp obtained for In_xGa_1−xN quasi-random alloys and close to the InN band gap pressure coefficient (28 meV GPa⁻¹). In contrast, dE_g/dp calculated for polar mInN/nGaN SLs [71] have significantly lower values, which decrease further with QW (m), and QB thickness (n). The gaps and their pressure coefficients for nonpolar SLs (especially those grown in the m direction) vary with composition in a similar fashion as those calculated for 'bulk' In_xGa_1−xN alloys.

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More details on the nonpolar SLs can be found in [97].

Significant efforts during the last 15 years have been devoted to studies of semipolar nitride quantum structures and devices [10, 70, 93, 98]. It is interesting to consider theoretically their electronic band structure and verify the expectation that their properties represent intermediate cases with respect to polar and nonpolar SLs.

There is a wide variety of the growth directions enabling realization of the semipolar type of structures. For the purpose of illustration, two selected cases will be used: SLs grown along the s2 and s6 directions. Figure 23 shows the hexagonal unit cell of the wurtzite structure with the s2-plane (a) and s6-plane (b) indicated. The s2-plane is defined by the orthogonal directions $\left[ 1\,\bar{1}\,0\,0 \right]$ and $\left[ 1\,1\,\bar{2}\,\bar{3} \right]$, corresponding to the vectors ${{\vec{\upsilon }}_{1}}={{\vec{a}}_{1}}-{{\vec{a}}_{2}}$ and ${{\vec{\upsilon }}_{2}}={{\vec{a}}_{1}}+{{\vec{a}}_{2}}-\vec{c}$ (in terms of the lattice vectors shown in figure 23(a)). The s2 growth direction is then given by the direction of ${{\vec{\upsilon }}_{1}}\times {{\vec{\upsilon }}_{2}}$, which is along ${{\vec{b}}_{1}}+{{\vec{b}}_{2}}+2{{\vec{b}}_{3}}$, in terms of the reciprocal lattice vectors ${{\vec{b}}_{i}}$. Similarly, the s6-plane, illustrated in figure 23(b), is defined by the orthogonal directions $\left[ 1\,\bar{2}\,1\,0 \right]$ and $\left[ 1\,0\,\bar{1}\,\bar{4} \right]$, which correspond to the vectors ${{\vec{w}}_{1}}=-{{\vec{a}}_{2}}$ and ${{\vec{w}}_{2}}=2{{\vec{a}}_{1}}+{{\vec{a}}_{2}}-4\vec{c}$, with the normal given by the direction of ${{\vec{w}}_{1}}\times {{\vec{w}}_{2}}=2\left(2{{{\vec{b}}}_{1}}+{{{\vec{b}}}_{3}} \right)$.

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The electronic structures of semipolar mInN/nGaN SLs have been calculated and compared to similar calculations for polar SLs (grown along the c-direction) and nonpolar SLs (grown along the a- and m-directions).

In figure 24, the relaxed atomic positions are shown through the MLs of the 1/5 SLs grown in the s2 (a) and s6 (b) directions. For the s2-SL, the constructed supercell contains two cations and two anions in one ML, whereas there are four cations and four anions in one ML in the s6-SL geometry, which lie at different coordinates along the growth direction leading to the formation of characteristic staggered interfaces [99, 100]. The average ML separation is indicated in figure 24, showing a local enhancement at the InN ML. In both cases (s2 and s6), the bonds in the vicinity of the interfaces A and B are generally shorter than in the corresponding binaries (InN: 2.15 Å, GaN: 1.95 Å). We observe also that the atomic relaxations are more complicated in the s6 case (the local bonds around the four cations within each ML are not identical).

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In the lower panel of figure 24, the valence and conduction band edges are traced through the SL showing typical band edge profiles of a SL with an internal electric field directed from the A interface towards the B interface in both QW and QB regions. The SL band gap ('indirect in real space') is the difference between the overall VBM (which is found at the InN layer at interface A) and the overall CBM (which is found at the InN layer at interface B). Its magnitude is E_g  =  2.9 eV for the s2-SL and E_g  =  2.4 eV for the s6-SL. The above results can be compared to the polar case where much stronger asymmetry in the bond lengths and charge distribution is leading to larger slopes of the band edge profiles and even smaller 'indirect' band gaps. Thus, semipolar SLs, with weaker influence of the electric field on the band structure than found for polar SLs, represent 'intermediate' case between polar and nonpolar SLs.

To further investigate the relation between the internal electric field and the band gap, the electric fields were estimated for several s2- and s6-SLs and compared with the values obtained for polar SLs. The electric field strengths were obtained by using the N-1s core-levels as reference energies by the procedure described in section 4.3.2. Examples of the obtained values of the electric field for semipolar and polar SLs are given in table 3. As it was already mentioned, it is not possible to find the electric field value for one or even two MLs. As it results from table 3, the electric field strengths are quite similar for the s2- and s6-SLs ranging from 1 MV cm⁻¹ to 4.9 MV cm⁻¹, but much smaller than those obtained for polar structures (between 3.6 MV cm⁻¹ and 15.6 MV cm⁻¹) for the same range of compositions.

The calculated band gaps, E_g, versus In content, x  =  m/(m  +  n), for semipolar m/n s2-SLs are compared in figures 25(a) and (b) with those obtained for polar and nonpolar SLs. The calculated band gaps for semipolar s2-SLs are seen to fall slightly below the band gaps of SLs grown in the nonpolar a-direction (figure 25(a)), whereas the band gaps for semipolar s6-SLs are lying below the band gaps of SLs grown in the nonpolar m-direction (figure 25(b)). In all the semipolar and nonpolar SLs, the band gaps increase with barrier thickness, n, and decrease with well thickness, m. This behavior is in agreement with the intuitive expectations. On the other hand, in polar SLs such a trend is seen only for m  =  1, the band gaps for m  > 1 decrease both with increasing m and n values (see figures 4(a) and 5).

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In conclusion, studying different types of SLs two aspects have to be taken into account: the influence of internal electric fields (comparison of polar, semipolar, and nonpolar SLs) and the dependence on lattice geometry (comparison of nonpolar SLs in the a- and m-direction, as well as semipolar SLs in the s2- and s6-direction).

Figure 26(a) shows the calculated band gaps, E_g, for sets of mIn_yGa_1−yN/nGaN SLs with three different compositions in the In_yGa_1−yN well: y  =  0.25, 0.33 (one set, 1/n) and y  =  0.5 (various choices of m/n). In figure 26(b) the gaps for mGaN/nGa_yAl_1−yN with the composition in the Ga_yAl_1−yN barrier: y  =  0.5 (selected sets m/n) and y  =  0.67 (one set, m/1) are presented. The band gaps are shown as functions of effective cation composition in order to compare to the E_g's of the corresponding random alloys (In_xGa_1−xN and Ga_xAl_1−xN). The calculated band gaps in both, the mInGaN/nGaN and mGaN/nGa_yAl_1−yN SLs increase with increasing barrier thickness and decrease with increasing well thickness. They deviate from the E_g's of the corresponding alloys, especially for wider QWs and QBs. This effect is more pronounced in mIn_yGa_1−yN/nGaN SLs.

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Analysing figure 26(a) we observe that the calculated band gaps for 1In_yGa_1−yN/nGaN SLs with lower y (y  =  0.25 and 0.33) are significantly larger compared to the analogical set (1/n) of SLs with y  =  0.5 and they are also lying much higher than the gaps of 1InN/nGaN SLs (see figure 4(a)). And even higher than the band gaps of the In_xGa_1−xN alloy. It can be explained by the fact, that for lower y, gaps are less affected by the internal electric field (decreasing the gaps) due to smaller lattice mismatch between QW and QB and more dominated by the hybridization effect which increases the E_g values.

The main factors influencing the band gap behavior will be discussed in more details in the next sections.

6.1.1. Comparison with experimental data.

The calculated band gaps are compared to PL energies measured on a set of 1In_0.33Ga_0.67N/nGaN samples with m/n  =  1/2, 1/3, 1/4, 1/10, and 1/40 (red dots in figure 26(a)) [11]. Intentionally, these samples were grown as 1InN/nGaN short-period SLs [2], and it was believed that they contain one ML of pure InN. Consequently, the previous theoretical works [72, 103, 104] on InN/GaN SL indicated on significant discrepancy between theory and experiment. The calculated 1InN/nGaN band gaps were about 1 eV lower than the PL energies measured on the mentioned samples.

Several hypotheses were proposed to explain this discrepancy, including the suggestion that optical transitions are due to GaN excitons partially localized in the InN region [11], or that the observed light emission originates from the recombination of carriers located in different spatial regions, i.e. from the GaN QB to the InN QW [25, 69] and that the observed discrepancy was caused by screening of the internal electric fields in the polar structures by free carriers originating from unintentional defects, an effect which was not included in the calculations [82]. The last hypothesis suggested that the inserted layer is not pure InN, but actually consists of an ternary InGaN alloy [70, 71, 99]. To check this hypothesis, quantitative high resolution transmission electron microscopy (TEM) studies were performed on the set of nominal 1InN/nGaN short-period SLs. It was revealed that the SLs consist of an In_xGa_1−xN ML with an indium content of x  =  0.33 instead of the intended x  =  1. Figure 26(a) shows that the calculated gaps of 1In_0.33Ga_0.67N/nGaN SLs are lying close to PL energies measured for the set of samples with In content corrected to the value y  =  0.33. Lower In incorporation may be a general property for all 'InN-ML' samples presented so far. To make a further comparison between the experiment and theory, under the assumption that the SL samples studied have an 1In_0.33Ga_0.67N/nGaN structure, we performed calculations of the E_g dependence on pressure. The results are compared in figure 27 to the experimental data obtained from the PL measurements under pressure for three investigated 1/n SL structures (1/3, 1/10, 1/40) [71]. The measured pressure coefficients, dE_PL/dp, range from 33.4 meV Gpa⁻¹ to 28.7 meV GPa⁻¹ and are in a good agreement with the band gap pressure coefficients, dE_g/dp, calculated for In_0.33G_0.67N/nGaN and In_0.4G_0.4N/nGaN SLs taking into account that the pressure coefficients are subject to appreciable error bars:  ±1.5 meV GPa⁻¹ in the experiment, and  ±2 meV GPa⁻¹ in the calculations. They show the same trend as observed in In_xGa_1−xN alloys approaching the InN band gap pressure coefficient (28 meV GPa⁻¹) for higher In content. In contrast, dE_g/dp calculated for 1InN/nGaN SLs. (see figure 27) have significantly lower values, which furthermore decrease with barrier thickness n.

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Another set of mIn_xGa_1−xN/nGaN samples, that we can compare to was grown recently [13] by MOVPE technique. QW thicknesses were 1–4 MLs and In content, x, was in the range from 0.30 to 0.33. The results of PL measurements brought a new information on SL properties since the reported so far experimental results were obtained on SLs with 1 or 2 MLs in QW, only. Figure 28 shows the calculated band gaps of mIn₀₃₃Ga_0.67N/nGaN versus QB thickness for different values of m in comparison with PL emission energies obtained on the fabricated samples. Good agreement between theoretical and the experimental gaps is observed.

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Analysing figure 28, we can see that the band gaps increase with increasing barrier thickness for all the considered well widths (up to  =  5). Contrary to the case of binary mInN/nGaN SLs (figure 5(a)), we do not observe the change of the increasing trend observed for m  =  1 to decreasing one, for m  > 1, however, E_g increases more slowly for higher m values. We can interpret it in terms of much weaker, than in mInN/nGaN SLs, built-in electric field, which is sensitive to indium content, x, in QW. Higher In content leads to higher lattice mismatch between InGaN and GaN layers, resulting in increases the piezoelectric polarization and consequently the electric field strength.

6.1.2. Influence of electric field and wave function hybridization on the band gaps.

Due to the possibility of comparison with the experimental data, we will concentrate in our analysis on the mIn_0.33Ga_0.67N/nGaN SLs.

Figure 29 illustrates the internal electric fields in wells (E_w) and barriers (E_B) of mIn_0.33Ga_0.67N/nGaN and mGaN/nAl_0.33Ga_0.67N SLs obtained from the model described in section 4.3.3. The fields are given as functions of the number of barrier MLs, n.

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Considering E_w and E_B dependence on n, one can distinguish two regions: thin QBs (n up to ~4 MLs) and thicker QBs (n  >  5 MLs). In the 1st region, absolute values of the electric field strongly increase with n in QW and strongly decrease in QB. Then, in the 2nd region the dependence of the electric field on n becomes quite weak. The electric fields in mIn_yGa_1−yN/nGaN are slightly higher than in mGaN/nGa_yAl_1−yN SLs, which relates to the larger lattice mismatch (stronger piezoelectric contribution), and they are significantly lower than in mInN/nGaN SLs (see figure 13(a)).

As it was already shown in section 4.5, the SL band gap E_g(SL) may be decomposed as a sum: E_g(SL)  =  E_g(well)  +  ΔE_g1  +  ΔE_g2, where E_g(well) denotes the band gap of the bulk QW material, ΔE_g1  =  eEd_w is the reduction of E_G due to the internal electric field, E_el, and ΔE_g2 include the effects of strains, local atomic relaxations, and hybridization of well and barrier wave functions. In figure 30(a), the energy gap shift, ΔE_g1, due to the internal electric field, is shown for the set of mIn_0.33Ga_0.67N/nGaN SLs. One can observe that ΔE_g1 depends strongly on m and n values. The largest values of ΔE_g1, are in SLs with wide QWs and QBs, where the influence of internal electric fields is dominant, whereas, in SLs with thin layers the hybridization of well and barrier wave functions is more significant. The highest estimated reduction of E_G due to the electric field, ΔE_g1  =  −0.57 eV, is found for 7/10 SL.

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By correcting the calculated band gaps for the effect of the electric field, one can obtain gaps, E_g(hyp), for the hypothetical case with the internal electric field 'switched off': E_g(hyp)  =  E_g(SL)  −  ΔE_g1. They are shown in figure 30(b) for three sets of mIn_0.33Ga_0.67N/nGaN SLs: 1/n, 3/n and 5/n (solid lines) in comparison with the SL gaps, E_g(SL) (dashed lines).

It can be concluded from figure 30(b) that the E_g(hyp) are lying even higher compared to the gaps of the equivalent random alloy In_0.33Ga₀₆₇N than E_g(SL), which effect can be explained by strong wave function-hybridization effect in these SLs.

6.1.3. Oscillator strength.

Figure 31 shows the oscillator strength ratio of SL and bulk GaN for the set of 1In_xGa_1−xN/nGaN SLs with In content x  =  0.33, 0.25 and 1. We observe that the oscillator strength ratio increases with decreasing In content in the QW, reaching the strength for 1In_0.25Ga_0.75N/1GaN SL almost equal 0.9. Also, the character of the dependence on barrier thickness changes drastically. Contrary to 1InN/nGaN SLs, in the 1In_0.25Ga_0.75N/1GaN and 1In_0.33Ga_0.67N/nGaN the dependence on layer thickness is quite weak for thin barriers and does not saturate so fast. In particular, the SLs 1/1 and 1/3 with x  =  0.25 are characterized by almost the same oscillator strength ratio. It can be explained by much stronger penetration of well wave functions into the barrier region (especially for thin barriers) than could occur in the case of binary 1InN/nGaN SLs and also by lower strength of the internal electric fields.

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6.1.4. Influence of atomic arrangement on the band gaps—cation clustering.

The band structure calculations discussed in the previous section were performed in a supercell geometry modelling a uniform distribution of cations in the GaAlN and InGaN layers of the SL. In practical sample preparation, it is a severe issue whether ideal homogeneity of the random ternary alloys can be achieved, or rather regions of varying relative cation concentration are formed during growth (cation clustering). This may have significant bearings on the predictions of the properties of the SLs, as clustering tends to decrease the effective band gap of the random alloy.

The effects of cation clustering in nitride alloys were studied earlier [66, 67] and a noticeable lowering of the alloy band gap was demonstrated. Here, we undertake a similar study for the SLs.

In figure 32, the calculated band gaps of polar 1In_0.33Ga_0.67N/nGaN SLs for uniform and clustered arrangements of the In atoms in the InGaN QW are compared. Generally, the clustering can be realized in different ways as was already discussed in [66]. Here, we simulate 1In_0.33Ga_0.67N/nGaN SLs by 1In₃Ga₆N₉/nGa₆N₉ supercell either by distributing three In atoms as uniformly as possible over the nine cation positions of the In₃Ga₆N₉ QW or by clustering them together in one corner of the supercell layer. The dashed lines in figure 32 represent comparisons of the calculated [66] band gaps of In_xGa_1−xN alloys with those of a uniform and a clustered indium distribution.

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The band gaps of the SLs with clustered In-arrangements in the QW are substantially smaller than those obtained for the SLs with uniformly distributed In atoms. This observation is in agreement with the band gap behavior in alloys. The effect was explained on the basis of lattice relaxation and strong hybridization of In and N wave functions when the number of In atoms binding to a particular nitrogen atom exceeds the statistical average (which seems to be the definition of clustering) [66].

The E_g difference between the two SL models is smaller than for the equivalent alloys, which corroborates the observation that for a 1 ML well, the gap is to a large extent governed by the barrier. A similar comparison performed for sets of 1In_0.25Ga_0.75N/nGaN SLs (simulated by the 1In₃Ga₉N₁₂/nGa₁₂N₁₂ supercell) shows that the band gaps for clustered arrangement of In ions are reduced by about 0.1 eV–0.15 eV with respect to those of the uniform arrangement.

It was suggested recently [106], that the uniform arrangement of In atoms in 1In_0.33Ga_0.67N/nGaN SL corresponds to a $\sqrt{3}\times \sqrt{3}R30{\hspace{0pt}}^\circ$ ordered structure. As observed by experiment, for certain growth conditions, In_xGa_1−xN monolayers may self-organize into laterally ordered and periodic structures, which were identified as a $\sqrt{3}\times \sqrt{3}R30{\hspace{0pt}}^\circ$ reconstruction with a stoichiometry of In_0.33Ga_0.67N. With this new compositional information, the calculated band gaps are in excellent agreement with the measured photoluminescence data [11].

Specific role of In cations was demonstrated above. The observation that the band gaps of SLs with clustered In-arrangements in alloys are substantially smaller than those for the SLs with uniformly distributed In atoms is similar to the trend found for In-containing alloys.

The InAlN band gap can be tuned from about 0.7 eV–6.2 eV. Due to such a large spectral range, quantum structures based on this material are very promising for applications. In this context, studies of band structure evolution in the system of the combined multilayers of In_yAl_1−yN and GaN forming SLs are interesting and important. The aim of this Section is to show that there is a number of physical problems which require a better understanding. Among them are: (i) expected strong internal strain and built-in electric field, both caused by large InN–AlN and InN–GaN lattice mismatches, (ii) interplay between spontaneous and piezoelectric polarizations (especially around y  =  0.17–0.18, where In_yAl_1−yN is lattice-matched to GaN), (iii) properties of mIn_yAl_1−yN/nGaN SLs with equal GaN and In_yAl_1−yN band gap values (3.5 eV) for y around 0.20; as a consequence, for y  <  0.2, we deal with nGaN/mIn_yAl_1−yN SL, whereas for y  >  0.2, it is mIn_yAl_1−yN/nGaN SL, according to the commonly used convention to denote QW/QB in a SL.

In figure 33, the calculated band gaps of mIn_yAl_1−yN/nGaN SLs for y  =  0.83, 0.67, 0.50, 0.33, and nGaN/mIn_yAl_1−yN SLs with y  =  0.17, 0.12 are shown for m  =  1, as functions of the effective Ga composition n. This way of presentation enables a comparison with the quaternary alloys In_y(1−x) Al_{(1−y)(1−x)}Ga_xN with the same effective contents of In, Al and Ga. More details can be found in [107], where the case m  =  3 is considered also.

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Analysing the band gap behavior of both types of SLs (y-value either larger or lower than 0.2) we observe that for higher values of n their E_g approaches the GaN band gap (3.5 eV) from the opposite directions (increasing or decreasing with n). An exception is the mIn_0.33Al_0.67N/nGaN SL, where the band gap increases first, exceeding the bulk values of both of the constituents for n  =  1, then decreases for higher n. As it is shown in figure 33, the band gaps of nGaN/mIn_yAl_1−yN SLs agree well with these In_y(1−x) Al_{(1−y)(1−x)}Ga_xN alloys with the same effective composition. On the other hand, E_g values of mIn_yAl_1−yN/nGaN SLs show, especially for thicker barriers, significant deviations from the alloy band gaps. These differences come from the characteristic features of SLs, as built-in strain, internal electric fields and the wave functions hybridization, which will be discussed in the next subsections.

The calculated gaps are compared in figure 33 with PL emission energy measured on the samples with the same effective Ga composition, but different layer thicknesses. They are: 3GaN/3In_0.12Al_0.88N [108] (blue dot), 1GaN/7In_0.18Al_0.82N [8] (blue star) and 3GaN/7In_0.18Al_0.82N [8] (blue triangle). Experimental data for the first two samples are lying close to the calculated curves for SL with the same (or similar) In composition. Observed deviation for the third sample can be explained by the slightly different In composition and by the fact that layer thicknesses in the calculations are different from those in the measured samples. Taking the above into account, one can conclude that the agreement between experiment and calculations is quite reasonable.

6.2.1. Influence of internal strain on the SLs band gaps.

To illustrate the role of internal strain, the band gaps are calculated for SLs in two strain modes; pseudomorphic growth on a GaN substrate (ps-GaN), and pseudomorphic growth on InAlN substrate (ps-InAlN).

The obtained results for 1In_yAl_1−yN/nGaN SLs and nGaN/1In_yAl_1−yN SLs are presented in figure 34. The most pronounced difference in E_g values between the two strain modes, ΔE_g  =  0.52 eV, is found for SL with the largest In content, y  =  0.67. It is caused by the large lattice mismatch between the In_0.67Al_0.33N alloy (a lattice constant of the alloy is about 3.393 Å) and GaN (a about 3.186 Å). Consequently, in the nearly lattice matched nGaN/In_0.17Al_0.83N and nGaN/In_0.12Al_0.88N SLs, the ΔE_g values are very small.

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We can conclude that the influence of internal strain on band gap values in the 1In_yAl_1−yN/nGaN SLs with high In content is quite pronounced and similar like in 'binary' InN/GaN SLs (ΔE_g around 0.5 eV). On the other hand it is much more significant than in GaN/AlN SLs (ΔE_g below 0.05 eV).

6.2.2. Influence of electric field and wave function hybridization on the band gaps.

An internal electric field of 3.64 MV cm⁻¹, due to spontaneous polarization, was reported as obtained from PL measurements performed on a set of nGaN/mIn_0.18Al_0.82N quantum structures [8].

The internal electric fields in wells of nGaN/mIn_yAl_1−yN and mIn_yAl_1−yN/nGaN SLs estimated from the semi-macroscopic model described in section 4.3.3, are illustrated in figures 35(a) and (b), respectively. The obtained electric fields are compared for selected SLs with the values calculated ab initio by the method described in section 4.3.2 (open circles in figure 35) and quite a good agreement is found, taking into account the model limitations. As it was already mentioned in section 4.3.3, the estimates from the model values of electric fields depend only on well to barrier thickness ratios (as follows from equations (1) and (2)), but not on the separate values of well and barrier thicknesses.

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Considering nGaN/mIn_yAl_1−yN SLs (figure 35(a)), strong internal fields are found for very small values of well-barrier thickness ratios, whereas for higher ratios (above 4), they change rather slowly, approaching zero for the ratios above 10. A similar behavior of the electric field was observed in wells of mIn_yAl_1−yN/nGaN SLs, with y  =  0.67 and y  =  0.50 (figure 33(b)). Comparing figures 33(a) and (b) we can conclude that the highest electric fields are obtained for SLs with highest In content, y (up to 9 MV cm⁻¹ for y  =  0.67) and weaker for SLs with smaller y values. An exception here is mIn_0.33Al_0.67N/nGaN SL where the electric field almost vanishes for all m/n values. It will be shown that this is due to the compensation of the spontaneous and the piezoelectric polarizations.

The semi-macroscopic model enables to analyze the electric fields in SLs in terms of contributions from the piezoelectric and spontaneous parts of polarization. The calculated spontaneous and piezoelectric components of polarization in nGaN/mIn_yAl_1−yN and mIn_yAl_1−yN/nGaN SLs, being differences in polarization between the InAlN and GaN layers, are shown in figure 36. We observe that both, spontaneous, P_sp, and piezoelectric, P_pz, parts of the polarization depend almost linearly on In content, y. Around y  =  0.17 (where well and barrier layers are lattice matched) the piezoelectric polarization vanishes, and only the spontaneous part contributes to the total polarization. The total polarization vanishes around y  =  0.33, at the crossing of P_sp and  −P_pz lines, so the model confirms the finding that in mIn_0.33Al_0.67N/nGaN SLs the internal electric field is close to zero.

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Summarizing, SLs containing In_yAl_1−yN alloys can be analysed in three different regions of y values:

• 1.  
  Low In content, y  < 0.33Wells and barriers of the SLs are nearly lattice-matched. The internal strains are small and the electric fields are mostly due to the spontaneous polarization. Their influence on the band gaps is compensated to some extent by the hybridization of wave functions. At y ~ 0.17, the piezoelectric polarization vanishes due to the matching of GaN and InAlN lattice constants and the internal strain also vanish. The band gaps are similar to the gaps of the quaternary alloys with equivalent compositions.
• 2.  
  y ~ 0.33The spontaneous and piezoelectric polarizations cancel and the built-in electric fields vanish. The band gap behavior is mostly due to other effects, as internal strain and hybridization of wave functions. The In_0.33Al_0.67N/1GaN SL represent somehow the border case between GaN/In_yAl_y−1N and In_yAl_y−1N/GaN SLs.
• 3.  
  Large In content, y  >  0.33Strong internal electric fields are observed in thin wells of the SLs with high In content (up to 9 MV cm⁻¹ in In_0.67Al_0.33N/1GaN SL). It results in band gaps much lower compared to the gaps of InAlGaN alloy with the equivalent composition.

Due to different MgO and ZnO crystallographic structures (rocksalt and wurtzite, respectively), Mg_xZn_1−xO alloys have a tendency to be metastable. Phase segregation has been reported to occur in Mg_xZn_1−xO for, x  >  33%, with a maximum E_g of about 4.0 eV [112, 113]. It is interesting to observe that in Mg_xZn_1−xO an increase of band gap with x is accompanied by a reduction of the lattice constant. This behavior is different from usually observed trends (see e.g. AlGaN case).

Our calculated band gaps for Zn_xMg_1−xO alloy in the wurtzite structure are compared in figure 38 with the band gaps obtained for Ga_xAl_1−xN (see figure 2(b)), see also [114]. Additionally, our results for Mg_xZn_1−xO are compared with the experimental data from the literature [112, 115–119] and a very good agreement is found.

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In both alloys we observe similar E_g dependence on composition, but different composition dependent band gap bowings, b(x), which are defined through equation (2). In a linear approximation: b(x)  =  α  +  βx.

The band gap bowing of Ga_xAl_1−xN alloy is small, b  =  0.71 eV for x  =  0.5 (α  =  0.75, β  =  −0.07), which likely results from almost the same covalent radii, 1.22 Å for Ga and 1.21 Å for Al and similar lattice parameters.

The calculated bowing parameter of Zn_xMg_1−xO is larger, and with increasing Zn content, x, changes from 1 to 3 eV, being equal to 2.6 eV for x  =  0.5 (α  =  0.8, β  =  3.4). Larger is also the difference in size of the cations. The covalent radius for Zn is equal to 1.22 Å and is 1.41 Å for Mg. In this respect, we can compare Zn_xMg_1−xO to In_xGa_1−xN where E_g bowing is also quite large and similar as in Zn_xMg_1−xO, ranging from 1.4 to 2.8 eV. Covalent radii of the cations are equal to 1.22 Å for Ga and 1.42 Å for In, the values are almost the same as for Zn and Mg, respectively.

Considering the band gap bowing in Zn_xMg_1−xO, an involvement of other factors should be also taken into account. We will mention here a very small value of c/a in MgO (1.55) and the high iconicity of this material resulting in a tendency to crystalize in rocksalt structure. All the above results in a large displacement of atoms by alloying. The above finding differ from the one obtained by Toporkov et al [115] where the E_g bowing for the Zn_xMg_1−xO alloy was found to be very small (b  =  0.237 eV). The experimental Mg_xZn_1−xO bowing parameter is unknown due to a limited range of available compositions (up to 40% of Mg), but the good agreement with the calculated band gaps (figure 38) indicates that the experimental E_g bowing should likely be also large.

To compare the short period SLs of the form: mGaN/nAlN and mZnO/nMgO, we choose the most typical growth direction along the wurtzite c-axis. Such a comparison is interesting because not only QWs (GaN and ZnO), but also QBs (AlN and MgO) are similar (Z(Al)  =  13, Z(Mg)  =  12).

In figure 39, energy band gaps versus number of barrier MLs, n, are presented for mZnO/nMgO (a) and mGaN/nAlN (b) SLs. Comparing figures 39(a) and (b), we observe surprisingly similar E_g behavior in both kinds of SLs. The range of the E_g variation is almost the same, and they exhibit the same trends with changing m and n. In both cases, with an increasing number of QW layers, m, E_g is significantly reduced. On the contrary, increasing the barrier width, i.e. n, leads to increase (smaller m) or decrease (larger m) of E_g.

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Analysing the contributions to the band gap evolution, we can estimate the influence of the internal strain as not substantial, due to similar lattice constants of quantum layers and substrates (GaN in mGaN/nAlN and ZnO in mZnO/nMgO). The interplay of the built-in electric field and wave functions hybridization should mostly influence the E_g behavior. The effect of the internal electric field will be discussed in the next section.

The electric fields in the wells (E_w) and barriers (E_B) of the mZnO/nMgO SL structures are estimated from the semi-macroscopic model described in section 4.3.3 and presented in figure 40. The values of the parameters for ZnO and MgO used in the model calculations are listed in table 4. Note that lattice matching to ZnO substrate is assumed.

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Figure 40 illustrates the internal electric fields in wells (E_w) and barriers (E_B) of mZnO/nMgO SLs as functions of the effective Zn content x. One can see that the built-in electric field in ZnO/MgO is weak, approaching 2 MV cm⁻¹. Comparing with mGaN/nAlN SLs (figure 11(b)), we observe a big difference in the electric fields strengths. The built-in electric field in mGaN/nAlN is strong, reaching 12 MV cm⁻¹. The above difference could be explained by the analysis of the spontaneous and piezoelectric contributions to the polarization. In mGaN/nAlN the signs of spontaneous and piezoelectric polarizations are the same (a(GaNsubstrate)  >  a(AlNbarrier)), contrary to the case of mZnO/nMgO where spontaneous and piezoelectric parts have opposite signs (a(ZnOsubstrate)  <  a(MgObarrier)), partially compensating, to result in a small total polarization and low strenght of the electric field.

In conclusion, comparing GaN–AlN and ZnO–MgO systems, we found many similarities, but also some differences. We found very similar behavior of the band gaps in Ga_xAl_1−xN and Zn_xMg_1−xO alloys; they cover almost the same range of values and exhibit similar trends, but there are substantial differences in the band gap bowings. Comparing mGaN/nAlN and mZnO/nMgO SLs, we observe a similar evolution of the band gaps with varying number of atomic MLs, whereas the strengths of the built-in electric fields, existing in these two families of SLs differ significantly.