Energetics of Interfaces and Strain Partition in GaN/AlN Pseudomorphic Superlattices

1. Introduction

Interfaces formed when homogeneous systems are in physical contact influence properties of the assembly, such as mechanical, electronic, or chemical. Composite assemblies of various crystalline constituents put in physical contact yield properties that differ from the ideal mechanical mixture of bulk constituents. Therefore, interfaces have been studied extensively in pure systems, metals, semiconductors, insulators, and alloys [1,2]. These studies have shown that interfaces perturb the structure of the constituent crystals over distances representative of the inter-crystalline global symmetry, resulting in excess energy and the modulation of interplanar spacings at the close neighborhood of the boundary. Interphase interfaces introduce effects additional to the perturbations mentioned above relating to the lattice mismatch, such as strains in defect-free heteroepitaxial structures [3] or threading dislocations and other interfacial structural defects accommodating the misfit [4,5,6,7,8,9]. In both cases, misfit-generated strains influence the properties of the assembly and deserve detailed investigations.

A specific case of heteroepitaxy refers to layers with various thicknesses in the form of multiple quantum wells (QWs). Such systems exhibit new structural, dynamic, and electronic properties. Multilayered structures, consisting of ultra-thin and unequal-in-thickness wells and barriers sharing the same in-plane lattice parameters of the crystalline layers, form strained structures known as superlattices. Superlattices are essential in manufacturing devices, e. g. optoelectronic, and present electronic properties highly conditioned by the strain partitioning in the superlattice. Above a critical layer thickness, the elastic energy stored therein is lowered via the spontaneous formation of interfacial misfit-dislocations [10,11,12,13,14,15,16,17,18]. It has been shown that the energy gaps of superlattices differ from these reached by ternary alloys of the constituents, and this offers a means for bandgap tuning [19,20,21,22]. GaN/AlN superlattices are promising structures for light-emitting devices operating in the ultraviolet spectral region and various optical devices with intersubband transitions in the infrared range, such as high-speed optical modulators, quantum cascade lasers, and photodetectors [23,24,25,26].

Electron microscopy (EM) observations are made on superlattices of thin GaN wells grown pseudomorphically on twice-thicker AlN barriers. A new methodology is proposed, based on optimizing the stored elastic energy of strained superlattices, for extracting structural data from high-resolution transmission electron microscopy (HRTEM) images. Unlike other experimental and theoretical studies of GaN/AlN multilayered structures [27,28,29,30], the present work focuses on a twofold experimental and systematic computational investigation of such superlattices. The pseudomorphic model that complies with the strain state condition determined experimentally is the so-called “strain-balance” model of GaN wells on AlN barriers, withholding the zero external stress condition, $\stackrel{̿}{\mathit{\sigma }}$ = 0. The computational study relies on density functional theory (DFT) atomistic simulations. These reveal compliance with the elastic theory predictions of strain partitioning and the contribution of the planar interfaces to the total energy of the superlattice. In addition, the electronic structure of the studied superlattices has been investigated to highlight the impact of interfaces on the energy gap values.

2. Materials and Methods

2.1. Experimental

The samples studied in this work have been kindly provided by Professor Ted Moustakas (Department of Physics and Department of Electrical and Computer Engineering, Boston University). These contain GaN/AlN multiple quantum wells (MQWs) grown by radio frequency plasma-assisted molecular beam epitaxy (rf-MBE) on (0001) AlN/Al₂O₃ templates [31]. The samples consisted of 35 periods of GaN/AlN short-period superlattice with 4c-GaN wells and 8c-AlN barriers (c: the lattice parameter along the common [0001] crystallographic direction) capped with 100 nm thick AlN.

The sandwich technique was used for preparing cross-sectional specimens. Mechanical grinding was followed by focused Ar+ ion milling to thin the specimens to electron transparency. HRTEM observations were performed along the [$\mathbf{1}\overline{\mathbf{2}}\mathbf{10}]$zone axis of the wurtzite structure, using a 200 kV JEOL JEM 2011 electron microscope, with a point-to-point resolution of 0.19 nm and a spherical aberration coefficient Cs = 0.5 mm. Processing and analyzing HRTEM images have been made using the software package GATAN Digital Micrograph Suite 3 [32].

2.2. Computational Method

2.2.1. Geometrical model

The experimental observations were used to prepare analogous atomistic supercells for use as the starting configurations for the numerical modeling of the interfacial structure and energy. Accordingly, reference axes x, y, and z spanned the crystallographic directions [$\overline{\mathbf{1}}\mathbf{010}]$, [$\mathbf{1}\overline{\mathbf{2}}\mathbf{10}]$and [0001] of the wurtzite structure, respectively. For all the DFT simulations, initial supercells were constructed with dimensions of a unit cell in the basal plane (one atom) and lengths L_bc = (m+8)c-AlN with m =1÷8. Within the initial supercells, the GaN wells were introduced by replacing 2m layers of the Al sublattice with 2m Ga layers. Figure 1 illustrates the projection of such a model along the $\left[\mathbf{1}\overline{\mathbf{2}}\mathbf{10}\right]$ zone axis. Here, the thicknesses of the constituent layers cope with the experimentally studied system, 4c-GaN/8c-AlN.

2.2.2. DFT calculations

Energetic and structural characteristics of GaN/AlN superlattices were investigated via DFT calculations with the Quantum Espresso ab-initio package [34, 35, 36] and the thermo_pw driver of routines [37] computing ab-initio material properties. The density functional of Wu et al. [38] was implemented using wc Projector Augmented Wave (PAW) pseudopotentials with non-linear core corrections [39]. This functional was chosen since, thereby, the lattice constants and the elastic moduli of the constitutive single crystalline compounds are realistically reproduced. The calculations used the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization algorithm [40] with cutoffs set respectively at E_c=140 Ry for the kinetic energy of plane waves and r_c=1260 Ry for the charge density. Moreover, the following convergence criteria have been adopted for the energy, $\mathit{\delta }\mathit{E}<{10}^{-8}\mathit{R}\mathit{y}$, the forces, $\mathit{\delta }\mathit{F}<{10}^{-8}\mathit{R}\mathit{y}/\mathit{B}\mathit{o}\mathit{h}\mathit{r}$ and stresses, $\mathit{\delta }\mathit{\sigma }<{10}^{-8}\mathit{k}$b. It can be seen in Table A1 and Table A2 that computed lattice constants and elastic moduli of the single crystalline binary compounds are in fair good agreement with their experimental counterparts, consistently with the above-given justification about the choice of using the specific wc, PAW pseudopotentials in this work.

Relaxed, strain-balanced bi-crystalline structures were obtained by minimizing the energy of initial configurations with full periodic boundary conditions and vanishingly small external stress ($\stackrel{̿}{\mathit{\sigma }}=\mathbf{0}$). The resulting values of the linear dimensions of the supercell, transversal and normal to the interfaces, respectively labeled, a_bc, L_bc, and the corresponding total potential energy, E_bc, are listed in Table A3.

For the reference purpose, total and projected densities of states, henceforth referred to as DOS and PDOS respectively, were calculated for the single crystalline compounds by including the Hubbard correction for strongly correlated systems implemented in the Quantum Espresso package (DFT+U) [41,42,43]. The values of calculated bandgaps (AlN:6.11 eV, GaN: 3.50) fit the experimental data (AlN:6.10 eV, GaN: 3.45 eV).

3. Results

3.1. TEM observations

Figure 2 (a) illustrates a dark field cross-section TEM (XTEM) image showing the 24 top QWs of the superlattice recorded using the 0002 reflection in two-beam conditions. The selected area electron diffraction (SAED) pattern in Figure 2 (b) corresponds to the $\left[\mathbf{1}\overline{\mathbf{2}}\mathbf{10}\right]$oriented zone axis of the wurtzite structure. The mismatch between the lattices of AlN and GaN amounts ≈4%. Consequently, two reflections should be visible in the SAED pattern originating from the two crystalline layers. The absence of such reflections indicates that GaN has grown pseudomorphically on AlN, whereas the satellite spots along the [0001] c-axis mark the periodicity of the superlattice (see the pattern in the inset).

Figure 3 is the HRTEM image of a 4c-thick GaN QW embedded in 8c-thick AlN adjacent barriers viewed along the [12 ̅10] zone axis. The stacking sequence of wurtzite is ‘AaBb’ where ‘A, B’, ‘a, b’ mark metal and nitrogen atoms respectively. This image vizualizes (0002) lattice planes of the wurtzite crystal structure. Fourier filtering, using the 101 ̅0 spatial frequency in a series of HRTEM images from several interfacial areas shows the absence of misfit dislocations, further strengthening the conclusion that GaN has grown pseudomorphically, as is shown in the inset of Figure 3. Accordingly, the in-plane lattice parameter is common to the two crystallites. A sampling methodology is developed and presented below, allowing the extraction and evaluation of the dimensional information from the HRTEM images. Thereby, the comparison between the experiment and the results of DFT modeling of analogous superlattices becomes possible. This methodology measures distances expressed in image pixels units between Nc successive AlN (0002) planes on both sides of any given GaN slab. As expected, these distances increase linearly with Nc, as shown in Figure 4. The linear least-squares fit on data collected from eight different sample regions yield the slope and the intercept with the vertical axis, respectively cAlN=28.453 and DGaN=118.31 (pixel units). The first represents the c-lattice parameter of AlN, and the second is the thickness of the GaN slab (Figure 4 (a)).

The apparent data dispersion is low and well inferior to the reasonable qualitative estimation of the maximum measurement error ±c^AlN/2, which expresses the deviation between image bright spots and atomic column positions in the sample, merging uncertainties from various causes. The error principally affects the intercept, whereas with increasing N_c, the slope of the linear fit is decreasingly influenced.

Figure 4 (b) displays the average distance, d (in pixels), between successive atomic columns along a [1 ̅010] row measured far from the interfaces and from eight different regions of the sample as functions of Nc. The slope of the linear least-squares fit, 15.59, relates to the a-lattice parameter of AlN, a=2/√3 15.59≈18.001. This last, combined with the above-found value cAlN=28.453 pixels (Figure 4 (a)), yields (c/a)AlN≈1.58. It is considered in the following that this value represents a trustable determination of this ratio. The value is lower than the experimental, c/a AlN≈1.6 [44], in the single crystal AlN at the mechanical equilibrium. It indicates that both sample constituents are deformed due to their pseudomorphic relationship. Moreover, the least-squares straight line does not pass precisely through the axes origin in this figure. This relates to the errors committed to measuring the distances of successive atomic columns,which affects very little the slope of the linear fit and its average value representative of the a-lattice constant of AlN barriers in the sample. It should be stressed here that lattice constants of the superlattice are likely to evolve with changing the thicknesses of the constituents. For further progress in the matters of strain partitioning and the comparison of the above findings with modeling, there is need to convert lattice constant measurements into real units, which has been done via the deconvolution methodology presented in the next section.

3.2. HRTEM data deconvolution

Distance measurements expressed in units of image pixels can be converted into real units (Å) since the correspondence is known between pixels and the image scale. However, appreciating the accuracy of the thereby emerging conversion factor, C_pA is not a simple task (pA stands for ‘pixels to Å’). Alternative to this ‘geometrical’ determination, the choice has been made to estimate C_pA and the corresponding lattice parameters and deformation states of the superlattice constituents via an optimization procedure guaranteeing that the total residual elastic energy stored in the sample is at the minimum. This assumption is qualitatively compatible with the strain-balanced condition that one can reasonably admit describing the deformation states of the experimental samples. It corresponds to states defined by the pseudomorphic relationship between the two compounds in mechanical equilibrium (vanishing external stresses). The path toward the minimal elastic energy condition consisted in simultaneously optimizing lattice constants and the elastic energy content of the sample by defining the following objective function and numerically localizing its lowest value (multi-dimensional minimization package MERLIN [45]):

$$f={\left(\frac{c_{pred}^{AlN}}{c_{TEM}^{AlN}}-1\right)}^2{+\left(\frac{a_{pred}^{AlN}}{a_{TEM}^{AlN}}-1\right)}^2+w_{el}\left\{E_{pred}^{el,AlN}{V}_{pred}^{AlN}+E_{pred}^{el,GaN}{V}_{pred}^{GaN}\right\}$$

(1)

Indexes ‘pred’ and ‘TEM’ mark respectively, quantities predicted and extracted from the TEM micrographs and $V_{pred}^{AlN}$, $V_{pred}^{GaN}$ are predicted volumes of the respective lattice cells at the mechanical equilibrium. The minimization algorithms included in the MERLIN package tentatively choose values of the function variables until the procedure repeatedly fails to further decrease the current value of f. During the minimization, the weight factor $w_{el}$, transforms the last term in eq. (1) into an a-dimensional quantity adjusting at a value comparable to the first two geometrical terms. Upon convergence of the procedure, optimal values of the conversion factor and the lattice constants are obtained, corresponding to the lowest value of the elastic energy stored in the sample.

$$\left\{\begin{array}{c}{a_{TEM}^{AlN}\left(\AA \right)=C}_{pa}{a}_{TEM}^{AlN}\left(pixels\right)\\ {c_{TEM}^{AlN}\left(\AA \right)=C}_{pa}{c}_{TEM}^{AlN}\left(pixels\right)\end{array}\right.$$

(2)

It is worth noting that the first two terms in eq. (1) refer solely to the lattice parameters of AlN, extracted from the least-square fits of Figure 4 (a)&(b). These values of the experimental lattice parameters represent ‘bulk’ AlN in the sample and are free from the interfacial perturbation, which is foreseen short-ranged for such interfaces. This analysis is the principal asset of the extraction methodology developed in the present work. Regarding the GaN lattice parameters, the pseudomorphic growth imposes $a_{TEM}^{GaN}=a_{TEM}^{AlN}$. Upon convergence to the minimum of the objective function f, the value C_pA=0.17684 Å/pixel is obtained. Consequently, the TEM data, transformed by using this value yield $D^{GaN}\approx 20.922\AA$, for the thickness of the GaN slab and $c^{GaN}\approx \frac{D^{GaN}}{4}\approx 5.23\AA$, $c^{AlN}=5.03\AA$. The values of the lattice parameters found via this optimization procedure, $a^{GaN}=a^{AlN}$, the corresponding parallel and normal deformations, the related zero-stress predictions, and residual total elastic energies in the two compounds are displayed in Table 1. The absolute values of the in-plane deformations in both compounds are comparable. Moreover, the values of ${\epsilon }_{\perp }$ are vanishingly small and compatible with the values expected holding the zero-stress condition. It follows that dimensions normal to the interfaces of the superlattice are about equal to these of the equimolar mechanical mixture, irrespective of the thicknesses of the bicrystals. The c-lattice parameter of AlN in Table 1 is about 1% lower the value deduced by transforming the experimental measurement in pixels via the conversion factor C_pa. This should be attributed to interfaces in the experimental sample, not accounted for by eq. (1).

3.3. Spatial extension and deformation states of interfaces

The relaxed configurations of the studied systems allow for extracting the perturbations of the interplanar distances d₀₀₀₂ due to the hetero-phase interfaces, information experimentally out of reach. Figure 5 (a) is the graph of the d[0002]-spacings of metallic and nitrogen successive layers of a strain-balanced pseudomorphic GaN on AlN (m=4, n=8) as a function of the z-position obtained from a relaxed configuration of a superlattice analogous to the experimental sample (Figure 2). As is qualitatively visible in the figure, only a few atomic layers are perturbed by the interfaces. Local, relative changes of d-spacings, are better highlighted by, ${\epsilon }^i$ defined by:

$${\epsilon }^i=\frac{z_{i+1}+z_{i-1}-2z_i}{z_i-z_{i-1}},$$

(3)

where the indexes i-1, i and i+1 refer to the same species of successive planes. The sum of the ${\epsilon }^i$values between planes locating respectively at z_n and z_m correspond to the total relative change of the spacing between them. Figure 5 (b) presents two representations for m=4 & n=8 and m=1 & n=8. It confirms that ‘bulk-like’ regions, corresponding to the “plateaus” in Figure 5 (a), separate the two interfaces and that perturbations induced by these are not evolving and are very short-ranged. An interesting result is also visible, namely that in the systems (m, n) = (m<2, n=8) the two inequivalent interfaces merge into a single interface.

Values of the a-lattice parameter and the length, $L_{bc}$, of the relaxed bicrystal prove that the deformation states corresponding to different GaN thicknesses are different (Figure 6). As expected, the variation of $L_{bc}$ is linear with m, whereas the slope of the linear least-squares fit the data yields the value of the GaN c-lattice parameter in the superlattice. This is found practically identical to the single crystal value, $c^{GaN}=5.1913\AA$ (Table A1). The intercept with the vertical axis is an estimation of the c-lattice parameter of AlN, $c^{AlN}=\frac{39.895}{8}\approx 4.9869\AA$, being this value identical to that obtained for the single crystal. This finding confirms the qualitative statement that interfacial perturbations are short-ranged and shows that the total thicknesses of the studied systems along the common c-axis are about equal to these of the equimolar mechanical mixture of the two compounds. However, by considering the precisely calculated values of L_bc, we may notice short elongations that should be attributed to the influence of interfaces. These almost negligible elongations are not visible in Figure 6 (b).

Taking as reference states the single crystalline binary compounds at mechanical equilibrium and the mechanical mixture, Figure 7 (a) displays the superlattice’s normal strain components and the unperturbed ‘bulk’ regions as functions of the GaN thickness. It is seen that the apparent deformation of the computational box along the c-direction is small and practically constant with increasing the GaN thickness, whereas, by comparison, the two nitrides undergo significant and opposed sign normal deformations as is expected. Remarkably, the overall normal deformation of the computational cell is vanishingly small. Optimal c-lattice parameters of the superlattice constituents and computed elastic moduli yield the normal deformations plotted in Figure 7 (b) as functions of the zero-stress prediction (dashed line in this figure). Minor and systematic deviations visible in the figure are likely related to the no-symmetric influence of the interfaces and have not been investigated in more detail.

The results above establish that the deformation state of the superlattice and of the two interfaces change with the evolving QW thickness. At this stage, it is foreseen that the interfacial excess energy should behave similarly in agreement with similar investigations in the literature [47,48,49].

3.4. Deformation states of relaxed superlattices and zero-stress

The total energy calculations have shown that withholding the condition, $\stackrel{̿}{\sigma }=0$ the tangential deformations, e_//, are functions of the QW thickness. Using the ab-initio values of the elastic moduli obtained by thermo_pw (Table A2), the elastic energy of the mechanical mixture of the two compounds is computed as a function of the common a-lattice parameter fixed by the pseudomorphic and the minimum elastic energy conditions. Figure 8 represents the results obtained for the different superlattices investigated in this work (m=1÷8, n=8) as functions of the common a-lattice parameter and, by using the experimental elastic moduli and equilibrium single crystal lattice parameters. The minima of the plots identify the equilibrium a-lattice parameter of the superlattices, whereas the corresponding c-lattice parameters can be derived via the zero-stress elastic condition.

Lattice constants are also available for the ‘bulk’ regions of the relaxed superlattices, which provides the basis of a comparison with the elastic predictions above and the experimental values for the system m=4, n=8 (Figure 9 (a)–(c)). The overall agreement for the a-lattice parameter is excellent. In contrast, differences are visible with the zero-stress predictions of the c-lattice parameters in both the constituents that amount to $\mathsf{\Delta }c$≈±0.01 Å. Moreover, fair agreement is obtained between the lattice constants extracted from the TEM images, the zero-stress, and the total energy evaluations. These deviations should be attributed to the interfacial d-spacing irregularities and the difference between experimental and calculated elastic constants.

The general conclusion is that the zero-stress condition is approximately fulfilled in the superlattice regions free from interfacial perturbations. It provides a useful tool for the experimentalist to predict the linear dimensions of superlattices without resorting to time-consuming total energy calculations. A similar method for predicting the common a-lattice parameters was presented in [50] with equivalent results.

3.5. Interfacial excess energy

Under zero external stresses, the deformation states of the investigated superlattices are functions of the thickness of the GaN QW along the c-axis direction (Figure 10 (a), (b)), which implies that the total interfacial energy content, γ, should also depend on the evolving QW thickness in the studied systems. This excess quantity is obtained from relaxed total energies of the superlattices with reference the mechanical assembly of unstrained AlN and GaN single crystals equimolar to the given superlattice constituents:

$$\gamma ={\gamma }_1{+\gamma }_2=\frac{1}{A_{bc}}\left[E_{bc}-(E_{mm}\left(m,n\right)+E_{el}^{GaN}\times V_0^{GaN}\left(m\right)+E_{el}^{AlN}\times V_0^{AlN}\left(n\right))\right]$$

(4)

where, ${\gamma }_1$, ${\gamma }_2$ are respectively the interfacial energies of the two inequivalent interfaces present in the studied superlattices, $A$ is the cross-sectional area of the computational box and $E_{bc}$ the total energy of the relaxed system. $E_{mm}\left(m,n\right)$ represents the total energy of the mechanical (ideal) mixture of equimolar, no-interacting single-crystalline compounds, i.e., without chemical bonding between them, made respectively of m-GaN and n-AlN lattice cells along the c-direction, normal to the interfaces, $V_0^{AlN}\left(n\right)$, $V_0^{GaN}\left(m\right)$ the volumes of the unstrained crystals. $E_{el}^{AlN}{,E}_{el}^{GaN}$ represent the respective elastic energies that were calculated by using, $a_c$, the common a-lattice parameter extracted from the total energy calculations with vanishing the external stress whereas the respective c-lattice parameters were determined using linear elasticity (Figure 9 (b),(c)). The ground states obtained for the successive GaN thicknesses correspond to different deformation states of the binary compounds and of their interfaces. The increment of the in-plane deformation of the interfaces between successive states (m±1,n), ${\epsilon }_{11}^s={\epsilon }_{22}^s$, for any value of m can be estimated by:

$${\epsilon }_{11}^s\left(m\pm 1\right)=ln\left(\frac{a_c\left(m\pm 1\right)}{a_c\left(m\right)}\right)$$

(5)

where ${\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}}_c\left(m\right)$ serves as the reference. Equation (5) holds because differences ${\delta a}_c$between neighboring states are small (Table A3). The obtained γ-values are displayed in Figure 10 (a) as functions of ${\epsilon }_{11}^s={\epsilon }_{22}^s$, whereas numerical labels identify the thickness in c-lattice parameter units of the QW each time used as the reference state. In this figure, the full line represents a parabolic fit to the numerical data corresponding to m=1. This functional form has been shown appropriate by recent theoretical work when the interface is tangentially deformed under vanishing the normal external stress component [48,49]. For the sake of clarity, parabolic fits for other m-values, i.e other reference states, have not been represented in the figure. The fits yield the values of the sums of the interfacial stresses $〈{\mathsf{\Gamma }}_{\alpha \beta }〉$and of the elasticity stiffnesses, $〈{\mathsf{\Gamma }}_{\alpha \beta \gamma \delta }〉$of the two dissimilar interfaces, quantities usually not provided by similar works in the literature [48,49]. The last reveals identical for all the eight reference states amounting to $〈{\mathsf{\Gamma }}_{\alpha \beta \gamma \delta }〉=171.53J/m^2$, whereas the former is found increasing linearly as a function of the in-plane deformation (Figure 10 (b)). The desirable comparison of the present findings with results from the literature is not possible, for the investigated systems differ from ours. However, the magnitudes of interfacial stress and elastic stiffness comply with these reported in the literature for a tilt grain boundary in copper [48,49]. Unlike other works, properties of the two dissimilar interfaces present in the studied superlattices are not individually addressable. However, γ-values displayed in Figure 10 (a) are remarkably lower than the experimental value of surface energy in AlN [51] and calculated values for AlN [52] and GaN [53,54,55], which complies with the experimentally known mechanical stability of such interfaces against decohesion.

3.6. Energy gap

The effect of strain on the band structure has been studied for a long time [56], and constitutes an essential ingredient of band gap engineering [57]. In the pseudomorphic superlattices considered here, the lattice mismatch between the two constituents is accommodated by parallel and normal to the interfaces strains in direct relation with the elastic properties of the ‘bulk’ constituents, provided the thicknesses of the GaN/AlN layers are small enough not to favor the formation of misfit dislocations. The variation of the valence and conduction band offsets with strain are mainly bulk effects [57]. It has been shown that strain-induced shifts of the conduction and the valence bands are proportional to the in-plane, parallel strain component, ${\epsilon }_{//}$ [56] holding the same for the energy gap between bands. Since, to our knowledge, the present work is the first systematic investigation of strain-balanced GaN/AlN superlattices with incrementally increasing the QW thickness, we computed the Al, Ga, and N PDOS, and for the reference purpose, these of the single crystals AlN and GaN (Figure 11 (a),(b)). In addition, these obtained for all the studied superlattices are presented in Figure B1 (a)÷(h).

Energy gaps extracted from these calculations are displayed in Figure 12 as functions of the parallel strain component. It is seen that the superlattice-related data satisfy the expected linear dependence of the gap on ${\epsilon }_{//}$ whereas a good agreement is found between experimental and calculated band gaps of the single crystalline compounds (see §2.2.2 above).

4. Discussion and conclusions

HRTEM images of superlattices may contain artifacts, image distortions due to various causes, structural defects other than the interfaces, and can be affected by point-to-point resolution limitations. To assess the influence of such distortions, we performed a statistical analysis of distance measurements between atomic columns in various areas of the experimental images to determine the lattice constants of the AlN barrier and the thickness of the GaN QWs. We found that the distance averages are affected by very low statistical noise proving the quality of the experimental sample. These values are expressed in image pixels, and their transformation in real units requires the determination of the conversion factor, C_pa. To this end, a new methodology is proposed based on the minimization of the total elastic energy stored in the experimental sample and has been approximated, evaluating the elastic energy of the equimolar mechanical mixture. Moreover, the hypothesis is made that the influence of interfacial perturbations is vanishingly small. The results establish that QWs and barriers are strained in agreement with the computational results, giving confidence to the proposed approach.

The innovating method adopted in this work for calculating the properties of pseudomorphic, GaN/AlN superlattices consisted in exploring computational analogous satisfying the mechanical equilibrium condition ($\stackrel{̿}{\sigma }=0$) for systems with a fixed thickness of the barrier and incrementally increasing that of the QW. This produces a variety of minimum energy configurations with evolving deformation states for the bulk constituents and their heterophase interfaces, unlike previous works investigating QWs and barriers for specific values of the respective thicknesses. In the present work, successive fundamental states of the superlattices are characterized by lattice constants and interfacial excess energies γ, all revealing functions of the GaN QW thickness. The observed trends have permitted us to reach the following conclusions:

(i)

The spatial extension of the interfaces is very short-ranged and remains practically unchanged when the thickness of the GaN QW changes.

(ii)

Energy-minimized configurations of the studied systems adopt different deformation states. It has been shown that the elastic prediction of the lattice constants of elastically strained bulk compounds reasonably agrees with the predictions of total energy minimizations.

(iii)

A no-trivial result is that the interfacial excess energy is a function of the thicknesses of the superlattice constituents, which relates to the evolving deformation states of the interfaces with changing the respective thicknesses. Works in the literature have not accounted for such effects.

(iii)

Similar to structural properties and the interfacial excess energy, the superlattice electronic properties are tightly deformation state dependent. Indeed, it has been established that the valence and conduction bands offsets in strained superlattices and the energy gaps as well are in a linear relationship with the tangential deformation component${\epsilon }_{//},$ of the bulk constituents [54]. We have verified that the energy gap obeys this theoretical prediction.