Free Access
Issue
Eur. Phys. J. Appl. Phys.
Volume 88, Number 1, October 2019
Article Number 10101
Number of page(s) 7
Section Semiconductors and Devices
DOI https://doi.org/10.1051/epjap/2019190119
Published online 17 December 2019

© EDP Sciences, 2019

1 Introduction

Among the group-III nitride semiconductors, aluminum nitride (AlN) has the largest band gap (6.2 eV) and many other prominent properties [1, 2], making it a promising key candidate in deep ultraviolet optoelectronic [3] and other applications [46]. As an important research topic in the field of AlN, the surface properties have also attracted more and more attentions in recent years [79], further expanding the applications. So far, a major challenge in the exploration of surface properties is the surface polarization arising from the non-centro-symmetric nature of the surface structure such as the typical (0001) surface. In particular, on the outermost layer of the surface, polarization is mainly induced by the remarkable charge and atom rearrangement due to the dangling bonds, which is much bigger than that of the other surface layers. This surface polarization is responsible for surface adsorption, reconstruction, and relaxation [10]. For example, some experiments and first principles studies have indicated that this polarization influences the molecular adsorption and reaction on the ferroelectric surface [1113]. Besides, a series of unusual structures such as nonorings and nanosprings have been attributed to the surface polarization [14]. It also plays an important role in the research of fast-growing thin film technology.

As we know, and are the usual low-index and steady surfaces of AlN, which are generally considered to be non-polar facets. Since bulk AlN possesses the largest spontaneous polarization which is nearly three times stronger than that of GaN and ZnO among nitride semiconductors [15], the induced electric field by polarization will drastically affect the electron distribution, causing band bending and reducing the luminescence efficiency [16, 17]. To avoid the influence of polarization on the UV LEDs and LDs based on AlN thin films, much effort has been paid to improve the epitaxial growth of high quality AlN films on these two surfaces [1821]. They are also the main components of the lateral facets of AlN nanostrctures, determining the surface properties of AlN nanoscale materials [22]. Although these two surfaces are usually regarded as non-polar facets, polarization may be induced by the charge and atom rearrangement in consequence of the surface dangling bonds. A similar finding has been reported in the study of SiC [23] which has the same surface structure as AlN. In contrast to the well-studied bulk polarization, theoretical study on the AlN surface polarization is insufficient. Therefore, to further understand the surface properties and improve the performance of device fabricated based on surface, it is much desired to study the surface polarization and how the polarization change in the structure evolution from a bulk to a surface.

In this work, we investigated polarization properties of AlN and surfaces in terms of local dipole (LD) obtained by maximally localized Wannier functions (MLWFs) [24]. It is found that polarization is introduced by dangling bonds in these two surfaces, and is much stronger than that of the bulk. Besides, we performed a layer-by-layer research of surface polarization, and found that polarization oscillates in the surface layers and quickly reaches the bulk value within about four layers. These polarization variations were further analyzed from the contribution of ionic and electronic part to the LD.

2 Calculation methods

Traditionally, polarization of a system is defined by the dipole moments per unit volume, which is applicable for bulk structure. However, for surface and nanostructure, the volumes remain uncertainty owing to the dangling bonds. Minor difference in the determination of volume can lead to a pronounced different conclusion of polarization for the same structure [25]. Moreover, according tothe modern polarization theory [26], surface and nanostructure are modeled by using the periodic system whose spontaneouspolarization is charactered by the polarization difference between the polar structure and a non-polar reference structure. Unfortunately, unlike the bulk that can easily build a non-polar centro-symmetric reference structure (e.g., for bulk wurtzite (WZ) AlN, the reference structure is bulk zinc-blend (ZB) AlN), it is impossible to construct such a non-polar reference system in form of surface or nanostructure owing to the ineluctable dangling bonds.

To solve these two puzzled problems and to quantify the concept of local polarization, Wu and co-workers [27] introduced a method by defining layer polarization based on MLWFs to analyze local polarization of ferroelectric perovskite superlattices layer-by-layer, which gives an insightful local description of the polarization. In their definition, polarization of layer j (pj) can be written as [27] (1)

where pj is along the epitaxial growth direction (z-axis). Here S is the basal cell area, Qτ and Rτ are the charge quantity and position of the τ-th ion in the layer j; is the position of the m-th Wannier center with charge -2e. Cicero et al. [28] further developed Wu’s method and expanded its application, by proposing an ingenious approach using average LD (p) of the minimum repeat unit of the structure, which can be expressed by (2)

Here N is the total number of minimum repeat unit of the structure, Pi is the LD of the i-th repeat unit displayed below by Figure 1b. This method is especially suitable for the polarization characterization of surface and nanostructure.

Specifically, since MLWFs method partitions the system to be localized Wannier function centers (WFCs) and ionic charges, one can define average LD of the minimal repeat unit of the system to discuss the polarity. Because computation of the average LD has included the bond length variation through out the whole system, the volume problem is then overcome naturally. Moreover, because bulk and surface both can be constructed by the same minimal repeat unit, spontaneous polarization of surface can be directly determined by the difference of average LD of polar surface and non-polar bulk structure. Thus it is no longer to build the impossible non-polar surface or nanostructure reference system. This indicates that the second problem is also overcome. Consequently, Cicero’s creative approach exhibits significant advantages in investigation of polarity properties of surface and nanostructure. Through the average LD, it is favorable to perform not only the comparison of different structures with different dimensions, but also the exploration of intrinsic contributions.

For AlN, the minimal repeat unit is N-Al pair. On the basis of MLWFs, AlN system can be partitioned into +5 N ions, +3 Al ions and −2 WFCs (see Fig. 1a). In our calculation, to detailedly determine the LD of N-Al pair, +5 N ion and +3 Al ion are further partitioned into four +1.25 N ions and four +0.75 Al ions, respectively. Each −2 WFC is further partitioned into −1.25 and −0.75 WFC. Then we considered that LD of a N-Al pair can be composed of two parts represented by the black arrows depicted in Figure 1b. One originates from the N atom and its four nearest WFCs, generating four local dipoles where each dipole can be evaluated by +1.25 N ion and −1.25 WFC. The other comes from the four WFCs and their neighboring Al atoms, where each dipole is computed by +0.75 Al ion and −0.75 WFC. The ultimate LD of a N-Al pair is the vector sum of the above eight local dipoles. Subsequently, polarization of the whole AlN structure is illustrated by the average LD of the total N-Al pairs.

Caro et al. [29] also proposed a method to calculate the local electric polarization. They have recognized that Berry phase method only gives an average value of polarization, not the local polarization at a microscopic scale for the system. Using classical dipole moment per unit volume of an ensemble of point charges, they proposed an approach to implement a position dependent value of the macroscopic polarization, namely, local polarization. Their purpose and method are both similar to that of our research. For instance, the dipole moment in atomic site is calculated within the region enclosed by the central atomic site 0 and each of its nearest neighbors, which is in analogy to the dipole moment of N-Al pair (shown in Fig. 1b) used in our study. The difference is that we utilize MLWFs to obtain the dipole moment while they employ Born effective charges calculated by Berry phase. Nevertheless, since Berry phase and MLWFs can both equivalently calculate the bulk spontaneouspolarization, and the macroscopic spontaneous polarization of AlN and GaN evaluated by local dipole moment of their research are in accordance with ours, we believe that these two local polarization methods are essentially identical.

The method employed by this research mainly involves calculation of MLWFs. As we know, Wannier functions are not unique due to the phase indeterminacy of the Bloch functions. By defining a measure of the spatial spread functional of the Wannier functions, Marzari and Vanderbilt [30, 24] proposed an approach that iteratively refines these degrees of freedom to determine the optimally localized set of generalized Wannier functions, the so called MLWFs. Thus, although there is arbitrariness in the initial generalized Wannier functions that one on the atom may be different from that on the bond, the final obtained MLWFs are unique. This is similar as that of the report by Wahn and Neugebauer [31], where there is only an optimum set of generalized Wannier functions to allow a reconstruction of the original band structures.

The calculations were performed employing the generalized gradient approximation of Perdew-Burke-Ernzerhof (PBE) as implemented in the VASP package [32, 33]. The electron-ion interaction was described by projected augmented wave method (PAW) [34]. The plane wave energy cutoff was set at 450 eV and a 9 × 6 × 1 K-point mesh is used for structural optimizations. The optimized lattice parameters of the primitive AlN cell are a = 3.1284 Å, c = 5.0146 Å and u = 0.3815. To perform a polarization analysis layer-by-layer and to research how many layers will be influenced by the surface, a large surface model is constructed containing 16 AlN bilayers in which two bilayers in the middle of the slabs are fixed during the structure relaxation. The other bilayers are fully relaxed until the force acting on each atom is reduced to be less than 0.02 eV/Å. Based on the test of energy convergence, the vacuum thickness between two neighbor slabs is set at 10 Å. The WFCs of the systems are calculated by using Wannier90 code [35].

thumbnail Fig. 1

(a) A regular minimal repeat unit of AlN with WFCs distribution. (b) Schematic diagram of the LD (represented by black arrows) of a N-Al pair. The N, Al and WFCs are displayed by the sapphire, light brown and small yellow balls, respectively.

3 Results and discussion

3.1 LD of bulk AlN

To verify the above approach, we firstly computed the spontaneous polarization of bulk WZ AlN in terms of average LD, and then compared the result with that calculated by traditional Berry phase method. According to the modern theory of polarization [26], spontaneous polarization of bulk WZ AlN can be defined as a polarization difference between the WZ and ZB structure. In our local dipole method, WZ and ZB is the minimal repeat unit (N-Al pair) of the corresponding primitive cell. After LD calculation of WZ and ZB structure, the spontaneous polarization can be determined. Note that LD is in unit of Debye (C⋅m), for comparison with Berry phase method, LD should be transformed to be in unit of C/m2 by , where V is volume of the N-Al pair.

The obtained average LD is listed in Table 1, in which the results of GaN and ZnO are also listed for comparison. The computed average LD for AlN and GaN are −0.5679 D and −0.2274 D, corresponding to the values of spontaneous polarization are −0.090 and −0.032 C/m2, respectively.These values coincide well with those obtained by the Berry phase method in the previous report [15]. The computed average LD of ZnO is −0.247 D, which is also well consistent with Cicero’s report (−0.24 D). [28] In particular, as shown in Table 1, the average LD of ZB AlN, GaN and ZnO is close to zero, illustrating the natural centro-symmetry of ZB structure and suggesting the intuition of the LD method. Consequently, the approach in use of LD is reasonable, and the following computation is proceeded within the frame of local dipole method.

Table 1

Average LD (in unit of Debye) of the minimal repeat unit of bulk ZB and WZ AlN, GaN and ZnO. ΔLD is the LD difference between the LD of WZ and ZB structure. Psp (in unit of C/m2) is the spontaneous polarization deduced by ΔLD.

3.2 Polarization of AlN surface

Since the LD is directly determined by the positions of WFCs and ions, it is important to carefully construct and compute MLWFs based on the surface band structure. Figure 2a represents the surface band structure of AlN calculated by the density functional theory (DFT). It is found that two surface states locate in the band gap due to the two dangling bonds within the surface unit cell. The lower one is fully occupied and the upper one is empty. Thus we considered all of the occupied valence bands as a composite group to construct the MLWFs with N-centered sp3 guiding functions, and obtained 128 WFCs after wannierization procedure. To assess the computed MLWFs, a comparison between the Wannier interpolation band and the traditional DFT results is shown in Figure 2a, where the two band structures are virtually indistinguishable, illustrating a good wannierization. Figure 2b shows the density of states (DOS) projected onto the selected surface WFC which evidently deviates from the bond direction. We can see that this WFC derives from the occupied surface state. It will be used to characterize the charge transfer on the surface in the following procedure.

Due to the dangling bonds on the surface, an apparent atomic rearrangement and charge transfer are induced in the surface layers, leading to a different structure compared to bulk. As shown in Figure 3a, the bond length of the outermost layer of the relaxed AlN surface is 7.28% shorter than that in bulk, with Al atom relaxing inward of about 0.25 Å, which is consistent with the previous report [22]. Moreover, small variations in bond length also occur in the other surface layers.

Charge transfer in AlN surface can be intuitively illustrated by distribution of the WFCs drawn in Figure 3a. Analyzing the positions of the WFCs on the outermost layer, they are found to be significantly different from those in bulk, revealing a charge redistribution due to the surface dangling bonds. Specifically, we can see that the location of the WFC (marked by red arrow) dramatically deviates from the N-Al bond and is much closer to the N atom than the other three ones. In light of the DOS curve, it indicated that the charge of Al dangling bond transfers to N atom in consequence of the large electro negativity difference between N and Al atoms. This charge transfer strengthens the Coulomb attraction and hence leads to the bond contraction. The result of charge transfer coincides well with previous report concluded by band theory [22]. In fact,the charge of N dangling bond also moves to N atom opposite the original orientation due to the strong attraction of N nucleus. Thus these two types of charge transfer determine the ultimate location of the peculiar WFC (where the red arrow is pointing). Owing to electrical charge repulsion, these four WFCs around N atom deviate from the original direction of the N-Al bond. The positions of the WFCs located on the other surface layers also have a slight deviation, elucidating that charge transfer is not just confined in the outermost layer.

The charge transfer and atomic rearrangement described above together result in surface dipole, which significantly influence the surface polarization properties. We performed a LD analysis in the direction perpendicular ( direction) and parallel ([0001] direction) tothe surface as a function of the distance of the AlN layer from the surface, and displayed the result in Figure 3b. It is found that the LD of the outermost layer exhibits a maximum value among the surface layers and is much greater than that of the bulk. Specifically, the LD parallel to the surface is about eleven times larger than the spontaneous polarization in bulk, and the LD perpendicular to the surface is about eight times larger. The existence of such strong polarization in non-polar surface is unexpected, but this phenomenon occurs mainly in the outermost layer. The LD oscillates in other surface layers and converges to the bulk value after about four layers (3.6 Å). In the direction, it is noted that the LD of the second and the third layer is also larger than that of the bulk, and the LD is reversed in some layers such as the third and the fifth layer. Conversely, in the [0001] direction, the LD of the second and third layer is smaller than that of the bulk. Similar result was also previously reported for SiC, where the surface LD is one order of magnitude larger than that of the bulk [23].

Therefore, the AlN surface is actually not a completely non-polar structure but rather a non-polar bulk covered by strong polar surface layers. This polarization can induce a localized electric field near the surface [3638] that brings dramatical influence on the surface. Specifically, according to the polarization orientation represented in Figure 3b, surface bands are bent upward which is similar to that of ZnO surface reported by Tisdale [39]. Meanwhile, the induced electric field not only promotes the separation of electrons and holes, but also the diffusion and distribution of carriers near the surface [40]. Therefore, although non-polar structure is usually used to circumvent the built-in electric field, it may still suffer from the consequent impact. Moreover, the non-polar structure often grows epitaxially on non-polar plane substrates such as sapphire and SiC, the dipole can be induced in the interface due to the lattice mismatch between them. This interface dipole will result in space charge region and thus may reduce the recombination efficiency of the light emitter based on the non-polar film [41, 42]. Similarly, interface dipole will also be introduced in heterostructures with non-polar orientations such as the non-polar GaN/AlN quantum dot, which cause a built-in field like the conventional heterostructures grown along the [0001] orientation and reduces the device efficiency [43].

Surface polarization is responsible for surface relaxation and reconstruction. The structure always tends to reduce the dipole moment to get lower surface energy through several ways, such as relaxation, reconstruction, adsorption, etc. [44]. Taking the relaxation as an example, we computed the variation of the dipole moment in the outermost surface layer before and after relaxation. It is found that the dipole moment decreases from 5.411 to 4.484 (in unit of Debye), demonstrating that surface relaxation reduces the polarization effect. Surface reconstruction is usually accompanied by a larger ionic and electronic rearrangement, so it may impose a stronger influence on the dipole moment than surface relaxation. On the contrary, surface passivation can fill the surface state and try to maintain a bulk-like electronic structure in the surface. That is, passivation prevents the charge rearrangement and may greatly decrease the surface dipole, thus reduces the polarization.

Surface polarization also affect the adsorption and reaction which implements on the surface. For example, in the study of molecular adsorption on the ferroelectric BaTiO3 and LiNbO3 surface [45], it is found that the reactive sticking coefficient is polarization dependent. Specifically, polar molecule such as ethanol is more likely to interact strongly with the polar surface since the surface polarization increases the Vander Waal interaction between the polar molecule and surface. Recently, Jin et al. performed a DFT study of the formaldehyde adsorption on ZnO surface which has the same surface structure as AlN [46]. They found that the electrostatic interactions between the polar formaldehyde molecule and the surface plays an crucial role in the molecular adsorption.

In addition, non-polar surface model is usually used to determine the natural band offset [47]. As described above, the induced electric field can produce an additional potential near the surface, consequently changes the bulk-vacuum potential difference and causes an indeterminacy. Thus in order to obtain the accurate natural band offset, this additional potential should be subtracted. However, since there was no effective method to calculate the absolutemagnitude of surface dipoles (responsible for the electric field) at that time, this influence was not considered in previous work [47]. Now, by using the method implemented in our research, this problem may be solved.

To further investigate the effect of surface relaxation on the polarization properties and explain the above LD variation in the two directions, we computed the changes of electronic and ionic contributions (ΔPel, ΔPion) to the LD compared with the bulk structure layer-by-layer, as illustrated in Figure 4. On the outermost layer, the ionic part has a larger value than that of the electronic part which presents an opposite sign in these two directions. As we have acquired from the surface relaxation, the evident compression and slight tilt of the N–Al bond in surface is accompanied by an apparent variation of ion position along the [0001] and direction. Contrary to the pronounced change of the ionic position, the electronic contribution mainly arises from the charge transfer of the dangling bonds (described by the location change of the WFC where the red arrow is pointing in Fig. 3a), which may cause a less impact on the LD than the ionic part. Thus atomic rearrangement plays an important role in the production of polarization of the outermost layer. It suggests that surface relaxation and reconstruction may be more efficient at adjusting the surface polarization than the other scenarios, such as the change of covalence in the surface layers, the filling of surface states [44].

Beneath the surface, both the ΔPion and ΔPel decrease rapidly, and nearly keep equal value with opposite sign after about four layers, leading to the counterbalance. In addition, as shown in Figure 4a, since ΔPion is larger than ΔPel in the third and fifth layer along the direction, the LD reversion can be intuitively explained. The above findings is of help to adjust the surface polarization.

thumbnail Fig. 2

(a) DFT (blue line) and Wannier interpolated (red line) band structure of AlN . (b) Density of states (DOS) projected onto the selected surface WFC. The Fermi level is set at zero.

thumbnail Fig. 3

(a) Side view of the relaxed AlN surface and WFCs (small yellow balls) distribution. N and Al atoms are depicted by sapphire and light brown balls, respectively. (b) Local dipole (illustrated by black ball and red Square) in the [0001] and direction. Each point corresponds to the average LD of a AlN layer away from the surface.

thumbnail Fig. 4

Variation of electronic and ionic contribution to the LD with respect to the bulk structure in the (a) direction and (b) [0001] direction, respectively.

3.3 Polarization of AlN surface

In the case of surface, the unit cell contains two N and two Al dangling bonds, leading to four surface states in the band gap where half of them are occupied, as shown in Figure 5a. The MLWFs computation is performed from all the occupied valence bands. Because of the excellent agreement between the DFT and the Wannier interpolation band structure, the wannierization is satisfactory. Through the DOS displayed in Figure 5b, we found that the two selected WFCs are projected from the two occupied surface states, which will be used to analyse the charge transfer of the dangling bonds.

The optimized structure and computed WFCs are shown in Figure 6a. Similar to the relaxation of surface, it is found that the Al atoms on the outermost layer have an apparent inward movement which is about 0.18 Å, smaller than that of the surface (0.25 Å). The N atoms slightly relaxed outward of about 0.007 Å. Then the induced bond length is shorter than that of bulk (1.807 Å instead of 1.901 Å). The electronic structure of the surface is analyzed through the WFCs distribution. We found that the WFC (marked by red arrow) is very close to N atom, elucidating that the charge of the Al dangling bonds transfer to N atom. It is also found that the WFCs on the surfaces deviate from the bond direction, and their distribution is very similar to that of the surface. This indicates the high similarity in electronic structure of these two surfaces.

The surface polarization analysis was also performed layer-by-layer and shown in Figure 6b. Since the most significant movement of the atoms and WFCs occurred on the outermost layer, it is found that the corresponding LD exhibits a pronounced value which is nearly eight times larger than that of the bulk in the direction perpendicular ( direction) to the surface. This behavior is almost the same as the surface. Along the [0001] direction, the LD is four times larger than that of the bulk, which is evidently smaller than that of the surface. These LD variation occurred in the two directions can be explained by the electronic and ionic contribution as shown in Figure 7. Along the direction, due to the similar surface relaxation, we found that the contribution from ionic part and electronic part are nearly identical to those of the surface. On the contrary, it is the electronic part that dominates the surface polarization along the [0001] direction, while the ionic part exhibits a little change when compared to the bulk. This is because the movement of Al atom and N atom is smaller than those of the in this direction. Interestingly, The LD is found to be reversed and tends to eliminate the polarization of the bulk on the two outer surface layers along the [0001] direction. This is attributed to the dominant electronic part which has an opposite sign. From the second layer, the LD decreases rapidly and approaches to the bulk value after about four layers. This also suggests that AlN surface is not a completely non-polar surface, which is similar to surface.

thumbnail Fig. 5

(a)DFT (blue line) and Wannier interpolated (red line) band structure of AlN surface. (b) Density of states (DOS) projected onto the selected surface WFC. The Fermi level is set at zero.

thumbnail Fig. 6

(a) Side view of the relaxed AlN surface and WFCs (small yellow balls) distribution. N and Al atoms are represented by sapphire and lightbrown balls, respectively. (b) Local dipole (illustrated by black ball and red Square) in [0001] and direction. Each point corresponds to a LD of an AlN layer away from the surface.

thumbnail Fig. 7

Variation of electronic and ionic contribution to the LD with respect to the bulk structure in the (a) direction and (b) [0001] direction, respectively.

4 Conclusion

The polarization properties of AlN and surface was investigated in terms of the LD obtained by MLWFs. The study indicated that an apparent polarization was produced due to the dangling bonds on these surfaces. The polarization on the outermost layer is much greater than that of the bulk and the orientation of LD is reversed in the [0001] direction. The polarization exhibits an oscillation and approaches to the bulk value after about four layers. Then a further polarization analysis was proceeded by the electronic and ionic part to explain the polarization reversal.

Acknowledgements

The authors gratefully acknowledge the financial support of the Natural Science Foundation of Education Bureau of Shannxi Province, China (No. 16JK2099), and the Fundamental Research Funds for the Central Universities.

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Cite this article as: Haibo Niu, Guangde Chen, Youzhang Zhu, Yelong Wu, Honggang Ye, Polarization properties of AlN (101̅0) and (112̅0) non-polar surfaces: maximally localized Wannier functions study, Eur. Phys. J. Appl. Phys. 88, 10101 (2019)

All Tables

Table 1

Average LD (in unit of Debye) of the minimal repeat unit of bulk ZB and WZ AlN, GaN and ZnO. ΔLD is the LD difference between the LD of WZ and ZB structure. Psp (in unit of C/m2) is the spontaneous polarization deduced by ΔLD.

All Figures

thumbnail Fig. 1

(a) A regular minimal repeat unit of AlN with WFCs distribution. (b) Schematic diagram of the LD (represented by black arrows) of a N-Al pair. The N, Al and WFCs are displayed by the sapphire, light brown and small yellow balls, respectively.

In the text
thumbnail Fig. 2

(a) DFT (blue line) and Wannier interpolated (red line) band structure of AlN . (b) Density of states (DOS) projected onto the selected surface WFC. The Fermi level is set at zero.

In the text
thumbnail Fig. 3

(a) Side view of the relaxed AlN surface and WFCs (small yellow balls) distribution. N and Al atoms are depicted by sapphire and light brown balls, respectively. (b) Local dipole (illustrated by black ball and red Square) in the [0001] and direction. Each point corresponds to the average LD of a AlN layer away from the surface.

In the text
thumbnail Fig. 4

Variation of electronic and ionic contribution to the LD with respect to the bulk structure in the (a) direction and (b) [0001] direction, respectively.

In the text
thumbnail Fig. 5

(a)DFT (blue line) and Wannier interpolated (red line) band structure of AlN surface. (b) Density of states (DOS) projected onto the selected surface WFC. The Fermi level is set at zero.

In the text
thumbnail Fig. 6

(a) Side view of the relaxed AlN surface and WFCs (small yellow balls) distribution. N and Al atoms are represented by sapphire and lightbrown balls, respectively. (b) Local dipole (illustrated by black ball and red Square) in [0001] and direction. Each point corresponds to a LD of an AlN layer away from the surface.

In the text
thumbnail Fig. 7

Variation of electronic and ionic contribution to the LD with respect to the bulk structure in the (a) direction and (b) [0001] direction, respectively.

In the text

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