Strain- and electric field-induced band gap modulation in nitride nanomembranes

The equilibrium structural parameters of the nitride nanomembranes are collected in table 1. Note that the fully optimized hexagonal configurations retain the symmetry of the graphene-like structure.

The calculated intraplanar bond length (i.e. nearest-neighbor distance, R) increases in going from BN to AlN to GaN. R_B−N of 1.45 Å is slightly larger than the typical value of 1.42 Å of the sp² hybridized carbon configuration, and is in agreement with the results of previous theoretical DFT studies [23]. The calculated lattice constants are 2.509, 3.159 and 3.301 Å for the BN, AlN and GaN nanomembranes, respectively. Interestingly, these 2D materials have similar lattice constants compared with their 3D counterparts, in spite of a different atomic coordination number. The lattice constants of the wurtzite BN, AlN and GaN were reported to be 2.54, 3.06 and 3.17 Å, respectively [34].

The cohesive energies per atom for BN, AlN, and GaN nanomembranes are found to be 7.81, 4.94, and 3.65 eV, respectively suggesting the B–N bond to be the strongest followed by the Al–N and Ga–N bonds. We define the cohesive energy as the difference between total energy of a nanomembrane and the sum of total energies of corresponding constituent atoms⁵.

The trend in the cohesive energy can be correlated with the nature of chemical bonds as demonstrated by the contour plots of the charge density (figure 2). For BN, the presence of directional bonds indicates their nature to be covalent with sharing of electrons among B and N-atoms with a considerable overlap of their electronic wavefunctions. Conversely, the bonds have more ionic character in AlN due to a charge transfer from Al to N, which leads to localized wavefunctions (shown as concentric spheres in figure 2) centered at the N-atoms with no significant overlap of the two neighboring spherical islands. For the GaN nanomembrane, we found localization of the electronic wavefunction, similar to the case of AlN, but with a smaller degree of distortion together with a smaller overlap in the wavefunction centered at Ga and N (figure 2). Consequently the bonding in GaN could be considered to be partly ionic and partly covalent.

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The electronic band structures of the nitride nanomembranes are presented in figure 3. The minimum energy gaps for BN, AlN and GaN are calculated to be 4.60, 3.06 and 1.70 eV, respectively. The valence band maximum (VBM) is located at the high-symmetry K point for all configurations whereas the conduction band minimum (CBM) is at K for BN, and Γ for AlN and GaN. Therefore, a direct bandgap is predicted for BN whereas indirect band gaps are predicted for AlN and GaN. This is in contrast to what has been reported for the bulk wurtzite nitrides with the gap being indirect for BN and direct for AlN and GaN [35]. The difference in the nature of the bandgap can be attributed to a shift in the position of the VBM—from Γ to K—when the dimensionality is reduced, whereas the character of the CBM remains unchanged. Nevertheless, the calculated magnitude of the minimum bandgap of the nanomembranes follows the same order as predicted for their bulk counterparts [35].

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The calculated density of states (DOS) and partial DOS are shown in figure 4. VBM is mainly composed of N-p orbitals, with a small contribution from the cation-p orbitals. The conduction band minimum of BN is composed of the B-p orbitals whereas that of AlN (GaN) is mainly composed of Al-s (Ga-s) orbitals.

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The most significant and advantageous change, which the nitride nanomembranes undergo due to the reduced dimensionality, is the enhancement of their electronic mobility. The electron effective mass is calculated to be 0.14 and 0.06 m₀ for AlN and GaN, respectively, where m₀ is the free electron mass. Note that the electron effective mass is formally the reciprocal of the second derivative of the energy–momentum relation. Compared with GaN, the relatively larger electron effective mass of AlN is directly related to the flatness of the energy curve around CBM (figure 3). On the other hand, the hole effective mass is calculated to be 0.43 and 0.36 m₀ for AlN and GaN, respectively. For comparison, the theoretical (experimental) values of the electron effective mass are reported to be 0.23 m₀ [37] (0.29–0.4 m₀ [37, 38]) and 0.14 m₀ [36] (0.22 m₀ [37]) for the wurtzite AlN and GaN, respectively.

We define a 'biaxial modulus' to characterize the in-plane isotropic stiffness of the considered nanomembranes. The so-called biaxial modulus is defined as, ${Y}_{2\mathrm{D}}=\frac{1}{A}(\frac{{\partial }^{2}E}{\partial {\epsilon }^{2}})\bigr \vert _{\epsilon =0}$, where E is the total energy, is the biaxial strain and A₀ the equilibrium area of the unit cell. The 'biaxial modulus' is analogous to the two-dimensional Young's modulus, which is related to the uniaxial strain [39].

In our case, the in-plane strain (ε) is simulated by varying the lattice constant from −8% to +8% away from its equilibrium value while all bond angles are kept fixed. Thus, a biaxial strain is considered whereby the entire lattice is uniformly stretched. This uniform strain could be obtained, for instance, by synthesizing the nanomembrane on top of a material that has a lattice mismatch. From our definition, 'negative' strain is considered as the strain by compression and 'positive' strain is considered as the tensile strain for the nanomembrane.

The corresponding total energy curves as a function of strain are shown in figure 5. The calculated values of the biaxial modulus are 384, 142 and 178 N m⁻¹ for BN, AlN and GaN nanomembranes, respectively. These results follow the same hierarchy as the strength of their interatomic bonds (table 1). For comparison, the bulk moduli follow a similar order with the values of 400, 207, and 245 GPa for BN [40], AlN [41], and GaN [42], respectively. Consequently, one can conclude that the nitride nanomembranes, specifically AlN and GaN are significantly more malleable than their bulk counterparts. We believe that the malleability of AlN and GaN together with the enhanced electron mobility make these systems attractive for flexible electronics [12].

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Considering that the application of strain can affect the electronic properties of a given system, we now investigate the effect of strain applied to the nanomembranes on their energy gaps. For the case of in-plane tensile strain (i.e. ε > 0), a consistent reduction in bandgap with increasing strain is predicted (figure 6).

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Elongation of the bond length under tensile strain modifies the orbital hybridization of neighboring atoms, thereby facilitating tunability of their band gaps. Both BN and AlN show a smaller degree of tunability with tensile strain. However, a direct–indirect band transition (from K–K to K–M) is noted for BN, which is related to the variation of orbital composition of the VBM states with the applied tensile strain. This is not the case with AlN where no such transition in the bandgap with the applied tensile strain is noted.

A significant tunability with the strain is predicted for GaN, which also goes through a semiconductor–semimetal transition at the value of +8%. This transition can be attributed to reduction of the overlap of the orbitals due to an increase in the Ga–N bond length in the strained lattice. In particular, the Ga-s-like conduction band around Γ shifts downwards until it eventually closes the gap (see supplementary figure S1 available at stacks.iop.org/JPhysCM/25/195801/mmedia). Our results are consistent with the previous results on graphitic GaN multi-layer films where the bandgap was reported to decrease linearly with the tensile strain [33].

Application of an in-plane compressive strain (i.e. ε < 0) to the nanomembranes increases their band gaps. A shortening of the bond length increases the repulsion between orbitals associated with the neighboring atoms resulting in the reorganization of the bands away from the Fermi level. For GaN, it leads to an indirect–direct gap transition at about −4% as VBM moves from K to Γ. The valence wavefunctions at Γ are initially doubly degenerate; one level is due to N-p_z whereas the other has an in-plane nitrogen p_x–p_y character. After the application of the compressive strain, p_z-orbitals shift down and the p_x–p_y orbitals move upward to become the CBM, consequently leading to the direct–indirect bandgap transition. The direct–indirect transition (from K–K to Γ–K) predicted for BN at higher compressive strains has a similar origin, i.e., from the splitting and subsequent upward shift of the levels at Γ. No change in the nature of the gap in AlN is predicted for either compressive or tensile strain applied to the system.

The effects of an electric field applied perpendicular to the nanomembranes on the electronic properties are shown in figure 7(a). A prominent variation of the bandgap is predicted for AlN as compared to that for BN or GaN. This is accompanied by a noticeable change in the dipole moment of AlN (figure 7(b)), and may be attributed to the nature of its chemical bond. A higher degree of electronic polarization in the electric field is facilitated by the ionic bond, which in turn, leads to a significant variation of the bandgap of AlN under the application of the external electric field.

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It is to be noted that the maximum value of 2 V Å⁻¹ considered for the perpendicular electric field appears to be high. However, a nanomembrane should be able to sustain such a high field since the electrical field is likely to induce changes in the charge density within a 2–3 Å around the nanomembrane. Furthermore, Britnell et al [43], found the breakdown voltage of ∼1 V nm⁻¹ (1 GV m⁻¹) for the system consisting of a few layers of h-BN sheet sandwiched between two gold electrodes. In experiments h-BN was used as a dielectric sandwiched between the metallic electrodes. The simulation model, on the other hand, employs a free-standing nanomembrane in vacuum which works as the dielectric. Thus, given the differences in configurations considered for experiment and simulation, the applied field of 2 V Å⁻¹ can be taken as the maximum value of the breakdown voltage for the 'isolated' nanomembrane.

A comparison of the effects of induced strain and applied electric field suggests that the strain-engineered tailoring of the bandgap is a more effective approach for the nitride nanomembranes. Application of the perpendicular electric field only shifts the states relative to the Fermi energy without changing the nature of the bands (i.e. the Stark effect). A tunability of the bandgap therefore depends upon the magnitude of dipole moments along the direction of the field. Introduction of the in-plain strain, however, changes the nature of the bands in addition to shifting them near the Fermi energy.