Correlation between band gap, dielectric constant, Young’s modulus and melting temperature of GaN nanocrystals and their size and shape dependences

With structural miniaturization down to the nanoscale, the detectable parameters of materials no longer remain constant but become tunable. For GaN nanocrystals example, the band gap increases while the dielectric constant, Young’s modulus and melting temperature decrease with decreasing the solid size. Herein, we developed the models to describe the size and shape dependences of these seemingly uncorrelated parameters for GaN nanocrystals, based on our established thermodynamic model for cohesive energy of metallic nanocrystals. Consistency between our theoretical predictions and the corresponding experimental or simulated results confirms the accuracy of the developed models and indicates the essentiality of cohesive energy in describing the effects of size and shape on the physicochemical properties of different low-dimensional systems.

where the subscripts 1 and 2 denote the nanocrystals with basal shape and other shapes. The validity of Eq. (1) has been verified by available experimental, molecular dynamics (MD) simulation and other theoretical results for metallic nanocrystals (e.g. Au, Ag and Ni), molecular nanocrystals (e.g. Ar) and covalent nanocrystals (e.g. Si) 34 . Thus, we assume that Eq. (1) is able to predict the melting behavior of GaN nanocrystals. Figure. 1 illustrates the dependence of the normalized melting temperature on the thickness t of hexagonal GaN nanotubes calculated by Eq. (1) where the inner radius r in of nanotube is set as 1.20 nm. As a comparison, available MD simulations results 15 are also listed. Since a periodic boundary condition has been applied in the axial direction 15 , the influence of two end faces is thus negligible. Moreover, the melting of nanotubes starts from the outer surface and proceeds towards the inner surface 15 , A 2 in Eq. (2) is thus equal to the outer surface area rather than the sum of outer and inner surface area. In this case, λ ≈ 2(r in + t) 2 /[3 1/2 t(2r in + t)] can be obtained in terms of Eq. (2). Note that the melting temperature of bulk GaN is not well known because of experimental difficulties related to the very high melting temperature and the overpressure of N 2 necessary to prevent decomposition before melting. Experiments in a high-pressure anvil cell indicated that GaN does not melt at a temperature up to 2573 K under 68 kbars of pressure 35 . Vasil' ev and Gachon correlated the melting temperature and enthalpy of formation  18 . Since the bulk solid-vapor transition entropy for GaN is unavailable, S b ≈ 13R is taken as a first-order approximation, which equals to that of the mean value of most elements (70-150 J/mol-K) 53 .
Scientific RepoRts | 5:16939 | DOI: 10.1038/srep16939 of III-V compounds and evaluated the melting point of bulk GaN to be 2570 K 36 . There are a few reports on MD simulations of the melting point of bulk GaN. Nord et al. simulated the melting temperature to be 3500 ± 500 K 37 . Using a single-phase or a two-phase MD simulation, the melting temperature of bulk GaN is determined to be 4200 and 3000 K, respectively 15 . Since there is no recognized value of the melting temperature for bulk GaN and the MD simulations results 15 are presented in Fig. 1 for comparison, the average value 3567 K of three reported MD simulation results is taken as a first-order approximation. As shown in Fig. 1, it is evident that the melting temperature of GaN nanotubes increases with the thickness of the nanotubes and agreements between our model predictions and MD results can be found, which confirms our assumption that Eq. (1) can also be used to predict the melting behavior of GaN nanocrystals.

Formula
To deduce the size dependent band gap of GaN, a well-known Arrhenius expression for the size and temperature dependent electrical conductivity μ(D,T) is introduced, where μ 0 denotes a pre-exponential constant. The activation energy for electrical migration is E a = E c − E F with E c and E F being the conduction-band energy and the Fermi energy. In many semiconductors, E F is near mid-gap and thus E a ≈ E g /2 38 . If the change of E g is supposed to be proportional to the change of E a , there is Δ E g (D)/E g (∞) = |Δ E a (D)/E a (∞)| where Δ denotes the change. Assuming that the electrical conductivity at the melting temperature is the same, independent of the melting temperature and therefore independent of the size, one can obtain the expression The dielectric constant results from electronic polarization or electron migration from the lower valence band to the upper conduction band. This process is subject to the selection rule of energy and momentum conservation, which determines the optical response of semiconductors and reflects how strongly the valence electrons couple with the excited conduction electrons 39 . Thus, the dielectric constant of a semiconductor is directly related to its band gap at room temperature. Extending the relationship of ε(∞) = χ(∞) + 1 and the approximation relation of χ(∞) ∝ [E g (∞)] −2 into nanoscale with χ being the electric susceptibility 39 It is known that the Young modulus is fundamentally related to the interatomic bonding and is thus influenced by modifications of the atomic environment. Semi-empirical methods to correlate the Young modulus and the surface thermodynamic properties are always possible. For example, the Young modulus is linked to the surface energy γ sv with the following expression of (∞) =    where L 0 denotes the equilibrium interplanar spacing normal to a surface of the solid 40 . Since the GaN nanocrystal remains the wurtzite structure which is the same as the corresponding bulk, the expression may thus be extended to nanometer size, namely Although the nanocrystal has lattice contraction induced by the large surface-to-volume ratio, ab initio density functional investigations of the atomic structure of GaN nanocrystals with diameters ranging from 2.28 to 1.1 nm found that the contraction of Ga-N bond length at the surface is only 0.7-0.9% in comparison with the bulk 20 . Considering that the Y(D) depression reaches about 40% when the cross-sectional size of square nanowires is reduced to 1 nm 23 , we can concluded that the contribution of h(D) on Y(D) is negligible in this case as a first order approximation to simplify the derivations and calculations. On the other hand, the surface energy denotes the bond energy difference between surface atoms and interior ones while the melting temperature is also directly proportional to the bond strength, the size dependence of the surface energy has thus been deduced to be the same as that of the melting temperature 41 . Combining the above discussions, we can have (1) and the computer simulation results 17,19,21 where agreements can be found, noted that the λ value for hexagonal GaN nanowires can be determined as 2/3 1/2 according to Eq. (2). It is known that different simulation methods result in different results, and we choose the DFT calculations employing the screened Heyd-Scuseria-Ernzerhof 06 hybrid functional rather than the local density approximation for comparison since the former gave more accurate prediction of E g (∞) than the latter 19 . The band gap of GaN nanocrystals is not only size-dependent but also shape-dependent. Figure. 2b shows the Δ E g (D) functions of hexagonal and triangular GaN nanowires in terms of Eq. (7) where the λ value for triangular nanowires can be determined as 3 1/2 according to Eq. (2). These two kinds of GaN nanowires show an increase in the band gap with the decreasing diameter and the band gap of triangular nanowires is larger than that of hexagonal nanowires for a given wire diameter. This trend is understandable since the triangular nanowires are less stable and thus the occupied edge-induced states are at a higher energy compared to the hexagonal nanowires. As a comparison, available DFT calculations employing the DMOL 3 rather than SIESTA code are listed since the predictions of DMOL 3 for bulk properties are closer to the experimental results than those of SIESTA 18 . Noted that the calculated band gap of bulk GaN by DMOL 3 is only 2.58 eV rather than the experimental result of 3.50 eV, E g (∞) = 2.58 eV is thus taken in our calculations. As shown in Fig. 2b, the model predictions are in good agreements with the corresponding DFT simulation results. Figure 2c presents the Δ E g (D) functions of spherical GaN nanoparticles and cylindrical nanowires in terms of Eq. (7) with λ = 1 according to the definition of the shape factor. As a comparison, available DFT simulations results 20 are also listed, which agree with the corresponding model predictions. As shown in Fig. 2c, the relative band gap of the nanowires is always smaller than that of the nanoparticles. Considering the mathematical relation of exp(− x) ≈ 1 − x when x is small enough, as a first order approximation, Eq. (7) combined with Eq. (1) can be simplified as Since D 0 = 2(3 − d)h with the dimensionality d being 0 and 1 for the nanoparticles and nanowires, the ratio of the relative band gap between the nanowires and the nanoparticles is about 2/3, which is similar to the results (0.584 for large size and 2/3 for small size) obtained from an effective mass approximation calculations 42 and 0.62 obtained from the experimental investigations 43 .
In Fig. 2a-c, the employed simulation results correspond to the nanowires with infinite length 17-23 . However, not only the diameter but also the length has an influence on how strong the size effect will be, when the length L of the nanowires is comparable to the diameter. In this case, A 2 = 2π (D/2) 2 + π DL and V 2 = π (D/2) 2 L for cylindrical nanowires with finite length L. While for cylindrical nanowires with infinite length L′ , the surface areas of the top and bottom of nanowire are negligible as a first order approximation due to the length L′ » D, namely A 1 ≈ π DL′ and V 1 = π (D/2) 2 L′ . In terms of Eq. (2), the shape factor λ can be determined as λ ≈ 1 + D/(2L). As shown in Fig. 2c, the Δ E g (D) functions of cylindrical GaN nanowires with L/D = 3 is presented (the blue line) based on Eq. (7). It can be found that the size effect of nanowires with finite length is stronger than that of nanowires with infinite length while smaller than that of nanoparticles when the diameter is the same.
In addition to the size, the band gap of semiconductors can be tuned by alloying. It is reported that the semiconductor nanoalloys have luminescent properties comparable to or even better than the best-reported binary semiconductor nanocrystals and they are promising materials for optoelectronic devices 5,22 . The experimental and theoretical results have found that the composition-dependent band gap E g (x,∞) of the bulk varies monotonically but not linearly with the composition x over the whole range of composition 5,22,32 . On the basis of assumption that a bulk ternary or pseudo-binary semiconductor compound alloy is a regular solution of components, E g (x,∞) of A x B 1-x C semiconductor is often where E g (AC,D) and E g (BC,D) can be determined by Eq. (7). Figure. 2d plots the E g (x) as a function of constituent stoichiometry x for Al x Ga 1-x N nanowires with D = 15 nm in terms of Eqs. (12) and (7), where the experimental data are also included for comparison 22 . It can be observed that the E g (x) plot has downward shift and E g (x) increases with the increasing x and the prediction are consistent with the experimental results. The agreement confirms the advantage of Eq. (12) in comparison with the empirical expression that there is no adjustable parameter in the equation and hence it substantially simplifies the calculation of E g (x,D) for semiconductor nanoalloys. As shown in Fig. 2a-d, both size and composition can increase the band gap of semiconductor. Thus, in order to raise the band gap of narrow-gapped semiconductor, applying alloying nanocrystals is a better way compared to semiconductor elements or compounds just by decreasing size.
Based on Eqs. (8) and (10), we calculate the reduced ε(D) and Y(D) functions of square GaN nanowires as shown in Fig. 3, where λ = 1 is determined for square GaN nanowires according to Eq. (2). Clearly, the calculated ε(D) and Y(D) values correspond to the MD simulations results 23 . In detail, ε(D) and Y(D) decreases with the decreasing cross-sectional size and the drop becomes significant when D < 2 nm. Similar to the blue line presented in Fig. 2c, the length effect on the dielectric constant and Young's modulus should also be include when the length L of the nanowires is comparable to the diameter. In this case, A 2 = 2D 2 + 4DL and V 2 = D 2 L for square nanowires with finite length L and then the shape factor λ can be determined as λ ≈ 1 + D/(2L) in terms of Eq. (2). As shown in Fig. 3

Discussions
The agreements shown in Figs 1-3 not only confirm the accuracy and validity of our developed models but also indicate that there must be the common origin of the size dependences of these seemingly uncorrelated parameters. Pauling and Goldschmidt indicated that, if the coordination number of an atom were reduced, the ionic or metallic diameter of the atom would shrink spontaneously 45 . A bond-order-length-strength correlation mechanism also indicated that imperfections in atom coordination would cause the remaining bonds of under-coordinated atom to contract in association with a gain in strength of single bonds 45 , noted that the coordination imperfection can originate from the changes of the size and shape. However, because of the bond-order loss, the cohesive energy (which is the sum of single-bond energies over all of the coordinates of a specific atom) of an under-coordinated atom will drop 45 . As the result, the cohesive energy suppression induced local quantum entrapment perturbs the Hamiltonian that determines the band gap (band-gap expansion) and hence the process of electron polarization consequently (dielectric constant suppression) since the dielectric suppression is thought to result from the breaking of surface polarizable bonds 46 . Zhou and Huang have proposed that whether the Young's modulus is softer or stiffer depends on the competition of bond loss and bond saturation 47 , where the bond loss originates from the cohesive energy suppression while the bond saturation is induced by the electron redistribution. The agreement between the model prediction and the corresponding MD simulation results shown in Fig. 3b indicates that the bond loss is too much to be compensated by the bond saturation, leading to the reduction of Young's modulus of GaN nanowires. Based on Lindemann's criterion of melting, the melting is related to the lattice vibration where the latter is determined by the cohesive energy 34 . Moreover, Li et al. ascribed the melting to diffusion of the local clusters, which would lead to some new defects in nanocrystals, noting that these defects would cause the decrease of the cohesive energy and thus accelerate the melting 48 . Together with the above discussions and the fact that Eq. (1) is established based on the size-dependent cohesive energy model, we can conclude it is the coordination-imperfection induced cohesive energy suppression responsible for the size and shape trends of the melting temperature, band gap, dielectric constant and Young's modulus.
The agreement between our developed models and the observations not only confirms the validity of our models but also provides guidelines for III-V semiconductor nanomaterials and device design. Together with previous works on size dependences of thermal stability and surface energy of nanoparticles and nanocavities 24,41,49,50 , surface tension of liquid droplets 49 , thermal conductivity and diffusivity of semiconductor nanocrystals 51 , and catalytic activation energy of metallic nanocrystals 52 , the work here indicates the essentiality of cohesive energy in describing the effects of size and shape on the physicochemical properties of different low-dimensional systems.