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Phase transition and band-structure tuning in InN through uniaxial and biaxial strains
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 025501 (http://iopscience.iop.org/0953-8984/26/2/025501) View the table of contents for this issue, or go to the journal homepage for more
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 025501 (9pp)
doi:10.1088/0953-8984/26/2/025501
Phase transition and band-structure tuning in InN through uniaxial and biaxial strains Yifeng Duan1 , Lixia Qin1 , Liwei Shi1 , Gang Tang1 and Hongliang Shi2 1
Department of Physics, China University of Mining and Technology, Xuzhou 221116, People’s Republic of China 2 Beijing Computational Science Research Center, Beijing 100084, People’s Republic of China E-mail: [email protected], [email protected] and [email protected] Received 23 October 2013, in final form 3 November 2013 Published 5 December 2013 Abstract
The phase transitions and band structure of InN under uniaxial and biaxial strains are systematically investigated using first-principles calculations. The main findings are summarized as follows: (I) although graphite-like phases are observed for both types of strain, the phase transitions are drastically different: second order for uniaxial strain and first order for biaxial strain. Furthermore, the second-order transition is driven by elastic and dynamical instabilities, whereas the first-order transition is driven only by elastic instability. (II) The wurtzite bandgap is always direct and that of the graphite-like phase is always indirect. Furthermore, the wurtzite bandgap is drastically enhanced by compressive uniaxial strain but reduced by tensile uniaxial strain. However, both biaxial strains greatly reduce the bandgap and eventually the semi-metallic phases are achieved. (Some figures may appear in colour only in the online journal)
1. Introduction
equibiaxial stress has reached the ideal tensile strength [19]. Therefore, the wurtzite-to-graphite-like transition needs to be further investigated for other wurtzite semiconductors. In this paper, our initial work is to systematically study the effects of uniaxial and biaxial strain on the crystal structure of InN in order to reveal its phase transitions and the underlying mechanism. The 0.7 eV Eg of ground-state InN has extended the direct Eg of group-III nitride alloys into the near-infrared range, which is experimentally crucial in optoelectronic applications [1]. In theory, the small Eg of InN is attributed to the lowest conduction-band minimum (CBM) among all group III–V semiconductors and the high valence-band maximum (VBM) due to the p–d repulsion [20, 21]. It is noteworthy that the band structure is sensitive to the interatomic distances and relative positions of atoms. In the previous works [12–14], the effects of both types of strain on the band structure have been systematically investigated for AlN and GaN using first-principles simulations and novel results have been reported. (I) AlN: the wurtzite and graphite-like Eg are always direct for uniaxial strain, whereas
InN, with its small direct bandgap (Eg ), is constantly under extensive investigations due to its abundant physical and electrical properties [1]. Strain engineering has been widely adopted to adjust the Eg to a desired value for particular optoelectronic applications by inducing intrinsic strain through lattice mismatching in epitaxial films or core/shell nanowires [2–4]. Such lattice mismatching leads either to biaxial strain in the (0001) plane or uniaxial strain along the [0001] direction. First of all, the hexagonal structure is easily modified by the strain: an intermediate graphite-like phase between wurtzite and rocksalt has been observed [5–16]. The new phase, with the P63 /mmc space group, shows fivefold coordination bonding (wurtzite shows fourfold and rocksalt sixfold coordination). This is consistent with widely accepted common sense [17, 18]: both types of strain show the equivalent effects on the structural and electronic properties. However, the graphite-like phase is only obtained in ZnS for uniaxial strain, although this high-symmetry phase is preferred for biaxial strain—before the phase transition, the 0953-8984/14/025501+09$33.00
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of unoccupied bands, 192 such bands are included to ensure good convergence. To find the strain-free lattice constants, the lattice vectors and atomic coordinates are fully relaxed until the Hellmann–Feynman force acting on each atom is reduced to ˚ −1 . In the presence of uniaxial (biaxial) less than 0.02 eV A strain (), relaxation is performed with the lattice constant c (a and b) fixed until the following conditions are satisfied within a small tolerance: σ11 = σ22 < 0.02 GPa (σ33 < 0.02 GPa) and σij = 0 for i 6= j, where σ11 and σ22 are the stresses in the (0001) plane, and σ33 along the [0001] direction. The σ33 (σ11 ) is adjusted by changing strain 33 (11 ) step by step to reach all desired values in the uniaxial (biaxial) case.
direct and indirect band structures are observed in either the wurtzite or graphite-like phase for biaxial strain. GaN: the wurtzite Eg is always direct and the graphite-like Eg always indirect for both types of strain. (II) AlN: uniaxial strain reduces the wurtzite Eg and enhances the graphite-like Eg , whereas the wurtzite and graphite-like Eg are greatly reduced by biaxial strain. GaN: both types of strain reduce the wurtzite and graphite-like Eg . Although the strains theoretically adopted are relatively large, appreciable tuning in Eg may be accomplished for modest and realistic strains achievable in nanowires or ultrathin films. As a member of group-III nitrides, although the ground-state Eg of InN is experimentally available, the accurate dependence of its Eg on uniaxial and biaxial strains remains unclear so far. In this work, our main aim is to systematically compare the effects of uniaxial and biaxial strains on the structural and electronic properties of InN in order to reveal the phase transition as well as its driving mechanism and to illustrate the accurate dependence of band structure on strain. The conventional density functional theory (DFT) is adopted to obtain the optimized structure at each special strain, due to its credibility in predicting the structural properties [22]. To overcome the deficiency of conventional DFT: where the Eg of ground-state InN is negative, the Heyd–Scuseria–Ernzerhof (HSE) functionals and G0 W0 approximation are adopted simultaneously [23–25]. (I) The HSE functionals employ a screened short-range Hartree–Fock (HF) exchange, which overestimates the Eg . (II) The G0 W0 approximation, based on many-body perturbation theory, offers a parameter-free option for the accurate prediction of the electronic structure [18].
3. Results and discussion
In the presence of σii , a new lattice parameter corresponds to a specified uniaxial out-of-plane given by 33 = (c − c0 )/c0 and a commensurate equibiaxial in-plane by 11 = 22 = (a − a0 )/a0 , where a0 and c0 are the strain-free lattice constants. Figure 1 summarizes the calculated dependence of σ , which displays the markedly different trends for uniaxial and biaxial . The tensile σ33 first increases and then decreases drastically with 33 , with a maximum of ∼21 GPa at 33 = 0.16. This indicates that the calculated ideal tensile strength along the c axis is ∼21 GPa, which is much smaller than the critical values of ∼45 GPa for AlN and of ∼34 GPa for GaN [14]. This is consistent with the ground-state elastic constant c33 of InN being much smaller than that of GaN, and the c33 of GaN being smaller than that of AlN [30, 31], emphasizing that the larger the ground-state c33 , the larger the ideal uniaxial tensile strength [19]. The compressive σ33 displays a similar trend to those of AlN, GaN and ZnS [14, 19]: the σ33 magnitude first increases for −0.10 < 33 < 0 and then decreases for −0.16 < 33 < 0.10, finally increasing once again for 33 < −0.16. More unusually, unlike the uniaxial results, the tensile σ11 first increases, until dropping to a low value at 11 = 0.10, and then increases as 11 further increases. Similar behaviors have been observed in AlN and GaN [12], but not in ZnS [19], where an obvious decrease in σ11 at high 11 is observed. As the compressive 11 increases, the σ11 magnitude always increases, which is well known and anticipated, consistent with previous theoretical findings [12, 19]. To reveal the detailed structural evolution, we systematically study the effects of uniaxial and biaxial on the other terms , lattice ratio c/a and internal lattice parameter u, as shown in figure 2. When 33 > −0.16 and 11 < 0.09, the calculated other terms and u display a nonlinear response with respect to , especially for uniaxial ; a linear behavior is obtained for c/a. When 33 < −0.16 and 11 > 0.09, c/a ' 1.20 and u = 0.50. Therefore, the structural transition to the graphite-like phase is achieved simultaneously for both uniaxial and biaxial . This is similar to the previous results of AlN and GaN [12], but different from those of ZnS [19], where the graphite-like phase is only obtained for uniaxial
2. Computational methods
The total energy and band-structure calculations are performed using the projector-augmented-wave method [26] implemented in the VASP code [27]. The projectoraugmented-wave (PAW) method as implemented in the VASP code is utilized to describe the interaction between the ionic cores and the valence orbitals. The Perdew–Burke–Ernzerhof generalized gradient approximation (PBE-GGA) is employed in the conventional DFT calculations, which are labeled hereafter as PBE. The N 2s2 2p3 and In 4d10 5s2 5p orbitals are explicitly included as valence electrons. The electronic wavefunctions are described using a plane wave basis set with an energy cutoff 600 eV, while the obtained fundamental Eg is almost identical to that of 700 eV. A Monkhorst–Pack k-point mesh of 6 × 6 × 6 is used throughout the calculations to obtain well-converged results. The phonon-dispersion calculations are performed with the direct method implemented in the PHONOPY package [28, 29]. This method uses the Hellmann–Feynman forces calculated for an optimized supercell through VASP [27]. In theory, the larger the supercell, the more accurate the dispersion curves obtained. The 2 × 2 × 3 supercell is adopted for all phonon calculations and an accuracy of 0.05 THz for the highest optical zone-center phonon frequency is achieved. In addition, since the G0 W0 calculations are also sensitive to the number 2
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Figure 1. Uniaxial out-of-plane stress σ33 and biaxial in-plane stress σ11 as a function of . The inserted panel shows the detailed 11 dependence of σ11 for 0.10 < 11 < 0.20.
Figure 2. In-plane strain 11 and out-of-plane strain 33 , lattice ratio c/a and internal parameter u as a function of . The structural
transition to the graphite-like phase occurs at 33 ∼ = −0.16 and 11 ∼ = 0.09 (dashed lines). The open symbols refer to the wurtzite phase and the solid symbols refer to the graphite-like phase.
. It is most likely to observe the transformation to the higher-symmetry phase in core/shell nanowires along the c axis or epitaxial films on lattice-mismatched substrates. The most unexpected is that 11 , c/a and u change continuously with uniaxial , whereas discontinuous behaviors are observed for biaxial at the phase transition.
To reveal the driving mechanisms of the phase transition, figure 3 shows the elastic constants cij as a function of uniaxial and biaxial . As shown in the left panel, c33 < 0 and c13 < 0 when 33 > 0.16, which is consistent with the structural instability after the σ33 reaching the ideal strength. When −0.16 < 33 < 0, the value of c33 always decreases 3
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Figure 3. Elastic constants cij as a function of uniaxial and biaxial . The open symbols refer to the wurtzite phase and the solid symbols refer to the graphite-like phase.
decreases until becoming negative at 11 = 0.09, followed by jumping to a high value, and then decreases and eventually becomes negative once again when 11 > 0.20. This suggests that the wurtzite phase becomes elastically unstable at 11 = 0.09, as does the graphite-like phase when 11 > 0.20. The anomalous stability–instability–stability behaviors have been observed in AlN and GaN [12, 14], but not in ZnS for biaxial [19]: where the graphite-like phase is not achieved due to the structural instability. Therefore, the elastic instability plays an important role in the wurtzite-to-graphite-like phase transition in the IIIA nitrides for both types of . To further verify the structural stability, four phonondispersion curves at different are plotted in figure 5 for uniaxial and biaxial , respectively. There are twelve branches per unit cell with four atoms: one longitudinal-acoustic (LA), two transverse-acoustic (TA), three longitudinal-optical (LO), and six transverse-optical (TO) branches, as shown in panel (c). Since the N atom is slighter than the In atom in mass, the internal vibrations of In atoms are mainly responsible for the low-frequency modes. When = 0 (see panel (c)), 33 = 0.06 (see panel (d)) and 11 = −0.08 (see panel (e)), the phonon modes always remain stable, which is consistent with the elastic stability for broad ranges of tensile uniaxial and compressive biaxial . When 33 = −0.20 (see panel (a)) and 11 = 0.14 (see panel (h)), dynamical stability is obtained for the graphite-like phase, consistent with the aforementioned elastic results. This suggests that this metastable phase can be stabilized by large compressive uniaxial and tensile biaxial . The most unexpected is that the soft LA modes are observed near the 0 point at 33 = −0.16 (see panel (b)), but not at 11 = 0.09 (see panel (g)). This is drastically different from that of the wurtzite structure becoming elastically instable for −0.16 < 33 < −0.10 and at 11 = 0.09, respectively. Therefore, although elastic instability plays an important role
and remains negative for a broad range of 33 , which leads to the abnormal behavior of σ33 for −0.16 < 33 < −0.10, similar to the results of AlN, GaN and ZnS [14, 19]. The increase in c33 for 33 < −0.16 leads to an increase in the magnitude of σ33 after the phase transition. As shown in the right panel, as the compressive 11 magnitude increases, the values of c11 , c12 , c13 and c33 increase monotonically, while c44 remains almost unchanged. This is different from the previous results of ZnS [19]: where c44 always decreases and eventually reaches zero. As the tensile 11 increases, c11 , c12 and c44 always decrease and c11 > c12 . A different trend is observed in ZnS for tensile 11 [19]: c11 < 0, c12 >0 with c44 increasing slightly with 11 , suggesting the structural unstability. It is noteworthy that the elastic responses of AlN and GaN to biaxial remain unclear so far. The mechanical stability of the hexagonal structure is judged from the following criteria: c11 −|c12 | > 0, c44 > 0 and (c11 +c12 )c33 > 2c213 (equivalently expressed as 1C = (c11 + c12 )c33 −2c213 > 0). To further compare the effects of uniaxial and biaxial on the structural stability, figure 4 displays c11 − |c12 |, c44 and 1C as a function of in detail. c11 − |c12 | and c44 show similar trends for uniaxial and biaxial , respectively. It is noteworthy that discontinuous behaviors of c11 − |c12 | and c44 are observed at the phase transition for biaxial , but not for uniaxial . The most interesting is the dependence of 1C: (I) 1C shows a similar trend to c33 for uniaxial . When 33 > 0.16 and −0.16 < 33 < −0.10, 1C < 0, indicating the structural instability. Therefore, the wurtzite structure only remains elastically stable for −0.10 < 33 < 0.16. On the other hand, the graphite-like 1C increases as the compressive 33 magnitude further increases. This indicates that the graphite-like elastic stability is achieved at large compressive 33 . (II) 1C increases drastically for a broad range of compressive 11 . As the tensile 11 increases, 1C first 4
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Figure 4. Mechanical stability criteria c11 − |c12 | (GPa), c44 (GPa) and 1C = (c11 + c12 )c33 − 2c213 (GPa2 ) as a function of uniaxial and
biaxial . The negative values (red symbols) indicate the elastic instability. The open symbols refer to the wurtzite phase and the solid symbols refer to the graphite-like phase.
Figure 5. Phonon-dispersion curves for different uniaxial and biaxial . Uniaxial: (a) 33 = −0.20, (b) 33 = −0.16, (c) 33 = 0.0 and
(d) 33 = 0.06; biaxial: (e) 11 = −0.08, (f) 11 = 0.0, (g) 11 = 0.09 and (h) 11 = 0.14. The crystal structures are graphite-like in (a) and (h), and wurtzite in (b)–(g).
in the wurtzite-to-graphite-like phase transitions for both types of , the underlying mechanisms should be drastically different.
To further reveal the phase-transition driving forces, figure 6 shows the unit-cell enthalpy H and volume V as a function of uniaxial and biaxial . It is well known that 5
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Figure 6. Unit-cell enthalpy H and volume V as a function of uniaxial and biaxial . The open symbols refer to the wurtzite phase and the solid symbols to the graphite-like phase. Furthermore, the red symbols refer to the elastically unstable structures and the other symbols to the elastically stable structures.
when the H = E + PV of different phases are equal, the phase transition occurs, where the E and V are the unit-cell total energy and volume at a special pressure P, respectively. As shown in top panels, the elastically stable wurtzite and graphite-like phases show different H trends for uniaxial and biaxial . The elastically unstable structures for −0.16 < 33 < −0.10 and 11 = 0.09 act as the ‘bridges’ between the two stable phases, which emphasizes the important role of elastic instability in the phase transitions. As shown in the bottom panels, V, the derivative of H with respect to P, changes continuously with uniaxial , whereas a discontinuous behavior is observed at the phase transition for biaxial . This suggests that the wurtzite structure transforms to the graphite-like structure via a second-order phase transition for uniaxial and a first-order transition for biaxial . Furthermore, the two phase transitions depend on different driving mechanisms: the second-order transition is driven simultaneously by the elastic and dynamical instabilities, but the first-order transition is driven only by the elastic instability. Figure 7 shows the unit-cell total energy E as a function of uniaxial and biaxial . For different types of applied , the relaxations are performed along the different paths to find the lowest-energy structures, eventually leading to a difference in the phase transition from wurtzite to graphite-like. The effects of uniaxial and biaxial on the band structure is systematically investigated using the HSE and G0 W0 methods, respectively. The G0 W0 band structure is plotted by interpolation using the MKWFs approach [32], starting
Figure 7. Unit-cell energy E as a function of uniaxial and biaxial .
To achieve the minimum energy (E), the values of c, u and V are relaxed fully while a and b are kept constant for biaxial ; the values of a, b, u and V are relaxed fully while c is kept constant for uniaxial .
from the G0 W0 calculations. According to the tight-binding model, the valence (conduction) band edge derives mainly from the bonding (antibonding) state of N and In p (s) atomic orbitals [33, 34]. In the interpolation, the s and p orbitals of N and In atoms are chosen for the initial projections. Table 1 lists the ground-state Eg at various levels of theory, together with experimental data for comparison. Since the CBM in InN is the lowest along all group III–V semiconductors; the 6
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Figure 8. Calculated HSE28 and G0 W0 (HSE28) band structures at different uniaxial and biaxial . Uniaxial: (a) 33 = −0.20,
(b) 33 = −0.08, (c) 33 = 0.0, (d) 33 = 0.04 and (e) 33 = 0.10; biaxial: (f) 11 = −0.12, (g) 11 = −0.06, (h) 11 = 0.04, (i) 11 = 0.10 and (j) 11 = 0.14. The structures are graphite-like in panels (a), (i) and (j), and wurtzite in the other panels. The calculations are performed at the PBE optimized structure. The inclusion of the p–d repulsion is described in the text. The VBM is set to zero in each figure. Table 1. Calculated Eg at various levels of theory for the ground-state structure together with the experimental data for comparison.
HSE Eg (eV)
G0 W0
PBE
0.25
0.28
0.30
0.35
0.40
−0.354
0.570
0.698
0.776
0.990
1.211 −0.126
VBM is greatly pushed up by the p–d repulsion because the In 4d level is the highest [34], the PBE Eg is negative. The standard exchange mixing, containing 25% HF and 75% PBE-GGA, is employed in the HSE06 hybrid functionals [35]. The HSE06 Eg is smaller than the experimental data. To display the influence of exchange mixing coefficient α on the band structure, the ground-state Eg at different α are listed and a linear increase of Eg with respect to α is obtained. As shown in table 1, α = 0.28 is the best choice, which is denoted hereafter as HSE28. Since the G0 W0 Eg depends much on the starting wavefunctions and eigenvalues, the G0 W0 (HSE28) Eg is larger than the G0 W0 (HSE06) data and the G0 W0 (PBE) value is still negative. The electronic structure is modified by both types of . Five band structures at different are plotted in figure 8 for uniaxial and biaxial , respectively, where the HSE28 and G0 W0 (HSE28) results are listed together. The HSE28 and G0 W0 (HSE28) valence bands are in good agreement for each . For the conduction bands, although the G0 W0 (HSE28) values are slightly larger than the HSE28 values at certain points, good agreement is obtained near the Fermi level. This is due to the atomic orbitals, mainly contributing to the CBM and VBM, being chosen for the initial projections of the MKWFs approach. For both types of , the stable wurtzite Eg always remains direct, with the VBM and CBM at 0, as shown in panels (b)–(h); the graphite-like Eg is always indirect, with the VBM at H and the CBM at 0, as
PBE
HSE06
HSE28 Exp. [34]
0.457
0.553
0.70
shown in panels (a), (i) and (j). Therefore, for uniaxial and biaxial , the strain-induced trends in the band structure of InN are similar to those of GaN, but different from those of AlN [12, 14]. The d orbitals in the heavy atoms should be responsible for the behaviors. Furthermore, the semi-metallic phase is observed only for tensile uniaxial (see panel (e)), whereas it is also observed for both compressive and tensile biaxial (see panels (f) and (j)). To describe the band-structure evolution in detail, figure 9 shows the HSE28 conduction-band minima (the top panels) and the valence-band maxima (the bottom panels) at different points as a function of uniaxial and biaxial . For the valence bands, N s and In s orbital hybridization is observed at 0, similar to the previous results of AlN and GaN [14, 36]. Furthermore, the N s orbital shows a greater contribution to the CBM in InN, which is consistent with the increasing ionicity from AlN to GaN and to InN [34]. The wurtzite CBM is enhanced by a compressive uniaxial and reduced by a tensile uniaxial , whereas the graphite-like CBM is insensitive to a uniaxial . On the other hand, the wurtzite and graphite-like CBM are reduced by a biaxial , with a discontinuity at the phase transition. It is noteworthy that the CBM of unstable wurtzite structures plays the role of connecting the values of stable wurtzite and graphite-like phases for both types of , similar to the aforementioned results for H. 7
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Figure 9. HSE28 uniaxial and biaxial dependence of the conduction-band minima (the top panels) and the valence-band maxima (the
bottom panels) at different points. The open symbols refer to the wurtzite phase and the solid symbols to the graphite-like phase. Furthermore, the red symbols refer to the elastically instable structures and the other symbols to the elastically stable structures. The Fermi level is set to zero in each case.
Eg . (II) Tensile and compressive biaxial both reduce the wurtzite and graphite-like Eg , with discontinuities at the phase transition. The maximum HSE28 Eg values are 0.821 eV and 0.706 eV at 33 = −0.08 and 11 = −0.02, respectively. The wurtzite semi-metallic phase is obtained for tensile uniaxial and compressive biaxial , and the graphite-like semi-metallic phase is obtained for tensile biaxial .
The bottom panels of figure 9 show the valence-band maxima of 0 and H points as a function of uniaxial and biaxial . The N p and In p, d orbital hybridizations are identified at the VBM, but the N p orbital plays the dominant role, consistent with the results of AlN and GaN [14, 36]. It is well known that the ground-state VBM is mainly occupied by the N px , py and pz orbitals, which overlap in a ‘degenerate’ manner, as shown in figure 8(c). The wurtzite VBM at 0 originates from the N pz orbital for the compressive uniaxial and tensile biaxial (see figures 8(b) and (h)), and from the N px and py orbitals for the tensile uniaxial and compressive biaxial (see figures 8(d)–(g)). The H state is occupied by the N px , py and pz orbitals. As the compressive uniaxial and tensile biaxial increase, the N px and py orbitals play increasingly important roles. More specially, the N px and py levels are greatly pushed up by the compressive uniaxial and tensile biaxial , whereas the N pz level is insensitive to both types of . This leads to an indirect band structure with the VBM shifting to H in the graphite-like phase for uniaxial and biaxial . Figure 10 shows Eg as a function of uniaxial and biaxial . The HSE06, HSE28 and G0 W0 (HSE28) are listed together for comparison and similar trends are observed. It is noteworthy that the unstable wurtzite Eg act as the ‘bridge’ between the values of stable wurtzite and graphite-like phases for both types of , similar to the H and CBM results. The Eg values show different dependences on both types of . (I) Tensile uniaxial reduces Eg , whereas compressive uniaxial enhances the stable wurtzite Eg and reduces the graphite-like
4. Summary
In summary, we have systematically compared the effects of uniaxial and biaxial strains on the phase transition and band structure of InN using the HSE and G0 W0 methods. Although the wurtzite-to-graphite-like transition is observed for both types of strain, the phase-transition properties are drastically different: second order for uniaxial strain and first order for biaxial strain. The second-order transition is driven by elastic and dynamical instabilities, whereas the first-order transition is driven only by elastic instability. Although the bandgap is direct in the stable wurtzite phase and indirect in the graphite-like phase, both types of strain show inequivalent effects on the bandgap: compressive uniaxial strain enhances and tensile uniaxial strain reduces the gap, whereas biaxial strain always reduces it. Although the theoretical strains are relatively large, this work has shed light on the further understanding of such phase transitions and identified opportunities for engineering the bandgap of wurtzite semiconductors through the strains achievable in 8
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Figure 10. Calculated HSE and G0 W0 Eg as a function of uniaxial and biaxial . The electronic-structure calculations are performed at the
PBE optimized structure. The open symbols refer to the wurtzite phase and the solid symbols to the graphite-like phase. Furthermore, the red symbols refer to the elastically instable structures and the other symbols to the elastically stable structures.
core/shell nanowires or epitaxial films on lattice-mismatched substrates. This should also stimulate more investigations into strain engineering in the field of semiconductors, thereby enabling a number of important technological applications with better performance in the fields of electronics and photoelectrics.
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Acknowledgments
The work is sponsored by the Fundamental Research Funds for Central Universities under Grant No 2013QNA30 and the Qing Lan Project. We are grateful to the Advanced Analysis and Computation Center of CUMT for the award of CPU hours to accomplish this work. References [1] Wu J, Walukiewicz W, Yu K M, Ager J W III, Haller E E, Lu H, Schaff W J, Saito Y and Nanishi Y 2002 Appl. Phys. Lett. 80 3967 [2] Duan Y, Qin L, Tang G and Shi L 2008 Eur. Phys. J. B 66 201 [3] Osbourn G C 1982 J. Appl. Phys. 53 1586 [4] Duan Y, Li J, Li S-S and Xia J-B 2008 J. Appl. Phys. 103 023705 [5] Limpijumnong S and Lambrecht W 2001 Phys. Rev. Lett. 86 91 [6] Limpijumnong S and Lambrecht W 2001 Phys. Rev. B 63 104103 [7] Limpijumnong S and Jungthawan S 2004 Phys. Rev. B 70 054104 [8] Sarasamak K, Kulkarni A J, Zhou M and Limpijumnong S 2008 Phys. Rev. B 77 024104 [9] Alahmed Z and Fu H 2008 Phys. Rev. B 77 045213 [10] Maznichenko I V et al 2009 Phys. Rev. B 80 144101 [11] Morgan B J 2010 Phys. Rev. B 82 153408 [12] Duan Y, Qin L, Shi L, Tang G and Shi H 2012 Appl. Phys. Lett. 100 022104 [13] Dong L, Yadav S K, Ramprasad R and Alpay S P 2010 Appl. Phys. Lett. 96 202106 9
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Phase transition and band-structure tuning in InN through uniaxial and biaxial strains
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 025501 (http://iopscience.iop.org/0953-8984/26/2/025501) View the table of contents for this issue, or go to the journal homepage for more
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 025501 (9pp)
doi:10.1088/0953-8984/26/2/025501
Phase transition and band-structure tuning in InN through uniaxial and biaxial strains Yifeng Duan1 , Lixia Qin1 , Liwei Shi1 , Gang Tang1 and Hongliang Shi2 1
Department of Physics, China University of Mining and Technology, Xuzhou 221116, People’s Republic of China 2 Beijing Computational Science Research Center, Beijing 100084, People’s Republic of China E-mail: [email protected], [email protected] and [email protected] Received 23 October 2013, in final form 3 November 2013 Published 5 December 2013 Abstract
The phase transitions and band structure of InN under uniaxial and biaxial strains are systematically investigated using first-principles calculations. The main findings are summarized as follows: (I) although graphite-like phases are observed for both types of strain, the phase transitions are drastically different: second order for uniaxial strain and first order for biaxial strain. Furthermore, the second-order transition is driven by elastic and dynamical instabilities, whereas the first-order transition is driven only by elastic instability. (II) The wurtzite bandgap is always direct and that of the graphite-like phase is always indirect. Furthermore, the wurtzite bandgap is drastically enhanced by compressive uniaxial strain but reduced by tensile uniaxial strain. However, both biaxial strains greatly reduce the bandgap and eventually the semi-metallic phases are achieved. (Some figures may appear in colour only in the online journal)
1. Introduction
equibiaxial stress has reached the ideal tensile strength [19]. Therefore, the wurtzite-to-graphite-like transition needs to be further investigated for other wurtzite semiconductors. In this paper, our initial work is to systematically study the effects of uniaxial and biaxial strain on the crystal structure of InN in order to reveal its phase transitions and the underlying mechanism. The 0.7 eV Eg of ground-state InN has extended the direct Eg of group-III nitride alloys into the near-infrared range, which is experimentally crucial in optoelectronic applications [1]. In theory, the small Eg of InN is attributed to the lowest conduction-band minimum (CBM) among all group III–V semiconductors and the high valence-band maximum (VBM) due to the p–d repulsion [20, 21]. It is noteworthy that the band structure is sensitive to the interatomic distances and relative positions of atoms. In the previous works [12–14], the effects of both types of strain on the band structure have been systematically investigated for AlN and GaN using first-principles simulations and novel results have been reported. (I) AlN: the wurtzite and graphite-like Eg are always direct for uniaxial strain, whereas
InN, with its small direct bandgap (Eg ), is constantly under extensive investigations due to its abundant physical and electrical properties [1]. Strain engineering has been widely adopted to adjust the Eg to a desired value for particular optoelectronic applications by inducing intrinsic strain through lattice mismatching in epitaxial films or core/shell nanowires [2–4]. Such lattice mismatching leads either to biaxial strain in the (0001) plane or uniaxial strain along the [0001] direction. First of all, the hexagonal structure is easily modified by the strain: an intermediate graphite-like phase between wurtzite and rocksalt has been observed [5–16]. The new phase, with the P63 /mmc space group, shows fivefold coordination bonding (wurtzite shows fourfold and rocksalt sixfold coordination). This is consistent with widely accepted common sense [17, 18]: both types of strain show the equivalent effects on the structural and electronic properties. However, the graphite-like phase is only obtained in ZnS for uniaxial strain, although this high-symmetry phase is preferred for biaxial strain—before the phase transition, the 0953-8984/14/025501+09$33.00
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c 2014 IOP Publishing Ltd Printed in the UK
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of unoccupied bands, 192 such bands are included to ensure good convergence. To find the strain-free lattice constants, the lattice vectors and atomic coordinates are fully relaxed until the Hellmann–Feynman force acting on each atom is reduced to ˚ −1 . In the presence of uniaxial (biaxial) less than 0.02 eV A strain (), relaxation is performed with the lattice constant c (a and b) fixed until the following conditions are satisfied within a small tolerance: σ11 = σ22 < 0.02 GPa (σ33 < 0.02 GPa) and σij = 0 for i 6= j, where σ11 and σ22 are the stresses in the (0001) plane, and σ33 along the [0001] direction. The σ33 (σ11 ) is adjusted by changing strain 33 (11 ) step by step to reach all desired values in the uniaxial (biaxial) case.
direct and indirect band structures are observed in either the wurtzite or graphite-like phase for biaxial strain. GaN: the wurtzite Eg is always direct and the graphite-like Eg always indirect for both types of strain. (II) AlN: uniaxial strain reduces the wurtzite Eg and enhances the graphite-like Eg , whereas the wurtzite and graphite-like Eg are greatly reduced by biaxial strain. GaN: both types of strain reduce the wurtzite and graphite-like Eg . Although the strains theoretically adopted are relatively large, appreciable tuning in Eg may be accomplished for modest and realistic strains achievable in nanowires or ultrathin films. As a member of group-III nitrides, although the ground-state Eg of InN is experimentally available, the accurate dependence of its Eg on uniaxial and biaxial strains remains unclear so far. In this work, our main aim is to systematically compare the effects of uniaxial and biaxial strains on the structural and electronic properties of InN in order to reveal the phase transition as well as its driving mechanism and to illustrate the accurate dependence of band structure on strain. The conventional density functional theory (DFT) is adopted to obtain the optimized structure at each special strain, due to its credibility in predicting the structural properties [22]. To overcome the deficiency of conventional DFT: where the Eg of ground-state InN is negative, the Heyd–Scuseria–Ernzerhof (HSE) functionals and G0 W0 approximation are adopted simultaneously [23–25]. (I) The HSE functionals employ a screened short-range Hartree–Fock (HF) exchange, which overestimates the Eg . (II) The G0 W0 approximation, based on many-body perturbation theory, offers a parameter-free option for the accurate prediction of the electronic structure [18].
3. Results and discussion
In the presence of σii , a new lattice parameter corresponds to a specified uniaxial out-of-plane given by 33 = (c − c0 )/c0 and a commensurate equibiaxial in-plane by 11 = 22 = (a − a0 )/a0 , where a0 and c0 are the strain-free lattice constants. Figure 1 summarizes the calculated dependence of σ , which displays the markedly different trends for uniaxial and biaxial . The tensile σ33 first increases and then decreases drastically with 33 , with a maximum of ∼21 GPa at 33 = 0.16. This indicates that the calculated ideal tensile strength along the c axis is ∼21 GPa, which is much smaller than the critical values of ∼45 GPa for AlN and of ∼34 GPa for GaN [14]. This is consistent with the ground-state elastic constant c33 of InN being much smaller than that of GaN, and the c33 of GaN being smaller than that of AlN [30, 31], emphasizing that the larger the ground-state c33 , the larger the ideal uniaxial tensile strength [19]. The compressive σ33 displays a similar trend to those of AlN, GaN and ZnS [14, 19]: the σ33 magnitude first increases for −0.10 < 33 < 0 and then decreases for −0.16 < 33 < 0.10, finally increasing once again for 33 < −0.16. More unusually, unlike the uniaxial results, the tensile σ11 first increases, until dropping to a low value at 11 = 0.10, and then increases as 11 further increases. Similar behaviors have been observed in AlN and GaN [12], but not in ZnS [19], where an obvious decrease in σ11 at high 11 is observed. As the compressive 11 increases, the σ11 magnitude always increases, which is well known and anticipated, consistent with previous theoretical findings [12, 19]. To reveal the detailed structural evolution, we systematically study the effects of uniaxial and biaxial on the other terms , lattice ratio c/a and internal lattice parameter u, as shown in figure 2. When 33 > −0.16 and 11 < 0.09, the calculated other terms and u display a nonlinear response with respect to , especially for uniaxial ; a linear behavior is obtained for c/a. When 33 < −0.16 and 11 > 0.09, c/a ' 1.20 and u = 0.50. Therefore, the structural transition to the graphite-like phase is achieved simultaneously for both uniaxial and biaxial . This is similar to the previous results of AlN and GaN [12], but different from those of ZnS [19], where the graphite-like phase is only obtained for uniaxial
2. Computational methods
The total energy and band-structure calculations are performed using the projector-augmented-wave method [26] implemented in the VASP code [27]. The projectoraugmented-wave (PAW) method as implemented in the VASP code is utilized to describe the interaction between the ionic cores and the valence orbitals. The Perdew–Burke–Ernzerhof generalized gradient approximation (PBE-GGA) is employed in the conventional DFT calculations, which are labeled hereafter as PBE. The N 2s2 2p3 and In 4d10 5s2 5p orbitals are explicitly included as valence electrons. The electronic wavefunctions are described using a plane wave basis set with an energy cutoff 600 eV, while the obtained fundamental Eg is almost identical to that of 700 eV. A Monkhorst–Pack k-point mesh of 6 × 6 × 6 is used throughout the calculations to obtain well-converged results. The phonon-dispersion calculations are performed with the direct method implemented in the PHONOPY package [28, 29]. This method uses the Hellmann–Feynman forces calculated for an optimized supercell through VASP [27]. In theory, the larger the supercell, the more accurate the dispersion curves obtained. The 2 × 2 × 3 supercell is adopted for all phonon calculations and an accuracy of 0.05 THz for the highest optical zone-center phonon frequency is achieved. In addition, since the G0 W0 calculations are also sensitive to the number 2
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Figure 1. Uniaxial out-of-plane stress σ33 and biaxial in-plane stress σ11 as a function of . The inserted panel shows the detailed 11 dependence of σ11 for 0.10 < 11 < 0.20.
Figure 2. In-plane strain 11 and out-of-plane strain 33 , lattice ratio c/a and internal parameter u as a function of . The structural
transition to the graphite-like phase occurs at 33 ∼ = −0.16 and 11 ∼ = 0.09 (dashed lines). The open symbols refer to the wurtzite phase and the solid symbols refer to the graphite-like phase.
. It is most likely to observe the transformation to the higher-symmetry phase in core/shell nanowires along the c axis or epitaxial films on lattice-mismatched substrates. The most unexpected is that 11 , c/a and u change continuously with uniaxial , whereas discontinuous behaviors are observed for biaxial at the phase transition.
To reveal the driving mechanisms of the phase transition, figure 3 shows the elastic constants cij as a function of uniaxial and biaxial . As shown in the left panel, c33 < 0 and c13 < 0 when 33 > 0.16, which is consistent with the structural instability after the σ33 reaching the ideal strength. When −0.16 < 33 < 0, the value of c33 always decreases 3
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Figure 3. Elastic constants cij as a function of uniaxial and biaxial . The open symbols refer to the wurtzite phase and the solid symbols refer to the graphite-like phase.
decreases until becoming negative at 11 = 0.09, followed by jumping to a high value, and then decreases and eventually becomes negative once again when 11 > 0.20. This suggests that the wurtzite phase becomes elastically unstable at 11 = 0.09, as does the graphite-like phase when 11 > 0.20. The anomalous stability–instability–stability behaviors have been observed in AlN and GaN [12, 14], but not in ZnS for biaxial [19]: where the graphite-like phase is not achieved due to the structural instability. Therefore, the elastic instability plays an important role in the wurtzite-to-graphite-like phase transition in the IIIA nitrides for both types of . To further verify the structural stability, four phonondispersion curves at different are plotted in figure 5 for uniaxial and biaxial , respectively. There are twelve branches per unit cell with four atoms: one longitudinal-acoustic (LA), two transverse-acoustic (TA), three longitudinal-optical (LO), and six transverse-optical (TO) branches, as shown in panel (c). Since the N atom is slighter than the In atom in mass, the internal vibrations of In atoms are mainly responsible for the low-frequency modes. When = 0 (see panel (c)), 33 = 0.06 (see panel (d)) and 11 = −0.08 (see panel (e)), the phonon modes always remain stable, which is consistent with the elastic stability for broad ranges of tensile uniaxial and compressive biaxial . When 33 = −0.20 (see panel (a)) and 11 = 0.14 (see panel (h)), dynamical stability is obtained for the graphite-like phase, consistent with the aforementioned elastic results. This suggests that this metastable phase can be stabilized by large compressive uniaxial and tensile biaxial . The most unexpected is that the soft LA modes are observed near the 0 point at 33 = −0.16 (see panel (b)), but not at 11 = 0.09 (see panel (g)). This is drastically different from that of the wurtzite structure becoming elastically instable for −0.16 < 33 < −0.10 and at 11 = 0.09, respectively. Therefore, although elastic instability plays an important role
and remains negative for a broad range of 33 , which leads to the abnormal behavior of σ33 for −0.16 < 33 < −0.10, similar to the results of AlN, GaN and ZnS [14, 19]. The increase in c33 for 33 < −0.16 leads to an increase in the magnitude of σ33 after the phase transition. As shown in the right panel, as the compressive 11 magnitude increases, the values of c11 , c12 , c13 and c33 increase monotonically, while c44 remains almost unchanged. This is different from the previous results of ZnS [19]: where c44 always decreases and eventually reaches zero. As the tensile 11 increases, c11 , c12 and c44 always decrease and c11 > c12 . A different trend is observed in ZnS for tensile 11 [19]: c11 < 0, c12 >0 with c44 increasing slightly with 11 , suggesting the structural unstability. It is noteworthy that the elastic responses of AlN and GaN to biaxial remain unclear so far. The mechanical stability of the hexagonal structure is judged from the following criteria: c11 −|c12 | > 0, c44 > 0 and (c11 +c12 )c33 > 2c213 (equivalently expressed as 1C = (c11 + c12 )c33 −2c213 > 0). To further compare the effects of uniaxial and biaxial on the structural stability, figure 4 displays c11 − |c12 |, c44 and 1C as a function of in detail. c11 − |c12 | and c44 show similar trends for uniaxial and biaxial , respectively. It is noteworthy that discontinuous behaviors of c11 − |c12 | and c44 are observed at the phase transition for biaxial , but not for uniaxial . The most interesting is the dependence of 1C: (I) 1C shows a similar trend to c33 for uniaxial . When 33 > 0.16 and −0.16 < 33 < −0.10, 1C < 0, indicating the structural instability. Therefore, the wurtzite structure only remains elastically stable for −0.10 < 33 < 0.16. On the other hand, the graphite-like 1C increases as the compressive 33 magnitude further increases. This indicates that the graphite-like elastic stability is achieved at large compressive 33 . (II) 1C increases drastically for a broad range of compressive 11 . As the tensile 11 increases, 1C first 4
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Figure 4. Mechanical stability criteria c11 − |c12 | (GPa), c44 (GPa) and 1C = (c11 + c12 )c33 − 2c213 (GPa2 ) as a function of uniaxial and
biaxial . The negative values (red symbols) indicate the elastic instability. The open symbols refer to the wurtzite phase and the solid symbols refer to the graphite-like phase.
Figure 5. Phonon-dispersion curves for different uniaxial and biaxial . Uniaxial: (a) 33 = −0.20, (b) 33 = −0.16, (c) 33 = 0.0 and
(d) 33 = 0.06; biaxial: (e) 11 = −0.08, (f) 11 = 0.0, (g) 11 = 0.09 and (h) 11 = 0.14. The crystal structures are graphite-like in (a) and (h), and wurtzite in (b)–(g).
in the wurtzite-to-graphite-like phase transitions for both types of , the underlying mechanisms should be drastically different.
To further reveal the phase-transition driving forces, figure 6 shows the unit-cell enthalpy H and volume V as a function of uniaxial and biaxial . It is well known that 5
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Figure 6. Unit-cell enthalpy H and volume V as a function of uniaxial and biaxial . The open symbols refer to the wurtzite phase and the solid symbols to the graphite-like phase. Furthermore, the red symbols refer to the elastically unstable structures and the other symbols to the elastically stable structures.
when the H = E + PV of different phases are equal, the phase transition occurs, where the E and V are the unit-cell total energy and volume at a special pressure P, respectively. As shown in top panels, the elastically stable wurtzite and graphite-like phases show different H trends for uniaxial and biaxial . The elastically unstable structures for −0.16 < 33 < −0.10 and 11 = 0.09 act as the ‘bridges’ between the two stable phases, which emphasizes the important role of elastic instability in the phase transitions. As shown in the bottom panels, V, the derivative of H with respect to P, changes continuously with uniaxial , whereas a discontinuous behavior is observed at the phase transition for biaxial . This suggests that the wurtzite structure transforms to the graphite-like structure via a second-order phase transition for uniaxial and a first-order transition for biaxial . Furthermore, the two phase transitions depend on different driving mechanisms: the second-order transition is driven simultaneously by the elastic and dynamical instabilities, but the first-order transition is driven only by the elastic instability. Figure 7 shows the unit-cell total energy E as a function of uniaxial and biaxial . For different types of applied , the relaxations are performed along the different paths to find the lowest-energy structures, eventually leading to a difference in the phase transition from wurtzite to graphite-like. The effects of uniaxial and biaxial on the band structure is systematically investigated using the HSE and G0 W0 methods, respectively. The G0 W0 band structure is plotted by interpolation using the MKWFs approach [32], starting
Figure 7. Unit-cell energy E as a function of uniaxial and biaxial .
To achieve the minimum energy (E), the values of c, u and V are relaxed fully while a and b are kept constant for biaxial ; the values of a, b, u and V are relaxed fully while c is kept constant for uniaxial .
from the G0 W0 calculations. According to the tight-binding model, the valence (conduction) band edge derives mainly from the bonding (antibonding) state of N and In p (s) atomic orbitals [33, 34]. In the interpolation, the s and p orbitals of N and In atoms are chosen for the initial projections. Table 1 lists the ground-state Eg at various levels of theory, together with experimental data for comparison. Since the CBM in InN is the lowest along all group III–V semiconductors; the 6
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Figure 8. Calculated HSE28 and G0 W0 (HSE28) band structures at different uniaxial and biaxial . Uniaxial: (a) 33 = −0.20,
(b) 33 = −0.08, (c) 33 = 0.0, (d) 33 = 0.04 and (e) 33 = 0.10; biaxial: (f) 11 = −0.12, (g) 11 = −0.06, (h) 11 = 0.04, (i) 11 = 0.10 and (j) 11 = 0.14. The structures are graphite-like in panels (a), (i) and (j), and wurtzite in the other panels. The calculations are performed at the PBE optimized structure. The inclusion of the p–d repulsion is described in the text. The VBM is set to zero in each figure. Table 1. Calculated Eg at various levels of theory for the ground-state structure together with the experimental data for comparison.
HSE Eg (eV)
G0 W0
PBE
0.25
0.28
0.30
0.35
0.40
−0.354
0.570
0.698
0.776
0.990
1.211 −0.126
VBM is greatly pushed up by the p–d repulsion because the In 4d level is the highest [34], the PBE Eg is negative. The standard exchange mixing, containing 25% HF and 75% PBE-GGA, is employed in the HSE06 hybrid functionals [35]. The HSE06 Eg is smaller than the experimental data. To display the influence of exchange mixing coefficient α on the band structure, the ground-state Eg at different α are listed and a linear increase of Eg with respect to α is obtained. As shown in table 1, α = 0.28 is the best choice, which is denoted hereafter as HSE28. Since the G0 W0 Eg depends much on the starting wavefunctions and eigenvalues, the G0 W0 (HSE28) Eg is larger than the G0 W0 (HSE06) data and the G0 W0 (PBE) value is still negative. The electronic structure is modified by both types of . Five band structures at different are plotted in figure 8 for uniaxial and biaxial , respectively, where the HSE28 and G0 W0 (HSE28) results are listed together. The HSE28 and G0 W0 (HSE28) valence bands are in good agreement for each . For the conduction bands, although the G0 W0 (HSE28) values are slightly larger than the HSE28 values at certain points, good agreement is obtained near the Fermi level. This is due to the atomic orbitals, mainly contributing to the CBM and VBM, being chosen for the initial projections of the MKWFs approach. For both types of , the stable wurtzite Eg always remains direct, with the VBM and CBM at 0, as shown in panels (b)–(h); the graphite-like Eg is always indirect, with the VBM at H and the CBM at 0, as
PBE
HSE06
HSE28 Exp. [34]
0.457
0.553
0.70
shown in panels (a), (i) and (j). Therefore, for uniaxial and biaxial , the strain-induced trends in the band structure of InN are similar to those of GaN, but different from those of AlN [12, 14]. The d orbitals in the heavy atoms should be responsible for the behaviors. Furthermore, the semi-metallic phase is observed only for tensile uniaxial (see panel (e)), whereas it is also observed for both compressive and tensile biaxial (see panels (f) and (j)). To describe the band-structure evolution in detail, figure 9 shows the HSE28 conduction-band minima (the top panels) and the valence-band maxima (the bottom panels) at different points as a function of uniaxial and biaxial . For the valence bands, N s and In s orbital hybridization is observed at 0, similar to the previous results of AlN and GaN [14, 36]. Furthermore, the N s orbital shows a greater contribution to the CBM in InN, which is consistent with the increasing ionicity from AlN to GaN and to InN [34]. The wurtzite CBM is enhanced by a compressive uniaxial and reduced by a tensile uniaxial , whereas the graphite-like CBM is insensitive to a uniaxial . On the other hand, the wurtzite and graphite-like CBM are reduced by a biaxial , with a discontinuity at the phase transition. It is noteworthy that the CBM of unstable wurtzite structures plays the role of connecting the values of stable wurtzite and graphite-like phases for both types of , similar to the aforementioned results for H. 7
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Figure 9. HSE28 uniaxial and biaxial dependence of the conduction-band minima (the top panels) and the valence-band maxima (the
bottom panels) at different points. The open symbols refer to the wurtzite phase and the solid symbols to the graphite-like phase. Furthermore, the red symbols refer to the elastically instable structures and the other symbols to the elastically stable structures. The Fermi level is set to zero in each case.
Eg . (II) Tensile and compressive biaxial both reduce the wurtzite and graphite-like Eg , with discontinuities at the phase transition. The maximum HSE28 Eg values are 0.821 eV and 0.706 eV at 33 = −0.08 and 11 = −0.02, respectively. The wurtzite semi-metallic phase is obtained for tensile uniaxial and compressive biaxial , and the graphite-like semi-metallic phase is obtained for tensile biaxial .
The bottom panels of figure 9 show the valence-band maxima of 0 and H points as a function of uniaxial and biaxial . The N p and In p, d orbital hybridizations are identified at the VBM, but the N p orbital plays the dominant role, consistent with the results of AlN and GaN [14, 36]. It is well known that the ground-state VBM is mainly occupied by the N px , py and pz orbitals, which overlap in a ‘degenerate’ manner, as shown in figure 8(c). The wurtzite VBM at 0 originates from the N pz orbital for the compressive uniaxial and tensile biaxial (see figures 8(b) and (h)), and from the N px and py orbitals for the tensile uniaxial and compressive biaxial (see figures 8(d)–(g)). The H state is occupied by the N px , py and pz orbitals. As the compressive uniaxial and tensile biaxial increase, the N px and py orbitals play increasingly important roles. More specially, the N px and py levels are greatly pushed up by the compressive uniaxial and tensile biaxial , whereas the N pz level is insensitive to both types of . This leads to an indirect band structure with the VBM shifting to H in the graphite-like phase for uniaxial and biaxial . Figure 10 shows Eg as a function of uniaxial and biaxial . The HSE06, HSE28 and G0 W0 (HSE28) are listed together for comparison and similar trends are observed. It is noteworthy that the unstable wurtzite Eg act as the ‘bridge’ between the values of stable wurtzite and graphite-like phases for both types of , similar to the H and CBM results. The Eg values show different dependences on both types of . (I) Tensile uniaxial reduces Eg , whereas compressive uniaxial enhances the stable wurtzite Eg and reduces the graphite-like
4. Summary
In summary, we have systematically compared the effects of uniaxial and biaxial strains on the phase transition and band structure of InN using the HSE and G0 W0 methods. Although the wurtzite-to-graphite-like transition is observed for both types of strain, the phase-transition properties are drastically different: second order for uniaxial strain and first order for biaxial strain. The second-order transition is driven by elastic and dynamical instabilities, whereas the first-order transition is driven only by elastic instability. Although the bandgap is direct in the stable wurtzite phase and indirect in the graphite-like phase, both types of strain show inequivalent effects on the bandgap: compressive uniaxial strain enhances and tensile uniaxial strain reduces the gap, whereas biaxial strain always reduces it. Although the theoretical strains are relatively large, this work has shed light on the further understanding of such phase transitions and identified opportunities for engineering the bandgap of wurtzite semiconductors through the strains achievable in 8
J. Phys.: Condens. Matter 26 (2014) 025501
Y Duan et al
Figure 10. Calculated HSE and G0 W0 Eg as a function of uniaxial and biaxial . The electronic-structure calculations are performed at the
PBE optimized structure. The open symbols refer to the wurtzite phase and the solid symbols to the graphite-like phase. Furthermore, the red symbols refer to the elastically instable structures and the other symbols to the elastically stable structures.
core/shell nanowires or epitaxial films on lattice-mismatched substrates. This should also stimulate more investigations into strain engineering in the field of semiconductors, thereby enabling a number of important technological applications with better performance in the fields of electronics and photoelectrics.
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Acknowledgments
The work is sponsored by the Fundamental Research Funds for Central Universities under Grant No 2013QNA30 and the Qing Lan Project. We are grateful to the Advanced Analysis and Computation Center of CUMT for the award of CPU hours to accomplish this work. References [1] Wu J, Walukiewicz W, Yu K M, Ager J W III, Haller E E, Lu H, Schaff W J, Saito Y and Nanishi Y 2002 Appl. Phys. Lett. 80 3967 [2] Duan Y, Qin L, Tang G and Shi L 2008 Eur. Phys. J. B 66 201 [3] Osbourn G C 1982 J. Appl. Phys. 53 1586 [4] Duan Y, Li J, Li S-S and Xia J-B 2008 J. Appl. Phys. 103 023705 [5] Limpijumnong S and Lambrecht W 2001 Phys. Rev. Lett. 86 91 [6] Limpijumnong S and Lambrecht W 2001 Phys. Rev. B 63 104103 [7] Limpijumnong S and Jungthawan S 2004 Phys. Rev. B 70 054104 [8] Sarasamak K, Kulkarni A J, Zhou M and Limpijumnong S 2008 Phys. Rev. B 77 024104 [9] Alahmed Z and Fu H 2008 Phys. Rev. B 77 045213 [10] Maznichenko I V et al 2009 Phys. Rev. B 80 144101 [11] Morgan B J 2010 Phys. Rev. B 82 153408 [12] Duan Y, Qin L, Shi L, Tang G and Shi H 2012 Appl. Phys. Lett. 100 022104 [13] Dong L, Yadav S K, Ramprasad R and Alpay S P 2010 Appl. Phys. Lett. 96 202106 9
