Reduced Hamiltonian for Electronic States of Dilute Nitride Semiconductors
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Reduced Hamiltonian for Electronic States of Dilute Nitride Semiconductors
Masato Morifuji and Fumitaro Ishikawa Graduate School of Engineering, Osaka University, 2-1 Yamada-oka, Suita, Osaka 565-0871, Japan Graduate School of Science and Engineering, Ehime University, 3 Bunkyo-cho, Matsuyama, Ehime 790-8577, Japan (Dated: October 16, 2018)We present a novel model to describe conduction band of GaN x As − x (GaNAs). As well known, GaNAsshows exotic behavior such as large band gap bowing. Although there are various models to describe theconduction band of GaNAs, origin of the band gap bowing is still under debate. On the basis of perturbationtheory, we show that the behavior of conduction band is mainly arising from intervalley mixing between Γ andL or X. By using renormalization technique and group theoretical treatment, we derive a reduced Hamiltonianwhich describes well the band gap shrinkage of GaNAs. I. INTRODUCTION
III-V compound semiconductors containing nitrogen havebeen extensively studied for their properties di ff erent fromthose of conventional semiconductors. In particular, behav-ior of conduction band edge of GaN x As − x (GaNAs) withsmall x attracts wide attention. As well known, the bandgap of GaNAs decreases with nitrogen concentration. This iscontrary to the conventional Begard’s law, that is, band gap ofa mixed compound is well described as a linear interpolationof band gaps of constituent materials.There have been various models to explain such behaviorof GaNAs.
However, origin of the band gap bowing isstill under debate. Band anticrossing model is widelyused for phenomenological explanation of experimental re-sults, however, its physical foundation is ambiguous. Bandtheories, which is a powerful tool to investigate electronicstates, also have been applied to GaNAs. The tight-bindingmodel, empirical pseudopotentials, and the first prin-ciple calculations have been carried out to reproduce ex-perimental results. Reliable results were obtained from thesecalculations, however, physical insight can be missed in han-dling the large matrix containing all the e ff ects. In addition,when nitrogen concentration is very low, band calculations re-quire much computational resources. As a result, calculationsbecome di ffi cult to carry out.In this study, in order to investigate the conduction band ofGaN x As − x with small x , we present a novel model derivedfrom perturbation calculations using wavefunctions of bulkGaAs as bases. Investigation on behavior of conduction bandhas revealed that intervalley mixing induced by lattice distor-tion around nitrogen plays an important role for the band gapreduction of GaNAs. Utilizing this result, along with renor-malization technique and symmetry considerations, we derivea simple equation to evaluate energy of the conduction bandof GaNAs. II. THEORYA. Overview of band calculation procedure
First, we briefly review the procedure to calculate elec-tronic states of bulk GaAs within the empirical pseudopoten- -4-2024 E ne r g y ( e V ) L Γ X Γ Γ Γ L L X X X L FIG. 1. Dispersion curves of bulk GaAs calculated by using planewave bases and empirical pseudopotentials. The lowest conductionband, which we consider in this study, is denoted by crosses. Originof energy axis is set to the conduction band edge. tial method to be used as the basis in the following calcula-tions.Hamiltonian of bulk GaAs is given by a summation of ki-netic energy and atomic potential energy V ( r ) as H = − ~ ∇ m + V ( r ) , (1)with V ( r ) = X i [ V Ga ( r − τ − R i ) + V As ( r − R i )] , (2)where V Ga ( V As ) is atomic pseudopotential of Ga (As) locatedin a unit cell specified by a lattice vector of the zinc blendestructure R i . τ = ( a / , a / , a /
4) with a the lattice constantis a vector which specifies position of Ga within a unit cell.Based on the empirical point of view, we regard that Coulombinteraction between electrons, exchange and correlation inter-actions, etc. are e ff ectively included in the atomic pseudopo-tentials. We neglected the spin-orbit interaction. First, wecalculate a Hamiltonian matrix H k + G i , k + G j ≡ h k + G i |H | k + G j i , (3)using plane wave basis functions h r | k + G i i = √ Ω e i ( k + G i ) · r , (4)with k a wavevector, G i a reciprocal lattice vector, and Ω thesystem volume, respectively. By diagonalizing the Hamilto-nian matrix, we can calculate a band energy ε n , k = h ψ n , k |H | ψ n , k ′ i δ k , k ′ , (5)and a wavefunction ψ n , k ( r ) = X i c n , k + G i | k + G i i , (6)where c n , k + G i is an eigenvector with an index n specifyingband. The superscript “0”indicates non-perturbed quantities.In Figure 1, we show dispersion curves of bulk GaAs evalu-ated using empirical pseudopotential, where zero of the en-ergy axis is set to the conduction band edge.Using the bulk wavefunctions, we carry out perturbationcalculations to evaluate energies of GaNAs. In what fol-lows, we consider only the lowest conduction band plottedby crosses because we are interested in behavior of the con-duction band edge labeled by Γ in Fig. 1. From now on, wethus omit the index n which specifies band. B. Perturbation matrix
Let us consider an N × N × N supercell in which one of Asatoms therein is replaced by a nitrogen atom. Although it ispossible to apply the present theory for a system containingmany nitrogen atoms, in this paper, we treat only the case ofa single nitrogen atom. This supercell contains 4 N primitivecells of the zinc blende structure.Introduction of an N atom gives rise to change in crystallinepotential. We take three factors into account: (i) change of theatomic potential from As to N, (ii) displacement of Ga atomsneighboring to the N atom, and (iii) displacement of As atomsat the second neighboring positions to the N atom. Then, theperturbation Hamiltonian is written as H ′ ( r ) = [ V N ( r − R I ) − V As ( r − R I )] + X j h V Ga ( r + τ − R j − ξ j ) − V Ga ( r + τ − R j ) i + X j ′ h V As ( r − R j ′ − η j ′ ) − V As ( r − R j ′ ) i , (7)In the right hand side of eq. (7), the first term denotes poten-tial change from that of As to N located at the position R I .The second and the third terms are arising from displacementof atoms neighboring to the nitrogen where ξ j and η j are thedisplacement of the first neighboring Ga and the second neigh-boring As, respectively. The indices j and j ′ run through sothat R j − τ + ξ j and R j ′ + η j ′ indicate the positions of thefirst neighboring four Ga atoms and the second neighboringtwelve As atoms, respectively. We set ξ j so that the Ga atoms FIG. 2. (Color Online) Matrix element |H ′′ k ′ , k | with k ′ = (0 , , k on the k z = k z = π/ a plane (right column) of the first Brillouin zone. From thetop to bottom, total value of the matrix elements, contribution fromthe factor (i), and contribution from the factor (ii) are plotted in theunit of eV. approach to the N atom by 0.38 Å. Similarly, η j ′ was deter-mined so that the second neighboring As atoms approach tothe N by 0.1 Å. These values of atom displacements were de-termined from total energies evaluated by the first principlecalculations using CASTEP package.We calculated matrix elements of the perturbation Hamil-tonian between the Bloch states of bulk GaAs taking the threefactors (i), (ii), and (iii) mentioned above into account. InFig. 2, we plot absolute value of the matrix elements H ′ k ′ , k ≡ ΩΩ uc ! h ψ k ′ |H ′ | ψ k i , (8)for k ′ = Γ as a function of k . Ω uc = a / |H ′ Γ , k | s are plotted onthe k z = Γ and X-points. On theright column, |H ′ Γ , k | s on the k z = π/ a plane (the L-point isincluded) are shown. Note that di ff erent scales are used forfigures in the left and right columns and that the values are inunit of eV. From top to bottom, total value, contribution fromthe factor (i), and contribution from the factor (ii) are plotted,respectively. We do not show contribution from the factor (iii)the position shift of second neighboring As, since this is muchsmaller than others. We note that the matrix elements are ba- FIG. 3. (Color Online) |H ′ k ′ , k | / | ε k ′ − ε k | with k ′ = (0 , ,
0) is plottedon the k z = k z = π/ a plane (right colum)of the first Brillouin zone. From the top to bottom, total value, con-tribution from the factor (i), and contribution from the factor (ii) areplotted. sically negative values, although we plot absolute values sincethey are complex quantities. We see that |H ′ Γ , k | takes a largevalue when k is X and L. We also see that the e ff ect of dis-placement of neighboring Ga atoms is larger than that of theN potential.Fig. 3 shows the quantity, |H ′ Γ , k | / | ε Γ − ε k | plotted as afunction of k . Similar to Fig. 2, total value, contribution fromnitrogen potential, and displacement of Ga atoms are plottedfrom top to bottom. It is seen that |H ′ Γ , k | / | ε Γ − ε k | is largestwhen k is the L-state, although the matrix element for the X-state is larger than that for the L-state. This is because en-ergy di ff erence between the L and Γ , | ε Γ − ε L | , is smaller than | ε Γ − ε X | . We also see that the N potential gives rise to mix-ing between the Γ -state and states in vicinity of Γ , whereasdisplacement of Ga atoms gives rise to the intervalley mixing.These results indicates that mixing between the Γ and L-statesand / or between the Γ and X-states is relevant to the band gapreduction. -0.5 -0.25 0 0.25 0.5-40-20020 re[ψ Γ (r)]re[ψ X (r)] W a v e f un c t i on ( a . u . ) P o t en t i a l ( e V ) Position along <111> (in unit of 3 a) As,N GaΔV(r)0
FIG. 4. Upper panel: wavefunctions of the Γ -state and X-state areplotted along the h i direction by solid and dashed curves, respec-tively. Lower panel: Perturbation potential along the h i directionis plotted by solid curve. The arrows show positions of As (N) andGa. The horizontal dotted line shows zero. C. Character of wavefunctions and matrix elements
We can discern the k -dependence of the perturbation ma-trix elements from wavefunctions of bulk GaAs. In the upperpanel of Fig. 4, the solid and dashed curves show ψ Γ ( r ) and ψ X ( r ) plotted along the h i direction. The As (or N) atomlocates at the position 0 .
0, and a Ga atom without displace-ment locates at + .
25 as indicated by arrows. It is seen thatthe Γ -state wavefunction consists of anti-bonding coupling be-tween an s -like orbital of As and an s -like orbital of Ga. Wealso observe that the wavefunction around Ga is largely ex-tended. The X-state consists of anti-bonding coupling be-tween s -like orbital of As and p -like orbital of Ga which hasa node at the Ga position. The L-state has character similar tothe X-state though it is not shown in the figure.In the lower panel, we plot perturbation potential along the h i direction. We observe that the N atom gives rise to neg-ative potential with s -like symmetry. On the other hand, per-turbation potential around the Ga position is anti-symmetricaround the Ga, that is, p -like symmetry.These curves of crystalline potential and wavefunctions en-able us to make qualitative interpretation on the matrix ele-ments shown in the previous section. First, we consider thediagonal element H ′ Γ , Γ This quantity is arising mainly fromN potential because ψ Γ has a large amplitude at the N position.On the other hand, shift of Ga contribute little to H ′ Γ , Γ . This isbecause ψ Γ has s -like character around Ga. As we have noted,potential change due to Ga displacement is of p -character. In-tegration | ψ Γ | × ∆ V around the Ga atom will make the matrixelement small. The coupling between Γ and L is also deter-mined in the similar mechanism.For the coupling between the Γ - and X-states H ′ Γ , X , shiftof Ga atoms has an important role. From Fig. 4, we see that ψ Γ has s -like symmetry around the Ga atom, whereas both ψ X and H ′ have p -like symmetry. Therefore, we anticipate thatmultiplication of these three quantities becomes even functionaround Ga, which enlarges the matrix element H ′ Γ , X .We noted that contribution from shift of the second neigh-boring As atoms is small. This is also understood from sym-metry. Although displacement of As atoms is about 1 / p -like symmetry. As seen from Fig. 4, both the Γ -state and X-state have s -like charge distribution around Asatoms. From a simple consideration on symmetry, we see that h ψ k |H ′ | ψ k ′ i with k and k ′ Γ or X has a small value.These results indicates that mixing between Γ and X or be-tween Γ and L induced by lattice distortion around N givesrise to band gap reduction of GaNAs. k x k x k y k y (a) N=4 (b) N=8 FIG. 5. Reciprocal lattice vectors used in perturbation calculationsfor (a) 4 × × × × D. Band gap shrinkage
The matrix elements of the perturbation Hamiltonian for asingle N atom in an N × N × N supercell are written as H ′ ( N ) k , k ′ = N h ψ k |H ′ | ψ k ′ i , (9)where 1 / N is a factor to be normalized over the supercell.The states k and k ′ at which H ′ ( N ) k , k ′ is evaluated are obtainedas follows: Since the perturbation potential has translationalsymmetry with a period Na in all the x -, y -, and z -directions, h ψ k |H ′ | ψ k ′ i must be unchanged when H ′ ( r ) is replaced by H ′ ( r + R ) with a lattice vector of the supercell R . From this,the wavevectors k and k ′ in eq. (9) must satisfy a relation k − k ′ = π Na ( n x , n y , n z ) , (10)with n x , n y , and n z integers. In Figs. 5 (a) and (b), the dotsindicate possible k − k ′ plotted on the first Brillouin zone of thezinc blende structure for N = N =
8, respectively. Note E ne r g y ( e V ) N concentration ( % ) DiagonalizedRef. [6] supercelleq. (11)
FIG. 6. (Color Online) Energies of conduction band edge are shown.Dashed curves show results of second order perturbation, evaluatedby eq. (11). Filled circles show results calculated by diagonalizingfull Hamiltonian matrix given by eq. (12). For comparison, theoret-ical data from Ref. [6] and results of supercell calculation are alsoplotted by cross and square, respectively. that some points on the border are equivalent. For example,2 π/ a (1 , ,
0) and 2 π/ a ( − , ,
0) are identical and thus one ofthem must be excluded, though both are plotted in the figure.Excluding such equivalent points, we have 4 N k -points inthe first Brillouin zone to be mixed due to the perturbationpotential; there are 256 points for N = N = k -points are the points that are folded onto the Γ -point in theBrillouin zone of the N × N × N supercell. We can calculateconduction band energy from H ′ ( N ) k , k ′ given by eq. (9) with bulkGaAs states shown in Fig. 5.We may evaluate energy of the conduction band edge ε Γ using the matrix elements by perturbation expansion. Up tothe second order term, the energy change is given by ε Γ = N h ψ Γ |H ′ | ψ Γ i + N ) ′ X k |h ψ Γ |H ′ | ψ k i| ε Γ − ε k , (11)where x = / N is nitrogen concentration. As we show inFig. 6 by dashed curve, the second order perturbation is insuf-ficient to explain experimental results, indicating that higherorder perturbation energies are necessary since the potentialchange due to nitrogen is far from moderate.Then, in order to evaluate energy of conduction band edge,we diagonalized a matrix H ( N ) k , k ′ ≡ h ψ k |H | ψ k ′ i + H ′ ( N ) k , k ′ . (12)Results are shown in Fig. 6 by filled circles. For comparison,we plot theoretical data from Ref. [6] by crosses. We alsoplot a result of supercell calculation with cuto ff energy 3.0Ryd. We see that the present perturbation calculations yieldreasonable results. III. REDUCED HAMILTONIAN
As we have shown in the previous section, proper ener-gies were evaluated by diagonalizing a Hamiltonian matrix of4 N × N size. Mixing between the Γ -state and other k -statesdue to nitrogen doping gives rise to the reduction of the con-duction band edge. Among the states, the mixing between Γ and L is the largest. This fact leads us to an idea that we mayhave an e ff ective Hamiltonian with only the Γ and L as bases.For this purpose, we applied L¨odin’s theory described in de-tail in appendix A. In the present case where bases are the Γ and four L-states, we can reduce the size of the perturbationHamiltonian down to 5 × Γ -state.Coe ffi cients for such a linear combination among the L-statesare obtained considering symmetry of the L-state. Followingstandard procedure to obtain normal modes of an irreduciblerepresentation of the point group T d , we symmetrize the linearcombinations among the four L-states, L ∼ L . In thisway, we have a singlet state ψ ˜ L = ψ L + ψ L + ψ L + ψ L , (13)and triplet states ψ L ′ = √ ψ L − √ ψ L ,ψ L ′′ = − √ ψ L + √ ψ L , ψ L ′′′ = ψ L − ψ L − ψ L + ψ L . However, we have to note that these coe ffi cients depend on sit-uations such as nitrogen position, choice of origin, and trivialphase of wavefunctions etc. It is necessary to consider situa-tions carefully in evaluating the coe ffi cients.Since the singlet state given by eq. 13 connects with the Γ -state and the triplet states do not, we can transform the 5 × U † ˜ H ( N ) k , k ′ U = ˜ ε Γ ˜ V Γ L V ∗ Γ L ˜ ε L − | ˜ V LL | ε L + | ˜ V LL | ε L + | ˜ V LL |
00 0 0 0 ˜ ε L + | ˜ V LL | , (14)with a matrix U in the form U = U L , (15)where 4 × U L is consisting of the coe ffi cients ofthe linear combinations mentioned above. From this reducedmatrix, we have energies of the Γ - and L-states as ε Γ = (cid:16) ˜ ε Γ + ˜ ε L − | ˜ V LL | (cid:17) − q(cid:16) ˜ ε Γ − ˜ ε L + | ˜ V LL | (cid:17) + | ˜ V Γ L | , (16a) ε − L = (cid:16) ˜ ε Γ + ˜ ε L − | ˜ V LL | (cid:17) + q(cid:16) ˜ ε Γ − ˜ ε L + | ˜ V LL | (cid:17) + | ˜ V Γ L | , (16b) ε + L = ˜ ε L + | ˜ V LL | . (16c)As seen in eqs. (A8) and (A9), the elements of the reducedmatrix ˜ ε Γ etc. depend on E . We evaluated the elements asfollows: In evaluating band egde energy ε Γ , we set E = ε Γ and in evaluating L-point energy ε ± L , we set E = ε L .In Fig. 7, we show energies calculated by the reducedHamiltonian. ε Γ and ε ± L are plotted by squares and triangles,respectively. For comparison, we also plot the energies eval-uated by diagonalization which were already shown in Fig. 6.For the Γ -state, the two methods give rise to almost the sameresults. This good agreement is indicate validity of renormal-ization procedure.As for the L-state energy, behavior of ε − L is similar to thatof E + ∆ transition . ε + L seems corresponding to the E + transition. See, for example, Fig. 4 of Ref. [6]. How-ever, further verification is necessary to apply the present the-ory to high energy states of GaNAs. In the present paper, wederived Γ L-reduced Hamiltonian, paying attention mainly toband gap reduction. However, many levels are observed inhigh energies in GaNAs, and thus only Γ and L might be in-su ffi cient for description of high energy states. It is possible toconstruct Γ X- or Γ XL-reduced Hamiltonian in the same way.To investigate higher states, inclusion of the X-state would benecessary.Band calculations using supercell can treat high energystates. For example, in Ref. [21] where the first principlecalculations were carried out, the E + transition is assignedto transition to the L-state. In such calculations, however,we have a di ffi culty in picking up the state under attention,because of a number of states accumulated in energy due tomultiply folded bands. In addition, as we have noted, someL(X)-states mix with the Γ -state and some do not, resultingin di ff erent dependence on N concentration. It is not straightforward to investigate high energy states by the supercell cal-culations. On the other hand, in the present theory, we caneasily obtain high energy states. -0.10.10.3 E ne r g y ( e V ) N concentration ( % ) Diagonalized ε L - ε Γ ε L + FIG. 7. (Color Online) Energies calculated by the reduced Γ -LHamiltonian. ε Γ , ε − L and ε + L evaluated by eqs. (16a) – (16c) are plottedby square, filled triangle, and open triangle, respectively. Filled cir-cles show energies evaluated by diagonalization of full Hamiltonian,which was already shown in Fig. 6. In Fig. 8, we show elements of the reduced Hamiltonian. ˜ ε Γ and ˜ ε L − | ˜ V Γ L | are plotted in the upper panel and ˜ V Γ L is plottedin the lower panel as functions of nitrogen concentration. Thesquares show the values for E = ε Γ used to evaluate Γ -stateenergy, whereas triangles show the value with for E = ε L forL-state calculation. Present scheme where Γ - and L-states areretained as bases of the reduced Hamiltonian is inapplicablewhen N the supercell dimension is an odd number, becausethe L-point is not included in the set of k -points necessaryfor perturbation calculation (see Fig. 5). Thus, calculationswere carried out only for N = , IV. CONCLUSIONS
We presented a model to describe the conduction band ofdilute nitride compound GaNAs. Using wavefunctions of con-duction band of bulk GaAs as bases, we carried out pertur-bation calculations. Calculated perturbation matrix elementsshow that Γ -L mixing and / or Γ -X mixing are impoprtant forthe band gap reduction. Though conventional second orderformula yields a poor result, diagonalization of the full Hamil-tonian matrix reveals that the present method brings about rea-sonable results. By remaining Γ - and L-states, we renormal-ized other states to derive e ff ective 2 × V Γ (cid:2) ( e V ) ε Γ ε (cid:0) -3|V (cid:1)(cid:3) | E ne r g y ( e V ) realimag N concentration ( % ) | | | | FIG. 8. (Color Online) Elements of reduced Hamiltonian are plottedas functions of nitrogen concentration. In the upper panel, ε Γ and ε L are plotted. In the lower panel, real and imaginary parts of V Γ L areplotted. Squares (triangles) are values to evaluate ε Γ ( ε Γ ). Note thatwhen nitrogen concentration is zero, these quantities take the valuesof bulk GaAs. Appendix A: Renomalization procedure to reduce Hamiltonian
We show the procedure to reduce size of the Hamiltonianby renormalizing states whose interaction with the Γ -state isweak. First, we divide basis functions into two groups: (A)states to be bases of the reduced Hamiltonian, and (B) theothers. The states of the group (A) are those that interact with Γ strongly. By rearranging order of the bases, we rewrite thematrix H ( N ) k , k ′ in the form H ( N ) k , k ′ = H A H AB H BA H B , (A1)where H A is a matrix consisting of the states belonging to thegroup (A), and so on. We omitted the indices k , k ′ and ( N ) inthe right hand side for simplicity.The secular equation is then written as H A − E H AB H BA H B − E c A c B = , (A2)where is a unit matrix, 0 a zero vector, and c A and c B arecolumn vectors with corresponding size. Owing to the choiceof states for the groups (A) and (B), we expect that elementsof the matrices H B , H AB and H BA are small, so that we cantreat these quantities within lower order terms of expansionseries.Multiplying block by block, eq. (A2) is written as( H A − E ) c A + H AB c B = , (A3)and H BA c A + ( H DB − E ) c B + H OB c B = , (A4)where H DB and H OB denote diagonal and o ff -diagonal parts of H B , respectively. We rewrite eq. (A4) in the form c B = ( E − H DB ) − H BA c A + ( E − H DB ) − H OB c B . (A5)This expression enables us to calculate c B recursively. Thelowest order expression for c B is readily obtained by setting c B = c B = c B in the right hand by the right hand side itself, we have thesecond expression to c B as c B = ( E − H DB ) − H BA c A + ( E − H DB ) − H OB ( E − H DB ) − H BA c A . (A6) In this way, we can express c B in a series until suitable accu-racy is obtained. Once we have c B , by inserting the expressionof c B into eq. (A3), we have a secular equation h H A + H AB ( E − H DB ) − H BA + H AB ( E − H DB ) − H OB ( E − H DB ) − H BA · · · − E i c A = . (A7)Dimension of the matrices in this equation is that of the group(A). We thus have the reduced Hamiltonian˜ H ( N ) k , k ′ = H A + H AB h g ( E ) + g ( E ) H OB g ( E ) + g ( E ) H OB g ( E ) H OB g ( E ) · · · i H BA , (A8)with g ( E ) = ( E − H DB ) − , (A9)in which e ff ects from states of group (B) are e ff ectively con-tained. M. Kondow, K. Uomi and T. Nozue, Jpn. J. Appl. Phys. , L1056(1994). M. Weyers, M. Sato and H. Ando, Jpn. J. Appl. Phys. , L853(1992). W. G. Bi and C. W. Tu, Appl. Phys. Lett. , 1608 (1997). C. Skierbiszewski, S .P. Lepkowski, P. Perlin, T. Suski,W. Jantsch, and J. Geisz, Physica E W. Shan, W. Walukiewicz, K. M. Yu, and J. W. Ager III,E. E. Haller, J. F. Geisz, D. J. Friedman, J. M. Olson, and SarahR. Kurtz, and C. Nauka, Phys. Rev. B , 4211 (2000). P. H. Tan, X. D. Luo, Z. Y. Xu, Y. Zhang, A. Mascarenhas,H. P. Xin, C. W. Tu, and W. K. Ge, Phys. Rev. B , 205205(2006). K. Uesugi, N. Morooka, and I. Suemune, Appl. Phys. Lett. ,1254 (1999). S. Noguchi, S. Yagi, D. Sato, Y. Hijikata, K. Onabe, S. Kuboya,and H. Yaguchi, IEEE J. Photovoltaics, , 1287 (2013). K. Sumiya, M. Morifuji, Y. Oshima, and F. Ishikawa, AppliedPhysics Express T. Fukushima, Y. Hijikata, H. Yaguchi, S. Yoshida, M. Okano,M. Yoshita, H. Akiyama, S. Kuboya, R. Katayama, K. Onabe,Physica E , 2529 (2010). Chuan-Zhen Zhao, Na-Na Li, Tong Wei, Chun-Xiao Tang, andKe-Qing Lu, Appl. Phys. Lett. , 142112 (2012). W. Shan, W. Walukiewicz, J. W. Ager III, E. E. Haller, J. F. Geisz,D. J. Friedman, J. M. Olson, and S. R. Kurtz, Phys. Rev. Lett. ,1221 (1999). J. Wu, W. Walukiewicz, K. M. Yu, J. W. Ager III, E. E. Haller, Y. G. Hong, H. P. Xin, and C. W. Tu, Phys. Rev. B , 241303(2002). A. Lindsay and E. P. O’Reilly, Phys. Rev. Lett. E. P. O’Reilly, A. Lindsay, S. Tomi´c and M. Kamal-Saadi, Semi-cond. Sci. Technol.
870 (2002). W. J. Fan, M. F. Li, and T. C. Chong, J. B. Xia, J. Appl. Phys.
188 (1996). L. Bellaiche, S.-H. Wei, and A. Zunger, Appl. Phys. Lett. (1997). L. Bellaiche, S.-H. Wei, and A. Zunger Phys. Rev. B , 17568(1996). Kurt A. Mader and A. Zunger, Phys. Rev. B , 17393 (1994). P. R. C. Kent and A. Zunger Phys. Rev. B, , 115208 (2001) V. Timoshevskii, M. Cˆot´e, G. Gilbert, and R. Leonelli, S. Tur-cotte, J.-N. Beaudry, P. Desjardins, S. Larouche, L. Martinu, andR. A. Masut, Phys. Rev. B , 165120 (2006). C. -K. Tan, J. Zhang, X. -H. Li, G. Liu, B. O. Tayo, and N. Tansu,J. Display Technol., , 272L (2013). Pre-Olov L¨odin, J. Chem. Phys. , 1396 (1951). G. Burns,
Introduction to group theory with applications , Aca-demic Press, INC, New York 1977. H. Jones,
The theory of Brillouin zones and electronc statesin crystals , Second, revised edition, North-Holland, Amsterdam1977. S. Francoeur, M. J. Seong, M. C. Hanna, J. F. Geisz, A. Mascaren-has, H. P. Xin, and C. W. Tu, Phys. Rev. B68