Unusual electronic properties of InN

Physics Letters A 372 (2008) 5377–5380

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Physics Letters A www.elsevier.com/locate/pla

Unusual electronic properties of InN Vinit Sharma, B.L. Ahuja ∗ Department of Physics, University College of Science, M.L. Sukhadia University, Udaipur, 313001 Rajasthan, India

a r t i c l e

i n f o

Article history: Received 8 May 2008 Received in revised form 15 June 2008 Accepted 17 June 2008 Available online 20 June 2008 Communicated by R. Wu

a b s t r a c t Electronic structure of zinc-blende and wurtzite InN using linear combination of atomic orbitals and the latest approach of generalised gradient approximation within full potential linearised augmented plane wave schemes is reported. An unusual small band gap and real space analysis of our first ever experimental Compton profile are discussed. © 2008 Elsevier B.V. All rights reserved.

PACS: 13.60.Fz 71.15.Ap 72.80.Ey 78.70.Ck Keywords: X-ray scattering Band structure calculation Density functional theory

The III–V nitride semiconductors have been of considerable interest in recent years. According to the band-gap-common-cation rule, semiconductors have direct band gap at Γ and band gap of semiconductor increases as the cation atomic number decreases. In case of InN, there is a controversy among researchers, where this common-cation rule may be violated. Its structural stability is based on the comparison between the zinc-blende (ZB) and wurtzite (WZ) phases. The low band gap of InN also provides a challenge to understand its chemical properties. Several fascinating properties such as high thermal conductivity, large bulk moduli and extreme hardness make InN very promising for optoelectronic device applications. Among earlier band calculations, several authors have used different kind of approximations to describe the ground state properties. For example, Foley and Tansley [1] have reported band structure calculations using pseudopotential (PP) method. They have compared their calculations with reflectance data. To calculate the energy bands of WZ-InN, Tsai et al. [2] have used exchange and correlation of Hedin–Lundquist within PP local density approximation (LDA) scheme. Employing plane wave pseudopotential (PP-PW) method, Wright and Nelson [3] have calculated structural properties of group-III nitrides. Vogel et al. [4] have used the self-interaction- and relaxation corrected PP method together with Gaussian basis sets to calculate the structural and

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electronic properties of group-III nitrides. Adopting the empirical pseudopotential method, Yeo et al. [5] have discussed band structure of WZ-InN. Stampfl and Van de Walle [6] have used PP-PW LDA and generalised gradient approximation (GGA) based density functional calculations to obtain electronic properties of group-III compounds. PP-LDA based calculations have also been performed by Wang and Ye [7]. Recent PP-calculations with a projector augmented wave potential [8] show a negative value of band gap using LDA and a very low positive band gap using LDA + U. A majority of the DFT calculations within GGA and PP-PW-LDA calculations for both the phases of InN have shown the negative value of band gap [3,4,6,8]. Using the orthogonalised linear combination of atomic orbitals within LDA approach, Xu and Ching [9] have reported the electronic structures of ten WZ compounds including InN. Christensen and Gorczyca [10] have used linear muffin-tinorbital (LMTO) method to report the optical and structural properties of III–V nitrides under pressure. The relativistic full-potential Korringa–Kohn–Rostoker method for WZ phase has also shown the negative value of band gap of WZ-InN [11]. Gavrilenko and Wu [12] have studied the linear and nonlinear optical properties of both phases of group-III nitrides using full potential linearised augmented plane wave (FP-LAPW) method within LDA and GGA approaches. To evaluate the band structure of semiconductors (including InN) and insulators, Kotani and Schilfgaarde [13] have employed GW (where G means Green function and W stands for screened coulomb interaction) approximation with mixed basis expansion based on the full-potential LMTO method. Using LDA based semiempirical method, Wei et al. [14] have studied the band

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structure of ZB and WZ phases of group-III nitrides. Some early experimental studies on absorption spectra [15–17] have suggested the band gap of WZ-InN equal to about 2 eV. Recent photoluminescence studies [18–20] on WZ-InN samples have predicted a positive band gap. Although a number of theoretical and experimental efforts have been made to describe the structural and the electronic properties of ZB and WZ InN, but still a consistent picture related to the energy of bands, the position of d bands, the position and width of the N(2p ) bands and the hybridisation of N(2s) with semi-core d states of In is lacking. To rectify the confusing array of electronic structure data, in this Letter, we present more accurate electronic structure of ZB and WZ InN using the full potential linearised augmented plane wave (FP-LAPW) method as embodied in Wien2k code [21]. The present relativistic FP-LAPW calculations have been undertaken employing the recent non-empirical GGA approach of Wu and Cohen [22], which is more accurate for solids than other GGA prescriptions [23,24] used by different workers. It is worth mentioning that present GGA functional is based on a diffused radial cutoff for the real space exchange hole. Moreover, there are no adjustable parameters and the constraining conditions of Perdew– Burke–Ernzerhof (PBE) GGA [24] remain unchanged. In the present computations, the applied basis set consists of 677 plane waves in the interstitial region for both the ZB and WZ structures. We have expanded the basis functions up to R MT K Max = 10, where R MT is the plane wave radius and K Max is the maximum modulus of the reciprocal lattice vector. The value of angular momentum up to lmax = 10 in the muffin-tin (MT) spheres was used for the wave functions. Accurate Brillouin zone (BZ) integrations for k points were performed using the modified-tetrahedron integration scheme [21]. For ZB and WZ phases, the respective meshes were 8 × 8 × 8 and 8 × 8 × 4. The integration points over the irreducible BZ were 43 and 40 k points for ZB and WZ phases, respectively. The MT radii adopted for In and N were 2.1 and 1.6 a.u., respectively. At present the Wien2k code does not include computation of the momentum densities. In present calculations, the lattice parameters for WZ-InN were a = 3.533 and c = 5.693 Å [3], while in case of the ZB structure the value of a was kept equal to 4.920 Å [3]. To compute the electronic structure of ZB- and WZ-InN, we have also used the self-consistent linear combination of atomic orbitals (LCAO) method [25]. In LCAO computations, the Bloch orbitals of the crystal (CO’s) are expanded using atom-centered Gaussian orbitals of s, p or d symmetry. In the present study the all electron Gaussian basis sets [25] for In and N were used. To improve the convergence of N (2p ) and In (4d) potentials, basis sets were optimised using BILLY software [25]. Following the standard truncation criteria for the CRYSTAL03 code, the SCF calculations have been performed at 133 k points in the irreducible Brillouin zone. In the DFT-LDA calculations, we have chosen the Dirac-Slater exchange [25] and the Perdew–Zunger correlation potential [26]. For the GGA, the exchange and correlation potential of Perdew– Wang [23] have been used. The small value of band gap is also explained with the help of real space analysis of our first ever experimental Compton profile. It is worthwhile to mention that the Compton profile method is a well-recognised tool to probe the accuracy of band structure models. Within the impulse approximation, the Compton profile J ( p z ) is a projection of electron momentum density ρ ( p ) and can be deduced from the spectral distribution of Compton scattered photons. In such experiments, one measures the double differential cross-section that is related to the Compton profile as follows [27]: d2 σ dΩ dω2

 ∝ J (pz) =

ρ (p ) dp x dp y ,

(1)

Table 1 The calculated band gaps ( E g ) of InN along with available data Method

 E g (eV) InN ZB phase

WZ phase

0.62 0.57 −0.12

0.52 0.51 0.10

(i) Present work LCAO FP-LAPW

DFT-LDA DFT-GGA GGA (Wu-Cohen)

(ii) Available Theory PP

OLCAO (LDA) SR-LMTO RFKKR FP-LAPW (with QP correction) LMTO

Semi-empirical (LDA)

OPW LDA PW-LDA PP SIRC-PP EPM PW-LDA PW-GGA PW-LDA LDA LDA + U

LDA GW GW ( Z = 1)

−0.35 [3]

–

−0.40 [6] −0.55 [6] 0.75 [7]

0.08 [10] −0.42 [11] 1.90 [12] −0.26 [13] 0.01 [13] 0.05 [13] −0.48 [14]

2.10 [1] 1.30 [2] −0.04 [3] −0.40 [4] 1.60 [4] 2.04 [5] −0.27 [6] −0.37 [6] +0.90 [7] −0.18 [8] 0.03 [8] 1.02 [9] 0.43 [10] 1.90 [12]

−0.34 [14]

(iii) Available Expt. 1.90 [17]

2.11 [15] 1.89 [16] 1.90 [17] 0.90 [19] 0.70 [18,20]

EPM: empirical pseudopotential method, QP: quasi-particle, RFKKR: relativistic full potential KKR.

where z-axis is along the scattering vector and ω2 is the energy of scattered photon. The isotropic experimental Compton profile of WZ-InN was measured using our 740 GBq (20 Ci) 137 Cs Compton spectrometer [28]. The 137 Cs radioisotope emits photons of energy 661.65 keV, which are scattered by electrons in the sample through an angle 160 ± 0.6◦ . The photons scattered from polycrystalline sample of InN (thickness 0.35 cm and dia 2.2 cm) at ambient temperature were analysed with a high purity Ge detector and associated electronics. The raw Compton data were accumulated by a 4096channel analyser (Canberra, Accuspec B) with a channel width of about 61 eV. The total momentum resolution of the present measurement was 0.38 a.u. (Gaussian, full width at half maximum), where 1 a.u. of momentum = 1.9929 × 10−24 kg m s−1 . The raw Compton data were corrected for the background (measured without sample), the instrumental resolution, and the energy dependent corrections like the detector efficiency, the photon absorption, the Compton cross-section and the multiple scattering. For implementation of these corrections, the computer code of Warwick group was used [29]. Since the bremsstrahlung background was very small in the present measurements [30], we have not incorporated it in the present data reduction. The deconvolution (instrumental resolution correction) was limited to stripping off the low energy tail from the data; therefore the theoretical profiles have to be convoluted with the experimental resolution. The band structure of ZB and WZ phases of InN along with the total and partial density of states (DOS) obtained by using the GGA scheme of FP-LAPW method are shown in Figs. 1–2, respectively. In case of ZB-InN (Fig. 1), the lower valence bands are due to hybridisation of In (4d) and N (2s) states. The In (4d) bands re-

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Fig. 1. Selected energy bands (E–k relation) along high symmetry directions of the first Brillouin zone of ZB-InN using GGA scheme within FP-LAPW method. On the right-hand side, the total and partial density of states (DOS) is shown. In the FP-LAPW code, as a reference level, the Fermi energy E F is shifted to a standard value 0 eV.

Fig. 2. Same as Fig. 1, except the sample that is WZ-InN.

sult near −13 eV, which are fully resonant near Γ point of N (2s) valence band. The DOS for ZB structure has single anion valence s orbitals and is thus expected to exhibit a single anion s peak in the DOS. However, the DOS reveals two anion s peaks about −11.5 and −15 eV that surround In (4d) bands in lower valence region. It is seen that the s–d hybridisation splits the N (2s) bands in two sections one above and one below the In (4d) bands and accordingly falsifies its dispersion. The upper valence band (VB) is dominated by N (2p ) states, but of course In (spd) contribution is also present. The conduction band (CB) is mainly contributed by N ( p ) states. In addition, the present calculation also depicts that the CB minimum at the Γ point is much lower than the conduction band edge at other points in k-space. From the energy bands of ZB-InN, it is seen that the VB width from the FP-LAPW calculation is 6.34 eV while from the LCAO based energy bands (not shown here) it comes to be 7.10 and 6.99 eV for DFT-LDA and DFT-GGA, respectively. The band gaps computed from the present FP-LAPW and LCAO-DFT calculations along with available data are compared in Table 1. Here, the band gap involves the transitions between the top of the VB (mainly from anion p valence states) and the bottom of the CB (derived mostly from the cation d states). The band structure of WZ-InN using FP-LAPW (Fig. 2) depicts that the upper region of VB consists of p states of N and also significant contribution of spd states of In. The lower VBs near −13 eV arise from In (d) and N (s) states. Fig. 2 also shows the

contribution of s states of In in CB minimum whereas the upper CB is almost equally predominated by s states of In and p states of N. In both the phases some prominent features like dominant character of p at the N sites are seen. Our FP-LAPW based band structure calculations reveal an unusual low conduction band minimum at the Γ point which results band gap of 0.10 eV. The present LCAO (DFT with LDA/GGA) calculations predict about 0.5 eV band gap as listed in Table 1. The d electrons in InN play a major role in the unusual band gap because the 4d semi core electrons of In extend further outside the core. The DOS in Fig. 2 shows that the valence bandwidth calculated by FP-LAPW method is 5.92 eV while from present LCAO calculations (figure not shown) within DFT-LDA and GGA schemes it comes out to be 7.08 and 6.70 eV. It is interesting to note that the present FP-LAPW electronic structure calculations using the most accurate GGA prescription also confirm an unusual low conduction band minimum at the Γ point of WZ-InN. Therefore, valence states exist very close to CB, which is a major cause of low band gap in InN. The negative band gap predicted by different workers as collated in Table 1 seems to be unphysical. From Table 1, it is seen that the GGA underestimates the band gap in case of InN. To overcome this situation, GW correction may be required. Recently, Kotani and Schilfgaarde [13] have demonstrated that the inclusion of GW correction in full potential LMTO calculation of InN changes the band gap from −0.26 eV to 0.01 eV.

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The electronic properties namely energy bands and DOS are presented using the FP-LAPW within the recent gradient approximation of Wu and Cohen and also the LCAO-DFT approach. The value of the Fermi momentum of WZ-InN derived from the real space analysis of experimental Compton profile of WZ-InN indicates the closeness of the valence band maximum and conduction band minimum. Acknowledgements This work is supported by Department of Science and Technology, New Delhi (Grant No. SR/S2/CMP-16/2004). We are also grateful to Prof. R. Dovesi and Prof. P. Blaha for providing the CRYSTAL03 and the WIEN2k packages, respectively. References

Fig. 3. (Color online.) The real part of the autocorrelation function, B ( z), for WZ-InN obtained from the experimental and free electron Compton profiles.

To reconfirm the low band gap of WZ-InN, we have deduced the Compton profile in real space. It may be noted that the Fourier transform (FT) of the Compton profile can also be used to interpret the momentum densities in real space; it circumvents the difficulties of experimental data like relatively poor resolution, multiple scattering, etc., [27,31]. The one-dimensional FT of the Compton profile, which provide an accurate information about the Fermi momentum ( p F ) of systems, is given by [31,32]

 B ( z) =

1

1/2 +∞

2π

J ( p z )e −ip z .z dp z .

(2)

−∞

[18]

For a free electron gas, the parabolic Compton profile has a zero value at the p F . For a unit area of the parabolic free electron profile, the B ( z) function is given by B ( z) =

3 p 2F z2



sin( p F z) pF z

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

 − cos( p F z) .

[19]

[20]

(3)

Therefore, for a free electron based Compton profile, the B ( z) is zero firstly at p F z = 4.493. It is also known that the inclusion of core contribution in the free electron Compton profile does not change the position of the zero passages of B ( z). In Fig. 3, we have plotted the B ( z) function derived from the valence experimental profiles along with free electron theory based Compton profiles of WZ-InN. In Fig. 3, it is seen that the experimental B ( z) curve cuts the p z axis at z = 4.157 a.u. From this cut, the p F value comes out to be 1.05 ± 0.03 a.u. which is close to theoretical value 1.02 a.u. obtained from the free electron model calculations. This result demonstrates that the effective mass of Fermi electrons and their angular distribution are close to the free electrons, which characterise the dispersion of the valence band. In addition, a close agreement between the free electron and the experimental p F indicates a very small difference between the valence band maximum and conduction band minimum in InN. It agrees with the present value of small band gap computed using FP-LAPW calculations and the work reported by other workers, see for example, [3,6,8]. It may be noted that the real space analysis approach was successful in determining the metal like character of β -tin [32].

[21]

[22] [23] [24] [25]

[26] [27]

[28] [29]

[30] [31] [32]

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