Electron momentum density and band structure calculations of α- and β-GeTe

Radiation Physics and Chemistry 80 (2011) 1316–1322

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Electron momentum density and band structure calculations of a- and b-GeTe Laxman Vadkhiya a, Gunjan Arora b, Ashish Rathor a, B.L. Ahuja a,n a b

Department of Physics, University College of Science, M.L. Sukhadia University, Udaipur 313001, Rajasthan, India Department of Physics, Techno India NJR Institute of Technology, Udaipur 313002, Rajasthan, India

a r t i c l e i n f o

a b s t r a c t

Article history: Received 10 December 2010 Accepted 14 July 2011 Available online 28 July 2011

We have measured isotropic experimental Compton profile of a-GeTe by employing high energy (662 keV) g-radiation from a 137Cs isotope. To compare our experiment, we have also computed energy bands, density of states, electron momentum densities and Compton profiles of a- and b-phases of GeTe using the linear combination of atomic orbitals method. The electron momentum density is found to play a major role in understanding the topology of bands in the vicinity of the Fermi level. It is seen that the density functional theory (DFT) with generalised gradient approximation is relatively in better agreement with the experiment than the local density approximation and hybrid Hartree–Fock/DFT. & 2011 Elsevier Ltd. All rights reserved.

Keywords: X-ray scattering Electron momentum density Band structure calculations Density functional theory

1. Introduction

! where rð p Þ is the EMD and is given by

It is well known that the spectrum of inelastically scattered photons by electrons is related to the electron momentum density (EMD) of the scatterer (see for examples, Cooper et al., 2004; Ahuja, 2010). In bulk materials, the Compton technique is assumed to be valid within the impulse approximation (IA), where the energy transferred in the scattering process is much larger than the binding energy of the electronic states involved. In Compton spectroscopy, we fix the direction of the incident g-rays and measure the energy distribution of the scattered photons at a fixed scattering angle y. The Compton profile (CP), J(pz), can be deduced from double differential cross-section using the relation

rð! p Þp

d2 s ¼ Fðoi , of , y, pz ÞJðpz Þ d O d of

ð1Þ

Here F is a function of incident and scattered photon energies (oi and of, respectively), scattering angle (y) and the electron momentum (pz) along the scattering vector direction. Within the IA, the CP is a 2-D integral over the momentum density. Theoretically, CP is determined as Jðpz Þ ¼

Z Z

rð! p Þdpx dpy

px py

n

Corresponding author. Fax: þ91 294 2411950. E-mail address: [email protected] (B.L. Ahuja).

0969-806X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.radphyschem.2011.07.007

ð2Þ

Z 

!

! !

2 

cn ð r Þexpði p  r Þdr

ð3Þ

! In Eq. (3), cn ð r Þ is the position space wave function for the electron in the nth state. Therefore, the computed CP provides a way to probe the performance of various electronic structure calculations used to generate solid-state wave functions. GeTe, which is a IV–VI semiconductor, consists of a-phase (rhombohedral, lower symmetric, ferroelectric, space group R3m) at room temperature and b-phase (cubic, rocksalt-type, highly symmetric, paraelectric, space group Fm3m) at high-temperature. The a and b structures are related by a ferroelectric type of transition at a critical temperature of 720 K (Chattopadhyay et al., 1987) that can be seen by considering the atomic coordinates. GeTe is an interesting material for industrial applications because when it is alloyed with Sb, its electrical and optical properties change dramatically due to the change in microscopic structure from crystalline to amorphous (Libera and Chen, 1993; Yamada, 1996). Ferroelectric behaviour of GeTe makes it a promising material for various optoelectronic applications. Moreover, GeTe is also suitable for infrared detectors and light emitting devices. Among theoretical band structure calculations, several authors have used various approaches to describe the ground state properties of a- and b-GeTe. Recently, Shaltaf et al. (2008) have reported the dynamical, dielectric and elastic properties of a phase using plane waves and norm-conserving pseudopotentials (PP) method. By employing fully relativistic projected augmented wave (PAW) method, Ciucivara et al. (2006) have compared the electronic structure of GeTe using local density approximation (LDA) and

L. Vadkhiya et al. / Radiation Physics and Chemistry 80 (2011) 1316–1322

generalised gradient approximation (GGA). Edwards et al. (2006) have employed various methodologies like relativistic plane wave DFT, non-relativistic DFT–LDA and quantum Monte Carlo calculations, to understand the electronic structure of intrinsic defects in a and b states of GeTe. Waghmare et al. (2003) have reported electronic structure using the linear muffin-tin orbital (LMTO) method in the atomic sphere approximation (ASA) and plane wave PP schemes. For the b phase of GeTe, the electronic and optical properties have been computed by Okoye (2002) using full potential linearised augmented plane wave (FP-LAPW) method. The author has used exchange and correlation of Perdew–Wang (PW) and Perdew–Burke–Ernzerhof for LDA and GGA, respectively. Rabe and Joannopoulos (1987) have employed ab initio scalar relativistic PP in LDA including spin–orbit coupling to find the structural properties of GeTe. Regarding experimental studies, Onodera et al. (1997) have reported the X-ray diffraction and electrical resistivity of GeTe. Although a number of theoretical and experimental efforts have been made to describe the structural, electronic and optical properties of GeTe, still studies related to the momentum densities and CPs are lacking. In this paper, we report the first ever CP of a-GeTe measured using our 20 Ci 137Cs Compton spectrometer. Due to non-availability of large size single crystals, we have measured the isotropic CP. To compare our experimental data, we have computed CPs using DFT and hybrid Hartree–Fock (HF)/DFT methods as embodied in the ab initio linear combination of atomic orbitals (LCAO) method developed by Torino group (Towler et al., 1996; Saunders et al., 2003). In addition, we report energy bands, density of states (DOS), band gaps and Mulliken’s population of both the a and b phases.

2. Experiment We have measured the isotropic experimental CP of a-GeTe at ambient temperature using our 20 Ci 137Cs Compton spectrometer (Ahuja and Sharma, 2005; Ahuja et al., 2006). The g-rays of energy 661.65 keV were scattered by polycrystalline GeTe (pellet of thickness 0.2 cm and diameter 2.46 cm) at an angle of 16070.61. The scattered photons were analysed by a high purity Ge detector (Canberra made, operated at bias of  700 V) and associated electronics (spectroscopy amplifier, analog to digital converter, etc.). The channel width of 4096-channel analyser (Canberra; Accuspec B) was about 61 eV. During the exposure time of 276 h about 2.2  107 counts were collected in the CP region. The overall momentum resolution of the present measurement, which includes the detector resolution and the geometrical broadening, was 0.38 a.u. (Gaussian, full width at half maximum). The raw Compton data were processed by the application of a series of corrections like background, detector response function, the variation of detector efficiency with energy, absorption, Compton scattering cross-section, multiple scattering within the sample, etc. (Cooper et al., 2004). The duly corrected profiles were normalised to the respective free atom CP area (Biggs et al., 1975) of 34.28 electrons over the momentum range 0–7 a.u.

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with LDA and GGA, and also posteriori Becke’s three parameter hybrid functional, i.e. B3LYP. In DFT–LDA, one electron exchange-correlation potential operator is defined as R ! ! ! @½Exc ðrÞ ¼ d r rð r Þexc frð r Þg ð4Þ n^ xc ð! r Þ¼ ! @rð r Þ where Exc is the exchange-correlation density functional energy ! and r is the electron density at point r , while in the case of GGA Z ! ! ! ! d r rð r Þexc frð r Þ,9rrð r Þ9g ð5Þ Exc ðrÞ ¼ Here the integrals are taken over the unit cell. In the DFT–LDA calculations, we have chosen the Dirac–Slater exchange (Saunders et al., 2003) and Perdew–Zunger correlation potentials (Perdew and Zunger, 1981), while for GGA the exchange and the correlation potential of Perdew–Wang (Perdew and Wang, 1986 and 1992) have been used. In the B3LYP (hybrid HF/DFT), we have used Becke’s gradient correction (Becke, 1988) to the exchange and the correlation functionals due to Lee–Yang–Parr (Lee et al., 1988) and Vosko–Wilk–Nusair (Vosko et al., 1980). In the present calculations, for a better prescription of a and b structures of GeTe, the all electron Gaussian basis sets for Ge and Te were taken from the website http://www.tcm.phy.cam.ac.uk/  mdt26/basis_sets. The basis sets were energy optimised using BILLY software (Saunders et al., 2003). Following the standard truncation criteria for the CRYSTAL03 code, we have performed the SCF calculations with 413 k points in the irreducible Brillouin ˚ while zone. For b-GeTe the lattice parameter was a ¼5.996 A, ˚ in case of a phase the structural parameters were a¼ 4.289 A,

a ¼ b ¼ g ¼58.071.

4. Results and discussion 4.1. Energy bands 4.1.1. b-GeTe Fig. 1(a,b) shows the energy bands along with the total and partial DOS of b-GeTe calculated using the DFT–GGA scheme of LCAO method. Other schemes as included in the CRYSTAL03 code (viz. DFT–LDA and B3LYP) show almost similar topologies of energy bands and DOS (hence not shown here). We discuss mainly the energy bands in the vicinity of Fermi energy (EF) which are very important to examine the electronic properties of materials. The non-degenerate valence band maximum (VBM) and the conduction band minimum (CBM; degenerate) located at Z point make it a direct band gap semiconductor. The direct band gap at Z defines the lowest gap M0 (0.148 eV), as the surfaces of constant energy separation are ellipsoidal in the neighbourhood of Z. The minimum energy separation at L point and D (G–X) branch define M1 (0.160 eV) and M2 (1.459 eV) gaps, respectively. Therefore, in addition to the direct gap at Z, VBM and CBM at L point and along D branch are more likely to be saddle points than absolute extrema. Except positions of non-degenerate states, fine structures and band gap values, the present energy bands show a similar topology as reported earlier (Okoye, 2002; Rabe and Joannopoulos, 1987).

3. LCAO calculations We have computed the electronic structure, including momentum densities, using self-consistent LCAO scheme as embodied in CRYSTAL03 program (Towler et al., 1996; Saunders et al., 2003). The code enables computation of electronic properties using DFT

4.1.2. a-GeTe The band structure of a-GeTe along with total and partial DOS obtained by using the DFT–GGA scheme of LCAO method is shown in Fig. 2(a,b). Unlike b-GeTe, a-GeTe has CBM at L point whereas VBM at Z point which gives an indirect band gap of

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Fig. 1. (a) Energy bands along high symmetry directions of the first Brillouin zone for b-GeTe using LCAO (GGA) method and (b) total and partial density of states (DOS) corresponding to energy bands shown in (a).

Fig. 2. (a,b) Same as in Fig. 1, except the sample which is a-GeTe.

0.33 eV. As seen in the case of b-GeTe, the overall shape of bands is in good agreement with the earlier reported data (Shaltaf et al., 2008; Ciucivara et al., 2006; Rabe and Joannopoulos, 1987). Now we discuss the DOS of both the phases of GeTe. In Fig. 1(b), the DOS for b-GeTe reveals a sharp peak around 10.5 eV (the lowest part of VB) due to the predominantly Te-5s character followed by 4s and p states of Ge. Secondly the top of VB is composed of Te-5p states, although there is a broad contribution of Ge-4sp states. From Fig. 1(a,b), it is seen that the bands in the energy range ( 9 to  6 eV) are dominated by Ge 4s states, while 5s and 5p states of Te also contribute in this range. Therefore, the p states of anion (Te) are somewhat hybridised with the cation (Ge) s states in VB. The contribution of d states of both the elements is found to be negligible and is not seen here. In the CB (0–6 eV), an overlapping between Ge-4p and Te-5p states is seen. Therefore, in GeTe the cation-p–anion-p covalency tends the system towards a narrow band gap and hence the metal like situation. There is a very small interaction between s states of Ge

and Te, whereas s–p interactions are found to be more probable in the fcc symmetry. The DOS obtained in the case of a-GeTe (Fig. 2(b)) are almost similar to those of b-GeTe (Fig. 1(b)). As the a phase represents a small perturbation in the b phase, the overall topology of band structure and DOS looks similar. 4.2. Energy band gap and charge transfer In Table 1, we have collated our computed band gaps along with available theoretical and experimental data. Our DFT based band gap values are in very good agreement with that reported by Edwards et al. (2006) (0.14 eV) using the Perdew–Zunger parameterization of the Ceperly–Alder within SEQQUEST. In the case of a-GeTe the fundamental gaps computed by DFT–LDA and GGA theories (0.270 and 0.330 eV respectively) are close to the experimental value (0.1–0.2 eV) reported by Bahl and Chopra (1970). Our B3LYP computations give the value of band gap

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Table 1 Calculated band gaps (DEg) in eV of a- and b-GeTe along with available data.

b-GeTe

a-GeTe

Direct

Direct

Indirect

DFT–LDA DFT–GGA B3LYP

0.132 0.148 0.486

0.409 0.490 0.757

0.270 0.330 0.672

DFT–LDA DFT–GGA DFT–LDA

0.399a, 0.244a (with SO) 0.340a, 0.201a (with SO) 0.040b (with SO) 0.140c 0.002c

0.400b (with SO) 0.522c 0.367c 0.700c 0.369d 0.685d 0.480e

0.209d 0.469d 0.280e

Method

(i) Present work (a) LCAO

(ii) Available theory (a) FP-LAPW (b) Scalar relativistic PP calculations (c) Non-relativistic DFT (LDA)–SEQQUEST (d) Relativistic ab initio plane wave DFT calculations (e) Non-relativistic QMC calculations (f) Fully relativistic PAW method using VASP code

DFT–LDA DFT–GGA

(g) DFT (ab initio) plane wave and PP calculations (iii) Available experiment (a) Tunnelling spectroscopy

0.100–0.200f, 0.200g

a

Okoye (2002). Rabe and Joannopoulos (1987). Edwards et al. (2006). d Ciucivara et al. (2006). e Shaltaf et al. (2008). f Bahl and Chopra (1970). g Vinet et al. (1987). b c

Table 2 Charge transfer in a and b phases of GeTe as computed by various approaches of LCAO. Overlap population is shown in the brackets. Sample

DFT (LDA)

DFT (GGA)

B3LYP

a-GeTe

0.101 (0.068)

0.116 (0.068)

0.177 (0.075)

b-GeTe

0.285 (0.052)

0.285 (0.050)

0.376 (0.043)

as 0.672 eV which is comparable to that from the PAW (GGA) method as embodied in VASP code (0.685 eV). Our DFT–LDA based values show a reasonable agreement with the data reported by Ciucivara et al. (2006) and Edwards et al. (2006). The overestimation of the gap in B3LYP scheme is due to hybridisation of HF theory with the DFT. Since the experimental band gap decreases with increasing temperature, as expected, the high temperature phase (b) shows smaller values of the gap as compared to the low temperature (a) phase. This trend is consistent with the earlier reported calculations (Rabe and Joannopoulos, 1987; Edwards et al., 2006). For a better experimental verification of band gap, more measurements on bulk materials are required. It is worth mentioning that the tunnelling data measurements in Table 1 are performed on the polycrystalline films (Vinet et al., 1987). Using Mulliken’s population (MP) analysis, one can analyse the nature of bonding between the elements participating in the bonding. Charge transfer and overlap population within different DFT approximations (based on MP analysis) are summarised in Table 2. The charge transfer in the case of b-GeTe is more than in a state, so one can infer that b phase is more ionic than a phase. 4.3. EMD and Compton profiles The EMD of b-GeTe with scattering vector along low-indexed directions namely [1 0 0], [1 1 0] and [1 1 1] using the DFT (GGA)

Fig. 3. Electron momentum density (EMD) distribution plots for b-GeTe along [1 0 0], [1 1 1] and [1 1 0] directions using DFT–GGA.

theory, is plotted in Fig. 3. Now we discuss the momentum densities in terms of energy bands along L [1 0 0], D [1 1 0] and S [1 1 1] directions which contribute to the fine structure in the momentum densities. Energy bands shown in Fig. 1(a) can explain the nature of EMD plots obtained for b-GeTe. In the case of G–L [1 0 0] branch, there are allowed states at L point having energy upto EF. This leads to higher value of momentum density at 0, 0.5, 0.9, 1.35 a.u. due to G–L distances (0.48n a.u.; n ¼0, 1, 2, 3, y). In the G–X [1 1 0] direction, there are three degenerate bands at G point, leading to a sharp EMD at pz ¼ 0 a.u. Due to dispersion of energy bands around 0.2 a.u. towards EF there is a bump at

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pz ¼0.2 a.u. Next positive oscillation at 0.45 a.u. is due to crossing of bands around 0.4 a.u. while going from G-X. A positive bump around 0.6 a.u. is due to G–X distances (0.55 a.u.), where we have two bands at X point. Along the G–Z [1 1 1] direction, higher value of EMD at pz ¼0 a.u. is due to degenerate states (and hence more electron density) at G point. The higher value of EMD is also expected at the integral multiples of G–Z distances (0.48 a.u.). The anisotropies in the directional Compton profiles computed for a- and b-GeTe within the framework of DFT (with LDA and GGA) and B3LYP are plotted in Fig. 4(a,b), respectively. Mathematically, these anisotropies are represented as follows:

DJðpz Þ ¼ Jhkl ðpz ÞJh0 k0 l0 ðpz Þ

ð6Þ

Here [h k l] and [h0 k0 l0 ] are principal crystallographic directions. It can be noted that except fine structures and amplitude, the general trend of oscillations in the theoretical Compton profile anisotropies in both phases is almost similar. As discussed in case of EMDs, the fine structures in J100, J110 and J111 reflect the degenerate states, cross-over of bands and number of allowed states in L, D and S branches, respectively. The theoretical unconvoluted anisotropies of a-GeTe (as shown in Fig. 4(b)) can be explained on the basis of corresponding E–k relations (Fig. 2(a)). At G point, there are more degenerate states along G–X [1 1 0] direction than G–Z [1 1 1] and G–L [1 0 0] directions which increases the momentum density in [1 1 0] direction in the vicinity of Compton peak. Therefore, it leads to a negative amplitude in J111–J110 and a positive amplitude in J110–J100 at pz ¼0. The positive oscillations near 0.4, 0.8 and 1.4 a.u. in J111–J110 are understandable in terms of closely spaced bands and degenerate states at Z point of G–Z branch, which lead to higher momentum density at G–Z distances (0.47 a.u., n¼ 1, 2, 3,y). In J111  J100 and J110 J100, we observe negative oscillations at 0.5 and 1.0 a.u. which are due to lift of degeneracy at L point of G–L [1 0 0] branch at 0.49n a.u.; n¼1, 2y) distances. Some fine structures that were seen in EMD curves (Fig. 3) are not visible in the anisotropies, which is due to cancellation of momentum densities in the directional differences. In the absence of directional experimental data, we compare the isotropic experimental Compton profiles of polycrystalline a-GeTe with the spherically averaged convoluted theoretical Compton profiles. The experimental profiles along with unconvoluted theoretical

line shapes computed within the frame-work of DFT (LDA and GGA) and B3LYP are listed in Table 3. Fig. 5 depicts the differences between the convoluted theory (at resolution of 0.38 a.u.) and the experiment. This figure shows that all the DFT based theoretical prescriptions show almost similar difference between the theory and experiment. It is noticed that all the theoretical profiles overestimate the momentum densities in the vicinity of Compton peak (at pz ¼0 a.u.) while a reverse trend is seen in the momentum range 0.8–3.8 a.u. In the high momentum side (pz 44 a.u.), the theoretical values agree well with the experiment. This is expected because this region is dominated by core electrons whose Compton profiles are well approximated by free atom profiles. The differences between present theoretical calculations and the experiment may be due to improper inclusion of the Lam–Platzman (LP) correlation effects in the DFT calculations. It is worth mentioning that the LP correlation Table 3 Spherically averaged Compton profiles computed for different LCAO schemes. Also included here is the experimental profile of a-GeTe. All the theoretical profiles are unconvoluted. pz (a.u.)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 6.0 7.0

a-GeTe DFT–LDA

DFT–GGA

B3LYP

Expt. 7 s (error)

15.269 15.100 14.820 14.413 13.864 13.169 12.345 11.429 10.499 8.964 8.092 7.599 7.177 6.734 6.273 4.111 2.752 2.034 1.596 1.304

15.249 15.081 14.802 14.398 13.852 13.161 12.341 11.430 10.505 8.975 8.100 7.603 7.180 6.736 6.275 4.111 2.752 2.034 1.596 1.304

15.262 15.093 14.814 14.413 13.864 13.167 12.342 11.433 10.511 8.977 8.095 7.597 7.177 6.736 6.272 4.111 2.752 2.034 1.596 1.303

14.6797 0.032 14.620 14.386 14.054 13.499 12.752 12.001 11.259 10.545 9.260 70.025 8.341 7.835 7.285 6.798 6.331 7 0.017 4.174 7 0.013 2.741 7 0.009 2.018 70.007 1.584 7 0.006 1.203 70.005

Fig. 4. Unconvoluted anisotropies in Compton profiles of GeTe using DFT–LDA, DFT–GGA and B3LYP theory for (a) b phase and (b) a phase.

L. Vadkhiya et al. / Radiation Physics and Chemistry 80 (2011) 1316–1322

ΔJ (pz) (e/a.u.)

5. Conclusions

DFT-LDA DFT-GGA B3LYP

0.6

In summary, the first-ever theoretical Compton profiles of a- and

0.4

b-GeTe along with energy bands, density of states, and Mulliken’s population are presented. It is seen that the b phase shows a direct

0.2 0.0 -0.2 -0.4

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0

1

2

3 pz (a.u.)

4

5

6

Fig. 5. Differences between the isotropic experimental and the convoluted theoretical (spherically averaged) profiles using DFT schemes (including hybrid HF/DFT, so called B3LYP) for a-GeTe. Solid lines are drawn to guide the eyes.

band gap at Z point while the a phase has an indirect band gap. The small value of band gap in b-GeTe shows that at high temperature (4720 K), GeTe behaves like a metal. Both the a and b phases show a close similarity in anisotropy in the Compton profiles of GeTe. The isotropic Compton profile of a-GeTe computed using DFT–GGA theory is closer to experimental data, in comparison to DFT–LDA and B3LYP (hybrid HF/DFT). On the basis of experimental equalvalence-electron-density (EVED) profiles, it is shown that a-GeTe is more covalent than the isoelectronic GeS. This is also in agreement with the LCAO–GGA based EVED profile of a-GeTe and GeS and Mulliken’s population analysis. High resolution directional Compton profiles may be helpful in verification of theoretical momentum densities in a better way.

Acknowledgments The authors are grateful to UGC, New Delhi, and DRDO, New Delhi, for financial support. We thank Prof. R. Dovesi for providing the CRYSTAL03 code. One of us (Laxman Vadkhiya) thanks CSIR, New Delhi for the grant of Junior Research Fellowship. References

Fig. 6. EVED profiles (on pz/pF scale) of isoelectronic compounds GeS and a-GeTe.

correction shifts the momentum density by raising a few electrons from a level just below the Fermi momentum (pF) to above the pF without shifting the discontinuity at pF. Hence this correction reduces the amplitude of theoretical profiles near pz ¼0. In view of the possibility of opposite signs of the LP correction in the intermediate region (say 1–3 a.u.), the theoretical values in this range would come closer to the experiment. Moreover, for pz 44 a.u. the agreement between theoretical and experimental values may also be improved due to normalisation of LP corrected theoretical profiles. Further, there may be a possibility of improvement in the quality of basis sets, which governs the wave functions and thereby the electron momentum densities (Eq. (3)). To compare the relative nature of bonding in the isoelectronic GeS and GeTe, we have scaled (on pz/pF scale) our experimental and convoluted DFT–GGA profiles of a-GeTe to equal-valenceelectron-density (EVED). Similarly, the data on GeS (Rathor, 2009) is also scaled to EVED conditions. Both the EVED profiles were normalised to five valence electrons in the momentum range 0–6 a.u. The similarity in the EVED profiles suggests that, to a first order, the wave functions for bonding electrons in the isoelectronic materials may be identical if written as a function of r/a (a being the lattice parameter). It is known that the covalent bonding results from the sharing of electrons, which increases the localisation of charge in the direction of bonding. Therefore, the materials with covalent bonding have sharper Compton profiles. The EVED profiles are shown in Fig. 6. From this figure, it is seen that near pz ¼0 a.u., both the theoretical and experimental EVED profiles of a-GeTe are higher than those of GeS. It clearly indicates that a-GeTe is more covalent than GeS, which is also supported by present MP analysis and that given by Rathor (2009) for GeS.

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