Electron Momentum Density Band Structure And Structural

Published Date: 02 Nov 2017

G. Sharma1,*, N. Munjal2, V. Vyas2, R. Kumar3, B.K. Sharma3, K.B. Johsi4,5

1Department of Pure & Applied Physics, University of Kota, Kota-324 010, India

2Department of Physics, Banasthali University, Banasthali-304 022, India

3Department of Physics, University of Rajasthan, Jaipur-302 004, India

4Department of Physics, MLS University, Udaipur-313 002, India

5Department of Physics, Central University of Himachal Pradesh, Dharamshala-176215, India

Abstract

Electron momentum density, electronic band structure and structural properties of SrS are presented in this paper. The isotropic Compton profile, anisotropies in the directional Compton profiles, electronic band structure and density of states are calculated employing the ab-initio periodic linear combination of atomic orbitals method utilizing CRYSTAL06 code. Structural parameters of SrS namely lattice constants, bulk moduli in the B1 and B2 phases are computed together with the transition pressure. The computed parameters are well in agreement with earlier investigations. To compare the calculated isotropic Compton profile, measurement on polycrystalline SrS is performed using 5 Ci - 241Am Compton spectrometer. Additionally, charge transfer is studied by means of the Compton profiles computed from the ionic model. Nature of bonding in the isovalent SrS and SrO compounds is compared on the basis of equal-valence-electron-density profiles and found that bonding in SrS is more covalent than SrO.

Key words: Momentum density, Electronic band structure, SrS, Phase transition, Gamma-ray spectrometry.

PACS: 71.15.Dx, 71.20.Nr, 78.70.-g

*Corresponding author. Tel.: +91 744 2471038, Fax: +91 744 2471038

E-mail address: [email protected] (G. Sharma)

1. Introduction

Strontium Sulfide (SrS), a II-VI compound, has been studied experimentally as well as theoretically [1-16] owing to technological applications in luminescent, magneto-optical and infrared sensitive devices [2,15]. Under normal conditions it crystallizes in the rock-salt (B1) structure and shows structural phase transition to the cesium chloride (B2) structure under pressure. A number of investigations on SrS are devoted to study phase transition, optical and electronic properties etc. (see, e.g., [1-16]).

Compton profile,, is an important observable to characterize momentum density distribution of electrons in solids [17,18]. The is defined as:

, (1)

where is the electron momentum density in solids [17]. It gives the projection of electron momentum density along the scattering vector direction (z-axis). It is one of the few observables which can be calculated as well as directly measured. It enables to unfold Fermi surface calipers in metals and related systems, nature of bonding, bond length and charge transfer in compounds and alloys [17,18]. Despite extensive investigations neither theoretical nor experimental attempt has been made to explore electron momentum density and thereby Compton profile in SrS.

In this work, therefore, a systematic theoretical and experimental Compton profile study of SrS is carried out. Ionic model is applied to examine the charge transfer in SrS by means of Compton profiles. Other electronic properties namely electronic band structure and density of states are also presented. Moreover, the structural properties namely equilibrium lattice constants and bulk moduli are computed for the B1 and B2 phases together with the transition pressure for the B1→B2 structural transition. All theoretical calculations are performed applying the periodic linear combination of atomic orbitals (LCAO) method implementing the CRYSTAL06 code. To compare the calculated Compton profile, a measurement on the polycrystalline sample of SrS has been done utilizing a 5Ci -Compton spectrometer based on the 241Am radioisotope.

The paper is organized as follows. In Section 2 a brief description of the experiment is given. The computational details are presented in Section 3. In Section 4 we present and discuss the results and conclusions are given in Section 5.

2. Experimental details

The Compton profile measurement on polycrystalline sample of SrS was carried out using 5Ci- 241Am gamma-rays spectrometer. The spectrometer offers modest resolution (Gaussian, full width at half maximum) around 0.6. a.u. The salient features of the experimental set-up are available elsewhere [19]. In the present measurement, the incident gamma-rays of 59.54 keV were scattered at an angle 16603.00 by the sample (pellet of 18 mm dia, 3.2 mm thickness and 1.627 gm/cm3 effective density). The scattered radiation was recorded using an HPGe detector (Canberra model, GL0110S) and associated electronics to collect 45,000 counts at the Compton peak. To achieve true Compton profile, the raw data were corrected for several systematic corrections like background, instrumental resolution, sample absorption, scattering cross section and multiple scattering etc. [20,21]. Finally, the corrected profile was normalized on the corresponding free atom [22] area i.e. 24.049 electrons within the momentum range 0–7 a.u. The 1s electrons of Sr were neglected since they do not contribute to the experimental Compton profile within 0-7 a.u. in the current setup.

3. Computational details

3.1 DFT-LCAO method

The structural and electronic properties of SrS were computed using the ab-initio LCAO method embodied in the CRYSTAL code [23]. In this method, the crystalline orbitals, ψi(r,k), are linear combination of Bloch functions, φμ(r,k), defined in terms of local functions, φμ(r), normally referred as atomic orbitals. The Gaussian basis sets were taken for Sr and S [24]. The exchange and correlation are treated under the generalized gradient approximation. The correlation functional proposed by Perdew, Burke and Ernzerhof (PBE) [25], which has been one of the reasonably successful correlation functional [26-28], is applied while exchange is considered by applying the Becke’s ansatz [29]. The self-consistent calculations were performed considering 145k points in the irreducible Brillouin zone with tight tolerances and the self-consistency was achieved within 10 cycles.

3.2 Ionic model

The ionic model based theoretical Compton profiles of SrS for various charge transfer configurations were determined by using the free atom profiles [22] of Sr and S. The valence profiles for various Sr+xS-x (0.0≤x≤2.0) configurations were derived by transferring x electron from 5s shell of Sr to the 3p shell of S and then these profiles were added to the core contributions to get the total ionic profiles. All these ionic profiles were then appropriately normalized to compare with the experimental data.

4. Results and Discussion

In the present study, we have computed directional and spherically averaged Compton profiles of SrS. In practice, calculations are not directly compared with the experimental data. Rather, these are convoluted by the residual instrumental function or a Gaussian function to include resolution effects and thereafter the convoluted data are used for comparison. Generally, here, convolution smears the calculated Compton profiles. The experimental and unconvoluted spherically averaged theoretical (DFT-LCAO) Compton profiles of SrS are given in Table 1. The ionic profiles based on various charge transfer configurations i.e. Sr+xS-x (0.0≤x≤2.0 in step of 0.5), are also given in the table. These data may be useful for comparison with experiment and calculations which may appear in future.

Study of anisotropies facilitates to examine the directional features in the electron momentum density. The anisotropies in Compton profiles are therefore derived from the directional Compton profiles computed along the [100], [110] and [111] principal crystallographic directions. The [100]-[110], [100]-[111] and [110]-[111] anisotropies are plotted in Fig. 1. In order to expect a magnitude of experimental anisotropy the three plotted DFT-LCAO anisotropies are derived from the convoluted directional Compton profiles. The anisotropies in the Compton profiles remain upto ~2 a.u. Thereafter, major contribution comes from the core electrons which have identical contribution in all directions and so anisotropies vanish. The figure depicts that the [110]-[111] anisotropy has the smallest magnitude in the entire range. The anisotropies [100]-[110] and [100]-[111] show similar trend at all momenta. The maximum anisotropy can be seen between [100] and [111] directions. It indicates larger role of [100] direction compared to the other two with regard to the momentum density distribution in SrS. Consequently specific features i.e. extremes in anisotropies are anticipated at 2Ï€/a =0.55, 1.1 a.u. The extremes in anisotropies related to the [100] direction can be clearly seen at these positions in Fig. 1. As expected, these features appear at albeit lower momentum in the [110]-[111] anisotropy. Moreover, the positive nature of all these anisotropies around pz=0. a.u. indicates larger occupied states along [100] direction with low momentum. To quantify these directional features, the directional Compton profile measurements on SrS would be valuable.

Now we compare the Compton profile calculations with our own measurement. Firstly; we compare Compton profiles computed from various ionic arrangements. The difference profiles (convoluted ionic-experiment) are shown in Fig. 2. A similar approach was applied to estimate the charge transfer in other compounds [30-32]. It is clear from the figure that the effect of variation of charge on Sr and S is visible only up to 1.5 a.u. All ionic arrangements show identical trend in the high momentum region. To estimate the charge transfer, we have computed and observed that the Sr+2.0S-2.0 configuration gives the best agreement with the measurement. Thus, the ionic model suggests transfer of 2.0 electrons from Sr to S. Secondly; we compare the Compton profile computed from the DFT-LCAO method. The difference profiles i.e. ΔJ=JTheory(pz)-JExpt.(pz) derived from convoluted DFT-LCAO and the best agreed ionic arrangement (Sr+2.0S-2.0) with the experiment are plotted in Fig. 3. The figure reveals that the difference curves derived from both the schemes show differences in the low momentum region (0.0≤pz≤2.0). It may be due to the inclusion solid state effects in the DFT-LCAO method because this is highly sensitive region where valence electrons contribute largely to the electron momentum density. The maximum differences shown by the DFT-LCAO and ionic model with the experimental J(0) value are about 3.76% and 4.52% respectively. The overall agreement, examined on the basis of , is shown by the DFT-LCAO with the measurement. Beyond 2.0 a.u. difference curves derived from both the schemes are overlapping each other. This is well expected as the contribution of core electrons, which are less affected by solid formation, dominates in this region. In the high momentum region i.e. 3-7 a.u. differences are beyond experimental error. This is probably due to the pellet which may have non-uniform thickness which gives rise to non uniform absorption correction. As this correction is isotropic in nature it will naturally get cancel when anisotropies are measured.

Now we examine the nature of bonding in isostructural and isovalent SrS and SrO compounds. For this, we compute equal valence electron density profile (EVED) profiles on the pz/pF scale, where pF is the Fermi momentum, from the experimental and theoretical Compton profiles of valence electron of SrS and SrO. The experimental valence profiles are deduced by subtracting the convoluted core from total experimental profiles. These valence electron profiles are normalized to 4.0 electrons and multiplied by the corresponding Fermi momentum (pF=0.862 and 1.006 a.u. for SrS and SrO respectively). The EVED profile scheme provides a way to understand the nature of bonding in isostructural and isovalent compounds [33,34]. The experimental EVED profiles of the two compounds are plotted in the left panel of Fig. 4. For SrO, data is taken from our own earlier measurement [35]. In the right panel, we show the EVED profiles derived from DFT-LCAO scheme for the two compounds. It is clear from the figure that the EVED profile of SrS is higher than SrO at the lower values of pz/pf. It indicates larger covalent and smaller ionic character of SrS as compared to SrO. This is well supported by Fig. 4, above, where complete ionic model deviate from the experiment more than the DFT-LCAO scheme. The less ionic character of SrS as compared to SrO is well supported by the ionicity factors fi proposed by J.C. Phillips [36].

The electronic band structure and density of states (DOS) for the B1 phase of SrS are plotted in Fig. 5. In Table 2, we give our energy band-gaps, calculated at the principal symmetry points, along with the experimental and theoretical available data in the literature. It is clear from the Table 2 that the present PBE-GGA electronic band gaps are in good agreement with experiment. From the calculated DOS for SrS as, shown in Fig. 5, we find that below and above the Fermi level the electronic states of sulphur dominate the density of states.

To determine the structural parameters, the total energies are calculated for rocksalt (B1) and cesium chloride (B2) phases of SrS at different volumes around the equilibrium primitive cell volume V0. The plots of calculated total energies versus volume for SrS in both structures are given in Fig. 6. The Birch-Murnaghan equations of state (EOS) [28,37] are represented by solid lines in the figure. It depicts that the energy of the lowest point of the B1 structure is below than that of the B2 structure. We have computed equilibrium lattice constant (a0), bulk modulus (B0) and its pressure derivatives (B0') for B1 and B2 phases of SrS by fitting the Birch-Murnaghan EOS [31,32] and results are summarized in Table 3. Our results obtained for both structures of SrS are in very good agreement with the experimental and earlier theoretical data. The structural phase stability and pressure induced transitions from B1 to B2 structure is studied by performing the enthalpy (H = E + PV) calculations. Variation of enthalpy with volume is shown in Fig. 7 for the two phases which clearly shows that SrS transforms from B1 to B2 phase at 18.77 GPa. The transition pressures (Pt) predicted by our calculation is in good agreement with the experimental data as well as other theoretical calculations listed in Table 3.

5. Conclusions

In this paper, calculations of isotropic and directional Compton profiles, electronic band structure, density of states together with lattice constant and bulk modulus of SrS are reported using the ab-initio LCAO method. The computed anisotropies, especially related to the [100] direction, very well reflect the directional features of momentum density distribution. The spherically averaged Compton profile is in good agreement with the first-ever measurement on SrS. The simple ionic calculation, which suggests transfer of 2.0 electrons from the valence s state of Sr to the p state of S atom, shows poor agreement with experiment than DFT-LCAO. On the basis of EVED profiles, it is found that bonding in SrS is less ionic or more covalent than SrO. The first-principles total energy calculations for the B1 and B2 phase are performed to compute lattice constant and bulk modulus. The results are in very good agreement with earlier investigations and suggest structural phase transition from B1 to B2 to occur at 18.77 GPa.

Acknowledgements

This work is financially supported by the University Grant Commission (UGC) through Emeritus Fellowship and SR/33-37/2007 to BKS and SR/39-982/2010 to GS. GS is also thankful to the Head, Department of Pure & Applied Physics, University of Kota, Kota for providing the computational facilities.