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Directional Compton profiles of CdTe using a low intensity 241Am source
Radiation Physics and Chemistry 87 (2013) 35–39
Contents lists available at SciVerse ScienceDirect
Radiation Physics and Chemistry journal homepage: www.elsevier.com/locate/radphyschem
Directional Compton profiles of CdTe using a low intensity
241
Am source
Veera Raykar, Jagrati Sahariya, B.L. Ahuja n Department of Physics, University College of Science, M.L. Sukhadia University, Udaipur 313001, Rajasthan, India
H I G H L I G H T S
Reported first-ever directional experimental Compton profile (CP) of CdTe. Interpreted experimental CP using theoretical CP within density functional theory. Reported energy bands and DOS of CdTe using DFT–GGA. Analyzed the anisotropy in CPs in terms of energy bands. Discussed potential of low intensity 241Am point source in measurement of CPs.
art ic l e i nf o
a b s t r a c t
Article history: Received 23 January 2013 Accepted 11 March 2013 Available online 18 March 2013
We report the first-ever experimental Compton profiles (CPs) of CdTe measured along [100] and [111] directions using 100 mCi 241Am Compton spectrometer. The experimental Compton data have been interpreted in terms of theoretical anisotropy in the momentum densities and also energy bands computed using density functional theory with local density approximation, generalized gradient approximation (GGA) and second order GGA. It is found that DFT–GGA based CPs are in marginally better agreement with the experimental data in comparison to other DFT based profiles. The anisotropy in CPs is found to be in tune with the energy bands. The bond length (5.2370.11 a.u.) deduced from the oscillation in the experimental anisotropy reconciles well with the theoretical Cd–Te bond length (5.30 a.u.) which establishes the role of γ-ray Compton measurements in determination of structural parameters. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Compton scattering Band structures calculations Density functional theory II–VI semiconductor
over all occupied state. The anisotropy in the directional CPs is defined as,
1. Introduction An energy spectrum of Compton scattered γ-rays provides information about the ground state electron momentum densities (EMDs) of the scatterer (Cooper, 1985; Cooper et al., 2004; Ahuja et al., 2004; Ahuja, 2010). Within the impulse approximation (Kaplan et al., 2003), the differential scattering cross-section is proportional to Compton profile (CP), J(pz), defined as projection of EMD along the scattering vector k (z-axis of Cartesian coordinate system). Mathematically, Jðpz Þ ¼ ∬ ρðpÞdpx dpy ,
ð1Þ
where ρðpÞis the electron momentum density which can be deduced from the Fourier transformation of real space wave function using the following equation, Z 2 1 ρðpÞ ¼ ψðrÞexpð−ip:rÞdr ∑ ð2Þ ð2πÞ3 occ here Ψ(r) is the electron wave functions and summation extends n
Corresponding author. Tel.: þ91 294 2423322; fax: þ91 294 2411950. E-mail address: [email protected] (B.L. Ahuja).
0969-806X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.radphyschem.2013.03.020
ΔJðpz Þ ¼ J hkl ðpz Þ–J h0 k0 l0 ðpz Þ
ð3Þ
where (hkl) and (h'k'l') are planes perpendicular to the scattering vector. Cubic zine-blende (ZB) CdTe, a II–VI semiconductor, has always been the center of attraction due to its vast technological importance in photovoltaic devices, solar panels, modern optoelectronic and spintronics devices, etc. (see for example, Sanchez-Almazan et al., 1996; Merad et al., 2005; Fleszar and Hanke, 2005; Reshak, 2006; Hosseini, 2008). Earlier, numerous studies on electronic and optical properties of CdTe have been reported by several workers. Markowski and Podgorny (1991) have reported the optical absorption of CdTe using the semi-relativistic linear muffin-tin orbital calculations with the local density approximation (LDA). The frequency dependent second- and third-harmonic response functions have been presented by Ghahramani et al. (1991), using the linear combination of Gaussian type orbitals. Huang and Ching (1993) have computed the electronic structures and linear optical dielectric functions of CdTe using the orthogonalized linear combination of atomic orbitals (LCAOs) method with the LDA. Zakharov et al. (1994) have discussed the quasiparticle band structure using
36
V. Raykar et al. / Radiation Physics and Chemistry 87 (2013) 35–39
ab-initio pseudopotential with the LDA and the GW approximation for both ZB and hexagonal wurtzite (WZ) phases of CdTe, while the structural and electronic properties of both phases using the firstprinciple calculations have been reported by Wei and Zhang (2000). Using the photoluminescence spectra, Fonthal et al. (2000) have measured the temperature dependent band gap of CdTe. The electronic structure within the GW approximation has been discussed by Fleszar and Hanke (2005). Merad et al. (2005) have used full-potential augmented plane wave with the local orbitals to compute the electronic and optical properties. Reshak (2006) has employed the full-potential linearized augmented plane wave (FPLAPW) method to study the electronic properties and first- and second-harmonic generations. The optical dielectric functions, reflectivity index, extinction coefficient and reflectivity of CdTe have been reported by Hosseini (2008) using FP-LAPW with the generalized gradient approximation (GGA). The electronic, lattice dynamics and mechanical properties of ZB-ATe (A¼ Cd, Zn, Mn, Mg, Hg) and their ternary alloys have been studied by Mnasri et al. (2009). Using the DFT–LDA, an empirical pseudopotential method and a full Brillouin zone k.p method, Penna et al. (2009) have studied the electronic structure of CdTe, HgTe and Cd1−xHgxTe. The electronic structure and phase stability of MgTe, ZnTe and CdTe in different crystalline structures have been reported by Yang et al. (2009). Ouendadji et al. (2011) have investigated the structural, electronic and thermal properties of CdX (X¼S, Se, Te) using the FPLAPW method. Very recently, Sarkar et al. (2012) have studied the structural and electronic properties of CdTe nanowires with hexagonal or triangular cross-sections using a self-consistent charge density functional tight binding method. Regarding the earlier Compton profile (CP) measurements of CdTe, Heda et al. (2007) have presented the γ-ray based experimental CP of polycrystalline CdTe. In the present work, we report the directional CPs of CdTe using a low intensity planar 100 mCi 241Am Compton spectrometer. The experimental CP data have been compared with the DFT–LDA, DFT–GGA and the second order GGA (SOGGA) based momentum densities and the anisotropic behavior in the Compton line shapes is discussed in terms of energy bands. Present directional CP measurements using a point source have the following advantages:
features of the spectrometer is shown in Fig. 1(a). For the present measurements, single crystals of the ZB-CdTe having dimensions 2 mm (thickness) 15 mm (diameter) were procured from M/s Cradely Crystals, Russia. The orientations of the each crystal were confirmed with their Laue patterns and were found to be aligned to within 711. The overall instrumental resolution of the spectrometer was having a Gaussian shape with a full width at half maxima (FWHM) of 0.55 a.u. (a.u., where 1 a.u.¼1.9929 10−24 kg m s−1). The spectra of Compton scattered photons were measured by a high purity Ge detector (Model GL0210P, Canberra). The raw data were processed through several systematic corrections like background, instrumental resolution, sample absorption, Compton scattering cross-section, etc. (Williams, 1977; Timms, 1989). To extract the true CP, the effect of double and triple scattering were simulated using the Monte Carlo procedure prescribed by Felsteiner et al. (1974). The correction for the instrumental resolution was limited to stripping off the low energy tail of the Compton spectra. True experimental CPs (high energy side), J111 and J100, after data reduction are shown in Fig. 1(b).
(i) In such type of directional CP measurements, many systematic errors arising due to background, bremsstrahlung and multiple scattering contributions, etc. which influence the individual profiles are eliminated while deducing the anisotropies in momentum densities (see, for example, Mathur and Ahuja, 2005; Ahuja et al., 2007, 2008). (ii) A use of the planar geometry enables a well defined scattering vector and minimum broadening in the scattering angle which facilitates accurate alignment (k || [hkl]) of crystalline samples in the scattering chamber. It may be noted that use of an annular 241Am isotope in the measurement of directional CPs is complicated because in the annular geometry the k may lie anywhere at the surface of a cone of semi-angle of the order of ð1=2Þð1801 −scattering angleÞ: In addition, the angular divergences of the beam cause scattering angle to be distributed on a family of cones with varying conic angles. Such variations in the k may not affect the isotropic measurements, but they introduce uncertainties in the directional measurements.
2. Experiment The CPs of CdTe were resolved along two major crystallographic directions [100] and [111] using the 100 mCi 241Am Compton spectrometer (Ahuja et al., 2007). A sketch consisting of important
Fig. 1. (a) Layout of the first-ever lowest intensity 100 mCi 241Am Compton spectrometer. Shown here are: 100 mCi 241Am disk source with active dimension 4 mm (dia.) and 1.2 mm (length) (1), sample (2), HPGe detector sensor (3), detector capsule (4), thin mylar foil to evacuate scattering chamber (5), scattering chamber made of brass (dia. 100 mm, length 300 mm and wall thickness 5 mm) (6), port for evacuation (7), lead shielding around detector and source (8) and pre-amplifier (9). Distances between source to sample and sample to detector are 57 mm and 88 mm, respectively. (b) Experimental CPs of CdTe after application of various corrections, along the [100] and [111] directions. The experimental errors are within the size of symbols used.
V. Raykar et al. / Radiation Physics and Chemistry 87 (2013) 35–39
Table 1 Parameters of Compton line shape measurements of CdTe along [100] and [111] directions using 100 mCi 241Am Compton spectrometer. Up to triple Direction Exposure Integrated time (h) Compton intensity scattering (–10 to þ 10 a.u.) ( 107)
Normalization of the profile (0–7 a.u. in e−)
[100] [111]
39.24
387 434
2.12 2.02
3.64% 3.64%
Since the binding energies of the K shell electrons in Cd and Te are quite large (26.71 keV for Cd and 31.81 keV for Te), these electrons do not contribute in the high-energy side of the Compton spectra as seen in 5d transition metals also (see for example, Pandya et al., 1997). Therefore, the experimental profiles (range pz ¼0–7 a.u.) were normalized to the free atom CP area (Biggs et al., 1975) after subtracting the contribution of 1s electrons of Cd and Te from the absolute free atom CP. Some parameters related to the present experiment are collated in Table 1.
3. Computation of electronic properties For the computation of CPs, energy bands and density of states (DOS) in CdTe, we have used the LCAO method as embodied in the CRYSTAL09 code (Dovesi et al., 2005, 2009). The LCAO includes various schemes like DFT–LDA, DFT–GGA, DFT–SOGGA, etc. In the LCAO method, one electron crystalline orbitals are the linear combination of Bloch functions which are given as m
ϕi ðr,kÞ ¼ ∑ ∑ aμi ðkÞϕμ ðr−Aμ −gÞexpðik:gÞ, μ¼1 g
ð4Þ
where the wave vector k is generating vector of irreducible representation of the group of crystal translations (g). The Bloch functions, which are the solutions of one electron equations, are built from local atoms by a linear combination of m Bloch functions built from local atom-centered (Aμ) Gaussian-type function ϕμ. In the present calculations, we have taken the exchange potential prescribed by Dirac-Slater (Dovesi et al., 2009) for the LDA and that by Wu and Cohen (2006) for the GGA. The correlation functional suggested by Perdew et al. (1996) has been used for the LDA, GGA and SOGGA. The all-electron Gaussian basis sets for Cd and Te, taken from w w w.tcm.phy.cam.ac.uk/~mdt26/basis_ sets, were optimized for the lowest total energy of the system. The self-consistent field calculations have been performed with 120 k points in the irreducible Brillouin zone (IBZ). The tolerance on the total-energy convergence in the iterative solution of the Kohn– Sham equations is set to 10−6 Hartree. To achieve a fast convergence, the BROYDEN scheme (Dovesi et al., 2009) has been used.
4. Results and discussion Fig. 2 shows the energy bands (E–k relations) and shell-wise DOS of CdTe using LCAO–DFT–GGA scheme. Except small differences in the eigen values and band gaps (2.14 eV for LCAO–DFT– LDA, 2.30 eV for GGA and 2.34 eV for SOGGA) overall shape of our energy bands are fairly in good agreement with the available data (Zakharov et al., 1994; Merad et al., 2005; Reshak, 2006; Heda et al., 2007; Hosseini, 2008; Penna et al., 2009). Since the energy bands and DOS computed using the DFT–LDA and DFT–SOGGA within the LCAO calculations are similar to the DFT–GGA based energy bands and DOS, these bands and DOS are not shown here.
37
In Fig. 2, the bands in energy range below –5.3 eV are mainly dominated by Cd-4d states with a very small contribution of Te-5sp states. Energy bands in the energy range –2.7 eV to Fermi level (EF) are formed due to hybridization of Cd-4d, Cd-5sp and Te-5sp states. The bands in the unoccupied states (above EF) are due to overlap of Cd-5sp, Cd-4d, Te-5sp and Te-4d states, as evident from the DOS curves. In Fig. 3, we have shown the anisotropies (J111–J100) in unconvoluted theoretical Compton profiles for DFT–GGA calculations along with those derived from the same theoretical directional profiles after convolution with Gaussian FWHM of 0.20 a.u. (typical resolution of synchrotron based high resolution Compton experiments) and the present instrumental resolution of 0.55 a.u. It is seen that the pz values for zeroamplitude of anisotropies is not affected by different experimental resolutions, while at other values of pz the anisotropy is smeared more in case of γ-ray measurement (0.55 a.u. resolution) than the high resolution data. Fig. 4(a, b) shows the difference between the theoretical and experimental CPs along [111] and [100] directions. It may be noted that before taking the differences, the theoretical CPs were convoluted with the instrumental resolution function of the spectrometer. It is observed that in the range momentum range pz 44.0 a.u., the theoretical profiles are close to respective experimental directional CP (Fig. 4), which depicts an accuracy of our data correction part. It is worth mentioning that in the high momentum region (pz 44.0 a.u.), CPs are dominated by the core electrons which are well defined by the free-atom wave functions. Therefore, accuracy in deriving the true CP from the raw data is ensured by a reasonable agreement between theory and experiment in the high momentum region. In the low momentum region (pz o 2.0 a.u.) the theoretical profiles show significant deviations from the experimental profiles. The overestimation of theoretical CPs in the low momentum region may be due to a poor quality of the basis sets which is an important ingredient in the present DFT calculations. Other conceivable source of inaccuracy in the density functional description within LCAO calculations is its independentparticle approximation (IPA). The IPA at the independent-particle level dictates that only some occupancies are different from zero, which are equal to one (Cooper et al., 2004). Such a deficiency has also been observed in case of directional CPs of ZnSe (Ahuja and Heda, 2007). The failure of LCAO calculations in predicting the momentum densities in the low momentum region may also be attributed to the non-inclusion of the Lam-Platzman (LP) correction for e−–e− correlation (Cooper, 1985). It is worth mentioning that the LP correction shifts momentum density below the Fermi momentum (pF) to above the pF, therefore it reduces the amplitude of the theoretical CPs near pz ¼ 0. To test the best agreement between the theory and experiment, we have also derived the χ2 which is given as, " #2 J Th ðpz Þ−J Exp ðpz Þ , sðpz Þ pz ¼ 0 7
χ2 ¼ ∑
ð5Þ
where s(pz) represents the statistical error in the experiment. On the basis of χ2 fitting, it is observed that DFT–GGA scheme is in marginally better agreement with the experimental profiles, in both directions, in comparison to other theoretical profiles. This is also evident from the inset of Fig. 4. Fig. 5 shows the directional differences (J111–J100) in the experimental and theoretical CPs of CdTe. It is seen that the position of various oscillations in the experimental anisotropies are in reasonable agreement with the theoretical data. Within the limitations of statistical precision of γ-ray experiments, the periodicity in anisotropies (J111–J100) can be interpreted in terms of the energy bands (Fig. 2). The basic concept to connect the energy bands and the momentum densities depends upon the presence of
38
V. Raykar et al. / Radiation Physics and Chemistry 87 (2013) 35–39
Fig. 2. Energy bands (E–k relation) of ZB-CdTe based on LCAO–DFT–GGA calculations. On the right hand side, the projected and total density of states are presented. The featured k points are L (1/4, 1/4, 1/4), Γ (0, 0, 0), X (0, 1/2, 0), K (3/8, 3/8, 0).
Fig. 3. Unconvoluted and convoluted (with Gaussian of FWHM 0.20 and 0.55 a.u.) anisotropy in theoretical Compton profiles (J111−J100) derived from DFT–GGA calculations. The solid lines are drawn to guide the eyes.
energy bands and degenerate states in the vicinity of EF (which leads to higher electron momentum density) in the concerned branch of the BZ (here: ΓX branch for [100] direction and ΓL branch for [111] direction). While going from Γ-X in the ΓX [100] branch (ΓX distance ¼ 0.45 a.u.), the energy band deviates to low energy side (with respect to EF) which reduces the possibility of electrons (near EF) at X point. This leads to a successive reduction of momentum density when one moves from Γ-X, which results a lower value of J100 (at pz ¼0.45 a.u.) than J111. This aspect is in tune with a positive oscillation near the momentum value of 0.5 a.u. in J111−J100 differences (Fig. 5). Other oscillations in the anisotropies are also consistent with the topology of energy bands. Due to cancellation effect of the momentum densities while taking the J111−J100 differences, few fine structures may diminish in the anisotropy curves. In the experimental anisotropic curve (Fig. 5), the oscillation near pz ¼1.2 a.u. is well explained by the bond oscillation (BO) principle. According to the BO principle (Williams, 1977), the electron momentum distributions and Compton line shapes
Fig. 4. Difference profiles between the convoluted theoretical and experimental Compton profiles of ZB-CdTe along (a) [111] direction and (b) [100] direction. In insets, difference profiles up to pz ¼ 0.5 a.u. are enlarged for clarity. Statistical errors are within the size of symbols used.
associated with the chemical bonds exhibits oscillation along the direction of bonding with a period equal to 2π divided by the bond length. From experimental anisotropy (Fig. 5), the bond length comes out to be 5.23 70.11 a.u. (equal to 2π/1.2 a.u.) which is close to theoretical value of Cd–Te bond length (5.30 a.u.) corresponding to the atomic positions Cd (0, 0, 0) and Te (1/4, 1/4, 1/4).
5. Conclusions The anisotropy in momentum densities measured between [111] and [100] directions are compared with the LCAO–DFT based
V. Raykar et al. / Radiation Physics and Chemistry 87 (2013) 35–39
Fig. 5. Anisotropy (J111−J100) in the Compton profiles of ZB-CdTe using DFT–LDA, DFT–GGA and DFT–SOGGA schemes within LCAO. Theoretical anisotropies are convoluted with the Gaussian function of FWHM 0.55 a.u. The statistical error ( 7 s) is also shown at few points.
approximations. The experimental anisotropies are found to be in a reasonable agreement with the LCAO calculations. The anisotropies in the Compton data are also interpreted in terms of energy bands. It is concluded that the absolute Compton profiles computed within DFT–GGA, DFT–LDA and DFT–SOGGA almost show similar deviations from the experimental data. The experimental bond length obtained from the anisotropy in experimental momentum densities is found to be close to that obtained from the atomic positions of Cd and Te. This confirms the potential of γ-ray Compton technique for deducing the structural parameters. High resolution measurements using synchrotron radiation based spectrometer may be helpful to confirm the present band structure calculations.
Acknowledgments The authors are thankful to the DST, New Delhi and BRNS, Mumbai for financial support. The authors acknowledge Prof. R. Dovesi for providing the CRYSTAL09 code. References Ahuja, B.L., 2010. Recent Trends in Radiation Physics Research. Himanshu Publications, New Delhi, India. Ahuja, B.L., Heda, N.L., 2007. Electron momentum density in ZnSe: theory and experiment. Radiat. Phys. Chem. 76, 921–928. Ahuja, B.L., Rathor, A., Sharma, V., Sharma, Y., Jani, A.R., Sharma, B.K., 2008. Electronic structure and Compton profiles of tungsten. Z. Naturforsch. 63a, 703–711. Ahuja, B.L., Sharma, V., Rathore, A., Jani, A.R., Sharma, B.K., 2007. Electronic structure of rhodium using Compton profiles: experiment and theory. Nucl. Instrum. Methods Phys. B, 262; 391–398. Ahuja, B.L., Sharma, M., Mathur, S., 2004. Electronic structure of liquid mercury using Compton scattering technique. Z. Naturforsch. 59a, 543–549.
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Biggs, F., Mendelsohn, L.B., Mann, J.B., 1975. Hartree–Fock Compton profiles. At. Data Nucl. Data Tables 16, 201–308. Cooper, M.J., 1985. Compton scattering and electron momentum determination. Rep. Prog. Phys. 48, 415–481, and references therein. Cooper, M.J., Mijnarends, P.E., Shiotani, N., Sakai, N., Bansil, A., 2004. X-ray Compton Scattering. Oxford Science Publications, Oxford University Press, New York, and references therein. Dovesi, R., Orlando, R., Civalleri, B., Roetti, C., Saunders, V.R., Zicovich-Wilson, C.M., 2005. CRYSTAL: a computational tool for the ab-initio study of the electronic properties of crystal. Z. Kristallogr., 220; 571–573. Dovesi, R., Saunders, V.R., Roetti, C., Orlando, R., Zicovich-Wilson, C.M., Pascale, F., Civalleri, B., Doll, K., Harrison, N.M., Bush, I.J., D'Arco, Ph., Llunell, M., 2009. CRYSTAL09 Users' Manual. University of Torino, Torino. Felsteiner, J., Pattison, P., Cooper, M.J., 1974. 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Contents lists available at SciVerse ScienceDirect
Radiation Physics and Chemistry journal homepage: www.elsevier.com/locate/radphyschem
Directional Compton profiles of CdTe using a low intensity
241
Am source
Veera Raykar, Jagrati Sahariya, B.L. Ahuja n Department of Physics, University College of Science, M.L. Sukhadia University, Udaipur 313001, Rajasthan, India
H I G H L I G H T S
Reported first-ever directional experimental Compton profile (CP) of CdTe. Interpreted experimental CP using theoretical CP within density functional theory. Reported energy bands and DOS of CdTe using DFT–GGA. Analyzed the anisotropy in CPs in terms of energy bands. Discussed potential of low intensity 241Am point source in measurement of CPs.
art ic l e i nf o
a b s t r a c t
Article history: Received 23 January 2013 Accepted 11 March 2013 Available online 18 March 2013
We report the first-ever experimental Compton profiles (CPs) of CdTe measured along [100] and [111] directions using 100 mCi 241Am Compton spectrometer. The experimental Compton data have been interpreted in terms of theoretical anisotropy in the momentum densities and also energy bands computed using density functional theory with local density approximation, generalized gradient approximation (GGA) and second order GGA. It is found that DFT–GGA based CPs are in marginally better agreement with the experimental data in comparison to other DFT based profiles. The anisotropy in CPs is found to be in tune with the energy bands. The bond length (5.2370.11 a.u.) deduced from the oscillation in the experimental anisotropy reconciles well with the theoretical Cd–Te bond length (5.30 a.u.) which establishes the role of γ-ray Compton measurements in determination of structural parameters. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Compton scattering Band structures calculations Density functional theory II–VI semiconductor
over all occupied state. The anisotropy in the directional CPs is defined as,
1. Introduction An energy spectrum of Compton scattered γ-rays provides information about the ground state electron momentum densities (EMDs) of the scatterer (Cooper, 1985; Cooper et al., 2004; Ahuja et al., 2004; Ahuja, 2010). Within the impulse approximation (Kaplan et al., 2003), the differential scattering cross-section is proportional to Compton profile (CP), J(pz), defined as projection of EMD along the scattering vector k (z-axis of Cartesian coordinate system). Mathematically, Jðpz Þ ¼ ∬ ρðpÞdpx dpy ,
ð1Þ
where ρðpÞis the electron momentum density which can be deduced from the Fourier transformation of real space wave function using the following equation, Z 2 1 ρðpÞ ¼ ψðrÞexpð−ip:rÞdr ∑ ð2Þ ð2πÞ3 occ here Ψ(r) is the electron wave functions and summation extends n
Corresponding author. Tel.: þ91 294 2423322; fax: þ91 294 2411950. E-mail address: [email protected] (B.L. Ahuja).
0969-806X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.radphyschem.2013.03.020
ΔJðpz Þ ¼ J hkl ðpz Þ–J h0 k0 l0 ðpz Þ
ð3Þ
where (hkl) and (h'k'l') are planes perpendicular to the scattering vector. Cubic zine-blende (ZB) CdTe, a II–VI semiconductor, has always been the center of attraction due to its vast technological importance in photovoltaic devices, solar panels, modern optoelectronic and spintronics devices, etc. (see for example, Sanchez-Almazan et al., 1996; Merad et al., 2005; Fleszar and Hanke, 2005; Reshak, 2006; Hosseini, 2008). Earlier, numerous studies on electronic and optical properties of CdTe have been reported by several workers. Markowski and Podgorny (1991) have reported the optical absorption of CdTe using the semi-relativistic linear muffin-tin orbital calculations with the local density approximation (LDA). The frequency dependent second- and third-harmonic response functions have been presented by Ghahramani et al. (1991), using the linear combination of Gaussian type orbitals. Huang and Ching (1993) have computed the electronic structures and linear optical dielectric functions of CdTe using the orthogonalized linear combination of atomic orbitals (LCAOs) method with the LDA. Zakharov et al. (1994) have discussed the quasiparticle band structure using
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V. Raykar et al. / Radiation Physics and Chemistry 87 (2013) 35–39
ab-initio pseudopotential with the LDA and the GW approximation for both ZB and hexagonal wurtzite (WZ) phases of CdTe, while the structural and electronic properties of both phases using the firstprinciple calculations have been reported by Wei and Zhang (2000). Using the photoluminescence spectra, Fonthal et al. (2000) have measured the temperature dependent band gap of CdTe. The electronic structure within the GW approximation has been discussed by Fleszar and Hanke (2005). Merad et al. (2005) have used full-potential augmented plane wave with the local orbitals to compute the electronic and optical properties. Reshak (2006) has employed the full-potential linearized augmented plane wave (FPLAPW) method to study the electronic properties and first- and second-harmonic generations. The optical dielectric functions, reflectivity index, extinction coefficient and reflectivity of CdTe have been reported by Hosseini (2008) using FP-LAPW with the generalized gradient approximation (GGA). The electronic, lattice dynamics and mechanical properties of ZB-ATe (A¼ Cd, Zn, Mn, Mg, Hg) and their ternary alloys have been studied by Mnasri et al. (2009). Using the DFT–LDA, an empirical pseudopotential method and a full Brillouin zone k.p method, Penna et al. (2009) have studied the electronic structure of CdTe, HgTe and Cd1−xHgxTe. The electronic structure and phase stability of MgTe, ZnTe and CdTe in different crystalline structures have been reported by Yang et al. (2009). Ouendadji et al. (2011) have investigated the structural, electronic and thermal properties of CdX (X¼S, Se, Te) using the FPLAPW method. Very recently, Sarkar et al. (2012) have studied the structural and electronic properties of CdTe nanowires with hexagonal or triangular cross-sections using a self-consistent charge density functional tight binding method. Regarding the earlier Compton profile (CP) measurements of CdTe, Heda et al. (2007) have presented the γ-ray based experimental CP of polycrystalline CdTe. In the present work, we report the directional CPs of CdTe using a low intensity planar 100 mCi 241Am Compton spectrometer. The experimental CP data have been compared with the DFT–LDA, DFT–GGA and the second order GGA (SOGGA) based momentum densities and the anisotropic behavior in the Compton line shapes is discussed in terms of energy bands. Present directional CP measurements using a point source have the following advantages:
features of the spectrometer is shown in Fig. 1(a). For the present measurements, single crystals of the ZB-CdTe having dimensions 2 mm (thickness) 15 mm (diameter) were procured from M/s Cradely Crystals, Russia. The orientations of the each crystal were confirmed with their Laue patterns and were found to be aligned to within 711. The overall instrumental resolution of the spectrometer was having a Gaussian shape with a full width at half maxima (FWHM) of 0.55 a.u. (a.u., where 1 a.u.¼1.9929 10−24 kg m s−1). The spectra of Compton scattered photons were measured by a high purity Ge detector (Model GL0210P, Canberra). The raw data were processed through several systematic corrections like background, instrumental resolution, sample absorption, Compton scattering cross-section, etc. (Williams, 1977; Timms, 1989). To extract the true CP, the effect of double and triple scattering were simulated using the Monte Carlo procedure prescribed by Felsteiner et al. (1974). The correction for the instrumental resolution was limited to stripping off the low energy tail of the Compton spectra. True experimental CPs (high energy side), J111 and J100, after data reduction are shown in Fig. 1(b).
(i) In such type of directional CP measurements, many systematic errors arising due to background, bremsstrahlung and multiple scattering contributions, etc. which influence the individual profiles are eliminated while deducing the anisotropies in momentum densities (see, for example, Mathur and Ahuja, 2005; Ahuja et al., 2007, 2008). (ii) A use of the planar geometry enables a well defined scattering vector and minimum broadening in the scattering angle which facilitates accurate alignment (k || [hkl]) of crystalline samples in the scattering chamber. It may be noted that use of an annular 241Am isotope in the measurement of directional CPs is complicated because in the annular geometry the k may lie anywhere at the surface of a cone of semi-angle of the order of ð1=2Þð1801 −scattering angleÞ: In addition, the angular divergences of the beam cause scattering angle to be distributed on a family of cones with varying conic angles. Such variations in the k may not affect the isotropic measurements, but they introduce uncertainties in the directional measurements.
2. Experiment The CPs of CdTe were resolved along two major crystallographic directions [100] and [111] using the 100 mCi 241Am Compton spectrometer (Ahuja et al., 2007). A sketch consisting of important
Fig. 1. (a) Layout of the first-ever lowest intensity 100 mCi 241Am Compton spectrometer. Shown here are: 100 mCi 241Am disk source with active dimension 4 mm (dia.) and 1.2 mm (length) (1), sample (2), HPGe detector sensor (3), detector capsule (4), thin mylar foil to evacuate scattering chamber (5), scattering chamber made of brass (dia. 100 mm, length 300 mm and wall thickness 5 mm) (6), port for evacuation (7), lead shielding around detector and source (8) and pre-amplifier (9). Distances between source to sample and sample to detector are 57 mm and 88 mm, respectively. (b) Experimental CPs of CdTe after application of various corrections, along the [100] and [111] directions. The experimental errors are within the size of symbols used.
V. Raykar et al. / Radiation Physics and Chemistry 87 (2013) 35–39
Table 1 Parameters of Compton line shape measurements of CdTe along [100] and [111] directions using 100 mCi 241Am Compton spectrometer. Up to triple Direction Exposure Integrated time (h) Compton intensity scattering (–10 to þ 10 a.u.) ( 107)
Normalization of the profile (0–7 a.u. in e−)
[100] [111]
39.24
387 434
2.12 2.02
3.64% 3.64%
Since the binding energies of the K shell electrons in Cd and Te are quite large (26.71 keV for Cd and 31.81 keV for Te), these electrons do not contribute in the high-energy side of the Compton spectra as seen in 5d transition metals also (see for example, Pandya et al., 1997). Therefore, the experimental profiles (range pz ¼0–7 a.u.) were normalized to the free atom CP area (Biggs et al., 1975) after subtracting the contribution of 1s electrons of Cd and Te from the absolute free atom CP. Some parameters related to the present experiment are collated in Table 1.
3. Computation of electronic properties For the computation of CPs, energy bands and density of states (DOS) in CdTe, we have used the LCAO method as embodied in the CRYSTAL09 code (Dovesi et al., 2005, 2009). The LCAO includes various schemes like DFT–LDA, DFT–GGA, DFT–SOGGA, etc. In the LCAO method, one electron crystalline orbitals are the linear combination of Bloch functions which are given as m
ϕi ðr,kÞ ¼ ∑ ∑ aμi ðkÞϕμ ðr−Aμ −gÞexpðik:gÞ, μ¼1 g
ð4Þ
where the wave vector k is generating vector of irreducible representation of the group of crystal translations (g). The Bloch functions, which are the solutions of one electron equations, are built from local atoms by a linear combination of m Bloch functions built from local atom-centered (Aμ) Gaussian-type function ϕμ. In the present calculations, we have taken the exchange potential prescribed by Dirac-Slater (Dovesi et al., 2009) for the LDA and that by Wu and Cohen (2006) for the GGA. The correlation functional suggested by Perdew et al. (1996) has been used for the LDA, GGA and SOGGA. The all-electron Gaussian basis sets for Cd and Te, taken from w w w.tcm.phy.cam.ac.uk/~mdt26/basis_ sets, were optimized for the lowest total energy of the system. The self-consistent field calculations have been performed with 120 k points in the irreducible Brillouin zone (IBZ). The tolerance on the total-energy convergence in the iterative solution of the Kohn– Sham equations is set to 10−6 Hartree. To achieve a fast convergence, the BROYDEN scheme (Dovesi et al., 2009) has been used.
4. Results and discussion Fig. 2 shows the energy bands (E–k relations) and shell-wise DOS of CdTe using LCAO–DFT–GGA scheme. Except small differences in the eigen values and band gaps (2.14 eV for LCAO–DFT– LDA, 2.30 eV for GGA and 2.34 eV for SOGGA) overall shape of our energy bands are fairly in good agreement with the available data (Zakharov et al., 1994; Merad et al., 2005; Reshak, 2006; Heda et al., 2007; Hosseini, 2008; Penna et al., 2009). Since the energy bands and DOS computed using the DFT–LDA and DFT–SOGGA within the LCAO calculations are similar to the DFT–GGA based energy bands and DOS, these bands and DOS are not shown here.
37
In Fig. 2, the bands in energy range below –5.3 eV are mainly dominated by Cd-4d states with a very small contribution of Te-5sp states. Energy bands in the energy range –2.7 eV to Fermi level (EF) are formed due to hybridization of Cd-4d, Cd-5sp and Te-5sp states. The bands in the unoccupied states (above EF) are due to overlap of Cd-5sp, Cd-4d, Te-5sp and Te-4d states, as evident from the DOS curves. In Fig. 3, we have shown the anisotropies (J111–J100) in unconvoluted theoretical Compton profiles for DFT–GGA calculations along with those derived from the same theoretical directional profiles after convolution with Gaussian FWHM of 0.20 a.u. (typical resolution of synchrotron based high resolution Compton experiments) and the present instrumental resolution of 0.55 a.u. It is seen that the pz values for zeroamplitude of anisotropies is not affected by different experimental resolutions, while at other values of pz the anisotropy is smeared more in case of γ-ray measurement (0.55 a.u. resolution) than the high resolution data. Fig. 4(a, b) shows the difference between the theoretical and experimental CPs along [111] and [100] directions. It may be noted that before taking the differences, the theoretical CPs were convoluted with the instrumental resolution function of the spectrometer. It is observed that in the range momentum range pz 44.0 a.u., the theoretical profiles are close to respective experimental directional CP (Fig. 4), which depicts an accuracy of our data correction part. It is worth mentioning that in the high momentum region (pz 44.0 a.u.), CPs are dominated by the core electrons which are well defined by the free-atom wave functions. Therefore, accuracy in deriving the true CP from the raw data is ensured by a reasonable agreement between theory and experiment in the high momentum region. In the low momentum region (pz o 2.0 a.u.) the theoretical profiles show significant deviations from the experimental profiles. The overestimation of theoretical CPs in the low momentum region may be due to a poor quality of the basis sets which is an important ingredient in the present DFT calculations. Other conceivable source of inaccuracy in the density functional description within LCAO calculations is its independentparticle approximation (IPA). The IPA at the independent-particle level dictates that only some occupancies are different from zero, which are equal to one (Cooper et al., 2004). Such a deficiency has also been observed in case of directional CPs of ZnSe (Ahuja and Heda, 2007). The failure of LCAO calculations in predicting the momentum densities in the low momentum region may also be attributed to the non-inclusion of the Lam-Platzman (LP) correction for e−–e− correlation (Cooper, 1985). It is worth mentioning that the LP correction shifts momentum density below the Fermi momentum (pF) to above the pF, therefore it reduces the amplitude of the theoretical CPs near pz ¼ 0. To test the best agreement between the theory and experiment, we have also derived the χ2 which is given as, " #2 J Th ðpz Þ−J Exp ðpz Þ , sðpz Þ pz ¼ 0 7
χ2 ¼ ∑
ð5Þ
where s(pz) represents the statistical error in the experiment. On the basis of χ2 fitting, it is observed that DFT–GGA scheme is in marginally better agreement with the experimental profiles, in both directions, in comparison to other theoretical profiles. This is also evident from the inset of Fig. 4. Fig. 5 shows the directional differences (J111–J100) in the experimental and theoretical CPs of CdTe. It is seen that the position of various oscillations in the experimental anisotropies are in reasonable agreement with the theoretical data. Within the limitations of statistical precision of γ-ray experiments, the periodicity in anisotropies (J111–J100) can be interpreted in terms of the energy bands (Fig. 2). The basic concept to connect the energy bands and the momentum densities depends upon the presence of
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V. Raykar et al. / Radiation Physics and Chemistry 87 (2013) 35–39
Fig. 2. Energy bands (E–k relation) of ZB-CdTe based on LCAO–DFT–GGA calculations. On the right hand side, the projected and total density of states are presented. The featured k points are L (1/4, 1/4, 1/4), Γ (0, 0, 0), X (0, 1/2, 0), K (3/8, 3/8, 0).
Fig. 3. Unconvoluted and convoluted (with Gaussian of FWHM 0.20 and 0.55 a.u.) anisotropy in theoretical Compton profiles (J111−J100) derived from DFT–GGA calculations. The solid lines are drawn to guide the eyes.
energy bands and degenerate states in the vicinity of EF (which leads to higher electron momentum density) in the concerned branch of the BZ (here: ΓX branch for [100] direction and ΓL branch for [111] direction). While going from Γ-X in the ΓX [100] branch (ΓX distance ¼ 0.45 a.u.), the energy band deviates to low energy side (with respect to EF) which reduces the possibility of electrons (near EF) at X point. This leads to a successive reduction of momentum density when one moves from Γ-X, which results a lower value of J100 (at pz ¼0.45 a.u.) than J111. This aspect is in tune with a positive oscillation near the momentum value of 0.5 a.u. in J111−J100 differences (Fig. 5). Other oscillations in the anisotropies are also consistent with the topology of energy bands. Due to cancellation effect of the momentum densities while taking the J111−J100 differences, few fine structures may diminish in the anisotropy curves. In the experimental anisotropic curve (Fig. 5), the oscillation near pz ¼1.2 a.u. is well explained by the bond oscillation (BO) principle. According to the BO principle (Williams, 1977), the electron momentum distributions and Compton line shapes
Fig. 4. Difference profiles between the convoluted theoretical and experimental Compton profiles of ZB-CdTe along (a) [111] direction and (b) [100] direction. In insets, difference profiles up to pz ¼ 0.5 a.u. are enlarged for clarity. Statistical errors are within the size of symbols used.
associated with the chemical bonds exhibits oscillation along the direction of bonding with a period equal to 2π divided by the bond length. From experimental anisotropy (Fig. 5), the bond length comes out to be 5.23 70.11 a.u. (equal to 2π/1.2 a.u.) which is close to theoretical value of Cd–Te bond length (5.30 a.u.) corresponding to the atomic positions Cd (0, 0, 0) and Te (1/4, 1/4, 1/4).
5. Conclusions The anisotropy in momentum densities measured between [111] and [100] directions are compared with the LCAO–DFT based
V. Raykar et al. / Radiation Physics and Chemistry 87 (2013) 35–39
Fig. 5. Anisotropy (J111−J100) in the Compton profiles of ZB-CdTe using DFT–LDA, DFT–GGA and DFT–SOGGA schemes within LCAO. Theoretical anisotropies are convoluted with the Gaussian function of FWHM 0.55 a.u. The statistical error ( 7 s) is also shown at few points.
approximations. The experimental anisotropies are found to be in a reasonable agreement with the LCAO calculations. The anisotropies in the Compton data are also interpreted in terms of energy bands. It is concluded that the absolute Compton profiles computed within DFT–GGA, DFT–LDA and DFT–SOGGA almost show similar deviations from the experimental data. The experimental bond length obtained from the anisotropy in experimental momentum densities is found to be close to that obtained from the atomic positions of Cd and Te. This confirms the potential of γ-ray Compton technique for deducing the structural parameters. High resolution measurements using synchrotron radiation based spectrometer may be helpful to confirm the present band structure calculations.
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