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A study of electronic structure of CdSe by Compton scattering technique
Physica B 405 (2010) 3537–3542
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A study of electronic structure of CdSe by Compton scattering technique M.S. Dhaka a, G. Sharma b, M.C. Mishra c, K.B. Joshi d, R.K. Kothari c, B.K. Sharma c,n a
Department of Physics, Engineering College Bikaner, Bikaner 334004, India Department of Physics, Banasthali University, Banasthali 304022, India Department of Physics, University of Rajasthan, Jaipur 302004, India d Department of Physics, University College of Science, M.L. Sukhadia University, Udaipur 313 002, India b c
a r t i c l e in fo
abstract
Article history: Received 10 February 2010 Received in revised form 27 March 2010 Accepted 13 May 2010
The electronic structure of CdSe through a Compton profile study is presented in this paper. Theoretical calculations are performed following the empirical pseudopotential method and the linear combination of atomic orbitals method. The measurement on a polycrystalline sample of CdSe is performed using 59.54 keV gamma-rays from 241Am radioisotope. The spherically averaged theoretical Compton profiles are in agreement with the measurement. The best agreement is, however, shown by the linear combination of atomic orbitals method based on the Hartree–Fock theory. The electron momentum density is also discussed in terms of theoretical anisotropies in the directional Compton profiles calculated from the linear combination of atomic orbitals method. On the basis of equal valence electron-density profiles, it is found that CdSe is less covalent compared to ZnSe. The superposition model suggests the transfer of 1.6 electrons from Cd to Se on compound formation. & 2010 Elsevier B.V. All rights reserved.
Keywords: Cadmium selenide Charge transfer Compton profile LCAO method
1. Introduction Cadmium selenide i.e. CdSe is an important II–VI group semiconductor [1]. It has applications in opto-electronic devices, nanosensing and high-efficiency solar cells [2–5]. Applications of CdSe in photoconductive materials, thin film transistors, optical-data recording and spintronics are also reported [6–10]. Structurally CdSe occurs in two phases namely the hexagonal wurtzite and the cubic zinc-blende (ZB) structures [11,12]. A number of workers reported the electronic, optical and structural properties of CdSe [13]. The band structure of CdSe has been reported by Bergstresser and Cohen [14] along different symmetry directions following the pseudopotential method. Alward and Fong [15] utilized these results in photoemission studies on CdSe. Electronic core levels in CdSe are studied by Vesely and Langer [16] using X-ray induced electron emission spectroscopy. Wei and Zhang [12] have studied the structural and electronic properties of binary semiconductors including CdSe in wurtzite and ZB phases using the first-principles calculations and observed larger band gap in wurtzite structure than the ZB counterparts. Among Compton profile studies on Cd chalcogenides, the isotropic and directional Compton profiles of wurtzite-CdS at a resolution of 0.6 a.u. have been reported by Perkkio et al. [17] using 241Am Compton spectrometer. They have also computed the directional Compton profiles along the [0 0 1], [1 0 0] and [1 1 0] directions and their spherical average using linear
n
Corresponding author. Tel.: + 91 141 2707728; fax: +91 141 2707728. E-mail address: [email protected] (B.K. Sharma).
0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.05.035
combination of atomic orbitals (LCAO) method. The anisotropies in Compton profiles of CdTe have been investigated by Nara et al. [18] using pseudopotential calculations. Recently, Heda et al. [19] reported the isotropic Compton profiles of ZB–CdS and CdTe using 137Cs Compton spectrometer and found that the isotropic experimental profiles were in agreement with the Hartree–Fock calculations. It is well known that the Compton scattering experiments are particularly sensitive to the momentum distribution of loosely bound electrons, and therefore provide an interesting way to probe the electronic structure of materials. Measurements of line shape yield the Compton profile, J(pz), which is the onedimensional projection of the electron momentum density along the scattering vector. Within impulse approximation Z þ1 Z þ1 rðpx ,py ,pz Þdpx dpy ð1Þ Jðpz Þ ¼ 1
1
where r(px, py, pz) is the electron momentum density and pz is the electron momentum component parallel to the photon scattering vector [20,21]. In order to gain insight into the electronic structure of CdSe, in the present paper, we report the theoretical and experimental Compton profile studies. The decision to measure the isotropic Compton profile was due to non-availability of large-size (diameter 18 mm, thickness 3 mm) single crystals of CdSe. To compare our experimental data, we have computed the Compton profiles using empirical pseudopotential method (EPM) and the linear combination of atomic orbitals (LCAO) method embodied in CRYSTAL code [22]. We also examine and compare the nature of
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bonding in isostructural and isovalent II–VI compounds CdSe and ZnSe by determining equal valence electrons density (EVED) profiles. Further, superposition model has been applied to estimate the charge transfer in CdSe on compound formation by performing new measurements on elemental solids Cd and Se. In this paper, unless stated, all quantities are in atomic units (a.u.) where e¼:¼ m¼1 and c¼ 137.036, giving unit momentum ¼1.9929 10 24 kg m s 1, unit energy ¼27.212 eV and unit length ¼5.2918 10 11 m.
momentum scale, a Monte Carlo simulation of the multiple scattering was performed [26]. To account for multiple scattering corrections, history of approximately 107 photons was considered. Thereafter, the profile was normalized to free atom Compton profile [27] in the momentum range from 0 to + 7 a.u. as given in Table 1. The contribution of 1s electrons was excluded as these do not contribute in the scattering. The experimentally determined absolute Compton profiles of CdSe, Cd and Se are shown in Fig. 1.
3. Theory and computational details 2. Experimental detail and data analysis
3.1. The empirical pseudopotential method 241
Am The present measurements were carried out employing gamma-ray Compton spectrometer [23]. Only a brief summary of the experiment is given here. The incident gamma-rays of 59.54 keV energy were scattered by the samples through a mean angle 16573.01. The samples having purity better than 99.99% were procured from Alfa Aesar, Johnson Matthey Co., USA. The powder samples were at room temperature (22 1C) in a circular cell with mylar windows on both the front as well as the back sides. A brass sample holder with a circular opening of 18 mm diameter, masked with lead, is used. To perform measurements, powder sample was placed in the sample holder and was covered by the lead masked brass holder to ensure that the irradiated area of the samples as seen by the detector at the sample position remains same for all materials. The sample chamber was evacuated to about 1.33 Pa of pressure with a rotary oil pump to reduce the contribution of air scattering. The scattered radiations were detected and analysed using a HPGe detector (Canberra model GL0110P) and associated electronics like spectroscopy amplifier (Canberra, 2020 model), an analogue to digital converter (Canberra, 8701 model) and a multi-channel analyzer (Canberra, S-100). To reduce pile-up contribution in the profile, the experiment was performed with 1 ms shaping time. As a confirmation of this, we measured the Compton profile of Cd metal. Our values were in close agreement with published data [24]. The spectrum was collected in a multi-channel analyzer (MCA) with 4096 channels having a channel width of 20 eV, which corresponds to about 0.03 a.u. of the momentum on electron momentum scale. The spectrometer had an overall momentum resolution of 0.6 a.u. (Gaussian FWHM), which includes the detector resolution and the geometrical broadening of the incident and the scattered radiations. The drift in the electronic system was checked by using a weak 241Am calibration source. Summary of the measurements on Cd, Se and CdSe is given in Table 1. To correct the background, measurement was made with empty sample holder only including lead mask. Thereafter, the measured background was subtracted from the raw data point by point after scaling it to the actual counting time. The measured profile was then corrected for the effect of detector response function, energy dependent absorption and scattering cross-section using computer code of the Warwick group [25]. The data reduction for the detector response function was restricted to striping-off the low energy tail of the resolution function and after converting the Compton profile into
To calculate the theoretical directional Compton profiles, the EPM has been employed. The EPM considers fitting of atomic form factors to the experimental data and one expands the crystal wave function in terms of plane waves. The detail scheme of the EPM can be found in ref. [28,29]. Calculations for CdSe were performed considering the ZB structure with experimental lattice constant 6.052 A˚ [11,12]. The potential parameters reported by Hannachi and Bouarissa [30] were considered to generate crystal potential within the EPM. To compute the k-summation within the Brillouin zone, the special points scheme of Chadi and Cohen [31] was employed. The convergence in the autocorrelation functions was ensured by observing an accuracy of 1 part in 104 at z¼30 a.u. Consequently, computation of B(z) beyond z¼30 a.u. does not improve the accuracy of the normalizing integral of the autocorrelation function and the Compton profile. The autocorrelation functions and the Compton profiles were calculated for the [1 0 0], [1 1 0], [1 1 1], [2 1 0], [2 1 1] and [2 2 1] directions. The valence Compton profile of polycrystalline CdSe was computed by taking the spherical average (6D-SPAV) of six directional Compton profiles using the formula [32] J6D-SPAV ðpz Þ ¼ ½0:1088J100 þ 0:0708J110 þ0:0162J111 þ 0:3526J210 þ 0:2877J211 þ0:1639J221
ð2Þ
The core contribution [27] was added to the spherically averaged valence profile to obtain the Compton profile of CdSe. Also, before comparing the theoretical profile with the experiment, the theoretical profile was convoluted with a Gaussian of 0.6 a.u. FWHM and normalized to free atom area in the momentum range 0 to + 7 a.u. 3.2. LCAO calculations We have also used ab initio LCAO method embodied in CRYSTAL code [22] for the computation of theoretical Compton profile of CdSe. In the LCAO technique, each crystalline orbital is built from the linear combination of Bloch functions. The Bloch functions are defined in terms of local functions constructed from the atom centred certain number of Gaussian functions. The Hamiltonian can be constructed from the Hartree–Fock (HF) and Kohn–Sham approaches [22]. Thus, one can perform the HF-LCAO and the DFT-LCAO calculations. Both HF-LCAO and
Table 1 Details of Compton profile measurements on Cd, Se and CdSe. Sample
Density (gm/cm3)
Sample thickness (mm)
Exposure time (h)
Integrated counts ( 10 to +10 a.u.)
Normalization (0–7.0 a.u.) (e-)
Cd Se CdSe
8.69 2.14a 2.29a
0.35 3.20 3.20
33 39 55
1.2 106 5.6 106 2.9 106
18.963 14.187 33.150
a
Effective density of the sample.
M.S. Dhaka et al. / Physica B 405 (2010) 3537–3542
Fig. 1. Experimental Compton profiles of CdSe, Cd and Se. Profiles of CdSe, Cd and Se are normalized to 33.150, 18.963 and 14.187 electrons, respectively in the given range of momentum.
DFT-LCAO calculations are performed for CdSe by considering the ZB structure, which belongs to the space group F4¯3m with experimental lattice parameters a¼ 6.052 A˚ [11,12]. The 976-311d631G and 976-311d51G basis sets [33] were considered for cadmium and selenium, respectively. The Kohn–Sham Hamiltonian was constructed following the prescription of Dovesi et al. [22]. As the correlation functional proposed by Perdew, Burke and Ernzerhof (PBE) [34] is one of the reasonably successful correlation functional employed to study the properties of 4d binary compounds [35], we use in our calculations to treat correlation together with the Becke’s ansatz for the exchange potential. In the CRYSTAL code, the level of numerical accuracy in evaluating the Coulomb and exchange series appearing in the SCF equations for periodic systems is controlled by five tolerance parameters [22]. The parameters estimate overlap or penetration for integral of Gaussian function on a different center, which define cutoff limits for series summation. In the present calculations, standard truncation parameters 10 6, 10 6,10 6,10 6 and 10 8 were undertaken. Following these truncation criteria to evaluate the bielectronic integrals, the calculations were carried ! out considering 256 k points in the irreducible wedge of the first Brillouin zone, taking advantage of the symmetry. To achieve selfconsistency ,25% mixing of successive cycles was considered and the self-consistency was achieved within 20 cycles.
4. Results and discussion In Table 2 we present the experimental Compton profiles for CdSe, Cd and Se. Also given here are the unconvoluted spherically averaged theoretical Compton profiles for CdSe computed from EPM and DFT-LCAO methods as discussed earlier. The present measurements on Cd and Se are in good agreement with the earlier data [24,36]. In case of CdSe, we could not compare the present measurement with the earlier data since numerical values were not reported [37]. Thus, the experimental data for CdSe reported here will be useful for other workers. In order to compare the theoretical values with the experiment, the theoretical values were convoluted with the Gaussian function and normalized to 33.15 electrons from 0 to +7 a.u. To examine
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the agreement between theory and experiment, we plot in Fig. 2 the difference profiles deduced from the convoluted theoretical profiles and the experiment. It is obvious from the figure that in the entire range, the HF and DFT schemes show similar trend and very close to each other. Differences in the momentum density computed by the two schemes are within experimental errors in the entire range. Moreover, it reveals that around pz ¼0.0 and within 0.5 rpz r1.6, the LCAO calculations mostly overestimate the profile. Thereafter the trend is reverse. The maximum differences shown by the EPM, HF-LCAO and DFT-LCAO with experimental J(0) are about 3.69%, 2.02% and 2.53%, respectively. However, in the high momentum region it can be seen that all difference curves show almost similar trend. This is the region dominated by core electrons and hence it is well expected that free atom Compton profiles of core electron give a fair description of the experiment. From w2 check and Fig. 2, it can be inferred that HF-LCAO gives the best agreement with the experiment but there are residual differences especially in the low momentum region. It hints that correlation schemes like LP correction [38] may also be useful in bringing the theory closer to the experiment below Fermi momentum. Also single crystalline measurements will be required wherein isotropic effects cancel out when anisotropies are considered. It is also worth noting that most of the theories do not correctly estimate the band gap in semiconductors indicating that reproduction of bands and bonds in semiconductors more accurately needs specific attention [39]. The larger disagreement of EPM with the experiment compared to the LCAO calculations is noteworthy. This is in contrast to our recent study on III–V semiconductor AlAs [40], where the EPM showed better description of momentum density in comparison to the LCAO calculations. This may possibly be due to the direct involvement of d electrons of Cd in bonding in CdSe. The EPM, attempted here for CdSe, considers distribution of only Cd 5s valence electrons along with 4s 4p electrons of Se. On the other hand, the LCAO considers Cd 4d electrons also besides the above valence electrons. It clearly points that the larger differences appearing around 0.2 and 0.9 a.u. may be due to Cd d electrons. It also affirms our findings on other II–VI compounds [41] that for ground state properties the role of d bands needs to be included. To examine the directional features theoretically, we consider the directional Compton profiles of CdSe along the [1 0 0], [1 1 0] and [1 1 1] directions. We present the anisotropies derived from HF-LCAO method in Fig. 3 as its isotropic profile gives the best agreement with our measurement. The two anisotropies are obtained while taking difference of the unconvoluted directional profiles. Fig. 3 shows that the [1 0 0]–[1 1 0] anisotropy is 0.2% of Jiso(0) and higher in the low momentum region. The [1 0 0]–[1 1 1] anisotropy is lower in magnitude showing opposite trend in the low momentum region. Moreover, it shows larger variation in the given momentum range. It may possibly be due to the [1 1 1] direction, which is the direction of bonding in ZB structures. The figure also depicts that anisotropies are observable only up to 4.0 a.u. Thereafter, isotropic contribution due to core electrons dominates in determining the momentum density and therefore anisotropies diminish. Measurements on single crystalline samples would be helpful to examine the directional features of bonding through anisotropies. Now, we compare the nature of bonding in isostructural and isovalent semiconducting compounds CdSe and ZnSe on the basis of the EVED profiles. In Fig. 4, we plot the experimental EVED profiles of the two compounds deduced from the experimental valence. For ZnSe, data are taken from our own earlier measurement [41]. We also plot the EVED profiles derived from EPM as it is available for the two compounds. The EVED profiles were derived by normalizing the valence electron profiles to 4.0
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Table 2 Unconvoluted theoretical Compton profiles of CdSe calculated from EPM, HF-LCAO and DFT-LCAO methods. The experimental Compton profiles of CdSe, Cd and Se are also given. All profiles are normalized to the area given in the last row. Statistical errors ( 7 s) in measurements are given at some points. pz (a.u.)
J(pz) (e/a.u.) CdSe
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
Cd
Se
EPM
HF-LCAO
DFT-LCAO
Experiment
Experiment
Experiment
13.834 13.751 13.574 13.323 12.966 12.506 12.031 11.545 10.962 9.755 8.798 8.001 7.232 6.515 5.881 3.758 2.646 1.977 1.521 1.185
14.338 14.280 14.097 13.792 13.334 12.771 12.111 11.389 10.663 9.379 8.428 7.693 7.037 6.442 5.898 3.806 2.654 1.969 1.509 1.171 Norm
14.421 14.361 14.166 13.842 13.369 12.793 12.096 11.304 10.511 9.236 8.381 7.687 7.032 6.428 5.882 3.806 2.668 1.986 1.526 1.188
13.810 7 0.064 13.835 13.710 13.365 12.840 12.255 11.693 11.143 10.556 9.374 7 0.054 8.494 7.714 7.077 6.568 6.063 7 0.042 3.869 7 0.032 2.680 7 0.026 2.050 70.022 1.568 7 0.019 1.238 7 0.016 33.150
7.658 7 0.060 7.650 7.570 7.412 7.198 6.957 6.699 6.420 6.112 5.456 7 0.050 4.923 4.555 4.244 3.941 3.587 7 0.039 2.078 7 0.028 1.472 7 0.023 1.175 7 0.020 0.949 7 0.018 0.768 7 0.016 18.963
6.575 7 0.021 6.554 6.441 6.235 5.964 5.657 5.325 4.967 4.598 3.921 7 0.016 3.367 2.988 2.788 2.624 2.474 7 0.012 1.749 7 0.010 1.177 7 0.007 0.8097 0.006 0.573 70.005 0.419 70.004 14.187
Fig. 2. Differences between convoluted theoretical Compton profiles and the experiment for CdSe. Theoretical profiles from EPM, HF-LCAO and DFT-LCAO schemes are convoluted with a Gaussian of 0.6 a.u. FWHM. Experimental errors (7 s) are also shown at some points.
electrons and scaling the resulting profiles by the Fermi momentum (pF). For CdSe and ZnSe, pF turned out to be 0.855 and 0.917 a.u., respectively using the expression (3p2n)1/3 where n is the valence electron density. According to Reed and Eisenberger [42], this scheme offers a way to understand the nature of bonding, to a first approximation, in isovalent and isostructural compounds. Fig. 4 depicts that the EVED profile corresponding to ZnSe is larger around the low momentum region compared to CdSe. As larger value around low momentum is attributed to larger covalent character, it points that CdSe is less covalent and hence more ionic than ZnSe. The larger ionicity of CdSe ( 0.737) compared to ZnSe ( 0.619) is well supported by the ionicity factors fi proposed by other workers [43,44].
Fig. 3. Anisotropies in the unconvoluted directional Compton profiles of CdSe obtained from the HF-LACO method.
The ionicity can also be examined by estimating charge transfer combining Compton experiments with the superposition model [40]. To determine the charge transfer on compound formation, experimental Compton profiles of Cd, Se and CdSe were used to obtain the valence electron profiles. We synthesized the valence profiles Jxs(pz) for CdSe through the following relation describing the superposition model Cd=e Se=e Jsx ðpz Þ ¼ ð1:0 7 xÞJExp,val ðpz Þ þ ð3:0 8xÞJExp,val ðpz Þ Cd=e-
ð3Þ
Se=e-
where JExp,val ðpz Þ and JExp, val ðpz Þ are the experimental valence profiles per electron for Cd and Se, respectively and x is the charge transfer. The positive sign in the first term indicates charge transferred from Se to Cd and the negative sign indicates charge transferred from Cd to Se. Here, we considered charge transfer
M.S. Dhaka et al. / Physica B 405 (2010) 3537–3542
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5. Conclusions In this paper, Compton profile measurement on polycrystalline CdSe is reported. The results are in better agreement with the LCAO calculations as compared to the EPM. The HF-LCAO calculation gives the best agreement with the experiment. Among the anisotropies computed, the [1 0 0]–[1 1 1] indicates larger variation in the momentum density compared to the [1 0 0]–[1 1 0]. It may possibly be due to the [1 1 1] direction, which is the direction of bonding in ZB structures. On the basis of EVED profiles, it is observed that bonding in CdSe is less covalent and more ionic than ZnSe. This is further supported by the superposition model, which suggests charge transfer of 1.6 electrons from Cd to Se on compound formation.
Acknowledgements
Fig. 4. The EVED profiles of CdSe and ZnSe (pF ¼0.855 and 0.917 a.u, respectively). The labels on top X-axis and right Y-axis belong to experiment and the bottom X-axis and left Y-axis belong to the EPM theory.
This work is financially supported by the University Grant Commission (UGC) through Emeritus Fellowship and SR/33-37/ 2007 to BKS and Post Doctoral Fellowship to RKK. MSD is thankful to Prof. M. P. Poonia, Principal, Engineering College, Bikaner for providing computational facilities. References
Fig. 5. The differences (DJ) between synthesized valence profiles Jsx ðpz Þðx ¼ 0:6, CdSe ðpz Þ of CdSe. 0:8, 1:0Þ and the experimental valence Compton profile JVal
CdSe (0.0rx r1.0) in step of 0.1. The differences (DJ ¼Jsx ðpz ÞJExp,val ðpz Þ)
between synthesized valence profiles, Jsx ðpz Þ, (x¼ 0.6, 0.8 and 1.0) and the experimental valence profile of CdSe are shown in Fig. 5. The configuration ,which gives the best overall agreement between the superposition model and the compound profile in the entire range, will obviously suggest the charge transfer. To quantify this, we computed w2, which indicates that the synthetic Cd Se profile ½ð0:2ÞJval ðpz Þ þ ð3:8ÞJval ðpz Þ corresponding to x ¼ 0.8 e shows the least differences with measured profile for CdSe. In the present measurement, the statistical error on the individual band occupancy is 70.05e . Thus, the study suggests transfer of 1.670.05 e from Cd-Se on compound formation. The large values of DJ up to 1 a.u. in Fig. 5 shows that the superposition model is not adequate to describe the reorganisation of momentum density on compound formation as the model considers only the charge, which is transferred. Nevertheless, fair amount of charge transfer indicates larger ionic character of bonding in CdSe as suggested by the EVED profiles also.
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Physica B journal homepage: www.elsevier.com/locate/physb
A study of electronic structure of CdSe by Compton scattering technique M.S. Dhaka a, G. Sharma b, M.C. Mishra c, K.B. Joshi d, R.K. Kothari c, B.K. Sharma c,n a
Department of Physics, Engineering College Bikaner, Bikaner 334004, India Department of Physics, Banasthali University, Banasthali 304022, India Department of Physics, University of Rajasthan, Jaipur 302004, India d Department of Physics, University College of Science, M.L. Sukhadia University, Udaipur 313 002, India b c
a r t i c l e in fo
abstract
Article history: Received 10 February 2010 Received in revised form 27 March 2010 Accepted 13 May 2010
The electronic structure of CdSe through a Compton profile study is presented in this paper. Theoretical calculations are performed following the empirical pseudopotential method and the linear combination of atomic orbitals method. The measurement on a polycrystalline sample of CdSe is performed using 59.54 keV gamma-rays from 241Am radioisotope. The spherically averaged theoretical Compton profiles are in agreement with the measurement. The best agreement is, however, shown by the linear combination of atomic orbitals method based on the Hartree–Fock theory. The electron momentum density is also discussed in terms of theoretical anisotropies in the directional Compton profiles calculated from the linear combination of atomic orbitals method. On the basis of equal valence electron-density profiles, it is found that CdSe is less covalent compared to ZnSe. The superposition model suggests the transfer of 1.6 electrons from Cd to Se on compound formation. & 2010 Elsevier B.V. All rights reserved.
Keywords: Cadmium selenide Charge transfer Compton profile LCAO method
1. Introduction Cadmium selenide i.e. CdSe is an important II–VI group semiconductor [1]. It has applications in opto-electronic devices, nanosensing and high-efficiency solar cells [2–5]. Applications of CdSe in photoconductive materials, thin film transistors, optical-data recording and spintronics are also reported [6–10]. Structurally CdSe occurs in two phases namely the hexagonal wurtzite and the cubic zinc-blende (ZB) structures [11,12]. A number of workers reported the electronic, optical and structural properties of CdSe [13]. The band structure of CdSe has been reported by Bergstresser and Cohen [14] along different symmetry directions following the pseudopotential method. Alward and Fong [15] utilized these results in photoemission studies on CdSe. Electronic core levels in CdSe are studied by Vesely and Langer [16] using X-ray induced electron emission spectroscopy. Wei and Zhang [12] have studied the structural and electronic properties of binary semiconductors including CdSe in wurtzite and ZB phases using the first-principles calculations and observed larger band gap in wurtzite structure than the ZB counterparts. Among Compton profile studies on Cd chalcogenides, the isotropic and directional Compton profiles of wurtzite-CdS at a resolution of 0.6 a.u. have been reported by Perkkio et al. [17] using 241Am Compton spectrometer. They have also computed the directional Compton profiles along the [0 0 1], [1 0 0] and [1 1 0] directions and their spherical average using linear
n
Corresponding author. Tel.: + 91 141 2707728; fax: +91 141 2707728. E-mail address: [email protected] (B.K. Sharma).
0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.05.035
combination of atomic orbitals (LCAO) method. The anisotropies in Compton profiles of CdTe have been investigated by Nara et al. [18] using pseudopotential calculations. Recently, Heda et al. [19] reported the isotropic Compton profiles of ZB–CdS and CdTe using 137Cs Compton spectrometer and found that the isotropic experimental profiles were in agreement with the Hartree–Fock calculations. It is well known that the Compton scattering experiments are particularly sensitive to the momentum distribution of loosely bound electrons, and therefore provide an interesting way to probe the electronic structure of materials. Measurements of line shape yield the Compton profile, J(pz), which is the onedimensional projection of the electron momentum density along the scattering vector. Within impulse approximation Z þ1 Z þ1 rðpx ,py ,pz Þdpx dpy ð1Þ Jðpz Þ ¼ 1
1
where r(px, py, pz) is the electron momentum density and pz is the electron momentum component parallel to the photon scattering vector [20,21]. In order to gain insight into the electronic structure of CdSe, in the present paper, we report the theoretical and experimental Compton profile studies. The decision to measure the isotropic Compton profile was due to non-availability of large-size (diameter 18 mm, thickness 3 mm) single crystals of CdSe. To compare our experimental data, we have computed the Compton profiles using empirical pseudopotential method (EPM) and the linear combination of atomic orbitals (LCAO) method embodied in CRYSTAL code [22]. We also examine and compare the nature of
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bonding in isostructural and isovalent II–VI compounds CdSe and ZnSe by determining equal valence electrons density (EVED) profiles. Further, superposition model has been applied to estimate the charge transfer in CdSe on compound formation by performing new measurements on elemental solids Cd and Se. In this paper, unless stated, all quantities are in atomic units (a.u.) where e¼:¼ m¼1 and c¼ 137.036, giving unit momentum ¼1.9929 10 24 kg m s 1, unit energy ¼27.212 eV and unit length ¼5.2918 10 11 m.
momentum scale, a Monte Carlo simulation of the multiple scattering was performed [26]. To account for multiple scattering corrections, history of approximately 107 photons was considered. Thereafter, the profile was normalized to free atom Compton profile [27] in the momentum range from 0 to + 7 a.u. as given in Table 1. The contribution of 1s electrons was excluded as these do not contribute in the scattering. The experimentally determined absolute Compton profiles of CdSe, Cd and Se are shown in Fig. 1.
3. Theory and computational details 2. Experimental detail and data analysis
3.1. The empirical pseudopotential method 241
Am The present measurements were carried out employing gamma-ray Compton spectrometer [23]. Only a brief summary of the experiment is given here. The incident gamma-rays of 59.54 keV energy were scattered by the samples through a mean angle 16573.01. The samples having purity better than 99.99% were procured from Alfa Aesar, Johnson Matthey Co., USA. The powder samples were at room temperature (22 1C) in a circular cell with mylar windows on both the front as well as the back sides. A brass sample holder with a circular opening of 18 mm diameter, masked with lead, is used. To perform measurements, powder sample was placed in the sample holder and was covered by the lead masked brass holder to ensure that the irradiated area of the samples as seen by the detector at the sample position remains same for all materials. The sample chamber was evacuated to about 1.33 Pa of pressure with a rotary oil pump to reduce the contribution of air scattering. The scattered radiations were detected and analysed using a HPGe detector (Canberra model GL0110P) and associated electronics like spectroscopy amplifier (Canberra, 2020 model), an analogue to digital converter (Canberra, 8701 model) and a multi-channel analyzer (Canberra, S-100). To reduce pile-up contribution in the profile, the experiment was performed with 1 ms shaping time. As a confirmation of this, we measured the Compton profile of Cd metal. Our values were in close agreement with published data [24]. The spectrum was collected in a multi-channel analyzer (MCA) with 4096 channels having a channel width of 20 eV, which corresponds to about 0.03 a.u. of the momentum on electron momentum scale. The spectrometer had an overall momentum resolution of 0.6 a.u. (Gaussian FWHM), which includes the detector resolution and the geometrical broadening of the incident and the scattered radiations. The drift in the electronic system was checked by using a weak 241Am calibration source. Summary of the measurements on Cd, Se and CdSe is given in Table 1. To correct the background, measurement was made with empty sample holder only including lead mask. Thereafter, the measured background was subtracted from the raw data point by point after scaling it to the actual counting time. The measured profile was then corrected for the effect of detector response function, energy dependent absorption and scattering cross-section using computer code of the Warwick group [25]. The data reduction for the detector response function was restricted to striping-off the low energy tail of the resolution function and after converting the Compton profile into
To calculate the theoretical directional Compton profiles, the EPM has been employed. The EPM considers fitting of atomic form factors to the experimental data and one expands the crystal wave function in terms of plane waves. The detail scheme of the EPM can be found in ref. [28,29]. Calculations for CdSe were performed considering the ZB structure with experimental lattice constant 6.052 A˚ [11,12]. The potential parameters reported by Hannachi and Bouarissa [30] were considered to generate crystal potential within the EPM. To compute the k-summation within the Brillouin zone, the special points scheme of Chadi and Cohen [31] was employed. The convergence in the autocorrelation functions was ensured by observing an accuracy of 1 part in 104 at z¼30 a.u. Consequently, computation of B(z) beyond z¼30 a.u. does not improve the accuracy of the normalizing integral of the autocorrelation function and the Compton profile. The autocorrelation functions and the Compton profiles were calculated for the [1 0 0], [1 1 0], [1 1 1], [2 1 0], [2 1 1] and [2 2 1] directions. The valence Compton profile of polycrystalline CdSe was computed by taking the spherical average (6D-SPAV) of six directional Compton profiles using the formula [32] J6D-SPAV ðpz Þ ¼ ½0:1088J100 þ 0:0708J110 þ0:0162J111 þ 0:3526J210 þ 0:2877J211 þ0:1639J221
ð2Þ
The core contribution [27] was added to the spherically averaged valence profile to obtain the Compton profile of CdSe. Also, before comparing the theoretical profile with the experiment, the theoretical profile was convoluted with a Gaussian of 0.6 a.u. FWHM and normalized to free atom area in the momentum range 0 to + 7 a.u. 3.2. LCAO calculations We have also used ab initio LCAO method embodied in CRYSTAL code [22] for the computation of theoretical Compton profile of CdSe. In the LCAO technique, each crystalline orbital is built from the linear combination of Bloch functions. The Bloch functions are defined in terms of local functions constructed from the atom centred certain number of Gaussian functions. The Hamiltonian can be constructed from the Hartree–Fock (HF) and Kohn–Sham approaches [22]. Thus, one can perform the HF-LCAO and the DFT-LCAO calculations. Both HF-LCAO and
Table 1 Details of Compton profile measurements on Cd, Se and CdSe. Sample
Density (gm/cm3)
Sample thickness (mm)
Exposure time (h)
Integrated counts ( 10 to +10 a.u.)
Normalization (0–7.0 a.u.) (e-)
Cd Se CdSe
8.69 2.14a 2.29a
0.35 3.20 3.20
33 39 55
1.2 106 5.6 106 2.9 106
18.963 14.187 33.150
a
Effective density of the sample.
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Fig. 1. Experimental Compton profiles of CdSe, Cd and Se. Profiles of CdSe, Cd and Se are normalized to 33.150, 18.963 and 14.187 electrons, respectively in the given range of momentum.
DFT-LCAO calculations are performed for CdSe by considering the ZB structure, which belongs to the space group F4¯3m with experimental lattice parameters a¼ 6.052 A˚ [11,12]. The 976-311d631G and 976-311d51G basis sets [33] were considered for cadmium and selenium, respectively. The Kohn–Sham Hamiltonian was constructed following the prescription of Dovesi et al. [22]. As the correlation functional proposed by Perdew, Burke and Ernzerhof (PBE) [34] is one of the reasonably successful correlation functional employed to study the properties of 4d binary compounds [35], we use in our calculations to treat correlation together with the Becke’s ansatz for the exchange potential. In the CRYSTAL code, the level of numerical accuracy in evaluating the Coulomb and exchange series appearing in the SCF equations for periodic systems is controlled by five tolerance parameters [22]. The parameters estimate overlap or penetration for integral of Gaussian function on a different center, which define cutoff limits for series summation. In the present calculations, standard truncation parameters 10 6, 10 6,10 6,10 6 and 10 8 were undertaken. Following these truncation criteria to evaluate the bielectronic integrals, the calculations were carried ! out considering 256 k points in the irreducible wedge of the first Brillouin zone, taking advantage of the symmetry. To achieve selfconsistency ,25% mixing of successive cycles was considered and the self-consistency was achieved within 20 cycles.
4. Results and discussion In Table 2 we present the experimental Compton profiles for CdSe, Cd and Se. Also given here are the unconvoluted spherically averaged theoretical Compton profiles for CdSe computed from EPM and DFT-LCAO methods as discussed earlier. The present measurements on Cd and Se are in good agreement with the earlier data [24,36]. In case of CdSe, we could not compare the present measurement with the earlier data since numerical values were not reported [37]. Thus, the experimental data for CdSe reported here will be useful for other workers. In order to compare the theoretical values with the experiment, the theoretical values were convoluted with the Gaussian function and normalized to 33.15 electrons from 0 to +7 a.u. To examine
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the agreement between theory and experiment, we plot in Fig. 2 the difference profiles deduced from the convoluted theoretical profiles and the experiment. It is obvious from the figure that in the entire range, the HF and DFT schemes show similar trend and very close to each other. Differences in the momentum density computed by the two schemes are within experimental errors in the entire range. Moreover, it reveals that around pz ¼0.0 and within 0.5 rpz r1.6, the LCAO calculations mostly overestimate the profile. Thereafter the trend is reverse. The maximum differences shown by the EPM, HF-LCAO and DFT-LCAO with experimental J(0) are about 3.69%, 2.02% and 2.53%, respectively. However, in the high momentum region it can be seen that all difference curves show almost similar trend. This is the region dominated by core electrons and hence it is well expected that free atom Compton profiles of core electron give a fair description of the experiment. From w2 check and Fig. 2, it can be inferred that HF-LCAO gives the best agreement with the experiment but there are residual differences especially in the low momentum region. It hints that correlation schemes like LP correction [38] may also be useful in bringing the theory closer to the experiment below Fermi momentum. Also single crystalline measurements will be required wherein isotropic effects cancel out when anisotropies are considered. It is also worth noting that most of the theories do not correctly estimate the band gap in semiconductors indicating that reproduction of bands and bonds in semiconductors more accurately needs specific attention [39]. The larger disagreement of EPM with the experiment compared to the LCAO calculations is noteworthy. This is in contrast to our recent study on III–V semiconductor AlAs [40], where the EPM showed better description of momentum density in comparison to the LCAO calculations. This may possibly be due to the direct involvement of d electrons of Cd in bonding in CdSe. The EPM, attempted here for CdSe, considers distribution of only Cd 5s valence electrons along with 4s 4p electrons of Se. On the other hand, the LCAO considers Cd 4d electrons also besides the above valence electrons. It clearly points that the larger differences appearing around 0.2 and 0.9 a.u. may be due to Cd d electrons. It also affirms our findings on other II–VI compounds [41] that for ground state properties the role of d bands needs to be included. To examine the directional features theoretically, we consider the directional Compton profiles of CdSe along the [1 0 0], [1 1 0] and [1 1 1] directions. We present the anisotropies derived from HF-LCAO method in Fig. 3 as its isotropic profile gives the best agreement with our measurement. The two anisotropies are obtained while taking difference of the unconvoluted directional profiles. Fig. 3 shows that the [1 0 0]–[1 1 0] anisotropy is 0.2% of Jiso(0) and higher in the low momentum region. The [1 0 0]–[1 1 1] anisotropy is lower in magnitude showing opposite trend in the low momentum region. Moreover, it shows larger variation in the given momentum range. It may possibly be due to the [1 1 1] direction, which is the direction of bonding in ZB structures. The figure also depicts that anisotropies are observable only up to 4.0 a.u. Thereafter, isotropic contribution due to core electrons dominates in determining the momentum density and therefore anisotropies diminish. Measurements on single crystalline samples would be helpful to examine the directional features of bonding through anisotropies. Now, we compare the nature of bonding in isostructural and isovalent semiconducting compounds CdSe and ZnSe on the basis of the EVED profiles. In Fig. 4, we plot the experimental EVED profiles of the two compounds deduced from the experimental valence. For ZnSe, data are taken from our own earlier measurement [41]. We also plot the EVED profiles derived from EPM as it is available for the two compounds. The EVED profiles were derived by normalizing the valence electron profiles to 4.0
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Table 2 Unconvoluted theoretical Compton profiles of CdSe calculated from EPM, HF-LCAO and DFT-LCAO methods. The experimental Compton profiles of CdSe, Cd and Se are also given. All profiles are normalized to the area given in the last row. Statistical errors ( 7 s) in measurements are given at some points. pz (a.u.)
J(pz) (e/a.u.) CdSe
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
Cd
Se
EPM
HF-LCAO
DFT-LCAO
Experiment
Experiment
Experiment
13.834 13.751 13.574 13.323 12.966 12.506 12.031 11.545 10.962 9.755 8.798 8.001 7.232 6.515 5.881 3.758 2.646 1.977 1.521 1.185
14.338 14.280 14.097 13.792 13.334 12.771 12.111 11.389 10.663 9.379 8.428 7.693 7.037 6.442 5.898 3.806 2.654 1.969 1.509 1.171 Norm
14.421 14.361 14.166 13.842 13.369 12.793 12.096 11.304 10.511 9.236 8.381 7.687 7.032 6.428 5.882 3.806 2.668 1.986 1.526 1.188
13.810 7 0.064 13.835 13.710 13.365 12.840 12.255 11.693 11.143 10.556 9.374 7 0.054 8.494 7.714 7.077 6.568 6.063 7 0.042 3.869 7 0.032 2.680 7 0.026 2.050 70.022 1.568 7 0.019 1.238 7 0.016 33.150
7.658 7 0.060 7.650 7.570 7.412 7.198 6.957 6.699 6.420 6.112 5.456 7 0.050 4.923 4.555 4.244 3.941 3.587 7 0.039 2.078 7 0.028 1.472 7 0.023 1.175 7 0.020 0.949 7 0.018 0.768 7 0.016 18.963
6.575 7 0.021 6.554 6.441 6.235 5.964 5.657 5.325 4.967 4.598 3.921 7 0.016 3.367 2.988 2.788 2.624 2.474 7 0.012 1.749 7 0.010 1.177 7 0.007 0.8097 0.006 0.573 70.005 0.419 70.004 14.187
Fig. 2. Differences between convoluted theoretical Compton profiles and the experiment for CdSe. Theoretical profiles from EPM, HF-LCAO and DFT-LCAO schemes are convoluted with a Gaussian of 0.6 a.u. FWHM. Experimental errors (7 s) are also shown at some points.
electrons and scaling the resulting profiles by the Fermi momentum (pF). For CdSe and ZnSe, pF turned out to be 0.855 and 0.917 a.u., respectively using the expression (3p2n)1/3 where n is the valence electron density. According to Reed and Eisenberger [42], this scheme offers a way to understand the nature of bonding, to a first approximation, in isovalent and isostructural compounds. Fig. 4 depicts that the EVED profile corresponding to ZnSe is larger around the low momentum region compared to CdSe. As larger value around low momentum is attributed to larger covalent character, it points that CdSe is less covalent and hence more ionic than ZnSe. The larger ionicity of CdSe ( 0.737) compared to ZnSe ( 0.619) is well supported by the ionicity factors fi proposed by other workers [43,44].
Fig. 3. Anisotropies in the unconvoluted directional Compton profiles of CdSe obtained from the HF-LACO method.
The ionicity can also be examined by estimating charge transfer combining Compton experiments with the superposition model [40]. To determine the charge transfer on compound formation, experimental Compton profiles of Cd, Se and CdSe were used to obtain the valence electron profiles. We synthesized the valence profiles Jxs(pz) for CdSe through the following relation describing the superposition model Cd=e Se=e Jsx ðpz Þ ¼ ð1:0 7 xÞJExp,val ðpz Þ þ ð3:0 8xÞJExp,val ðpz Þ Cd=e-
ð3Þ
Se=e-
where JExp,val ðpz Þ and JExp, val ðpz Þ are the experimental valence profiles per electron for Cd and Se, respectively and x is the charge transfer. The positive sign in the first term indicates charge transferred from Se to Cd and the negative sign indicates charge transferred from Cd to Se. Here, we considered charge transfer
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5. Conclusions In this paper, Compton profile measurement on polycrystalline CdSe is reported. The results are in better agreement with the LCAO calculations as compared to the EPM. The HF-LCAO calculation gives the best agreement with the experiment. Among the anisotropies computed, the [1 0 0]–[1 1 1] indicates larger variation in the momentum density compared to the [1 0 0]–[1 1 0]. It may possibly be due to the [1 1 1] direction, which is the direction of bonding in ZB structures. On the basis of EVED profiles, it is observed that bonding in CdSe is less covalent and more ionic than ZnSe. This is further supported by the superposition model, which suggests charge transfer of 1.6 electrons from Cd to Se on compound formation.
Acknowledgements
Fig. 4. The EVED profiles of CdSe and ZnSe (pF ¼0.855 and 0.917 a.u, respectively). The labels on top X-axis and right Y-axis belong to experiment and the bottom X-axis and left Y-axis belong to the EPM theory.
This work is financially supported by the University Grant Commission (UGC) through Emeritus Fellowship and SR/33-37/ 2007 to BKS and Post Doctoral Fellowship to RKK. MSD is thankful to Prof. M. P. Poonia, Principal, Engineering College, Bikaner for providing computational facilities. References
Fig. 5. The differences (DJ) between synthesized valence profiles Jsx ðpz Þðx ¼ 0:6, CdSe ðpz Þ of CdSe. 0:8, 1:0Þ and the experimental valence Compton profile JVal
CdSe (0.0rx r1.0) in step of 0.1. The differences (DJ ¼Jsx ðpz ÞJExp,val ðpz Þ)
between synthesized valence profiles, Jsx ðpz Þ, (x¼ 0.6, 0.8 and 1.0) and the experimental valence profile of CdSe are shown in Fig. 5. The configuration ,which gives the best overall agreement between the superposition model and the compound profile in the entire range, will obviously suggest the charge transfer. To quantify this, we computed w2, which indicates that the synthetic Cd Se profile ½ð0:2ÞJval ðpz Þ þ ð3:8ÞJval ðpz Þ corresponding to x ¼ 0.8 e shows the least differences with measured profile for CdSe. In the present measurement, the statistical error on the individual band occupancy is 70.05e . Thus, the study suggests transfer of 1.670.05 e from Cd-Se on compound formation. The large values of DJ up to 1 a.u. in Fig. 5 shows that the superposition model is not adequate to describe the reorganisation of momentum density on compound formation as the model considers only the charge, which is transferred. Nevertheless, fair amount of charge transfer indicates larger ionic character of bonding in CdSe as suggested by the EVED profiles also.
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