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Study on momentum density in magnetic semiconductor MnTe by positron annihilation

Study on momentum density in magnetic semiconductor MnTe by positron annihilation






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    Study on momentum density in magnetic semiconductor MnTe by positron annihilation Study on momentum density in magnetic semiconductor MnTe by positron annihilation Document Transcript

    • Oct. 31 IJASCSE Vol 1, Issue 3, 2012 Study on momentum density in magnetic semiconductor MnTe by positron annihilation N.Amrane and M. Benkraouda United Arab Emirates University Faculty of Science Abstract--- Electron and positron excluded not only, from the ion charge densities are calculated as cores but also to a considerable a function of position in the unit degree from the valence bonds. cell for MnTe. Wave functions are Electron-positron momentum derived from pseudopotential densities are calculated for band structure calculations and (001,110) planes. The results are the independent particle used to analyze the positron approximation (IPM), respectively, effects in MnTe. for the electrons and the positrons. It is observed that the Keywords: band structure, positron density is maximum in positron charge density, the open interstices and is momentum density. I. INTRODUCTION The family of manganese (MLs) [4, 5] were investigated. chalcogenides (MnS, MnSe, MnTe) Recently, new heterostructures have and pnictides (MnP, MnAs, MnSb) is of been developed in which fractional great experimental and theoretical MLs of magnetic ions are introduced interest because of the digitally within a semiconductor nonstandardmagnetic and electronic quantum well [6]. These structures are behaviour of these materials (Allen of special interest due to the possibility era1 1977, Motizuki and Katoh 1984, to tailor the spin splitting in addition to Neitzel and Barner 1985). Zinc-blende the electronic eigenstates [6,7]. (ZB) MnTe is a prototype of an fcc Recently several calculations were Heisenberg system with strongly done for the ground-state properties dominating antiferromagnetic of MnTe. The present study extends nearestneighbour interactions. While these investigations of the electronic bulk grown crystals of MnTe exhibit the structure of MnTe using positrons. The hexagonal NiAs crystal structure [1], by investigation of the electronic structure nonequilibrium growth techniques like of solids using positrons occupies a molecular beam epitaxy (MBE) single place of increasing importance in solid crystals of MnTe can be synthesized state physics [8,9]. The recent growth also in the ZB phase [2]. in positron studies of defect trapping in In previous works, mainly semiconductors [10,11,12,13] suggests epilayers of ZB MnTe [3,4] and the desirability of an improved superlattices containing MnTe layers theoretical understanding of the with a thickness of several monolayers annihilation parameters for such www.ijascse.in Page 1
    • Oct. 31 IJASCSE Vol 1, Issue 3, 2012 much as 50-100%. The LDA, also systems. Although there has been overestimates the positron annihilation some attempt to study the behavior of rate in the low-momentum regime, thus the positron wave function in giving rise to shorter positron lifetimes compound semiconductors than the experimental values. [14,15,16,17] , so far no calculation Moreover, the LDA overestimates the has been reported on the angular cohesive energy in electronic structure correlation of positron annihilation calculations, for reasons connected radiation (ACPAR) lineshapes for with the shape of the correlation hole MnTe. This has prompted us to take close to the nucleus. The empirical up such a calculation. methods [22,23,24], while simple in The theoretical calculations of nature , and with the drawback that a the lineshapes are carried out large number of fitting parameters are employing the pseudopotential band required , are very accurate and model for the computation of the produce electronic and positronic wave electron wave function. The positron functions that are in good agreement wave function is evaluated under with experiments. This approach was the point core approximation ( the encouraged by the work of Jarlborg et independent particle model) . The al who discovered that the empirical crystal potential experienced by a pseudopotentials gave a better positron differs from that experienced agreement with the experimental by an electron. Since we assume that electronic structures than the first- there is at most one positron in the principles calculations [25]. crystal at any time, there are no We remark, at this point, that positron-positron interactions, i-e. while a positron in a solid state is a exchange or corrections. Thus positron part of the system with important potential results from a part due to the many-body interactions, the quantum nuclei and another part due to the independent model (IPM) is often very electrons, both components being useful. Positron annihilation techniques purely coulombic in nature. have resulted in very useful The density functional theory information on the electron behavior (DFT) combined with the local density in semiconductors and alloys . The approximation (LDA) or with the positron initially with a large energy (1 generalized gradient approximation MeV) rapidly loses energy in the (GGA) [ 18,19,20] is one of the most sample mostly through ionization and efficient methods for electron-structure excitation processes, when the calculations, it has also been used for positron is in thermal equilibrium with positrons states in bulk metals in order the sample, annihilation occurs with a to determine the momentum valence electron yielding two  rays. distribution of the annihilating positron- The positron lifetime measurements electron pairs [21]. However those yield information [26] on the electron calculations are technically difficult and density at the position of the positron. computationally time consuming. It is In addition, the angular correlation of well known that electronic structure the two γ-rays resulting from the most based on the DFT calculations probable decay process can be underestimates the band gaps by as measured. The two photons arising www.ijascse.in Page 2
    • Oct. 31 IJASCSE Vol 1, Issue 3, 2012 to the positron annihilation is well from the annihilation are nearly known to give a powerful means of collinear because of the conservation sampling the occupied states and of momentum. Since these photons gives direct information of the are created by positron annihilation geometry of the Fermi surfaces. For with electrons in a solid and the semiconductors, however, it is not momentum distribution of the photons clear what kind of information could be thus corresponds to that of the obtained, one may expect by analogy electrons, this gives information on with metals to obtain the geometry of the momentum distribution of the the occupied k-space, namely the first annihilating positron-electron pair . Brillouin zone. Experimental results There have been experimental in this approach are not yet investigations on several reported for semiconductors. In order semiconductors, among them are to investigate the electronic states of GaN, AlN [14], this work provides bonds, we applied the LCW theorem to the complementary theoretical data the positron annihilation. The details of to show the power of the calculations are described in section independent particle approximation. 2 of the present paper. The results for In the case of metals or alloys, MnTe are discussed in section 3. the LCW folding theorem [27] applied II-FORMALISM In the independent particle assuming that the positron is fully approximation the probability of thermalized, we regard p as the annihilation of the electron-positron momentum of the valence electron. pair with momentum p is given by: The counting rate measured by the standard parallel slit apparatus is proportional to ( px , p y )   dpz ( px , p y , pz ) 2 occ. (p)  const    nk (r )(r ) exp( ipr )dr ………(2) n k  ..(1) We define the function N(p) by folding (p) with respect to all reciprocal lattice where  nk is the Bloch wave function of vectors G as follows: the valence electron with wave vector k N (p)   (p  G) in the n-th band, and  is the Bloch G ….(3) wave function of the thermalized positron . The integration is performed over the whole volume of the crystal and the summation is taken over the We have exactly occupied electronic states. By 1 N (p)  const   dk(k  p  G ) 2  U nk (r) V (r) dr 2  …(4) www.ijascse.in Page 3
    • Oct. 31 IJASCSE Vol 1, Issue 3, 2012 Where U nk and V are the periodic parts of 1  U nk (r) V (r) dr  Cnk (R )Cnk (R)D(G )D(G)k kG G 2 2 the wave function of valence electron and  R R G G positron, respectively, and the r-integration is performed over the unit cell with volume …………(8) . the C nk (R) s and D(G) s were determined In the folded function N(p) , each k -point in the following energy band calculations . in the momentum space occupied by the electrons is mapped by the  -function in The object of each band structure the weight of the electron-positron overlap calculation, be it for an electron or a in their densities. Corresponding to the positron, is to solve the Schrödinger experimental condition, N(p) is one equation for a crystal potential V(r) , dimensionally integrated along the For the valence electrons we have Hnk (r)  Enk (r) direction towards a fixed detector of  - …….(9) rays as p 2 H  Vpseudo N ( px , p y )   dpz N ( px , p y , pz ) 2m ………….(10) …..(5) the mapping of the N(p x , p z ) on the p x  p y where the Vpseudo is the empirical pseudo- plane gives an information of the occupied k -space . potential determined by Kobayashi [32] . If the positron wave function is The form factors used in our calculations assumed to be constant (namely a uniform were taken from [33]. distribution of positrons ), we obtain the exact geometry of the occupied k -space For the positron we have along the direction of integration, namely H(r)  E(r) ……….(11) the projection of the first Brillouin zone, for semiconductors the real non-uniform p2 distribution of positrons deforms the H   V…………(12) 2m geometry, according to the weight of the electron-positron overlap . vionic core + V valence electrons, where For the calculation of the weight the Vionic core is the crystal ionic function, we adopted the pseudo-potential potential given by method, where the periodic parts U nk and V(r) are expanded in terms of the plane  v(r  Ri  t j ) waves, (r)= i j …………(13) U (r)   Cnk exp(iRr ) R …..(6) Here, in the point core approximation we for valence electrons adopted V (r)   D(G) exp (iGr) Ze2 G ……(7) V (r )  r ………….(14) for positrons , Where R’s and G’s are the and the potential due to the valence reciprocal lattice vectors . The weight electrons is function is expressed as follows : (r )dr V valence electrons = e2  r - r … (15) www.ijascse.in Page 4
    • Oct. 31 IJASCSE Vol 1, Issue 3, 2012 where G i is the i-th reciprocal lattice The density of the valence electrons  (r ) vector defined within the first Brillouin is evaluated by using  nk (r) as zone . Using Block’s theorem, n (k ) can be (r )  2 nk (r ) 2 described as: n(k )  const  ( EF  En, k )   n, k (r)(r)dr 2 n …….. (16) k n ….(20) The wave function of the fully thermalized where EF is the Fermi energy and positron  is given, in good ( EF  En,k ) approximation, by the wave function is a step function as follows : 1 EF  En, k  n 1,k 0 , i.e. the wave function at the ( EF  En, k )   EF  En, k 0 ………(21) bottom of the positron energy band. For the metallic material , the two photon The two-photon momentum density  2 (p) momentum distribution exhibits breaks at for positron annihilation is given, in the the Fermi momentum p=k and also another IPM, by: at p=k+G. 2  (p)   n (k )  d 3r exp( ipr ) nk(r) 2 n, k ………(17) However, in the long slit angular where  n (k) is the occupation number correlation experiment one measures a component of the pair momentum density equal to 1 for the occupied states and zero as given by: for the empty states. For a periodic potential at zero temperature Eq. (17) will N ( pz )   2  (p)dpx dp y …….. (22) be reduced to: 2  (p)   n (k ) An, k (G) (p  k  G) 2 It contains two sets of information. The n, k G ….(18) sharp breaks in N(p x , p y ) reveal the where A n,k (G) are the Fourier coefficients topology and size of the Fermi surface (FS) of the positron-electron wave function N ( px , p y ) product. while the shape of reflects more It is usual to perform a “Lock-Crisp-West” details of the wave functions of the (LCW) zone folding [27] of the various electron and the positron. The parameters extended zone components of  (p) into used for this calculation are listed in table the first Brillouin zone, thus forming the 1, the calculated Fourier coefficients of the zone-reduced momentum density: valence charge densities for MnTe are n(k )   (p  G i ) given in table 2. Gi ………..(19) the lattice constant for MnTe. The resulting III-RESULTS Fourier coefficients are used to generate the corresponding positron wave function In the first step of our calculations, we using the IPM. have computed the Fourier coefficients of The positron band structure for the valence charge densities using the MnTe is displayed in figure 1, we note the empirical pseudopotential method (EPM). astonishing similarity with its electron This method has been proved to be largely counterpart, with the exception that the sufficient to describe qualitatively the positron energy spectrum does not exhibit realistic charge densities. As input, we a band gap. This is consistent with the fact have introduced the form factors (the that these bands are all conduction bands. symmetric and anti-symmetric parts) and www.ijascse.in Page 5
    • Oct. 31 IJASCSE Vol 1, Issue 3, 2012 An oversimplified explanation of this the profile along the <001> direction is similarity has been presented elsewhere seen to be flat as observed in Ge and Si [28], in terms of the electron and positron [29]. Compared to this, the profile along potential. The calculated positron charge the <110> direction is sharply peaked. densities in the (110) plane and along the However, the valleys and dips observed in <111> direction are displayed in Figures (p) for MnTe are very shallow as (2a ,2b), it is seen that the positron is compared with those of Si and Ge. This located in the interstitial region and that the fact clearly tells us that the momentum probability is low around the positions of (p) is very much the nuclei. The positron is repelled by the dependence of positively charged atomic cores and tend to different between elemental and compound move in the interstitial regions. The semiconductors. In the case of Si, the 7 maximum of the charge is located at the symmetry is O h which contains 48 tetrahedral site. From a quantitative point symmetry operations including glide and of view, there is a difference of charge in screw, in the case of MnTe, the symmetry the interstitial regions, the positron 7 2 distribution is more pronounced in the is lowered from O h to Td : the two atoms neighborhood of the Te anion than in that in each unit cell are in-equivalent and the of the Mn cation. These differences in number of symmetry operations thus profiles are immediately attributable to the decreases from 48 to 24. Since the glide cell which contains the larger valence and and the screw operations are not included the larger ion core. We are considering the in this space group, this crystal is implications of this in regard to the symmorphic. It is emphasized that the propensity for positron trapping and the symmetry lowering from Oh to Td revives anisotropies that might be expected in the some of the bands which are annihilation momentum densities for both free and inactive in the case of Si. If this symmetry trapped positron states. We should point lowering effect is large enough, the ratio in out that the good agreement of the band the annihilation rate of the [110] line to the structure and charge densities were used as [001] one becomes small since the bands an indication of both the convergence of become annihilation active for both ridge our computational procedure and the [110] and valley [001] lines. From the correctness of the pseudopotential calculations performed by Saito et al. [30] approach using the adjusted form factors, in GaAs, it was found that the contribution these latter as well as the lattice constant of these revived bands to the annihilation have been adjusted to the experimental rate is small. The sharp peaking along the data before the calculations. <110> direction and the flatness of the Let us now discuss the results of peak along the <001> direction could also the calculated 2D-electron-positron be understood in terms of the contribution momentum density for MnTe, obtained by of σ and π* orbitals to the ideal sp3 integration of the appropriate plane along hybrid ones. Since the electronic the <110> and <001> directions (Figures 3 configuration of Manganese is [Ar] 4s23d5 and 4), the first obvious observation is that and that of Tellerium is [Kr]5s2p44d10 the the profiles exhibit marked departures from interaction between second neighbour σ simple inverted parabola, suggesting that bonds is equivalent to a π antibonding for MnTe the electrons behave as nearly interaction between neighbouring atoms. free (NFE). At the low momentum region, The explanations are in good agreement with an earlier analysis based on group www.ijascse.in Page 6
    • Oct. 31 IJASCSE Vol 1, Issue 3, 2012 theory [8]. distortion is expected to be observed since The calculated electron-positron both of the 2p and 3p set of electrons momentum density (contour maps and possess a perfect point symmetry. But it bird’s eye view of reconstructed 3D can be seen that for MnTe, the degree of momentum space density) in the (110- distortion is smaller than in Si. Compared 001) plane is displayed in Figs. 5(a) and to this result, the number of contour lines 5(b). There is a good agreement in the is smaller and the space between the qualitative feature between our results and contour lines is wider in MnTe system. experimental data obtained by Berko and Figure 6 gives the calculated LCW co-workers for carbon [31], one can notice folded distribution for MnTe. The that there is a continuous contribution, i.e. momentum distribution in the extended there is no break, thus all the bands are zone scheme is represented by n(k) in the full. The contribution to the electron- reduced zone scheme. We can deduce from positron momentum density are at various the map that the electronic structure p=k+G. In case of elemental consists entirely of full valence bands, semiconductors like Si, a set of bonding since the amplitude variation in the LCW electrons is composed of 3p electrons, the folded data is merely constant. TABLE 1: THE ADJUSTED SYMMETRIC AND ANTISYMMETRIC FORM FACTORS (IN RY), AND THE LATTICE CONSTANT AO (IN ATOMIC UNITS) FOR MNTE USED IN THESE CALCULATIONS. TABLE I. compound Adjusted Experimental Adjusted form Experimental lattice lattice factors form factors constant ao constant [34] ao [33] MnTe 6.3278826 6.3198220 Vs(3)=-0.20011 Vs(3)=-0.19886 Vs(8)=0.00473 Vs(8)=0.00398 Vs(11)=0.07342 Vs(11)=0.06598 Va(3)=0.14135 Va(3)=0.13987 Va(4)=0.08659 Va(4)=0.08095 Va(11)=0.01801 Va(11)=0.01455 TABLE 2: THE CALCULATED FOURIER COEFFICIENTS OF THE VALENCE CHARGE DENSITIES FOR MNTE a Fourier coefficients (e/Ω) G( 2 ) for MnTe 000 8.0000 0.0000 111 0.2487 -0.4398 220 0.0484 0.0339 311 -0.0289 -0.0219 222 0.0000 -0.1498 400 0.0000 0.0342 331 -0.0122 0.0078 www.ijascse.in Page 7
    • Oct. 31 IJASCSE Vol 1, Issue 3, 2012 IV.CONCLUSION we have shown that by performing the electron-positron momentum densities, a deep insight into the electronic In the present paper we have properties can be achieved. More reported positronic distributions for importantly, because of its relatively MnTe calculated within the few assumptions, the present theory pseudopotential formalism and yields a reliable single-particle employing the independent particle description of positron annihilation. As a model (IPM).These distributions are consequence it represents an excellent found to be strongly influenced by the starting point for a systematic many- actual symmetry of the orbitals taking particle description of the process. part in bonding, therefore, it is expected that the positron-annihilation technique is an effective tool and a sensitive microscopic probe of semiconductors; V.REFERENCES [1] R.K. Sharma, A.C. Rastogi, G. Singh, Technol. 15 (2000) 506–510. Mater. Chem. Phys. 84 (2004) 46. [8] M.Saito, A. Oshiyama, S.Tanigawa, [2] J.B.C.E. Efrem DSa, P.A. Bhobe, K.R. Phys.Rev.B. 44 (1991) 10601 Priolkar, A. Das, S.K. Paranjpe, R. B. Prabhu, P.R. Sarode, J. Magn. Magn. [9] B.K.Panda, D.P.Mahapatra, H.C.Padhi, Mater. 285 (2005) 267. Phys.Stat.Sol.(b) 169 (1992) 89 [3] Ando K, Takahashi K, Okuda T and [10] N.Yu. Arutyunov, V.V.Emstev, Umehara M 1992 Phys. Rev. B 46 12 289 A.V.Mikhailin, and V.Yu.Davidov , Physica B 308-310 (2001) 110-113 [4] E. Przeździecka , E. Dynowska , W. Paszkowicz , W. Dobrowolski , H. Kępa , [11] M.J.Puska et al. Physical Review B, C.F. Majkrzak , T.M. Giebultowicz , E. vol. 52, p. 10947 (1995); Janik , J. Kossut. Thin Solid Films Accepted 2007 (in Press). [12] S. Lin, J. N. Sun, D. W. Gidley, J. T. Wetzel, K. A. Monnig, E. T. Ryan, S. [5] Pohlt M, Herbst W, Pascher H, Jang, D. Yu and M. S. Liang, Materials Faschinger W and Bauer G 1998 Phys. Research Society Symposium Proceeding Rev. B 57 9988. 686, A9.7, (2002). [6] N.G. Szwacki, E. Przezdziecka, E. [13] R.Krause-Rehberg and H.S.Leipner, Dynowska, P. Boguslawski, J. Kossut, Positron annihilation in semiconductors Acta Phys. Pol. A 106 (2004) 233. Springer series in solid state sciences [7] G Prechtl et al, Semicond. Sci. Vol.127 (Springer Verlag, Berlin 1999). www.ijascse.in Page 8
    • Oct. 31 IJASCSE Vol 1, Issue 3, 2012 33, Issue 3, pp. 293-296. [14] N.Benosman, N.Amrane, S.Mecabih, H.Aourag, Journal. Phys. And Chem. Of [25] Jarlborg T, Manuel A A, Peter M, Solids 61 (2000) 1727-1733 Sanchez D, Singh A K, Stephan J-L and Walker E 1988 Positron Annihilation ed L [15] Na.Amrane, B.Soudini, N.Amrane, Dorikens-Vanpraet, M Dorikens and D H.Aourag, Mater. Sci. and Eng. B40 Segers (Singapore: World Scientific) p (1996) 119-125. 266 [16] M.J. Puska and R.M. Nieminen, [26] C.Corbel, M.Stucky, P.Hautojarvi, Reviews of Modern Physics, vol. 66 K.Saarinen and P.Moser, Phys.Rev.B53 (1994) p. 841. (1988) 8192 [17] Z.Nabi, B.Abbar, N.Amrane, [27] D.G.Lock, V.H.C.Crisp and H.Aourag, Mater. Chem. And Phys. 60 R.N.West, J.Phys., F3 (1973) 561. (1999) 256-261 [28] H.Aourag, B.Khelifa, A.Belaidi, [18] W.Kohn, K.Sham, Phys.Rev.140A A.Tadjer, M.Rezki and M.Gamoudi, (1965) 1133-1138 Phys.Stat.Sol. (b) 160 (1990) 193. [19] J.P.Perdew,K.Burke, M.Ernzerhof, [29] K.Fujiwara, T,Hyodo, J.Phys.Soc. Phys.Rev.Lett. 77 (1996) 3865-3868 Jpn,35 (1973) 1133. [20] W.Liming, B.K.Panda, S.Fung and [30] M.Saito, A. Oshiyama, S.Tanigawa, C.D.Beiling, J.Phys:Cond.Matter 9 (1997) Private communication. 8147-8154. [31] W.Liu, S.Berko and A.P.Mills Jr., [21] T.Leguey, M.A.Monge, R.Pareja and Positron annihilation, Matter.Sci.Forum, J.M.Riveiro, J.Phys:Cond.Matter 7 (1995) Szombachely, 743 (1992). 6179-6185. [32] Y.F.Tsay, A.Vaidyanathan and [22] James R. Chelikowsky and Marvin L. M.S.Mitra, Phys. Rev. B19 (1979) 5422. Cohen, Phys. Rev. B 14, 2 (1976). [23] P. Friedel, M. S. Hybertsen, and M. [33] Fukuma Y, Murakami T, Asada H Schlüter, Phys. Rev. B 39, 7974–7977 and Koyanagi T 2001 Physica E 10 273 (1989) [34] A.Zaoui, M.Ferhat, J.P.Dufour, Phys. [24] ] V.G.Deibuk, Ya.I.Viklyuk and Stat.Sol. (b) 185, 163 (1994). I.M.Rarenko, Semiconductors (1999) Vol. www.ijascse.in Page 9
    • Oct. 31 IJASCSE Vol 1, Issue 3, 2012 Positronic Band structure MnTe 20 8 MnTe Positron charge density (arb.units) 15 7 10 6Energy (eV) 5 5 4 0 3 -5 2 Mn Te -10 1 -0.4 -0.2 0.0 0.2 0.4 -15  20  Atomic position (at.units) X W 40 L 60 80 K 100 X K points Figure 1: Positron energy band structure along principal Figure 2a: The thermalized positron charge density in MnTe at the symmetry lines for MnTe 1 point along >111< direction. MnTe <001> direction 6.1 6.1 5.5 0.4 4.9 100 MnTe 4.2 3.6 Momentum density (arb. units) 5.52.9 4.9 1.6 4.2 3.6 80 0.2 2.3 Position (arb.units) 60 0.0 40 -0.2 1.6 20 4.9 2.9 2.3 4.9 -0.4 4.2 0 4.2 3.6 0 20 40 60 80 -0.4 -0.2 0.0 0.2 0.4 angle (mrad) Figure 2b: ThePosition (arb. units) thermalized positron charge density in Figure 3: The integrated electron-positron momentum density in MnTe along the >001< MnTe at 1 point in the (110) plane. direction. www.ijascse.in Page 10
    • Oct. 31 IJASCSE Vol 1, Issue 3, 2012 <110> direction 120 MnTe (001-110) plane 70 MnTe 100 Momentum density (arb. units) 60 0.22 50 80 0.510.15 Angle (mrad) 40 0.37 60 0.66 30 0.29 0.59 0.44 40 0.073 20 10 20 0 0 20 40 60 80 100 120 10 20 30 40 50 60 70 80 Angle (mrad) Angle (mrad) Figure 5a: The calculated electron -positron Figure 4: The integrated electron-positron momentum densities for MnTe in the (001-110) momentum density in MnTe along the >110< direction. plane Contour maps MnTe 30 2.3 2.3 2.3 2.3 25 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 20 2.3 2.3 2.3 2.3 2.32.3 2.3 Py (mrad) 2.3 2.3 2.3 2.3 15 2.3 2.3 2.3 10 2.3 2.3 120 2.3 2.3 100 2.3 2.3 2.3 2.3 2.3 2.3 80 2.3 2.3 60 5 2.3 2.3 2.3 d) 10 2.3 2.3 20 ra 30 40 (m 40 2.3 2.3 2.3 50 20 e Angle 60 gl 70 2 Figure 6:6 calculated electron-positron 18 20 4 The 8 10 12 14 16 An (mra 80 Figure 5b: The d) calculated electron -positron momentum density after LCW folding in MnTe. momentum densities for MnTe in the (001-110) P (mrad) x plane bird’s eye view www.ijascse.in Page 11