Paper MnTe by IJASCSE

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									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

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                                                         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

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                                                                       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)


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          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)
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                                                                               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
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          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

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          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


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                     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;


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          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                                                                                                                                                             6
Energy (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      4     The 8 10 12 14 16
                                                                                                                                         Figure 6:6 calculated electron-positron 18 20
                                                                                              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




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