1. 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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2. 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
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3. 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)
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4. 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)
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5. 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
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6. 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
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7. 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
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8. 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;
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10. 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.
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11. 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
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