Study on momentum density in magnetic semiconductor MnTe by positron annihilation

IJASCSE Vol 1, Issue 3, 2012

www.ijascse.in Page 1

Oct. 31

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 charge densities are calculated as a function of position in the unit cell for MnTe. Wave functions are derived from pseudopotential band structure calculations and the independent particle approximation (IPM), respectively, for the electrons and the positrons. It is observed that the positron density is maximum in the open interstices and is

excluded not only, from the ion cores but also to a considerable degree from the valence bonds. Electron-positron momentum densities are calculated for (001,110) planes. The results are used to analyze the positron effects in MnTe. Keywords: band structure, positron charge density, momentum density.

I. INTRODUCTION

The family of manganese

chalcogenides (MnS, MnSe, MnTe) and pnictides (MnP, MnAs, MnSb) is of great experimental and theoretical interest because of the nonstandardmagnetic and electronic behaviour of these materials (Allen era1 1977, Motizuki and Katoh 1984, Neitzel and Barner 1985). Zinc-blende (ZB) MnTe is a prototype of an fcc Heisenberg system with strongly dominating antiferromagnetic nearestneighbour interactions. While bulk grown crystals of MnTe exhibit the hexagonal NiAs crystal structure [1], by nonequilibrium growth techniques like molecular beam epitaxy (MBE) single crystals of MnTe can be synthesized also in the ZB phase [2].

In previous works, mainly epilayers of ZB MnTe [3,4] and superlattices containing MnTe layers with a thickness of several monolayers

(MLs) [4, 5] were investigated. Recently, new heterostructures have been developed in which fractional MLs of magnetic ions are introduced digitally within a semiconductor quantum well [6]. These structures are of special interest due to the possibility to tailor the spin splitting in addition to the electronic eigenstates [6,7]. Recently several calculations were done for the ground-state properties of MnTe. The present study extends these investigations of the electronic structure of MnTe using positrons. The investigation of the electronic structure of solids using positrons occupies a place of increasing importance in solid state physics [8,9]. The recent growth in positron studies of defect trapping in semiconductors [10,11,12,13] suggests the desirability of an improved theoretical understanding of the annihilation parameters for such



Oct. 31

systems. Although there has been some attempt to study the behavior of the positron wave function in compound semiconductors [14,15,16,17] , so far no calculation has been reported on the angular correlation of positron annihilation radiation (ACPAR) lineshapes for MnTe. This has prompted us to take up such a calculation.

The theoretical calculations of the lineshapes are carried out employing the pseudopotential band model for the computation of the electron wave function. The positron wave function is evaluated under the point core approximation ( the independent particle model) . The crystal potential experienced by a positron differs from that experienced by an electron. Since we assume that there is at most one positron in the crystal at any time, there are no positron-positron interactions, i-e. exchange or corrections. Thus positron potential results from a part due to the nuclei and another part due to the electrons, both components being purely coulombic in nature.

The density functional theory (DFT) combined with the local density approximation (LDA) or with the generalized gradient approximation (GGA) [ 18,19,20] is one of the most efficient methods for electron-structure calculations, it has also been used for positrons states in bulk metals in order to determine the momentum distribution of the annihilating positron-electron pairs [21]. However those calculations are technically difficult and computationally time consuming. It is well known that electronic structure based on the DFT calculations underestimates the band gaps by as

much as 50-100%. The LDA, also overestimates the positron annihilation rate in the low-momentum regime, thus giving rise to shorter positron lifetimes than the experimental values. Moreover, the LDA overestimates the cohesive energy in electronic structure calculations, for reasons connected with the shape of the correlation hole close to the nucleus. The empirical methods [22,23,24], while simple in nature , and with the drawback that a large number of fitting parameters are required , are very accurate and produce electronic and positronic wave functions that are in good agreement with experiments. This approach was encouraged by the work of Jarlborg et al who discovered that the empirical pseudopotentials gave a better agreement with the experimental electronic structures than the first-principles calculations [25].

We remark, at this point, that while a positron in a solid state is a part of the system with important many-body interactions, the quantum independent model (IPM) is often very useful. Positron annihilation techniques have resulted in very useful information on the electron behavior in semiconductors and alloys . The positron initially with a large energy (1 MeV) rapidly loses energy in the sample mostly through ionization and excitation processes, when the positron is in thermal equilibrium with the sample, annihilation occurs with a

valence electron yielding two rays. The positron lifetime measurements yield information [26] on the electron density at the position of the positron. In addition, the angular correlation of the two γ-rays resulting from the most probable decay process can be measured. The two photons arising



Oct. 31

from the annihilation are nearly collinear because of the conservation of momentum. Since these photons are created by positron annihilation with electrons in a solid and the momentum distribution of the photons thus corresponds to that of the electrons, this gives information on the momentum distribution of the annihilating positron-electron pair . There have been experimental investigations on several semiconductors, among them are GaN, AlN [14], this work provides the complementary theoretical data to show the power of the independent particle approximation. In the case of metals or alloys, the LCW folding theorem [27] applied

to the positron annihilation is well known to give a powerful means of sampling the occupied states and gives direct information of the geometry of the Fermi surfaces. For semiconductors, however, it is not clear what kind of information could be obtained, one may expect by analogy with metals to obtain the geometry of the occupied k-space, namely the first Brillouin zone. Experimental results in this approach are not yet reported for semiconductors. In order to investigate the electronic states of bonds, we applied the LCW theorem to the positron annihilation. The details of calculations are described in section 2 of the present paper. The results for MnTe are discussed in section 3.

II-FORMALISM

In the independent particle approximation the probability of annihilation of the electron-positron pair with momentum p is given by:

2.

)exp()()()( n

occ

k

n diconst rprrrp k

..(1)

where kn is the Bloch wave function of

the valence electron with wave vector k in the n-th band, and is the Bloch

wave function of the thermalized positron . The integration is performed over the whole volume of the crystal and the summation is taken over the occupied electronic states. By

assuming that the positron is fully thermalized, we regard p as the momentum of the valence electron. The counting rate measured by the standard parallel slit apparatus is proportional to

),,(),( zyxzyx pppdppp ………(2)

We define the function )(N p by folding

)(p with respect to all reciprocal lattice

vectors G as follows:

G

Gpp )()(N

….(3)

We have exactly

rrrGpkkp k dVUdconstN n

22

)()(1

)()(

…(4)



Oct. 31

Where knU and V are the periodic parts of

the wave function of valence electron and

positron, respectively, and the r-integration

is performed over the unit cell with volume

.

In the folded function )(N p , each k -point

in the momentum space occupied by the

electrons is mapped by the -function in

the weight of the electron-positron overlap

in their densities. Corresponding to the

experimental condition, )(N p is one

dimensionally integrated along the

direction towards a fixed detector of -

rays as

),,(),( zyxzyx pppNdpppN …..(5)

the mapping of the )p,p(N zx on the yx pp

plane gives an information of the occupied

k -space .

If the positron wave function is

assumed to be constant (namely a uniform

distribution of positrons ), we obtain the

exact geometry of the occupied k -space

along the direction of integration, namely

the projection of the first Brillouin zone,

for semiconductors the real non-uniform

distribution of positrons deforms the

geometry, according to the weight of the

electron-positron overlap .

For the calculation of the weight

function, we adopted the pseudo-potential

method, where the periodic parts knU and

)(V r are expanded in terms of the plane

waves,

)exp(C)( n RrrR

k iU …..(6)

for valence electrons

G

)(i)D(V GrGr exp)(

……(7)

for positrons , Where R’s and G’s are the

reciprocal lattice vectors . The weight

function is expressed as follows :

…………(8)

the s)'(Cn Rk and s)'(D G were determined

in the following energy band calculations .

The object of each band structure

calculation, be it for an electron or a

positron, is to solve the Schrödinger

equation for a crystal potential V(r) ,

For the valence electrons we have

)()( rr kk nn EH …….(9)

pseudoVm

pH

2

2

………….(10)

where the pseudoV is the empirical pseudo-

potential determined by Kobayashi [32] .

The form factors used in our calculations

were taken from [33].

For the positron we have

)()( rr EH ……….(11)

…………(12)

vionic core +V valence electrons, where

the Vionic core is the crystal ionic

potential given by

(r)=

)( ji tRr i j

v

…………(13)

Here, in the point core approximation we

adopted

r

ZeV

2

)( r………….(14)

and the potential due to the valence

electrons is

V valence

electrons = … (15)

G'Gk'k

'

nk

22

)()()()C'()()(1

GG'RRrrrR' R G G

kk DDCdVU nn

r'-r

rr' d)(e2

Vm

pH

2

2



Oct. 31

The density of the valence electrons )(r

is evaluated by using )(n rk as 2

)(2)( n k

nk rr

…….. (16)

The wave function of the fully thermalized

positron is given, in good

approximation, by the wave function

0k,1n , i.e. the wave function at the

bottom of the positron energy band.

The two-photon momentum density )(2p

for positron annihilation is given, in the

IPM, by: 2

3

,

2 )()exp()()( rprrkp

nk

kn

n id

………(17)

where )k(n is the occupation number

equal to 1 for the occupied states and zero

for the empty states. For a periodic

potential at zero temperature Eq. (17) will

be reduced to:

)()()()(2

,

,

2GkpGkp

kn G

knn A

….(18)

where )(A k,n G are the Fourier coefficients

of the positron-electron wave function

product.

It is usual to perform a “Lock-Crisp-West”

(LCW) zone folding [27] of the various

extended zone components of )(p into

the first Brillouin zone, thus forming the

zone-reduced momentum density:

)()( iG

kn iGp

………..(19)

where iG is the i-th reciprocal lattice

vector defined within the first Brillouin

zone . Using Block’s theorem, )k(n can be

described as: 2

,, )()()()( rrr dEEconstkn knkn

n

F

….(20)

where EF is the Fermi energy and

)( ,knF EE is a step function as follows :

knF

knF

knFEE

EEEE

,

,

,0

1)(

………(21)

For the metallic material , the two photon

momentum distribution exhibits breaks at

the Fermi momentum p=k and also another

at p=k+G.

However, in the long slit angular

correlation experiment one measures a

component of the pair momentum density

as given by:

yxz dpdppN )()( 2p

…….. (22)

It contains two sets of information. The

sharp breaks in )p,p(N yx reveal the

topology and size of the Fermi surface (FS)

while the shape of ),( yx ppN reflects more

details of the wave functions of the

electron and the positron. The parameters

used for this calculation are listed in table

1, the calculated Fourier coefficients of the

valence charge densities for MnTe are

given in table 2.

III-RESULTS

In the first step of our calculations, we

have computed the Fourier coefficients of

the valence charge densities using the

empirical pseudopotential method (EPM).

This method has been proved to be largely

sufficient to describe qualitatively the

realistic charge densities. As input, we

have introduced the form factors (the

symmetric and anti-symmetric parts) and

the lattice constant for MnTe. The resulting

Fourier coefficients are used to generate

the corresponding positron wave function

using the IPM.

The positron band structure for

MnTe is displayed in figure 1, we note the

astonishing similarity with its electron

counterpart, with the exception that the

positron energy spectrum does not exhibit

a band gap. This is consistent with the fact

that these bands are all conduction bands.



Oct. 31

An oversimplified explanation of this

similarity has been presented elsewhere

[28], in terms of the electron and positron

potential. The calculated positron charge

densities in the (110) plane and along the

<111> direction are displayed in Figures

(2a ,2b), it is seen that the positron is

located in the interstitial region and that the

probability is low around the positions of

the nuclei. The positron is repelled by the

positively charged atomic cores and tend to

move in the interstitial regions. The

maximum of the charge is located at the

tetrahedral site. From a quantitative point

of view, there is a difference of charge in

the interstitial regions, the positron

distribution is more pronounced in the

neighborhood of the Te anion than in that

of the Mn cation. These differences in

profiles are immediately attributable to the

cell which contains the larger valence and

the larger ion core. We are considering the

implications of this in regard to the

propensity for positron trapping and the

anisotropies that might be expected in the

momentum densities for both free and

trapped positron states. We should point

out that the good agreement of the band

structure and charge densities were used as

an indication of both the convergence of

our computational procedure and the

correctness of the pseudopotential

approach using the adjusted form factors,

these latter as well as the lattice constant

have been adjusted to the experimental

data before the calculations.

Let us now discuss the results of

the calculated 2D-electron-positron

momentum density for MnTe, obtained by

integration of the appropriate plane along

the <110> and <001> directions (Figures 3

and 4), the first obvious observation is that

the profiles exhibit marked departures from

simple inverted parabola, suggesting that

for MnTe the electrons behave as nearly

free (NFE). At the low momentum region,

the profile along the <001> direction is

seen to be flat as observed in Ge and Si

[29]. Compared to this, the profile along

the <110> direction is sharply peaked.

However, the valleys and dips observed in

( ) p for MnTe are very shallow as

compared with those of Si and Ge. This

fact clearly tells us that the momentum

dependence of ( ) p is very much

different between elemental and compound

semiconductors. In the case of Si, the

symmetry is 7

hO which contains 48

symmetry operations including glide and

screw, in the case of MnTe, the symmetry

is lowered from 7

hO to

2

dT: the two atoms

in each unit cell are in-equivalent and the

number of symmetry operations thus

decreases from 48 to 24. Since the glide

and the screw operations are not included

in this space group, this crystal is

symmorphic. It is emphasized that the

symmetry lowering from Oh to Td revives

some of the bands which are annihilation

inactive in the case of Si. If this symmetry

lowering effect is large enough, the ratio in

the annihilation rate of the [110] line to the

[001] one becomes small since the bands

become annihilation active for both ridge

[110] and valley [001] lines. From the

calculations performed by Saito et al. [30]

in GaAs, it was found that the contribution

of these revived bands to the annihilation

rate is small. The sharp peaking along the

<110> direction and the flatness of the

peak along the <001> direction could also

be understood in terms of the contribution

of σ and π* orbitals to the ideal sp3

hybrid ones. Since the electronic

configuration of Manganese is [Ar] 4s23d5

and that of Tellerium is [Kr]5s2p44d10 the

interaction between second neighbour σ

bonds is equivalent to a π antibonding

interaction between neighbouring atoms.

The explanations are in good agreement

with an earlier analysis based on group



Oct. 31

theory [8].

The calculated electron-positron

momentum density (contour maps and

bird’s eye view of reconstructed 3D

momentum space density) in the (110-

001) plane is displayed in Figs. 5(a) and

5(b). There is a good agreement in the

qualitative feature between our results and

experimental data obtained by Berko and

co-workers for carbon [31], one can notice

that there is a continuous contribution, i.e.

there is no break, thus all the bands are

full. The contribution to the electron-

positron momentum density are at various

p=k+G. In case of elemental

semiconductors like Si, a set of bonding

electrons is composed of 3p electrons, the

distortion is expected to be observed since

both of the 2p and 3p set of electrons

possess a perfect point symmetry. But it

can be seen that for MnTe, the degree of

distortion is smaller than in Si. Compared

to this result, the number of contour lines

is smaller and the space between the

contour lines is wider in MnTe system.

Figure 6 gives the calculated LCW

folded distribution for MnTe. The

momentum distribution in the extended

zone scheme is represented by n(k) in the

reduced zone scheme. We can deduce from

the map that the electronic structure

consists entirely of full valence bands,

since the amplitude variation in the LCW

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

lattice

constant ao

Experimental

lattice

constant

ao [33]

Adjusted form

factors

Experimental

form factors

[34]

MnTe 6.3278826 6.3198220 Vs(3)=-0.20011

Vs(8)=0.00473

Vs(11)=0.07342

Va(3)=0.14135

Va(4)=0.08659

Va(11)=0.01801

Vs(3)=-0.19886

Vs(8)=0.00398

Vs(11)=0.06598

Va(3)=0.13987

Va(4)=0.08095

Va(11)=0.01455

TABLE 2: THE CALCULATED FOURIER COEFFICIENTS OF THE VALENCE CHARGE DENSITIES

FOR MNTE

G(a

2 ) Fourier coefficients (e/Ω)

for MnTe

000

111

220

311

222

400

331

8.0000 0.0000

0.2487 -0.4398

0.0484 0.0339

-0.0289 -0.0219

0.0000 -0.1498

0.0000 0.0342

-0.0122 0.0078



Oct. 31

IV.CONCLUSION

In the present paper we have

reported positronic distributions for

MnTe calculated within the

pseudopotential formalism and

employing the independent particle

model (IPM).These distributions are

found to be strongly influenced by the

actual symmetry of the orbitals taking

part in bonding, therefore, it is expected

that the positron-annihilation technique

is an effective tool and a sensitive

microscopic probe of semiconductors;

we have shown that by performing the

electron-positron momentum densities,

a deep insight into the electronic

properties can be achieved. More

importantly, because of its relatively

few assumptions, the present theory

yields a reliable single-particle

description of positron annihilation. As a

consequence it represents an excellent

starting point for a systematic many-

particle description of the process.

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Oct. 31

Figure 2a

20 40 60 80 100

-15

-10

-5

0

5

10

15

20

En

erg

y (

eV

)

K points

X W L K X

Positronic Band structure MnTe

3.6

2.9 2.3

1.6

1.6

2.3

2.9

3.6

3.64.2

4.2

4.9

4.9

4.2

4.2

5.5

5.5

4.94.9

6.1 6.1

-0.4 -0.2 0.0 0.2 0.4

-0.4

-0.2

0.0

0.2

0.4

Po

sitio

n (

arb

.un

its)

Position (arb. units)

MnTe

-0.4 -0.2 0.0 0.2 0.4

1

2

3

4

5

6

7

8

Pos

itron

cha

rge

dens

ity (

arb.

units

)

Atomic position (at.units)

Mn Te

MnTe

Figure 1: Positron energy band structure along principal

symmetry lines for MnTe

Figure 2a: The thermalized positron charge density in MnTe at the

1 point along >111< direction.

0 20 40 60 80

0

20

40

60

80

100

Mom

entu

m d

ensi

ty (a

rb. u

nits

)

angle (mrad)

MnTe

<001> direction

Figure 2b: The thermalized positron charge density in

MnTe at 1 point in the (110) plane.

Figure 3: The integrated electron-positron momentum density in MnTe along the >001<

direction.



Oct. 31

0 20 40 60 80 100 120

0

10

20

30

40

50

60

70

Mo

me

ntu

m d

en

sity (

arb

. u

nits)

Angle (mrad)

<110> direction

MnTe

0.073

0.15

0.22

0.29

0.37

0.44

0.51

0.59

0.66

10 20 30 40 50 60 70 80

20

40

60

80

100

120

An

gle

(m

rad

)Angle (mrad)

(001-110) planeMnTe

Figure 4: The integrated electron-positron

momentum density in MnTe along the >110< direction.

Figure 5a: The calculated electron -positron

momentum densities for MnTe in the (001-110)

plane Contour maps

1020

3040

5060

7080

20

40

60

80100

120

Ang

le (m

rad)

Angle (mrad)

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.32.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2 4 6 8 10 12 14 16 18 20

5

10

15

20

25

30

Py (

mra

d)

Px (mrad)

MnTe

Figure 5b: The calculated electron -positron momentum densities for MnTe in the (001-110)

plane bird’s eye view

Figure 6: The calculated electron-positron

momentum density after LCW folding in MnTe.

Education

Study on momentum density in magnetic semiconductor MnTe by positron annihilation