Positron annihilation in boron nitride
Electron and positron charge densities are calculated as a function of position in the unit cell for boron nitride. Wave functions are derived from pseudopotential band structure calculations and the independent particle approximation (IPM), respectively, for electrons and positrons. It is observe...
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| Zitieren: | Positron annihilation in boron nitride / N. Amrane // Condensed Matter Physics. — 2006. — Т. 9, № 4(48). — С. 763–772. — Бібліогр.: 42 назв. — англ. |
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irk-123456789-1214522017-06-15T03:05:30Z Positron annihilation in boron nitride Amrane, N. Electron and positron charge densities are calculated as a function of position in the unit cell for boron nitride. Wave functions are derived from pseudopotential band structure calculations and the independent particle approximation (IPM), respectively, for electrons and positrons. It is observed that the positron density is maximum in the open interstices and is excluded not only from ion cores but also to a considerable degree from valence bonds. Electron-positron momentum densities are calculated for (001,110) planes. The results are used in order to analyse the positron effects in BN. Електроннi i позитроннi зарядовi густини є обчисленi як функцiї положення одиничної комiрки для нiтриду бору. Хвильовi функцiї отриманi, вiдповiдно, з обчислень псевдопотенцiальної зонної структури i наближених незалежних частинок (IPM) для електронiв i позитронiв. Спостережено, що позитронна густина є максимальною у вiдкритих щiлинах, i є обмеженою не тiльки iонним кором, але також, значною мiрою, валентними зонами. Густини електрон-позитронного моменту обчисленi для площин (001,110). Результати використанi для аналiзу позитронних ефектiв у BN. 2006 Article Positron annihilation in boron nitride / N. Amrane // Condensed Matter Physics. — 2006. — Т. 9, № 4(48). — С. 763–772. — Бібліогр.: 42 назв. — англ. 1607-324X PACS: 78.70.Bj, 71.15.Dx DOI:10.5488/CMP.9.4.763 http://dspace.nbuv.gov.ua/handle/123456789/121452 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Electron and positron charge densities are calculated as a function of position in the unit cell for boron nitride.
Wave functions are derived from pseudopotential band structure calculations and the independent particle
approximation (IPM), respectively, for electrons and positrons. It is observed that the positron density is maximum
in the open interstices and is excluded not only from ion cores but also to a considerable degree from
valence bonds. Electron-positron momentum densities are calculated for (001,110) planes. The results are
used in order to analyse the positron effects in BN. |
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Amrane, N. |
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Amrane, N. Positron annihilation in boron nitride Condensed Matter Physics |
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Amrane, N. |
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Amrane, N. |
| title |
Positron annihilation in boron nitride |
| title_short |
Positron annihilation in boron nitride |
| title_full |
Positron annihilation in boron nitride |
| title_fullStr |
Positron annihilation in boron nitride |
| title_full_unstemmed |
Positron annihilation in boron nitride |
| title_sort |
positron annihilation in boron nitride |
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Інститут фізики конденсованих систем НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/121452 |
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Positron annihilation in boron nitride / N. Amrane // Condensed Matter Physics. — 2006. — Т. 9, № 4(48). — С. 763–772. — Бібліогр.: 42 назв. — англ. |
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Condensed Matter Physics |
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AT amranen positronannihilationinboronnitride |
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2025-07-08T19:55:45Z |
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Condensed Matter Physics 2006, Vol. 9, No 4(48), pp. 763–772
Positron annihilation in boron nitride
N.Amrane
United Arab Emirates University Faculty of Science, Physics Department,
Al Ain P.O.Box17551 United Arab Emirates
Received January 24, 2006, in final form May 20, 2006
Electron and positron charge densities are calculated as a function of position in the unit cell for boron nitride.
Wave functions are derived from pseudopotential band structure calculations and the independent particle
approximation (IPM), respectively, for electrons and positrons. It is observed that the positron density is max-
imum in the open interstices and is excluded not only from ion cores but also to a considerable degree from
valence bonds. Electron-positron momentum densities are calculated for (001,110) planes. The results are
used in order to analyse the positron effects in BN.
Key words: positron annihilation, momentum density, angular correlation
PACS: 78.70.Bj, 71.15.Dx
1. Introduction
Boron nitride can be synthesized in hexagonal (h-BN) or in cubic form (c-BN), according to
the growth conditions. Both of them present numerous attractive properties, such as high chemical
inertness, low density, wide range of transparency, and good thermal conductivity. However, the
hexagonal form is mechanically softer than the cubic phase, and above all is anisotropic. This
anisotropy is evidenced in all its physical properties, such as mechanical strength, electrical resi-
stivity, thermal conductivity [1] or optical indices [2]. Therefore, in thin films growth, the control
or the determination of the c-axis orientation is of major importance. In the publications it is
reported that this axis can be oriented in the plane of the growing film as well as in all the di-
rections up to the normal of the sample. This orientation can be determined by interpreting the
infrared transmittance spectroscopy measurements performed at non-normal incidence, based on
the optical model of an uniaxial medium developed by Schubert et al. [2].
Boron nitride is a wide band-gap (gap ∼5.8 eV) III-nitride semiconductor which is transparent
in a wide range of wavelengths, from the ultraviolet to the infrared region. It has been widely
used as a coating material for tribological applications and as the passivation layer and insulating
dielectric in microelectronic devices. In fact, since a piezoelectric surface layer like AlN or ZnO is
often required to generate surface acoustic waves, h-BN belonging to the 6 mm symmetry class
can be an excellent substrate for acousto-optic devices. Its high velocity of sound suggests possible
applications in surface acoustic wave devices (SAW).
Most of all, its low chemical re-activity with transition-metal alloys makes it an unequalled
substitute to diamond for the machining of ferrous materials [3–6]. So, the synthesis of cubic boron
nitride has been motivated.
It is well known that cubic boron nitride can be synthesized under high pressures and high
temperatures from the graphite-like hexagonal modification (hBN) or in the pressure of a suitable
catalyst-solvent. However, even with the use of catalysts, the pressure is higher than 4.0 GPa [7–9].
Besides, it has been found that cBN formation could be obtained at pressures down to 2.5 GPa
using amorphous boron nitride as starting materials [10]. Recently, Hao et al. reported that cBN
was synthesized at low pressure.
For the past few years, the ease of fabrication of this material using new processes such as
plasma enhanced chemical vapor deposition (PECVD), reactive ion plating (RIP), sputtering, ion
c© N.Amrane 763
N.Amrane
beam deposition techniques [11,12] has opened a large area of applications. Materials such as ZnO,
AlN and sputtered LiNbO3 have been widely investigated for a long time.
Recently several calculations were done for the ground-state properties of both cBN and hBN
[13–15]. The present study extends these investigations of the electronic structure of BN using
positrons.
The investigation of the electronic structure of solids using positrons occupies a place of in-
creasing importance in solid state physics [16,17]. The recent growth in positron studies of defect
trapping in semiconductors [18–21] suggests the desirability of an improved theoretical under-
standing of the annihilation parameters for such systems.
Although there has been some attempt to study the behavior of the positron wave function
in compound semiconductors [22–25], so far no calculations have been reported on the angular
correlation of positron annihilation radiation (ACPAR) lineshapes for BN. This has prompted us
to take up such calculations.
The theoretical calculations of the lineshapes are carried out employing a pseudopotential band
model for 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 is partly caused by the nuclei and partly by 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) [26–28] is one of the most efficient techniques
of electron-structure calculations. It has also been used for positron states in bulk metals in order
to determine the momentum distribution of the annihilating positron-electron pairs [29]. 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 positron lifetimes shorter 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 [30–32], 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 [33].
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 initially large energy positron (1 MeV) rapidly loses energy in
the sample mostly through ionization and excitation processes, when the positron is in thermal
equilibrium with the sample, and annihilation occurs with a valence electron yielding two γ rays.
The positron lifetime measurements yield information [34] on the electron density at the position of
the positron. Moreover, the angular correlation of the two γ-rays resulting from the most probable
decay process can be measured. The two photons arising from the annihilation are nearly collinear
due to 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 provides information on the momentum distribution of the annihilating positron-
electron pair. There have been experimental investigations on several semiconductors, such as GaN,
AlN [18]. This work provides complementary theoretical data to show the power of the independent
particle approximation.
In the case of metals or alloys, the LCW folding theorem [35] 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
764
Positron annihilation in boron nitride
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 BN are discussed in section 3.
2. Calculations
The electron and positron wave functions are essential ingredients in the calculation of the
electron-positron k-space densities. We therefore focus our attention on the evaluation of electron
wave function derived from band structure calculations. One of the central problems in the band
theory of solids is to find the propagating solution of a Schrodinger equation
{
p2
2m
+ V (r)
}
ψnk(r) = Enkψnk(r) (1)
in which the potential has the periodicity of the lattice. Exact solutions of this problem are in
general not possible, and so a number of approximation methods have been used so far.
In our case, we have used the empirical pseudo-potential method (EPM), which involves a direct
fit of the atomic form factors V(G) to the experimental band structure. Therefore, the first step
in this calculation is to choose the best possible set of form factors, which will allow us to obtain
the theoretical band structure. The experimentally known energy gaps at Γ, X and L points of the
Brillouin zone are taken as a reference.
Let us define our empirical pseudo-potential parameters (EPP) of a semiconductor as a super-
position of the pseudo-atomic potential of the form V (r) = VL(r) + VNL(r), where VL and VNLare
local and nonlocal parts, respectively. In these calculations we have omitted the nonlocal part. We
regard the Fourier components of VL(r) as the EPP local parameters.
We determine the EPP parameters by a nonlinear least squares method, in which all the param-
eters are simultaneously optimized under a defined criterion of minimizing the root-mean square
(rms) deviation. The experimental electronic band structure data are used at normal pressure.
Our nonlinear least squares method requires that the rms deviation of the calculated level
spacing (LS) from the experimental ones defined by
δ =
[
∑
(
∆E(i,j)
)2
m−N
]1/2
(2)
should be minimum.
∆E(i,j) = E(i,j)
exp − E
(i,j)
calc , (3)
where E
(i,j)
exp and E
(i,j)
calc are the observed and calculated LSs between the i-th state at the wave
vector k = ki and the j-th state at k = kj , respectively, in the m chosen pairs (i,j). N is the number
of the EPP parameters. The calculated energies given by solving the EPP secular equation depend
nonlinearly on the EPP parameters.
The valence electron density ρ(r)is defined as
ρe(r) = 2
∑
n
∑
k
|Ψnk(r)|
2
, (4)
where Ψnk is the wave function of the valence electron with the wave vector k in the n-th
valence band. The summations are taken over the occupied states (we have used about 1200
k-points).
The pseudopotential method used here starts with that used by Bergstresser and Cohen in their
well known treatment of cubic binary compounds [36]. The pseudopotential Hamiltonian contains
765
N.Amrane
an effective potential which is expanded as Fourier series in a reciprocal lattice space. For a binary
compound the expansion is written in two parts which are symmetric and antisymmetric with
respect to an interchange of two atoms about their midpoint:
V (r) =
∑
G
[
SS(G)V S
G + SA(G)V A
G
]
exp(iG.r), (5)
G are the reciprocal lattice vectors. The structure and form factors are given by:
SS(G) = cosG.τ, SA(G) = sinG.τ, (6)
V
S,A
G (1, 2) =
1
Ω1,2
∫
1
2
[V1(r) ∓ V2(r)]1,2 exp(−iG.r)d3r. (7)
Here τ = a/8(111) is half the vector between the two atoms contained in the unit cell, a is the
lattice parameter and V S,A
G (1, 2) are the pseudopotential form factors of the individual atoms.
This can be simplified through the use of transferability approximation for atomic pseudopo-
tentials as discussed by Phillips [37], by writing:
VG(J) =
1
Ωj
∫
Vi(r) exp(−iG.r)d3r, (8)
where Ωj is the volume per atom of the monoatomic solid consisting of atoms of type j.
The charge density is eρ(r), where e is the electron charge. This has the Fourier transform
given by
ρ(G) =
1
Ω
∫
ρ(r)eiGrd3r. (9)
We follow the approach of Aourag et al. [23] to evaluate the positron wave function; the total
positron potential can be expressed as
Vp(r) = Vi(r) + Vc(r) + Vep(r), (10)
where Vi(r), Vc(r), and Vep(r) are the ionic, Coulomb, and electron-positron correlation potentials,
respectively.
Since Vi(r) is a periodic potential and the zincblende structure involves non-primitive lattice
translations, it is expressed as
Vi(r) =
∑∑
vi(r −Rn − Ti), (11)
where Rn denotes the set of all Bravais lattices vectors, it consists of all points with position vectors
Rn,where the n’s range all through positive and negative integer values and Ti is a non-primitive
vector of a two-atom basis. The set of vectors T describes the so called translational symmetry of
a lattice.
The nuclear charge can be expressed as a δ-function at the origin of each cell and in the core
approximation:
vi(r) =
Ze2
r
. (12)
On the other hand, the electron-positron coulomb potential is expressed as:
Vc(r) = −2
∫
ρe(r
′)d3r
|r − r′| , (13)
where ρe(r) is the charge density of the valence electrons for the binary semiconductor. ρe(r) has
been calculated using the empirical pseudopotential scheme (EPM). The electron-positron potential
is a slow function of the electron density (only one positron). It is generally flat in the interstitial
region and swamped by the Vi(r) and Vc(r) in the ion core region. Hence it is not considered here.
This gives good results.
766
Positron annihilation in boron nitride
The positron density is evaluated by using Ψnk(r) as in equation (4). In our calculations, a
fully thermalized positron is assumed to be, in good approximation, at the bottom of the positron
band with k = 0 and n = 1
ρp(r) = |Φn=1,k=0|2 . (14)
The EPM scheme employs an extended plane wave basis, where the pseudo-wave function is
expressed as an expansion over an arbitrary large number of plane waves,
Ψnk(r) =
∑
G
ψn(k +G)ei(k+G).r. (15)
In our calculations the expansion is cut off at 14 Ry, thus including 136 plane waves.
The wave function of the thermalized positron in the reciprocal space is given by:
Ψ+(r) =
1√
Ω
∑
A(G)eiGr. (16)
The coefficients A(G) are found out by solving the secular equation for the positron. The
positron can be described by a band model with one positron per unit cell independent-particle
model (IPM). The probability of annihilating the e− − e+pair with momentum p is proportional
to the pair momentum density:
ρ2γ(p) =
∑
n,k
ηn(k)
∣
∣
∣
∣
∫
d3re−iprΨn,k(r).Ψ+(r)
∣
∣
∣
∣
2
, (17)
where Ψn,k(r) and Ψ+(r) are the electron and positron Block wave functions and ηn(k) is the
occupation number. In the long-slit angular correlation experiment one measures a component of
the pair momentum density as given by
N(pz) =
∫∫
ρ2γ(p)dpxdpy . (18)
It is usual to perform a “Lock-Crisp-West” (LCW) zone folding [35] of the various extended
zone components of ρ(p) into the first Brillouin zone, thus forming the zone-reduced momentum
density:
n(k) =
∑
Gi
ρ(p+Gi), (19)
where Gi is the i-th reciprocal lattice vector defined within the first Brillouin zone. Using Block’s
theorem, n(k)can be described as:
n(k) = const
∑
n
θ (EF − En,k)
∫
|ψn,k(r)φ(r)dr|2, (20)
where EF is the Fermi energy and θ(EF − En,k) is a step function as follows:
θ(EF − En,k) =
{
1, EF 6 En,k ,
0, EF > En,k .
(21)
The parameters used for this calculation are listed in table 1, the calculated Fourier coefficients
of the valence charge densities for BN are given in table 2.
3. 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 proved to be
quite sufficient to qualitatively describe the realistic charge densities. As input, we have introduced
767
N.Amrane
Table 1. The adjusted symmetric and antisymmetric form factors (in Ry), and the lattice con-
stant ao (in atomic units) for BN used in these calculations.
compound Adjusted lattice
constant ao
Experimental
lattice constant
ao [42]
Adjusted form
factors
Experimental form
factors [42]
BN 6.8333 6.8241 Vs(3)=–0.7706
Vs(8)=0.1985
Vs(11)=0.1330
Va(3)=0.2912
Va(4)=0.1250
Va(11)=0.0251
Vs(3)=–0.755
Vs(8)=0.182
Vs(11)=0.133
Va(3)=0.1755
Va(4)=0.1250
Va(11)=0.0251
Table 2. The calculated Fourier coefficients of the valence charge densities for BN.
G(a/2π
) Fourier coefficients (e/Ω) for BN
000
111
220
311
222
400
331
8.0000
0.2356
0.0112
–0.0201
0.0000
0.0000
–0.0037
0.0000
–0.4266
0.0189
–0.0431
–0.1628
0.0160
0.0031
Figure 1. Positron energy band structure along principal symmetry lines for BN.
the form factors (the symmetric and antisymmetric parts) and the lattice constant for BN. The
resulting Fourier coefficients are used to generate the corresponding positron wave function using
the IPM.The positron band structure for BN is displayed in figure 1. We note the astonishing similarity
to 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. An oversim-
plified explanation of this similarity has been presented elsewhere [38], 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 the quantitative point of view, there is a
768
Positron annihilation in boron nitride
(a) (b)
Figure 2. (a) The thermalized positron charge density in BN at the Γ1 point along 〈111〉 direc-
tion. (b) The thermalized positron charge density in BN atΓ1 point in the (110) plane.
difference of charge in the interstitial regions, the positron distribution being more pronounced in
the neighborhood of the N anion than in that of the B cation. These differences in profiles are
immediately attributable to the cell which contains a larger valence and a 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 a good agreement of the band structure and charge densities was
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 prior to the calculations.
Figure 3. The integrated electron-positron
momentum density in BN along the 〈001〉
direction.
Figure 4. The integrated electron-positron
momentum density in BN along the 〈110〉
direction.
Let us now discuss the results of the calculated 2D-electron-positron momentum density for
BN, 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 BN 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
[39]. Compared to this, the profile along the 〈110〉 direction is sharply peaked. However, the valleys
and dips observed in ρ(p) for BN are very shallow as compared with those of Si and Ge. This fact
clearly tells us that the momentum dependence of ρ(p) is very much different between elemental
769
N.Amrane
and compound semiconductors. In the case of Si, the symmetry is O7
h which contains 48 symmetry
operations including glide and screw, while in the case of BN, the symmetry is lowered from O7
h to
T 2
d : the two atoms in each unit cell are inequivalent and the number of symmetry operations thus de-
creases 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 cal-
(a) (b)
Figure 5. The calculated electron -positron momentum densities for BN in the (001–110) plane:
(a) contour maps; (b) bird’s eye view.
Figure 6. The calculated electron-positron momentum density after LCW folding in BN.
culations performed by Saito et al. [40] 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 boron is 1s22s22p1 and
that of nitrogen is 1s22s22p3, the interaction between second neighbour σ bonds is equivalent to a π
antibonding interaction between neighbouring atoms. As a consequence, there is a strong (2p,2p)σ
bond along 〈110〉 direction and an admixture of (2p,2p)σ and (2p,2p)π∗ bonds along 〈001〉 direc-
tion, the explanations are in good agreement with an earlier analysis based on group theory [16].
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 figures 5(a) and
770
Positron annihilation in boron nitride
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 [41], 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 BN, 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 BN system.
Figure 6 gives the calculated LCW folded distribution for BN. 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.
4. Conclusion
In the present paper we have reported positronic distributions for BN calculated within the
pseudopotential formalism and employing the independent particle model (IPM).These distributi-
ons are found to be strongly effected 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 sensi-
tive 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 impor-
tantly, due to its relatively few assumptions, the present theory yields a reliable single-particle
description of positron annihilation. Thus, it represents an excellent starting point for a systematic
many-particle description of the process.
References
1. Slack G.A., J. Phys. Chem. Solids, 1973, 34, 321.
2. Schubert M., Rheinlander B., Franke E., Neumann H., Hahn J., Roder M., Richter F., Appl. Phys.
Lett., 1997, 70, 1819.
3. Will G., Perkin P.G., Mater. Lett., 1999, 40, 1–4.
4. Vel L., Demazeau G., J. Etourneau, Mater. Sci. Eng. B, 1991, 10, 149–151.
5. Jiang L., Fitzgerald A.G., Rose M.J., et al., Appl. Surf. Sci., 2000, 167, 89–93.
6. Bocqillon G., Lorewrs-Susse C., J. Loriers, J. Mater. Sci., 1993, 28, 3547–3556.
7. Solozhenko V.L., Turkevich V.Z., Holzapfel W.B., J. Phys. Chem. B, 1999, 103, 8137–8140.
8. Kagamida M., Kanda H., Akaishi M., et al., J. Cryst. Growth, 1989, 94, 261–269.
9. Saito T., Endo T., Kashima S., et al., J. Mater. Sci., 1983, 18, 3054–3062.
10. Singh B.P., Solozhenko V.L., Will G., Diam. Relat. Mater., 1995, 4, 1193–1195.
11. Davis R.F., Proc. IEEE, 1991, 79, 7–11.
12. Boudiombo J., Baehr O., Boudrioua A., Thevenin P., Loulergue J.C., Bath A., Mat. Sci. Eng. B, 1997,
46, 96.
13. Susumu Okada, Susumu Saito and Atsushi Oshiyama, Physica B, 2002, 323, 224–226.
14. Ferhat M., Zaoui A., Certier M., Aourag H., Physica B, 1998, 252, 229–236.
15. Gorczyca I., Christensen N.E., Physica B, 1994, 185, 410–414.
16. Saito M., Oshiyama A., Tanigawa S., Phys. Rev. B, 1991, 44, 10601.
17. Panda B.K., Mahapatra D.P., Padhi H.C., Phys. Stat. Sol. (b), 1992, 169, 89.
18. Arutyunov N.Yu., Emstev V.V., Mikhailin A.V., Davidov V.Yu., Physica B, 2001, 308–310, 110–113.
19. Puska M.J. et al., Phys. Rev. B, 1995, 52, 10947.
20. Lin J.N., Sun J.N., Gidley D.W., Wetzel J.T., Monnig K.A., Ryan E.T., Jang S., Liang D.Yu,
Liang M.S. Materials Research Society Symposium Proceeding, 2002, 686, A9.7.
21. Krause-Rehberg R., Leipner H.S. Positron annihilation in semiconductors Springer series in solid state
sciences 127. Springer Verlag, Berlin 1999.
22. Benosman N., Amrane N., Mecabih S., Aourag H., J. Phys. and Chem. Of Solids, 2000, 61, 1727–1733.
23. Amrane Na., Soudini B., Amrane N., Aourag H., Mater. Sci. and Eng. B, 1996, 40, 119–125.
24. Puska M.J., Nieminen R.M., Reviews of Modern Physics, 1994, 66, 841.
25. Nabi Z., Abbar B., Amrane N., Aourag H., Mater. Chem. And Phys., 1999, 60, 256–261.
771
N.Amrane
26. Kohn W., Sham K., Phys. Rev. A, 1965, 140, 1133–1138.
27. Perdew J.P., Burke K., Ernzerhof M., Phys. Rev. Lett., 1996, 77, 3865–3868.
28. Liming W., Panda B.K., Fung S., Beiling C.D., J. Phys: Cond. Matter, 1997, 9, 8147–8154.
29. Leguey T., Monge M.A., Pareja R., Riveiro J.M., J. Phys: Cond. Matter, 1995, 7, 6179–6185.
30. Chelikowsky J.R, Cohen M.L., Phys. Rev. B., 1976, 14, 2.
31. Friedel P., Hybertsen M.S., Schlüter M., Phys. Rev. B., 1989, 39, 7974–7977.
32. Deibuk V.G., Viklyuk Ya.I., Rarenko I.M., Semiconductors, 1999, 33, No. 3, 293–296.
33. Jarlborg T., Manuel A.A., Peter M., Sanchez D., Singh A.K., Stephan J.-L., Walker E. Positron Anni-
hilation (eds. L.Dorikens-Vanpraet, M.Dorikens and D.Segers). World Scientific, Singapore, 266, 1988.
34. Corbel C., Stucky M., Hautojarvi P., Saarinen K., Moser P., Phys. Rev. B, 1988, 53, 8192.
35. Lock D.G., Crisp V.H.C., West R.N., J. Phys. F, 1973, 3, 561.
36. M.L.Cohen, T.K.Bergstresser, Phys. Rev., 1966, 141, 789.
37. Phillips J.C., J. Phys. Soc. Japan, 1966, 21 (Supp.), 3.
38. Aourag H., Khelifa B., Belaidi A., Tadjer A., Rezki M., Gamoudi M., Phys. Stat. Sol. (b), 1990,
160, 193.
39. Fujiwara K., Hyodo T., J. Phys. Soc. Jpn, 1973, 35, 1133.
40. Saito M., Oshiyama A., Tanigawa S., Private communication.
41. Liu W., Berko S., Mills A.P. Jr. Positron annihilation, Matter., 1992, Sci. Forum, Szombachely, 743.
42. Tsay Y.F., Vaidyanathan A., Mitra M.S., Phys. Rev. B, 1979, 19, 5422.
Позитронна анiгiляцiя в нiтридi бору
Н.Амране
Унiверситет Об’єднаних Арабських Емiратiв, природознавчий факультет, Об’єднанi Арабськi
Емiрати
Отримано 24 сiчня 2006 р., в остаточному виглядi – 20 травня 2006 р.
Електроннi i позитроннi зарядовi густини є обчисленi як функцiї положення одиничної комiрки для
нiтриду бору. Хвильовi функцiї отриманi, вiдповiдно, з обчислень псевдопотенцiальної зонної стру-
ктури i наближених незалежних частинок (IPM) для електронiв i позитронiв. Спостережено, що по-
зитронна густина є максимальною у вiдкритих щiлинах, i є обмеженою не тiльки iонним кором, але
також, значною мiрою, валентними зонами. Густини електрон-позитронного моменту обчисленi для
площин (001,110). Результати використанi для аналiзу позитронних ефектiв у BN.
Ключовi слова: позитронна анiгiляцiя, густина iмпульсу, кутова кореляцiя
PACS: 78.70.Bj, 71.15.Dx
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