Positron states in materials: density functional and quantum monte carlo studies
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electron and positron states and annihilation characteristics in materials in order
to support the experimental research.
2.
POSITRON ANNIHILATION
To conserve energy and momentum, electrons and positrons usually annihilate by a second order process in which two photons are emitted [3, 4]. The
process is shown in Fig. 1. At the first vertex the electron emits a photon, at
the second vertex it emits a second photon and jumps into a negative energy
state (positron). This phenomenon is analogous to Compton scattering and the
calculation proceeds very much as the Compton scattering calculation [5]. The
annihilation cross-section for a pair of total momentum is given by
where
is the classical electron radius and
Positron states in materials: DFT and QMC studies.
In the non-relativistic limit
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this gives
where is the relative velocity of the colliding particles. The first derivation
of the positron annihilation cross-section formula was done by Dirac [6]. The
annihilation rate
is obtained on multiplying
by the flux density
where
is the density of positron-electron pairs with total momentum p. In
the non-relativistic limit, the product
is a constant, therefore
and
are proportional.
Second-quantized many-body formalism can be used to study positron annihilation in an electron gas and the electron-positron interaction can be discussed
in terms of a Green function [7, 8, 9]. The density
of positron-electron
pairs with total momentum p can be written as
where
and
are plane wave annihilation operators for the electron and
the positron respectively and V is the volume of the sample. In terms of the
corresponding point annihilation operators
and
one has
Substituting one obtains
This formula can also be expressed as
and the two-particle electron-positron Green’s function defined by
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where is a four-vector and T is the time-ordering operator. In the nonrelativistic limit, the annihilation rate
is given by
The total annihilation rate is obtained by integrating over p
Therefore the effective density
is given by
The inverse of the total annihilation rate yields the positron lifetime
which an important quantity in positron annihilation spectroscopies.
One can go from the second-quantization representation to the configuration
space, using the many body wave function
The vector
is the
positron position,
is an electron position and stands for the remaining
electron coordinates
One can show that
is also given by
After integrating over
functions
as
Eq. (15) can be expressed with two particle wave
The summation is over all electron states and is the occupation number of the
electron state labeled .
is the two-particle wave function when the
positron and electron reside in the same point.
can be further written
with the help of the positron and electron single particle wave functions
and
respectively, and the so-called enhancement factor
The enhancement factor is a manifestation of electron-positron interactions
and it is always a crucial ingredient when calculating the positron lifetime. The
Positron states in materials: DFT and QMC studies.
131
independent particle model (IPM) assumes that there is no correlation between
the positron and the electrons and that
This approximation is justified
only when the spectrum
reflects quite well the momentum density of the
system in absence of the positron.
Many-body calculations for a positron in a homogeneous electron gas (HEG)
have been used to model the electron-positron correlation. Kahana [8] used
a Bethe-Golstone type ladder-diagram summation and predicted that the annihilation rate increases when the electron momentum approaches the Fermi
momentum
as shown in Fig. 2. This momentum dependence is explained
by the fact that the electrons deep inside the Fermi liquid cannot respond as
effectively to the interactions as those near the Fermi surface. According to
the many-body calculation by Daniel and Vosko [10] for the HEG, the electron
momentum distribution is lowered just below the Fermi level with respect to
the free electron gas. This Daniel-Vosko effect would oppose the increase of
the annihilation rate near the Fermi momentum
To describe the Kahana
theory, it is convenient to define a momentum-dependent enhancement factor
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where
is the IPM partial annihilation rate. Stachowiak [11] has proposed a phenomenological formula for the increase of the enhancement factor
at
given by
where
is the electron gas parameter given by
and is the electron density. This behavior of
is quite sensitive to the
construction of the many-body wave function. Experimentally, the peaking of
at
should in principle be observable in alkali metals [12].
The Kahana theory in the plane-wave representation (corresponding to single particle wave functions in the HEG) can be generalized by using Bloch
wave functions for a periodic ion lattice. This approach has been reviewed by
Sormann [13]. An important conclusion is that the state dependence of the
enhancement factor is strongly modified by the inhomogeneity and the lattice
effects. Therefore in materials, which are not nearly-free-electron like, the
Kahana momentum dependence of
is probably completely hidden.
The plane wave expansions used in the Bethe-Golstone equation can be
slowly convergent to describe the cusp in the screening cloud. Choosing more
appropriate functions depending on the electron-positron relative distance
may provide more effective tools to deal with the problem. The Bethe-Golstone
equation is equivalent to the Schrödinger equation for the electron-positron pair
wave function
where V is a screened Coulomb potential. The Pluvinage approximation [14]
for
consists in finding two functions
and
such
that
and such that the Schrödinger equation
becomes separable.
describes the orbital motion of the two particles ignoring each other, and
describes the correlated motion. The
correlated motion depends strongly on the initial electron state (without the
presence of the positron). Obviously, the core and the localized and valence
electrons are less affected by the positron than the
valence orbitals. On
the basis of the Pluvinage approximation, one can develop a theory for the momentum density of annihilating electron-positron. In practice, this leads to a
scheme in which one first determines the momentum density for a given electron state within the IPM. When calculating the total momentum density this
contribution is weighted by
Positron states in materials: DFT and QMC studies.
133
where is the partial annihilation rate of the electron state and
is the
same quantity in the IPM. This means that a state-dependent enhancement factor
substitutes
in Eq. (17). The partial annihilation rate is obtained as
where
and
are the electron density for the state the total
electron density and the state independent enhancement factor, respectively. If
this theory is applied to the HEG it leads to the same constant enhancement factor
to all electron states, i.e. there is no Kahana-type momentum dependence in
the theory. In a HEG, the enhancement factor
can be obtained by solving a radial Schrödinger-like equation [15, 16, 17] for
an electron-positron pair interacting via an effective potential W
Multiplying by
and integrating gives
This result shows that the enhancement factor is proportional to the expectation
value of the effective electric
The potential W can be determined
within the hypernetted chain approximation (HNC) [16,17]. The bosonization
method by Arponen and Pajanne [18] is considered to be superior over the HNC.
The parametrization of their data, shown in Fig. 3, reads as [19]
The only fitting parameter in this equation is the factor in the front of the square
term. The first two terms are fixed to reproduce the high-density RPA limit [20]
and the last term the low-density positronium (Ps) atom limit. There is an upper
bound for i.e. [15]
where
is the enhancement factor in the case of a proton and
is the reduced mass of the electron-positron system. Eq. (28) is called the
scaled proton formula and it is truly an upper bound, because we cannot expect
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a greater screening of a delocalized positron than that of a strongly localized
proton. The positron annihilation rate in the HEG is given by the simple relation
and the lifetime
is shown in Fig. 4 for several electron densities. One
can notice that saturates to the lifetime of Ps atom in free space (about 500
ps).
3.
TWO-COMPONENT DFT
The DFT reduces the quantum-mechanical many-body problem to a set of
manageable one-body problems [21]. It solves the electronic structure of a
system in its ground state so that the electron density
is the basic quantity.
The DFT can be generalized to positron-electron systems by including the
positron density
as well; it is then called a 2-component DFT [22, 23].
The enhancement factor is treated as a function of the electron density
in
the local density approximation (LDA) [22]. However, quite generally, the LDA
underestimates the positron lifetime. In fact one expects that the strong electric
field due to the inhomogeneity suppresses the electron-positron correlations in
Positron states in materials: DFT and QMC studies.
135
the same way as the Stark effect decreases the electron-positron density at zero
distance for the Ps atom [18]. In the generalized gradient approximation (GGA)
[19, 24] the effects of the nonuniform electron density are described in terms
of the ratio between the local length scale
of the density variations and
the local Thomas-Fermi screening length
The lowest order gradient
correction to the LDA correlation hole density is proportional to the parameter
This parameter is taken to describe also the reduction of the screening cloud
close to the positron. For the HEG
whereas in the case of rapid density
variations approaches infinity. At the former limit the LDA result for the
induced screening charge is valid and the latter limit should lead to the IPM
result with vanishing enhancement. In order to interpolate between these limits,
we use for the enhancement factor the form
Above
has been set so that the calculated and experimental lifetimes
agree as well as possible for a large number of different types of solids.
The effective positron potential is given by the total Coulomb potential plus
the electron-positron correlation potential [22, 23]. The electron-positron po-
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tential per electron due to a positron impurity can be obtained via the HellmannFeynman theorem [25] as
where
is the screening cloud density and Z is the electronpositron coupling constant. Let us suppose that the electron-positron correlation
for an electron gas with a relevant density is mainly characterized by a single
length Then for the electron-positron correlation energy,
is
constant and the normalization factor of the screening cloud scales as
with
for the dimension of space. Compared to the IPM result the electronpositron correlation increases the annihilation rate as
which is proportional to the density of the screening cloud at the positron. Consequently,
we have the following scaling law [26]
The values of the correlation energy calculated by Arponen and Pajanne [18]
obey the form of Eq. (33) quite well and the coefficient
has a relatively small
value of 0.11 Ry. Therefore, one can use in the practical GGA calculations the
correlation energy
which is obtained from the HEG result
by
the scaling
where
and
are the annihilation rates in the LDA model and in the
GGA model, respectively. One can use for the correlation energy
the
interpolation form of Ref. [23] obtained from Arponen and Pajanne calculation
[18].
4.
DFT RESULTS
4.1.
Positron Affinity
The positron affinity is an energy quantity defined by
where and
are the electron and positron chemical potentials, respectively
[27]. In the case of a semiconductor,
is taken from the position of the top
of the valence band. The affinity can be measured by positron re-emission
spectroscopy [28]. The comparison of measured and calculated values for
Positron states in materials: DFT and QMC studies.
137
different materials is a good test for the electron-positron correlation potential.
The Ps atom work function is given by [28]
Since the Ps is a neutral particle,
is independent of the surface dipole. The
LDA shows a clear tendency to overestimate the magnitude of
[19]. This
overestimation can be traced back to the screening effects. In the GGA, the
value of
is improved with respect to experiment by reducing the screening
charge. The calculated positron affinities within LDA and GGA against the
corresponding experimental values for several metals are shown in Fig. 5.
Kuriplach et al. [29] calculated
for different polytypes of SiC and showed
that the GGA agrees better with the experimental values than the LDA. Panda
et al. showed that the computed affinities depend crucially on the electronpositron potential used in the calculation (LDA or GGA) and on the quality
of the wave function basis set [30]. The result with a more accurate basis set
for valence electrons and within GGA gives –3.92 eV for 3C-SiC, which is
surprisingly close to the experimental
[30].
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4.2.
Positron Lifetime
The LDA underestimates systematically the positron lifetime in real materials. Sterne and Kaiser [31] suggested to use a constant enhancement factor of
unity for core electrons. Plazaola et al. [32] showed that the positron lifetimes
calculated for II-VI compound semiconductors are too short due to the LDA
overestimation of the annihilation rate with the uppermost atom-II d electrons.
Puska et al. [33] introduced a semiempirical model in order to decrease the
positron annihilation rate in semiconductors and insulators. In the GGA these
corrections are not necessary. In general, the agreement for the GGA with the
experiment is excellent, as shown in Fig. 6. Moreover, Ishibashi et al. [34] have
shown that the GGA reproduces the experimental values much better than the
LDA even for the low-electron-density systems such as the molecular crystals
of
TTF-TCNQ and
The GGA can also be safely
applied to the calculation of annihilation characteristics for positrons trapped
at vacancies in solids [24].