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Positron states in materials: density functional and quantum monte carlo studies

Positron states in materials: density functional and quantum monte carlo studies

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128

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



129



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



130

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



132

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



134



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-



136

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].



138

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].



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