Positron States in Materials: Dft and QMC Studies
ABSTRACT Firstprinciples approaches based on density functional theory (DFT) for calculating positron states and annihilation characteristics
in condensed matter are presented. The treatment of electronpositron correlation effects is shown to play a crucial role
when calculating affinities and annihilation rates. A generalized gradient approximation (GGA) takes the strong inhomogeneities
of the electron density into account and is particularly successful in describing positron characteristics in various materials
such as metals, semiconductors, cuprate superconductors and molecular crystals. The purpose of Quantum Monte Carlo (QMC) simulations
is to provide highly accurate benchmark results for positronelectron systems. In particular, a very efficient QMC technique,
based on the Stochastic Gradient Approximation (SGA), can been used to calculate electronpositron correlation energies.
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Article: Relativistic Quantum Theory
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ABSTRACT: This book is concerned with the qualitative and asymptotic properties of stochastic approximation algorithms in many diverse forms in which they arise in applications. The first three chapters deal with the classical RobbinsMonro and KieferWolfowitz methods and their modifications motivated by practical considerations. Many examples of applications to learning, state dependent noise, queueing as well as in signal processing and adaptive control are described. The five following chapters give the mathematical background concerning the convergence theorems of the algorithms (convergence in probability as well as weak convergence). The proofs of convergence and investigation of the rate of convergence of the applications mentioned before follow. The problem of selecting a good step size sequence for the algorithms is dealt with in the last but one chapter. The closing chapter concerns the decentralized and asynchronous forms of the stochastic approximation algorithms.  [Show abstract] [Hide abstract]
ABSTRACT: Positron decay curves in various semiconducting elements and in the semimetals As, Sb, and Bi have been measured with a delayed coincidence system with 0.35 nsec full width at halfmaximum prompt time resolution. The annihilation in ideal, pure semiconductors is found to be characterized by a simple exponential decay: The second lowintensity decay component may be entirely due to spurious effects. In nonintrinsic Si and Ge powders, however, a second decay component with considerable abundance is observed whose presence is assigned to unclear extrinsic processes. The intrinsic decay component in diamond, Ge, and Si can be interpreted as due to annihilation with the valence electrons, taking into account the screening effect through an evaluation of the dielectric constant. In S, Se, and Te the effect of the electron charge distribution is dominant over the effect of electron polarizability.Physical Review  PHYS REV X. 01/1968; 175(2):383388.
Page 1
POSITRONSTATESINMATERIALS:DFTANDQMC
STUDIES.
Bernardo Barbiellini
Physics Department, Northeastern University, Boston, Massachusetts 02115
bba@neu.edu
Abstract
Firstprinciples approaches based on density functional theory (DFT) for cal
culating positron states and annihilation characteristics in condensed matter are
presented. Thetreatmentofelectronpositroncorrelationeffectsisshowntoplay
a crucial role when calculating affinities and annihilation rates. A generalized
gradient approximation (GGA) takes the strong inhomogeneities of the electron
density into account and is particularly successful in describing positron charac
teristics in various materials such as metals, semiconductors, cuprate supercon
ductors and molecular crystals. The purpose of Quantum Monte Carlo (QMC)
simulations is to providehighly accurate benchmark results for positronelectron
systems. In particular, a very efficient QMC technique, based on the Stochas
tic Gradient Approximation (SGA), can been used to calculate electronpositron
correlation energies.
1.INTRODUCTION
Positron spectroscopygives valuable information on the electronic and ionic
structures of condensed media [1]. For example, experimental methods based
onpositronannihilation[2]canidentifypointdefectsinsemiconductorsincon
centrationsaslowas0.1ppm. Theexperimentaloutputis,however,indirecte.g.
intheformofthepositronlifetimeorofdatarelatedtothemomentumcontentof
theannihilatingelectronpositron pairin aspecific environment. Moreover, the
positron causes a local perturbation in the surrounding material. Therefore it is
important to distinguish to which extent the annihilation characteristics depend
either on the original unperturbed electronic properties of the material or on the
electronpositron correlations. Clearly, the interpretation of the experimental
data calls for theoretical methods with quantitative predicting power. On the
other hand, the positron annihilation measurements give unique experimental
data to be used in comparing the results of manybody theories for electron
electron and electronpositron interactions. The goal of the present work is
is to introduce some of the computational approaches developed for predicting
1
Page 2
2
electronandpositronstatesandannihilationcharacteristicsinmaterialsinorder
to support the experimental research.
2. POSITRON ANNIHILATION
To conserve energy and momentum, electrons and positrons usually annihi
late 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
e+
e−
γ
γ
Figure 1.
Feynman diagram for the electron positron annihilation.
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 crosssection for a pair of total momentum
?
is given by
???
????????????
?
?
?
?
?????
?
???
?
???
???????
???????! #"
?
?
"
?????
"
?$?
"
??????%
?
?
?
?
?
?'&
?
?
?$?(?
???*)
(1)
where
?
?
?,+
?'?.0/'?is the classical electron radius and
?
?
?
?21
?43
.
?
/65
?
?
?
/
?
.0/
?
?87
(2)
Page 3
Positron states in materials: DFT and QMC studies.
3
In the nonrelativistic limit (
?:9;?), this gives
?
?
?<?
?
?
/
=
)
(3)
where
of the positron annihilation crosssection formula was done by Dirac [6]. The
annihilation rate
=
is the relative velocity of the colliding particles. The first derivation
>
?@?
?is obtained on multiplying
???
?A?by the flux density
B
?C?
?D?E=GF
?@?
?
>
?C?
?D?
???
???
B
?@?
?H?
???
???I=GF
?C?
?*)
(4)
where
the nonrelativistic limit, the product
are proportional.
Secondquantized manybody formalism can be used to study positron anni
hilationinanelectrongasandtheelectronpositroninteractioncanbediscussed
in terms of a Green function [7, 8, 9]. The density
pairs with total momentumcan be written as
F
?@?
?is the density of positronelectron pairs with total momentum
?. In
?
=is a constant, therefore
>
?@?
?and
F
?@?
?
F
?C?
?of positronelectron
?
F
?C?
???
1
JLKM?N$KM?OQPSR
is the volume of the sample. In terms of the
?and
TVU
M?NXW
U
M?O6W
M?O
T
M?N
R?Y;Z
M?N
U
M?O2[M
)
(5)
where
the positron respectively and
corresponding point annihilation operators
T
M
N
and
W
M
O
are plane wave annihilation operators for the electron and
J
\
?
?@]?^
\<_
?C]
?
?one has
T
M?N
?
1
"
Ja`cb
]?^ed6fVgh?
?ji
??^?]?^
?
\
?
?@]?^
?*)
(6)
W
M?N
?
1
"
Ja`cb
]
?
d'f?gh?
?ji
?
?
]
?
?
\?_
?C]
?
?87
(7)
Substituting one obtains
F
?@?
???
1
J
`cb
]
b
]lk8d'f?gh?
?ji
?m?C]
?
]lk
?2?
PnR
\
?
?@]
?
Uo\?_
?C]
?
Uh\?_
?C]lk
?
\
?
?C]lk
?
RGY
7
(8)
This formula can also be expressed as
F
?@?
?D?
?
i
?
?
J
`cb
]
b
]pk8d'f?go?
?qi
?
?C]
?
]pk
???sr
?
_
?C]lt
)
]lt*uX]pkvt
U
)
]lkwt
U
?8)
(9)
and the twoparticle electronpositron Green’s function defined by
r
?
_
?Cx<^
)
x
?
u2xyk
^
)
xzk
?
?D?
i
?
PnR{
?
\
?
?xh^
?
\?_
?Cx
?
?
\}U
_
?xzk
?
?
\}U
?
?Cxyk
^
???
RGY
)
(10)
Page 4
4
where
tivistic limit, the annihilation rate
x
is a fourvector and
{
is the timeordering operator. In the nonrela
>
?@?
?is given by
>
?@?
?D?~?<?
?
?
/
i
?
J
`b
]
b
]lk*d'fVgh?
?ji
??@]
?
]lk
?2?sr
?
_
?@]lt
)
]pt8uX]pkwt
U
)
]lkwt
U
?87
(11)
The total annihilation rate is obtained by integrating over
?
?
K
M
>
?@?
?D?
?<?
?
?
/D
?p
7
(12)
Therefore the effective density
?p is given by
?p
?
i
?
J
`cb
]
r
?
_
?C]lt
)
]lt*uX]lt
U
)
]lt
U
?87
(13)
The inverse of the total annihilation rate yields the positron lifetime
?
1
)
(14)
which an important quantity in positron annihilation spectroscopies.
One can go from the secondquantization representation to the configuration
space, using the many body wave function
positron position,
electron coordinates
?@]
U
)
]?^
)2??. The vector
]
U
is the
]?^is an electron position and
stands for the remaining
?C]
?
)'77?7)
]l
?. One can show that
>
?@?
?is also given by
>
?C?
?D?
???
?
?
/
J
`cb
$
`cb
]od6fVgh?
?ji
?s]
?
?@]
)
]
)????
?
7
(15)
After integrating over
functions
, Eq. (15) can be expressed with two particle wave
\
?
_
as
>
?@?
???~???
?
?
/
JK
`b
]d'f?go?
?ji
?o]
?
\
?
_
?C]
)
]
??
?
7
(16)
Thesummationisoverallelectronstatesand
electron state labeled
positron and electron reside in the same point.
with the help of the positron and electron single particle wave functions
and
istheoccupationnumberofthe
i.
\
?
_
?@]
)
]
?is the twoparticle wave function when the
\
?
_
?C]
)
]
?can be further written
\
U
?@]
?
\
?@]
?, respectively, and the socalled enhancement factor
?C]
?,
\
?
_
?C]
)
]
?D?
\
U
?@]
?
\
?@]
?'&
?C]
?*7
(17)
The enhancement factor is a manifestation of electronpositron interactions
and it is always a crucial ingredient when calculating the positron lifetime. The
Page 5
Positron states in materials: DFT and QMC studies.
5
independent particle model (IPM) assumes that there is no correlation between
thepositronandtheelectronsandthat
only when the spectrum
system in absence of the positron.
Manybodycalculationsforapositroninahomogeneouselectrongas(HEG)
have been used to model the electronpositron correlation. Kahana [8] used
a BetheGolstone type ladderdiagram summation and predicted that the an
nihilation rate increases when the electron momentum approaches the Fermi
momentum
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
?@]
???
1. Thisapproximationisjustified
>
?@?
?reflects quite well the momentum density of the
?z, as shown in Fig. 2. This momentum dependence is explained
00.2 0.4 0.60.811.2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Kahana theory
p/pF
γ(p)
Figure 2.
Kahana theory for HEG with
X?0 .
the manybody 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 DanielVosko effect would oppose the increase of
the annihilation rate near the Fermi momentum
theory, it is convenient to define a momentumdependent enhancement factor
?z. To describe the Kahana
?
?A?D?EF
?
?A?

F???
?
?A??)
(18)
Page 6
6
where
posed a phenomenological formula for the increase of the enhancement factor
at
given by
F
??
?
?A?is the IPM partial annihilation rate. Stachowiak [11] has pro
?z
?
?
?
?
?@
?
?
?
?
7
1
?l
)
(19)
where
?
is the electron gas parameter given by
?l
?
?
?
??
?
^
)
(20)
and
construction of the manybody wave function. Experimentally, the peaking of
should in principle be observable in alkali metals [12].
The Kahana theory in the planewave representation (corresponding to sin
gle 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 nearlyfreeelectron like, the
Kahana momentum dependence of
The plane wave expansions used in the BetheGolstone equation can be
slowly convergent to describe the cusp in the screening cloud. Choosing more
appropriate functions depending on the electronpositron relative distance
mayprovidemoreeffectivetoolstodealwiththeproblem. TheBetheGolstone
equationisequivalenttotheSchr¨ odingerequationfortheelectronpositronpair
wave function
is the electron density. This behavior of
?
?A?is quite sensitive to the
?
?A?at
?z
?@?
?is probably completely hidden.
?
^
?
B
?@]?^
)
]
?
?
?
1
?A¡
?
^
?
1
?A¡
?
?
?
J
?
R
]?^
?
]
?
R
?¢
B
?@]?^
)
]
?
?D?
?
?
?l.
B
?@]?^
)
]
?
?*)
(21)
where
for
that
becomes separable.
cles ignoring each other, and
correlated motion depends strongly on the initial electron state
presenceofthepositron). Obviously, thecoreandthelocalized
electrons are less affected by the positron than the
the basis of the Pluvinage approximation, one can develop a theory for the mo
mentum density of annihilating electronpositron. In practice, this leads to a
scheme in which one first determines the momentum density for a given elec
tron statewithin the IPM. When calculating the total momentum density this
contribution is weighted by
J
is a screened Coulomb potential. The Pluvinage approximation [14]
B
?@]
^
)
]
?
?consists in finding two functions
r
?@]
^
)
]
?
?and
?C]
^
)
]
?
?such
B
?@]
^
)
]
?
?£?¤r
?@]
^
)
]
?
?
?@]
^
)
]
?
?and such that the Schr¨ odinger equation
r
?@]?^
)
]
?
?describes the orbital motion of the two parti
?@]?^
)
]
?
?describes the correlated motion. The
i
(without the
valence
b and
¥
?type valence orbitals. On
i
?

2A
)
(22)
Page 7
Positron states in materials: DFT and QMC studies.
7
where
samequantityintheIPM.Thismeansthatastatedependentenhancementfactor
is the partial annihilation rate of the electron state
i
and
??
is the
substitutes
?@]
?in Eq. (17). The partial annihilation rate
is obtained as
?
???
?
?
/
`cb
]
?@]
?
U
?C]
?
?C]
?8)
(23)
where
density and the state independent enhancement factor, respectively. If this
theory is applied to the HEG it leads to the same constant enhancement factor
?@]
?,
U
?@]
?,and
?@]
?aretheelectrondensityforthestate
i,thepositron
to all electron states, i.e. there is no Kahanatype momentum dependence in
the theory. In a HEG, the enhancement factor
?~
?
R
\
?
?
R
?
)
(24)
can be obtained by solving a radial Schr¨ odingerlike equation [15, 16, 17] for
an electronpositron pair interacting via an effective potential
¦
?
b
?
b
?
?
?
¦
?
?
?
%
?
\
?
?
???
7
(25)
Multiplying by
b

b
?
?
?
\
?
?
?2?and integrating gives
R
\
?
?
R
?
?
?
`¨§
©
?
?
R
\
?
?
?
R
?
b
¦
b
?
b
?
7
(26)
This result showsthat the enhancement factor is proportionalto the expectation
valueoftheeffectiveelectricfield
within the hypernetted chain approximation (HNC) [16, 17]. The bosonization
methodbyArponenandPajanne[18]isconsideredtobesuperiorovertheHNC.
The parametrization of their data, shown in Fig. 3, reads as [19]
?
b
¦

b
?. Thepotential
¦
canbedetermined
?
1
?
1
7
?
?
?
7
?ª
?
???
?
?
1
«
?
7
(27)
Theonlyfittingparameterinthisequationisthefactorinthefrontofthesquare
term. Thefirsttwotermsarefixed toreproducethehighdensityRPA limit[20]
andthelasttermthelowdensitypositronium(Ps)atomlimit. Thereisanupper
bound for
, i.e. [15]
?
?
l¬
?#
A®?¯z°±¯
²?
?
??)
(28)
where
is the reduced mass of the electronpositron system. Eq. (28) is called the
scaled proton formula and it is truly an upper bound, because we cannot expect
A®?¯z°±¯
istheenhancement factorin thecaseof aprotonand
¬
?
1
??
Page 8
8
012345678910
0
20
40
60
80
100
120
140
160
180
200
Enhancement factor in the HEG
rs (au)
γ
Arponen and Pajanne
Scaled Proton
Figure 3.
and Pajanne theory. The dashed line is the scaled proton (M. J. Puska, private communication).
Enhancement factor for several electron densities. The solid line is the Arponen
a greater screening of a delocalized positron than that of a strongly localized
proton. ThepositronannihilationrateintheHEGisgivenbythesimplerelation
?
1
?
?
?³y´6µ
^
?*7
(29)
and the lifetime
can notice that
ps).
?
1

is shown in Fig. 4 for several electron densities. One
saturates to the lifetime of Ps atom in free space (about 500
3.TWOCOMPONENT DFT
The DFT reduces the quantummechanical manybody problem to a set of
manageable onebody problems [21]. It solves the electronic structure of a
system in its ground state so that the electron density
The DFT can be generalized to positronelectron systems by including the
positron density
The enhancement factor is treated as a function of the electron density
thelocaldensityapproximation(LDA)[22]. However,quitegenerally,theLDA
underestimates the positron lifetime. In fact one expects that the strong electric
field due to the inhomogeneity suppresses the electronpositron correlations in
?@]
?is the basic quantity.
U (r) as well; it is then called a 2component DFT [22, 23].
?@]
?in
Page 9
Positron states in materials: DFT and QMC studies.
9
0123456789 10
0
50
100
150
200
250
300
350
400
450
500
Lifetime in the HEG
rs (au)
τ (ps)
Figure 4.
Positron lifetime in HEG for several electron densities.
the same way as the Stark effect decreases the electronpositron density at zero
distanceforthePsatom[18]. Inthegeneralizedgradientapproximation(GGA)
[19, 24] the effects of the nonuniform electron density are described in terms
of the ratio between the local length scale
the local ThomasFermi screening length
correction to the LDA correlation hole density is proportional to the parameter
o
R
¡
R of the density variations and
1
·¶
°
. The lowest order gradient
¸
?
R
¡
R
?

?
<¶
°
D?
?
?
R
¡
?
³
R
?
·¶
?
°
7
(30)
This parameter is taken to describe also the reduction of the screening cloud
close to the positron. For the HEG
variations
induced screening charge is valid and the latter limit should lead to the IPM
resultwithvanishingenhancement. Inordertointerpolatebetweentheselimits,
we use for the enhancement factor the form
¸= 0, whereas in the case of rapid density
¸approaches infinity. At the former limit the LDA result for the
V¹?¹±º
?
1
?
?
V»G¼Hº
?
1
?
d'f?go?
?¾½
¸
?87
(31)
Above
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 electronpositron correlation potential [22, 23]. The electronpositron po
½
?
7
???has been set so that the calculated and experimental lifetimes
Page 10
10
tentialperelectronduetoapositronimpuritycanbeobtainedviatheHellmann
Feynman theorem [25] as
J:¿ÀIÁÁ
?C]
?D?
?
`
^
©
b·ÂÃ`cb
]lk
?@]
k
?
?ÄA?@]
)
]
k
)
Â
?
?
1
?
R
?
?
?
k
R
)
(32)
where
positroncouplingconstant. Letussupposethattheelectronpositroncorrelation
for an electron gas with a relevant density is mainly characterized by a single
length
constant and the normalization factor of the screening cloud scales as
?
?
k
?
?ÅÄ??C]
)
]
k
)
Â
?
?
1
?isthescreeningclouddensityand
Â istheelectron
T . Then for the electronpositron correlation energy,
b
J
¿@ÀÁÁ

b
?21

T
?is
TGÆ with
b
?
for the dimension of space. Compared to the IPM result the electron
positron correlation increases the annihilation rate as
portional to the density of the screening cloud at the positron. Consequently,
we have the following scaling law [26]
?
?? , which is pro
J£¿@ÀÁÁ
?
/
^p?
?
2A
?
^I
Æ
?
/
?
7
(33)
The values of the correlation energy calculated by Arponen and Pajanne [18]
obeythe formof Eq. (33) quite welland the coefficient
value of 0.11 Ry. Therefore, one can use in the practical GGA calculations the
correlation energy
the scaling
/
?
has arelativelysmall
J
¿@ÀÁÁ
¹?¹±º , which is obtained from the HEG result (
J
¿@ÀÁÁ
»G¼Hº ) by
JÇ¿ÀÁÁ
¹?¹±º
?C]
?È?
J:¿ÀIÁÁ
»?¼Hº
?
µ
?C]
?2?
¹?¹±º
?
2A
»G¼Hº
?
2A
¢
^
?
J:¿ÀIÁÁ
»?¼Hº
?
µ
?C]
?2?+
µeÉ?Ê
?
)
(34)
where
GGA model, respectively. One can use for the correlation energy
interpolationformof Ref. [23]obtainedfrom ArponenandPajanne calculation
[18].
»G¼Hº and
DFT RESULTS
Positron Affinity
¹?¹±º are the annihilation rates in the LDA model and in the
J
¿@ÀÁÁ
»G¼Hº
the
4.
4.1.
The positron affinity is an energy quantity defined by
Ë
U
?
¬
µ
?
¬
U
)
(35)
where
[27]. In the case of a semiconductor,
of the valence band. The affinity can be measured by positron reemission
spectroscopy [28]. The comparison of measured and calculated values for
¬
µand
¬
U aretheelectronandpositronchemicalpotentials,respectively
is taken from the position of the top
¬
µ
Page 11
Positron states in materials: DFT and QMC studies.
11
different materials is a good test for the electronpositron correlation potential.
The Ps atom work function is given by [28]
¸
?
?
Ë
U
?
«
7
?
eV
7
(36)
Since the Ps is a neutral particle,
¸
is independent of the surface dipole. The
−5.5 −5−4.5 −4
Experiment (eV)
−3.5 −3−2.5−2 −1.5
−5.5
−5
−4.5
−4
−3.5
−3
−2.5
−2
−1.5
Cu
V
Fe
MoW
Positron Affinities
Theory (eV)
Figure 5.
LDA results as a function of the experimental ones, respectively. The solid line corresponds to
the perfect agreement between the theoretical and experimental results from refs. [28].
Positron affinities for several metals. The solid and open circles give the GGA and
LDA shows a clear tendency to overestimate the magnitude of
overestimation can be traced back to the screening effects. In the GGA, the
value of
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
that the GGA agrees better with the experimental values than the LDA. Panda
et al. showed that the computed affinities depend crucially on the electron
positron 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
surprisingly close to the experimental value
Ë
U [19]. This
Ë
U is improved with respect to experiment by reducing the screening
Ë
U for different polytypes of SiC and showed
?
7ÍÌ
? eV for 3CSiC, which is
eV [30].
?
7
?
jÎÏ
7
??Ð
Page 12
12
4.2. Positron Lifetime
The LDA underestimates systematically the positron lifetime in real materi
als. 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 IIVI compound semiconductors are too short due to the LDA
overestimation of the annihilation rate with the uppermost atomII 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
experimentisexcellent,asshowninFig. 6. Moreover,Ishibashietal. [34]have
180200220 240260280300
180
200
220
240
260
280
300
Si
Ge
GaAs
InP
CdTe
Positron Lifetime
Theory (ps)
Experiment (ps)
Figure 6.
GGA and LDA results as a function of the experimental ones, respectively. The solid line
corresponds to the perfect agreement between the theoretical and experimental results. The
experimental data are collected in ref. [19].
Positron lifetimes for some semiconductors. The solid and open circles give the
shown that the GGA reproduces the experimental values much better than the
LDA even for the lowelectrondensity systems such as the molecular crystals
ofC
applied to the calculation of annihilation characteristics for positrons trapped
at vacancies in solids [24].
Ñ
©, TTFTCNQand (BEDTTTF)
?Cu(NCS)
?. TheGGAcan alsobesafely
Page 13
Positron states in materials: DFT and QMC studies.
13
4.3.Momentum Density
The Doppler broadening technique [35] and the twodimensional angular
correlation of the annihilation radiation (2DACAR) [36, 37] are useful spec
troscopies for measuring projections of the momentum density
ACAR spectrum, given by
F
?@?
?. The 2D
Ò
?
?yÓ?)@?eÔ·?D?
`cb
?yÕ#F
?@?
?*)
(37)
hasbeensuccessfulfordeterminationof theFermiSurface(FS)inmany metal
licsystems. Forinstance,theoreticalandexperimentalstudiesofmaterialssuch
asvalent metal), Cu [38] (transition metal) and
(heavy fermion compound) have made FS determination possible, and have
also provided insight in electronpositron correlation. FS 3dimensional recon
structions as shown in Fig. 7 can be obtained from 2DACAR experiments
[41]. Similar studies of the highT
Sn [39] (
¥
?
Ö
d
Ö×
?ØGÙÚ?
[40]
¿ oxides are more difficult since important
Figure 7.
Fermi Surface 3dimensional reconstruction for Cu.
positronwavefunctionandcorrelationeffectsovershadowthesmallerFSsignal
[42]. However, a more favorable case is provided by the
R=Y, Dy, Ho or Pr), where the 1dimensional ridge FS has a twofold symme
try which distinguishes it from important fourfold symmetry wave function
effects [43]. The 2dimensional FS of the CuO plane has been observed in the
ÛqÜmÝ
?
Ö#×
pÞÇß
µeà (with
Page 14
14
Nd
with the CuO plane.
The Doppler broadening technique [45] and the 2DACAR [46] can be used
for identifying point defects in semiconductors. For instance, calculations for
momentum densities of electronpositron pairs annihilating at vacancy clusters
in Si have been performed [47]. The theoretical 2DACAR spectra are found
to be isotropic if the positron is trapped by a small vacancy cluster. The re
sults indicate that the Doppler profile narrows as the size of the vacancy cluster
increases in agreement with experimental data. Moreover, one can notice that
vacancies and vacancy clusters decorated with impurities can produce signif
icantly different lineshapes and therefore the method can be used in chemical
analysisofdefects. Thesetheoreticalresultshaveallowedtoidentify structures
of vacancyimpurity complexes in highly Asdoped Si samples [48].
?
µ
ÓCe
ÓCuO
5
µeà [44]. In this compound the positron has a larger overlap
5.
5.1.
QUANTUM MONTE CARLO
Positronic Systems
Theoretical approaches that accurately predict the effect of correlations in
electronpositron systems are needed in order to understand the detailed infor
mation provided by positron annihilation spectroscopies. We have seen that
the bound state of positrons in the vicinity of defects is a subject of interest
for the current positron annihilation experiments. The binding of positrons to
finite atomic and molecular systems is a related problem, relevant for positron
and positronium chemistry. The simplest examples of these systems are the
Ps
ion [53, 54], the Psmolecule [55, 56], the HPs molecule [58, 57] and
positronic water [59]. All these problems constitute important benchmarks to
study the role of correlation effects in positron physics. The Quantum Monte
Carlo (QMC) approach turns out to be an appropriate tool to study them rigor
ously. In particular, the Diffusion Quantum Monte Carlo (DQMC) [49, 50, 51]
is based on the property that in the asymptotic imaginary time limit (
the Euclidean evolution operator acting on a trial state,
ground state,
fiesthegeneralizeddiffusionequation(inHartreeatomicunits)withappropriate
boundary conditions
µ
?
t
9Lá
)
° , projects out the
, with a component in
° . The distribution of walkers
satis
â
â
t
?
1
?A¡
?
?
¡
?
¡
?
³h?
°
???
??ã}ä
À¿
)
(38)
where
of
Choosing a trial state close enough to the ground state
in the computed. One can use the Variational QMC (VQMC) method, with
the Stochastic Gradient Approximation (SGA) [52], to construct optimum trial
ã}ä
À¿
?
µ
^
°æå
° , with
å
the Hamiltonian operator. The average value
å
at steadystate conditions (
9
:
° ) gives the ground state energy
reduces the variance
ã
.
ã
Page 15
Positron states in materials: DFT and QMC studies.
15
wave functions of the JastrowSlater form
°
?çæè
êé
d'ë
Rì
?
?*í
?
R
)
(39)
where
up(down)positrons),
is the product of twobody correlation factors. In a good trial function the sin
gularities in the kinetic energy must cancel those of the potential one. For the
Coulomb potential this leads to the Kato cusp condition [60] of the twobody
correlation factor. The choice of onebody functions depends on the physical
systemoneistryingtoinvestigate. Forfiniteatomicsystems, wehavechosen
to be a linear combination of Slater functions
were used for HEG.
A very simple system that can bind positronium is a point charge
This system has been studied by DQMC [51]. We will assume that the positron
is repulsed by the point charge (
and electron coordinates is equivalent to the case for a point charge
stabletheenergyofthesystemshouldliebelowitslowestdissociationthreshold,
which corresponds to the more negative of the Ps and the hydrogenlike atom
(
is given by
model[51],oneconsiderstheelectronlocalizedclosetothepointchargeandthe
positron surrounding the hydrogenlike atom (
charge of strength
by
¥ runs over the different spin species (spinup(down) electrons, spin
ì ’saretheonebodywavefunctionsand
ç0?
d'f?gh?ïî
Cð
í?ñ
í
?
ì
?lò
d6fVgh?
?¾ó
?
?, while plane waves
Â
[27].
Â
Y
), but the interchange of the positron
?
Â . To be
Â
+
µ) binding energies. The separation between the two possible thresholds
Â
?
3
1
??, for which the system is the most stable. In the onion
Â
+
µ), feeling only an effective
Â
?
1. The total energy of the system in this model is given
ã
?
?
1
?
Â
?
?
1
?
?
Â
?
1
?
?
7
(40)
This model is exact for
unstable and the positron escapes away. On the other hand, for
the system becomes more stable by reducing the charge. Therefore, one can
expect that the Ps escapes from the point charge. However, in this regime,
the onion model is not accurate due to the overlap of the electron and positron
wave functions, and we need a more careful investigation using the DQMC. In
the limit
differ whenis smaller. In this regime, electronpositron correlations become
more important and the VQMC method does not provide the exact correlation
energy. Moreover, the DQMC calculations confirm the onion model prediction
that the Ps escapes from the point charge at a critical value
of correlations effects enlarge the range of stability and we find
The point
au. For
hydrogenlike system.
Â
9
1. It is clear that for
Â
?
1the system becomes
Â
P
1
??
Â
9
1, VQMC and DQMC give almost the same energy, but they
Â
Â?ô . The inclusion
Â?ôæõ
7
?GÐ.
Â
?
3
1
??gives the greatest stability, the binding energy being
7
Ð
Â
P
3
1
??, the energy is closer to the energy of a Ps than a
Â
+
µ
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