Introduction The history of Positron Annihilation starts with the positron, that was theoretically predicted by Dirac in 1928, and found experimentally by Anderson in 1932. Positron annihilation is a result of an encounter of the electron with its antiparticle - Positron. The energy released by the annihilation forms two highly energetic gamma photons, which travel in opposite direction. These gamma rays provide a useful analysis tool which has found many practical applications in physics, chemistry and medicine.
Outlook <ul><li>Methods of positron annihilation </li></ul><ul><li>Positron trapping into defects </li></ul><ul><li>Positron lifetime spectroscopy </li></ul><ul><li>Doppler broadening spectroscopy </li></ul><ul><li>Coincidence Doppler broadening spectroscopy </li></ul>
<ul><li>Questions of semiconductor industry </li></ul><ul><li>Defect types? </li></ul><ul><li>Defect charge states? </li></ul><ul><li>Defect concentrations? </li></ul><ul><li>Answers of positron annihilation </li></ul><ul><li>Vacancy-like defects and defect complexes </li></ul><ul><li>Size of a vacancy (mono-, di-, vacancy cluster) </li></ul><ul><li>Neutral or negatively charged vacancy-complexes </li></ul><ul><li>Positively charged defects are invisible </li></ul><ul><li>Sensitivity limits 10 14 -10 19 cm -3 </li></ul>
Basic positron physics The positron has positive charge +e, spin 1/2, and the same mass m e as the electron. The positron is emitted in the β + decay of radioactive nuclei such as 22 Na It rapidly associates itself with one of the electrons of the material and forms a bound system called positronium. In less than 10 -7 s the positron and electron annihilate to produce two gamma rays in the charge-conserving reaction e + + e - --> 2 2 m e c 2 = 2 E E e = m e c 2 = 0.511 eV The emission of two gamma rays of the same energy is required by linear momentum conservation. Pair annihilation into a single photon is not permitted since that photon would have to carry energy 2m e c 2
<ul><li>Positrons diffusing through matter may be captured in special trapping sites. </li></ul><ul><li>These trapping centers are crystal imperfections, e.g. vacancies and dislocations. </li></ul><ul><li>The wave function of a positron captured in such a defect is localized until it annihilates with an electron of the immediate surrounding into g-rays. Since the local electron density and the electron momentum distribution are changed with respect to the defect-free crystal, the annihilation radiation can be utilized to obtain information on the localization site. Hence, the character and the concentration of the trapping centers, i.e. lattice defects, can be investigated by positrons. </li></ul><ul><li>The different positron techniques are based on the analysis of the annihilation radiation. It appears through the mass-energy transformation of electron-positron pairs, mostly into two -quanta of 511 keV. In special cases, e.g. at surfaces, a positron may form a bound state with an electron, which is called positronium (Ps). </li></ul>Broad overview
Positrons are repelled by positive atom cores Vacancy represents a positron trap due to the missing nuclei (potential well for a positron)
Positron Sources <ul><li>In addition to 22 Na, other isotopes ( 64 Cu, 58 Co, etc.) can be used, but are less common. </li></ul><ul><li>Positron has energy distribution up to an energy of 540 keV and can penetrate deep into a sample. </li></ul>
<ul><li>Sources of weak activity are needed for conventional positron lifetime and Doppler-broadening measurements. Even activities as low as 8X10 5 Bq (20 µCi) are sufficient. (1 Ci = 3.7 x 10 10 becquerel ) </li></ul><ul><li>The sources are usually prepared by evaporating a solution of a 22 Na salt on a thin metal or polymer foil. </li></ul><ul><li>The positron beam and angular correlation techniques require much stronger sources. Source capsules of about 4 GBq (100 mCi) are commercially available, </li></ul>
Methods of positron Annihilation <ul><li>Positron wave-function can be localized in the attractive potential of a defect. </li></ul><ul><li>• annihilation parameters </li></ul><ul><li>change in the localized state. </li></ul><ul><li>• e.g. positron lifetime increases in a vacancy. </li></ul><ul><li>• lifetime is measured as time difference between 1.27 and 0.511 MeV quanta. </li></ul><ul><li>• defect identification and </li></ul><ul><li>quantification possible. </li></ul>
Positron Lifetime Spectroscopy The positron lifetime ז is a function of the electron density at the annihilation site. The annihilation rate , which is the reciprocal of the positron lifetime ז , is given by the overlap of the positron density n + (r) = | + (r)| 2 and the electron density n - (r) (Nieminen and Manninen 1979), r 0 is the classical electron radius, c the speed of light, and r the position vector. The correlation function = [ n - (r)] = 1 + n - / n - describes the increase n - in the electron density due to the Coulomb attraction between a positron and an electron.
<ul><li>When positrons are trapped in open-volume defects, such as in vacancies and their agglomerates, the positron lifetime increases with respect to the defect-free sample. </li></ul><ul><li>This is due to the locally reduced electron density of the defect. Thus, a longer lifetime component, which is a measure of the size of the open volume, appears. </li></ul><ul><li>The strength of this component, i.e. its intensity, is directly related to the defect concentration. </li></ul><ul><li>In principle, both items of information, i.e. the kind and concentration of the defect under investigation, can be obtained independently by a single measurement. This is the major advantage of positron lifetime spectroscopy compared with angular correlation of annihilation radiation or Doppler-broadening spectroscopy with respect to defect issues. </li></ul>Several trapping centers exist • Shallow traps visible in combination with vacancies in low temperature region • Charge state defines the trapping rate
The time-dependent positron decay spectrum D ( t ) in the sample is given by k different defect types. If no positron traps are present in the sample, is reduced to where t b is the positron lifetime in the defect-free bulk of the sample. The positron lifetime spectrum N ( t ) is the absolute value of the time derivative of the positron decay spectrum D ( t ), Data Treatment
Momentum distribution technique As a result of momentum conservation during the annihilation process, the momentum of the electron–positron pair, p , is transferred to the photon pair. The momentum component p z in the propagation direction z of the g-rays results in a Doppler shift E of the annihilation energy of 511 keV, which amounts approximately to E = p z c /2 Since numerous annihilation events are measured to give the complete Doppler spectrum, the energy line of the annihilation is broadened due to the individual Doppler shifts in both directions, ± z . This effect is utilized in Doppler-broadening Spectroscopy.
Doppler broadening of Plastically deformed (blue) and as grown GaAs (red)
Angular correlation of annihilation radiation Precise measurements of the exact collinearity of the two annihilation gamma rays can be used to test momentum conservation
Observed angular breadth of the coincidence peak is entirely due to the rather large, finite solid angles subtended by both detectors and is not due to a breakdown of our most important conservation law Two detectors has been employed to detect gamma rays At = 90 o or 270 o coincidence counting rate is not exactly zero. This results from the finite resolving time of the electronic coincidence circuit.
Basics of Positron Annihilation in Semiconductors The interaction processes of positrons with solids comprise Backscattering Channeling Thermalization Diffusion Possible Trapping in lattice defects: Vacancies Shallow positron traps Dislocations Voids and interaction with surfaces, interfaces or grain boundaries: Precipitates Surfaces Interfaces
<ul><li>Determination of absolute defect concentrations is important. </li></ul><ul><li>This is always achieved by the determination of the positron trapping rate k which is proportional to the defect concentration (c). </li></ul><ul><li>K = C; is the trapping coefficient. </li></ul><ul><li>two approaches to obtain the quantity k : </li></ul><ul><li>the measurement of the annihilation parameters, viz. the positron lifetime ז or the line shape parameter S of Doppler-broadening spectroscopy, or </li></ul><ul><li>positron diffusion experiments, i.e. the determination of the positron diffusion length. In case of lifetime analysis, quantitative information on positron traps can be obtained by fitting of experimental data with different Trapping Models , which may include : </li></ul><ul><li>One Defect </li></ul><ul><li>Several Independent Defects </li></ul><ul><li>Combined Positron Traps </li></ul><ul><li>The proportionality constant between trapping rate and defect concentration, trapping coefficient µ, requires a reference method which yields the defect concentration independently. </li></ul>
<ul><li>Vacancies are the most important positron traps. </li></ul><ul><li>Positron annihilation parameters, e.g. the positron lifetime, change in a defined manner when positrons are trapped in a vacancy. </li></ul><ul><li>The trapping process is based on an attractive potential for the positrons which originates basically from the missing repulsing force of the absent nucleus. </li></ul><ul><li>A charge of the vacancy is included by superimposing the square-well potential on a long-range Coulomb potential. This results in an additional repulsion or attraction in case of a positive or negative charge, respectively. </li></ul>Positron potential V + for negative, neutral, and positive vacancies in silicon (Puska et al.1990).
<ul><li>In addition to the deep bound states of the square-well potential, a series of Rydberg states is induced. </li></ul><ul><li>For positively charged vacancies, the repulsion at low temperatures is extremely strong so that the positrons are repelled and trapping is practically prohibited. </li></ul><ul><li>Higher temperatures may induce a thermal excitation over the potential barrier, so that the trapping coefficient of positively charged vacancies V+ may be only one order of magnitude smaller than for neutral vacancies. </li></ul><ul><li>The additional attractive Coulomb potential of the negative vacancy V- is especially effective at low temperatures </li></ul><ul><li>The coefficient µ is one order of magnitude larger than that for the neutral vacancy at room temperature. </li></ul><ul><li>Because Rydberg states "collect" the positron in a more extended volume compared with a neutral vacancy. Positrons may be trapped in the Rydberg states with a high transition rate from the free delocalized state. The final trapping state at the bottom of the vacancy square-well potential is reached very fast. As a result of this two-stage process, the trapping becomes more efficient and the total trapping coefficient is larger than for a neutral vacancy. </li></ul><ul><li>The positron binding energy at the Rydberg states amounts to some 10 meV. Thus, the trapping coefficient decreases with increasing temperature for negatively charged vacancies </li></ul>
<ul><li>Rydberg series is replaced by one single level. </li></ul><ul><li>Positrons are trapped into the Rydberg state with the trapping rate k R or directly into the deep ground state of the vacancy with the trapping rate k t , respectively. </li></ul><ul><li>Positrons trapped in the Rydberg state may escape thermally stimulated with the detrapping rate δ R or they can pass over to the deep level with the transition rate R . </li></ul><ul><li>Positrons annihilating from spatially extended Rydberg states are expected to feel the electron density of the bulk. Thus, nearly the same annihilation rate b is supposed there. </li></ul>A simple two-state model
The thermally stimulated re-escape of positrons from shallow Rydberg states is characterized by the detrapping rate K R , calculated by Manninen and Nieminen (1981) as v is the vacancy density, E R the positron binding energy to the Rydberg state. k t << b and R >> b Rydberg states is combined in a single energy level. A net trapping rate of such a two-step trapping, i.e. positrons are first trapped in a shallow Rydberg state as a precursor of a deep state, can be approxi-mated by The trapping in negatively charged vacancies is then given with
In case of negative vacancies, positron trapping is not transition limited but diffusion limited, because of the large spatial extension of the Rydberg states. This leads to an additional term as a function of temperature for the trapping rate. The temperature dependence of the diffusion to the defect is mostly determined by scattering at acoustic phonons, leading to a dependence of the diffusion constant according to . Consequently, the trapping rate k R should have the same temperature dependence, k R = k R0 T -1/2 . k R0 is the trapping rate at a certain low temperature, e.g. at 20 K. The overall trapping rate is Positron trapping rate k in negatively charged gallium vacancies determined in semi-insulating gallium arsenide as a function of temperature T . Different symbols stand for different samples
Surfaces Positrons implanted into a solid can be backscattered into the vacuum or can thermalize and start to diffuse. When the implantation energy amounts to only a few keV, positrons have a high probability of reaching the surface during diffusion. In contrast to electrons, the positron work function of surfaces may have negative values, i.e. positrons may be spontaneously emitted and they have a kinetic energy corresponding to the thermally spread work function.
Fine-Grained Material and Diffusion Trapping Model Positron spends a finite time:100 ps – 300 ps, prior the annihilation by randomly walking with thermal energies. In the conventional experiments the diffusion process is also present but well visible in fine grain samples where the size of the grain is comparable with the positron diffusion length defined as follows: where D + is the positron diffusion coefficient and t is the positron lifetime. In metals L + it is about 0.1 mm and in semiconductors 0.2 mm. The transition rate from the free to the localized state is described by the parameter equal to the width of the boundary times the trapping rate parameter which is related with the cross section for absorption of positrons by the grain surface.
Assumptions During thermalization process the positrons are located in two distinct regions of a sample, e.g., grain and its boundary, in which they next annihilate, therefore, only the volume ratio of the two regions is important. The main assumption of DTM: grain boundary is a perfect sink for positrons in which they are localized and then annihilate with the rate: b =1/ b < f , (Smoluchowski b.c.). In the interior of the grain positrons may randomly walk and annihilate with the rate: f =1/ f , where f is the positron lifetime in a free state. The number of trapped positrons at the grain boundary, denoted as n b , is a function of time. The same is with the local positron concentration within the grain: . Both functions must fulfil the equations (Dryzek et al. 1998): where Σ is the grain boundary. The first equation is a diffusion equation for positrons which can also annihilate in the grain interior. The second one is the rate equation for the trapped positrons, and the Third one exhibits the fact that only the positrons which pass through the surface are able to be localized there.
Predictions of DTM For the spherical grain of radius R the mean positron lifetime Similar for the value of the S-parameter Langevin function Sf and Sb are the S-parameter for annihilation in the free and bound state mean positron lifetime and the intensity of the longest lifetime component depend upon the radius of the grain or the positron diffusion length L+ . Also positron lifetime spectrum contains an infinity number of lifetime components because of the infinity radius of the grain or small value of the positron diffusion length.
<ul><li>Confirmation of the prediction of the DTM has been found by Dryzek 2002 using the multilayer stacking system of copper obtained by the electrodeposition process. </li></ul><ul><li>Changing the thickness of the copper layer on the stacking sequence one could control the grain size. </li></ul><ul><li>By fitting it was found the value of the positron diffusion length L+=94± 20 nm and the transition rate: ז f /L+=(2.83± 1.2)x10 3 . It corresponds well with the positron diffusion length obtained for the polycrystalline copper using the slow positron beam experiment: L+=121± 7 nm (Dryzek et al. 2001). </li></ul>
DTM and nano-crystallites <ul><li>If the grain size is getting much smaller than the diffusion length L+ then the positrons attain the grain boundary of the small crystallites with high probability and lifetimes longer than that of the “free” delocalized state are observed. </li></ul><ul><li>Thus the DTM is not valid any more and the interpretation of the positron characteristics by positron trapping in interfacial free volumes of the nanocrystalline grains is sufficient. </li></ul><ul><li>For so called nanocrystalline solids in the positron lifetime spectra even three or four lifetime components all higher than f can be resolved. </li></ul><ul><li>The production procedure has effect significantly on the measured values of positron lifetimes even for the same material, also the temperature treatment and pressure ( Schaefer (1992)). </li></ul><ul><li>Nonocrystalline materials seems to be interesting for the surface positron studies. </li></ul>Schaefer et al. 1988 reported in nanometer-size Fe polycrystals (the average size about 10 nm) four lifetime components. 1=180 15 ps is attributed to the free volumes of approximately the size of a monovacancy in the crystallite interfaces, 2=360 30 ps is attributed to the free volumes at the intersections of two or three crystallites interfaces, such volume can be a cluster of 10-15 vacancies with a spherical diameter of about 0.6 nm, 3 1300 ps and 4 4000 ps indicate the formation of o- Ps at the internal surfaces of larger voids