Detailed description of the Positron Annihilation Technique.

1. Study of semiconductor with Positron

2. 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.

5. 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

7. Positron in condensed matter

8. Positrons are repelled by positive atom cores Vacancy represents a positron trap due to the missing nuclei (potential well for a positron)

12. Techniques of Positron Lifetime Spectroscopy

13. 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.

15. 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

16. Data Treatment

17. Data Treatment

18. 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.

19. Doppler broadening of Plastically deformed (blue) and as grown GaAs (red)

21. Angular correlation of annihilation radiation Precise measurements of the exact collinearity of the two annihilation gamma rays can be used to test momentum conservation

22. 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.

24. 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

26. Types of point defects or vacancies

30. 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

31. 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

32. 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.

33. 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.

34. 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.

35. 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.