Postron Lfetme Spectroscopy Postron Lfetme Spectroscopy The postron lfetme τ s a functon of the electron densty at the annhlaton ste. The annhlaton rate λ, whch s the recprocal of the postron lfetme τ, s gven by the overlap of the postron densty n + (r) = ψ + (r) 2 and the electron densty n (r) (Nemnen and Mannnen 979), z 2 2 + λ= = πr0 c ψ ( r) n ( r) γd r. () τ r 0 s the classcal electron radus, c the speed of lght, and r the poston vector. The correlaton functon γ = γ[n (r)] = + n /n descrbes the ncrease n n the electron densty due to the Coulomb attracton between a postron and an electron. Ths effect s called enhancement. When postrons are trapped n open-volume defects, such as n vacances and ther agglomerates, the postron lfetme ncreases wth respect to the defect-free sample. Ths s due to the locally reduced electron densty of the defect. Thus, a longer lfetme component, whch s a measure of the sze of the open volume, appears. The strength of ths component,.e. ts ntensty, s drectly related to the defect concentraton. In prncple, both tems of nformaton,.e. the knd and concentraton of the defect under nvestgaton, can be obtaned ndependently by a sngle measurement. Ths s the major advantage of postron lfetme spectroscopy compared wth angular correlaton of annhlaton radaton or Doppler-broadenng spectroscopy wth respect to defect ssues.. Bascs of the Measurement The conventonal postron lfetme measurement s possble snce a γ-quantum wth an energy of.27 MeV s emtted almost smultaneously wth the postron n the 22 Na source. The postron energy, whch extends up to 540 kev, decreases n the sample wthn a few pcoseconds by non-elastc nteractons. The mean postron penetraton depth of ths so-called thermalzaton process s of the order of 00 µm. The thermalzaton tme usually amounts to a few pcoseconds. It s thus small compared wth the postron lfetme and can be neglected. On reachng thermal energes, the postron dffuses n the perodc lattce potental before t s possbly trapped n a lattce defect. The dffuson length s n of order of 00 nm. Ths dstance determnes the number of atoms to be probed for postron traps durng the postron lfetme. Hence, the dffuson length strongly determnes the senstvty of the postron methods to detect defects. The postron lfetme of a sngle event can be measured by detectng the tme dfference between the brth γ-quantum of the β + -decay n the source and one of the annhlaton γ-quanta of an energy of 5 kev. The actvty of the source must be suffcently low n order to ensure that on average only one postron s n the sample. Ths avods the ntermxng of start and stop quanta orgnatng from dfferent annhlaton events. A specal sandwch arrangement of fol source, samples, and detectors guarantees that all postrons emtted from the source are penetratng the sample materal. The γ-rays are converted by scntllator photomultpler detectors nto analog electrcal pulses. The pulses are processed by dscrmnators. Ther output pulses start and stop a tme-to-ampltude converter as an electronc stopwatch. The ampltude of the output pulse s proportonal to the tme dfference between the brth and the annhlaton γ-quanta and, thus, represents a measure of the postron lfetme. The sngle annhlaton event s stored after analog dgtal converson n the memory of a mult-channel analyzer. The channel numbers represent the tme scale. In order to obtan the complete lfetme spectrum, more than 0 6 annhlaton events must be recorded.
Postron Lfetme Spectroscopy 2 The scheme of the postron lfetme measurement s shown n Fg. BaF 2 or plastc scntllators and photomultplers wth a short pulse rse-tme are used to obtan a hgh tme resoluton. The dscrmnators suppress nose and generate standard tmng pulses by the constant-fracton dscrmnaton prncple. Ths prncple s favored over leadng-edge dscrmnaton n order to ensure stable tme markers ndependent of the pulse heght. Another task s to guarantee that the.27-mev and 0.5-MeV quanta are accepted only n the approprate channels. The dscrmnators are of dfferental type (sngle-channel analyzer) and accept nput pulses wthn an adjustable energy wndow. The tmng pulses are used to start and stop the chargng of a capactor n the tme-to-ampltude converter (TAC). The tme lnearty s ensured there by constant-current chargng that s stopped at the arrval of the stop pulse orgnatng from the annhlaton γ-quantum. The stop pulse s coax-cable delayed n order to shft the tme spectrum nto a lnear regon of the TAC. The spectrum s stored n a mult-channel analyzer. Ths expermental arrangement s called fast fast concdence setup. The term s related to the fact that the tme measurement as well as the energy selecton s performed n a fast channel. A slow channel was used for energy selecton when fast dfferental dscrmnators were not avalable at the begnnng of postron lfetme experments. Ths arrangement s called a fast slow setup. Inexpensve mult-channel plug-n boards for personal computers wth about 2000 channels are suffcent for storng the spectra. The tme resoluton of the spectrometer s determned manly by the scntllator multpler part and ranges between 80 and 280 ps. The practcal consequence of ths relatvely poor resoluton s the lmtaton of the determnaton of postron lfetme components larger than about 50 ps. The determnaton of postron lfetmes can, however, be carred out wth an accuracy of about ps. 2. Data Treatment The tme-dependent postron decay spectrum D(t) n the sample s gven by k + t D( t) = I exp τ = F HG I KJ. (2) k dfferent defect types contrbutng to the postron trappng are related to k + components n the spectra wth the ndvdual lfetmes τ and ntenstes I. If no postron traps are present n the sample, (2) s reduced to D(t) = exp( t/τ b ), where τ b s the postron lfetme n the defect-free bulk of the sample. The postron lfetme spectrum N(t) s the absolute value of the tme dervatve of the postron decay spectrum D(t), Fg.. Scheme of the postron lfetme experment n fast fast concdence. The lfetme s measured as the tme dfference between the appearance of the start and stop γ-quanta (PM photomultpler, SCA sngle-channel analyzer). The ampltude of the tme-toampltude converter (TAC) analog output pulse s proportonal to ths tme dfference. The whole lfetme spectrum N(t) s stored n a mult-channel analyzer (MCA).
Postron Lfetme Spectroscopy 3 0 6 As grown Cz S Plastcally deformed S 0 5 τ 2 = 320 ps Counts 0 4 0 3 τ b = 28 ps τ 3 = 520 ps 0 2 3 4 5 Tme [ns] Fg. 2. Expermental postron lfetme spectra obtaned n as-grown and n plastcally deformed Czochralsk-grown (Cz) slcon (Hübner et al. 997b). The curve of the deformed sample (3 % stran, deformaton temperature 775 C), s located sgnfcantly hgher, ndcatng the occurrence of long-lved lfetme components. After the decomposton of the upper spectrum, the obtaned lfetme components τ 2 and τ 3 are added as straght lnes n the sem-logarthmc plot for llustraton. The τ component s not ndcated as a straght lne (τ = 20 ps). Only one lfetme component correspondng to the postron bulk lfetme τ b s found n the as-grown sample. The devatons from the straght lne at hgher tmes are due to annhlatons n the source and the background contrbuton. The Gaussan-lke shape of the left part of the curve s manly caused by the resoluton functon. k + I t N( t) = exp τ τ = F HG I KJ. (3) The spectrum s shfted on the tme scale by the delay cable (Fg. ) to tme zero t 0, and consequently the tme t n (2) and (3) has to be replaced by t t 0. Typcal postron lfetme spectra obtaned n as-grown slcon and n S after plastc deformaton are shown n Fg. 2. The expermentally obtaned spectra dffer from the analytcal descrpton (3) manly by the convoluton wth the tme resoluton functon F t (t). In general, t s approxmated by a sum of several Gaussans or one Gaussan wth exponental tals. In the case of the use of plastc scntllators, t s often suffcent to take a sngle Gaussan G(t) centered at t 0, L NM F HG t t0 G( t) = exp σ π σ I K J 2 O QP. (4) The tme resoluton s characterzed by the full wdth at half maxmum, FWHM, FWHM = 2σ ln 2. (5) s wth the standard devaton σ s. The convoluton of the spectra (2) wth F t (t) gves the convolved decay spectrum Df ( t) = z D( t t' ) Ft ( t' )d t'. (6)
Postron Lfetme Spectroscopy 4 If F t (t) = G(t),.e. a sngle Gaussan s taken as the resoluton functon, the postron decay spectrum s gven as k + 2 I t t s / t t Df ( t) = exp 0 σ 4τ erf 2 τ 2σ τ σ = L NM a f O L QP F IO 0 NM HG s s KJ. (6) QP Standard computer programs based on Gauss Newton non-lnear fttng routnes are avalable for the decomposton of the spectra (e.g. LIFSPECFIT by Puska 978, POSITRONFIT or PATFIT by Krkegaard et al. 989). A model spectrum wth a gven number of decay components and a gven resoluton functon s used for the least squares ft to the measured spectrum by the varaton of the parameters of the lfetme components. The lnear parameters,.e. the ntenstes and the background, are ftted ndependently of the non-lnear ones, whch are the lfetmes τ and the tme-zero channel t 0. Only a one-component lfetme spectrum s obtaned n a defect-free semconductor. The deformed sample of Fg. 2 contans two defect types wth open volume, vz. vacances and vods, and thus, three dstnct components appear as a superposton of straght lnes n the sem-logarthmc plot, whch are to be folded by the tme resoluton functon to gve the lfetme spectrum. The spectra addtonally contan background contrbutons as well as annhlaton events n the postron source. The source correcton has to be performed after background subtracton. Ths means subtractng the characterstc lfetme spectrum of the source. The determnaton of ths source spectrum s rather complcated and usually carred out n such a way that a one-component spectrum of a defect-free sample s analyzed. The source components are vared usng a sngle-component ft to get the best ft,.e. the smallest varance. The problem of the source correcton was nvestgated n detal on measured and smulated spectra (Staab et al. 996). It was found that sources prepared from Al fols provde a three-component source spectrum. The fracton of the annhlatons n the source s not only determned by the source tself but t s also a functon of the atomc number of the sample and ncreases due to multple postron backscatterng through the source. The determnaton of the number of components n samples wth unknown defect populatons s started by a one-component ft of the spectrum. Addtonal components are added as long as the varance of the ft decreases. However, the possblty of obtanng a relable mult-component ft s restrcted due to the lmted tme resoluton of the spectrometer and the statstcs of the measurements. A number of one to four components can be resolved, dependng on the ntensty and the mutual separaton of the ndvdual lfetme components, the tme resoluton, and the statstcs. These effects were studed wth Monte-Carlo-smulated as well as expermentally obtaned spectra (Somesk et al. 996). It was found that the possblty of decomposng the postron lfetme spectra depends essentally on the number of components n the spectrum, the dstance of the ndvdual lfetmes, and the statstcs of the measurement. At least fve mllon events should be collected for a relable lfetme separaton, even f only two components are present. If only the average postron lfetme s to be measured, a lower number of counts s suffcent. The decomposton of spectra descrbed above s related to spectra havng dscrete lfetme components, as s expected n most cases for semconductors. A nearly contnuous dstrbuton of lfetme components, as found n polymers (Jean 995), may occur n semconductors f vacancy clusters of dfferent szes are present. In ths case, and f the number of components s too hgh to be resolved ndvdually, a contnuous dstrbuton of lfetmes can be ftted to the expermental data and an ntensty versus lfetme plot s obtaned. Such a data analyss s based on Laplace transformaton of the measured spectrum (CONTIN program, Provencher 982).
Postron Lfetme Spectroscopy 5 Intensty [arbtrary unts] 00 000 Postron lfetme [ps] Fg. 3. Result of a maxmum entropy of lfetme (MELT) analyss of the postron lfetmes of plastcally deformed Czochralsk-grown slcon (Hübner et al. 997b). The sample s the same as n Fg. 2 (3.5 % deformaton at 775 C). The peaks obtaned correspond well to the dscrete lfetmes from the decomposton of the spectrum n Fg. 2, τ = 20 ps, τ 2 = 320 ps, and τ 3 = 520 ps. Another approach has been made by Shukla et al. (993) usng lnear flterng and the method of maxmum entropy (MELT program). Prncpally, both methods can also be appled to dscrete lfetme spectra. The output s a graph dsplayng the ntensty versus lfetme. Knowledge of the number of components s not requred. Ths s an advantage for mult-component spectra, but t has not found a broad applcaton to semconductors up to now. The result of a MELT analyss of a plastcally deformed slcon sample s shown n Fg. 3. The same lfetmes were obtaned from the decomposton by a non-lnear ft accordng to (3) takng nto account the resoluton functon. Maxmum entropy of lfetme.
Postron Lfetme Spectroscopy 6