Abstract
Positron annihilation spectroscopy (PAS) was used to examine the effect of defined Cd-rich and Te-rich annealing on point defects in Cl-doped CdTe and Ge-doped CdZnTe semi-insulating single crystals. The as-grown crystals contain open-volume defects connected with Cd vacancies 
. It was found that the Cd vacancies agglomerate into clusters coupled with Cl in CdTe:Cl, and in CdZnTe:Ge they are coupled with Ge donors. While annealing in Cd pressure reduces of the 
density, subsequent annealing in Te pressure restores 
. The CdTe:Cl contains negatively-charged shallow traps interpreted as Rydberg states of 
A-centres and representing the major positron trapping sites at low temperature. Positrons confined in the shallow traps exhibit lifetime, which is shorter than the CdTe bulk lifetime. Interpretation of the PAS data was successfully combined with electrical resistivity, Hall effect measurements and chemical analysis, and allowed us to determine the principal point defect densities.
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Introduction
Cadmium telluride (CdTe) and Cadmium zinc telluride (CdZnTe) are important materials for multiple applications in solar cells1,2,3, X-ray and gamma-ray room-temperature detectors4,5, electro-optic modulators6, and substrates for CdHgTe infra-red detector epitaxy7. In case of room-temperature semiconductor detectors these compounds overcome convention materials (Si, Ge, GaAs) by optimum conjunction of principal parameters - large atomic number of elements, large enough band gap energy, convenient electron mobility and attainable high resistivity and low enough trap density - necessary for successful detector manufacture8.
In spite of the high interest and expectation on CdTe/CdZnTe-detectors, till nowadays they have been hardly commercialized due to high expenses for fabrication high quality crystals. A CdTe crystal with a high resistivity 
is required for a good detector in order to reduce the dark current. Moreover a large mobility-lifetime product is desired to ensure that carriers generated by radiation in the whole detector volume can be collected by electrodes. Lattice defects form energy levels in the band gap and affect thereby the resistivity and the mobility lifetime product9. CdTe can be alloyed with Zn forming CdZnTe ternary compound. Alloying of CdTe with Zn strengthens the lattice and increases the band gap and thereby also maximum achievable resistivity10.
The electric properties are affected by lattice defects forming energy levels in the band gap9. The determination of the properties of native defects and their complexes is particularly important to understand their role in crystal growth and compensating of extrinsic defects4. In spite of extensive investigations the defects attached to distinct levels have in most cases not been completely identified. High purity CdTe/CdZnTe crystals are usually grown under Te-rich conditions because the Cd pressure required for Cd-rich conditions is as high as several atmospheres11. The CdTe/CdZnTe crystals grown under Te-rich conditions are typically characterized by low resistivity and p-type conductivity4 due to Cd vacancies 
which are divalent acceptors with two ionization levels 
and 
located below the mid-gap level4,12. To achieve high resistivity 
acceptors and residual impurities must be electrically compensated usually by doping using group III (Al, In, Ga) or group VIII (Cl) donors4,12.
Positron annihilation spectroscopy (PAS) including Positron lifetime (LT) and Coincidence Doppler broadening (CDB) spectroscopy was usually used for the study of the structure of point defects in CdTe/CdZnTe in the past13,14,15,16,17,18, but the link between defects and controlled annealing treatment has never been studied before.
This work reports on an investigation of the influence of defined annealing in Cd or Te vapour on the defect structure of Cl-doped CdTe and Ge-doped 
semi-insulating single crystals. The defect structure is characterized by PAS before and after each annealing step. The results are compared with galvanomagnetic Hall effect measurements in the Van der Pauw configuration19.
The paper is organized as follows. In Section Experimental we introduce samples, show the impurity content and summarize technical details of experiments. In Section Theory, we introduce methods for lifetime calculations and discuss their limits. Section Results and Discussion presents galvanomagnetic, positron back-diffusion measurements and theoretical calculations of lifetimes of positrons captured in different point defects. The subsection Lifetime spectroscopy reports the results of temperature dependent positron lifetime spectroscopy and their analysis based on three-state simple trapping model. This subsection is divided to the three parts describing for the undoped CdTe, Cl-doped CdTe and Ge-doped CdZnTe. The next subsection contains the results of CDB spectroscopy and their interpretation.
Experimental
The Cl-doped CdTe (CdTe:Cl) and Ge-doped 
, 
(CdZnTe:Ge) single crystals were grown by the vertical gradient freeze method20 (using 6N purity source elements). The annealing was done in the two zone furnace at 700 °C while the Cd or Te source temperature was 600 °C, which corresponds to Cd or Te pressure of 111 or 7.7 mbar, respectively. After annealing, the samples were cooled down at a cooling rate of 1 °C min−1. Nominally undoped CdTe (u-CdTe) single crystal grown by the vertical gradient freeze method at fairly increased Cd overpressure 1.6 atm was used as a reference sample.
LT measurements were carried out using a digital spectrometer21 with time resolution of 145 ps. 22Na activity of 1 MBq deposited on a 2 μm thick Ti foil was used as a positron source. The LT measurements were performed at 295 K (room temperature - RT) and at 123 K (low temperature), respectively. In both temperatures typically, 107 positron annihilation events were collected in LT spectra, which were subsequently decomposed into exponential components. The contribution of positron annihilations in the source spot and in the covering foils were always subtracted. In addition, temperature-dependent LT measurements were carried out in the temperature range of 123–295 K at a heating rate of ~1 K hour−1. Here the statistics of 
events was collected at each temperature step.
CDB spectroscopy22 was employed for characterization of local chemical environment of vacancies. The CDB studies were performed at RT using the same positron source as in the LT measurements. A digital CDB spectrometer23 equipped with two HPGe detectors and characterized by the energy resolution of 0.9 keV at the annihilation line and the peak-to -background ratio 
was employed for the CDB studies. At least 
annihilations were collected in each two-dimensional CDB spectrum. Subsequently the CDB spectra were reduced into one-dimensional cuts representing the resolution function of the spectrometer and the Doppler broadened annihilation peak. Normalized Doppler broadened peaks were divided by the normalized peak for the u-CdTe reference sample. Hence, in this paper the CDB results are presented as ratio curves with respect to u-CdTe.
Positron back diffusion measurement was performed on a continuous magnetically guided slow positron beam with energy of incident positrons adjustable in the range from 0.05 to 30 keV. Doppler broadening of the annihilation photopeak was measured by a HPGe detector with the relative efficiency of 35% and the energy resolution of 
keV at 511 KeV. Shape of Doppler broadened annihilation photopeak was characterized using the S (sharpness) parameter24. The dependence of of the S parameter on positron energy was fitted using the VEPFIT code25.
Glow Discharge Mass Spectrometry (GDMS) was used for the chemical analysis. The concentrations of elements exceeding 
for u-CdTe, CdTe:Cl and CdZnTe:Ge presented in Table 1 in units 1015 cm−3. A 30% error of GDMS analysis was declared.
Theory
Positron properties were calculated using density functional theory (DFT) within so-called standard scheme26. In this approximation positron density is assumed to be everywhere vanishingly small and not affecting the bulk electron structure. At first electron density 
in the material is solved without the positron. Subsequently, the effective potential for positron is constructed as
where 
is the Coulomb potential produced by the charge distribution of electrons and nuclei and 
is the zero positron density limit of the electron-positron correlation potential26.
Positron wave functions 
were calculated by solution of a single particle Schrödinger equation
where 
is the energy eigenvalue for i-th positron state. In the present calculations we considered the positron ground state only.
The positron annihilation rate (i.e. the inverse of positron lifetime) is determined using the expression
where 
is the classical electron radius, c is the speed of light, and γ denotes the electron enhancement factor describing the pileup of electrons at the positron site26.
The electron-positrons correlation, i.e. the correlation potential 
and the enhancement factor γ, were treated by two approaches:
-
i
local density approximation (LDA) utilizing the parametrization by Boroński and Nieminen27 and taking into account incomplete positron screening28 using a high frequency dielectric constant of
29 and -
ii
generalized gradient approach (GGA) within the approach introduced by Barbiellini et al.30.
The electron density 
was constructed by superposition of atomic electronic densities calculated by a relativistic atomic code31. This approach called atomic superposition (ATSUP)32 neglects the charge transfer, but it is computationally feasible and can be used even for very large supercells retaining full 3-D geometry of the problem. In the following text the approach employing LDA and GGA scheme with superimposed electron density is denoted ATSUP-LDA and ATSUP-GGA, respectively.
The ATSUP-LDA and ATSUP-GGA calculations of positron parameters were performed on 
supercells containing 2048 Cd and 2048 Te atoms. Defects were modelled by removing the corresponding number of atoms from the supercell. Integration over the Brillouin-zone described in ref. 33 was used in calculations of positron parameters for defects.
It is well known that LDA overestimates positron annihilation rates especially with d electrons30 and the lifetimes calculated within the LDA approach are often shorter than the values measured in experiment. This shortcoming is to some extend compensated when LDA approach is used with electron density constructed by ATSUP32,34. On the other hand, the GGA approach when used with ATSUP electron density gives often lifetimes which are longer than the experimental values35. Hence, the lifetimes calculated using the LDA and the GGA approach can be considered as a lower and an upper bound of the interval where the actual lifetime falls. The GGA scheme is more sensitive to details of electronic structure than LDA30 and the lifetimes calculated using the GGA approach are it the best agreement with experiment when used with a self-consistent electron density. For these reasons we performed also self-consistent electron density calculations for selected positron states. The self consistent valence electron density was calculated by the plane wave code VASP (Vienna ab-inito simulation package)36,37 using projector augmented wave (PAW) potentials38. The 
, 
Cd electrons and 
, 
Te electrons were considered as valence electrons in the VASP calculations. The calculations were performed using 216 atoms based supercells and 
k-point grids generated using the Monkhorst-Pack scheme39. The wave functions were expanded in a plane wave basis with the cut-of energy of 277 eV. The calculated CdTe lattice constant a = 6.60 Å is in a reasonable agreement with the experimental value of 6.48 Å. Ionic relaxations were not considered at the present stage of calculations. In construction of the positron potential the frozen core electron orbitals were added to the self-consistent valence electron density calculated by VASP. In the following text this approach is denoted VASP-GGA.
The momentum distribution of annihilating pairs was calculated employing the ATSUP-based approach described in refs 40 and 41. The electron-positron correlations were treated within the GGA approach. The contribution 
from the i–th atom and a shell characterized by principal (n) and orbital (l) quantum numbers is calculated by the formula
where 
denotes the number of electrons in the 
shell, 
is the spherical Bessel function and 
and 
denotes the electron and positron radial wave functions. The symbol 
stands for the state-dependent positron enhancement factor41. The momentum distribution of the annihilating electron-positron pairs is obtained by summing the partial contributions 
over all occupied atomic sites and corresponding electron shells. Core electrons localized in atomic shells are practically not affected by crystal bonding. Hence, the ATSUP-based approach describes well the high momentum part of the momentum distribution where the contribution of positrons annihilated by core electrons dominates. But since it neglects the charge transfer it fails to describe properly the low momentum part of the momentum distribution defined by positrons annihilated by low momentum valence electrons.
In order to mimic the effect of the finite resolution of the experimental setup, the theoretical momentum distribution curves were convoluted with a Gaussian with FWHM of 
. To highlight the high momentum part of momentum distributions where the contribution of core electrons dominates the calculated momentum distributions are presented as ratio curves related to a perfect CdTe crystal.
Results and Discussion
Galvanomagnetic measurements
The galvanomagnetic properties of as-grown u-CdTe, CdTe:Cl and CdZnTe:Ge and annealed CdTe:Cl and CdZnTe:Ge samples are summarized in Table 2. The properties of u-CdTe accord with its growth at Cd overpressure, where the VCd-related defects are suppressed and the electron density is afforded by residual noncompensated donors. The as-grown CdTe:Cl and CdZnTe:Ge samples exhibit rather high resistivity, which is typical for the growth of Te-rich materials, where excess donors dominating above shallow acceptors are compensated by VCd8,42.
Annealing in Cd vapour leads to a decrease in resistivity for both samples. CdTe:Cl exhibited n-type conductivity due to the suppressed VCd and dominating shallow donors ClTe. The free carrier concentration of ~1017 cm−3 therefore gives a lower limit for the concentration of Cl dopants in the sample43. Cd vapour annealing of CdZnTe:Ge yield p-type conductivity contrary to CdTe:Cl. Such a behaviour is well known in annealed undoped or weakly doped CdTe/CdZnTe44, where acceptor-like impurities are dominant after annealing. In our CdZnTe:Ge, the extrinsic shallow donor ((B) + (Al)) and shallow acceptor ((P) + (Au) + (Cu)) densities are similar, and weak excess acceptor contamination is assumed. Subsequent annealing in Te vapour produced p-type conductivity in both samples. This is in agreement with expected VCd formation.
Calculated positron lifetimes
Positron lifetimes, calculated using ATSUP-LDA, ATSUP-GGA and VASP-GGA approaches for various positron states in CdTe are listed in Table 3. Theoretical positron lifetimes available in literature are shown in Table 3 as well. The positron lifetimes calculated using the ATSUP-LDA and ATSUP-GGA schemes in the present work are in an excellent agreement with the lifetimes calculated in refs 15,45. One can see in Table 3 that positron lifetimes calculated using various approaches for the electron-positron correlation, i.e. the enhancement factor and the electron-positron correlation potential, differ. The positron lifetimes calculated using the LDA approach are always lower than those obtained by the GGA scheme. Moreover, calculated positron lifetimes vary depending whether superimposed or self-consistent electron density was used in the calculation. The bulk positron lifetime for CdTe calculated using the ATSUP-LDA approach is 
ps, while the ATSUP-GGA scheme yielded higher bulk lifetime 
ps. For comparison experimental lifetimes reported in literature for CdTe are listed in Table 4. The experimental bulk positron lifetimes for CdTe reported in literature fall into the range from 280 to 291 ps13,14,15,16,18,46,47. The VASP-GGA scheme which can be considered as the most precise approach used here yielded the CdTe bulk positron lifetime 
ps which is comparable with the experimental values.
In order to compare positron lifetimes for various defects with the experimental values we use ratios 
of the calculated lifetime 
to the bulk lifetime 
calculated by the same approach. These ratios are listed in Table 5. The differences between various theoretical schemes are to a large extend cancelled using 
ratios and the 
values exhibit relatively low sensitivity to the approach used for electron-positron correlation30,33.
While 
is believed to be positively charged in CdTe, 
is either neutral 
or negatively charged 
depending on the Fermi level position12. Hence, positrons are repelled by 
while 
represent trapping sites for positrons. This is illustrated in Fig. 1 presenting the positron density in the (001) plane calculated by the ATSUP-GGA approach. Figure 1a shows the positron density in a perfect CdTe crystal where positron is de-localized in the lattice. The positron density calculated for a CdTe crystal containing 
is plotted in Fig. 1b. Obviously the positron is localized in 
. Experimental evidence for positron trapping in 
has been reported by many authors14,15,16,18,46. The lifetimes of positrons trapped 
determined experimentally fall into the range from 320 to 325 ps which corresponds to 
falling into the interval 1.11–1.15, see Table 4. The ATSUP-LDA and ATSUP-GGA calculations for positrons trapped at 
resulted in 
, which is remarkably lower than the experimental values. On the other hand, the VASP-GGA approach yielded 
which is close to the values measured in experiment. This indicates that charge transfer which was neglected in ATSUP calculations leads to a slight decrease of electron density in 
and consequently an increase of the lifetime of trapped positrons.
Positron density in the (001) plane calculated using the ATSUP-GGA approach (a) perfect CdTe crystal, (b) CdTe crystal containing VCd in 1, 1, 0 position, (c) CdTe crystal containing AC defect consisting of VCd and ClTe located in 1, 1, 0 and 0.25, 1.75, 0.25 position, respectively. The co-ordinates are expressed in the units of the CdTe lattice constant a. Positron density is given in the atomic units.
Positron lifetime was calculated also for 
associated with Te anti-site 
and with various impurities. One can see in Table 3 that 
causes shortening of positron lifetime. A 
donor associated with 
acceptor forms 
complex which is electrically neutral. Theoretical calculations revealed that positrons trapped at 
complexes exhibit practically the same lifetime as 
. Hence, these two kinds of defects cannot be distinguished by measurement of positron lifetime.
In Cl doped CdTe 
may couple with 
shallow donors forming negatively charged 
complexes called A-centres 
or neutral 
complexes. Results in Table 3 indicate that replacement of Te nearest neighbours of 
by Cl atoms increases the lifetime of trapped positron. This effect is caused by smaller size of Cl atoms compared to Te ones. Hence replacement of a Te nearest neighbour of 
by a Cl impurity increases the open volume of vacancy. This is illustrated in Fig. 1c which presents calculated positron density in the (001) plane for a positron trapped at AC defect. One can see in the figure that positron density in AC becomes asymmetric since it expanded towards the Cl nearest neighbour.
Lifetime spectroscopy
The results of LT measurements, i.e. lifetimes 
and relative intensities 
resolved in LT spectra for u-CdTe, CdTe:Cl and CdZnTe:Ge samples, are summarized in Table 6 for temperatures 295 and 123 K, respectively. The temperature dependences of the mean positron lifetime 
are presented in Fig. 2.
The lines represent the fit using 3-STM.
Undoped CdTe
The u-CdTe crystal exhibits a single component LT spectrum (except of the source contribution) with lifetime of ~290 ps, which fits in the CdTe bulk lifetimes reported in literature, see Table 4. Hence, the concentration of positron traps in the u-CdTe crystal is very low (below the sensitivity threshold of LT spectroscopy) and virtually all positrons are annihilated in the de-localized state. Low temperature LT measurements of the u-CdTe crystal yielded practically the same results as the measurement at RT.
Results of positron back-diffusion measurement, i.e. dependence of the S parameter on the energy E of incident positrons in plotted in Fig. 3. The positron penetration depth increases with increasing energy of implanted positrons. Hence, at very low energies virtually all positrons are annihilated at the surface. With increasing energy positrons penetrate deeper into the sample and the fraction of positrons diffusing back to the surface gradually decreases. Finally at high energies virtually all positrons are annihilated in the bulk and the S parameter approaches a plateau. One can see in Fig. 3 that the 
curve exhibits a maximum at energy around 2 keV. This indicates that the sample contains a thin surface layer of native oxides most probably a mixture of 
, 
and CdO48,49. The 
curve for the u-CdTe sample was fitted by VEPFIT25 using a two layer model: (i) native oxide layer on the surface and (ii) CdTe bulk region. The model curve calculated by VEPFIT is plotted in Fig. 3 by a solid line and is obviously in good agreement with the experimental points. The thickness of the native oxide layer ≈30 nm was fitted.
Solid lines show model curves calculated by VEPFIT.
The positron diffusion length determined in the CdTe region is 
nm. This value is comparable with the positron diffusion lengths in defect-free semiconductors24. Hence, the u-CdTe crystal contains very low density of positron traps (except of the thin surface oxide layer) and almost all positrons are annihilated from the free state in accordance with the results of positron lifetime characterization. Assuming that the u-CdTe sample contains no positron traps one can calculate the positron diffusion coefficient for u-CdTe from the expression 
, where 
ps is the bulk u-CdTe lifetime. This yields the positron diffusion coefficient for u-CdTe 
cm2s−1 which is in very good agreement with the value 
cm2s−1 reported for CdTe by Neretina et al.50.
The u-CdTe crystal exhibits very low density of dislocations which does not exceeds 109 m−2 as determined by etch pitting technique. We performed an additional independent test whether positrons in u-CdTe crystal are annihilated in the free state and the lifetime of ~290 ps measured in this sample can be indeed considered as the bulk lifetime. The u-CdTe sample was mechanically grinded by F600 SiC paste with grain size 9 μm. Plastic deformation caused by grinding introduced dislocations into the sample. The grinded u-CdTe sample exhibits two-component LT spectrum with lifetimes 
ps and 
ps and corresponding relative intensities 
% and 
%. Hence, it is clear that grinding introduced defects into the sample. The lifetime 
of the second component is comparable with the lifetime of positrons trapped at 
(see next section). This is typical for positrons trapped at dislocations24,51. Thus, the component with lifetime 
can be attributed to positrons trapped at dislocations introduced by grinding. The lifetime 
of the first component was shortened with respect to the as-grown u-CdTe sample. This is in accordance with the two-state simple trapping model24,52. When positrons are trapped at defects then the lifetime 
of the free positron component becomes shorter than that in the defect-free material and the quantity
equals to the positron bulk lifetime, i.e. 
24,52. For the grinded u-CdTe Eq. (5) yielded 
ps which is in an excellent agreement with the lifetime measured in the as-grown u-CdTe. This gives strong evidence that the lifetime of ~290 ps measured in the as-grown u-CdTe comes indeed from annihilations of free positrons. If the lifetime measured in the as-grown u-CdTe would come from positrons trapped at some defects, then introduction of additional defects (here dislocations) by grinding would not cause any shortening of the lifetime 
.
CdTe:Cl
The LT spectra of the CdTe:Cl crystal generally consist of up to three components (except of the source contribution) corresponding to the three positron states: (i) free positrons (lifetime τ1), (ii) positrons trapped in shallow traps (lifetime 
, trapping rate 
, and (iii) positrons trapped in deep traps (lifetime 
, trapping rate 
. The kinetics of positron trapping for these defects is described by the three-state simple trapping model (3-STM)24. The rate equations for such system read24
with the initial conditions 
and 
. The symbols 
, 
and 
denote the probabilities that a positron is at time t de-localized in the free state or trapped in a shallow or deep trap, respectively. 
and 
stand for the positron trapping rates for the shallow and the deep traps, respectively, and 
is the de-trapping rate from the shallow traps. The lifetimes of positrons trapped at the shallow and the deep traps, i.e. inverse of the annihilation rates of trapped positrons, are denoted 
and 
, respectively. Solution of the differential equations (6) gives the decay spectrum of positrons
where
and
The LT spectrum 
is the number of positrons annihilated in various times and is obtained as a negative time derivative of the decay spectrum 
Hence the lifetimes 
, 
, 
of the exponential components resolved in experimental LT spectra and their relative intensities 
, 
, 
can be interpreted within 3-STM using Eqs (8. Note that for a high concentration of shallow or deep positron traps the lifetime 
becomes extremely short and its intensity 
diminishes. In such case it is not possible to resolve the first component in LT spectrum due to limited resolution of LT spectrometer. This situation happened here in the case of low temperature LT spectra for CdTe:Cl. The short lived component could not be resolved in LT spectra due to very high density of shallow traps in this sample as will be shown in the following text.
The probability that a positron trapped in a shallow trap will escape by thermal excitation decreases with falling temperature. Hence the positron de-trapping rate from shallow traps 
depends on temperature53
where 
is the effective positron mass26, 
the Boltzmann constant, [c] the concentration of shallow traps, and 
the positron binding energy to the shallow traps. The temperature dependence of the mean lifetime 
was fitted using 3-STM with 
, 
, [c] and 
as fitting parameters. If the sample contains shallow traps, then the mean positron lifetime 
varies with temperature due to the temperature dependence of 
. Figure 2 shows that 
for CdTe:Cl decreases with temperature, which testifies to the presence of shallow traps in this sample. The temperature dependence of 
can be explained by the formation of series of attractive shallow Rydberg states connected with negatively-charged defects54,55. Since the positron binding energy of the shallow Rydberg states is rather small (typically ≤0.1 eV), they act as efficient positron traps at low temperatures only where the positron may not escape by thermal excitation54.
Hence, to describe LT spectra measured at RT 3-STM can be simplified by elimination of the shallow traps, i.e. by setting 
, 
in Eqs (6. This leads to the following rate equations
This simplified trapping model called two-state simple trapping model (2-STM) was used for analysis of the LT spectra measured at RT. The solution of 2-STM with the initial conditions 
and 
is two-component decay spectrum of positrons
where
The LT spectrum 
predicted by 2-STM is
Decomposition of LT spectrum of the as-grown CdTe:Cl measured at RT revealed two exponential components, see Table 6. The shorter component with lifetime 
ps represents a contribution of free positrons. Note that 
is shorter than the bulk positron lifetime for CdTe due to parallel positron annihilation and trapping in the deep traps, c.f. Eq. (14). The lifetime 
ps, 
, of the longer component is significantly higher than the positron lifetime for single vacancies in CdTe, see Table 3. Hence, we deduce that positrons in the as-grown CdTe:Cl are trapped in larger point defects with the open volume comparable with that of several vacancies. The positron trapping rate to these vacancy clusters can be calculated within 2-STM from Eqs. (14)
The concentration of vacancy clusters is directly proportional to the trapping rate 
determined by 2-STM, 
. The specific positron trapping rate for the vacancy clusters was estimated as 
, where 
is the specific positron trapping rate for neutral vacancies in semiconductors24 and 
is the average number of vacancies constituting the cluster. The latter value was estimated by comparison of the ratio 
determined experimentally with the theoretical calculations of clusters of the neutral 
complexes of various sizes in Table 5.
Extended investigations of CdTe:Cl with chlorine content from 100 to 3000 ppm performed by Krause-Rehberg et al.13 revealed two components in LT spectra measured at RT: (i) the shorter component, with a lifetime of 330(10) ps, 
, which was attributed to 
and (ii) a long-lived component with a lifetime of 450(15) ps, 
, which was assigned to clusters of 
neutral complexes. The concentration of both defects increased with Cl content, testifying that they are associated with chlorine. Our CdTe:Cl sample exhibits a smaller lifetime 
than 450 ps. This could be due to significantly lower Cl concentration leading to the formation of 
clusters of smaller size.
The analysis of the LT spectrum of the as-grown CdTe:Cl measured at 123 K revealed a new component with lifetime 
ps, which is shorter than the bulk positron lifetime (290 ps) and dominates the spectrum with 94% intensity, see Table 4. Taking into account the temperature dependence of 
in Fig. 2, this component obviously comes from positrons trapped in the shallow Rydberg states associated with negatively-charged A-centres 
. Indeed, the positron binding energy 
eV determined from fitting of the temperature dependence of 
falls into the range 
eV expected for the Rydberg states52. Evidently, 
. The concentration 
obtained from fitting of LT data by 3-STM is presented in Table 7. From the comparison of 
with 
we can estimate the specific positron trapping rate for the shallow traps 
s−1. This value falls into the expected range for 
calculated in ref. 54.
Note that Rydberg states are characterized by a lifetime which is close to the bulk lifetime54,55. Depending on the local electron density the lifetime of positrons confined at low temperatures in shallow traps can be either higher or lower than the bulk positron lifetime. The latter case has been recently reported for Cu in-diffused GaAs:Te crystals56. Low temperature LT studies of Cu in-diffused GaAs:Te revealed shallow traps characterized by a lifetime of 220 ps, which is about of 8 ps shorter than the GaAs bulk lifetime of 228 ps56. The shallow traps in Cu-indiffused GaAs:Te were identified as 
ions coupled with Ga vacancies 
. At RT positrons in the Cu-indiffused GaAs:Te sample are trapped at deep traps identified as 
complexes and characterized by a positron lifetime of 280 ps. At low temperatures positrons are confined predominantly in Rydberg states associated with 
ions and characterized by a positron lifetime of 220 ps, which is shorter than the bulk lifetime56. Thus, it seems that positron trapping in the CdTe:Cl sample is to some extent analogous to that in Cu-indiffused GaAs:Te: at RT positrons are trapped at deep traps (vacancy clusters) while at low temperatures positrons are confined at Rydber states associated with Coulomb field around negatively charged A-centers and characterized by a positrons lifetime 
ps, which is about of 
ps shorter than the CdTe bulk lifetime of 
ps measured on the u-CdTe sample, see Table 4. We assume that the reduced lifetime of the positron in the Rydberg state comparing to the bulk lifetime may stem from a lattice relaxation and enhanced mass density in the vicinity of the vacancy, at the radius about 1 nm, where the positrons in Rydberg state mostly occur.
Annealing the CdTe:Cl sample in Cd vapour led to a significant reduction of the intensity of the long-lived defect component and shortening of its lifetime down to 
ps at RT. The ratio 
is close to the value calculated for 
or single 
complex The reduction of this component is clearly caused by the removal of 
by Cd annealing and the reduction in cluster size. Ensuing analysis within 2-STM proves the depression of the density of 
related defects, see Table 7. The Cd-annealed CdTe:Cl is characterized by very low resistivity and n-type conductivity, caused by 
shallow donors which in contrast to the as-grown sample are not compensated by 
shallow acceptors. In spite of significant 
depreciation the sample measured at low temperature again exhibits a remarkable contribution of 
, see Table 6. The temperature dependence of 
plotted in Fig. 2 is similar to the as-grown sample just shifted to lower values due to the suppressed contribution from neutral clusters. Since the lifetime of positrons localized in the shallow traps is lower than the CdTe bulk lifetime, 
decreases below the bulk lifetime value at low temperatures where almost all positrons are confined in the shallow traps. Referring to similar concentration of defects at RT and at low temperature, see Table 7, we deduce that 
in the Cd-annealed CdTe:Cl sample is associated with 
only.
Subsequent annealing of a CdTe:Cl sample in Te vapour leads to the appearance of a defect component with lifetime 
ps, which corresponds to negatively-charged 
and neutral 
defects. Unlike the as-grown CdTe:Cl crystal, the 
complexes formed after Te-annealing are isolated and do not agglomerate into clusters. This could be caused by the fact that during Te-annealing the sample was kept at a lower temperature than during crystal growth, and the mobility of the 
complexes was therefore lower. The analysis of LT spectra measured at 123 K revealed that the positron trapping in 
dominates at low temperatures in a similar way as in the Cd-annealed crystal. The temperature dependence of 
presented in Fig. 2 again shows a drop with decreasing temperature in a similar manner as for the Cd-annealed sample.
Referring to the concentration of defects (Table 7), we can deduce that the defects observed at RT in the Te-annealed sample are associated with 
and 
at the total density 
. Similarly, the low temperature measurements allow us to establish 
, which yields 
. Consequently, the calculation of the total Cl density in 
-related complexes provides 
. This result is very close to the concentration of chlorine determined by GDMS 
. Since GDMS analysis is affected by up to 30% error, we consider this finding as an excellent confirmation of our assignment of defects.
Similar estimation can be done also for the as-grown CdTe:Cl sample where the concentration of 
clusters of 
was estimated in Table 7. Assuming that each cluster consists on average of 4 
complexes (as deduced from the ratio 
the concentration of Cl contained in 
clusters is 
. At the same time the low temperature measurement of the same sample yielded 
. Hence, the total concentration of Cl in 
-related complexes in the as-grown CdTe:Cl is 
. Again this value is in very reasonable agreement with the total Cl density determined by GDMS.
The as-grown CdTe:Cl crystal was characterized also by positron back-diffusion measurement on the slow positron beam. The dependence of the S-parameter on the energy E of incident positrons for the as-grown CdTe:Cl crystal is plotted in Fig. 3. Obviously the 
curve for the as-grown CdTe:Cl sample exhibits similar features as the curve for the u-CdTe crystal, namely a maximum at low energies (≈2 keV) followed by a gradual decrease down to a plateau value at high energies where virtually all positrons are annihilated inside the CdTe:Cl region. However the plateau value at high energies for the as-grown CdTe:Cl sample is remarkably higher than that for the u-CdTe. The 
curve was fitted in the similar manner as the curve for u-CdTe sample, i.e. using a two-layer model consisting of a thin surface oxide layer and the bulk CdTe:Cl region. The model curve calculated by VEPFIT25 is plotted by solid line in Fig. 3 and describes the experimental points accurately. The thickness of the oxide layer of ≈20 nm obtained from fitting is comparable with that measured in the u-CdTe. The positron diffusion length in the CdTe:Cl region is 
nm. This value is shorter than the positron diffusion length 
nm determined for the u-CdTe. Shortened positron diffusion length and higher S parameters at high energies give clear evidence that the as-grown CdTe:Cl sample exhibits higher density of defects than the u-CdTe. The concentration of defects [c] in the as-grown CdTe:Cl sample can be estimated from shortening of the positron diffusion length using the expression57
where v is the specific positron trapping rate to defects. From LT studies we know that the as-grown CdTe:Cl sample contains 
clusters consisting on average of four 
. Assuming 
one gets 
which is in very reasonable agreement with the concentration of 
clusters 
determined by LT spectroscopy in Table 7.
CdZnTe:Ge
At RT the CdZnTe:Ge as-grown crystal exhibits a defect component with lifetime 
ps falling into the range 
ps commonly attributed to 
shallow acceptors or complexes of 
associated with donors, see Table 4. The lifetime 
at 123 K is practically the same as in RT, and 
remains constant apart from the statistical scattering in the whole temperature range of 123–295 K, see Fig. 2. A similar result was observed17 for undoped CdZnTe. The temperature independence of 
observed in CdZnTe:Ge crystal testifies that (i) the defects responsible for the lifetime 
in this sample are mostly electrically neutral, and (ii) unlike CdTe:Cl, the CdZnTe:Ge crystal does not contain shallow traps in detectable concentration.
Taking into account electrical neutrality of defects and positron lifetime comparable to 
the component with the lifetime 
can be attributed to positrons trapped at electrically neutral 
complexes which are likely formed in CdZnTe:Ge. Table 7 shows the concentration of 
complexes determined as 
, where 
is the trapping rate calculated within 2-STM by Eq. (16) and 
is the specific positron trapping rate for neutral vacancies in semiconductors24.
Annealing of CdZnTe:Ge crystal in Cd vapour led to the disappearance of the defect component with lifetime 
, and the LT spectrum became single-component with the lifetime 
ps, which is close to the bulk lifetime of 292(1) ps measured in the u-CdTe, see Table 6. Hence, virtually all positrons in the Cd-annealed CdZnTe:Ge sample are annihilated in the de-localized state. This result proves that 
annihilation is due to Cd-annealing. Slightly higher value of the bulk positron for CdZnTe:Ge compared to that for u-CdTe could be caused by lower electron density in CdZnTe compared to that in CdTe. Temperature-dependent measurements did not reveal any temperature dependence of LT results for the Cd-annealed CdZnTe:Ge sample, see Fig. 2 and Table 6, which is in accordance with the assignment of the 
component to the free positrons.
The subsequent annealing of CdZnTe:Ge in Te vapour resulted in the restoration of the defect component with lifetime 
ps, which shows the re-emergence of 
. The LTs obtained after annealing are very similar to those measured in the as-grown CdZnTe:Ge, including the temperature independence of 
in Fig. 2, which testifies to positron trapping in electrically neutral 
complexes.
CDB spectroscopy
The CDB spectroscopy was employed in order to obtain direct information about chemical environment of 
in CdTe:Cl and CdZnTe:Ge crystals. Figure 4 shows the momentum distribution of annihilating electron-positron pairs measured in the u-CdTe reference sample. The calculated momentum distribution for a perfect CdTe crystal is plotted in the figure as well. One can see in the figure that the momentum distribution calculated using the ATSUP-GGA approach agrees well with the experimental points in the high momentum range 
where the contribution of positrons annihilated by core electrons dominates. This is caused by the fact that core electrons in the inner shells retain their atomic character are almost not influenced by crystal bonding. On the other hand the ATSUP-based approach is not able to describe accurately the momentum distribution in the the low momentum region 
where the contribution of positrons annihilated by valence electrons becomes dominating. For this reason the calculated momentum distributions will be compared with the experimental data in the high momentum region only.
The positron annihilation momentum density P(pz) measured by CDB in the undoped CdTe sample and calculated for a perfect CdTe crystal using the ATSUP-GGA approach.
Figure 5a shows the experimental CDB ratio curves related to the u-CdTe reference for the CdTe:Cl sample in the as-grown state and after annealing in Cd and Te vapour, respectively. The calculated CDB curves related to a perfect CdTe crystal for 
, 
and 
defects are plotted in Fig. 5b. From inspection of Fig. 5 it becomes clear that the CDB curve for CdTe:Cl annealed in Cd vapour is almost flat (except of statistical scattering) and close to unity in the whole momentum range testifying that the momentum distribution for the Cd-annealed CdTe:Cl is very similar to that in the u-CdTe reference. This confirms the picture that positron traps were removed by annealing in Cd vapour and virtually all positrons in the Cd-annealed sample are annihilated in the free state. On the other hand, the CDB ratio curves for the as-grown CdTe:Cl and the sample annealed in Te vapour clearly differ from unity and exhibit a distinct peak in the high momentum range at 
. Similar peak is observed in the CDB curves calculated for positrons trapped in 
and 
defects in Fig. 5b. This testifies to positron trapping in defects associated with chlorine in the as-grown and Te-annealed CdTe:Cl sample.
The CDB ratio curves (related to defect-free u-CdTe) (a) experimental data for CdTe:Cl in the as-grown state and after annealing in Cd and Te vapour; (b) calculated ratio curves for VCd, AC and VCd2ClTe defects. The calculated curves are plotted by dashed lines in the low momentum range (pz < 15 × 10−3m0c) where the accuracy of the momentum distribution calculated by ASTUP is poor.
The CDB results for CdZnTe:Ge samples are presented in Fig. 6. The experimental CDB curves for the as-grown and Cd or Te-annealed CdZnTe:Ge are plotted in Fig. 6a while Fig. 6b shows the CDB ratio curves calculated for positrons trapped at 
and 
defects. The CDB ratio curve for Cd-annealed CdZnTe:Ge sample is again almost flat and close to unity indicating that the momentum distribution in this sample is similar to that in the u-CdTe reference. A slight enhancement in the momentum range 
is most probably due to positrons annihilated by electrons belonging to Zn. Annealing in Te vapour modified the momentum distribution of CdZnTe:Ge sample. Shape of the ratio curve for the Te-annealed sample is very similar to the curves calculated for positrons trapped at 
and 
defects. Hence, in accordance with the results of LT spectroscopy the CDB data in Fig. 6 indicate that positrons in CdZnTe:Ge annealed in Te vapour are trapped at defects containing 
. Very similar CDB curve was observed also in the as-grown sample. However, CDB spectroscopy does not allow to discern positrons trapped in isolated 
from those trapped in 
complexes due to very similar CDB ratio curves for both defects, see Fig. 6b.
The CDB ratio curves (related to defect-free u-CdTe) (a) experimental data for CdZnTe:Ge in the as-grown state and after annealing in Cd and Te vapour; (b) calculated ratio curves for VCd and VCdGeCd defects. The calculated curves are plotted by dashed lines in the low momentum range (pz < 15 × 10−3 m0c) where the accuracy of the momentum distribution calculated by ASTUP is poor.
In spite of extensive research, the basic properties of native point defects in CdTe/CdZnTe are still far from agreed understanding. Both formation and ionization energies are not settled and analyses based on various defect models or experimental data produce incompatible defect characteristics8. Nowadays, it is generally assumed that the difficulties in detecting of native point defects ensue from their reactivity and fast diffusion at elevated temperature. During cooling to room temperature native defects are supposed to migrate and interact with charged impurities. As a result, various types of associates and precipitates are formed58,59. The remaining isolated 
mostly annihilate during cooling at surface or dislocations and the final room temperature density of 
comes out very low42. Evidently, such premise is in pure agreement with evaluated defect structure both in CdTe:Cl and CdZnTe:Ge. Analysis of PAS data suggests that all detected 
are incorporated in various complexes stabilizing them in the lattice.
Detailed characterization of defect structure of CdTe:Cl and CdZnTe:Ge samples summarized in Table 7 raises a query on the unambiguity of the presented defect model. We believe that the comprehensive exploration of CdTe:Cl and CdZnTe:Ge samples, which were studied both in Cd- and in Te-rich state by multiple techniques, LT, CDB and galvanomagnetic measurements, and characterized by GDMS, allowed us the interpreting of measured data by rather unique way. In view of the fact that the defect densities in Te-rich state significantly exceed the densities of obvious residual extrinsic acceptor, we may conclude that 
is the only candidate for the acceptor defect at these conditions.
Similarly, the intentional doping by Cl or Ge donors supplied the only extrinsic elements with the large enough concentration to arrange the detected density of revealed A-centres and complexes. Nevertheless, suggested interpretation of the defect structure of complexes does not represent the only possibility to create neutral or acceptor-type complex in CdTe and CdZnTe. An analogous defect might be created also without participation of extrinsic atoms taking only native point defects into account. That is the complex of Cd vacancy and Te anti-site defect 
predicted in ref. 60. Such complex was not identified in experiments yet and its electrical properties are not known. The reason, why we favour the structure of complex related to extrinsic Cl or Ge donors rises from the proximity of the evaluated complex densities in Te-annealed samples and respective element concentrations revealed by GDMS. The density of 
complexes appears significantly below extrinsic-donor-related complexes and does not manifest in presented measurements. This picture is supported also by theoretical calculations in Table 3 since the 
complex is characterized positron lifetime which is slightly lower that the lifetime for isolated 
. But such trend was not observed in experimental lifetimes.
Conclusions
Point defects in CdTe:Cl and CdZnTe:Ge crystals have been characterized by LT and CDB spectroscopies combined with galvanomagnetic measurements. The as-grown CdTe:Cl crystal exhibits larger point defects with open volume comparable with several vacancies. The low temperature LT studies in CdTe:Cl have proved the existence of negatively-charged shallow traps interpreted as Rydberg states associated with Cl-related A-centres. These shallow traps are able to confine positrons at low temperatures only when thermal de-trapping becomes sufficiently small. The lifetime of positrons localized in the shallow traps is shorter than the CdTe bulk lifetime. The as-grown CdZnTe:Ge crystal contains 
shallow acceptors forming neutral complexes with 
deep donors. It has been confirmed that Cd-rich annealing of both CdTe:Cl and CdZnTe:Ge crystals suppresses the concentration of 
, which are created again during subsequent Te-rich annealing.
Additional Information
How to cite this article: Šedivý, L. et al. Positron annihilation spectroscopy of vacancy-related defects in CdTe:Cl and CdZnTe:Ge at different stoichiometry deviations. Sci. Rep. 6, 20641; doi: 10.1038/srep20641 (2016).
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Acknowledgements
This work was supported by the Czech Science Foundation (project P108-11-0958), by the Grant Agency of Charles University (2052214/2014) and by Charles University (student project SVV-2014-260091).
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All authors reviewed the manuscript and approved the final version of the paper. L.S. carried out the sample preparation, the annealing and the galvanomagnetic measurements under the supervision of E.B. J.C. and O.M. carried out the PAS spectroscopy. J.C. and R.G. performed the theoretical calculations.
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Šedivý, L., Čížek, J., Belas, E. et al. Positron annihilation spectroscopy of vacancy-related defects in CdTe:Cl and CdZnTe:Ge at different stoichiometry deviations. Sci Rep 6, 20641 (2016). https://doi.org/10.1038/srep20641
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