Optical discrimination of deep trap contribution to carrier recombination in semi-insulating crystals

Optical discrimination of deep trap contribution to carrier recombination in semi-insulating crystals
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  Optical discrimination of deep trap contribution to carrier recombinationin semi-insulating crystals A. Kadys, 1,a  K. Jaraši ū nas, 1 and D. Verstraeten 2 1  Institute of Materials Science and Applied Research, Vilnius University, Sauletekio Avenue 9 Bld. 3, LT-10222 Vilnius, Lithuania 2 Centre Spatial de Liège, Université de Liège, Avenue du Pré Aily, 4031 Angleurs, Belgium  Received 18 March 2009; accepted 28 May 2009; published online 2 July 2009  We demonstrate a novel application of light-induced transient grating technique for discriminationof deep trap contribution to carrier recombination in compensated semiconductors. This applicationis based on photoexcitation of deep impurity levels by light interference pattern and subsequentoptical monitoring of the recharged state dynamics. The spatially modulated deep trap occupationleads to changes in a probe beam absorption and formation of a transient diffraction grating.Employment of light diffraction on a short-period reflection grating allowed to realize conditionswhen the absorption modulation in deep traps dominates over the coexisting refractive indexmodulation. Selectivity of a transient reflection grating configuration solely to processes in deeptraps was proven experimentally and confirmed by numerical modeling. In this way, wediscriminated the deep vanadium impurity governed carrier recombination rate from the othercoexisting recombination channels in as-grown and annealed semi-insulating CdTe:V crystals.©  2009 American Institute of Physics .   DOI: 10.1063/1.3158054  I. INTRODUCTION Optical and electronic applications of semi-insulatingsemiconductors are based on proper defect engineering,which allows tailoring of a material properties and obtaininghigh resistivity, homogeneity of compensation, photosensi-tivity, and long carrier lifetime. Number of techniques canprovide steady-state parameters of deep impurities, as theirdensity, energetic position in a bandgap, photoionizationcross sections, etc. Defect-related photoelectric propertieshave been studied by using the electrical and optical tech-niques, as photocurrent spectroscopy, 1–3 photoluminescence, 4–6 optical or microwave absorption byfree carriers   FCs  . 7–9 Among relevant time-resolved charac-terization techniques, the light-induced transient gratingtechnique deserves deeper attention, as it bridges the photo-electrical and optical phenomena through optical nonlinear-ity and opens a possibility to monitor carrier generation,transport, and recombination processes without electricalcontacts. 10–12 The latter technique is based on recording and recon-struction of a dynamic hologram in a crystal 13,14 and there-fore requires two steps in its implementation:   i   spatial andtemporal modulation of optical properties of a semiconductorby proper illumination, that is considered as a recording of atransient grating   i.e., generation of nonequilibrium carrierperiodic distribution  N    x  , t   =   N   t   1+cos  Kx   , which leadsto modulation of refractive and/or absorption indices    n  and  k   , and   ii   monitoring the dynamics of the optically in-duced changes by diffraction of a delayed probe beam on thecreated transient grating. Up to now, picosecond FC gratingshave been mainly explored for studies of carrier dynamics insemiconductors because of a simple relationships betweenthe carrier density modulation    N    x  , t    and refractive indexchange   n   x  , t     N    x  , t   . 13 The instantaneous value of aprobe beam diffraction efficiency on the grating,     t      n  t   2 , sensitively revealed the spatial and temporalchanges of carrier modulation. 10,12,15 Consequently, nonequi-librium carrier dynamics was investigated in   sub  nanosec-ond time domain and provided recombination and diffusionparameters at the surface or in the bulk of a crystal, 16–18 insemiconductor heterostructures, 19,20 dependence of carrierplasma parameters on nonequilibrium carrier density, latticetemperature, 12,21,22 as well on growth- or radiation-induceddefect density. 11,23,24 A huge metrological potential of thisnonlinear optical technique is based on interdisciplinaryknowledge in the fields of nonlinear optics, holography, andsemiconductor physics. 10,13,25 In this paper we demonstrate a novel application of non-linear transient grating technique for investigation of deeptrap governed carrier recombination processes. An advantageof used novel optical configuration—a transient reflection  TR   grating allowed an optical discrimination of the domi-nant deep trap contribution to carrier recombination. Experi-mental study of carrier dynamics in nanosecond time domainwas performed by using the standard picosecond FC gratingtechnique and compared with the recovery rate of photoex-cited deep traps, seen in TR grating kinetics. Numericalanalysis of FC and TR grating decay was performed andcompared with the experimental data in bulk CdTe:V crys-tals. We confirmed that FC grating decay reveals contributionof all defects to recombination process, while the absorptiveTR grating is sensitive only to the dominant vanadium deeptraps even the trap-determined capture rate is lower than thecontribution of other coexisting defects. In this way, applica-bility of reflection grating technique for optical discrimina- a  Electronic mail: JOURNAL OF APPLIED PHYSICS  106 , 013704   2009  0021-8979/2009/106  1   /013704/7/$25.00 © 2009 American Institute of Physics 106 , 013704-1 Downloaded 03 Jul 2009 to Redistribution subject to AIP license or copyright; see  tion and direct monitoring of electron capture rate by midgapvanadium donor was demonstrated in differently processedsemi-insulating CdTe crystals. II. EXPERIMENTAL TECHNIQUESA. Dominant mechanisms of optical nonlinearities inbulk semiconductors Different mechanisms of refractive index modulationhave been explored up to now for transient grating recording:due to nonlinear polarizability by excess FC density   i.e., viarecording of FC grating  13,14 or due to carrier diffusion cre-ated space-charge   SC   electric field and subsequent Pockelseffect   recording of a photorefractive   PR   grating  . 26 Pico-second FC and PR grating techniques have been used todetermine important electrical parameters of semi-insulatingGaAs, InP, and CdTe crystals, as nonequilibrium carrier dif-fusion coefficient, lifetime, surface recombination velocity,and carrier diffusion length. 17,27,28 Time-resolved   sub  nano-second carrier transport through spatially modulated carrierand SC field structure provided a sign of carriers   electronsor holes  , photogenerated from deep impurity level in differ-ently compensated crystals of InP and CdTe. 12,29,30 In recentyears, FC gratings were applied for studies of nonradiativeand radiative recombination rates in GaN and InGaNheterostructures, 31–33 freestanding GaN crystals 11 and syn-thetic diamonds 20 at conditions of interband carrier photoex-citation.Effective deep trap density as well as the charge state of a dominant defect in semi-insulating GaAs and CdTe can bedetermined by time-integrated PR grating technique   i.e., bycontrolling the amplitude of SC field, built in quasistationaryconditions of excitation  . 34 Time-resolved monitoring of light diffraction on a picosecond short-period diffractiongrating allowed determination of a deep trap occupationratio. 35 In the latter case, the modulation of probe beam ab-sorption was separated from the coexisting refractive FC andPR gratings by using Bragg diffraction on a reflection grat-ing. Experimental measurements and modeling of light dif-fraction efficiency on TR grating provided the deep EL2 do-nor compensation ratio in a semi-insulating GaAs wafer. 35 We note that the TR grating technique for optical evalu-ation of deep trap occupation ratio was based on recording of the amplitude grating in deep traps and monitoring its dif-fraction efficiency in nanosecond time domain. Density of optically recharged neutral    N  0  process  N  0 →  N  +   and ion-ized    N  +  process  N  + →  N  0   deep traps resulted a spatialmodulation of ionized deep trap concentration  N  1+   x   =   N  0   x   −   N  +   x   . Thereby the modulation  N  1+ created anamplitude grating with modulated absorption coefficient fora probe beam at wavelength   ,      x    = −   S  n  −  S   p   N  1+   x    = 4    k    x   /  .   1  The probe beam diffraction efficiency on the amplitudegrating,     t      k   t   2 , revealed the instantaneous density of recharged traps. For donorlike deep traps   S  e  S  h , here  S   isthe photoionization cross section of the trap;  S  e  is for elec-trons and  S  h  for holes  , their initial charge state and subse-quent changes under illumination directly determined theconcentration of generated electrons and holes:  N  e = S  e  N  0  I  0 /  h     and  N  h = S  h  N  +  I  0 /  h     here  I  0  is the excitationenergy density  . Therefore, the diffraction efficiency on am-plitude grating varied in accordance with the deep trap occu-pation ratio   see Fig. 6 in Ref. 35   and provided a way fordetermination of the steady-state occupation ratio.On the other hand, dynamics of refractive index modu-lation by FCs photogenerated from deep impurity levels alsoallows to control deep trap contribution to carrier generation,SC field formation, and carrier capture processes. 12 Never-theless, direct access to the amplitude grating in rechargeddeep traps is not possible in these conditions, as the diffrac-tion efficiency on FC grating is usually much stronger thanthat on the amplitude grating. To reveal the latter, we se-lected very small grating period   =  / 2 n  0.2    m, re-corded by nearly counterpropagating beams   here    is thewavelength of excitation beam and  n  is the refractive indexof the crystal  , that resulted in ultrafast decay of carriermodulation. As the grating vector  k  =2   /  =99    m −1 ishigher than the Debye wave vector for the carriers in thephotoexcited crystal  k  0 =   N  + e 2 /   0 k   B T   1 / 2 =7–23    m −1 at  N  + =10 15 –10 16 cm −3 , the amplitude of light-created SC fieldbecomes proportional to recharged deep trap concentrationintegrated over the grating period:  E    N  1+  . At condition  k   k  0  the partial screening of SC field takes place, and itsamplitude becomes weaker than the diffusion-governed SCfield, which is dominant at  k   k  0 . Moreover, to avoid contri-bution of coexisting PR grating to diffraction, we used thepolarization anisotropy of light diffraction on PR grating   seebelow  .Therefore, in this study we used two optical configura-tions of transient grating techniques, which allowed either tostudy dynamics of refractive index modulation   FC grating  or reveal absorption index modulation in optically rechargeddeep traps   TR grating  . B. Configurations of transient grating techniques Two complementary configurations of time-resolved pi-cosecond transient grating technique   Fig. 1   enabled us tostudy deep impurity related carrier generation, transport, andrecombination processes in a wide excitation range. A pecu-liarity of these schemes is the way of grating recording,which provides essentially different grating periods. In thefirst case   Fig. 1  a  , the grating was recorded by two co-propagating beams intersecting in the sample at a small angle  . Here the grating period   1 =  /  2 sin   / 2   varied from  10 to few micrometers. In the second case, the counter- FIG. 1. Optical schemes for recording of transient FC grating   a   and TRgrating   b  , here  I  0 / 2 are the recording beams,  I   p  is the probe beam,  I  tr  is thetransmitted, and  I  dif   is the diffracted part of the probe beam. 013704-2 Kadys, Jaraši ū nas, and Verstraeten J. Appl. Phys.  106 , 013704   2009  Downloaded 03 Jul 2009 to Redistribution subject to AIP license or copyright; see  propagating recording beams entered the sample from theopposite sides, thus the grating period   2 =  / 2 n  was equalto 190 nm   here  n =2.82 is the refractive index of CdTe crys-tal at 1064 nm wavelength  . The Bragg diffraction condi-tions were fulfilled in both cases, therefore diffraction of theprobe beam was detected in transmission   Fig. 1  a   or inreflection   Fig. 1  b  . The orthogonal polarization of thepump and probe beams allowed quite easy separation of thediffracted beam by a Glan prism, inserted into a path of therecording beam. The consequence of essentially smaller   2 value was an ultrafast diffusive decay time     D  of FC grating,    D =  2 /  4   2  D   here  D  is the diffusion coefficient   whichvalue became two orders of magnitude smaller in this geom-etry and even shorter than the laser pulse duration     L , i.e.,    D  1 ps     L .Nonequilibrium carriers were created in a sample bylight interference pattern  I    x   =  I  0  1+cos  2    x  /    of two co-herent 23 ps-duration laser beams at   =1064    m wave-length. The spatially modulated electrical and optical prop-erties were monitored by the delayed up to 2 ns probe beam  I   p  of the same wavelength. Kinetics of diffraction efficiency    t    exp  −2 t  /   G   of the grating   i.e., a ratio of the dif-fracted probe beam  I  dif   intensity to the transmitted one  I  tr  were measured at various excitation beam energy densities  I  0 and grating periods and provided the grating decay time    G .In the case of light diffraction on FC grating   Fig. 1  a  , thedecay time reflected the recombination rate and diffusiontime     D  along the grating vector, according to a simple rela-tionship 1 /   G =1 /    R +1 /    D . Under conditions of deep trapphotoionization, the diffusive grating decay leads to spatialredistribution of carriers and creation of the SC field, leadingto PR grating. This field also created a drift current whichopposed the diffusive one, thus the subsequent decay of theFC grating was governed by carrier recombination. For TRgrating   Fig. 1  b  , the fast diffusive decay leads to verysmall response of the FC grating, thus the absorptive gratingin recharged deep traps became dominant. Therefore, TRgrating decay time constant     revealed the recovery time of the recharged deep traps, which was governed by the carriercapture rate.The measurements were carried out in vanadium-dopedCdTe samples after the different technological treatments.The crystals were grown in four-zone furnace by the Bridg-man technique under excess of cadmium   3  10 18 cm −3   andhad 10 19 cm −3 vanadium atom concentration. The electricallyactive vanadium concentration in the crystals was few orderslower, approximately of about 10 16 –10 17 cm −3 . 27 One of thesamples was annealed under Cd atmosphere for 10 h at700 °C. The samples were 5  6  2 mm 3 in size and cutalong the crystallographic axes   110  ,   11 ¯  0  , and   001  .Beams for grating recording and probing entered the crystalthrough the   001   polished faces.In order to separate contribution of the amplitude gratingfrom the coexisting phase ones, we used the feature of lightdiffraction anisotropy on PR grating. The change of dif-fracted beam polarization for “PR-cut” crystal takes place  i.e., when the grating vector is oriented along the crystalaxis   110   or   11 ¯  0  , while the diffraction on absorption andFC gratings is isotropic. 27,35,36 In Fig. 2  a   the diffractioncomponents of coexisting FC, PR, and amplitude grating insemi-insulating GaAs crystal are shown. The anisotropic dif-fraction on PR grating was detected if the  s -polarized probebeam was used and the rotated  p -polarized diffraction com-ponent was measured. In this case, the build up of SC fieldwas observed with its subsequent slow decay. Rotation of thecrystal allowed to extract the contributions of different non-linearities and evaluate which one is the dominant. After ro-tation of the crystal by +45° or −45° around the   001   axis,the diffraction on PR grating became isotropic and over-lapped with the coexisting absorption grating   i.e.,  p -polarized probe beam was not changing its polarizationstate in diffraction  . We note that contribution of isotropicabsorption grating is either added or subtracted from the PRgrating signal, therefore a symmetry of the diffraction kinet-ics is seen   Fig. 2  a  .In PR-cut crystal, the contribution of absorption gratingalso adds to the isotropic FC grating at probing by  p -polarized beam. The measurements at low excitations haveshown that there is only a negligible contribution of FC grat-ing to diffraction with respect to the diffraction on absorptiongrating. The fast peaks in the curves correspond to two-photon absorption grating which exist only during the actionof the laser pulse. 35 The described features of light diffraction were used forproper alignment of the optical setup for recording the reflec-tion gratings in CdTe crystals. The analogous measurementsof anisotropic diffraction revealed the presence of PR gratingin CdTe:V crystals with   001   entrance face   Fig. 2  b  . Thefurther measurements were performed for PR-cut crystal ori-entation by monitoring only isotropic diffraction of   p -polarized probe beam on FC and absorption gratings. III. EXPERIMENTAL RESULTS The measurements were performed at relatively low ex-citation energy densities, when the electron generation fromthe neutral midgap vanadium levels at  E  c  0.67–0.88 eV  Refs. 37 and 38   dominates. For the used quantum energy  h   =1.17 eV  , the transfer of electrons to conduction bandfrom the deep donor is more efficient rather than the holegeneration   S  e  S  h  .The diffraction kinetics of FC grating in as-grown CdTecrystal is shown in Fig. 3  a  . Grating decay time    G µ = µ FIG. 2. Kinetics of the TR grating for the different crystal orientations andprobe beam polarizations in GaAs   Ref. 35   a   and CdTe:V   b   crystals.Origin of optical nonlinearity is given at the curves, where FC, PR, and ABSstand for FC, PR, or absorptive gratings. 013704-3 Kadys, Jaraši ū nas, and Verstraeten J. Appl. Phys.  106 , 013704   2009  Downloaded 03 Jul 2009 to Redistribution subject to AIP license or copyright; see  =1.4–1.8 ns was found weakly dependent on excitation en-ergy density. After annealing,    G  value increased by an orderof magnitude   up to 15 ns, Fig. 4  a   at low excitation con-ditions, while this value gradually decreased with increasingthe excitation.The observed features of carrier dynamics can by ex-plained by a model of light diffraction on a FC grating,    t      n  t   2 , when the refractive index modulation dynam-ics   n  t     N   t    is governed by nonequilibrium carrier re-combination, diffusion, and drift in light-created space-charge field. For the given grating period of 10    m, thediffusive decay is expected to be compensated by electrondrift in SC field, established between the mobile carriers  photoexcited electrons   and ionized donors. Therefore, forevaluation of diffusive decay time     D =  2 /  4   2  D   in thepresence of SC field, the  D  value must be replaced by  D eff  ,which is much smaller that the bipolar one  D a =4 cm 2 / s. 12 For the given excitation range   from 0.3 up to 2.7 mJ / cm 2  ,the measured  D eff   values varied from 0.2 to 0.7 cm 2 / s   Fig.5  , and the corresponding     D  values were as long as40–170 ns. Consequently, the observed grating decay timesin Figs. 2  a   and 3  a   can be attributed to carrier recombina-tion process. Nearly linear increase in carrier recombinationrate with excitation density, 1 /   G   I  0   see Fig. 3  a  , indi-cated the increasing density of ionized deep vanadium traps   N  +   I  0 , which acted as the electron trapping centers:1 /   G =   e   N  t  =0+ +   N  +  , here    e  is the electron recombinationcross section to the trap. This qualitative description is sup-ported by the given below numerical analysis   Sec. IV  .We note that shorter and weakly dependent on excitationcarrier lifetimes before annealing     G =1.4–1.8 ns, Fig. 3  a  indicated quite high number of electron trapping centers inthe as-grown crystal. The observed increase in carrier life-time by order of magnitude after annealing can be explainedby diminishing density of residual acceptorlike defect traps,typical for as-grown CdTe. 27 After annealing, the crystallinequality increased due to healing the Te precipitates, filling of Cd vacancies, reducing internal stress, 12,39 and thus diminish-ing defect density that resulted in increase in carrier lifetime.Diffraction kinetics on TR gratings provided direct in-sight into the electron capture by deep vanadium traps. Thegrating decay times    G  in both as-grown and annealed crys-tals were found quite similar and varied from   20 ns to  2 ns with increasing the excitation density   Figs. 3  b   and4  b  . This peculiarity we explain by the recovery rate of theabsorption grating, recorded in spatially recharged deep va-nadium traps. Diffraction efficiency on the amplitude gratingis related to the instantaneous concentration of the rechargeddeep trap modulation,     t      N  +1  t   2 , see Eq.   1  , which de-creased with time due to their filling by electrons. In thisway, the decay of TR grating directly revealed only onemechanism of carrier recombination, i.e., via deep vanadiumtraps. This approach is valid for both as-grown and annealedcrystals, as the deep vanadium impurity is determining thelight absorption in the infrared spectral region of CdTe.Comparison of the determined carrier lifetimes by usingtwo transient grating schemes is presented in Figs. 3  c   and4  c  . In the as-grown sample, the    G  values determined byFC grating technique are much shorter than those measuredby TR grating because the decay of carrier density modula-tion reveals an impact of all defects contributing to carriercapture, while the decay of amplitude grating is sensitiveonly to one mechanism—carrier trapping at deep vanadiumcenters. After sample annealing, the determined values of carrier lifetime and deep trap recovery became very closeand indicated that only one recombination center dominatesin carrier trapping after annealing   Fig. 4  c  . These experi-mental results clearly demonstrated a possibility to distin-guish the contribution of the deep traps in the presence of other, even faster recombination pathways. Numerical analy-sis of optical recharge of deep traps and their recovery isgiven in Sec. IV. IV. NUMERICAL MODELING For deeper understanding and successful application of this innovative technique, the numerical modeling of carrier ±±= µ ± ±±µ FIG. 3. Kinetics of diffraction efficiency for FC   a   and TR gratings   b   inas-grown CdTe crystals at different excitation energy densities. The ex-tracted grating decay times are shown at the curves and the determinedcarrier lifetimes are compared in   c  . ±0.2±0.3µ ±±±µ FIG. 4. Diffraction kinetics for FC   a   and TR gratings   b   at differentexcitation energy densities for the annealed CdTe crystals. The extractedgrating decay times are shown at the curves and the determined carrierlifetimes are compared in   c  .FIG. 5. Dependence of the effective diffusion coefficient on excitation en-ergy density in as-grown and annealed CdTe:V crystals. 013704-4 Kadys, Jaraši ū nas, and Verstraeten J. Appl. Phys.  106 , 013704   2009  Downloaded 03 Jul 2009 to Redistribution subject to AIP license or copyright; see  dynamics under deep trap photoexcitation was performed.Processes of carrier generation, transport, and recombinationare described by a set of equations, which account for elec-tron and hole generation from/via deep impurity level, carriertransport in SC field, as well as carrier capture by deeptraps. 13 Contribution of processes which do not require thepresence of deep impurity levels is presented by interbandtwo photon carrier generation and bimolecular recombina-tion at high excitation power densities, 12    N  e   t  = S  e  IN  0 h   +    I  2 2 h   +1 e    j e    x  −   e  N  e  N  + −   eh  N  e  N  h ,   2     N  h   t  = S  h  IN  + h   +    I  2 2 h   −1 e    j h    x  −   h  N  h  N  0 −   eh  N  e  N  h ,   3   j e  =  eN  e   e  E  SC +   e k   B T     N  e    x  ,   4   j h  =  eN  h   h  E  SC −   h k   B T     N  h    x  ,   5     N  +   t  = S  e  IN  0 h   − S  h  IN  + h   −   e  N  e  N  + +   h  N  h  N  0 ,   6     E  SC    x  =−  e  0    N  e  +  N   A  −  N  h  −  N  +  .   7  Here  S  e ,  S  h ,    e ,    h  are the deep trap parameters, provid-ing electron and hole photoexcitation cross sections fromdeep traps and carrier capture coefficients by the neutral    N  0  or ionized    N  +   deep traps,    =25 cm / GW is the two photoncoefficient for CdTe, 40 and    eh =2  10 −8 cm 3 / s is the bimo-lecular recombination rate,  N   A  is the shallow acceptor con-centration. These equations have been numerically solvedunder excitation by light interference pattern of two picosec-ond pulses  I    x   =  I  0  1+cos  2    x  /    and provided spatialprofiles of electron and hole density, carrier redistributiondynamics in diffusion created SC field, and instantaneouschanges in deep trap occupation. The refractive and absorp-tion index modulation was deduced as well as diffractionefficiencies on these gratings were calculated. This modelusually accounts for a single deep donor trap of density  N  T  1 ,which can be in neutral    N  0   and ionized    N  +   state:  N  T  1 =  N  0 +  N  + . For numerical modeling we used well knownCdTe:V optical and electrical parameters   S  e , S  h ,   e ,   h  which are listed in Ref. 41–43. In the given modeling, we also introduced an additional deep trap with lower density   N  T  2   N  T  1  , acting as an efficient recombination center    e 2 /   e 1  6  , but not contributing to carrier generation   thisassumption is valid for Cd vacancies, as electron-killingcenters 44  .The electron and hole generation rates strongly dependon occupation of deep donor impurity   or compensation ratio  R =  N  + /  N  T    and excitation energy density. In low excitationregime, a linear generation rate of electrons dominates fromthe occupied vanadium levels  N  e = S  e  N  0  I  0 /  h    , and total ex-cess carrier concentration is lower than the electrically activedeep trap density:  N  e +  N  h   N  T  . The hole generation requiresa presence of the ionized state, which is either present of created after the electron transition to the conduction band.Thus the hole generation is intensity dependent:  N  h = S  h  N  +  I  0 /  h    = S  h S  e  N  0  I  02 /  h    2 . For donor-type deep center,the concentration of photogenerated electrons is much higherthan that of holes    N  h   N  e   N  T   , therefore electron gratingdominates. Its decay time depends on electron capture rateand the ionized trap density   here both the initial density of ionized centers,  N  + t  =0 =  N   A  and the light created one    N  + ,given by Eq.   6   must be taken into account  .At high excitation conditions, the electron-hole pairs arecreated by two-step and two-photon transitions, thus condi-tion  N  h   N  e   N  T   may take place. The processes of bipolardiffusion, governed by SC field between the mobile carriers,as well as interband recombination will mask the deep traprelated processes. Therefore, the high excitation regime isnot favorable for investigation of deep trap effects. A. A single trap model and modeling results For simulation of carrier dynamics and deep trap re-charge in the annealed CdTe:V sample, we used a singlemidgap donor model. The deep vanadium trap total density  N  T  1 =4  10 16 cm −3 and the ionized part  N  + T  1 =4  10 15 cm −3 were assumed. 41,43 The modeling results of FCgrating decay   Fig. 6  a   were found very close to those mea-sured experimentally. The grating erasure time decreasedfrom 15 to 3.5 ns with the increase in excitation intensityand followed the electron recombination rate 1 /   G    e  N  T  1+  Fig. 6  a  . Here the electron capture cross section    e 1 =5.4  10 −9 cm 3 / s was used for fitting. The value of     e 1  wasfound similar to one   1.8  10 −9 cm 3 / s   given in Refs. 41and 43.The modeling was also performed for light diffraction onTR grating with 0.19    m period. It has been shown thatamplitude grating dominates in diffraction at low excitations  up to   1 mJ / cm 2  , 35 but at higher excitations the contribu-tion of isotropic FC grating may add. Therefore, we calcu-lated the diffraction efficiency     of TR grating for the currentexperimental conditions, as contribution of both the absorp-tion grating          and the FC one      n  ,   =      d  4   2 +      nd     2 .   8  µµµ µ ∆ FIG. 6. Numerical modeling of FC grating diffraction kinetics for differentexcitation energy densities  I  0 , assuming a single deep donor  N  T  1   a  . Com-parison of the calculated FC grating    =10    m   and TR grating    =0.19    m   decay times at different excitation levels   circles and trianglespresent the experimental data   b  . The doted and dashed lines correspond todiffraction on the absorption grating only or on both the absorption and FCgratings. 013704-5 Kadys, Jaraši ū nas, and Verstraeten J. Appl. Phys.  106 , 013704   2009  Downloaded 03 Jul 2009 to Redistribution subject to AIP license or copyright; see
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