Review Article Optical Measurement Techniques of Recombination … · 2019. 7. 31. · of the recombination process in the semiconductors theory. In fact the recombination lifetime

Review ArticleOptical Measurement Techniques of Recombination LifetimeBased on the Free Carriers Absorption Effect

Martina De Laurentis and Andrea Irace

Dipartimento di Ingegneria Elettrica e delle Tecnologie dellrsquoInformazione Universita degli Studi di Napoli ldquoFederico IIrdquoVia Claudio 21 80125 Napoli Italy

Correspondence should be addressed to Andrea Irace andreairaceuninait

Received 15 November 2013 Revised 21 February 2014 Accepted 3 April 2014 Published 24 June 2014

Academic Editor George Cirlin

Copyright copy 2014 M De Laurentis and A IraceThis is an open access article distributed under the Creative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

We review successful measurement techniques for the evaluation of the recombination properties in semiconductor materialsbased on the optically induced free carrier absorption All the methodologies presented share the common feature of exploiting alaser beam to excite electron-hole pairs within the volume of the sample under investigation while the probing methods can varyaccording to the different methodology analyzed As recombination properties are of paramount importance in determining theproperties of semiconductor devices (ie bipolar transistor gain power devices switching features and solar cells efficiency) theirknowledge allows for better understanding of experimental results and robust TCAD simulator calibration Being contactless andapplicable without any particular preparation of the sample under investigation they have been considered attractive to monitorthese parameters inline or just after production of many different semiconductor devices

1 Introduction

The minority carriers recombination lifetime is one of themost important parameters as it both characterizes thesemiconductor materials and it strongly influences devicesproperties As more than 95 of all the produced electronicdevices are fabricated through CMOS process nowadays themajority of the studies on recombination lifetime concernspower electronics diodes IGBTsmdashwhere lifetime killingmethodologies are mostly employedmdashor solar cells whererecombination parameters are directly related to the conver-sion capability of these devices

Solar cells without doubt occupy an important rolein the energy worldwide scenario so the industry andresearch interest in their production and characterizationare increasing As said their efficiency in converting theenergy from solar to electrical critically depends on therecombination process by means of two parameters the bulkrecombination lifetime 120591119887 which accounts for the recombi-nation of electrons and holes inside the silicon crystal andthe surface recombination velocity 119878119877119881 which is stronglydependent on the interface between the material and itsboundaries the former accounts for material quality while

the latter usually depends on the fabrication process Theseparameters are strongly related to the presence of defectswithin the semiconductor forbidden gap (which directlyaffects devices performance) and to the surface quality Theyalso depend on the semiconductor growth technique on thedoping on the surface condition and on the free carrierdensity injected in the material under operating conditionsIn the last decades the researchers efforts were addressedto develop contactless and totally compatible methods thatallow for monitoring these quantities during the productionprocess without interfering with other fabrication steps It isthus obvious that methods based on optical or microwaveradiations have been preferred

In this paper we present a review of two importantcontactless techniques used to determine the semiconductormaterials lifetime the Pump and Probe method (PP) [1]and the Infrared Lifetime Mapping method (ILM) otherwiseknown as Carriers Density Imaging (CDI) [2 3] Thesetechniques are based on the free carriers absorption thatoccurs in the semiconductormaterials As the name suggeststhis is the contribution to the absorption of the free carrierspresent in thematerial that involves photons with wavelengthbelonging to the NIRmiddle infrared region corresponding

Hindawi Publishing CorporationJournal of Solid State PhysicsVolume 2014 Article ID 291469 19 pageshttpdxdoiorg1011552014291469

2 Journal of Solid State Physics

in terms of energy to the energy lower than the materialband-gap In fact the free carriers concentration (and itsvariation) inside the material determines a variation of thematerial optical absorption coefficient This implies that avariation of transmittance can be detected and observedwhen a sub-band-gap radiation is transmitted by thematerialThe generation of the excess free carriers can be opticallyinduced by illuminating the sample with a laser pulse whosephotons energy is higher than the sample band gap

The PP technique belongs to the class of transient meth-ods These methods are attractive because they give a resultdirectly related to the velocity of the recombination processIn fact the signal detected is the time dependent signaltransmitted by the sample The PP allows for characterizingmaterials with very low lifetime too and usually it doesnot require any calibration procedure Moreover the actualadvantage of themethod is the possibility to separate the bulkfrom surface effect which means to simultaneously measurethe bulk lifetime and the surface recombination velocity

The ILMCDI belongs to the steady-state methods classThese methods measure a physical quantity related to thecarrier density It allows for a fast and high spatially resolvedmeasurement of the lifetime in each point of the waferwithout performing a scanning of it since an infrared CCDcamera is used to detect the transmitted infrared radiationIts greater attractive feature is the capability to perform thelifetime mapping in few minutes which could provide incalibrated condition an online monitoring feature

The paper is organized as follows in the first two sec-tions we briefly recall the concept of recombination lifetimetogether with the main recombination mechanisms in semi-conductor materials in the subsequent sections we describein detail of the selected techniques having previously recalledthe analytical tools necessary to their understanding

2 The Concept of Recombination Lifetime

The term ldquolifetimerdquo is used in physics in a totally intuitivemanner to indicate the temporal interval between the gen-eration and the death of a particle Nevertheless often theldquodeathrdquo is only a change of the particle state like in the caseof the recombination process in the semiconductors theoryIn fact the recombination lifetime refers to the time in whichthe electron returns from an excited state to the equilibriumstate The excitation can be obtained for example by aphoton absorption or an electrical injection process in theforward-biased p-n junction The electron comes back inthe equilibrium state occupying the vacancy left by theexcitation In the semiconductors theory this vacancy isdescribed like a particle named hole with the same electroncharge but differentmass [4 5] In this way the electron decayto the equilibrium state is described as its recombinationwith the holeThis recombination process destroys the chargeconstituted by the pair and the recombination lifetime thenrefers to the electron-hole pair what ceases to exist is thischarged pair and it ceases to exist because the electron of thepair recombines with the hole of the pair In other wordsthe recombination process is the destruction (or decay)

of the excess carriers (the electron-hole pairs) generated inthe semiconductors as consequence of an excitation Thetemporal interval during which the excess carriers decay isdefined as recombination lifetime 120591rec

Mathematically in a first order approximation 120591rec isdefined as the ratio

120591rec =Δ119899

R(1)

between the excess carriers density Δ119899 [cmminus3] and the recom-bination rate R = minus119889(Δ119899)119889119905 [sminus1] of the excess carriersitself

From (1) named Δ119899(0) the excess carriers concentrationat the instant 119905 = 0when the excitation source is turned off itis derived that 120591 characterizes the excess carriers decay

Δ119899 (119905) = Δ119899 (0) exp(minus 1

120591rec) (2)

In steady-state conditions the excitation source is at alltimes turned on constant or with assigned time dependenceand an equilibrium condition between the generation rateand recombination rate is reached G = R so that it ispossible to define the recombination lifetime as

120591rec =Δ119899

G (3)

In general it must be clearly remarked that the recombi-nation phenomenon can be comprehensively treated only bymeans of quantum mechanical calculations Since it involvestransitions between energetic states it is implicit that therecombination rate depends on occupation probability of theenergy levels and thus on the energy as well as on thetemperature on the electrons and holes concentration andit is in general a nonlinear function of Δ119899 Consequently therecombination lifetime depends on them and thus it must beconsidered that in general it is 120591rec = 120591rec(119864 119879 119899 119901 Δ119899 Δ119901)(where 119899 and 119901 are the concentration of the electrons andholes resp) that means among other things that 120591rec is afunction of the injection level Δ119899119899 (Δ119901119901)

21 Recombination Process The total recombination rate isa superposition of several recombination mechanisms Assummarized in (4) among these the main are the Shockley-Read-Hall the radiative and the Auger recombination (thefirst one linearly dependent on the excess carrier generationonly at lower injection level the second one and the thirdresp proportional to the square and to the cube of the excessfree carriers concentration)

R = RSRH +RRad +RAug (4)

The sketch in Figure 1 gives an intuitive 119864 minus 119896 pictureof these processes while in the following we give a shortdescription of themWe invite the readers to refer to [6 7] andespecially to [8] for all the analytical and theoretical details

Journal of Solid State Physics 3

k

Phonon

Conduction bandE

Photon

Valence band

Ec

Eg

E

Et

⟨111⟩ ⟨100⟩

(a)

k

E

Photon

Conduction band

Valence band

Ec

E

⟨111⟩ ⟨100⟩

(b)

k

EC band

Valence band

Ec

E

⟨111⟩ ⟨100⟩

(c)

Figure 1 Main Recombination Mechanisms in semiconductors (a) Trap-assisted recombination (b) Direct Recombination (c) AugerRecombination

211 Shockley-Read-Hall Recombination Shockley Readand Hall in 1952 were the first to note that in the indirectband-gap semiconductors like silicon and germanium therecombination lifetime is sensitive to the material propertyand that the recombination rate varies linearly with thecarriers concentration on a wide range of concentrationsand temperatures [9 10] Since this cannot be explainedwith a two-bodies collision mechanism adopted to explainthe interaction between the electrons and which implies aquadratic dependence they supposed and demonstrated thatthe process takes place by means of the material impuritiesthat introduce some intermediate energy levels in the forbid-den gap These levels act as recombination centers (or trapsfrom which the process is otherwise depicted as trap assistedrecombination) and an intermediate step is introduced in therecombination process

Therefore the electron in a first step falls down to theintermediate levels then it falls down to the valence bandin the same way Nevertheless since the process is observed

in the materials called indirect semiconductors because ofnoncorrespondence in the momentum-space (119896-119904119901119886119888119890) [4 5]between themaximumof the valence band and theminimumof the conduction band an electron in the conduction bandcannot simply decay to the valence band releasing its energybut it must also change its momentumThis can happen withthe assistance of the crystal lattice the electron gives thelattice its energy in the form of heat and simultaneously thisinteraction allows it to change its momentum 119896 producinglattice vibration In the quantum model this is described asthe interaction of the electron with the lattice vibrationalquanta the phononsThis fictitious particles are defined withan own momentum [4] so that when the electron interactswith them it can change its momentum to preserve the totalmomentum of the whole particle

The second step can be equivalently described as anhole that ascends to the intermediate level (equivalentlydescribed as an ldquohole capturerdquo of the trap) recombining withan electron Really depending on the nature of the trapping


process one of the two steps can be a radiative process so thatthe electron losses its energy emitting a photon This processdoes not imply a momentum change

It must be noted that the trap assisted recombination is avery complex phenomenon that can be properly describedby means of the quantum mechanics and we remark itcan be assumed linearly dependent on the excess carriersconcentration only on a limited even if wide range ofconcentrations and temperatures The readers can find acomplete and exhaustive treatment in [8 11]

212 Radiative Recombination In direct band-gap semi-conductors where the minimum of the conduction bandcoincideswith themaximumof the valence band the electronband to band transition can happen without involving thecrystal lattice but only with the emission or absorption ofa photon depending on the kind of process (if it is anabsorption or a decay) This is possible because the changeof momentum is not required and thus only the energyconservation must be preserved

Since the process simultaneously involves both thecharges electrons and holes the probability that it occursis proportional to the square of the population of the twoinvolved levels and likewise the recombination rate too

213 Auger Recombination In 1955 Pincherle published abrief letter on the Proceeding of Physical Society [12] inwhich the inverse proportionality of the minority carriersdensity lifetime to the square of the majority carriers densityexperimentally observed by Moss in impure lead sulfide(PbS) [13] in 1953 and by Hornbeck and Haynes in silicon(Si) in 1955 [14] was justified and predicted by means of theAuger effect [15] In the subsequent years between 1956 and1958 Landsberg et al [16ndash18] developed the first theory on theAuger recombination

The Auger recombination in semiconductors involvesthree carriers when an electron recombines with a hole theenergy and the momentum variation are transferred to athird carrier either an electron or a hole The conductionband Auger recombination (case ldquocrdquo of the figure) whichinvolves two electrons in the conduction band and a hole inthe valence band is sketched in Figure 1 In a similar waythe valence band Auger recombination involves one electronin the conduction band and two holes in valence band TheAuger recombination can involve trap levels too [8 19ndash21]The process is strongly dependent on the carriers injectionlevel and it becomes dominant at high carriers concentration

From this short summary on the recombination mecha-nism and recalling (1) and (4) it is possible to derive that thatthe recombination lifetime can be expressed as

1

120591rec=

1

120591SRH+

1

120591rad+

1

120591Auger (5)

In indirect band-gap semiconductors like Si the SRHis dominant at moderate carrier densities (up to 1018 cmminus3)while theAuger recombination dominates under high dopingand high injection condition (see eg [22])This implies that120591rad is enough higher than 120591SRH and 120591Auger and thus in (5) the

second term is usually negligible Radiative recombinationdominates in direct band-gap materials like GaAs and InP

22 Surface Recombination Lifetime and Surface Recom-bination Velocity The materialsrsquo samples and the devices(or the active region of them) are space limited by theirsurfaces which constitute an interface between two differentmedia like the silicon-air surface or the silicon-oxide surfaceLikewise to the impurities and to the defects these surfacesmdashwhich are discontinuities of the periodical crystal structuremdashintroduce allowed energy states within the forbidden bandThese states affect the recombination process in a way thatis very similar to the Shockley-Read-Hall model for thebulk recombination This kind of recombination is knownas surface recombination and it is characterized by an ownrecombination lifetime 120591119878

Usually all the experimental techniques determine aparameter that is a combination of the bulk contribute 120591119887 andof the surface one called effective lifetime

1

120591eff=

1

120591119887

+

1

120591119878

(6)

Nevertheless to characterize the recombination at thesurface rather than 120591119878 another parameter is usually usedthe so-called surface recombination velocity or SRV Anywaywithout entering in detail for which we refer to [8 23ndash27]to define SRV we have to consider that since the carriers thatrecombine at the surfaces of the sample can be described as acurrent-like flow outside the latest it is much easier to modelthe presence of the recombination centers on the surfaceby means of appropriate boundary conditions that bind thevalues of the carriers concentration and its gradient at theboundaries This leads to the following SRV definition

SRV = 119863[

120597119899120597119909

119899(119905)

]

boundary (7)

if 119909 is the space coordinate and a 1-119863 problem is underconsideration being 119863 the carriers ambipolar or monopolardiffusion coefficient as determined by the carriers injectionlevel

Even if the majority of the experimental cases are wellexplained with a constant SRV a more complicated theorycan be taken into account when a nonnegligible band-bending is present at the surface [26]

3 Free Carriers Absorption

In semiconductor materials the injected or generated freecarriers contribute to the absorption process by means ofintraband transitions It is known that three absorptionmechanisms happen in these kinds of materials We brieflyremember them

(i) Intrinsic or Band-gap Absorption occurs in the intrin-sic semiconductors when photons with energy 119864ph ge119864119892 (119864119892 is the material band gap) irradiate the mate-rial This absorption determines a transition from the


valence band to the conduction oneThe conventionalvisible and NIR photodetectors are based on thiseffect

(ii) Impurity Level to Band Absorption occurs in extrinsicmaterials like in the extrinsic infrared photodetec-tors when photons with energy 119864119894 le 119864ph le 119864119892

impinges on the material The photons absorptiondetermines transition from the impurity intrabandenergy level to the band as the name suggestsThis phenomenon is used for example to tune thewavelength response of the IR detectors by choosingimpurities with different energy level belonging to thematerial band-gap

(iii) Free Carriers Absorption The free carriers presentin the semiconductor materials contribute to theabsorption process absorbing photons with energy119864ph le 119864119892 (that corresponds to the middle or farinfrared region) and determining intraband transi-tion This phenomenon can degrade the detectorsresponse since the absorption increases as powerfunction of the wavelength Nevertheless it is usefulto measure the carriers concentration with this effectbeing proportional to them

The FCA has been widely studied in the middle of the20th century by means of both classical Drude model andquantum one (see eg [28ndash39])

The absorption of a photon with energy smaller thanthe energy band-gap of the semiconductor material allowsthe free carrier to have a transition from a lower energeticstate of the band to a higher energetic state of the samebandNevertheless since the two involved states belong to thesame band the transition can occur only with a momentumchange This change is possible by means of interactions ofthe free carriers with the material lattice (lattice vibrationdescribed by means of phonons) or by means of scatteringfrom ionized impurities The classical theory based on theDrude model describes the free carriers in the solid like agas of particles with a density 119899 subjected to collisions withthe relatively fixed ions of the lattice in which it is movingIt leads to a dependence of the free carriers absorptioncoefficient on the carriers concentration and on the squareof the wavelength of the absorbed photons

120572fc =11989911989021205822

119898119890120598012057841205871198883

1

120591

(8)

where 119890 is the electron charge119898119890 is the carrier effective mass1205980 is the vacuum permittivity 120578 is the material refractiveindex 119888 is the light speed 120582 is the wavelength of the absorbedphotons and 120591 is the relaxation time which represent theaverage time between two subsequent collisions or in otherwords the free motion time of the gas particle before anothercollision with the lattice ions happens

In the previous expression the relaxation constant isassumed independent of the energy but really it is a com-plicated function of the energy and contains the details ofthe collisionmechanismThese details have been investigatedby means of the quantum theory studying the interaction

mechanisms between the carriers and the semiconductorslattice which are described as interaction between the carriersand the vibrational modes of the lattice Moreover thequantum theory of the free carriers absorption justifies theoften observed deviation from the experimentally observed1205822 dependence in somematerials and compounds In general

without entering the details of the calculation and of thephenomenon description we can synthesizemdashlike in [40]mdashthat the main scattering process involving the free carriersin the semiconductors materials contributes with a resultantdependence of the absorption coefficient on the wavelengthThe latest can thus be expressed by a weighted sum of fewterms like

120572119904 = 119860 119904120582119901119904

(9)

where the label 119904 refers to a particular kind of scatteringprocess and each 119860 119904 is constants dependent on the carriersconcentration and on the temperature as well on otherparameters like the carriers effective mass The impuritiesconcentration in thematerial determines the dominantmodeof scattering and the relative value of the exponent 119901 ofthe wavelength Moreover the value should increase withthe doping Typical values are between 15 and 35 Someexamples are given in literature by studying the differentkind of scattering and considering different materials andcompounds For example Fan and Becker [29] reported avalue 119901 = 15 for the scattering by the acoustic phonons insilicon and germanium while in [35] the quantum mechani-cal calculation of the free carriers absorption considering thescattering by the optical phonons in the IIIndashV compounds ispresented and experimentally verified finding a value 119901 =

25 Furthermore [33] shows that the scattering by ionizedimpurities can lead to 119901 = 3 or 119901 = 35

The details of the calculations are reported in the citedpapers and also in many books like in [41] where it isremembered that a tabulation of the expression for 120572fc for theseveral scattering mechanisms is given in [42]

An experimental determination of the free carrierabsorption coefficient for the silicon at fixed temperatureof 350K was given by Schroder et al in 1978 [39] Inaccordance with the theoretical classical model the followingdependence on the free electrons and holes densities 119899 and 119901in the n-type and p-type material respectively was found at

120572119899 ≃ 1 times 10minus181205822119899

120572119901 ≃ 27 times 10minus181205822119901

(10)

where the wavelength 120582 is measured in cmThe experimental dependence on the wavelength at dif-

ferent doping levels in the n-type silicon and p-type at 300Kcan be found in [34] and [37] respectively while the freecarrier absorption versus wavelength for high purity Si atdifferent temperatures was reported in [38]


4 Free Carriers Absorption Methods toMeasure and Map the RecombinationLifetime in Semiconductors Materials

The intrinsic bond between the free carriers concentrationand the optical absorption change as previously seen hadoffered the possibility to directly monitor the former mea-suring the latter Moreover the great appeal in the use ofthis effect was and is the possibility to measure the quan-tities related to the carriers density (like the recombinationlifetime) and to measure them in an absolutely not invasiveand contactless way so that the sample integrity is totallypreserved

On this principle many of techniques that allow formeasuring the recombination lifetime in semiconductorsmaterials are based In practice these techniques operatewith a common principle that makes use of a sub-band-gapradiation to monitor the absorption variation in the sampledue to the excitation of an ultra-band-gap radiation So thatthe sub-band-gap radiation constitutes the sensitive elementin the measurement that is the probe while adopting thehabitual nomenclature used in optics to name an excitingsource the ultra-band-gap radiation constitutes the pump Inthis way all the techniques based on FCA can be classifiedas Pump and Probe We can refer to them as the FCA PPtechniques One of the first applications can be seen in twoworks published in 1966 by Chiarotti and Grassano [43 44]In 1970 Gauster and Bushnell [45] used the technique toderive the silicon absorption cross section and suggested thatthe method can be useful to investigate a variety of processessuch as the multiphonon absorption the recombination ofthe carriers and the measurement of the carriers lifetimeInteresting in their work is the use of a pulsed pump beam

FCA PP techniques can be either steady-statemethods ordynamic (we will refer to the former with Steady-State PP oreasily SS PP methods while to the latter with Transient PPor TPP) In the SS PP case the measurement is performedwith the pump on or with a modulated pump (so that thelock-in detection can be used to increase the signal to noiseratio) and what is directly observed in the measurement isthe distribution of the free carriers when the steady state hasbeen established The recombination lifetime is defined bythe relation (3) where the generation rate G is proportionalto the pump intensity This requires a calibration performedcomparing the distribution without the pump excitation andwith the pump excitation so that the variation of the carriersdensity can be evaluated and then the related quantities likethe actual recombination lifetime ILM and the ModulatedFree Carriers Absorption [46ndash50] belong to the Steady-StatePP methods In the case of the TPP to infer the value ofthe recombination lifetime in the sample a pulsed pumpis used In fact by recording the signal of the detectedprobe beam just in the transient time after the turn offof the pumb beam the sample recombination lifetime canbe estimated We will describe it in detail in the followingsections together with the ILMThey are in our opinion andin accordance with our experience the most consolidate andefficient being also perfectly complementary regarding theirrange of application

We will see that the major advantage of the TPP tech-nique is the possibility to discriminate the bulk contributionfrom the surface contribution with only one measurementwhereas on the other hand as the name said the ILM cangive in one measurement the mapping of the actual lifetimemapping of thewhole sample (the use of the adjective ldquoactualrdquowill be clear in the Section 7) Both techniques can operate inlow injection regime

5 Transient Pump and ProbeTechnique to Measure the RecombinationLifetime in Semiconductors

The experimental Transient Pump and Probe scheme tomeasure the recombination lifetime was well settled by Linget al in 1987 [1] but only with subsequent works [51ndash54]based on the analytical analysis of the interaction of a pulselaser beam with a semiconductor wafer performed by Lukeand Cheng [55] the potentialities of the technique wereclear These consist in the capability to discriminate betweenthe surface contribution to the recombination and the bulkone with only one measurement or in other words tomeasure simultaneously the bulk recombination lifetime andthe surface recombination velocity Moreover performingmore than one measurement with little changes in someparameters other characteristic electrical parameters of thematerials can be retrieved

Two schemes of Pump and Probe setup have beenproposed depending on the angle between the pump andprobe beams the transverse configuration and the parallelconfiguration The parallel configuration is more suitable forwafer lifetime mapping obtained by scanning the samplealong the plane orthogonal to the beams incidence plane[56 57] In this case the diffusion length of the free carriersin the sample 119871 = radic119863120591 limits the resolution of the systemIn fact in order to have a right interpretation of the signalthe spot size of the probe beam must be more than orequal to the expected diffusion length Otherwise if the spotsize is smaller the observed signal is more influenced bythe diffusion than by the recombination Nevertheless if theprobe beam is highly focused on the sample the desiredexperimental condition can be easily obtained

In the perpendicular configuration the sensitivity of themeasurement is substantially increased bymaking a long pathof interaction between pump and probe beam This path isin principle only limited by the spot size of the pump beam[58] Since there are no stringent requirements concerningthe state of wafer surface the use of a transverse probe is alsoattractive if the conditions of the sample do not allow for anefficient coupling of the probe radiation with the crystal asin the case of a sandblasted wafer surface or if special surfacetreatments are present and need to be characterized or cannotbe removed for any reason In both cases as previously saidthe pump photons energy is higher than the wafer band-gap(120582119901 gt 120582gap) but its intensity is not so high to determine a highinjection level (Figure 2) In this condition all parametersinvolved in the recombination process can be assumed to beconstant during the time decay


Conduction band

Valence band

Photon

Eph = h

Ec

Eg

E

(a) 119864ph gt 119864119892

Conduction band

Valence band

Photon

Eph = h

Ec

Eg

E

Ei

(b) 119864119894 lt 119864ph lt 119864119892

Conduction band

Valence band

phonon

Eph = h Ec

Eg

E

(c) 119864ph lt 119864119892

Figure 2 (a) Intrinsic or band-to-band absorption occurs when the energy of the absorbed photon is greater than the material band-gap (b)Extrinsic or impurity level to band absorption occurs in extrinsic materials when the photon energy is greater than the impurities energy butsmaller than the energy band-gap In the figure the process is sketched in the case of a p-type semiconductors (c) Intraband or free carriersabsorption can occur when the photon energy is smaller than the energy band-gap

Oscilloscope

NdYAG pulsed laser

Lens

InGaAsphotodiode

Trigger signal

Diode laserwith driver Optical fiber

Sample holderxy stage

Lens

Sample

Probe beam

Pump pulse

y

x

120572

120582 gt 120582gap

120582p lt 120582gap

Figure 3 Pump and Probe setup in the parallel configuration The pump pulsed beam and the probe beam impinge on the sample surfaceThe probe beam is focused on the sample surface attenuated by the sample and then detected by the InGaAs photodiode The diffusionlength determines the lateral resolution for the lifetime mapping so that the probe spot size must be chosen to be more than or at least equalto it [56]

A sketch of the parallel and transverse configuration ofTPP setup is shown in Figures 3 and 4 respectively Thecontinuous probe laser beam (usually a laser diode beam1550 nm) is detected by a photodiode after the propagationinside the sample (parallel to sample surface in the case oftransverse configuration and orthogonal to it in the othercase) Triggered on the pulse pump the decay curve acquiredby the InGaAs detector

119868 (119905) = 1198680119890minus1205720(120582)119871

(1 minus 119890minus120572fc(120582119905)119871

) (11)

is composed of two contributions the initial material absorp-tion coefficient 1205720(120582) and the time dependent free carriersabsorption coefficients at the probe wavelength 120572fc(120582 119905)

In Figure 5 a typical signal of the transmitted probe radia-tion in which the temporal dependence of the absorption canbe observed once the pump is turned off is shown

To clarify how the Transient PPmethodworks it is usefulto recall the analysis of the interaction of a laser pulse witha semiconductors sample We report in the following theanalysis as presented in [52] where the study of Luke andCheng [55] is extended to the case of different SRV on thetwo surfaces of the wafer

51 Modeling of the Analysis of the Interaction of a LaserPulse with a Silicon Wafer Let us consider a semiconductorwafer with a thickness 119889 and a laser pulse impinging on itssurface 1 as shown in Figure 6 Let119863 be theminority carriersdiffusion coefficient in the material 1198781 and 1198782 the surfacerecombination velocity on the surfaces 1 and 2 respectivelyand let us consider the same coordinate system as reported inthe cited figure


Sample

Oscilloscope

Pulsed laser


xyz stage

Lens

Trigger signal

Pump pulse

Probe beam

InGaAsphotodiode

x

120582 gt 120582gap

120582p lt 120582gap

Figure 4 Pump and Probe setup in the transverse configuration The pump pulsed beam impinges on the sample surface while the probebeam is orthogonal and propagates inside the sample

5

0

minus5

minus10

minus15

minus200 05 1 15 2 25 3 35

VminusV0

(mV

)

t (120583s)

Figure 5 A typical signal of TPPThe attenuation of the probe beamcan be observed just after the turn off of the pump pulse (the signalis normalized to the continuous level)

Pump pulse

Probe beam

0

y

z

x

S1 S2

d2minusd2

Figure 6 Coordinate system and wafer position adopted for theanalysis of the interaction of a laser pulse with a silicon wafer

If the duration of the pump pulse is much shorter thanthe expected recombination lifetime the analysis can becarried out for an ideal delta-pulse 119866(119905) = 1198730120575(119905) (1198730 is thenumber of photons and 120575(119905) the unitary Dirac distribution)and then it can be easily extended to the other cases [52 55]Moreover let us consider a pulse beam with a uniform spotsize impinging on the sample surface along the propagationdirection of the probe beam and similarly a probe spot sizeon the sample thickness In this condition the interactionvolume between the pump and the probe is extended tothe whole thickness (we are approximating the interactionvolume to a parallelogram) and the distribution of the excessfree carriers inside it can be considered changing only alongthe propagation direction of the pump denoted with 119909 Thedistribution of the excess free carriers generated along thewafer thickness for time 119905 gt 0 in low injection regime andneglecting the electric field can be considered governed bythe 1-119863 diffusion equation

120597119899 (119909 119905)

120597119905

= 119863

1205972119899 (119909 119905)

1205971199092

minus

119899 (119909 119905)

120591119887

(12)

with the boundary conditions

119863[

120597119899(119909 119905)

120597119905

]

119909=minus1198892

= 1198781119899(minus

119889

2

119905) (13)

minus119863[

120597119899(119909 119905)

120597119905

]

119909=1198892

= 1198782119899(

119889

2

119905) (14)

and initial condition given by the multiple reflections of thepulse along the sample thickness like in the Fabry-Perotcavity

119899 (119909 0) = 1198730 (1 minus 119877) 120572

119890minus120572(119909+1198892)

+ 119877119890minus120572119889

119890120572(minus119909+1198892)

(1 minus 1198772119890minus2120572119889

)

(15)


with 119877 and 120572 being the reflection coefficient on the wafersurfaces and the silicon absorption coefficient at the pumpwavelength respectively

The most general solution of (12) can be expressed as

119899 (119909 119905) =

infin

sum

119896=1

119890minus(1120591

119887+1198862

119896119863)119905

(119860119896 cos 119886119896119909 + 119861119896 sin 119886119896119909) (16)

where the coefficients 119886119896 119887119896 and 119861119896 can be found usingboundary conditions (13) and initial condition (15) respec-tively In particular the coefficients 119886119896 are the solutions of thecharacteristic equation

119886119896119889 = arctan 1198781

119886119896119863

+ arctan 1198782

119886119896119863

+ 119896120587 (17)

while if

119911119896 = 119886119896119889

1198920 =

(1 minus 119877) 120572

1 minus 119877119890minus2120572119889

1198611015840

119896=

(41198861198961198920120572) 119890minus1205721198892

[1198872

119896(119886119896119889 + sin 119886119896119889) + (119886119896119889 minus sin 119886119896119889)] (1205722 + 1198862119896)

(18)

it results in

119887119896 equiv

119860119896

119861119896

= minus

119886119896119863 cos 119911119896 + 1198781 sin 119911119896119886119896119863 sin 119911119896 minus 1198781 cos 119911119896

119861119896 = 1198611015840

119896[119887119896 (cos 119911119896 sinh

120572119889

2

+

119886119896

120572

sin 119911119896 cosh120572119889

2

)

times (1 + 119877119890minus120572119889

) ]

+ 1198611015840

119896[(

119886119896

120572

cos 119911119896 sinh120572119889

2

minus sin 119911119896 cosh120572119889

2

)

times (1 minus 119877119890minus120572119889

) ]

(19)

Since the measurement determines the absorption due to allthe free carriers generated along the excited volume of thesample by the pump pulse we have to consider the averageof the free carriers in that volume (at the beginning of thesection we have called it interaction volume 119881int)

119899av (119905) =1

119881intint

119881int

119899 (119909 119905) 119889119881 (20)

with 119860 int being the section of the interaction volume in thehorizontal planeThe latest approximation in (20) as pointedout at the beginning of the section subsists considering apump pulse that uniformly illuminates the sample along thepropagation direction of the probe beam This condition canbe easily obtained experimentally bymeans of suitable opticslike a cylindrical lens

If the diameter of the probe beam is larger than thesample thickness in a way that the uniform illumination of

the sample cross section is assured the interaction volumeis extended to the whole sample thickness so that since theexcess free carriers distribution depends only by the spatialcoordinate 119909 along which the pump pulse is propagating theexpression (20) becomes

119899av (119905) prop int

1198892

minus1198892

119899 (119909 119905) 119889119909

= sum

119896

119861119896119887119896119868119896 (minus

119889

2

119889

2

) 119906119896 (119905)

= sum

119896

119861119896119887119896

sin 119911119896119911119896

119906119896 (119905)

(21)

where we have defined

119906119896 (119905) = 119890minus(1120591

119887+1198862

119896119863)119905

120601119896 (119909) = 119887119896 cos 119886119896119909 + sin 119886119896119909

119868 (119909119894 119909119891) = int

119909119891

119909119894

120601 (119909) 119889119909

(22)

In themore general case ofGaussian pump pulse centeredat instant 1198790 and with width 120590

119866119892 (119905) =

1198730

radic2120587120590

119890minus(119905minus119879

0)21205902

(23)

119899av can be derived from the convolution of 119899av given by (20)with the pulse

119899gav = int119905

0

119899av (120591) 119866119892 (119905 minus 120591) 119889120591

= sum

119896

119861119896119887119896119868119896 (minus

119889

2

119889

2

)119864119896 (119905)

(24)

where

119864119896 (119905) = int

119905

0

119906119896 (120591) 119866119892 (119905 minus 120591) 119889120591

= 119906119896 (119905) 119890119865119896((1205902

2)119865119896+1198790)

times [erf ( 120590

radic2

119865119896 +

1198790

radic2120590

) minus erf ( 120590

radic2

119865119896 minus

119905 minus 1198790

radic2120590

)]

(25)

with

119865119896 =

1

120591119887

+ 1198862

119896119863 (26)

The semilog plot of 119899gav(119905) allows for defining an instanta-neous observed lifetime 1205910 given by

1

1205910

= minus

119889

119889119905

ln [119899gav (119905)] =1

120591119887

+ 119877119904 (119905) (27)

where from (20) (24) and (25) it is clear that 119877119904 is atime dependent ratio between two series and it depends on


the physical wafer parameters 119889 120591119887 119863 1198781 and 1198782 (see also(17)) Nevertheless it can be demonstrated [52 55] that itsasymptotic value is a constant

119877119904 (119905)

119905rarrinfin

997888rarr 119877119904119886 = 1198861119863(28)

The previous expression shows that the asymptotic value ofthe inverse of the instantaneous observed lifetime can beexpressed as the sum of two independent terms

1

1205910119886

=

1

120591119887

+ 119877119904119886 =

1

120591119887

+

1

1205911199041

(29)

the first of which is equal to the inverse of the bulk lifetimeand the second one is independent of latest and dependenton the surface effects as 1198861 depends on 1198781 1198782 and 119889 (thesubscript 119904 refers to this dependence of the asymptotic valueof 119877)

6 TPP Parameter Extraction

Therelation (29) is the core of the TPPmethod In accordancewith the sample properties and the set-up configurationdifferent information can be extracted by the measurementIn this way the TPP presents several versions

61 Single Slope Method When the surface effects can beneglected or when 1205911199041 is known from (29) the bulk lifetimecan be directly derived Luke and Cheng [55] plotted 1205911199041

versus the wafer thickness 119889 for different values of the surfacerecombination velocity (from 10 to 106 cms) supposing 1198782 =1198781 and showed that for 119878 gt 105 when 119889 increases the valueof 1205911199041 does not change Really the product 119878119889 is determinantsince when it increases the root 1198861119889 of the transcendentalequation (17) approaches a limit value This suggests that for119878119889 rarr infin the surface effects can be neglected and the bulklifetime can be easily derived as

120591119887 =

12059101198861205911199041

1205911199041 minus 1205910119886

(30)

with 1205911199041 = 11198861119863 being analytically computed and 1205910119886 derivedfrom the experimental data The simulations reported in [55]show that in practice the influence of 119878 can be neglected for119878119889 gt 120 cm2s

62 Dual Slope Method If the surface effects cannot beneglected two values of the instantaneous observed lifetimecan be used to derive from the experimental data the bulk life-time and the surface recombination velocities the asymptoticvalue (defined in (29)) and the maximum value (defined inthe same manner where obviously 119877119904(119905) is replaced with itsmaximum value max[119877119904(119905)] equiv 119877119904119898)

In this way if 1198782 = 1198781 = 119878 to determine the two unknownquantities 120591119887 and 119878 we have a system with two equations

1

1205910119886

=

1

120591119887

+ 119877119904119886

1

1205910119898

=

1

120591119887

+ 119877119898

(31)

so that

(119877119904119898 minus 119877119904119886)anal = (1

1205910119898

minus

1

1205910119886

)

exper(32)

can be used to evaluate 119878 and

120591119887 =

(1205910119886)anal1 minus (119877119904119886)exper(1205910119886)anal

(33)

gives 120591119887 The subscripts ldquoanalrdquo and ldquoexperrdquo indicate thatthe quantities are analytically computed or experimentallydetermined Luke and Cheng wisely suggest that to derive 120591119887it is better to use 1205910119886 since it can be better determined and thatthe error computed to neglect the dependence of 119877119904 on 120591119887 isabout 01 per cent over the range of 119878 = 1 to 106 cms or ratherfor 40 le 119878119889 le 10

6 cm2s This shows the complementary ofthe single slope and dual slop methods

63 Two Wafer Method In the region not covered by theprevious two methods Luke and Cheng proposed to utilizethe 119877119904 dependence on the wafer thickness In fact using twowafers which differ only in thickness 1198891 and 1198892 the formersystem (33) can be replaced with

(

1

1205910119886

)

1

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(

1

1205910119886

)

2

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(34)

Really the method works well even for large value of119878119889 so that more than complementary to the previous onescan substitute them Nevertheless Irace et al [59] observedthat since the two measurements of 1205910 are done on the samematerial and at most averaged on an area of about 1 cm2equal to the probe beam spot it can be safely assumedthat the bulk recombination lifetime remains unchangedOn the other hand since the separation and calculation ofthe surface effects rely on the hypothesis that also SRV hasto remain unchanged there is the problem of assessing asurface treatment procedure that is well reproducible andsets equal surface recombination velocities on both sides ofeach sample and in both samples These considerations ledthem to perform an analysis of the error sources of thisprocedure and of the uncertainties that they can lead toThey showed the effect of different values of SRVs on thetwo measurements using a graphical approach Thereforefor each measured value of the instantaneous lifetime it ispossible to draw a line on the 120591119887 minus 119878-plane that of course willspan all the allowed values of 120591119887 and 119878 On the other handbecause two measurements are available the two solutionsintersect in a single point corresponding to the looked for(120591119887 119878) pair If there are uncertainties Δ120591119894 (119894 = 1 2) on thequantities a region can be identified in the plane bounded bythe lines 12059101plusmnΔ1205911 and 12059102plusmnΔ1205912 respectively for each sampleIntersecting these two regions the area where the solutionof the (34) lies can be identified It contains the informationabout the uncertainty on the surface recombination velocity


1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

10minus3

10minus4

10minus5

10minus6

120591b

(s)

S (cms)

120591eff = 5 120583s

120591eff = 6 120583s

120591eff = 15 120583s

120591eff = 12 120583s

120591eff 2 = 115 120583s120591eff 1 = 55 120583sd1 = 500 120583m

d2 = 200 120583m

Figure 7 120591119887minus SRV plane as reported in [59]

or can take into account the effects of the experimentaluncertainties on the measured 1205910 value An example of theabove procedure is given in Figure 7

64 Dual Slope Method on Both Surface If 1198780 = 1198781 a furtherequation is needed to determine all the three parame-ters Kousik et al in 1991 [52] proposed to perform themeasurement flipping the wafer to have one measurementwith the pump beam impinging on the wafer surface withrecombination surface velocity 1198781 and one complementarymeasurement with the pump impinging on the other surface

The available equations are thus

1

1205910119898

=

1

120591119887

+ 119877119898 for (1198781 1198782) or (1198782 1198781)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198781 1198782)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198782 1198781)

(35)

where the notation (119878119894 119878119895) (119894 = 119895) means that the pumpimpinges on the surface 119878119894

65 Multiwavelength Method The bulk lifetime and thesurface recombination velocity do not depend on the pumpwavelength but from (27) we can see that the instantaneousobserved lifetime 1205910 depends on it In particular the depen-dence is all included in the initial part while the asymptoticpart is unaffected This means that for 119905 rarr infin ln(119899av(120582 119905))has the same slope 11205910119886 (see (24)ndash(29)) In other wordsthe asymptotic separation between two decay curves relativeto two different pump wavelengths 119899av(1205821 119905) minus 119899av(1205822 119905)approaches a constant value This observation suggested toBuczkowski and coworkers [62] in 1991 (when the dual slopemethod was not proposed yet) that the surface effects canbe extracted from this value They verified the idea with adifferent technique called two-laser microwave reflection in

the hypothesis of equal surface recombination velocities Thetechnique uses the reflection or absorption of a microwaveradiation on wafer surface as probe signal

From 2002 to 2005 Irace and coworkers [63ndash68] trans-posed and tested the idea with a pump and probe setup usinga laser diode 1550 nm as probe beam and the fundamentaland the second harmonic of a 7 ns pulsedNd-YAG laser (1205821 =1064 nm and 1205822 = 532 nm resp) to pump the sample

Defining

120575119886 = lim119905rarrinfin

120575 (Δ120582 119905) = ln(119873rel (1205821 119905)

119873rel (1205822 119905)) (36)

with119873rel = 119899av(120582 119905)119899av(120582 0) and Δ120582 = 1205821 minus120582 it is clear that(36) is a function of Δ120582119863 119889 1198781 1198782

If 1198781 = 1198782 = 119878 the experimental determination of 120575119886for a fixed pair of wavelength is enough to determine 119878Successively 120591119887 can be determined by means of (29)

For any pair of wavelengths 120575 is an increasing function of119878 For small value of 119878 where the surface effects are negligiblewith respect to the bulk ones it is difficult to distinguishbetween two cases with different absorption therefore thevalues assumed by 120575 are small When 119878 increases theseparation between the decay curves increases reaching asaturation value for high 119878 This limits the applicability of themethod to 119878 lt 105 while the lower limit is determined by thesignal-to-noise ratio

If 1198781 = 1198782 having only two pump wavelengths 1205821 and 1205822it is possible to determine both the surface recombinationvelocities by means of measurements with the pump imping-ing one time on one of the wafer surface and another timewith the pump impinging on the other wafer surface likein the dual slope method In this manner 1198781 and 1198782 can bedetermined from the experimental decay curve solving thesystem

(12057511988612)exper = 119891 (Δ120582 1198781 1198782) for pump on surface 1

(12057511988621)exper = 119892 (Δ120582 1198782 1198781) for pump on surface 2(37)

(with obvious meaning of the subscripts 12 and 21) Then 120591119887is computed using (33) with 119877119904119886 relative to one of the twomeasurements

The main advantage of the technique is that it gives thevalues of 1198781 1198782 and 120591119887 even when the sample thicknessis comparable to the minority carriers diffusion length forwhich the effects of recombination at the two interfaces arecoupled

66 Cross-Sectional Profiling Technique The previouslyexposed methods are based on the assumption that the otherparameters of the sample are well known (see eg (17) and(28) (29) depending on 119863) but it could not be the real caseand moreover some other experimental variables could notbe exactly controlled Mathematically this means that theused equation systems require additional equations to besolved

Moreover they work in conditions such that the interac-tion volumebetween the pump andprobe iswide as thewhole


sample thickness In this condition depending on the samplethickness and on the diffusion length the bulk contributionto the recombination lifetime could be dominant respect tothe surface contribution or vice versa To overcome thesedifficulties Gaubas et al [69 70] proposed an alternativemethod based on the simultaneous analysis of the time andamplitude characteristics of the excess free carriers decay byscanning the sample thickness with a probe beam spot sizesmaller than the wafer thickness In this way the interactionvolume between the pump (illuminating the whole samplealong the propagation direction of the probe) and the probeis limited to a narrow portion along the sample thicknessThey showed (with simulation and experimental results) thatthe transient decay shape changes with the depth along thesample thickness with that meaning as the distance of theprobe beam from the pump incident surface on the sampleincreases In particular they observed that the decay showsconcavity change which passes to concave to convex whenthe depth approaches the middle of the sampleThis happensbecause the surface effects decrease and the observed fractionof the excess free carriers in that region is more governed bythe bulk recombinationThe same behavior is observed with-out the scanning but changing only the probe beam spot size

Recently an improved version of the technique has beenreproposed where a detailed analysis of different probe beamshapes has been evidenced (ie both the case of collimatedand divergent probe beam) [71 72] The improvement is thecombination of it with the multiwavelength method In theparticular case the pump is a tunable source in the range500ndash1150 nm constituted by an Optical Parametric Oscillator(OPO) pumped with a third harmonic of a NDYAG laser Inthis way the method results more robust and accurate sinceit has both the advantages of of the cross-sectional profilingtechnique and of the multiwavelength one

About the latest it must be observed that a tunable sourceallows for having more than two wavelengths to perform themeasurements so that instead of the system (37) relationslike (36) are used for each pair of wavelengths This avoidsthemeasurements with the pump impinging one time on onesurface and one time on the other surface which can leadto some errors Moreover the desired quantities are betterand more accurately retrieved by fitting simultaneously a setof equations similar to (27) obtained nevertheless with amore accurate theoretical model of the best fit curve Themeasurement is then performed from the a-priori knowledgeof the other sample parameters involved and allows for abetter the determination of them

At the beginning of this section we stated that the parallelconfiguration is more suitable for wafer lifetime mapping(see eg [56]) nevertheless it must be observed that thelifetime mapping cannot be done with a unique acquisitionThe samplemust be scanned so that depending on the pumppulse frequency on the probe beam spot size and sampledimension the measurement requires an amount of time thattypically is at least of order of minutes For a more efficientand advantageous lifetime mapping an array of detectors canbe used to photograph the whole sample in one time Thetechnique that we present in the next sections allows exactlyfor this

Moreover it must be noted that in the transient lifetimemeasurements could happen that the effective measuredlifetime can differ from the actual lifetime by a factor equalto 2 or 3 [61]This inconvenience is not present in the steady-state techniques We will understand this in the next sectionbefore presenting the ILM technique

7 Differential and Actual Lifetime

In the paper we have several times mentioned that therecombination parameters in general depend on the excesscarriers concentration thus on the injection level When theinjection level is high that is when the generated excesscarriers concentration Δ119899 is higher than the majority carriersconcentration really the observed transient decay is not asingle-exponential and should require for data extraction amore complicated treatment Nevertheless Aberle et al [73ndash76] acutely noted that the situation can be treated in smallsignal regime They suggested that it can be actuated addinga constant bias of suitable intensity to the pulse pump togenerate a constant background carriers concentration 119899119887a lot higher than the excess carriers concentration inducedby the pump pulse 119899119901(119905) So that the excess carriers can bewritten as

Δ119899 (119909 119905) = 119899119887 (119909) + 119899119901 (119909 119905) (38)

In this way they showed that the measured lifetime is notthe actual but a differential lifetime related to the former bythe relation

120591 (119899119887) =

1

119899119887

int

119899119887

0

120591diff119889Δ119899 (39)

with a similar expression for the surface recombinationvelocity The differential and the actual lifetime are equalonly in the particular case of a constant bulk lifetimeThis situation is often verified in the case of low-injectioncondition At high-injection conditions when for examplethe Auger recombination becomes dominant it can be easilydemonstrated that the small signal approach predicts a valueof the differential lifetime three times smaller than the actualone For the radiative recombination the reduction factor is 2[76]

The steady-state methods where the generation rate andthe excess carriers density are controlled simultaneouslypresent the advantage to measure directly the actual lifetimeIn fact in the steady-state conditions since the generationrate and the recombination rate are equal and proportional tothe power density of the bias light on the approximation thatthe excess carriers concentration is approximately constantthroughout the sample the result is [75]

120591 (119899119887) =

1

119875

int

119875

0

120591diff (119875) 119889119875 (40)

Consequently as the integration is performed by the detectorthe actual lifetime is directly measured This is a furtheradvantage of the ILM method which is detailed in the nextsections


8 Infrared Camera Lifetime MappingCarrierDensity Imaging

The Infrared Lifetime Mapping (ILM) method was presentedthe first time in 2000 by Bail et al [2] Subsequently in 2001Riepe and coworkers [3] presented a variation on the themecalled Carrier Density Imaging (CDI) improved with thelock-in detection technique [77]

The method is also based on the free carriers absorp-tion Like in the pump and probe method the sub-band-gap radiation is used as a probe propagating through thesample to monitor the absorption variation (or rather thetransmittance variation) due to the free carriers generated bythe above band gap pump photons Nevertheless a black bodyradiation is used as probe instead of a laser This radiationbelonging to the 5ndash25 120583m range of the IR spectrum ispartially transmitted by the silicon wafer and can be detectedby an infrared CCD (charge coupled device) camera like theones used in IR thermography setups [77 78] So that whenthe sample is not excited by the laser pump beam (pump off )the CCD camera returns an image with a reference intensitylevel 119878off This image corresponds to the currents intensitiesproduced by each pixel sensor illuminated by the black bodyinfrared radiation that had passed through the silicon waferIn others words it is a matrix of119872times119873 elements 119878119894119895 as manyas the number of CCD pixels When the pump impinges onthe sample (pump on) the absorption of photons with energygreater than the energy band gap generates an excess of freecarriers (hole-electron pairs) that modifies the absorptioncoefficient and consequently causes an intensity variation ofthe transmitted black body radiation The camera snapshotreflects this variation with a different intensity currents map119878on

The authors showed that it is easy to derive the excesscarriers density variation per unit area Δ119899119882 (where 119899 is thefree carriers density and119882 is the sample thickness) from thecamera contrast

119870 = 119878off minus 119878on (41)

once it is calibrated and successively the effective actuallifetime This is the reason of the adopted name for thetechnique CDI given by [3] The effective actual lifetime isderived by means of the steady-state solution of the diffusionequation

120591eff =Δ119899119882

119866

(42)

where 119866 is the generation rate equal to the pump densityphotons flow per unit time (which has the dimension of[lenghtminus3times timeminus1]) reduced by the relative quantity reflectedby the wafer incident surface

119866 = Φ(1 minus 119877119891) (43)

with 119877119891 being the silicon wafer front side reflectance at pumpwavelength

The ILM schematic setup is shown in Figure 8

Optical fiber

Lens

Diode laser

Pump beam

Black body radiation

Wafer

Black body

CCD camera

Tb

Tw

120582 lt 120582gap

120582 gt 120582gap

Figure 8 In the ILMCDI schematic setup

81 The Detected Black Body Radiation The camera signal isproportional to the black body radiation transmitted throughthe wafer (of thickness 119882 and reflectance on the incidentsurface 119877) and integrated on the camera spectral range[120582min 120582max]

119878 = Γint

120582max

120582min

(1 minus 119877)2119890minus120572fc(120582)119882

119868 (120582 119879119887) 119889120582

asymp Γint

120582max

120582min

(1 minus 119877)2120572fc (120582)119882119868 (120582 119879119887) 119889120582

(44)

where Γ is a proportionality constant that takes into accountthe CCD response and the sample geometry

119868 (120582det 119879119887) =2120587ℎ119888

2

1205825

1

119890(ℎ119888120582119896

119861119879119887)minus1

asymp

120574

1205825119890minusΨ120582119879

119887 (45)

(with 120574 = 2120587ℎ1198882 and Ψ = ℎ119888119896119861) is the black body spectral

density given by Planckrsquos Law approximated in the case ofblack body temperature 119879119887 greater than hundreds kelvin (ℎis the Planck constant 119888 is the vacuum light velocity and 119896119861is the Boltzmann constant) 119877 is the silicon wafer reflectanceat the black bodywavelength and we remember120572fc is the freecarriers absorption coefficient (cmminus1) [2 34 37ndash39]

82 EmissionMode Since each body which is heated behaveslike a black body the emitted infrared radiation can bedetected increasing the wafer temperature 119879119908 and like in theabsorption mode the optical pumping determines a variationof the carriers intensity that leads to a variation of the emittedradiation From the Kirchhoff Law at thermal equilibriumthe emission coefficient 120576 must be equal to the absorptioncoefficient 120572

In the emission mode the wafer is heated above the blackbody temperature so that the black body radiation of thewafer on the background is dominant In fact asmade explicitin [79] the camera detects a signal that is the superposition


200

180

160

140

120

100

80

60

40

20

0

100

80

60

40

20

0

100806040200

120591ap

p(120583

s)

x (mm)

y(m

m)

Figure 9 Typical ILM image as reported in [60]

of the black body radiation given by (44) and of a blackbody radiation emitted just by the sample since at thermalequilibrium the free carriers also emit infrared radiationDepending on thewafer temperaturewith respect to the blackbody temperature (and vice versa) the emission componentcan dominate on the transmitted one Denoted with 120576fc(120582)the emissivity of the wafer due to the free carriers theemission component of the signal can be written like in (44)using Planckrsquos law (45) for nonideal black bodies (120576(120582) lt 1)

119878em = Γint

120582max

120582min

120576fc (120582)120574

1205825119890minusΨ120582119879

119908

119889120582 (46)

The total signal detected by the camera is thus

119878 = 119878em minus 119878119886119887 (47)

where 119878em and 119878119886119887 are given by (44) and (52) respectively Itis clear now that if 119879119887 ≪ 119879119908 the emission signal dominates(in the experiments 119879119887 gt 350K [2 79 80])

83 ILMCDI Calibration The calibration can be performedcomparing camera image ofwell-characterizedwafer (usuallya p-type) that differs from a reference one only in dopantconcentrations 119873 and at the most in the thickness 119882

(Figure 9) In fact the plot of the camera contrast relative toeach pixel individuated by the coordinates in the 119909-119910 plane119870119862(119909 119910) = 119878119873

119894119882119894

(119909 119910) minus 11987811987311198821

(119909 119910) obtained subtractingthe image of the 119894th wafer to that one of the referencewafer labeled 1 as function of the free carriers density perunit area relative to the same wafers and to the same pixelΔ(119873119882)(119909 119910) = 119873119894(119909 119910)119882119894 minus1198731(119909 119910)1198821 gives the followingexperimental relation

119870119862 (119909 119910) = 119898Δ (119873119882) (119909 119910) (48)

that suitably corrected gives the final conversion rulebetween the free carriers density per unit area variation andcamera contrast

Δ (119899119882) (119909 119910) =

119870 (119909 119910)

119898120590

120581 (119909 119910) (49)

The correction factor 120590 takes in account the fact that onlywafers with one type of dopant are used in the calibration [2]while the absorbed pump photons generate excess electronsand holes that bring about a variation in the absorbed blackbody radiation This correction factor in the case of p-typecalibration wafers must be

120590 = (1 +

120572119899

120572119901

) (50)

where 120572119899 and 120572119901 are the absorption coefficients of the freeelectrons and holes respectively at the absorbed radiationwavelength 120582 (the black body detected radiation) [2 34 37ndash39] Obviously the absorption coefficients must be invertedin (50) if the calibration is performed with n-type wafersThe factor 120581(119909 119910) suggested by [81] takes into account boththe individual pixel sensor responsivity (the variation of thesensor response to the variation of the incident radiation)and the CCD sensor geometrical configuration It must beconsidered that the central pixels of the CCD array see thesample under a larger steradian than the pixels near the arrayedge [81] so that the incident IR power flow is different The120581(119909 119910) factor is specific to each CCD array comparing thepixels response respect to the response of a reference pixel

84 ILMCDI Sensitivity To quantify the ILMCDI sensi-tivity a noise equivalent lifetime (NEL) is defined [2 81] asthe effective minority carrier lifetime that the sample musthave to generate an infrared camera contrast that equals thecamera noise This quantity is related to camera detectionlimit (orminimum resolvable contrast) that is the minimumvariation of the incident radiation that the camera candistinguish For the infrared camera it is expressed in termsof theminimum appreciable variation of temperature the so-called NETD (noise equivalent temperature difference) sincethey are usually used to detect the temperature dependentblack body radiation like in the ILMCDI and thermographyapplications Nevertheless as observed in [82] the nominalvalue of the NEDT cannot be considered since it is definedby measuring a black body at room temperature and usingspecific camera settings and specific f -number of the opticswhile the experimental condition is usually different It mustbe measured for the specific setup and its results are usuallyhigher than the nominal one

Using the (45) theminimum contrast can be written as

119870119889119897 = Γ (119868 (120582det 119879119887 +NETD) minus 119868 (120582det 119879119887))

asymp Γ

120574

1205825

det(119890minusΨ(119879

119887+NETD)

minus 119890minusΨ119879

119887

)

asymp Γ

120574

1205825

det

ΨNEDT1198792

119887

119890minusΨ119879

119887

(51)

where we remember 120574 = 2120587ℎ1198882 and we have redefined

Ψ = ℎ119888119896119861120582det The latest approximation derives from theconsideration that is NETD ≪ 119879119887 (for state-of-art camerasNEDT sim10mK)

When the camera ILMCDI signal equals the minimumcamera contrast then 120591eff = NEL Neglecting the reflectance


(in the experiment it is possible using an antireflectioncoating on the back of thewafer) supposing that the radiationis optically filtered at wavelength 120582det (as often it is inpractice) we can approximate (44) as

119878 asymp Γ120573Δ119899119882

120574

1205823

119889119890119905

119890minusΨ119879

119887

(52)

so that from (42) (43) (52) and (51) we obtain

NEL = ℎ119888

1198961198611198792

1198871205822

det

1

120573Φ (1 minus 119877119891)

NETDradic119873

(53)

where119873 is the number of averaged images [2 81 82]The sensitivity of the ILMCDI in the emission mode

can be derived in the same manner obtaining at the endan expression similar to the previous one where 119879119908 appearsinstead of 119879119887

Equation (53) shows that the setup sensitivity increases(i) when the black body temperature in the absorption

mode and the wafer temperature in the emission modeincrease This suggests that the latest mode resultsare more sensitive In fact in accordance with theKirkoff Law at thermal equilibrium the absorbedinfrared radiation is equal to the emitted infraredradiation but in spite of the smaller emissivity ofthe wafer with respect to the quasi-ideal black bodyused the dependence of the black body radiationon temperature must be considered Therefore whilerelatively intense signals are reached in emissionmodeheating the wafer in the other configuration the blackbody must be cooled determining a reduction of theemitted infrared intensity

(ii) when a detection wavelength around 8 120583m is usedThis wavelength corresponds to the maximum ofthe black body spectral emission Cameras with themaximum response in the MIR must be chosen

(iii) with the increasing of the camera sensitivity (decreas-ing of the NEDT) and camera speed (the cameraframe rate increases and thus the number of averagedimages or rather the averaging time) The introduc-tion of the lock-in technique [77] already suggestedin [2] and tested the first time in the CDI apparatusby Riepe et al [3] allows for reducing the NEL by 2orders of magnitude

85 Lateral Resolution The lateral resolution of the ILMdepends on the optical set up and on the camera resolutionWith a 384times288 pixel array detector and an optics that allowsfor focusing on an area varying between 15times15 100times100mmthe lateral resolution (given by the ratio between the focusedarea and the number of lateral pixels) of about 50 and 350120583mrespectively can be reached [81] With the modern cameraincreasing the number of pixel and with different optics theresolution can be increased to few tens of microns

9 Dynamic Carrier Lifetime Imaging

We have noted that being the ILM a steady-state techniqueone of its disadvantages is the necessity of an accurate

calibration Ramspeck and coworkers recently developed adynamic version of the technique (dynamic-ILM) [61] thatpresents the advantages of the transient technique to avoidthe calibration and the advantages of the ILM to yield a veryfast and high spatially resolved data acquisition

The setup is similar to the common ILM techniquebut the applied excitation source is a square-wave-shapedillumination and it implements a lock-in technique for thedata processing (this not only to improve the signal-to-noise-ratio as we will see)The pump period is chosen such that thesample reaches a stationary condition in the semiperiodwhenthe pump is on In this way four images are recorded thefirst one immediately after switching on the pump the secondwhen the steady-state condition is achieved the third imme-diately after switching off of the pump and the fourth whenthe new steady state is achieved with the pump off The latestimage records the background illumination thatmust be usedto correct the other images If the camera integration time ischosen sufficiently short that means of the order of the sam-ple expected effective lifetime 119905int asymp 120591eff the first and the thirdimages record the transient rise and decay of the excess freecarriers and in this condition a sufficient contrast betweenthe signals relative to the the images of the transients respectthe signal relative to the image of the steady-state is achieved

In Figure 10 the illustration of the images acquisitiontiming with respect to the pump illumination is reported asshown in the original work [61]

The authors observe that the ratio between the back-ground corrected images relative to the steady-state andto the rise transient respectively (the first image and thesecond one) depends only on 119905int and 120591eff so that known theformer the second one can be evaluated without requiringfurther calibration Nevertheless using the lock-in detectionthe correction of the background results are automatic Infact the acquisition is composed by two subsequent steps inthe first one the four images recorded during one excitationperiod are multiplied with coefficient of a sine and a cosinefunction in phase with the excitation in the second onethe four images are summed up The results are the cosine-correlation function 119878cos and the sine-correlation function119878sin that give respectively the difference between the firstand the third images and between the second and the fourthimages

119878cos = int119905int

0

Δ119899 (119905) 119889119905 minus int

1198792+int

1198792

Δ119899 (119905) 119889119905

= 119878st-st 119905int minus 2120591119890119891119891 [1 minus exp(minus119905int120591eff

)]

119878sin = int1198794+119905int

1198794

Δ119899 (119905) 119889119905 minus int

31198794+int

31198794

Δ119899 (119905) 119889119905

= 119878st-st

times 119905int minus 2120591eff [exp(minus119879

4120591eff) minus exp(minus

119879 + 4119905int4120591eff

)]

(54)


10

08

06

04

02

000 T

Time t (120583s)

1st image 2nd image 3rd image 4th image

Excess carrierdensity Δn(t)

Imageacquisitiontime tint

Generationrate 119970

Nor

mal

ized

carr

ier d

ensit

y Δn(t) a

nd g

ener

atio

n ra

te 119970

(a)

tint = 300 120583s

tint = 1100 120583s

tint = 2000 120583s

120

105

90

75

60

450 1000

Lifetime 120591eff (120583s)Ph

ase Φ

(∘)

(b)

Figure 10 (a) Schematic of the generation rate G excess carriers density Δ119899(119905) and image acquisition during one measurement period oflength 119879 as reported in the original work of Ramspeck et al [61] (b) Relationship between lifetime 120591eff and phase lock-in phaseΦ for severalimage acquisition times as reported in the original work of Ramspeck et al [61]

where 119878st-st = G120591eff represents the steady-state signal thatwould be recorded if any backgroundwould be present while119879 is the lock-in period The authors observe that the lock-in phase defined as Φ = arctan(119878sin119878cos) is an increasingfunction of 120591eff depending on the parameters 119905int and 119879that for small value of 119905int is enough linear for low lifetimeFor measurements performed with 119905int = 300 120583s and 119879 =

25ms the function results are linear for lifetime smaller than200120583s with a sensitivity of 025∘ times 120583sminus1 (see the right side ofFigure 10(a))They conclude that if a shorter integration timeis chosen the sensitivity increases and the linear range shiftstoward the lower lifetime values

10 Conclusion

In this paper we have examined all two optical techniquesto measure the recombination lifetime in the semiconductormaterials the Transient Pump and Probe and the InfraredLifetime Mapping Both techniques are based on the FreeCarriers Absorption and can be classified as Pump and Probemethods since they make use of the common operationprinciple that uses a sub-band-gap radiation as probe tomonitor the absorption variation in a sample induced bythe generation of excess free carriers by means of a pumpradiation

In the TPP method the signal where the interestingquantities are retrieved is the transient evolution of theexcess carrier density This is generated by the pump pulsejust after the turn off of the latter Its main merit is thepossibility to directly determine from the measurement asimultaneous evaluation of the bulk recombination lifetime

and of the surface recombination velocity Moreover we haveshown that little variations in the setup configuration give themethod the flexibility to measure other electrical parametersof the sample Furthermore it allows formeasuring very shortlifetime of the order of few nanosecond (depending on thepump pulse width)

The ILM on the other hand allows a very fast mappingof the actual recombination lifetime on the sample In factthis method uses an infrared camera sensor to detect thefree carriers absorption so that the local information on thecarrier density can be derived in a single acquisition withoutscanning the sample Having additional knowledge aboutthe carrier generation rate 119866 the effective carrier lifetime120591eff = Δ119899119866 of the sample is calculated In spite of beinga steady-state method based on the measurement of thefrequency response of the semiconductor when excited bya modulated laser beam it presents the disadvantages thatthe correct determination of the lifetime critically dependson the accuracy of the calibration procedure applied andon the accurate accounting of the lateral inhomogeneities inthe optical properties of the sample We have illustrated asthis disadvantage is surmounted by the its dynamic versionwhich releases themeasurements from the calibration so thatthe methods results are to be perfectly complementary withrespect to the TPP

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


References

[1] Z G Ling P K Ajmera M Anselment and L F DimauroldquoLifetimemeasurements in semiconductors by infrared absorp-tion due to pulsed optical excitationrdquo Applied Physics Lettersvol 51 no 18 pp 1445ndash1447 1987

[2] M Bail J Kentsch R Brendel and M Schulz ldquoLifetimemapping of Si wafers by an infrared camera [for solar cell-production]rdquo in Proceedings of the 28th Photovoltaic SpecialistsConference pp 99ndash103 IEEE Anchorage Alaska USA 2000

[3] S Riepe J Isenberg C Ballif S Glunz and W Warta ldquoCarrierdensity and lifetime imaging of silicon wafers by infraredlock-in thermographyrdquo in Proceedings of the 17th EuropeanPhotovoltaic Solar Energy Conference pp 1597ndash1599 2001

[4] N Ashcroft and N Mermin Solid State Physics CengageLearning Singapore 1976

[5] C Kittel and P McEuen Introduction to Solid State PhysicsWiley New York NY USA 1996

[6] JNelsonThePhysics of Solar Cells Imperial College Press 2003[7] V K Khanna ldquoPhysical understanding and technological con-

trol of carrier lifetimes in semiconductor materials and devicesa critique of conceptual development state of the art andapplicationsrdquo Progress in Quantum Electronics vol 29 no 2 pp59ndash163 2005

[8] P Landsberg Recombination in Semiconductors CambridgeUniversity Press 1991

[9] R N Hall ldquoElectron-hole recombination in germaniumrdquo Phys-ical Review vol 87 no 2 p 387 1952

[10] W Shockley andW T Read ldquoStatistics of the recombinations ofholes and electronsrdquo Physical Review vol 87 no 5 pp 835ndash8421952

[11] V K Khanna ldquoCarrier lifetimes and recombination-generationmechanisms in semiconductor device physicsrdquo European Jour-nal of Physics vol 25 no 2 pp 221ndash237 2004

[12] L Pincherle ldquoAuger effect in semiconductorsrdquoProceedings of thePhysical Society B vol 68 no 5 article 108 pp 319ndash320 1955

[13] T S Moss ldquoPhotoelectromagnetic and photoconductive effectsin lead sulphide single crystalsrdquo Proceedings of the PhysicalSociety B vol 66 no 12 article 301 pp 993ndash1002 1953

[14] J A Hornbeck and J R Haynes ldquoTrapping of minority carriersin silicon I P-type siliconrdquo Physical Review vol 97 no 2 pp311ndash321 1955

[15] P Auger ldquoThe compound photoelectric effectrdquo Journal dePhysique et le Radium vol 6 no 6 article 205 1925

[16] P T Landsberg and T S Moss ldquoRecombination theory forindium antimoniderdquo Proceedings of the Physical Society B vol69 no 6 article 310 pp 661ndash669 1956

[17] P T Landsberg ldquoLifetimes of excess carriers in InSbrdquo Proceed-ings of the Physical Society B vol 70 no 12 article 109 pp 1175ndash1176 1957

[18] A Beattie and P Landsberg ldquoAuger effect in semiconductorsrdquoProceedings of the Royal Society of London A Mathematical andPhysical Sciences vol 249 no 1256 pp 16ndash29 1959

[19] A B Grebene ldquoComments on auger recombination in semi-conductorsrdquo Journal of Applied Physics vol 39 no 10 pp 4866ndash4868 1968

[20] P Lal C Rhys-Roberts and P LandsbergAuger RecombinationInto Traps Ft Belvoir Defense Technical Information Center1964

[21] P T Landsberg ldquoTrap-Auger recombination in silicon of lowcarrier densitiesrdquoApplied Physics Letters vol 50 no 12 pp 745ndash747 1987

[22] J Dziewior andW Schmid ldquoAuger coefficients for highly dopedand highly excited siliconrdquo Applied Physics Letters vol 31 no 5pp 346ndash348 1977

[23] D J Fitzgerald and A S Grove ldquoSurface recombination insemiconductorsrdquo Surface Science vol 9 no 2 pp 347ndash369 1968

[24] P T Landsberg ldquoSome general recombination statisticsfor semiconductor surfacesrdquo IEEE Transactions on ElectronDevices vol ED-29 no 8 pp 1284ndash1286 1982

[25] D E Aspnes ldquoRecombination at semiconductor surfaces andinterfacesrdquo Surface Science vol 132 no 1ndash3 pp 406ndash421 1983

[26] G J Rees ldquoSurface recombination velocity a useful conceptrdquoSolid State Electronics vol 28 no 5 pp 517ndash519 1985

[27] D K Schroder ldquoCarrier lifetimes in siliconrdquo IEEE Transactionson Electron Devices vol 44 no 1 pp 160ndash170 1997

[28] H B Briggs ldquoInfra-red absorption in siliconrdquo Physical Reviewvol 77 no 5 pp 727ndash728 1950

[29] H Fan and M Becker ldquoInfra-red optical properties of siliconand germaniumrdquo in Semi-Conducting Materials Proceedingsof a Conference Held at the University of Reading under theAuspices of the International Union of Pure and Applied Physicsin cooperation with the Royal Society p 132 ButterworthsScientific Publications 1951

[30] H B Briggs and R C Fletcher ldquoNew infrared absorption bandsin p-type germaniumrdquo Physical Review vol 87 no 6 pp 1130ndash1131 1952

[31] H B Briggs and R C Fletcher ldquoAbsorption of infrared light byfree carriers in germaniumrdquo Physical Review vol 91 no 6 pp1342ndash1346 1953

[32] A H Kahn ldquoTheory of the infrared absorption of carriers ingermaniumand siliconrdquoPhysical Review vol 97 no 6 pp 1647ndash1652 1955

[33] H Y Fan W Spitzer and R J Collins ldquoInfrared absorption inn-type germaniumrdquoPhysical Review vol 101 no 2 pp 566ndash5721956

[34] W Spitzer andH Y Fan ldquoInfrared absorption in n-type siliconrdquoPhysical Review vol 108 no 2 pp 268ndash271 1957

[35] S Visvanathan ldquoFree carrier absorption due to polar modes inthe III-V compound semiconductorsrdquo Physical Review vol 120no 2 pp 376ndash378 1960

[36] W P Dumke ldquoQuantum theory of free carrier absorptionrdquoPhysical Review vol 124 no 6 pp 1813ndash1817 1961

[37] H Hara and Y Nishi ldquoFree carrier absorption in p-type siliconrdquoJournal of the Physical Society of Japan vol 21 no 6 p 12221966

[38] W Runyan Technology Semiconductor Silicon 1966[39] D Schroder R Thomas and J Swartz ldquoFree carrier absorption

in siliconrdquo IEEE Transactions on Electron Devices vol 25 pp254ndash261 1978

[40] J I Pankove Optical Processes in Semi-Conductors DoverPublications 1971

[41] P BasuTheory of Optical Processes in Semiconductors Bulk andMicrostructures Oxford University Press New York NY USA1997

[42] B Nag Electron Transport in Compound SemiconductorsSpringer 1980

[43] G Chiarotti and U M Grassano ldquoThe excited states of the Fcenter investigated by means of modulated optical absorptionrdquoIl Nuovo Cimento B Series 10 vol 46 no 1 pp 78ndash92 1966

[44] G Chiarotti and U M Grassano ldquoModulated F-center absorp-tion in KClrdquo Physical Review Letters vol 16 no 4 pp 124ndash1271966


[45] W B Gauster and J C Bushnell ldquoLaser-induced infraredabsorption in siliconrdquo Journal of Applied Physics vol 41 no 9pp 3850ndash3853 1970

[46] E J Conway ldquoLight-inducedmodulation of broad-band opticalabsorption in CdSrdquo Journal of Applied Physics vol 41 no 4 pp1689ndash1693 1970

[47] F Sanii R J Schwartz R F Pierret and W M Au ldquoMeasure-ment of bulk and surface recombination bymeans ofmodulatedfree carrier absorptionrdquo in Proceedings of the 20th IEEE Photo-voltaic Specialists Conference pp 575ndash580 September 1988

[48] F PGiles F Sanii R J Schwartz and J L Gray ldquoNondestructivecontactless measurement of bulk lifetime and surface recom-bination using single pass infrared free carrier absorptionrdquo inProceedings of the 22nd IEEE Photovoltaic Specialists Conferencepp 223ndash228 October 1991

[49] SWGlunzA B SproulWWarta andWWettling ldquoInjection-level-dependent recombination velocities at the Si-SiO2 inter-face for various dopant concentrationsrdquo Journal of AppliedPhysics vol 75 no 3 pp 1611ndash1615 1994

[50] S W Glunz and W Warta ldquoHigh-resolution lifetime mappingusing modulated free-carrier absorptionrdquo Journal of AppliedPhysics vol 77 no 7 pp 3243ndash3247 1995

[51] Z G Ling and P K Ajmera ldquoMeasurement of bulk lifetime andsurface recombination velocity by infrared absorption due topulsed optical excitationrdquo Journal of Applied Physics vol 69 no1 pp 519ndash521 1991

[52] G S Kousik Z G Ling and P K Ajmera ldquoNondestructivetechnique to measure bulk lifetime and surface recombinationvelocities at the two surfaces by infrared absorption due topulsed optical excitationrdquo Journal of Applied Physics vol 72 no1 pp 141ndash146 1992

[53] T Otaredian ldquoSeparate contactless measurement of the bulklifetime and the surface recombination velocity by the harmonicoptical generation of the excess carriersrdquo Solid-State Electronicsvol 36 no 2 pp 153ndash162 1993

[54] Z G Ling P K Ajmera and G S Kousik ldquoSimultaneousextraction of bulk lifetime and surface recombination velocitiesfrom free carrier absorption transientsrdquo Journal of AppliedPhysics vol 75 no 5 pp 2718ndash2720 1994

[55] K L Luke and L-J Cheng ldquoAnalysis of the interaction of a laserpulse with a silicon wafer determination of bulk lifetime andsurface recombination velocityrdquo Journal of Applied Physics vol61 no 6 pp 2282ndash2293 1987

[56] J Linnros ldquoCarrier lifetime measurements using free carrierabsorption transients I principle and injection dependencerdquoJournal of Applied Physics vol 84 no 1 pp 275ndash283 1998

[57] J Linnros ldquoCarrier lifetime measurements using free carrierabsorption transients II Lifetime mapping and effects ofsurface recombinationrdquo Journal of Applied Physics vol 84 no1 pp 284ndash291 1998

[58] R Bernini A Cutolo A Irace P Spirito and L Zeni ldquoCon-tactless characterization of the recombination process in siliconwafers separation between bulk and surface contributionrdquoSolid-State Electronics vol 39 no 8 pp 1165ndash1172 1996

[59] A Irace L Sirleto G F Vitale et al ldquoTransverse probe opticallifetimemeasurement as a tool for in-line characterization of thefabrication process of a silicon solar cellrdquo Solid-State Electronicsvol 43 no 12 pp 2235ndash2242 1999

[60] M C Schubert S Riepe S Bermejo and W Warta ldquoDetermi-nation of spatially resolved trapping parameters in silicon withinjection dependent carrier density imagingrdquo Journal of AppliedPhysics vol 99 no 11 Article ID 114908 2006

[61] K Ramspeck S Reissenweber J Schmidt K Bothe and RBrendel ldquoDynamic carrier lifetime imaging of silicon wafersusing an infrared-camera-based approachrdquo Applied PhysicsLetters vol 93 no 10 Article ID 102104 2008

[62] A Buczkowski Z J Radzimski G A Rozgonyi and F ShimuraldquoBulk and surface components of recombination lifetime basedon a two-laser microwave reflection techniquerdquo Journal ofApplied Physics vol 69 no 9 pp 6495ndash6499 1991

[63] L Sirleto A Irace G F Vitale L Zeni and A CutololdquoSeparation of bulk lifetime and surface recombination velocityby multiwavelength techniquerdquo Electronics Letters vol 38 no25 pp 1742ndash1743 2002

[64] L Sirleto A Irace G F Vitale L Zeni and A CutololdquoSeparation of bulk lifetime and surface recombination velocityobtained by transverse optical probing and multi-wavelengthtechniquerdquo Optics and Lasers in Engineering vol 38 no 6 pp461ndash472 2002

[65] L Sirleto A Irace G Vitale L Zeni and A Cutolo ldquoAll-opticalmultiwavelength technique for the simultaneous measurementof bulk recombination lifetimes and frontrear surface recom-bination velocity in single crystal silicon samplesrdquo Journal ofApplied Physics vol 93 no 6 pp 3407ndash3413 2003

[66] A Irace L Sirleto P Spirito et al ldquoOptical characterizationof the recombination process in silicon wafers epilayers anddevicesrdquoOptics and Lasers in Engineering vol 39 no 2 pp 219ndash232 2003

[67] A Irace F Sorrentino and G F Vitale ldquoMulti-wavelengthtransverse probe lifetime measurement for the characterizationof recombination lifetime in thin mc-Si samples for photo-voltaic industry userdquo Solar EnergyMaterials and Solar Cells vol84 no 1ndash4 pp 83ndash92 2004

[68] A Irace F Sorrentino and G F Vitale ldquoMulti-wavelength alloptical measurement for the characterization of recombinationprocess in thin mc-Si samplesrdquo Solar Energy vol 78 no 2 pp251ndash256 2005

[69] E Gaubas and J Vanhellemont ldquoA simple technique for theseparation of bulk and surface recombination parameters insiliconrdquo Journal of Applied Physics vol 80 no 11 pp 6293ndash62971996

[70] EGaubas J Vaitkus E Simoen C Claeys and J VanhellemontldquoExcess carrier cross-sectional profiling technique for determi-nation of the surface recombination velocityrdquoMaterials Sciencein Semiconductor Processing vol 4 no 1ndash3 pp 125ndash131 2001

[71] M De Laurentis A Irace and G Breglio ldquoAccurate modellingof the pump-probe spatial interaction in an all-optical recom-bination lifetime measurement setuprdquo in Proceedings of the 1stMediterranean Photonics Conference p 40 Ischia Italy 2008

[72] MDe Laurentis A Irace andG Breglio ldquoDetermination of thesurface electrical activity in silicon wafers with a laser pump-probe measurementrdquo in Proceedings of the 6th InternationalConference on Photo-Excited Processes and Applications p 1212008

[73] AGAberle J Schmidt andR Brendel ldquoOn the data analysis oflight-biased photoconductance decay measurementsrdquo Journalof Applied Physics vol 79 no 3 pp 1491ndash1496 1996

[74] J Schmidt and A G Aberle ldquoAccurate method for the deter-mination of bulk minority-carrier lifetimes of mono- andmulticrystalline silicon wafersrdquo Journal of Applied Physics vol81 no 9 pp 6186ndash6199 1997

[75] F M Schuurmans A Schonecker A R Burgers and W CSinke ldquoSimplified evaluation method for light-biased effective


lifetime measurementsrdquo Applied Physics Letters vol 71 no 13pp 1795ndash1797 1997

[76] J Schmidt ldquoMeasurement of differential and actual recombi-nation parameters on crystalline silicon wafersrdquo IEEE Transac-tions on Electron Devices vol 46 no 10 pp 2018ndash2025 1999

[77] O Breitenstein and M Langenkamp Lock-in ThermographyBasics and Use for Functional Diagnostics of Electronic Compo-nents Springer 2003

[78] WWarta ldquoAdvanced defect and impurity diagnostics in siliconbased on carrier lifetime measurementsrdquo Physica Status Solidi(A) Applications and Materials Science vol 203 no 4 pp 732ndash746 2006

[79] M C Schubert J Isenberg and W Warta ldquoSpatially resolvedlifetime imaging of silicon wafers by measurement of infraredemissionrdquo Journal of Applied Physics vol 94 no 6 pp 4139ndash4143 2003

[80] J Isenberg S Riepe SWGlunz andWWarta ldquoCarrier densityimaging (CDI) a spatially resolved lifetime measurement suit-able for in-line process-controlrdquo in Proceedings of the 29th IEEEPhotovoltaic Specialists Conference pp 266ndash269 May 2002

[81] J Isenberg S Riepe S W Glunz and W Warta ldquoImagingmethod for laterally resolved measurement of minority carrierdensities and lifetimes measurement principle and first appli-cationsrdquo Journal of Applied Physics vol 93 no 7 pp 4268ndash42752003

[82] P Pohl and R Brendel ldquoTemperature dependent infrared cam-era lifetimemapping (ILM)rdquo in Proceedings of the 19th EuropeanPhotovoltaic Solar Energy Conference pp 46ndash49 Paris France2004

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in terms of energy to the energy lower than the materialband-gap In fact the free carriers concentration (and itsvariation) inside the material determines a variation of thematerial optical absorption coefficient This implies that avariation of transmittance can be detected and observedwhen a sub-band-gap radiation is transmitted by thematerialThe generation of the excess free carriers can be opticallyinduced by illuminating the sample with a laser pulse whosephotons energy is higher than the sample band gap

The PP technique belongs to the class of transient meth-ods These methods are attractive because they give a resultdirectly related to the velocity of the recombination processIn fact the signal detected is the time dependent signaltransmitted by the sample The PP allows for characterizingmaterials with very low lifetime too and usually it doesnot require any calibration procedure Moreover the actualadvantage of themethod is the possibility to separate the bulkfrom surface effect which means to simultaneously measurethe bulk lifetime and the surface recombination velocity

The ILMCDI belongs to the steady-state methods classThese methods measure a physical quantity related to thecarrier density It allows for a fast and high spatially resolvedmeasurement of the lifetime in each point of the waferwithout performing a scanning of it since an infrared CCDcamera is used to detect the transmitted infrared radiationIts greater attractive feature is the capability to perform thelifetime mapping in few minutes which could provide incalibrated condition an online monitoring feature

The paper is organized as follows in the first two sec-tions we briefly recall the concept of recombination lifetimetogether with the main recombination mechanisms in semi-conductor materials in the subsequent sections we describein detail of the selected techniques having previously recalledthe analytical tools necessary to their understanding

2 The Concept of Recombination Lifetime

The term ldquolifetimerdquo is used in physics in a totally intuitivemanner to indicate the temporal interval between the gen-eration and the death of a particle Nevertheless often theldquodeathrdquo is only a change of the particle state like in the caseof the recombination process in the semiconductors theoryIn fact the recombination lifetime refers to the time in whichthe electron returns from an excited state to the equilibriumstate The excitation can be obtained for example by aphoton absorption or an electrical injection process in theforward-biased p-n junction The electron comes back inthe equilibrium state occupying the vacancy left by theexcitation In the semiconductors theory this vacancy isdescribed like a particle named hole with the same electroncharge but differentmass [4 5] In this way the electron decayto the equilibrium state is described as its recombinationwith the holeThis recombination process destroys the chargeconstituted by the pair and the recombination lifetime thenrefers to the electron-hole pair what ceases to exist is thischarged pair and it ceases to exist because the electron of thepair recombines with the hole of the pair In other wordsthe recombination process is the destruction (or decay)

of the excess carriers (the electron-hole pairs) generated inthe semiconductors as consequence of an excitation Thetemporal interval during which the excess carriers decay isdefined as recombination lifetime 120591rec

Mathematically in a first order approximation 120591rec isdefined as the ratio

120591rec =Δ119899

R(1)

between the excess carriers density Δ119899 [cmminus3] and the recom-bination rate R = minus119889(Δ119899)119889119905 [sminus1] of the excess carriersitself

From (1) named Δ119899(0) the excess carriers concentrationat the instant 119905 = 0when the excitation source is turned off itis derived that 120591 characterizes the excess carriers decay

Δ119899 (119905) = Δ119899 (0) exp(minus 1

120591rec) (2)

In steady-state conditions the excitation source is at alltimes turned on constant or with assigned time dependenceand an equilibrium condition between the generation rateand recombination rate is reached G = R so that it ispossible to define the recombination lifetime as

120591rec =Δ119899

G (3)

In general it must be clearly remarked that the recombi-nation phenomenon can be comprehensively treated only bymeans of quantum mechanical calculations Since it involvestransitions between energetic states it is implicit that therecombination rate depends on occupation probability of theenergy levels and thus on the energy as well as on thetemperature on the electrons and holes concentration andit is in general a nonlinear function of Δ119899 Consequently therecombination lifetime depends on them and thus it must beconsidered that in general it is 120591rec = 120591rec(119864 119879 119899 119901 Δ119899 Δ119901)(where 119899 and 119901 are the concentration of the electrons andholes resp) that means among other things that 120591rec is afunction of the injection level Δ119899119899 (Δ119901119901)

21 Recombination Process The total recombination rate isa superposition of several recombination mechanisms Assummarized in (4) among these the main are the Shockley-Read-Hall the radiative and the Auger recombination (thefirst one linearly dependent on the excess carrier generationonly at lower injection level the second one and the thirdresp proportional to the square and to the cube of the excessfree carriers concentration)

R = RSRH +RRad +RAug (4)

The sketch in Figure 1 gives an intuitive 119864 minus 119896 pictureof these processes while in the following we give a shortdescription of themWe invite the readers to refer to [6 7] andespecially to [8] for all the analytical and theoretical details


k

Phonon

Conduction bandE

Photon

Valence band

Ec

Eg

E

Et

⟨111⟩ ⟨100⟩

(a)

k

E

Photon

Conduction band

Valence band

Ec

E

⟨111⟩ ⟨100⟩

(b)

k

EC band

Valence band

Ec

E

⟨111⟩ ⟨100⟩

(c)














1

120591rec=

1

120591SRH+

1

120591rad+

1

120591Auger (5)





1

120591eff=

1

120591119887

+

1

120591119878

(6)


SRV = 119863[

120597119899120597119909

119899(119905)

]

boundary (7)













120572fc =11989911989021205822

119898119890120598012057841205871198883

1

120591

(8)





120572119904 = 119860 119904120582119901119904

(9)





120572119899 ≃ 1 times 10minus181205822119899

120572119901 ≃ 27 times 10minus181205822119901

(10)














Conduction band

Valence band

Photon

Eph = h

Ec

Eg

E

(a) 119864ph gt 119864119892

Conduction band

Valence band

Photon

Eph = h

Ec

Eg

E

Ei

(b) 119864119894 lt 119864ph lt 119864119892

Conduction band

Valence band

phonon

Eph = h Ec

Eg

E

(c) 119864ph lt 119864119892


Oscilloscope

NdYAG pulsed laser

Lens

InGaAsphotodiode

Trigger signal



Lens

Sample

Probe beam

Pump pulse

y

x

120572

120582 gt 120582gap

120582p lt 120582gap



119868 (119905) = 1198680119890minus1205720(120582)119871

(1 minus 119890minus120572fc(120582119905)119871

) (11)






Sample

Oscilloscope

Pulsed laser


xyz stage

Lens

Trigger signal

Pump pulse

Probe beam

InGaAsphotodiode

x

120582 gt 120582gap

120582p lt 120582gap


5

0

minus5

minus10

minus15

minus200 05 1 15 2 25 3 35

VminusV0

(mV

)

t (120583s)


Pump pulse

Probe beam

0

y

z

x

S1 S2

d2minusd2



120597119899 (119909 119905)

120597119905

= 119863

1205972119899 (119909 119905)

1205971199092

minus

119899 (119909 119905)

120591119887

(12)


119863[

120597119899(119909 119905)

120597119905

]

119909=minus1198892

= 1198781119899(minus

119889

2

119905) (13)

minus119863[

120597119899(119909 119905)

120597119905

]

119909=1198892

= 1198782119899(

119889

2

119905) (14)


119899 (119909 0) = 1198730 (1 minus 119877) 120572

119890minus120572(119909+1198892)

+ 119877119890minus120572119889

119890120572(minus119909+1198892)

(1 minus 1198772119890minus2120572119889

)

(15)




119899 (119909 119905) =

infin

sum

119896=1

119890minus(1120591

119887+1198862

119896119863)119905

(119860119896 cos 119886119896119909 + 119861119896 sin 119886119896119909) (16)


119886119896119889 = arctan 1198781

119886119896119863

+ arctan 1198782

119886119896119863

+ 119896120587 (17)

while if

119911119896 = 119886119896119889

1198920 =

(1 minus 119877) 120572

1 minus 119877119890minus2120572119889

1198611015840

119896=

(41198861198961198920120572) 119890minus1205721198892

[1198872

119896(119886119896119889 + sin 119886119896119889) + (119886119896119889 minus sin 119886119896119889)] (1205722 + 1198862119896)

(18)

it results in

119887119896 equiv

119860119896

119861119896

= minus


119861119896 = 1198611015840

119896[119887119896 (cos 119911119896 sinh

120572119889

2

+

119886119896

120572

sin 119911119896 cosh120572119889

2

)

times (1 + 119877119890minus120572119889

) ]

+ 1198611015840

119896[(

119886119896

120572

cos 119911119896 sinh120572119889

2

minus sin 119911119896 cosh120572119889

2

)


) ]

(19)


119899av (119905) =1

119881intint

119881int

119899 (119909 119905) 119889119881 (20)




119899av (119905) prop int

1198892

minus1198892

119899 (119909 119905) 119889119909

= sum

119896

119861119896119887119896119868119896 (minus

119889

2

119889

2

) 119906119896 (119905)

= sum

119896

119861119896119887119896

sin 119911119896119911119896

119906119896 (119905)

(21)


119906119896 (119905) = 119890minus(1120591

119887+1198862

119896119863)119905

120601119896 (119909) = 119887119896 cos 119886119896119909 + sin 119886119896119909

119868 (119909119894 119909119891) = int

119909119891

119909119894

120601 (119909) 119889119909

(22)


119866119892 (119905) =

1198730

radic2120587120590

119890minus(119905minus119879

0)21205902

(23)


119899gav = int119905

0

119899av (120591) 119866119892 (119905 minus 120591) 119889120591

= sum

119896

119861119896119887119896119868119896 (minus

119889

2

119889

2

)119864119896 (119905)

(24)

where

119864119896 (119905) = int

119905

0

119906119896 (120591) 119866119892 (119905 minus 120591) 119889120591

= 119906119896 (119905) 119890119865119896((1205902

2)119865119896+1198790)

times [erf ( 120590

radic2

119865119896 +

1198790

radic2120590


radic2

119865119896 minus

119905 minus 1198790

radic2120590

)]

(25)

with

119865119896 =

1

120591119887

+ 1198862

119896119863 (26)


1

1205910

= minus

119889

119889119905

ln [119899gav (119905)] =1

120591119887

+ 119877119904 (119905) (27)




119877119904 (119905)

119905rarrinfin

997888rarr 119877119904119886 = 1198861119863(28)


1

1205910119886

=

1

120591119887

+ 119877119904119886 =

1

120591119887

+

1

1205911199041

(29)






120591119887 =

12059101198861205911199041

1205911199041 minus 1205910119886

(30)




1

1205910119886

=

1

120591119887

+ 119877119904119886

1

1205910119898

=

1

120591119887

+ 119877119898

(31)

so that

(119877119904119898 minus 119877119904119886)anal = (1

1205910119898

minus

1

1205910119886

)

exper(32)


120591119887 =


(33)




(

1

1205910119886

)

1

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(

1

1205910119886

)

2

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(34)



1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

10minus3

10minus4

10minus5

10minus6

120591b

(s)

S (cms)

120591eff = 5 120583s

120591eff = 6 120583s

120591eff = 15 120583s

120591eff = 12 120583s

120591eff 2 = 115 120583s120591eff 1 = 55 120583sd1 = 500 120583m

d2 = 200 120583m





1

1205910119898

=

1

120591119887

+ 119877119898 for (1198781 1198782) or (1198782 1198781)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198781 1198782)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198782 1198781)

(35)





Defining

120575119886 = lim119905rarrinfin

120575 (Δ120582 119905) = ln(119873rel (1205821 119905)

119873rel (1205822 119905)) (36)



















Δ119899 (119909 119905) = 119899119887 (119909) + 119899119901 (119909 119905) (38)


120591 (119899119887) =

1

119899119887

int

119899119887

0

120591diff119889Δ119899 (39)



120591 (119899119887) =

1

119875

int

119875

0

120591diff (119875) 119889119875 (40)







119870 = 119878off minus 119878on (41)


120591eff =Δ119899119882

119866

(42)


119866 = Φ(1 minus 119877119891) (43)



Optical fiber

Lens

Diode laser

Pump beam


Wafer

Black body

CCD camera

Tb

Tw

120582 lt 120582gap

120582 gt 120582gap



119878 = Γint

120582max

120582min

(1 minus 119877)2119890minus120572fc(120582)119882

119868 (120582 119879119887) 119889120582

asymp Γint

120582max

120582min

(1 minus 119877)2120572fc (120582)119882119868 (120582 119879119887) 119889120582

(44)


119868 (120582det 119879119887) =2120587ℎ119888

2

1205825

1

119890(ℎ119888120582119896

119861119879119887)minus1

asymp

120574

1205825119890minusΨ120582119879

119887 (45)






200

180

160

140

120

100

80

60

40

20

0

100

80

60

40

20

0

100806040200

120591ap

p(120583

s)

x (mm)

y(m

m)



119878em = Γint

120582max

120582min

120576fc (120582)120574

1205825119890minusΨ120582119879

119908

119889120582 (46)


119878 = 119878em minus 119878119886119887 (47)




119894119882119894

(119909 119910) minus 11987811987311198821


119870119862 (119909 119910) = 119898Δ (119873119882) (119909 119910) (48)


Δ (119899119882) (119909 119910) =

119870 (119909 119910)

119898120590

120581 (119909 119910) (49)


120590 = (1 +

120572119899

120572119901

) (50)





asymp Γ

120574

1205825

det(119890minusΨ(119879

119887+NETD)


119887

)

asymp Γ

120574

1205825

det

ΨNEDT1198792

119887

119890minusΨ119879

119887

(51)






119878 asymp Γ120573Δ119899119882

120574

1205823

119889119890119905

119890minusΨ119879

119887

(52)


NEL = ℎ119888

1198961198611198792

1198871205822

det

1

120573Φ (1 minus 119877119891)

NETDradic119873

(53)














119878cos = int119905int

0

Δ119899 (119905) 119889119905 minus int

1198792+int

1198792

Δ119899 (119905) 119889119905


)]

119878sin = int1198794+119905int

1198794

Δ119899 (119905) 119889119905 minus int

31198794+int

31198794

Δ119899 (119905) 119889119905

= 119878st-st



119879 + 4119905int4120591eff

)]

(54)


10

08

06

04

02

000 T

Time t (120583s)





Nor

mal

ized

carr

ier d

ensit

y Δn(t) a

nd g

ener

atio

n ra

te 119970

(a)

tint = 300 120583s

tint = 1100 120583s

tint = 2000 120583s

120

105

90

75

60

450 1000


ase Φ

(∘)

(b)




10 Conclusion








References



























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




k

Phonon

Conduction bandE

Photon

Valence band

Ec

Eg

E

Et

⟨111⟩ ⟨100⟩

(a)

k

E

Photon

Conduction band

Valence band

Ec

E

⟨111⟩ ⟨100⟩

(b)

k

EC band

Valence band

Ec

E

⟨111⟩ ⟨100⟩

(c)














1

120591rec=

1

120591SRH+

1

120591rad+

1

120591Auger (5)





1

120591eff=

1

120591119887

+

1

120591119878

(6)


SRV = 119863[

120597119899120597119909

119899(119905)

]

boundary (7)













120572fc =11989911989021205822

119898119890120598012057841205871198883

1

120591

(8)





120572119904 = 119860 119904120582119901119904

(9)





120572119899 ≃ 1 times 10minus181205822119899

120572119901 ≃ 27 times 10minus181205822119901

(10)














Conduction band

Valence band

Photon

Eph = h

Ec

Eg

E

(a) 119864ph gt 119864119892

Conduction band

Valence band

Photon

Eph = h

Ec

Eg

E

Ei

(b) 119864119894 lt 119864ph lt 119864119892

Conduction band

Valence band

phonon

Eph = h Ec

Eg

E

(c) 119864ph lt 119864119892


Oscilloscope

NdYAG pulsed laser

Lens

InGaAsphotodiode

Trigger signal



Lens

Sample

Probe beam

Pump pulse

y

x

120572

120582 gt 120582gap

120582p lt 120582gap



119868 (119905) = 1198680119890minus1205720(120582)119871

(1 minus 119890minus120572fc(120582119905)119871

) (11)






Sample

Oscilloscope

Pulsed laser


xyz stage

Lens

Trigger signal

Pump pulse

Probe beam

InGaAsphotodiode

x

120582 gt 120582gap

120582p lt 120582gap


5

0

minus5

minus10

minus15

minus200 05 1 15 2 25 3 35

VminusV0

(mV

)

t (120583s)


Pump pulse

Probe beam

0

y

z

x

S1 S2

d2minusd2



120597119899 (119909 119905)

120597119905

= 119863

1205972119899 (119909 119905)

1205971199092

minus

119899 (119909 119905)

120591119887

(12)


119863[

120597119899(119909 119905)

120597119905

]

119909=minus1198892

= 1198781119899(minus

119889

2

119905) (13)

minus119863[

120597119899(119909 119905)

120597119905

]

119909=1198892

= 1198782119899(

119889

2

119905) (14)


119899 (119909 0) = 1198730 (1 minus 119877) 120572

119890minus120572(119909+1198892)

+ 119877119890minus120572119889

119890120572(minus119909+1198892)

(1 minus 1198772119890minus2120572119889

)

(15)




119899 (119909 119905) =

infin

sum

119896=1

119890minus(1120591

119887+1198862

119896119863)119905

(119860119896 cos 119886119896119909 + 119861119896 sin 119886119896119909) (16)


119886119896119889 = arctan 1198781

119886119896119863

+ arctan 1198782

119886119896119863

+ 119896120587 (17)

while if

119911119896 = 119886119896119889

1198920 =

(1 minus 119877) 120572

1 minus 119877119890minus2120572119889

1198611015840

119896=

(41198861198961198920120572) 119890minus1205721198892

[1198872

119896(119886119896119889 + sin 119886119896119889) + (119886119896119889 minus sin 119886119896119889)] (1205722 + 1198862119896)

(18)

it results in

119887119896 equiv

119860119896

119861119896

= minus


119861119896 = 1198611015840

119896[119887119896 (cos 119911119896 sinh

120572119889

2

+

119886119896

120572

sin 119911119896 cosh120572119889

2

)

times (1 + 119877119890minus120572119889

) ]

+ 1198611015840

119896[(

119886119896

120572

cos 119911119896 sinh120572119889

2

minus sin 119911119896 cosh120572119889

2

)


) ]

(19)


119899av (119905) =1

119881intint

119881int

119899 (119909 119905) 119889119881 (20)




119899av (119905) prop int

1198892

minus1198892

119899 (119909 119905) 119889119909

= sum

119896

119861119896119887119896119868119896 (minus

119889

2

119889

2

) 119906119896 (119905)

= sum

119896

119861119896119887119896

sin 119911119896119911119896

119906119896 (119905)

(21)


119906119896 (119905) = 119890minus(1120591

119887+1198862

119896119863)119905

120601119896 (119909) = 119887119896 cos 119886119896119909 + sin 119886119896119909

119868 (119909119894 119909119891) = int

119909119891

119909119894

120601 (119909) 119889119909

(22)


119866119892 (119905) =

1198730

radic2120587120590

119890minus(119905minus119879

0)21205902

(23)


119899gav = int119905

0

119899av (120591) 119866119892 (119905 minus 120591) 119889120591

= sum

119896

119861119896119887119896119868119896 (minus

119889

2

119889

2

)119864119896 (119905)

(24)

where

119864119896 (119905) = int

119905

0

119906119896 (120591) 119866119892 (119905 minus 120591) 119889120591

= 119906119896 (119905) 119890119865119896((1205902

2)119865119896+1198790)

times [erf ( 120590

radic2

119865119896 +

1198790

radic2120590


radic2

119865119896 minus

119905 minus 1198790

radic2120590

)]

(25)

with

119865119896 =

1

120591119887

+ 1198862

119896119863 (26)


1

1205910

= minus

119889

119889119905

ln [119899gav (119905)] =1

120591119887

+ 119877119904 (119905) (27)




119877119904 (119905)

119905rarrinfin

997888rarr 119877119904119886 = 1198861119863(28)


1

1205910119886

=

1

120591119887

+ 119877119904119886 =

1

120591119887

+

1

1205911199041

(29)






120591119887 =

12059101198861205911199041

1205911199041 minus 1205910119886

(30)




1

1205910119886

=

1

120591119887

+ 119877119904119886

1

1205910119898

=

1

120591119887

+ 119877119898

(31)

so that

(119877119904119898 minus 119877119904119886)anal = (1

1205910119898

minus

1

1205910119886

)

exper(32)


120591119887 =


(33)




(

1

1205910119886

)

1

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(

1

1205910119886

)

2

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(34)



1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

10minus3

10minus4

10minus5

10minus6

120591b

(s)

S (cms)

120591eff = 5 120583s

120591eff = 6 120583s

120591eff = 15 120583s

120591eff = 12 120583s

120591eff 2 = 115 120583s120591eff 1 = 55 120583sd1 = 500 120583m

d2 = 200 120583m





1

1205910119898

=

1

120591119887

+ 119877119898 for (1198781 1198782) or (1198782 1198781)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198781 1198782)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198782 1198781)

(35)





Defining

120575119886 = lim119905rarrinfin

120575 (Δ120582 119905) = ln(119873rel (1205821 119905)

119873rel (1205822 119905)) (36)



















Δ119899 (119909 119905) = 119899119887 (119909) + 119899119901 (119909 119905) (38)


120591 (119899119887) =

1

119899119887

int

119899119887

0

120591diff119889Δ119899 (39)



120591 (119899119887) =

1

119875

int

119875

0

120591diff (119875) 119889119875 (40)







119870 = 119878off minus 119878on (41)


120591eff =Δ119899119882

119866

(42)


119866 = Φ(1 minus 119877119891) (43)



Optical fiber

Lens

Diode laser

Pump beam


Wafer

Black body

CCD camera

Tb

Tw

120582 lt 120582gap

120582 gt 120582gap



119878 = Γint

120582max

120582min

(1 minus 119877)2119890minus120572fc(120582)119882

119868 (120582 119879119887) 119889120582

asymp Γint

120582max

120582min

(1 minus 119877)2120572fc (120582)119882119868 (120582 119879119887) 119889120582

(44)


119868 (120582det 119879119887) =2120587ℎ119888

2

1205825

1

119890(ℎ119888120582119896

119861119879119887)minus1

asymp

120574

1205825119890minusΨ120582119879

119887 (45)






200

180

160

140

120

100

80

60

40

20

0

100

80

60

40

20

0

100806040200

120591ap

p(120583

s)

x (mm)

y(m

m)



119878em = Γint

120582max

120582min

120576fc (120582)120574

1205825119890minusΨ120582119879

119908

119889120582 (46)


119878 = 119878em minus 119878119886119887 (47)




119894119882119894

(119909 119910) minus 11987811987311198821


119870119862 (119909 119910) = 119898Δ (119873119882) (119909 119910) (48)


Δ (119899119882) (119909 119910) =

119870 (119909 119910)

119898120590

120581 (119909 119910) (49)


120590 = (1 +

120572119899

120572119901

) (50)





asymp Γ

120574

1205825

det(119890minusΨ(119879

119887+NETD)


119887

)

asymp Γ

120574

1205825

det

ΨNEDT1198792

119887

119890minusΨ119879

119887

(51)






119878 asymp Γ120573Δ119899119882

120574

1205823

119889119890119905

119890minusΨ119879

119887

(52)


NEL = ℎ119888

1198961198611198792

1198871205822

det

1

120573Φ (1 minus 119877119891)

NETDradic119873

(53)














119878cos = int119905int

0

Δ119899 (119905) 119889119905 minus int

1198792+int

1198792

Δ119899 (119905) 119889119905


)]

119878sin = int1198794+119905int

1198794

Δ119899 (119905) 119889119905 minus int

31198794+int

31198794

Δ119899 (119905) 119889119905

= 119878st-st



119879 + 4119905int4120591eff

)]

(54)


10

08

06

04

02

000 T

Time t (120583s)





Nor

mal

ized

carr

ier d

ensit

y Δn(t) a

nd g

ener

atio

n ra

te 119970

(a)

tint = 300 120583s

tint = 1100 120583s

tint = 2000 120583s

120

105

90

75

60

450 1000


ase Φ

(∘)

(b)




10 Conclusion








References



























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics











1

120591rec=

1

120591SRH+

1

120591rad+

1

120591Auger (5)





1

120591eff=

1

120591119887

+

1

120591119878

(6)


SRV = 119863[

120597119899120597119909

119899(119905)

]

boundary (7)













120572fc =11989911989021205822

119898119890120598012057841205871198883

1

120591

(8)





120572119904 = 119860 119904120582119901119904

(9)





120572119899 ≃ 1 times 10minus181205822119899

120572119901 ≃ 27 times 10minus181205822119901

(10)














Conduction band

Valence band

Photon

Eph = h

Ec

Eg

E

(a) 119864ph gt 119864119892

Conduction band

Valence band

Photon

Eph = h

Ec

Eg

E

Ei

(b) 119864119894 lt 119864ph lt 119864119892

Conduction band

Valence band

phonon

Eph = h Ec

Eg

E

(c) 119864ph lt 119864119892


Oscilloscope

NdYAG pulsed laser

Lens

InGaAsphotodiode

Trigger signal



Lens

Sample

Probe beam

Pump pulse

y

x

120572

120582 gt 120582gap

120582p lt 120582gap



119868 (119905) = 1198680119890minus1205720(120582)119871

(1 minus 119890minus120572fc(120582119905)119871

) (11)






Sample

Oscilloscope

Pulsed laser


xyz stage

Lens

Trigger signal

Pump pulse

Probe beam

InGaAsphotodiode

x

120582 gt 120582gap

120582p lt 120582gap


5

0

minus5

minus10

minus15

minus200 05 1 15 2 25 3 35

VminusV0

(mV

)

t (120583s)


Pump pulse

Probe beam

0

y

z

x

S1 S2

d2minusd2



120597119899 (119909 119905)

120597119905

= 119863

1205972119899 (119909 119905)

1205971199092

minus

119899 (119909 119905)

120591119887

(12)


119863[

120597119899(119909 119905)

120597119905

]

119909=minus1198892

= 1198781119899(minus

119889

2

119905) (13)

minus119863[

120597119899(119909 119905)

120597119905

]

119909=1198892

= 1198782119899(

119889

2

119905) (14)


119899 (119909 0) = 1198730 (1 minus 119877) 120572

119890minus120572(119909+1198892)

+ 119877119890minus120572119889

119890120572(minus119909+1198892)

(1 minus 1198772119890minus2120572119889

)

(15)




119899 (119909 119905) =

infin

sum

119896=1

119890minus(1120591

119887+1198862

119896119863)119905

(119860119896 cos 119886119896119909 + 119861119896 sin 119886119896119909) (16)


119886119896119889 = arctan 1198781

119886119896119863

+ arctan 1198782

119886119896119863

+ 119896120587 (17)

while if

119911119896 = 119886119896119889

1198920 =

(1 minus 119877) 120572

1 minus 119877119890minus2120572119889

1198611015840

119896=

(41198861198961198920120572) 119890minus1205721198892

[1198872

119896(119886119896119889 + sin 119886119896119889) + (119886119896119889 minus sin 119886119896119889)] (1205722 + 1198862119896)

(18)

it results in

119887119896 equiv

119860119896

119861119896

= minus


119861119896 = 1198611015840

119896[119887119896 (cos 119911119896 sinh

120572119889

2

+

119886119896

120572

sin 119911119896 cosh120572119889

2

)

times (1 + 119877119890minus120572119889

) ]

+ 1198611015840

119896[(

119886119896

120572

cos 119911119896 sinh120572119889

2

minus sin 119911119896 cosh120572119889

2

)


) ]

(19)


119899av (119905) =1

119881intint

119881int

119899 (119909 119905) 119889119881 (20)




119899av (119905) prop int

1198892

minus1198892

119899 (119909 119905) 119889119909

= sum

119896

119861119896119887119896119868119896 (minus

119889

2

119889

2

) 119906119896 (119905)

= sum

119896

119861119896119887119896

sin 119911119896119911119896

119906119896 (119905)

(21)


119906119896 (119905) = 119890minus(1120591

119887+1198862

119896119863)119905

120601119896 (119909) = 119887119896 cos 119886119896119909 + sin 119886119896119909

119868 (119909119894 119909119891) = int

119909119891

119909119894

120601 (119909) 119889119909

(22)


119866119892 (119905) =

1198730

radic2120587120590

119890minus(119905minus119879

0)21205902

(23)


119899gav = int119905

0

119899av (120591) 119866119892 (119905 minus 120591) 119889120591

= sum

119896

119861119896119887119896119868119896 (minus

119889

2

119889

2

)119864119896 (119905)

(24)

where

119864119896 (119905) = int

119905

0

119906119896 (120591) 119866119892 (119905 minus 120591) 119889120591

= 119906119896 (119905) 119890119865119896((1205902

2)119865119896+1198790)

times [erf ( 120590

radic2

119865119896 +

1198790

radic2120590


radic2

119865119896 minus

119905 minus 1198790

radic2120590

)]

(25)

with

119865119896 =

1

120591119887

+ 1198862

119896119863 (26)


1

1205910

= minus

119889

119889119905

ln [119899gav (119905)] =1

120591119887

+ 119877119904 (119905) (27)




119877119904 (119905)

119905rarrinfin

997888rarr 119877119904119886 = 1198861119863(28)


1

1205910119886

=

1

120591119887

+ 119877119904119886 =

1

120591119887

+

1

1205911199041

(29)






120591119887 =

12059101198861205911199041

1205911199041 minus 1205910119886

(30)




1

1205910119886

=

1

120591119887

+ 119877119904119886

1

1205910119898

=

1

120591119887

+ 119877119898

(31)

so that

(119877119904119898 minus 119877119904119886)anal = (1

1205910119898

minus

1

1205910119886

)

exper(32)


120591119887 =


(33)




(

1

1205910119886

)

1

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(

1

1205910119886

)

2

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(34)



1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

10minus3

10minus4

10minus5

10minus6

120591b

(s)

S (cms)

120591eff = 5 120583s

120591eff = 6 120583s

120591eff = 15 120583s

120591eff = 12 120583s

120591eff 2 = 115 120583s120591eff 1 = 55 120583sd1 = 500 120583m

d2 = 200 120583m





1

1205910119898

=

1

120591119887

+ 119877119898 for (1198781 1198782) or (1198782 1198781)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198781 1198782)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198782 1198781)

(35)





Defining

120575119886 = lim119905rarrinfin

120575 (Δ120582 119905) = ln(119873rel (1205821 119905)

119873rel (1205822 119905)) (36)



















Δ119899 (119909 119905) = 119899119887 (119909) + 119899119901 (119909 119905) (38)


120591 (119899119887) =

1

119899119887

int

119899119887

0

120591diff119889Δ119899 (39)



120591 (119899119887) =

1

119875

int

119875

0

120591diff (119875) 119889119875 (40)







119870 = 119878off minus 119878on (41)


120591eff =Δ119899119882

119866

(42)


119866 = Φ(1 minus 119877119891) (43)



Optical fiber

Lens

Diode laser

Pump beam


Wafer

Black body

CCD camera

Tb

Tw

120582 lt 120582gap

120582 gt 120582gap



119878 = Γint

120582max

120582min

(1 minus 119877)2119890minus120572fc(120582)119882

119868 (120582 119879119887) 119889120582

asymp Γint

120582max

120582min

(1 minus 119877)2120572fc (120582)119882119868 (120582 119879119887) 119889120582

(44)


119868 (120582det 119879119887) =2120587ℎ119888

2

1205825

1

119890(ℎ119888120582119896

119861119879119887)minus1

asymp

120574

1205825119890minusΨ120582119879

119887 (45)






200

180

160

140

120

100

80

60

40

20

0

100

80

60

40

20

0

100806040200

120591ap

p(120583

s)

x (mm)

y(m

m)



119878em = Γint

120582max

120582min

120576fc (120582)120574

1205825119890minusΨ120582119879

119908

119889120582 (46)


119878 = 119878em minus 119878119886119887 (47)




119894119882119894

(119909 119910) minus 11987811987311198821


119870119862 (119909 119910) = 119898Δ (119873119882) (119909 119910) (48)


Δ (119899119882) (119909 119910) =

119870 (119909 119910)

119898120590

120581 (119909 119910) (49)


120590 = (1 +

120572119899

120572119901

) (50)





asymp Γ

120574

1205825

det(119890minusΨ(119879

119887+NETD)


119887

)

asymp Γ

120574

1205825

det

ΨNEDT1198792

119887

119890minusΨ119879

119887

(51)






119878 asymp Γ120573Δ119899119882

120574

1205823

119889119890119905

119890minusΨ119879

119887

(52)


NEL = ℎ119888

1198961198611198792

1198871205822

det

1

120573Φ (1 minus 119877119891)

NETDradic119873

(53)














119878cos = int119905int

0

Δ119899 (119905) 119889119905 minus int

1198792+int

1198792

Δ119899 (119905) 119889119905


)]

119878sin = int1198794+119905int

1198794

Δ119899 (119905) 119889119905 minus int

31198794+int

31198794

Δ119899 (119905) 119889119905

= 119878st-st



119879 + 4119905int4120591eff

)]

(54)


10

08

06

04

02

000 T

Time t (120583s)





Nor

mal

ized

carr

ier d

ensit

y Δn(t) a

nd g

ener

atio

n ra

te 119970

(a)

tint = 300 120583s

tint = 1100 120583s

tint = 2000 120583s

120

105

90

75

60

450 1000


ase Φ

(∘)

(b)




10 Conclusion








References



























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics










120572fc =11989911989021205822

119898119890120598012057841205871198883

1

120591

(8)





120572119904 = 119860 119904120582119901119904

(9)





120572119899 ≃ 1 times 10minus181205822119899

120572119901 ≃ 27 times 10minus181205822119901

(10)














Conduction band

Valence band

Photon

Eph = h

Ec

Eg

E

(a) 119864ph gt 119864119892

Conduction band

Valence band

Photon

Eph = h

Ec

Eg

E

Ei

(b) 119864119894 lt 119864ph lt 119864119892

Conduction band

Valence band

phonon

Eph = h Ec

Eg

E

(c) 119864ph lt 119864119892


Oscilloscope

NdYAG pulsed laser

Lens

InGaAsphotodiode

Trigger signal



Lens

Sample

Probe beam

Pump pulse

y

x

120572

120582 gt 120582gap

120582p lt 120582gap



119868 (119905) = 1198680119890minus1205720(120582)119871

(1 minus 119890minus120572fc(120582119905)119871

) (11)






Sample

Oscilloscope

Pulsed laser


xyz stage

Lens

Trigger signal

Pump pulse

Probe beam

InGaAsphotodiode

x

120582 gt 120582gap

120582p lt 120582gap


5

0

minus5

minus10

minus15

minus200 05 1 15 2 25 3 35

VminusV0

(mV

)

t (120583s)


Pump pulse

Probe beam

0

y

z

x

S1 S2

d2minusd2



120597119899 (119909 119905)

120597119905

= 119863

1205972119899 (119909 119905)

1205971199092

minus

119899 (119909 119905)

120591119887

(12)


119863[

120597119899(119909 119905)

120597119905

]

119909=minus1198892

= 1198781119899(minus

119889

2

119905) (13)

minus119863[

120597119899(119909 119905)

120597119905

]

119909=1198892

= 1198782119899(

119889

2

119905) (14)


119899 (119909 0) = 1198730 (1 minus 119877) 120572

119890minus120572(119909+1198892)

+ 119877119890minus120572119889

119890120572(minus119909+1198892)

(1 minus 1198772119890minus2120572119889

)

(15)




119899 (119909 119905) =

infin

sum

119896=1

119890minus(1120591

119887+1198862

119896119863)119905

(119860119896 cos 119886119896119909 + 119861119896 sin 119886119896119909) (16)


119886119896119889 = arctan 1198781

119886119896119863

+ arctan 1198782

119886119896119863

+ 119896120587 (17)

while if

119911119896 = 119886119896119889

1198920 =

(1 minus 119877) 120572

1 minus 119877119890minus2120572119889

1198611015840

119896=

(41198861198961198920120572) 119890minus1205721198892

[1198872

119896(119886119896119889 + sin 119886119896119889) + (119886119896119889 minus sin 119886119896119889)] (1205722 + 1198862119896)

(18)

it results in

119887119896 equiv

119860119896

119861119896

= minus


119861119896 = 1198611015840

119896[119887119896 (cos 119911119896 sinh

120572119889

2

+

119886119896

120572

sin 119911119896 cosh120572119889

2

)

times (1 + 119877119890minus120572119889

) ]

+ 1198611015840

119896[(

119886119896

120572

cos 119911119896 sinh120572119889

2

minus sin 119911119896 cosh120572119889

2

)


) ]

(19)


119899av (119905) =1

119881intint

119881int

119899 (119909 119905) 119889119881 (20)




119899av (119905) prop int

1198892

minus1198892

119899 (119909 119905) 119889119909

= sum

119896

119861119896119887119896119868119896 (minus

119889

2

119889

2

) 119906119896 (119905)

= sum

119896

119861119896119887119896

sin 119911119896119911119896

119906119896 (119905)

(21)


119906119896 (119905) = 119890minus(1120591

119887+1198862

119896119863)119905

120601119896 (119909) = 119887119896 cos 119886119896119909 + sin 119886119896119909

119868 (119909119894 119909119891) = int

119909119891

119909119894

120601 (119909) 119889119909

(22)


119866119892 (119905) =

1198730

radic2120587120590

119890minus(119905minus119879

0)21205902

(23)


119899gav = int119905

0

119899av (120591) 119866119892 (119905 minus 120591) 119889120591

= sum

119896

119861119896119887119896119868119896 (minus

119889

2

119889

2

)119864119896 (119905)

(24)

where

119864119896 (119905) = int

119905

0

119906119896 (120591) 119866119892 (119905 minus 120591) 119889120591

= 119906119896 (119905) 119890119865119896((1205902

2)119865119896+1198790)

times [erf ( 120590

radic2

119865119896 +

1198790

radic2120590


radic2

119865119896 minus

119905 minus 1198790

radic2120590

)]

(25)

with

119865119896 =

1

120591119887

+ 1198862

119896119863 (26)


1

1205910

= minus

119889

119889119905

ln [119899gav (119905)] =1

120591119887

+ 119877119904 (119905) (27)




119877119904 (119905)

119905rarrinfin

997888rarr 119877119904119886 = 1198861119863(28)


1

1205910119886

=

1

120591119887

+ 119877119904119886 =

1

120591119887

+

1

1205911199041

(29)






120591119887 =

12059101198861205911199041

1205911199041 minus 1205910119886

(30)




1

1205910119886

=

1

120591119887

+ 119877119904119886

1

1205910119898

=

1

120591119887

+ 119877119898

(31)

so that

(119877119904119898 minus 119877119904119886)anal = (1

1205910119898

minus

1

1205910119886

)

exper(32)


120591119887 =


(33)




(

1

1205910119886

)

1

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(

1

1205910119886

)

2

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(34)



1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

10minus3

10minus4

10minus5

10minus6

120591b

(s)

S (cms)

120591eff = 5 120583s

120591eff = 6 120583s

120591eff = 15 120583s

120591eff = 12 120583s

120591eff 2 = 115 120583s120591eff 1 = 55 120583sd1 = 500 120583m

d2 = 200 120583m





1

1205910119898

=

1

120591119887

+ 119877119898 for (1198781 1198782) or (1198782 1198781)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198781 1198782)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198782 1198781)

(35)





Defining

120575119886 = lim119905rarrinfin

120575 (Δ120582 119905) = ln(119873rel (1205821 119905)

119873rel (1205822 119905)) (36)



















Δ119899 (119909 119905) = 119899119887 (119909) + 119899119901 (119909 119905) (38)


120591 (119899119887) =

1

119899119887

int

119899119887

0

120591diff119889Δ119899 (39)



120591 (119899119887) =

1

119875

int

119875

0

120591diff (119875) 119889119875 (40)







119870 = 119878off minus 119878on (41)


120591eff =Δ119899119882

119866

(42)


119866 = Φ(1 minus 119877119891) (43)



Optical fiber

Lens

Diode laser

Pump beam


Wafer

Black body

CCD camera

Tb

Tw

120582 lt 120582gap

120582 gt 120582gap



119878 = Γint

120582max

120582min

(1 minus 119877)2119890minus120572fc(120582)119882

119868 (120582 119879119887) 119889120582

asymp Γint

120582max

120582min

(1 minus 119877)2120572fc (120582)119882119868 (120582 119879119887) 119889120582

(44)


119868 (120582det 119879119887) =2120587ℎ119888

2

1205825

1

119890(ℎ119888120582119896

119861119879119887)minus1

asymp

120574

1205825119890minusΨ120582119879

119887 (45)






200

180

160

140

120

100

80

60

40

20

0

100

80

60

40

20

0

100806040200

120591ap

p(120583

s)

x (mm)

y(m

m)



119878em = Γint

120582max

120582min

120576fc (120582)120574

1205825119890minusΨ120582119879

119908

119889120582 (46)


119878 = 119878em minus 119878119886119887 (47)




119894119882119894

(119909 119910) minus 11987811987311198821


119870119862 (119909 119910) = 119898Δ (119873119882) (119909 119910) (48)


Δ (119899119882) (119909 119910) =

119870 (119909 119910)

119898120590

120581 (119909 119910) (49)


120590 = (1 +

120572119899

120572119901

) (50)





asymp Γ

120574

1205825

det(119890minusΨ(119879

119887+NETD)


119887

)

asymp Γ

120574

1205825

det

ΨNEDT1198792

119887

119890minusΨ119879

119887

(51)






119878 asymp Γ120573Δ119899119882

120574

1205823

119889119890119905

119890minusΨ119879

119887

(52)


NEL = ℎ119888

1198961198611198792

1198871205822

det

1

120573Φ (1 minus 119877119891)

NETDradic119873

(53)














119878cos = int119905int

0

Δ119899 (119905) 119889119905 minus int

1198792+int

1198792

Δ119899 (119905) 119889119905


)]

119878sin = int1198794+119905int

1198794

Δ119899 (119905) 119889119905 minus int

31198794+int

31198794

Δ119899 (119905) 119889119905

= 119878st-st



119879 + 4119905int4120591eff

)]

(54)


10

08

06

04

02

000 T

Time t (120583s)





Nor

mal

ized

carr

ier d

ensit

y Δn(t) a

nd g

ener

atio

n ra

te 119970

(a)

tint = 300 120583s

tint = 1100 120583s

tint = 2000 120583s

120

105

90

75

60

450 1000


ase Φ

(∘)

(b)




10 Conclusion








References



























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics














Conduction band

Valence band

Photon

Eph = h

Ec

Eg

E

(a) 119864ph gt 119864119892

Conduction band

Valence band

Photon

Eph = h

Ec

Eg

E

Ei

(b) 119864119894 lt 119864ph lt 119864119892

Conduction band

Valence band

phonon

Eph = h Ec

Eg

E

(c) 119864ph lt 119864119892


Oscilloscope

NdYAG pulsed laser

Lens

InGaAsphotodiode

Trigger signal



Lens

Sample

Probe beam

Pump pulse

y

x

120572

120582 gt 120582gap

120582p lt 120582gap



119868 (119905) = 1198680119890minus1205720(120582)119871

(1 minus 119890minus120572fc(120582119905)119871

) (11)






Sample

Oscilloscope

Pulsed laser


xyz stage

Lens

Trigger signal

Pump pulse

Probe beam

InGaAsphotodiode

x

120582 gt 120582gap

120582p lt 120582gap


5

0

minus5

minus10

minus15

minus200 05 1 15 2 25 3 35

VminusV0

(mV

)

t (120583s)


Pump pulse

Probe beam

0

y

z

x

S1 S2

d2minusd2



120597119899 (119909 119905)

120597119905

= 119863

1205972119899 (119909 119905)

1205971199092

minus

119899 (119909 119905)

120591119887

(12)


119863[

120597119899(119909 119905)

120597119905

]

119909=minus1198892

= 1198781119899(minus

119889

2

119905) (13)

minus119863[

120597119899(119909 119905)

120597119905

]

119909=1198892

= 1198782119899(

119889

2

119905) (14)


119899 (119909 0) = 1198730 (1 minus 119877) 120572

119890minus120572(119909+1198892)

+ 119877119890minus120572119889

119890120572(minus119909+1198892)

(1 minus 1198772119890minus2120572119889

)

(15)




119899 (119909 119905) =

infin

sum

119896=1

119890minus(1120591

119887+1198862

119896119863)119905

(119860119896 cos 119886119896119909 + 119861119896 sin 119886119896119909) (16)


119886119896119889 = arctan 1198781

119886119896119863

+ arctan 1198782

119886119896119863

+ 119896120587 (17)

while if

119911119896 = 119886119896119889

1198920 =

(1 minus 119877) 120572

1 minus 119877119890minus2120572119889

1198611015840

119896=

(41198861198961198920120572) 119890minus1205721198892

[1198872

119896(119886119896119889 + sin 119886119896119889) + (119886119896119889 minus sin 119886119896119889)] (1205722 + 1198862119896)

(18)

it results in

119887119896 equiv

119860119896

119861119896

= minus


119861119896 = 1198611015840

119896[119887119896 (cos 119911119896 sinh

120572119889

2

+

119886119896

120572

sin 119911119896 cosh120572119889

2

)

times (1 + 119877119890minus120572119889

) ]

+ 1198611015840

119896[(

119886119896

120572

cos 119911119896 sinh120572119889

2

minus sin 119911119896 cosh120572119889

2

)


) ]

(19)


119899av (119905) =1

119881intint

119881int

119899 (119909 119905) 119889119881 (20)




119899av (119905) prop int

1198892

minus1198892

119899 (119909 119905) 119889119909

= sum

119896

119861119896119887119896119868119896 (minus

119889

2

119889

2

) 119906119896 (119905)

= sum

119896

119861119896119887119896

sin 119911119896119911119896

119906119896 (119905)

(21)


119906119896 (119905) = 119890minus(1120591

119887+1198862

119896119863)119905

120601119896 (119909) = 119887119896 cos 119886119896119909 + sin 119886119896119909

119868 (119909119894 119909119891) = int

119909119891

119909119894

120601 (119909) 119889119909

(22)


119866119892 (119905) =

1198730

radic2120587120590

119890minus(119905minus119879

0)21205902

(23)


119899gav = int119905

0

119899av (120591) 119866119892 (119905 minus 120591) 119889120591

= sum

119896

119861119896119887119896119868119896 (minus

119889

2

119889

2

)119864119896 (119905)

(24)

where

119864119896 (119905) = int

119905

0

119906119896 (120591) 119866119892 (119905 minus 120591) 119889120591

= 119906119896 (119905) 119890119865119896((1205902

2)119865119896+1198790)

times [erf ( 120590

radic2

119865119896 +

1198790

radic2120590


radic2

119865119896 minus

119905 minus 1198790

radic2120590

)]

(25)

with

119865119896 =

1

120591119887

+ 1198862

119896119863 (26)


1

1205910

= minus

119889

119889119905

ln [119899gav (119905)] =1

120591119887

+ 119877119904 (119905) (27)




119877119904 (119905)

119905rarrinfin

997888rarr 119877119904119886 = 1198861119863(28)


1

1205910119886

=

1

120591119887

+ 119877119904119886 =

1

120591119887

+

1

1205911199041

(29)






120591119887 =

12059101198861205911199041

1205911199041 minus 1205910119886

(30)




1

1205910119886

=

1

120591119887

+ 119877119904119886

1

1205910119898

=

1

120591119887

+ 119877119898

(31)

so that

(119877119904119898 minus 119877119904119886)anal = (1

1205910119898

minus

1

1205910119886

)

exper(32)


120591119887 =


(33)




(

1

1205910119886

)

1

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(

1

1205910119886

)

2

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(34)



1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

10minus3

10minus4

10minus5

10minus6

120591b

(s)

S (cms)

120591eff = 5 120583s

120591eff = 6 120583s

120591eff = 15 120583s

120591eff = 12 120583s

120591eff 2 = 115 120583s120591eff 1 = 55 120583sd1 = 500 120583m

d2 = 200 120583m





1

1205910119898

=

1

120591119887

+ 119877119898 for (1198781 1198782) or (1198782 1198781)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198781 1198782)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198782 1198781)

(35)





Defining

120575119886 = lim119905rarrinfin

120575 (Δ120582 119905) = ln(119873rel (1205821 119905)

119873rel (1205822 119905)) (36)



















Δ119899 (119909 119905) = 119899119887 (119909) + 119899119901 (119909 119905) (38)


120591 (119899119887) =

1

119899119887

int

119899119887

0

120591diff119889Δ119899 (39)



120591 (119899119887) =

1

119875

int

119875

0

120591diff (119875) 119889119875 (40)







119870 = 119878off minus 119878on (41)


120591eff =Δ119899119882

119866

(42)


119866 = Φ(1 minus 119877119891) (43)



Optical fiber

Lens

Diode laser

Pump beam


Wafer

Black body

CCD camera

Tb

Tw

120582 lt 120582gap

120582 gt 120582gap



119878 = Γint

120582max

120582min

(1 minus 119877)2119890minus120572fc(120582)119882

119868 (120582 119879119887) 119889120582

asymp Γint

120582max

120582min

(1 minus 119877)2120572fc (120582)119882119868 (120582 119879119887) 119889120582

(44)


119868 (120582det 119879119887) =2120587ℎ119888

2

1205825

1

119890(ℎ119888120582119896

119861119879119887)minus1

asymp

120574

1205825119890minusΨ120582119879

119887 (45)






200

180

160

140

120

100

80

60

40

20

0

100

80

60

40

20

0

100806040200

120591ap

p(120583

s)

x (mm)

y(m

m)



119878em = Γint

120582max

120582min

120576fc (120582)120574

1205825119890minusΨ120582119879

119908

119889120582 (46)


119878 = 119878em minus 119878119886119887 (47)




119894119882119894

(119909 119910) minus 11987811987311198821


119870119862 (119909 119910) = 119898Δ (119873119882) (119909 119910) (48)


Δ (119899119882) (119909 119910) =

119870 (119909 119910)

119898120590

120581 (119909 119910) (49)


120590 = (1 +

120572119899

120572119901

) (50)





asymp Γ

120574

1205825

det(119890minusΨ(119879

119887+NETD)


119887

)

asymp Γ

120574

1205825

det

ΨNEDT1198792

119887

119890minusΨ119879

119887

(51)






119878 asymp Γ120573Δ119899119882

120574

1205823

119889119890119905

119890minusΨ119879

119887

(52)


NEL = ℎ119888

1198961198611198792

1198871205822

det

1

120573Φ (1 minus 119877119891)

NETDradic119873

(53)














119878cos = int119905int

0

Δ119899 (119905) 119889119905 minus int

1198792+int

1198792

Δ119899 (119905) 119889119905


)]

119878sin = int1198794+119905int

1198794

Δ119899 (119905) 119889119905 minus int

31198794+int

31198794

Δ119899 (119905) 119889119905

= 119878st-st



119879 + 4119905int4120591eff

)]

(54)


10

08

06

04

02

000 T

Time t (120583s)





Nor

mal

ized

carr

ier d

ensit

y Δn(t) a

nd g

ener

atio

n ra

te 119970

(a)

tint = 300 120583s

tint = 1100 120583s

tint = 2000 120583s

120

105

90

75

60

450 1000


ase Φ

(∘)

(b)




10 Conclusion








References



























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Conduction band

Valence band

Photon

Eph = h

Ec

Eg

E

(a) 119864ph gt 119864119892

Conduction band

Valence band

Photon

Eph = h

Ec

Eg

E

Ei

(b) 119864119894 lt 119864ph lt 119864119892

Conduction band

Valence band

phonon

Eph = h Ec

Eg

E

(c) 119864ph lt 119864119892


Oscilloscope

NdYAG pulsed laser

Lens

InGaAsphotodiode

Trigger signal



Lens

Sample

Probe beam

Pump pulse

y

x

120572

120582 gt 120582gap

120582p lt 120582gap



119868 (119905) = 1198680119890minus1205720(120582)119871

(1 minus 119890minus120572fc(120582119905)119871

) (11)






Sample

Oscilloscope

Pulsed laser


xyz stage

Lens

Trigger signal

Pump pulse

Probe beam

InGaAsphotodiode

x

120582 gt 120582gap

120582p lt 120582gap


5

0

minus5

minus10

minus15

minus200 05 1 15 2 25 3 35

VminusV0

(mV

)

t (120583s)


Pump pulse

Probe beam

0

y

z

x

S1 S2

d2minusd2



120597119899 (119909 119905)

120597119905

= 119863

1205972119899 (119909 119905)

1205971199092

minus

119899 (119909 119905)

120591119887

(12)


119863[

120597119899(119909 119905)

120597119905

]

119909=minus1198892

= 1198781119899(minus

119889

2

119905) (13)

minus119863[

120597119899(119909 119905)

120597119905

]

119909=1198892

= 1198782119899(

119889

2

119905) (14)


119899 (119909 0) = 1198730 (1 minus 119877) 120572

119890minus120572(119909+1198892)

+ 119877119890minus120572119889

119890120572(minus119909+1198892)

(1 minus 1198772119890minus2120572119889

)

(15)




119899 (119909 119905) =

infin

sum

119896=1

119890minus(1120591

119887+1198862

119896119863)119905

(119860119896 cos 119886119896119909 + 119861119896 sin 119886119896119909) (16)


119886119896119889 = arctan 1198781

119886119896119863

+ arctan 1198782

119886119896119863

+ 119896120587 (17)

while if

119911119896 = 119886119896119889

1198920 =

(1 minus 119877) 120572

1 minus 119877119890minus2120572119889

1198611015840

119896=

(41198861198961198920120572) 119890minus1205721198892

[1198872

119896(119886119896119889 + sin 119886119896119889) + (119886119896119889 minus sin 119886119896119889)] (1205722 + 1198862119896)

(18)

it results in

119887119896 equiv

119860119896

119861119896

= minus


119861119896 = 1198611015840

119896[119887119896 (cos 119911119896 sinh

120572119889

2

+

119886119896

120572

sin 119911119896 cosh120572119889

2

)

times (1 + 119877119890minus120572119889

) ]

+ 1198611015840

119896[(

119886119896

120572

cos 119911119896 sinh120572119889

2

minus sin 119911119896 cosh120572119889

2

)


) ]

(19)


119899av (119905) =1

119881intint

119881int

119899 (119909 119905) 119889119881 (20)




119899av (119905) prop int

1198892

minus1198892

119899 (119909 119905) 119889119909

= sum

119896

119861119896119887119896119868119896 (minus

119889

2

119889

2

) 119906119896 (119905)

= sum

119896

119861119896119887119896

sin 119911119896119911119896

119906119896 (119905)

(21)


119906119896 (119905) = 119890minus(1120591

119887+1198862

119896119863)119905

120601119896 (119909) = 119887119896 cos 119886119896119909 + sin 119886119896119909

119868 (119909119894 119909119891) = int

119909119891

119909119894

120601 (119909) 119889119909

(22)


119866119892 (119905) =

1198730

radic2120587120590

119890minus(119905minus119879

0)21205902

(23)


119899gav = int119905

0

119899av (120591) 119866119892 (119905 minus 120591) 119889120591

= sum

119896

119861119896119887119896119868119896 (minus

119889

2

119889

2

)119864119896 (119905)

(24)

where

119864119896 (119905) = int

119905

0

119906119896 (120591) 119866119892 (119905 minus 120591) 119889120591

= 119906119896 (119905) 119890119865119896((1205902

2)119865119896+1198790)

times [erf ( 120590

radic2

119865119896 +

1198790

radic2120590


radic2

119865119896 minus

119905 minus 1198790

radic2120590

)]

(25)

with

119865119896 =

1

120591119887

+ 1198862

119896119863 (26)


1

1205910

= minus

119889

119889119905

ln [119899gav (119905)] =1

120591119887

+ 119877119904 (119905) (27)




119877119904 (119905)

119905rarrinfin

997888rarr 119877119904119886 = 1198861119863(28)


1

1205910119886

=

1

120591119887

+ 119877119904119886 =

1

120591119887

+

1

1205911199041

(29)






120591119887 =

12059101198861205911199041

1205911199041 minus 1205910119886

(30)




1

1205910119886

=

1

120591119887

+ 119877119904119886

1

1205910119898

=

1

120591119887

+ 119877119898

(31)

so that

(119877119904119898 minus 119877119904119886)anal = (1

1205910119898

minus

1

1205910119886

)

exper(32)


120591119887 =


(33)




(

1

1205910119886

)

1

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(

1

1205910119886

)

2

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(34)



1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

10minus3

10minus4

10minus5

10minus6

120591b

(s)

S (cms)

120591eff = 5 120583s

120591eff = 6 120583s

120591eff = 15 120583s

120591eff = 12 120583s

120591eff 2 = 115 120583s120591eff 1 = 55 120583sd1 = 500 120583m

d2 = 200 120583m





1

1205910119898

=

1

120591119887

+ 119877119898 for (1198781 1198782) or (1198782 1198781)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198781 1198782)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198782 1198781)

(35)





Defining

120575119886 = lim119905rarrinfin

120575 (Δ120582 119905) = ln(119873rel (1205821 119905)

119873rel (1205822 119905)) (36)



















Δ119899 (119909 119905) = 119899119887 (119909) + 119899119901 (119909 119905) (38)


120591 (119899119887) =

1

119899119887

int

119899119887

0

120591diff119889Δ119899 (39)



120591 (119899119887) =

1

119875

int

119875

0

120591diff (119875) 119889119875 (40)







119870 = 119878off minus 119878on (41)


120591eff =Δ119899119882

119866

(42)


119866 = Φ(1 minus 119877119891) (43)



Optical fiber

Lens

Diode laser

Pump beam


Wafer

Black body

CCD camera

Tb

Tw

120582 lt 120582gap

120582 gt 120582gap



119878 = Γint

120582max

120582min

(1 minus 119877)2119890minus120572fc(120582)119882

119868 (120582 119879119887) 119889120582

asymp Γint

120582max

120582min

(1 minus 119877)2120572fc (120582)119882119868 (120582 119879119887) 119889120582

(44)


119868 (120582det 119879119887) =2120587ℎ119888

2

1205825

1

119890(ℎ119888120582119896

119861119879119887)minus1

asymp

120574

1205825119890minusΨ120582119879

119887 (45)






200

180

160

140

120

100

80

60

40

20

0

100

80

60

40

20

0

100806040200

120591ap

p(120583

s)

x (mm)

y(m

m)



119878em = Γint

120582max

120582min

120576fc (120582)120574

1205825119890minusΨ120582119879

119908

119889120582 (46)


119878 = 119878em minus 119878119886119887 (47)




119894119882119894

(119909 119910) minus 11987811987311198821


119870119862 (119909 119910) = 119898Δ (119873119882) (119909 119910) (48)


Δ (119899119882) (119909 119910) =

119870 (119909 119910)

119898120590

120581 (119909 119910) (49)


120590 = (1 +

120572119899

120572119901

) (50)





asymp Γ

120574

1205825

det(119890minusΨ(119879

119887+NETD)


119887

)

asymp Γ

120574

1205825

det

ΨNEDT1198792

119887

119890minusΨ119879

119887

(51)






119878 asymp Γ120573Δ119899119882

120574

1205823

119889119890119905

119890minusΨ119879

119887

(52)


NEL = ℎ119888

1198961198611198792

1198871205822

det

1

120573Φ (1 minus 119877119891)

NETDradic119873

(53)














119878cos = int119905int

0

Δ119899 (119905) 119889119905 minus int

1198792+int

1198792

Δ119899 (119905) 119889119905


)]

119878sin = int1198794+119905int

1198794

Δ119899 (119905) 119889119905 minus int

31198794+int

31198794

Δ119899 (119905) 119889119905

= 119878st-st



119879 + 4119905int4120591eff

)]

(54)


10

08

06

04

02

000 T

Time t (120583s)





Nor

mal

ized

carr

ier d

ensit

y Δn(t) a

nd g

ener

atio

n ra

te 119970

(a)

tint = 300 120583s

tint = 1100 120583s

tint = 2000 120583s

120

105

90

75

60

450 1000


ase Φ

(∘)

(b)




10 Conclusion








References



























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Sample

Oscilloscope

Pulsed laser


xyz stage

Lens

Trigger signal

Pump pulse

Probe beam

InGaAsphotodiode

x

120582 gt 120582gap

120582p lt 120582gap


5

0

minus5

minus10

minus15

minus200 05 1 15 2 25 3 35

VminusV0

(mV

)

t (120583s)


Pump pulse

Probe beam

0

y

z

x

S1 S2

d2minusd2



120597119899 (119909 119905)

120597119905

= 119863

1205972119899 (119909 119905)

1205971199092

minus

119899 (119909 119905)

120591119887

(12)


119863[

120597119899(119909 119905)

120597119905

]

119909=minus1198892

= 1198781119899(minus

119889

2

119905) (13)

minus119863[

120597119899(119909 119905)

120597119905

]

119909=1198892

= 1198782119899(

119889

2

119905) (14)


119899 (119909 0) = 1198730 (1 minus 119877) 120572

119890minus120572(119909+1198892)

+ 119877119890minus120572119889

119890120572(minus119909+1198892)

(1 minus 1198772119890minus2120572119889

)

(15)




119899 (119909 119905) =

infin

sum

119896=1

119890minus(1120591

119887+1198862

119896119863)119905

(119860119896 cos 119886119896119909 + 119861119896 sin 119886119896119909) (16)


119886119896119889 = arctan 1198781

119886119896119863

+ arctan 1198782

119886119896119863

+ 119896120587 (17)

while if

119911119896 = 119886119896119889

1198920 =

(1 minus 119877) 120572

1 minus 119877119890minus2120572119889

1198611015840

119896=

(41198861198961198920120572) 119890minus1205721198892

[1198872

119896(119886119896119889 + sin 119886119896119889) + (119886119896119889 minus sin 119886119896119889)] (1205722 + 1198862119896)

(18)

it results in

119887119896 equiv

119860119896

119861119896

= minus


119861119896 = 1198611015840

119896[119887119896 (cos 119911119896 sinh

120572119889

2

+

119886119896

120572

sin 119911119896 cosh120572119889

2

)

times (1 + 119877119890minus120572119889

) ]

+ 1198611015840

119896[(

119886119896

120572

cos 119911119896 sinh120572119889

2

minus sin 119911119896 cosh120572119889

2

)


) ]

(19)


119899av (119905) =1

119881intint

119881int

119899 (119909 119905) 119889119881 (20)




119899av (119905) prop int

1198892

minus1198892

119899 (119909 119905) 119889119909

= sum

119896

119861119896119887119896119868119896 (minus

119889

2

119889

2

) 119906119896 (119905)

= sum

119896

119861119896119887119896

sin 119911119896119911119896

119906119896 (119905)

(21)


119906119896 (119905) = 119890minus(1120591

119887+1198862

119896119863)119905

120601119896 (119909) = 119887119896 cos 119886119896119909 + sin 119886119896119909

119868 (119909119894 119909119891) = int

119909119891

119909119894

120601 (119909) 119889119909

(22)


119866119892 (119905) =

1198730

radic2120587120590

119890minus(119905minus119879

0)21205902

(23)


119899gav = int119905

0

119899av (120591) 119866119892 (119905 minus 120591) 119889120591

= sum

119896

119861119896119887119896119868119896 (minus

119889

2

119889

2

)119864119896 (119905)

(24)

where

119864119896 (119905) = int

119905

0

119906119896 (120591) 119866119892 (119905 minus 120591) 119889120591

= 119906119896 (119905) 119890119865119896((1205902

2)119865119896+1198790)

times [erf ( 120590

radic2

119865119896 +

1198790

radic2120590


radic2

119865119896 minus

119905 minus 1198790

radic2120590

)]

(25)

with

119865119896 =

1

120591119887

+ 1198862

119896119863 (26)


1

1205910

= minus

119889

119889119905

ln [119899gav (119905)] =1

120591119887

+ 119877119904 (119905) (27)




119877119904 (119905)

119905rarrinfin

997888rarr 119877119904119886 = 1198861119863(28)


1

1205910119886

=

1

120591119887

+ 119877119904119886 =

1

120591119887

+

1

1205911199041

(29)






120591119887 =

12059101198861205911199041

1205911199041 minus 1205910119886

(30)




1

1205910119886

=

1

120591119887

+ 119877119904119886

1

1205910119898

=

1

120591119887

+ 119877119898

(31)

so that

(119877119904119898 minus 119877119904119886)anal = (1

1205910119898

minus

1

1205910119886

)

exper(32)


120591119887 =


(33)




(

1

1205910119886

)

1

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(

1

1205910119886

)

2

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(34)



1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

10minus3

10minus4

10minus5

10minus6

120591b

(s)

S (cms)

120591eff = 5 120583s

120591eff = 6 120583s

120591eff = 15 120583s

120591eff = 12 120583s

120591eff 2 = 115 120583s120591eff 1 = 55 120583sd1 = 500 120583m

d2 = 200 120583m





1

1205910119898

=

1

120591119887

+ 119877119898 for (1198781 1198782) or (1198782 1198781)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198781 1198782)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198782 1198781)

(35)





Defining

120575119886 = lim119905rarrinfin

120575 (Δ120582 119905) = ln(119873rel (1205821 119905)

119873rel (1205822 119905)) (36)



















Δ119899 (119909 119905) = 119899119887 (119909) + 119899119901 (119909 119905) (38)


120591 (119899119887) =

1

119899119887

int

119899119887

0

120591diff119889Δ119899 (39)



120591 (119899119887) =

1

119875

int

119875

0

120591diff (119875) 119889119875 (40)







119870 = 119878off minus 119878on (41)


120591eff =Δ119899119882

119866

(42)


119866 = Φ(1 minus 119877119891) (43)



Optical fiber

Lens

Diode laser

Pump beam


Wafer

Black body

CCD camera

Tb

Tw

120582 lt 120582gap

120582 gt 120582gap



119878 = Γint

120582max

120582min

(1 minus 119877)2119890minus120572fc(120582)119882

119868 (120582 119879119887) 119889120582

asymp Γint

120582max

120582min

(1 minus 119877)2120572fc (120582)119882119868 (120582 119879119887) 119889120582

(44)


119868 (120582det 119879119887) =2120587ℎ119888

2

1205825

1

119890(ℎ119888120582119896

119861119879119887)minus1

asymp

120574

1205825119890minusΨ120582119879

119887 (45)






200

180

160

140

120

100

80

60

40

20

0

100

80

60

40

20

0

100806040200

120591ap

p(120583

s)

x (mm)

y(m

m)



119878em = Γint

120582max

120582min

120576fc (120582)120574

1205825119890minusΨ120582119879

119908

119889120582 (46)


119878 = 119878em minus 119878119886119887 (47)




119894119882119894

(119909 119910) minus 11987811987311198821


119870119862 (119909 119910) = 119898Δ (119873119882) (119909 119910) (48)


Δ (119899119882) (119909 119910) =

119870 (119909 119910)

119898120590

120581 (119909 119910) (49)


120590 = (1 +

120572119899

120572119901

) (50)





asymp Γ

120574

1205825

det(119890minusΨ(119879

119887+NETD)


119887

)

asymp Γ

120574

1205825

det

ΨNEDT1198792

119887

119890minusΨ119879

119887

(51)






119878 asymp Γ120573Δ119899119882

120574

1205823

119889119890119905

119890minusΨ119879

119887

(52)


NEL = ℎ119888

1198961198611198792

1198871205822

det

1

120573Φ (1 minus 119877119891)

NETDradic119873

(53)














119878cos = int119905int

0

Δ119899 (119905) 119889119905 minus int

1198792+int

1198792

Δ119899 (119905) 119889119905


)]

119878sin = int1198794+119905int

1198794

Δ119899 (119905) 119889119905 minus int

31198794+int

31198794

Δ119899 (119905) 119889119905

= 119878st-st



119879 + 4119905int4120591eff

)]

(54)


10

08

06

04

02

000 T

Time t (120583s)





Nor

mal

ized

carr

ier d

ensit

y Δn(t) a

nd g

ener

atio

n ra

te 119970

(a)

tint = 300 120583s

tint = 1100 120583s

tint = 2000 120583s

120

105

90

75

60

450 1000


ase Φ

(∘)

(b)




10 Conclusion








References



























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119899 (119909 119905) =

infin

sum

119896=1

119890minus(1120591

119887+1198862

119896119863)119905

(119860119896 cos 119886119896119909 + 119861119896 sin 119886119896119909) (16)


119886119896119889 = arctan 1198781

119886119896119863

+ arctan 1198782

119886119896119863

+ 119896120587 (17)

while if

119911119896 = 119886119896119889

1198920 =

(1 minus 119877) 120572

1 minus 119877119890minus2120572119889

1198611015840

119896=

(41198861198961198920120572) 119890minus1205721198892

[1198872

119896(119886119896119889 + sin 119886119896119889) + (119886119896119889 minus sin 119886119896119889)] (1205722 + 1198862119896)

(18)

it results in

119887119896 equiv

119860119896

119861119896

= minus


119861119896 = 1198611015840

119896[119887119896 (cos 119911119896 sinh

120572119889

2

+

119886119896

120572

sin 119911119896 cosh120572119889

2

)

times (1 + 119877119890minus120572119889

) ]

+ 1198611015840

119896[(

119886119896

120572

cos 119911119896 sinh120572119889

2

minus sin 119911119896 cosh120572119889

2

)


) ]

(19)


119899av (119905) =1

119881intint

119881int

119899 (119909 119905) 119889119881 (20)




119899av (119905) prop int

1198892

minus1198892

119899 (119909 119905) 119889119909

= sum

119896

119861119896119887119896119868119896 (minus

119889

2

119889

2

) 119906119896 (119905)

= sum

119896

119861119896119887119896

sin 119911119896119911119896

119906119896 (119905)

(21)


119906119896 (119905) = 119890minus(1120591

119887+1198862

119896119863)119905

120601119896 (119909) = 119887119896 cos 119886119896119909 + sin 119886119896119909

119868 (119909119894 119909119891) = int

119909119891

119909119894

120601 (119909) 119889119909

(22)


119866119892 (119905) =

1198730

radic2120587120590

119890minus(119905minus119879

0)21205902

(23)


119899gav = int119905

0

119899av (120591) 119866119892 (119905 minus 120591) 119889120591

= sum

119896

119861119896119887119896119868119896 (minus

119889

2

119889

2

)119864119896 (119905)

(24)

where

119864119896 (119905) = int

119905

0

119906119896 (120591) 119866119892 (119905 minus 120591) 119889120591

= 119906119896 (119905) 119890119865119896((1205902

2)119865119896+1198790)

times [erf ( 120590

radic2

119865119896 +

1198790

radic2120590


radic2

119865119896 minus

119905 minus 1198790

radic2120590

)]

(25)

with

119865119896 =

1

120591119887

+ 1198862

119896119863 (26)


1

1205910

= minus

119889

119889119905

ln [119899gav (119905)] =1

120591119887

+ 119877119904 (119905) (27)




119877119904 (119905)

119905rarrinfin

997888rarr 119877119904119886 = 1198861119863(28)


1

1205910119886

=

1

120591119887

+ 119877119904119886 =

1

120591119887

+

1

1205911199041

(29)






120591119887 =

12059101198861205911199041

1205911199041 minus 1205910119886

(30)




1

1205910119886

=

1

120591119887

+ 119877119904119886

1

1205910119898

=

1

120591119887

+ 119877119898

(31)

so that

(119877119904119898 minus 119877119904119886)anal = (1

1205910119898

minus

1

1205910119886

)

exper(32)


120591119887 =


(33)




(

1

1205910119886

)

1

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(

1

1205910119886

)

2

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(34)



1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

10minus3

10minus4

10minus5

10minus6

120591b

(s)

S (cms)

120591eff = 5 120583s

120591eff = 6 120583s

120591eff = 15 120583s

120591eff = 12 120583s

120591eff 2 = 115 120583s120591eff 1 = 55 120583sd1 = 500 120583m

d2 = 200 120583m





1

1205910119898

=

1

120591119887

+ 119877119898 for (1198781 1198782) or (1198782 1198781)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198781 1198782)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198782 1198781)

(35)





Defining

120575119886 = lim119905rarrinfin

120575 (Δ120582 119905) = ln(119873rel (1205821 119905)

119873rel (1205822 119905)) (36)



















Δ119899 (119909 119905) = 119899119887 (119909) + 119899119901 (119909 119905) (38)


120591 (119899119887) =

1

119899119887

int

119899119887

0

120591diff119889Δ119899 (39)



120591 (119899119887) =

1

119875

int

119875

0

120591diff (119875) 119889119875 (40)







119870 = 119878off minus 119878on (41)


120591eff =Δ119899119882

119866

(42)


119866 = Φ(1 minus 119877119891) (43)



Optical fiber

Lens

Diode laser

Pump beam


Wafer

Black body

CCD camera

Tb

Tw

120582 lt 120582gap

120582 gt 120582gap



119878 = Γint

120582max

120582min

(1 minus 119877)2119890minus120572fc(120582)119882

119868 (120582 119879119887) 119889120582

asymp Γint

120582max

120582min

(1 minus 119877)2120572fc (120582)119882119868 (120582 119879119887) 119889120582

(44)


119868 (120582det 119879119887) =2120587ℎ119888

2

1205825

1

119890(ℎ119888120582119896

119861119879119887)minus1

asymp

120574

1205825119890minusΨ120582119879

119887 (45)






200

180

160

140

120

100

80

60

40

20

0

100

80

60

40

20

0

100806040200

120591ap

p(120583

s)

x (mm)

y(m

m)



119878em = Γint

120582max

120582min

120576fc (120582)120574

1205825119890minusΨ120582119879

119908

119889120582 (46)


119878 = 119878em minus 119878119886119887 (47)




119894119882119894

(119909 119910) minus 11987811987311198821


119870119862 (119909 119910) = 119898Δ (119873119882) (119909 119910) (48)


Δ (119899119882) (119909 119910) =

119870 (119909 119910)

119898120590

120581 (119909 119910) (49)


120590 = (1 +

120572119899

120572119901

) (50)





asymp Γ

120574

1205825

det(119890minusΨ(119879

119887+NETD)


119887

)

asymp Γ

120574

1205825

det

ΨNEDT1198792

119887

119890minusΨ119879

119887

(51)






119878 asymp Γ120573Δ119899119882

120574

1205823

119889119890119905

119890minusΨ119879

119887

(52)


NEL = ℎ119888

1198961198611198792

1198871205822

det

1

120573Φ (1 minus 119877119891)

NETDradic119873

(53)














119878cos = int119905int

0

Δ119899 (119905) 119889119905 minus int

1198792+int

1198792

Δ119899 (119905) 119889119905


)]

119878sin = int1198794+119905int

1198794

Δ119899 (119905) 119889119905 minus int

31198794+int

31198794

Δ119899 (119905) 119889119905

= 119878st-st



119879 + 4119905int4120591eff

)]

(54)


10

08

06

04

02

000 T

Time t (120583s)





Nor

mal

ized

carr

ier d

ensit

y Δn(t) a

nd g

ener

atio

n ra

te 119970

(a)

tint = 300 120583s

tint = 1100 120583s

tint = 2000 120583s

120

105

90

75

60

450 1000


ase Φ

(∘)

(b)




10 Conclusion








References



























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119877119904 (119905)

119905rarrinfin

997888rarr 119877119904119886 = 1198861119863(28)


1

1205910119886

=

1

120591119887

+ 119877119904119886 =

1

120591119887

+

1

1205911199041

(29)






120591119887 =

12059101198861205911199041

1205911199041 minus 1205910119886

(30)




1

1205910119886

=

1

120591119887

+ 119877119904119886

1

1205910119898

=

1

120591119887

+ 119877119898

(31)

so that

(119877119904119898 minus 119877119904119886)anal = (1

1205910119898

minus

1

1205910119886

)

exper(32)


120591119887 =


(33)




(

1

1205910119886

)

1

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(

1

1205910119886

)

2

=

1

120591119887

+ (119877119904119886 (1198891 119878 119863))1

(34)



1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

10minus3

10minus4

10minus5

10minus6

120591b

(s)

S (cms)

120591eff = 5 120583s

120591eff = 6 120583s

120591eff = 15 120583s

120591eff = 12 120583s

120591eff 2 = 115 120583s120591eff 1 = 55 120583sd1 = 500 120583m

d2 = 200 120583m





1

1205910119898

=

1

120591119887

+ 119877119898 for (1198781 1198782) or (1198782 1198781)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198781 1198782)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198782 1198781)

(35)





Defining

120575119886 = lim119905rarrinfin

120575 (Δ120582 119905) = ln(119873rel (1205821 119905)

119873rel (1205822 119905)) (36)



















Δ119899 (119909 119905) = 119899119887 (119909) + 119899119901 (119909 119905) (38)


120591 (119899119887) =

1

119899119887

int

119899119887

0

120591diff119889Δ119899 (39)



120591 (119899119887) =

1

119875

int

119875

0

120591diff (119875) 119889119875 (40)







119870 = 119878off minus 119878on (41)


120591eff =Δ119899119882

119866

(42)


119866 = Φ(1 minus 119877119891) (43)



Optical fiber

Lens

Diode laser

Pump beam


Wafer

Black body

CCD camera

Tb

Tw

120582 lt 120582gap

120582 gt 120582gap



119878 = Γint

120582max

120582min

(1 minus 119877)2119890minus120572fc(120582)119882

119868 (120582 119879119887) 119889120582

asymp Γint

120582max

120582min

(1 minus 119877)2120572fc (120582)119882119868 (120582 119879119887) 119889120582

(44)


119868 (120582det 119879119887) =2120587ℎ119888

2

1205825

1

119890(ℎ119888120582119896

119861119879119887)minus1

asymp

120574

1205825119890minusΨ120582119879

119887 (45)






200

180

160

140

120

100

80

60

40

20

0

100

80

60

40

20

0

100806040200

120591ap

p(120583

s)

x (mm)

y(m

m)



119878em = Γint

120582max

120582min

120576fc (120582)120574

1205825119890minusΨ120582119879

119908

119889120582 (46)


119878 = 119878em minus 119878119886119887 (47)




119894119882119894

(119909 119910) minus 11987811987311198821


119870119862 (119909 119910) = 119898Δ (119873119882) (119909 119910) (48)


Δ (119899119882) (119909 119910) =

119870 (119909 119910)

119898120590

120581 (119909 119910) (49)


120590 = (1 +

120572119899

120572119901

) (50)





asymp Γ

120574

1205825

det(119890minusΨ(119879

119887+NETD)


119887

)

asymp Γ

120574

1205825

det

ΨNEDT1198792

119887

119890minusΨ119879

119887

(51)






119878 asymp Γ120573Δ119899119882

120574

1205823

119889119890119905

119890minusΨ119879

119887

(52)


NEL = ℎ119888

1198961198611198792

1198871205822

det

1

120573Φ (1 minus 119877119891)

NETDradic119873

(53)














119878cos = int119905int

0

Δ119899 (119905) 119889119905 minus int

1198792+int

1198792

Δ119899 (119905) 119889119905


)]

119878sin = int1198794+119905int

1198794

Δ119899 (119905) 119889119905 minus int

31198794+int

31198794

Δ119899 (119905) 119889119905

= 119878st-st



119879 + 4119905int4120591eff

)]

(54)


10

08

06

04

02

000 T

Time t (120583s)





Nor

mal

ized

carr

ier d

ensit

y Δn(t) a

nd g

ener

atio

n ra

te 119970

(a)

tint = 300 120583s

tint = 1100 120583s

tint = 2000 120583s

120

105

90

75

60

450 1000


ase Φ

(∘)

(b)




10 Conclusion








References



























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

10minus3

10minus4

10minus5

10minus6

120591b

(s)

S (cms)

120591eff = 5 120583s

120591eff = 6 120583s

120591eff = 15 120583s

120591eff = 12 120583s

120591eff 2 = 115 120583s120591eff 1 = 55 120583sd1 = 500 120583m

d2 = 200 120583m





1

1205910119898

=

1

120591119887

+ 119877119898 for (1198781 1198782) or (1198782 1198781)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198781 1198782)

1

1205910119886

=

1

120591119887

+ 119877119904119886 for (1198782 1198781)

(35)





Defining

120575119886 = lim119905rarrinfin

120575 (Δ120582 119905) = ln(119873rel (1205821 119905)

119873rel (1205822 119905)) (36)



















Δ119899 (119909 119905) = 119899119887 (119909) + 119899119901 (119909 119905) (38)


120591 (119899119887) =

1

119899119887

int

119899119887

0

120591diff119889Δ119899 (39)



120591 (119899119887) =

1

119875

int

119875

0

120591diff (119875) 119889119875 (40)







119870 = 119878off minus 119878on (41)


120591eff =Δ119899119882

119866

(42)


119866 = Φ(1 minus 119877119891) (43)



Optical fiber

Lens

Diode laser

Pump beam


Wafer

Black body

CCD camera

Tb

Tw

120582 lt 120582gap

120582 gt 120582gap



119878 = Γint

120582max

120582min

(1 minus 119877)2119890minus120572fc(120582)119882

119868 (120582 119879119887) 119889120582

asymp Γint

120582max

120582min

(1 minus 119877)2120572fc (120582)119882119868 (120582 119879119887) 119889120582

(44)


119868 (120582det 119879119887) =2120587ℎ119888

2

1205825

1

119890(ℎ119888120582119896

119861119879119887)minus1

asymp

120574

1205825119890minusΨ120582119879

119887 (45)






200

180

160

140

120

100

80

60

40

20

0

100

80

60

40

20

0

100806040200

120591ap

p(120583

s)

x (mm)

y(m

m)



119878em = Γint

120582max

120582min

120576fc (120582)120574

1205825119890minusΨ120582119879

119908

119889120582 (46)


119878 = 119878em minus 119878119886119887 (47)




119894119882119894

(119909 119910) minus 11987811987311198821


119870119862 (119909 119910) = 119898Δ (119873119882) (119909 119910) (48)


Δ (119899119882) (119909 119910) =

119870 (119909 119910)

119898120590

120581 (119909 119910) (49)


120590 = (1 +

120572119899

120572119901

) (50)





asymp Γ

120574

1205825

det(119890minusΨ(119879

119887+NETD)


119887

)

asymp Γ

120574

1205825

det

ΨNEDT1198792

119887

119890minusΨ119879

119887

(51)






119878 asymp Γ120573Δ119899119882

120574

1205823

119889119890119905

119890minusΨ119879

119887

(52)


NEL = ℎ119888

1198961198611198792

1198871205822

det

1

120573Φ (1 minus 119877119891)

NETDradic119873

(53)














119878cos = int119905int

0

Δ119899 (119905) 119889119905 minus int

1198792+int

1198792

Δ119899 (119905) 119889119905


)]

119878sin = int1198794+119905int

1198794

Δ119899 (119905) 119889119905 minus int

31198794+int

31198794

Δ119899 (119905) 119889119905

= 119878st-st



119879 + 4119905int4120591eff

)]

(54)


10

08

06

04

02

000 T

Time t (120583s)





Nor

mal

ized

carr

ier d

ensit

y Δn(t) a

nd g

ener

atio

n ra

te 119970

(a)

tint = 300 120583s

tint = 1100 120583s

tint = 2000 120583s

120

105

90

75

60

450 1000


ase Φ

(∘)

(b)




10 Conclusion








References



























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics











Δ119899 (119909 119905) = 119899119887 (119909) + 119899119901 (119909 119905) (38)


120591 (119899119887) =

1

119899119887

int

119899119887

0

120591diff119889Δ119899 (39)



120591 (119899119887) =

1

119875

int

119875

0

120591diff (119875) 119889119875 (40)







119870 = 119878off minus 119878on (41)


120591eff =Δ119899119882

119866

(42)


119866 = Φ(1 minus 119877119891) (43)



Optical fiber

Lens

Diode laser

Pump beam


Wafer

Black body

CCD camera

Tb

Tw

120582 lt 120582gap

120582 gt 120582gap



119878 = Γint

120582max

120582min

(1 minus 119877)2119890minus120572fc(120582)119882

119868 (120582 119879119887) 119889120582

asymp Γint

120582max

120582min

(1 minus 119877)2120572fc (120582)119882119868 (120582 119879119887) 119889120582

(44)


119868 (120582det 119879119887) =2120587ℎ119888

2

1205825

1

119890(ℎ119888120582119896

119861119879119887)minus1

asymp

120574

1205825119890minusΨ120582119879

119887 (45)






200

180

160

140

120

100

80

60

40

20

0

100

80

60

40

20

0

100806040200

120591ap

p(120583

s)

x (mm)

y(m

m)



119878em = Γint

120582max

120582min

120576fc (120582)120574

1205825119890minusΨ120582119879

119908

119889120582 (46)


119878 = 119878em minus 119878119886119887 (47)




119894119882119894

(119909 119910) minus 11987811987311198821


119870119862 (119909 119910) = 119898Δ (119873119882) (119909 119910) (48)


Δ (119899119882) (119909 119910) =

119870 (119909 119910)

119898120590

120581 (119909 119910) (49)


120590 = (1 +

120572119899

120572119901

) (50)





asymp Γ

120574

1205825

det(119890minusΨ(119879

119887+NETD)


119887

)

asymp Γ

120574

1205825

det

ΨNEDT1198792

119887

119890minusΨ119879

119887

(51)






119878 asymp Γ120573Δ119899119882

120574

1205823

119889119890119905

119890minusΨ119879

119887

(52)


NEL = ℎ119888

1198961198611198792

1198871205822

det

1

120573Φ (1 minus 119877119891)

NETDradic119873

(53)














119878cos = int119905int

0

Δ119899 (119905) 119889119905 minus int

1198792+int

1198792

Δ119899 (119905) 119889119905


)]

119878sin = int1198794+119905int

1198794

Δ119899 (119905) 119889119905 minus int

31198794+int

31198794

Δ119899 (119905) 119889119905

= 119878st-st



119879 + 4119905int4120591eff

)]

(54)


10

08

06

04

02

000 T

Time t (120583s)





Nor

mal

ized

carr

ier d

ensit

y Δn(t) a

nd g

ener

atio

n ra

te 119970

(a)

tint = 300 120583s

tint = 1100 120583s

tint = 2000 120583s

120

105

90

75

60

450 1000


ase Φ

(∘)

(b)




10 Conclusion








References



























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








119870 = 119878off minus 119878on (41)


120591eff =Δ119899119882

119866

(42)


119866 = Φ(1 minus 119877119891) (43)



Optical fiber

Lens

Diode laser

Pump beam


Wafer

Black body

CCD camera

Tb

Tw

120582 lt 120582gap

120582 gt 120582gap



119878 = Γint

120582max

120582min

(1 minus 119877)2119890minus120572fc(120582)119882

119868 (120582 119879119887) 119889120582

asymp Γint

120582max

120582min

(1 minus 119877)2120572fc (120582)119882119868 (120582 119879119887) 119889120582

(44)


119868 (120582det 119879119887) =2120587ℎ119888

2

1205825

1

119890(ℎ119888120582119896

119861119879119887)minus1

asymp

120574

1205825119890minusΨ120582119879

119887 (45)






200

180

160

140

120

100

80

60

40

20

0

100

80

60

40

20

0

100806040200

120591ap

p(120583

s)

x (mm)

y(m

m)



119878em = Γint

120582max

120582min

120576fc (120582)120574

1205825119890minusΨ120582119879

119908

119889120582 (46)


119878 = 119878em minus 119878119886119887 (47)




119894119882119894

(119909 119910) minus 11987811987311198821


119870119862 (119909 119910) = 119898Δ (119873119882) (119909 119910) (48)


Δ (119899119882) (119909 119910) =

119870 (119909 119910)

119898120590

120581 (119909 119910) (49)


120590 = (1 +

120572119899

120572119901

) (50)





asymp Γ

120574

1205825

det(119890minusΨ(119879

119887+NETD)


119887

)

asymp Γ

120574

1205825

det

ΨNEDT1198792

119887

119890minusΨ119879

119887

(51)






119878 asymp Γ120573Δ119899119882

120574

1205823

119889119890119905

119890minusΨ119879

119887

(52)


NEL = ℎ119888

1198961198611198792

1198871205822

det

1

120573Φ (1 minus 119877119891)

NETDradic119873

(53)














119878cos = int119905int

0

Δ119899 (119905) 119889119905 minus int

1198792+int

1198792

Δ119899 (119905) 119889119905


)]

119878sin = int1198794+119905int

1198794

Δ119899 (119905) 119889119905 minus int

31198794+int

31198794

Δ119899 (119905) 119889119905

= 119878st-st



119879 + 4119905int4120591eff

)]

(54)


10

08

06

04

02

000 T

Time t (120583s)





Nor

mal

ized

carr

ier d

ensit

y Δn(t) a

nd g

ener

atio

n ra

te 119970

(a)

tint = 300 120583s

tint = 1100 120583s

tint = 2000 120583s

120

105

90

75

60

450 1000


ase Φ

(∘)

(b)




10 Conclusion








References



























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




200

180

160

140

120

100

80

60

40

20

0

100

80

60

40

20

0

100806040200

120591ap

p(120583

s)

x (mm)

y(m

m)



119878em = Γint

120582max

120582min

120576fc (120582)120574

1205825119890minusΨ120582119879

119908

119889120582 (46)


119878 = 119878em minus 119878119886119887 (47)




119894119882119894

(119909 119910) minus 11987811987311198821


119870119862 (119909 119910) = 119898Δ (119873119882) (119909 119910) (48)


Δ (119899119882) (119909 119910) =

119870 (119909 119910)

119898120590

120581 (119909 119910) (49)


120590 = (1 +

120572119899

120572119901

) (50)





asymp Γ

120574

1205825

det(119890minusΨ(119879

119887+NETD)


119887

)

asymp Γ

120574

1205825

det

ΨNEDT1198792

119887

119890minusΨ119879

119887

(51)






119878 asymp Γ120573Δ119899119882

120574

1205823

119889119890119905

119890minusΨ119879

119887

(52)


NEL = ℎ119888

1198961198611198792

1198871205822

det

1

120573Φ (1 minus 119877119891)

NETDradic119873

(53)














119878cos = int119905int

0

Δ119899 (119905) 119889119905 minus int

1198792+int

1198792

Δ119899 (119905) 119889119905


)]

119878sin = int1198794+119905int

1198794

Δ119899 (119905) 119889119905 minus int

31198794+int

31198794

Δ119899 (119905) 119889119905

= 119878st-st



119879 + 4119905int4120591eff

)]

(54)


10

08

06

04

02

000 T

Time t (120583s)





Nor

mal

ized

carr

ier d

ensit

y Δn(t) a

nd g

ener

atio

n ra

te 119970

(a)

tint = 300 120583s

tint = 1100 120583s

tint = 2000 120583s

120

105

90

75

60

450 1000


ase Φ

(∘)

(b)




10 Conclusion








References



























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119878 asymp Γ120573Δ119899119882

120574

1205823

119889119890119905

119890minusΨ119879

119887

(52)


NEL = ℎ119888

1198961198611198792

1198871205822

det

1

120573Φ (1 minus 119877119891)

NETDradic119873

(53)














119878cos = int119905int

0

Δ119899 (119905) 119889119905 minus int

1198792+int

1198792

Δ119899 (119905) 119889119905


)]

119878sin = int1198794+119905int

1198794

Δ119899 (119905) 119889119905 minus int

31198794+int

31198794

Δ119899 (119905) 119889119905

= 119878st-st



119879 + 4119905int4120591eff

)]

(54)


10

08

06

04

02

000 T

Time t (120583s)





Nor

mal

ized

carr

ier d

ensit

y Δn(t) a

nd g

ener

atio

n ra

te 119970

(a)

tint = 300 120583s

tint = 1100 120583s

tint = 2000 120583s

120

105

90

75

60

450 1000


ase Φ

(∘)

(b)




10 Conclusion








References



























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




10

08

06

04

02

000 T

Time t (120583s)





Nor

mal

ized

carr

ier d

ensit

y Δn(t) a

nd g

ener

atio

n ra

te 119970

(a)

tint = 300 120583s

tint = 1100 120583s

tint = 2000 120583s

120

105

90

75

60

450 1000


ase Φ

(∘)

(b)




10 Conclusion








References



























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




References



























































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics

















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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