Patent Application: US-79760704-A

Abstract:
the present invention relates to metrologic methodologies and instrumentation , in particular laser - frequency domain infrared photocarrier radiometry , for contamination and defect mapping and measuring electronic properties in industrial si wafers , devices and other semiconducting materials . in particular the invention relates to the measurement of carrier recombination lifetime , τ , carrier diffusivity , d , surface recombination velocities , s , carrier diffusion lengths , l , and carrier mobility , μ , as well as heavy metal contamination mapping , ion implantation mapping over a wide range of dose and energy , and determination of the concentration of mobile impurities in sio 2 layers on semiconductor substrates . the present invention provides a method and complete photocarrier radiometric apparatus comprising novel signal generation and analysis techniques as well as novel instrumental hardware configurations based on the physical principle of photocarrier radiometry . the method comprises optical excitation of the sample with a modulated optical excitation source and detection of the recombination - induced infrared emission while filtering any planck - mediated emissions . the present invention provides an instrumental method for detecting weak inhomogeneities among semiconducting materials that are not possible to detect with conventional single - ended photocarrier radiometry . the method comprises irradiating both sides of the sample with modulated optical excitation sources that are 180 degrees out of phase with respect to one another and monitoring the diffusion of the interfering , separately generated carrier waves through the corresponding recombination - induced ir emissions for pcr detection , or the use of an alternative detection scheme that monitors a sample property dependent on the carrier wave transport in the sample .

Description:
a schematic diagram of a first embodiment of a novel single - ended photocarrier radiometric instrument for laser pcr for semiconductor characterization is shown at 10 in fig3 . the excitation source is a laser 12 capable of producing photons of energy greater than the bandgap of the sample material ( hv & gt ; e g ). an acousto - optic modulator 14 is used to modulate the laser emissions resulting in a harmonic energy source or beam 16 that is directed using mirrors 18 and focused onto the sample 20 . a pair of reflecting objectives or other suitable infrared optics , such as two off - axis paraboloidal mirrors , or one paraboloidal mirror collimator and a focusing lens 22 are aligned with the focal point coincident with that of the laser beam and used to collect emitted ir photons from the sample . the collected ir emissions are focused onto a detector 24 after being passed through a filter with a narrow spectral window that ensures the planck - mediated thermal infrared emission band ( 7 – 12 μm ) will be completely excluded from the detection range , while encompassing almost the entire emission band from the free carriers , found to be below 3 μm [ 26 ]. the signal from the detector 24 is demodulated using a lock - in amplifier 26 . the entire data acquisition process is controlled using a personal computer 27 , which is also connected to an xyz motor assembly 28 to control sample positioning . while a preferred way of filtering out the planck - mediated thermal infrared emission band ( 7 – 12 μm ) is by way of the above - mentioned filter , it will be appreciated that one could also used a detector designed to have a sufficiently low sensitivity to the planck mediated thermal ir emissions but a high sensitivity to the pcr wavelengths . the optical excitation source 12 in apparatus 10 may be either a pulsed or a modulated optical excitation source 32 . while fig3 shows the system configured with the excitation source being modulated using the aom 14 , the system may be readily modified for operation in the pulsed mode whereby the aom 14 is removed and instead the laser 12 is operated in the pulsed mode producing a train of pulses triggered by its internal circuit or by use of external electronics with pulse duration in the sub - microsecond to sub - nanosecond range and repetition rate depending on the type of laser . when operating in pulsed mode , the signal processing is performed using one of three approaches : 1 ) a transient scope replaces the lockin amplifier 26 and the averaged pulse data will be stored in the scope / computer for later analysis ; 2 ) in a hybrid mode , the lock - in amplifier 26 remains and utilizes the trigger to the pulsed laser &# 39 ; s periodic firing of pulses as its reference and displays the fundamental fourier component of the time - domain signal ; or 3 ) in ultrafast applications , an optical delay line and auto - correlation signal processing are used to monitor the relaxation time of carriers . while the apparatus 10 uses a laser as the source of optical excitation , it will be understood by those skilled in the art that any other excitation source with enough energy to excite carriers in the semiconductor or optical material under examination may be used . detector 24 may be an imaging array sensor to rapidly image a large surface area . one could enlarge the energy beam and use the array to monitor 1 - d pcr signals within each pixel , with spatial resolution determined by the imager array technology . however , one could also use the array detector with a small beam and monitor the pcr emissions as a function of position . in these area imaging applications , parallel lock - in detection schemes involving capturing the full image at least 4 times per period and performing in - phase and quadrature operations , or suitable alternative lock - in schemes , will be used . a schematic diagram of an alternative configuration for the single - ended photocarrier radiometric instrument used to perform the measurements on the industrial grade silicon wafers is shown generally at 30 in fig3 a . the excitation source is a laser 32 and is capable of producing photons of energy greater than the bandgap of the sample material ( hv & gt ; e g ). the laser spot size of the exciting beam 34 is between 1 - to - 5 microns and controlled by using a reflective objective 36 . the beam intensity at the surface of the sample is between 10 - to - 30 mw . a current modulator circuitry 38 is used to modulate the laser emissions resulting in a harmonic energy source that is directed using a beam splitter 40 and focused onto the sample 42 . the sample 42 is mounted on a x - y automated stage 44 for sample positioning , mapping or scanning purposes . the reflecting objective or other suitable infrared optics 36 are aligned with the focal point coincident with that of the laser beam 34 and used to collect emitted ir photons from the sample . the collected ir emissions are directed to the spectrally matched beam splitter 40 optimized for transmission within the specific spectral emission range and focused by a suitable infrared lens 48 onto a detector 50 equipped with a suitable low - noise preamplifier and a narrow spectral window so that the combination of the spectral bandwidth of detector and filter ensures the planck - mediated thermal infrared emission band ( 7 – 12 μm ) and leakage from the optical source are completely excluded from the detection range , while encompassing almost the entire emission band from the free carriers , found to be below 3 μm [ 26 ]. the signal from the detector is demodulated using a lock - in amplifier 52 . as with the apparatus 10 in fig3 , the optical excitation source 32 in apparatus 30 may be either a pulsed or modulated optical excitation source 32 . the entire data acquisition and signal generation process is controlled using a personal computer 54 , which is also connected to a ccd camera 56 and beam splitter 58 that slides in position to locate the beam spot on the sample 42 . a customised microscope tube 60 is used to hold the various optics , reflective objective 36 and ir detector 50 . this microscope tube and laser 32 are attached through arm 64 to the focus block 66 . the photodiode 68 , beam splitter 70 and focus block 66 are used to perform auto focusing by measuring the reflection of the laser beam and adjusting the sample 42 focal distance to the reflective objective 36 . the basic components of the interferometric pcr instrument are similar to the single - ended instrument with a few significant additional components is shown generally at 80 in fig4 . the emissions from a single laser source 12 are split by a non - polarizing beam splitter 82 . one beam 84 follows the same path as the beam in the singled ended apparatus 10 ( fig3 ) and is focused on the front surface of the sample 20 . the second beam 86 is directed using a separate series of two mirrors 88 , modulated using a second acousto - optic modulator 90 , and focused onto the back surface of the sample 20 as beam 91 . a dual function waveform generator 92 is used to produce two waveforms of identical frequency but with one phase shifted 180 degrees with respect to the other . one of the waveforms is sent to the modulator 14 for the front surface excitation beam and the other to the modulator 90 for the beam 91 directed to the back surface of the sample 20 . this results in the two laser beams being modulated at identical frequencies with one having a phase lag of 180 degrees so that when one beam has maximum intensity the other has the minimum intensity and vice versa . the intensity of the beam 84 on the front surface is adjusted using a linear intensity attenuator 96 to ensure that the destructive interference of the two interfering carrier density waves results in a zero baseline pcr signal . the ir collection , data acquisition and sample positioning are identical to the single - ended pcr instrument . as with the apparatus 10 in fig3 , the optical excitation source 12 in apparatus 80 may be either a pulsed or modulated optical excitation source 12 . instrumental filtering of all thermal infrared emission contributions allows for all planck - mediated terms to be eliminated from equation ( 4 ) yielding p ⁡ ( ω ) = ⁢ ∫ λ 2 λ 1 ⁢ ⁢ ⅆ λ ⁡ [ 1 - r 1 ⁡ ( λ ) ] ⁢ { 1 + r b ⁡ ( λ ) [ 1 + ⁢ r 1 ⁡ ( λ ) ] ) ⁢ η r ⁢ w er ⁡ ( λ ) ⁢ ∫ 0 l ⁢ ɛ fc ⁡ ( z , ω ; λ ) ⁢ ⁢ ⅆ z ( 5 ) the absorption ( and , equivalently , assuming kirchhoff &# 39 ; s law is valid , the emission coefficient ) depends on the free - carrier density as [ 27 ] ɛ fc ⁡ ( z , ω ; λ ) = α irfc ⁡ ( z , ω ; λ ) = q ⁢ ⁢ λ 2 4 ⁢ π 2 ⁢ ɛ od ⁢ c 3 ⁢ n ⁢ ⁢ m * 2 ⁢ μ ⁢ δ ⁢ ⁢ n ⁡ ( z , ω ; λ ) = c ⁡ ( λ ) ⁢ δ ⁢ ⁢ n ⁡ ( z , ω ; α ) ( 6 ) for relatively low cw densities . here q is the elementary charge , ε od is the dielectric constant , c is the speed of light in the medium , n is the refractive index , m * is the effective mass of the carrier ( electron or hole ) and μ is the mobility . this allows the pcr signal to be expressed in the form p ⁡ ( ω ) ≈ f ⁡ ( λ 1 , λ 2 ) ⁢ ∫ 0 l ⁢ δ ⁢ ⁢ n ⁡ ( z , ω ) ⁢ ⁢ ⅆ z ( 7 ) f ⁡ ( λ 1 , λ 2 ) = ∫ λ 2 λ 1 ⁢ [ 1 - r 1 ⁡ ( λ ) ] ⁢ ( 1 + r b ⁡ ( λ ) ⁡ [ 1 + r 1 ⁡ ( λ ) ] ) ⁢ η r ⁢ w er ⁡ ( λ ) ⁢ c ⁡ ( λ ) ⁢ ⁢ ⅆ λ ( 8 ) the pcr signal is the integration of equation ( 7 ) over the image of the detector on the sample and thus is directly proportional to the depth integral of the carrier density in the sample . consequently , the relative lateral concentration of any defects that affect the carrier density , either by enhancing recombination or altering diffusion coefficients , can be determined by scanning the surface of the wafer with the pcr microscope . in addition , frequency scan techniques can be used with the appropriate carrier diffusion model to obtain quantitative values for the four transport parameters [ 5 ]. this quantitative technique can be combined with the lateral maps to provide quantitative imaging of the semiconductor sample . the optical excitation source 12 may be either a pulsed or modulated optical excitation source . pulsed refers to a single burst of light of short duty cycle over the laser pulse repetition period , whereas modulated is essentially a repetition of square - wave pulses and a certain frequency at approx . 50 % duty cycle or of a harmonic ( sinewave ) nature . typically , when using a pulsed excitation source one measures response as a function of time , i . e . time domain , ( essentially watching the signal decay after the short light pulse has been terminated ). for modulated experiments the surface is irradiated using a repeating excitation at a given frequency ( the modulation frequency ) and one monitors the signal response only at this frequency , i . e . frequency domain . pulsed responses can also be obtained using a lock - in amplifier referenced to the pulse repetition period , which monitors the fundamental fourier coefficient of the sample response . when using a pulsed , rather than a modulated , excitation source the pcr signal is obtained by integrating the inverse temporal fourier transform of equation 7 over the surface area ( image ) of the detector [ 14 ]. quantitative information obtained from observation of the time response of the pcr signal can then be combined with lateral maps to provide quantitative imaging of the semiconductor sample at discrete time intervals after the cessation of the laser pulse . the experimental implementation of laser infrared photo - carrier radiometry is similar to the typical ptr set - up for semiconductors [ 4 – 9 ], with the crucial difference being that the spectral window of the ir detector and / or optical filter , and the modulation frequency response of the preamplifier stage , must be tailored through spectral bandwidth matching to a combination of carrier recombination emissions and effective filtering of the planck - mediated thermal infrared emission band and of the synchronously modulated optical source . conventional ptr utilizes photoconductive liquid - nitrogen - cooled hgcdte ( mct ) detectors with spectral bandwidth in the 2 – 12 μm range . this includes the thermal infrared range , 7 – 12 μm , and only part of the electronic emission spectrum at shorter wavelengths . unfortunately , the spectral detectivity responses of mct detectors are heavily weighed toward the thermal - infrared end of the spectrum [ 28 ]. in addition , the physics of ptr signal generation involves a substantial contribution from the thermal - wave component resulting from direct absorption by the lattice and by non - radiative recombinations of photo - excited carriers [ 16 ]. the result is an infrared signal with unequal superposition of recombination and thermal emission responses with a larger weight of the thermal infrared component . from preliminary studies with several ir detectors and bandpass optical filters it has been observed that emissive infrared radiation from electronic cw recombination in si is centered mainly in the spectral region below 3 μm [ 26 ]. among those , ingaas detectors with integrated amplifiers , a visible radiation filter and a spectral response in the & lt ; 1800 - nm range , was found to be most suitable , exhibiting 100 % efficient filtering of the thermal infrared emission spectrum from si as well as maximum signal - to - noise ratio over ingaas detectors with separate amplifiers and inas detectors . therefore , infrared pcr was introduced using an optimally spectrally matched room - temperature ingaas photodetector ( thorlabs model pda255 ) for our measurements , with a built - in amplifier and frequency response up to 50 mhz . the active element area was 0 . 6 mm 2 with a spectral window in the 600 – 1800 nm range with peak responsivity 0 . 95 a / w at 1650 nm . the incident ar - ion laser beam size was 1 . 06 mm and the power was 20 – 24 mw . the detector was proven extremely effective in cutting off all thermal infrared radiation : preliminary measurements using non - electronic materials ( metals , thin foils and rubber ) showed no responses whatsoever . comparison with conventional ptr results was made by replacing the ingaas detector with a judson technologies liquid - nitrogen - cooled model j15d12 mct detector covering the 2 – 12 μm range with peak detectivity 5 × 10 10 cmhz 1 / 2 w − 1 . fig5 shows two frequency responses from a test algaas quantum well array on gaas substrate using both the mct and the ingaas detectors . the mct response is characteristic of thermal - wave domination of the ptr signal throughout the entire modulation frequency range of the lock - in amplifier . on the other hand , the pcr signal from the ingaas detector / preamplifier exhibits very flat amplitude , characteristic of purely carrier - wave response and zero phase lag up to 10 khz , as expected from the oscillation of free carriers in - phase with the optical flux which excites them ( modulated pump laser ). the apparent high - frequency phase lag is associated with electronic processes in the sample . the ptr signals were normalized for the instrumental transfer function with the thermal - wave response from a zr alloy reference , whereas the pcr signals were normalized with the response of the ingaas detector to a small fraction of the exciting modulated laser source radiation at 514 nm . regarding the well - known non - linearity of ptr signals with pump laser power [ 29 ], fig6 shows a non - linear response from the ptr system at laser powers & gt ; 5 mw . the pcr system , however , exhibits a fairly linear behavior for powers & gt ; 15 mw and up to 35 mw , within the range of the present experiments . the non - linear behavior below 15 mw is due to surface state annihilation associated with the semiconductor sample used for these measurements . unlike the readily available thermal infrared emissions from well - controlled reference samples for the purpose of instrumental signal normalization in semiconductor ptr [ 30 ], the quest for suitable reference samples for pcr is a much more difficult problem because of the absence of detector response in the thermal infrared spectral region . an indirect normalization method was introduced as shown in fig7 . furthermore , a normalization procedure using a small fraction of the excitation laser beam may also be suitable , as the ingaas detector is extremely sensitive to light intensity and its wavelength and some scattered optical source light may be allowed to leak into the detector and its intensity modulation frequency scanned to obtain the system transfer function . therefore , frequency scans on a si wafer with a large degree of signal variation across its surface were obtained from two such locations with very different responses , using both the mct and the ingaas detectors . then the amplitude ratios and phase differences between the two locations using the same detector were plotted and the amplitude ratios were further normalized at 100 khz , fig7 a . these self - normalized data are independent of the instrumental transfer function and depend only on differences among electronic parameters ( pcr ) or combinations of thermal and electronic parameters ( ptr ) at the two coordinate locations . upon superposition of the self - normalized signals it was found that both amplitude and phase curves essentially overlapped within the electronic region . this implies that both detectors monitor the same electronic cw phenomena at high frequencies and thus the instrumental normalization of the pcr signal can be performed by 1 ) using the ptr signal from a high - electronic - quality reference si wafer , normalized by a simple one - dimensional thermal - wave frequency scan of a homogeneous metallic solid [ 5 ]; 2 ) mathematically extracting the electronic component of the ptr signal [ 5 ] and adjusting the pcr signal to this component ; and 3 ) using the pcr amplitude and phase frequency correction functions for all other signal normalizations . this indirect scheme was proven satisfactory . it will be seen in part b ) of this section , however , that the small differences in the self - normalized high - frequency signals of fig7 are indicative that the thermal - wave component of the ptr signal can be present even at the highest modulation frequencies and , without independent knowledge of the electronic properties of the reference semiconductor , it can affect their “ true ” values significantly , a conclusion we also reached about photomodulated thermoreflectance [ 22 ]. normalizing the pcr signals with a small scattered portion of the incident optical source remains by far the easiest and most straightforward method , provided other instrumental complications do not arise . a small area of the back surface of the si wafer which was used for the signal linearity studies was very slightly damaged through gentle rubbing with sandpaper . pcr frequency scans were obtained from outside and inside the region with the back - surface defect . then line scans and 2 - dimensional images at fixed frequency were obtained covering the defect area . the wafer was suspended in air using a hollow sample holder , or was supported by a black rubber or by a mirror backing . fig8 shows pcr frequency scans for all three backings . the pcr technique resolves the amplitudes from the three backings in the order s m & gt ; s a & gt ; s r , ( m : mirror , a : air , r : rubber ). to understand the origins of the signal changes in the presence of a backing support , a highly reflecting aluminum - foil - covered backing was placed at a variable distance from the back - surface of the si wafer across from an intact region and pcr signals were monitored , fig9 . it is observed that the pcr amplitude remains constant for all three frequencies up to a distance of ˜ 1 mm away from contacting the back surface , where it starts to increase . the curves are normalized to their value on the surface to show that the rate of increase is independent of frequency . the pcr phase remains essentially flat throughout . to determine the origin of pcr signal variations with backing ( whether due to ir photon internal reflections or backing emissivity changes [ 31 ]) the laser was turned off and a mechanical chopper was placed at some distance away from the ir detector . the lockin amplifier signals from the ingaas detector nearly vanished at ˜ 5 μv , a baseline value that remained constant for all combinations of wafer , chopper , and the three substrate materials . these dc emissivity experiments with the ingaas detector in place are clear evidence that its spectral bandwidth lies entirely outside the thermal ir ( planck ) emission range of the si wafer with or without substrate . therefore , the pcr amplitude enhancement for mirrored and rubber backings , fig8 a , is consistent with simple reflection of exiting ( transmitted ) cw - generated ir photons at the surface of the backing , with no possibility for thermal infrared emissivity contributions from the backing itself . the order of the pcr amplitude curves indicates that the surface of highest reflectivity ( mirror ) yields the strongest signal . it appears the si - air interface is a more efficient back - scatterer of ir photons than the si - black rubber interface , where these photons are expected to be more readily absorbed by the rubber . from eq . ( 5 ) it is expected that the ratio of pcr signals with mirror and black rubber backings should be approximately [ 2 + r 1 ( λ )]/[ 1 + r b ( λ )[ 1 + r 1 ( λ )]]≈ 1 . 94 . the measured ratio from the low - frequency end in fig8 a is 1 . 8 . fig1 shows line scans with the excitation laser beam scanning the front ( polished ) surface of a 20 ωcm p - type si wafer and the ir detector on the same side . based on the backing results , for maximum signal strength the sample was resting on a mirror . both ptr and pcr amplitude and phase scans were obtained and both show sensitivity to the deep defect on the back surface scratched region . however , at 100 khz imaging can be performed only with the pcr signal . at all three selected modulation frequencies , the pcr amplitude decreases when the laser beam scans over the defect region , consistent with the expected cw density decrease as the back - surface defect efficiently traps carriers and removes them from further diffusion and potential radiative recombination . the pcr phase scan remains essentially constant at 10 hz , fig1 b , as the diffusion - wave centroid is solely determined by the ac carrier - wave diffusion length [ 12 ] l a ⁢ ⁢ c ⁡ ( ω ) = d * τ 1 + ⅈ ⁢ ⁢ ω ⁢ ⁢ τ ( 9 ) where τ is the lifetime and d * is the ambipolar carrier diffusion coefficient . this particular wafer was measured to have τ ≅ 1 ms and d *≅ 12 cm 2 / s , which yields an | l ac ( 10 hz )|≅ 1 . 1 mm . therefore , the cw centroid lies well beyond the thickness of the wafer (˜ 630 μm ) and no phase shift can be observed . at the intermediate frequency of 1360 hz , | l ac |≅ 373 μm , well within the bulk of the wafer . in this case , a phase lead appears within the defective region . this occurs because the cw spatial distribution across the body of the wafer in the defective region is weighed more heavily toward the front surface on account of the heavy depletion occurring at , and near , the back surface . as a result , the cw centroid is shifted toward the front surface , manifested by a phase lead . finally , at 100 khz , | l ac |≅ 44 μm . nevertheless , fig1 a shows that there is still pcr amplitude contrast at that frequency , accompanied by a small phase lead , fig1 b . for the ptr scans , fig1 c shows that the overall amplitude is controlled by the cw component at 10 and 1360 hz , and there is a small contrast at 100 khz . the ptr phase contrast within the region with the back - surface defect first appears as a lag at the lowest frequency of 10 hz , as expected from a shift away from the front surface of the diffusion - wave centroid in the presence of a remote thermal - wave source which is added to the combined ptr signal . at that frequency the thermal - wave diffusion length [ 14 ] is l t /( ω )=( 2d t / ω ) 1 / 2 ≅ 1 . 7 mm , that is , the back surface is in thermal conductive communication with the front surface . therefore , the thermal wave , rather than the carrier wave , controls the overall diffusion - wave ptr behavior of the si wafer at 10 hz . at 1360 hz , however , l t ≅ 148 μm , therefore , there is no thermal contact with the back surface . the only signal component affected by the remote defect is the cw , and the phase behaves as in the pcr case , exhibiting a net lead within the defective region . at 100 khz there is no ptr phase sensitivity to the defect ; only a vestigial amplitude contrast , fig1 c , d . to maximize pcr and ptr imaging contrast , differences in amplitudes and phases as a function of frequency were obtained outside and inside the defective region . it is with the help of this type of analysis that the 1360 hz frequency was chosen for both techniques as one with the highest contrast in phase ( but not in amplitude ). it is clear that while the ptr contrast is generally higher at low frequencies due to the cooperative trends in both thermal - wave and carrier - wave components , however , pcr imaging contrast becomes superior above ca . 1 khz and retains its contrast even at the highest frequency of 100 khz . fig1 shows images of the back - surface defect obtained through front - surface inspection using both techniques at the optimum contrast frequency of 1360 hz . fig1 shows the same scan at 100 khz . at this frequency , the ptr image is dominated by noise and is unable to produce any contrast between the intact and defective regions , whereas the pcr image clearly shows the highest spatial resolution of the back - surface defect possible . the pcr phase , fig1 b , shows details of the central defect as well as the radially diverging defect structures at the base of the central defect , like a “ zoomed in ” version of the 1360 hz image , fig1 b . both pcr images clearly reveal internal sub - structure of the central defect , which was invisible at 1360 hz . in a manner reminiscent of conventional propagating wavefields , image resolution increases with decreasing carrier wavelength , | l ac |. similar images to fig1 and 12 were obtained with air or rubber backing of the same wafer , with marginally diminished detail and contrast . the contrast for pcr imaging at 100 khz , fig1 b , is about 11 % for amplitude ( fig1 a ) while the phase difference is only 1 ° ( fig1 b ). the very high sensitivity of pcr imaging to defect identification is apparent : despite this very small variation in phase , the defect can be clearly delineated . in the case of ptr at 100 khz , the contrast for amplitude is about 28 % ( taking the sharp peak in fig1 c into account ). the phase difference is about 10 °. an examination of fig1 c and 10 d at 100 khz shows that this “ higher contrast ” is caused by fluctuations of the signal , as the ptr signal - to - noise ratio ( snr ) is relatively poor , resulting in the disappearance of the back - surface defect from the images fig1 c , d . the pcr images exhibit much higher snr and clearly reveal the defects structure . under front - surface inspection and precise depth profilometric control by virtue of the pcr modulation - frequency - adjustable carrier - wave diffusion length , eq . ( 9 ), fig1 and 12 show for the first time that with today &# 39 ; s high - quality , long - lifetime industrial si wafers , one can observe full images of sharp carrier - wave density contrast due to underlying defects very deep inside the bulk of a si wafer . specifically , high frequency pcr imaging reveals so far unknown very long - range effects of carrier interactions with deep sub - surface defect structures and the detrimental ability of such structures to decrease the overall free photoexcited - carrier density far away from the defect sites at or near the front surface where device fabrication takes place . this phenomenon may be important toward device fabrication improvement through careful selection of substrate wafers with regard to deep bulk growth and manufacturing defects which were heretofore not associated with device performance . further pcr imaging experiments with shorter lifetime si wafers have shown that it may be beneficial to use lower quality starting substrates in order to avoid the full effects of deep sub - surface defects on the electronic quality of the upper ( device - level ) surface . the structure of eq . ( 4 ), the expression for the total emitted power from a semiconductor crystal at the fundamental frequency across the field of view of the ir detector , shows depth dependence of the spatial integrals on the equilibrium ir emission coefficient ε o ( λ ) of the semiconductor . if this parameter is larger than 1 – 5 cm − 1 , it introduces a weighting factor e − εo ( λ ) z under the integral signs of the compact expression for the total ir emission , eq . ( 4 ), as well as for pure pcr emission , eqs . ( 5 ) and ( 7 ). to estimate the effect of such a factor on the pcr signal , especially in the case of low - resistivity , high - residual infrared absorption si wafers , a simulation was performed using the pcr eq . ( 7 ) in the three - dimensional form p ⁡ ( r , ω ; λ 1 , λ 2 ) ≈ ⁢ ∫ λ 2 λ 1 ⁢ [ 1 - r 1 ⁡ ( λ ) ] ⁢ ( 1 + r b ⁡ ( λ ) ⁡ [ 1 + r 1 ⁡ ( λ ) ] ) ⁢ η r ⁢ w er ⁡ ( λ ) ⁢ c ⁡ ( λ ) ⁢ ⅆ λ ⨯ ∫ 0 l ⁢ δ ⁢ ⁢ n ⁡ ( r , z , ω ) ⁢ ⅇ - ɛ o ⁡ ( λ ) ⁢ z ⁢ ⁢ ⅆ z ( 10 ) the equation for δn ( r , z , ω ), the 3 - d extension of δn ( z , ω ) is the solution to the photo - carrier - wave boundary - value problem . it was obtained from ref . [ 14 ], chap . 9 , eq . ( 9 . 106 ), and it is reproduced here : δ ⁢ ⁢ n ⁡ ( r , z , ω ) = η q ⁢ p o ⁢ α 2 ⁢ π ⁢ ⁢ hvd * ⁢ ∫ 0 ∞ ⁢ ⅇ - k 2 ⁢ w 2 / 4 ( α 2 - ξ e 2 ) ⁢ [ ( g 2 ⁢ g 1 - g 1 ⁢ g 2 ⁢ ⅇ - ( ξ e + α ) ⁢ l g 2 - g 1 ⁢ ⅇ - 2 ⁢ ξ e ⁢ l ) ⁢ ⅇ - ξ e ⁢ z - ⅇ - a ⁢ ⁢ z + ( g 2 ⁢ g 1 - g 1 ⁢ g 2 ⁢ ⅇ - ( ξ e + α ) ⁢ l g 2 - g 1 ⁢ ⅇ - 2 ⁢ ξ e ⁢ l ) ⁢ ⅇ - ξ e ⁡ ( 2 ⁢ l - z ) ] ⁢ j o ⁡ ( kr ) ⁢ k ⁢ ⅆ k ( 11 ) where ⁢ ⁢ g 1 ⁡ ( k ) ≡ d * α + s 1 d * ξ e ⁡ ( k ) + s 1 ; ⁢ g 2 ⁡ ( k ) ≡ d * α - s 2 d * ξ e ⁡ ( k ) - s 2 ( 12 ⁢ a ) with ⁢ ⁢ g 1 ⁡ ( k ) ≡ d * ξ e ⁡ ( k ) - s 1 d * ξ e ⁡ ( k ) + s 1 ; ⁢ g 2 ⁡ ( k ) ≡ d * ξ e ⁡ ( k ) + s 2 d * ξ e ⁡ ( k ) - s 2 ( 12 ⁢ b ) ξ e ( k )≡√{ square root over ( k 2 + σ e 2 )} ( 12c ) here , k stands for the hankel variable of radial integration , w is the gaussian laser beam spotsize , s 1 and s 2 are the front - and back - surface recombination velocities , l is the thickness of the semiconductor slab , α is the optical absorption coefficient at the excitation wavelength λ vis = c o / v . η q is the quantum yield for optical to electronic energy conversion and p o is the laser power . the carrier wavenumber is defined as σ e ⁡ ( ω ) ≡ 1 + ⅈ ⁢ ⁢ ωτ d * ⁢ τ = 1 l ac ⁡ ( ω ) ( 13 ) in the simulations that follow and in the theoretical fits to the experimental data , the variable r was integrated over the surface of the ir detector [ 4 ]. fig1 shows simulations of the pcr frequency dependence for p - si of ( what amounts to ) different resistivity with the equilibrium ir absorption coefficient as a ir - wavelength - independent ( average ) parameter . from kirchhoff &# 39 ; s law , ε o = α iro . the curves show a decrease in amplitude , especially at low frequencies , in the carrier - diffusion - wave thin regime (| l ac ( ω )|& gt ; l ), as emissions throughout the bulk of the crystal are gradually impeded with increasing background carrier density ( and thus ir absorption coefficient ) due to self - absorption of the ir recombination photons by the background free carrier - wave density . at high frequencies , in the carrier - diffusion - wave thick regime (| l ac ( ω )|& lt ;& lt ; l ), little attenuation of the backward emitted ir recombination photon flux occurs because the ir - opaque sub - surface layer involved in the cw - generated emission is very thin . therefore , all amplitude curves converge . pcr phase lags show sensitivity at high frequencies ; they decrease with increasing frequency because the contributing cw centroid moves closer to the front surface with increasing ir opacity of the semiconductor . fig1 shows that for typical α iro ranges of 1 – 2 cm − 1 [ 32 ] the effect of self - reabsorption of ir photons due to background free carrier - wave densities is minimal and therefore the approximate eqs . ( 4 ), ( 5 ), ( 7 ), and ( 8 ) are justified . the pcr image contrast of fig1 and 12 can , in principle , be quantified by use of the cw term in eq . ( 7 ), appropriately modified to accommodate the defective region : δ ⁢ ⁢ p ⁡ ( ω ) ≈ f 2 ⁡ ( λ 1 , λ 2 ) ⁡ [ ∫ 0 l ⁢ δ ⁢ ⁢ n ⁡ ( z , ω ) ⁢ ⁢ ⅆ z - ∫ 0 l ⁢ δ ⁢ ⁢ n d ⁡ ( z , ω ) ⁢ ⁢ ⅆ z ] ( 14 ) where δp ( ω ) is the difference in signal between the intact and defective regions . this is a complex quantity , so it can be separated out into amplitude and phase components . the apparent simplicity of this expression is due to the fact that the sub - surface defects considered here are on the back surface of the wafer and their presence mostly impacts the value of s 2 in eq . ( 11 ), while the bulk parameters and the terms comprising the prefactor f ( λ 1 , λ 2 ), eq . ( 8 ), remain essentially unaltered , including c d ( τ )≈ c ( λ ) for a thin damage layer in an otherwise homogeneous semiconductor . if these conditions are not fulfilled , then a more complete expression of the carrier recombination related emissions must be used to quantify pcr contrast due to distributed sub - surface electronic defect structures . the mild mechanical defect on the back surface of the p - type si wafer that generated the images of fig1 and 12 proved to be too severe for our sensitive ingaas photodetector : upon scanning the affected surface the pcr signal vanished within the region of the defect , apparently due to the highly efficient trapping of the photogenerated free carriers by the high density of near - surface electronic defect states . therefore , a different region of the same wafer was chosen to create a visually undetectable defect by simply touching the back surface of the wafer with paper . then both pcr frequency scans were performed on both sides of the material , outside and inside the defect region . fig1 shows the pcr frequency scan amplitudes and phases for all four spots , as well as theoretical fits to the experimental data . the signal normalization was performed by extracting the cw component of the ptr signal , i . e . the depth integral over δn ( r , z , ω ), eq . ( 11 ), associated with the prefactor c p in the front intact region , and making it the reference pcr signal for the same region . the thus obtained pcr amplitude and phase transfer functions were subsequently used for all other measurements . the d * values those outside the defect remain constant for both sides of the wafer , however , the d * value from the back inside the defect region is relatively low . the higher sensitivity of the ingaas detector to the electronic state of the inspected surface is probably responsible for this discrepancy , as the theoretical phase fit is poor at high frequencies (& gt ; 1 khz ) within that region , an indication of near - surface depth inhomogeneity of transport properties . fig1 and the resultant theoretical fits show that pcr signals are very sensitive to the electronic state of the probed semiconductor surface and bulk . the ability to measure the diffusion coefficient d * also allows for the calculation of the conductivity mobility , μ , through the use of the einstein relation d =( kt / q ) μ where k is the boltzmann constant , t is the temperature , and q is the elementary charge . the measured conductivity mobility of a n - type silicon wafer with resistivity ρ = 10 − 15 ωcm , n ˜ 8 × 10 14 cm − 3 , and a 980 angstrom thermally grown oxide layer is presented in fig1 . the temperature dependence of the conductivity mobility was found to have a relationship similar to that measured using electrical techniques [ 33 ]. in an embodiment of the method the semiconductor material is suitably and rapidly heated by a contacting thermal source and the pcr signal ( controlled by the thermal emissions from the recombination - induced infrared emission ) is monitored at a suitable pcr frequency such that thermal emissions occur from a defect or impurity state in the material produce a peak in the temperature scan when the material temperature is such that the thermal energy forces trapped carriers to evacuate their trap states at a rate simply related to the pcr frequency . in this manner the energy of the impurity or defect deep level is extracted from the pcr peaks in a series of temperature scans at fixed frequencies using a simple boltzmann factor , and the pcr signal magnitude is a measure of the occupation density of the level . alternatively , the pcr frequency is scanned for different ( fixed ) temperatures and the energy of the level is obtained from an arrhenius plot of the logarithm of the ( modulation period p max of the lock - in in - phase signal where a pcr peak occurs at each temperature t j times t j 2 ) vs . 1 / t j . this metrology method can be suitably called pcr deep - level thermal spectroscopy and can be used for identification of electronic impurity species and / or contamination ions and for estimating their concentration in a semiconductor . the demonstrated sensitivity to surface defects [ 19 ] allows for the use of pcr to monitor ion implant dose . fig1 shows the pcr amplitude dependence as a function of dose for ( 100 ) oriented p - type silicon wafers with a thermally grown oxide layer of 200 å implanted at room temperature at an angle of 7 ° to suppress channelling with fluences from 10 10 to 10 16 cm − 2 with the following species and energy combinations : 11 b + ( 10 kev , 50 kev , 180 kev ), 75 as + ( 80 kev , 150 kev ), 31 p + ( 30 kev , 80 kev , 285 kev ) and bf 2 + ( 30 kev , 50 kev ). inspection of fig1 ( a ) through ( d ) shows that the pcr signal dependence on dose can be broken down roughly into four regions with the actual dose defining the transition of each region depending on the mass of the implanted ion . in region i the amplitude decreases rapidly with increasing dose as the degree of damage to the lattice structure increases and electronic integrity of the surface region is compromised resulting in carrier trapping , increased surface recombination velocities , and decreased diffusivities and lifetimes . in region 11 the electronic sensitivity to dose begins to saturate and the pcr amplitude decreases slightly as the size of the damaged region increases with dose [ 34 ]. the production of amorphous phase si brings the onset of sensitivity to the optical properties in region iii resulting in another dose range of rapidly decreasing amplitude as the absorption coefficient increases and results in a greater percentage of the photogenerated carriers being created in a region of compromised electronic integrity . for the more massive as + implants a fourth region is visible as the onset of optical saturation occurs near 10 16 cm − 2 and the pcr sensitivity to dose again experiences a rapid decline . saturation of the electrical sensitivity prior to the onset of the optical sensitivity is a result of the dependence of the carrier - diffusion - wave on electrical percolation paths and the dependence of the optical properties of the sample on relatively large volumes [ 35 ]. a key feature of the results presented in fig1 is the monotonic behavior over a large range of implant dose . the only exceptions for the wafers studied were the b + and p + implanted samples that exhibited slightly non - monotonic behavior in the 5 × 10 12 to 10 13 cm − 2 region at intermediate energy levels and the as + implanted samples that had non - monotonic behaviour above 5 × 10 15 cm − 2 . this monotonic behaviour is an advantage over photothermal techniques such as photomodulated reflectance which exhibit non - monotonic signals over this dose range due to the competing thermal - wave and carrier - wave components that generate them [ 37 ]. several other features of note in fig1 are the pcr dependence on energy and on excitation wavelength . in general , the pcr amplitude decreases with implant energy as the depth of the damaged region increases and consumes a greater portion of the photo - generation volume . similarly , for a given energy , the pcr amplitude decreases with the excitation wavelength as the increasing absorption coefficient results in a photo - generation volume closer to the implanted region . both of these phenomena are the result of a modification of the weighting of the contributions to the pcr signal from the ( damaged ) surface region and the bulk of the sample [ 38 , 39 ]. this increased dependence on the damaged region of the sample results in an increasing sensitivity of the pcr signal to dose with decreasing absorption depth ( i . e . wavelength ) of the excitation source [ 40 ]. all previous photothermal and optical approaches to characterizing semiconductors have been based a single excitation source focused onto one surface of the sample . this results in a baseline signal from the homogeneous bulk that can be large relative to the contrast signal ( i . e . signal variation ) from any inhomogeneities . recently , two approaches have been taken in efforts to eliminate this baseline signal from photothermal experiments in order to improve the sensitivity of the instrument . a purely thermal - wave interferometric approach has been applied to photo - pyroelectric measurements of trace gas elements [ 41 , 42 ]. also , the introduction of a dual - pulse wave form into a single excitation beam has been used to create a differential technique that amounts to a common - mode reject scheme [ 43 , 44 ]. both approaches have shown that suppression of the baseline signal improves the sensitivity and dynamic range compared to the conventional single - ended equivalent . the double - ended interferometric invention , while similar in principle to the ppe interferometric approach , is a completely novel approach to the characterization of solid - state samples , and in particular semiconductors , relying on the interference of the separately generated carrier - density waves in the sample as opposed to other suggested photothermal interferometric techniques which utilize the interference patterns of optical beams interacting with the sample [ 45 ]. the instrumentation for interferometric pcr of a semiconductor sample is described above with respect to fig4 . two laser beams modulated at identical frequencies with one phase shifted 180 degrees with respect to the other are focused onto opposite sides of the sample to generate two separate carrier - density waves . the pcr signal is described by equation ( 10 ) with the carrier density δn ( r , z , ω ) being the solution to the photo - carrier - wave boundary - value problem similar to equation ( 11 ) but with two excitation sources . the intensity of the laser focused on the front surface of the sample is adjusted to ensure destructive interference of the two waves and thus a zero baseline signal . as the wafer is scanned laterally any inhomogeneities or defects present in the material alter the diffusion of one or both of the photogenerated carrier density waves which no longer interfere destructively and thus produce a non - zero signal . this approach of using a zero baseline signal improves the dynamic range of the instrument and the sensitivity to inhomogeneities and thus provides enhanced imaging contrast of lateral contamination or other inhomogeneities compared to the single - ended pcr . from these scans , maps can be produced of any inhomogeneities or defects that affect carrier density , either by enhancing recombination or altering diffusion coefficients . the theoretical model includes an effective carrier diffusion model to obtain quantitative values for the electronic and transport parameters , and combining quantitative results of the theoretical model with maps produced from spatially scanning across at least one surface of the material to provide quantitative imaging of the material . besides the theoretical fits , maps produced from scanning at least one surface of the material can be combined with calibration curves to provide quantitative imaging of the material . the calibration curves are obtained by measuring the pcr signal from reference samples with known composition , structure and material properties . the calibration curves allow for direct correlation between the pcr signals and the material property and / or industrial process to be monitored . this approach of using a zero baseline signal improves the dynamic range of the instrument and the sensitivity to inhomogeneities and thus provides enhanced imaging contrast of lateral contamination or other inhomogeneities compared to the single - 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