Optical properties of phosphorus-dopedpolycrystalline silicon layers
`G. Lubberts, B. C. Burkey, F. Moser, and. A. Trabka
`Research Laboratories, Eastman Kodak Company, Rochester, New York 14650
`
`(Received 4 June 1981; accepted for publication 6 July 1981)
`
`The index ofrefraction and the absorption coefficient of low-pressure chemical vapor deposited
`polycrystalline silicon (poly Si) implanted with various doses of phosphorus were obtained by
`meansoftransmittance and reflectance measurements. Therefractive index ofpolySi is relatively
`insensitive to doping in thevisible region of the spectrum and agrees quite well with published
`values for single-crystal Si. The absorption coefficientof thesefilms decreases with doping in the
`visible. Nevertheless, the absorption coefficient corresponding to the highest-doped film
`(N=3 x 107°/cm’) is almost twice the published valuesfor lightly doped single-crystal Si. In the
`infrared, the index of refraction decreases systematically and the absorption coefficient increases
`systematically with doping. This dependenceofthe optical parameters on dopingis attributed to
`the presence offree carriers. A free-carrier dispersion/absorption model is applied to the
`measured refractive index in the region 1.2-1.8 zm andto the reflectance measurements in the
`region 2.5—7.5 zm. The model gives carrier concentrations for the two types of measurements
`which differ by 50%. Possible explanationsfor this difference are discussed. The carrier
`concentrations and mobilities obtained from optical and Hall-effect measurements are compared.
`
`PACSnumbers: 78.20.Dj, 78.65.Jd, 85.30.De
`
`|. INTRODUCTION
`
`Heavily doped (=~ 10?°/cm*)polycrystalline silicon
`(poly Si) layers are often used as transparentelectrodesin
`metal-oxide-semiconductor (MOS) imagingarrays.’ Since
`the optical transmission of such layers determines the spec-
`tral quantum efficiency of these devices,it is of interest to
`determinethe index of refraction n, and the absorption coef-
`ficient a as functions of wavelength.” The active area ofa
`MOSphotosensing device is usually covered with a multi-
`layer structure, doped poly Si being one ofthe layers.
`A numberofpapersin theliterature deal with the mod-
`eling of the transmittance of such multilayer structures.*~°
`In these papers, the index of refraction and the absorption
`coefficient of single-crystal Si are used in the computa-
`tions.’”-? Kuhl, Schlotterer, and Schwidefsky have shown
`that in particular the extinction coefficient ofsingle-crystal
`Si is significantly different from that of undopedpoly Si.'° In
`a previous publication,'' the index of refraction and the ab-
`sorption coefficient of poly Si, undoped and doped with
`phosphorus atomsto 2.9 x 107°/cm?, were determinedin the
`range 0.4-0.75 zm. Using these optical constants, we calcu-
`lated the transmittance of a four-phase charge-coupled de-
`vice. The calculated transmittance agreed quite accurately
`with the experimental measurements."’
`In this paper we measuredthe optical constants of low-
`pressure chemical vapor deposited (LPCVD)poly Siin the
`range of 0.4~1.8 zm with doping density as parameter. The
`optical parameters of poly Si varied systematically with
`phosphorus doping. Theindex ofrefraction of these films
`has been determined by meansofthe interference extremaof
`both transmittance and reflectance measurements. Mea-
`surementof the minimum in transmittancein the infrared
`allowed the determination of the index of refraction and the
`thickness!” of the undoped poly Si layer where absorptionis
`negligible, as discussed in Sec. III. A new method using
`contours of constant 7,,,, and contours of constant
`
`Trin + Rmax iS introduced, which allows the determination
`of the index of refraction and the extinction coefficient of
`doped layers where absorption is present. The thickness of
`the poly Si layers is then obtained from the interference
`condition.
`The dependenceof the absorption coefficient on wave-
`length was obtained from a numerical fit of transmittance
`measurementsby use of the matrix method of Heavens. '?
`(Scattering of light in these films is negligible in the wave-
`length rangeofinterest.) The systematic changesin the index
`of refraction and in the absorption coefficient with doping in
`the nearinfrared are attributed to free-carrier interaction
`with the radiation. Mishima, Hirose, and Osaka"* haveesti-
`mated the index of refraction for heavily doped poly Si near
`the band-gap energy. Their index values showlittle disper-
`sion and are assumed constant for wavelengths larger than
`1.2 zm. Their results do not agree with our data. Dispersion
`in the index ofrefraction is expected in the infrared for heav-
`ily doped polySi films as a result of the presence offree
`carriers.
`.
`;
`.
`.
`A free-carrier modelis used in the 1.2-1.8 zm region. A
`linear relationship between nj and A ” predicted by the model
`is verified experimentally. The slope of this curve yields the
`carrier concentration, and the extrapolation to the nj axis
`gives the index of refraction of the undoped poly Si. The
`application of the same free-carrier model to obtain a nu-
`merical fit to the measured reflectance of these layers in the
`2.5-7.5 um range gave a 50% lowercarrier concentration,
`suggesting that the free-carrier modelis not a complete de-
`scription of these heavily doped layers.
`(Il. EXPERIMENTAL PROCEDURES
`A. Sample preparation
`The preparation of the poly Si layers for our optical
`studies is schematically shownin Fig. 1. Poly Si layers were
`deposited on both sides of fused-quartz (SiO,) substrates in
`an LPCVDsystem by decomposition ofsilane gas at 620 °C.
`
`6870
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`J. Appl. Phys. 52(11), November 1981
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`0021-8979/81/116870-09$01.10
`
`6870
`© 1981 AmericanInstitute of Physics
`HANWHA1059
`
`HANWHA 1059
`
`

`

`implant
`oe
`
`(a)
`poly
`ZZfused quartz
`EE Ze Sj
`
`FOSSSeed
`
`FIG. 1. Preparationofpoly Si layers for optical studies. (a) Deposit LPCVD
`poly Si on both sides of a fused-quartz wafer. (b) Phosphorus implant one
`side. (c) Anneal 950 °C, 30 min in N). (d) Pattern poly Si on both sides of
`wafer,
`.
`
`The substrates were polished and cleaned quartz wafers3 in.
`in diameter and 0.020in. thick. Subsequently, the poly Si
`film was implanted with 150-keV phosphorus,resulting in
`= 90% incorporation in the poly Si layer. Implant doses of
`1x 10'5, 5x 10'5, 1 10'®, and 2 x 10'°/cm? were used.
`Both the implanted and the unimplanted samples were an-
`nealed at 950 °C in nitrogen for 30 min, resulting in a nearly
`homogeneousdoping distribution. Finally, a pattern was
`etched in the poly Si on bothsides of the substrate [Fig. 1(d)].
`Since only one side was implanted, we have a doped and an
`undopedpoly Si layer of equal thickness on each substrate.
`Theresulting poly Si layers are smooth and uniformly 0.27
`pm thick. The thickness ofthe undoped layers was accurate-
`ly determined from a measurementofthe minimumin trans-
`mittancein the infrared, as discussed in Sec. III. The thick-
`ness was independently measured, with excellent agreement,
`by a method which involved the motion of a mechanical
`stylus over a step in the poly Si layer.
`
`B. Reflectance and transmittance measurements
`
`Optical transmittance andreflectance spectra were ob-
`tained on all poly Sifilms. In all cases the light was incident
`on the polySi film. Several instruments were used to cover
`the range 0.4-7.5 4m andto provide a check on the spectral
`and photometric accuracy. To obtain the thickness, the in-
`dex of refraction, and the absorption coefficientof these lay-
`ers, we measuredspecularspectra (collimated light incident,
`collimated transmission or reflection measured) on Cary
`14RI and Cary 17D recording spectrophotometersin the
`region 0.4-2.0 um. The measured transmittance andreflec-
`tance spectra at 0.4-2.0 um are shownin Fig. 2 for undoped
`poly Si. The spectra of doped poly Si are similar to those
`shownin Fig. 2, with the extremashifted to shorter wave-
`lengths in the infrared. Comparisonof specular andtotal
`transmittance measurements madeat 0.4-2.0 um showsthat
`scattering in thesefilmsis negligible. To study theeffect of
`interaction of infrared radiation with free carriers, we mea-
`sured reflectance at 2.5-7.5 4m using the Beckman IR20A
`recording spectrophotometer. The measuredreflectance of
`the poly Si films implanted with various doses of phosphorus
`is shownin Fig. 3. The reflectance is characterized by a mini-
`
`
`
`Wavelength (jm)
`
`aounjiwsudsy
`
`Reflectance
`
`FIG.2. Reflectance and transmittance spectra of 0.27-~m-thick, undoped
`poly Si layers on a fused-quartz substrate.
`
`mum arising from free-carrier dispersion. Note that the dop-
`ing in Curve A is too low for a minimum to occurin the 2.5-—
`7.5 wm range. Reflectance measurements were not made be-
`yond 7.5 um becausethe quartz substrate showsa strong
`dispersion band at 9 zm.
`
`Ul. THEORY
`A. Matrix method—Nonabsorbing,semi-infinite
`substrate
`
`The optical transmittance andreflectance of a multi-
`layer structure for normally incident radiation are readily
`calculated by a matrix method using Fresnel coefficients. '?
`Although the method is applicable to any numberoflayers,
`wewill apply this method to a single layerofpoly Si of thick-
`ness d, on a semi-infinite layer of quartz. Subsequently,in
`Sec. III B, the method will be modified to arrive at the calcu-
`lated transmittance for our samples, which consist of a
`quartz substrateoffinite thickness. As Fig. 4 shows,the radi-
`ation of wavelength A is incident on the poly Si from theair
`side. The indexofrefraction ofair is Nj = ny = 1, the index
`of dopedpoly Si is complex andis given by NV, =n, —Jk,
`and the index of fused quartz is N, = 7, a real quantity.'5
`Thedispersion relation for n, given by Malitson wasused.'°
`
`1.0
`
`0.8
`
`2 062 0.
`8
`g
`304e
`
`0.2
`
`D
`
`Cc
`A
`
`B
`
`oD
`
`3
`
`4
`
`6
`5
`Wavelength (ym)
`
`7
`
`8
`
`FIG.3. Infrared reflectancespectra of 0.27-42m-thick, phosphorus-implant-
`ed poly Si layers on a quartz substrate. Implanted dose: (A) 1 x 10'5/cm?, (B)
`5x 10!5/em?, (C) 1 x 10'*/em?, (D) 2x 10'°/cm?.
`
`6871
`
`J. Appi. Phys., Vol. 52, No. 11, November 19814
`
`Lubberts et a/.
`
`6871
`
`

`

`flectance R, given by Eq.(7) do not directly predict the trans-
`mittance and reflectance of our samples. This case has been
`treated by Goodman.’” His methodinvolves expressing the
`total transmittance Tofthe sample in termsof the quantities
`T,, T,, R{, and R, (Fig. 5), using the assumption that the
`substrate thickness is sufficiently large and/or nonuniform
`so that light traversing it loses phase coherence andintensi-
`ties can be added. Let R, be the reflectanceat the poly Si
`surface andlet 7, be the transmittance into the substrate
`whenthe radiation is normally incident on the air—poly Si
`interface as shownin Fig. 5. Similarly, R | and T} are the
`comparable quantities for light normally incident on the
`poly Si from the substrate side. Furthermore, we let R, and
`T, be the reflectance and transmittanceat the substrate-air
`interface for radiation normally incident from the substrate
`side.
`
`The transmittance T can be calculated by considering
`the progress ofa waveasit is incidenton the poly Si from the
`air side. If 7, is the incident intensity, [,7, is transmitted
`through the polySi layer into the quartz substrate and J,R, is
`reflected. Since the substrate is nonabsorbing, 7, is also
`incident on the back surface or quartz-air interface. At this
`surface a fraction /,7',R, is reflected and JT, T,is transmit-
`ted. Thefraction reflected becomesincident on the poly Si
`layer from the substrate side, and /,7,R,T | is transmitted
`through the poly Si, becomingpartofthetotal reflectance;
`ITRR §
`is reflected back into the substrate of which
`I,T,R,R | T, is transmitted out of the substrate, becoming
`part of the total transmittance.
`By continuingthis process, the total transmitted and
`reflected intensities are obtained by adding all the
`contributions
`
`Typ =4(T\T, + T\T,RoR | + 7T)T,R,R {RR j...)
`(8)
`
`Nat tN
`Tp =1(R, + T,R.T) + 7,R.R (RT;
`+T7,R RR Ri RTs +...).
`bo Nt
`Summingtheinfinite series, we obtain
`Nm t Nin
`
`p= NeatNmform =1,2, (3)
`
`and
`
`form=12, 1
`
`(9)
`
`and
`
`Sm, 1 =(27/A Ny — 14m — 1
`
`form =1,2.
`
`(5)
`
`[ratio
`fe,
`LLLes
`Fis
`
`Since the radiation exits in the semi-infinite quartz medium,
`we have no negative-going wave in this medium and E >
`= 0. Furthermore, to implement these equations, weset
`5, = 0. The transmittance andreflectanceof this structure
`can nowbecalculated from the relations
`T, = (n/n) |Z /E ¢ 7,
`
`(6)
`
`quartz substrate
`No, ka=O0
`
`aE
`
`FIG.5. Poly Si—quartz layer structure for transmittance and reflectance
`model. Transmittance and reflectance are considered for radiation incident
`at three different interfaces as shown.
`
`and
`
`(7)
`R,= |E> /ES |’.
`This formulation of the transmittance and reflectance prob-
`lem was programmed in a computer with complex arithme-
`tic capability.
`
`B. Finite-thickness substrate
`
`Since the quartz substrate in our sampleshasfinite
`thickness, the transmittance 7, given by Eq. (6) and the re-
`
`6872
`
`J. Appl. Phys., Vol. 52, No. 11, November 1981
`
`Lubberts eta/.
`
`6872
`
`No=No=l
`
`air
`
`
`No=Nno
`Eo*
`Jez-0 quartz
`
`FIG.4. Electric vectors of electromagnetic waves traveling in positive and
`negative directions in a layer structure consisting of a thin polySi film of
`thickness d, on a semi-infinite quartz substrate. The radiation is incident on
`the polySi film from the air side.
`
`Experimental data of Malitson'® and Philipp’’ fit this dis-
`persion relation from 0.4 to 7.5 4zm, the region ofinterest in
`this paper.
`In the matrix method," the electric vectors of the posi-
`tive- and negative-going electromagnetic wavesin the inci-
`dent medium (subscript0) are related to those in the quartz
`(subscript 2) by
`+
`
`(5) = icy) ( ),
`
`0
`
`tl,
`
`2
`
`Es
`
`(1)
`
`where
`
`. Pia 7
`reo dl
`Cn = ie ems
`
`form = 1,2,
`
`(2)
`
`with
`
`

`

`T = [T,T,/(1 — R{R,)]
`
`(10)
`
`and
`
`(11)
`R=R,+ [7,7 ,R/(1 —R{R,)]).
`The transmittance T;of light incident on the quartz-air in-
`terface is given by’*
`(12)
`T, = Angn,/(ng + 12)’,
`and R, + T, = 1. In Sec. III A, the quantities R, and 7,
`were calculated by using a beam of light normally incident
`on the air—poly Si interface. Similarly, R | and T; are cal-
`culated by meansof the matrix method by using a wave nor-
`mally incident on the poly Si and originating in the quartz
`medium. Thus,the total transmittance and reflectance can
`be calculated for an absorbing poly Si film on a transparent
`substrate by meansof Eqs. (10) and(11), respectively. The
`cases of no absorption (k, = 0) and low absorption (k, <n)
`are ofspecialinterest, since under these conditions the index
`of refraction and the thickness of the poly Si film can be
`determined.
`
`1, Nonabsorbing poly Si
`
`In the infrared, the undopedpolySi films becometrans-
`parent. Underthis condition k, = 0. Performing the matrix
`operations in Eq. (1), we obtain for the transmittance T, as
`defined by Eq.(6)
`
`(13)
`T, = (no/no)[t7t3/1 + AZ + 2rir, cos 26,)],
`where r,, rz, ¢, f,and 6, are given by Eqs.(3), (4), and (5). For
`a nonabsorbing film we also writeR,+7,;=1,R,+T}
`= | (see Fig. 5). In addition, as'is evident from Eq.(13), we
`have 7, = T{ and thus R, = R ;. Equation (10) can then be
`written as
`
`T= (T)T,/(T, + T, — 7T,T,)).
`
`(14)
`
`Substituting Eq. (13) into Eq. (14) and making use of
`Eqs. (3) and (4), we find that the minimum ofthe transmit-
`tance of the poly Si—quartz substrate structure is given by
`Trin = ANN} M2/(ng + ni \(nt +73),
`(15)
`which occurs when
`
`(16)
`2n,d,=hA, where h= 1/2, 3/2, 5/2...,
`underthe condition that n, <n, and, <n,, which applies to
`our sample. A maximum in the total transmittance occurs
`when
`
`(17)
`2nd, =(h + 1/2)A,
`Equation (15) is quadratic in 1? , which can besolved in
`termsof T,,,,. The refractive index of air ny = 1. Hence, a
`measurementof 7,,,,, in the infrared permits the determina-
`tion of the index 7, of the undopedpoly Si layer at the wave-
`length under consideration. Knowledge of the approximate
`layer thickness establishes the order h, and thus an accurate
`thickness is determined from Eq.(16). Having established
`the thickness, the index as a function of wavelength can now
`be determined from the maxima and minimain transmit-
`tance. Furthermore,since a reflectance maximum coincides
`with a transmittance minimum andviceversa,the index can
`also be determined from the reflectance extrema. In fact,
`
`6873
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`J. Appl. Phys., Vol. 52, No. 11, November 1981
`
`Lubberts e¢ a/.
`
`6873
`
`FIG.6. Contourplots of and R + Tat a transmittance minimum(reflec-
`tance maximum)with A = 3/2 and A = 1.20 um,correspondingto a 0.27-
`pm layerof poly Si implanted with a phosphorusdose of 1 x 10!°/cm?. The
`min
`heavier contours correspond to measured values of Tryin ANd Tri, + Trax:
`
`reflectance extrema can be determined more accurately than
`transmittance extremain the visible region of the spectrum,
`where thepolySifilm absorbs strongly. We found that the
`thickness of the doped poly Si on one side of the quartz sub-
`strate is equalto that of the undopedpoly Si on the otherside
`of the substrate. This means that Eqs. (16) and (17) can also
`be used for determining n, of the doped poly Si as a function
`of wavelength.
`Whenabsorptiontakesplace, analytical expressionsfor
`T,,R,,T 1, and R { are uselessly cumbersomeand numerical
`techniques haveto be used.
`
`2. Absorbing poly Si—contourplots
`
`Heavily doped polySi filmswill absorb radiation, even
`in the infrared. When undopedpoly Sifilms are not available
`for thickness determination, the methodofSec.III B cannot
`be used to determine n, as a function of 1. However, when
`R+T<1andk,€n,
`the location of the transmittance mini-
`ma and reflectance maximaarestill accurately determined
`by 22,d, = AA with A again an odd half integer. Then d, is
`known when J and A are determined from experiment, and
`n, is determined by meansofthe following procedure. Using
`the matrix formulation of Sec. III A, we determine T7,,,,, and
`Rynax for sets of values n, and k,. By plotting k, vs n,, we
`obtain contours of constantT,,,;,, and contours of constant
`Riuax- We found it convenient to consider contours of con-
`stant 7,,;, and contours of constant T.,,,, + Rimax- Phe point
`at which a contourof constant 7,,,, and a contour of con-
`stant T,,;, + Rmax Cross determines unique values of 7, and
`k,. The intersection of the contours corresponding to the
`experimentalvalues of T_,;, and Tinin + Rmax then gives the
`desired value of 2, and k,. An example of such contourplots
`is shownin Fig. 6, correspondingto a poly Si film implanted
`with a phosphorusdose of 1 x 10'°/cm?.
`
`0.08
`
`0.07
`
`0.06
`
`2°oa
`
`2oO £
`
`Extinction
`
`3.2
`
`3.3
`
`3.4
`
`Index of
`
`3.8
`
`3,9
`
`4,0
`
`coefficient,k, $0 3.1
`
`35
`
`3.6
`
`37
`refraction, n,
`
`

`

`C. Free-carrier absorption
`
`Thereflectance R of the poly Si film on a nonabsorbing
`quartz substrate can be calculated by meansof the matrix
`methodof Sec. III A. The real and imaginary partsof the
`complex index of refraction of the poly Si film need to be
`specified for this purpose. Accordingto the classical theory
`of free-carrier dispersion, the real part n, and the imaginary
`part k, of the index of refraction can be expressed by the
`relations'®
`
`Sec. III A. Variation of 7 and N /m* permits us to obtain a
`“best” fit of R to the measured reflectance spectrum of a
`given dopedlayerof poly Si on a quartz substrate.
`
`IV. RESULTS AND DISCUSSION
`
`Theindex of refraction as determined from the wave-
`length locationofthe reflectance of transmittance extremais
`shownin Fig. 7 for the undoped and doped samples. The
`methodoutlined in Sec. III B was used. The refractive indi-
`(18a)
`ni —ki =e, — [wo°/(1+o'7)],
`ces oflightly doped single-crystal Si measured by several
`authors and tabulated by Huen’ are shownascircles. In the
`(18b)
`2n,k, =€, + [w7,7/or(l+077r)],
`visible region of the spectrum the refractive indices of the
`where e, and ¢,are the real and imaginaryparts ofthe dielec-
`undoped and dopedfilm are approximately equal, with the
`tric constantof the undopedpolySi, 7 is the relaxation time,
`dopedlayer having a slightly lower index. Only the indices of
`is the angular frequencyof the incidentradiation, and w ,
`refraction of the undoped (curve A) and the most heavily
`is the plasma frequency given (in MKSunits) by
`doped (curve E) samples are shownin thevisible. The data
`(19)
`0, = VGN /egm*).
`correspondingto the intermediate dopinglevelsfall between
`these curves.
`Hereg is the magnitudeof the electronic charge, N is carrier
`The systematic variation ofthe indexof refraction with
`concentration, €,
`is the permittivity of free space, and m* is
`dopingin the infrared (Fig. 7) is attributed to the interaction
`the effective mass of the carriers. Note that the second terms
`of the radiation with free carriers. Plotting nj asa function of
`onthe right-handside of Eqs. 18(a) and (b) represent the
`A*results in a straight-line relationship, as predicted by Eq.
`contribution ofthe free carriers to the dielectric constant.
`(23). Such plots are shownin Fig. 8. The intercept with the ni
`axis gives n?,, the square of the index of undopedpoly Si,
`whichis essentially constant in the infrared. This value
`agrees very well with the experimental index of 3.50 mea-
`sured in the infrared (see Fig. 7, curve A). Assuming m* is
`0.28 times the massofa free electron, '? we obtain the carrier
`
`1. Wavelength region 1.2-1.8 um
`In this wavelength region, Eq. 18(a) can be simplified.
`The imaginarypart k, ofthe refractive index, also referred to
`as the extinction coefficient, is related to the absorption coef-
`ficient @ as
`
`(20)
`k, =a/4n,
`where J is the wavelength of the radiation in free space. In
`Sec. IV wefind that a < 10* cm™! and 7, >3 in this wave- |
`length region, so that n7 >k 7. Furthermore,in calculating +
`from the mobility (see Sec. IV), we can conclude that @?77> 1.
`With these two simplifications, Eq. 18(a) can be written as
`ny =Nio —(@,/a)’,
`(21)
`since for undopedpoly Si €, = nj), which is assumed to be
`constant in this wavelength region. The angular frequency
`can be expressed as
`o=2re/A,
`
`(22)
`
`wherec is the speed of light. By means of Eqs. (19) and (22),
`we can now write Eq. (21) as
`(23)
`ne = ni, — (g°NA 2/42€,c?m"*).
`Thus, a straight-line relationship is predicted between nj
`and A”.
`
`2. Wavelength region 2.5-7.5 um
`
`refraction
`Indexof
`
`3.6
`
`3.2
`
`5.2
`
`5.0
`
`4.8
`
`4.6
`
`4.4
`
`4.2
`
`
`4.0
`
`3.8
`
`
`
`
`3.4
`
`
`
`
`
`0
`° 04
`O06
`O08 O12 La
`16
`18
`2.0
`Wavelength (jm)
`
`In this wavelength region we can set €, = 0. Further-
`more, we set V = 0 for undoped poly Si. This meansthat
`k, =Oand nj =nj, =€,,
`as is evident from Eqs. 18(a) and
`(b). The value 7, is determined from experimentin Sec. [IV
`FIG.7. Index of refraction as a function of wavelength for undoped and
`phosphorus-doped poly Si. Implanted dose: (A} none, (B) 1 x 10'°/cm?,(C}
`and is assumed constantin this wavelength range. By speci-
`5x 10'5/em?, (D) 1 Xx 10'°/em?, (E) 2 Xx 10'°/cm”.Thecircles correspond to
`fying 7 and N /m*, Eqs. 18(a) and (b) can be solved simulta-
`single-crystal Si. The squares are independently obtained from contour
`neously for 7, and k, as a function ofw or A. Thereflectance
`plots of Tand R +7at transmittance minima(reflectance maxima)of a
`R can then be calculated by using the matrix formulation of
`poly Si film implanted with a dose of 1 x 10'®/cm?.
`
`6874
`
`J. Appl. Phys., Vol. 52, No. 11, November 1981
`
`Lubberts eta/.
`
`6874
`
`

`

`3.2
`
`2.8
`
`Ud>
`
`
`
`
`
`3.26 x 102%m°>
`
`N=2.66 x 10'Yom>
`
`145 x 102%m>
`
`2.81 x 10O&°%em?
`
`3
`
`4
`
`0
`
`|
`
`2
`v2 (107 l2 m?)
`FIG.8. Plots of 1? vs A ? for phosphorus-implanted polySi films. Implanted
`dose: (A) 1x 10'5/cm?, (B) 5x 10'S/cm?, (C) 1x 10'*/cm?,(D)
`2x 10'°/cm?. The correspondingcarrier concentrations N are indicated.
`
`concentration N from theslopeofthe lines in Fig. 8 accord-
`ing to Eq. (23).
`A comparison of the measured carrier concentration
`and the implanted dosegives the fraction of the implanted
`phosphorusthatis electrically active. In Fig. 9 (curve A), we
`plot the carrier concentration ofeach dopedpoly Si layer asa
`function of the implanted dose of phosphorus atoms. Note
`that a linear relationship is obtained. However,at the highest
`implant doses, the carrier concentrationlevels off since the
`solid solubility is approached. From theslopeofthe linear
`portion, wefind that 75% of the implanted phosphorusis
`electrically active.
`Asindicated in Sec. III C, the carrier concentration can
`also be deduced from the measuredreflectance of these thin
`poly Si films on quartz substrates in the range 2.5-7.5 um by
`using the samefree-carrier theory. Measuredreflectance
`curves of four doped poly Si films are shownin Fig. 3. Note
`that the minimumin reflectance shifts systematically to
`shorter wavelengths as the dopingincreases. This is consis-
`tent with the increase in plasma frequency with carrier den-
`sity, as predicted by Eq. (19). The reflectance can be calculat-
`ed by specifying 7 and N,as outlined in Sec.III C. Since the
`mobility « and 7 are related by
`
`(24)
`B= qr/m*,
`wefoundit convenient to specify uz as defined above. Similar
`calculations have been reported by Lavineet al., who consid-
`ered the reflectance of poly Si deposited on oxidized single-
`crystal Si substrates.”° For a “best” fit to the measuredre-
`flectance data, N and uw were varied. We foundthatthe value
`
`Carrierconcentration(IO2%cm73) ipa
`
`YX°
`
`2°@
`
`0.4
`0.2 2
`
`0
`
`2.0
`1.6
`12
`0.8
`04
`Phosphorus implant dose (l0'Scm-2)
`
`FIG.9. Carrier concentration vs phosphorus-implant dose for 0.27-4m-
`thick poly Si films. A. Index of refraction analysis in the 1.2-1.8 zm region.
`B. Analysis of reflectance data in the 2.5-7.5 44m range. C. Hall-effect
`measurements.
`
`ofN primarily determines the wavelength at which the mini-
`mum in the reflectance occurs and thaty affects the shape of
`the curve. An example of such a calculation is shownin Fig.
`10. Values of N = 1.90 10?°/cm? and « = 45 cm?/V sec
`were used in this calculation. Changes of 10% in either N or
`i. cause significant changesin the reflectance. In Fig. 9
`(curve B), the values ofN obtained in this mannerare plotted
`
`1.0
`
`0.8
`
`9a
`
`oOb
`
`Reflectance
`
`3
`
`4
`
`5
`
`6
`
`7
`
`8
`
`Wavelength (jm)
`FIG. 10. Experimental(solid curve) and calculated (circles) reflectance of
`0.27-um-thick layer of poly Si on a fused quartz substrate. The sample was
`implanted with a phosphorus dose of 1 x 10'°/cm?.In the calculation
`N = 1.9X 10°/em*, uw = 45 cm?/V sec, ny) = 3.50, and m* = 0.28 times
`the free-electron mass.
`
`6875
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`J. Appl. Phys., Vol. 52, No. 11, November 1981
`
`Lubberts et a/.
`
`6875
`
`

`

`for the doped samplesas a function of the implanted dose of
`phosphorus atoms. Similar to curve A,a linear relationship
`is obtained at the lower implant levels. However,in this case
`50% of the implanted phosphorusiselectrically active as
`comparedto the 75% electrical activity obtained previously,
`even though the samefree-carrier theory expressed by Eqs.
`18(a) and (b) is applied to the wavelength ranges of 1.2-1.8
`fem and 2.5-7.5 um.
`Possible explanationsfor the difference between curves
`A and B in Fig.9 include(1) the inadequacy of the simple
`free-carrier dispersion (Drude) model and (2) the occurrence
`of an additional optical transition which depends on the
`presenceoffree carriers. In this case, interconduction band
`transitions are believed to occurin heavily doped n-typesili-
`con. An absorption bandin the infrared is associated with
`these transitions.*'~?? A refractive index change accompa-
`nies such an absorption band, and the measured index no
`longer is accurately described by Eqs. 18(a) and (b). Since our
`poly Si films are thin and the absorption in the near infrared
`is small, we were unable to identify an additional absorption
`band arising from interconduction bandtransitions.
`
`After all optical measurements were completed, Hall-
`effect measurements using a van der Pauw pattern were
`madeon these samples to determine the Hall mobility and
`carrier concentration. The carrier concentrations obtained
`from Hall-effect measurements are included in Fig. 9 {curve
`C). The carrier concentrations obtained from Hall-effect
`measurements compare morefavorably with the carrier den-
`sities derived from reflectance measurements than with
`those obtained from the analysis of the index of refraction.
`The mobilities obtained from fitting the reflectance
`data and those obtained from Hall measurements are shown
`in Fig. 11. The mobility derived from the Irvin curveis in-
`cluded for comparison. Also included are experimental Hall
`
`data reported by Shibataet a/.** of phosphorus-implanted,
`1000 °C thermally annealed polySi layers 0.18 zm thick.
`The lower mobility in poly Si has been attributed to grain
`boundary barriers.** The Hall mobilities of our samples
`agree very well with those obtained by Shibataet a/.4 These
`dataare consistent with a grain size of 0.33 zm,as deter-
`mined by Shibataef al.”4
`Onthe other hand, the mobility obtained from infrared
`reflectance data is a high-frequency mobility. In this case,
`mostof the carriers are not transported across a grain
`boundary during a cycle of the electric field. Therefore, the
`reflectance mobility is expected to be larger than the Hall
`mobility, in agreement with observation.
`The absorptioncoefficient a was obtained by meansofa
`least-squaresfit to the experimental transmittance. A typical
`fit is shown in Fig. 12 for a poly Si film implanted with a
`phosphorusdose of 1 X 10'°/cm?. Other implant doses gave
`equally good results. The experimentally determined index
`of refraction and thicknessof the poly Sifilm are used in the
`transmittance calculation. The transmittance can then be
`calculated as discussed in Sec. III B if either the extinction
`coefficient k, or a of the poly Sifilm is specified. We repre-
`sented log a by cubic splines with knots at several wave-
`lengths.” The valuesof a were then determined bya nonlin-
`ear regression to give an optimumfit to the measured
`transmittance over the entire wavelength rangeofinterest.
`Asseen in Fig. 12, an excellent fit is obtained. The absorp-
`tion coefficient is shownin Fig. 13 over the visible region of
`the spectrum for all doping levels considered.
`It is seen that at the shorter wavelengths a generally
`decreases with doping. Nevertheless, a correspondingto the
`most heavily doped layer can be as muchasa factor of two
`larger than that of lightly doped single-crystal Si For ease of
`
`@3°NS°°°
`Mobility(cm®/Vsec) a °
`
`
`
`Transmittance
`
`oO oy
`
`0.2
`
`o.8b
`
`°a————
`
`TT
`
`0
`
`6
`
`8 10?!
`6
`4
`2
`8 1020
`3)Carrier concentration (cm
`
`FIG. 11. Mobility of 0.27-um-thick poly Sifilms as a function of carrier
`concentration. A, Hall-effect measurements; O, mobilities determined from
`reflectance measurements; U, Hall mobilities for polysilicon as given in Ref.
`24. Thesolid line represents single-crystal Si mobility calculated from the
`Irvin curve.
`
`O%4
`
`0.6
`
`1.0
`0.8
`Wavelength (zum)
`
`1.2
`
`14
`
`FIG. 12. Nonlinear-regressionfit of the transmittance of a 0.27-4m-thick
`poly Sifilm on a quartz substrate to determinethe extinction and absorption
`coefficients. The poly Si was implanted with a phosphorusdose of
`1x 10'*/cm?. Solid curve: experiment. Circles: calculation.
`
`6876
`
`J. Appl. Phys., Vol. 52, No. 11, November 1981
`
`Lubbertseta/.
`
`6876
`
`

`

`coefficient,a(cm!) 108
`Absorption
`
`040
`
`045
`
`050
`
`055
`
`060
`
`0.65
`
`0.70
`
`075
`
`10°
`
`10%
`
`
`
`
`
`Absorptioncoefficient,a(cm7'}
`
`1070.4
`
`0.5
`
`0.6
`
`07
`
`0.8
`
`O39
`
`1.0
`
`Lt
`
`12
`
`13
`
`Wavelength (j:m)
`FIG. 14. Absorption coefficient as a function of wavelength in the visible
`and infrared regionsof the spectrum fora poly Si film implanted with a
`phosphorusdoseof 1 x 10'°/cm?. The squares are independently obtained
`from contourplots.
`
`are in good agreement with curve D. The computed values of
`a@ are shownas squaresin Fig, 14. Also, in this case, the two
`methodsare in good agreement.
`
`Vv. CONCLUSIONS
`
`The indexof refraction and the absorption coefficientof
`undoped and phosphorus-dopedpoly Si films were mea-
`sured in the visible and infrared regions of the spéctrum.
`Both the index of refraction and the absorption coefficient
`have been tabulated for the visible region for easy reference.
`In the infrared, the index of refraction systematically de-
`creases with doping and showsconsiderable wavelength de-
`pendence. A free-carrier absorption model explains the es-
`sential features of dispersion in the index in the 1.2-1.8 um
`range. The samefree-carrier model wasalso used to calcu-
`late the reflectance in the 2.5—7.5 um region. The calculated
`and measuredreflectances were in good agreement. The car-
`rier concentrations obtained by applying the free-carrier
`model to both spectral regions differ by 50%. This implies
`that the free-carrier model is an incomplete, although useful,
`description of the optical behavior of heavily doped poly Si
`films.
`
`ACKNOWLEDGMENTS
`
`Wearegrateful to ourcolleaguesin the Silicon Process-
`ing Facility of the Kodak Research Laboratoriesfor their
`help and advice in the sample preparations. We thank M.
`Castelluzzo for dicing and bonding the Hall-effect samples
`a