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JOURNAL OF THE OPTICAL SOCIETY OF AMERICA
Effects of component imperfections on ellipsometer calibration
William Ralph HunterIBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598
(Received 7 December 1972)
The accuracy of determination of film properties from ellipsometer measurements is determined by theaccuracy with which 'p and A can be measured. This paper examines the effects of off-diagonal elements inthe Jones matrices on calibration measurements in P -A, P -C -A, and P -S -A configurations. Based onthese results, specific calibration procedures for determining instrument constants are given that willminimize the effects of off-diagonal elements. In particular, the method of McCrackin et aL fordetermining the polarizer-prism offset is found to be influenced by off-diagonal elements, and a simplermethod based on extinction readings is presented. The advantages of using a sample with p = ± j duringP -S -A calibration measurements and a subsequent P -C -S -A check are pointed out. Polarizer-prism beamdeviation in combination with photomultiplier sensitivity variations has been found to be a source of error.However, a specific procedure for determining the relative phase retardation 8' of the compensator in theP -C -A configuration is described, which eliminates beam-deviation effects.
Index Heading: Ellipsometry.
Recently, Azzam and Basharal examined the effects ofoff-diagonal elements in the Jones matrices of the opti-cal components of an ellipsometer on one-, two-, andfour-zone measurements. They introduced quantitiescalled residuals, which can be applied as correctionfactors to one- and two-zone measurements. Determi-nation of these residuals constitutes calibration of theellipsometer.
Other methods of calibration have been described inthe literature,2 -4 but the effects of nonzero off-diagonalelements in the Jones matrices have been neglected.The purpose of this paper is to study these effects. Asecond purpose is to discuss other possible instrumentimperfections and their effects. Sample-cell windows, 7
are not treated; the Muller Nebraska convention isused.8 The notation used is similar to that used inRef. 4.
I. THEORY
Monochromatic polarized light undergoes a changeof polarization upon transmission through or reflectionfrom a component along the optical path of the ellip-someter. This polarization change can be characterizedby the 2 X2 Jones matrix in the principal frame of thecomponent. For the ith component, the general formfor the 2 X2 Jones matrix is
/Ti1Tli T12Z'i=C ,o \
tT21i T22J 1
where C, S, and A stand for compensator, sample, andanalyzer, respectively.
Table I lists the elements Tjk1 for each of the compo-nents in their respective principal frames. In Table I,T is the relative amplitude transmission of the com-pensator, 5 is the relative phase retardation of the com-pensator, and .'-a--T. Also, Vf and A are the usualellipsometric parameters of the sample. The compensa-tor off-diagonal elements jlc and 12c, which cause light
incident along one of the principal axes to be scatteredinto light along the other principal axis, can be due tostrain birefringence, optical activity, or physical defects.The sample off-diagonal elements #Is and :2S can becaused by surface optical activity, surface roughness,etc. No attempt to relate the IO's to actual mechanismswill be made here. They will be treated only as smallcomplex numbers, and their effects studied. With theexception of Azzam and Bashara's work,' ellipsometrytheory in the literature has dealt only with the case inwhich all O3's are zero.
The electric-field vector transmitted by the analyzeris given by
EAo=TAR(A)TsR(-C)TcR(C-P)Epo, (2)
where Epo, the electric-field vector transmitted by thepolarizer, can be written9
(j tanX)(3)
and the rotation matrix of argument a is defined by
R(a) =cosa sina\e
-sina cosa/(4)
In Eq. (3), X is the residual polarizer ellipticity.2 4 Theirradiance I detected by, e.g., an ideal photomultiplier
TABLE I. Jones-matrix elements.
Compensator Sample Analyzeri=C i=S i=A
T11i 1 p = tan 6ei3A 1T12i #iIC UIS g1A
T21' p2C #2S ~ 2A
T22' Te-i 8=-jTe-i" 1 #3A
(5-r+8')
951
VOLUME 63, NUMBER 8 AUGUST 1973
WILLIAM RALPH HUNTER
where the subscripts R and I indicate real and imagi-(5) nary parts. The near-extinction relations are
where t denotes hermitian conjugate and the conditionfor extinctions is given by
al aI=- =0, (6)
OA OP
where A and P are the analyzer and polarizer azimuthalangles. Angles A and P satisfying Eq. (6) will be calledextinction values with subscript e. In deriving extinc-tion angles, we will retain only terms up to first orderin small quantities.
Suppose, however, that P= Pe+SP, where SP issmall and nonzero. The angle A can still be varied forminimum irradiance so that I/OA = 0. ThenA =A,+6A, where AA is nonzero. We will be interestedin such situations, in which one of the components isvaried for a minimum detected irradiance while theother components are near their extinction settings.The equations satisfied by the angles P, C, and A forthese situations will be referred to as the near-extinctionrelations.
We will list the extinction results and near-extinctionrelations in three commonly used calibration configu-rations; only first-order terms are kept. Discussion ofthe application of these results is deferred to Sec. II.
A. P-A Measurements
With the compensator removed and no samplepresent, the incident arm and reflected arm are madecollinear. At extinction,
A e= P.--21731AR, (7)where the subscript R denotes the real part of a complexnumber. Because there are only two components, thereare no near-extinction relations. Without loss of gen-erality, we will consider only the case where /1AR is
zero in all subsequent discussions" and hence Eq. (7)becomes the usual condition of crossed polarizer andanalyzer.
B. P-C-A Measurements
With the instrument arms remaining collinear, thecompensator is mounted. For a given compensatorazimuthal angle C, there are two distinct extinctionsfor Pe and Ae labeled 1 and 2, which satisfy
C-Pel1+7rl 17r T3AZ 020rT
A e1-C=F!7r=XT+J62CR,
C- P&2F2r = -/ICR - V/31A,
Ae 2 -C+h'7r-411r= 31Ci-XT
(8a)
0I-=0:
oP
ai=0:
OCai0I
-=0:oA
or
22-0
01-=0:OP
aI-=0:ac
01-=0:aA
8'Al = P-+/ 3
1A I /32CI,T
P1 = 1.T 2 +fl2cR
+T(132cr-x-01AI),
v1 =LT8'+32CR+XT,
Z2 ==v 2TY' IcR -P1iArT,
12 1P2 =-+-f13C1+313AI-X]
T2 T
l13cR
T2'
5' 1P2 =12±+-[flCI-XI,
T T
(1Oa)
(lob)
(lOc)
(Ila)
(lib)
(tic)
where
pA. (C-P,2+7r4z-p2r), v1==(A, -C=F27r),
a2- (C-P 2 F 2),and
V2 (A2-C+274I).
C. P-S-A MeasurementsThe compensator is removed, and the arms set for
reflection from a suitable sample. The alignment of thesample is important and a sample alignment procedurethat can be reproduced in future measurements shouldbe used. The two distinct extinctions are given by
cosAPel = 2 7r 217r-fl2SR+(X+fl2 SI)-
sinAtankp
-931AF,sinA
1 X+1 3 2SI /31Ar COSA
tant sinA sinAor
Pl $13 1sr+AI X COSAPe2±ita+ -__sA
tarnkt sinA sinA(8b) cosA(9a) A e2-2 7r=F2r= -sR+(#isr+I3AIr)i
sinA
x tankt
sinA
(12a)
(12b)
(13a)
952 Vool. 63
is given by,(= (EAO)NEA,0),
(9b) (13b)
EFFECTS OF ELLIPSOMETER IMPERFECTIONS
The near-extinction relations are
/ui+Yj tank/ cosAP -0:i P
A0:aA
=/3
2SR+fl1AI tank/ sinA,
/.' cosA+pi tangk
= (X+f2S.1) sinA+1 2SR COSA,
/U2 tan41'+v 2 cosA = -#1sR cosA
- (flS1+/31A1) sinA,
A2 tanOb COSA+v 2
= -,31sR-X tanV/ sinA,
(14a)
mined. If this is repeated for several values of As from00 to 360° (an increment of 30° should be sufficient)and the quantity (PSe2-PSei::F7) is plotted vs Pse,,then Eqs. (7) and (16a) imply that the data can befitted by
(PSe2-PSel:Fr) = 2cxp Cos(Pse,+0P)- (17)
This permits determination of ap and 3 p.14 Similarly,aA and /
3A can be determined if Ase, and A Se2 (where
A sei-A seiz[ 7r) are measured for each of several values(14b) of Ps ranging from 00 to 3600 in increments of 300.
The azimuthal offset a" can be experimentally de-termined by substituting Eqs. (16a) and (16c) forEq. (7) and solving for
a" = Pse+ap cos (Pse+8P)
(15a)
(15b)
whereIAJ=- (-Pil +-ri 27),v,_(A 1=F- 2), A2-(-P2=127r),
andV2- (A 2+r±S=2r).
II. DISCUSSION AND APPLICATION
Section I described the principal-frame parameters(e.g., x, T, 5') associated with the optical components.We must also know the orientation of the principalframe for a given divided-circle scale reading. Param-eters describing this orientation will be introduced be-low. This section also discusses the application of theextinction and near-extinction relations of Sec. I tospecific calibration procedures.
The relations between actual angles P, C, A anddivided-circle scale readings Ps, Cs, A s may be written12
P= Ps+aep cos (Ps+p) +p', (16a)
C=Cs+ac cos(Cs+#c)+c"+p', (16b)
A=A S+aA cos(AS+13A)+a"+p'. (16c)
Here p', c", and a" are the azimuthal offsets of thecomponents in their divided circles, and the a's and O3'saccount for eccentricity of the mounting of the rotatingcircle."3
All angles at minimum irradiance are assumed to beobtained by averaging angles that give rise to equalirradiance on both sides of the minimum.
A. P-A Measurements
For a given As, the two polarizer-extinction scalereadings Ps,, and PS8 2 (where Ps822zPsliar) aremeasured and the quantity (Pse 2-PSelF7r) is deter-
-ASe-aA COS(AS e+#3A)-I2 r, (18)
where the same data used to determine ap, fp, iA, /3JAcan be used to determine an average a".
B. P-C-A Measurements
This subsection discusses the practical application ofthe P-C-A extinction relations, Eqs. (8) and (9), andthe P-C-A near-extinction relations, Eqs. (10) and (11).For definiteness, let us assume that two-zone measure-ments are made with CL+-4r, and let Cs+ be an ap-proximate value of Cs that makes Cf+{7r. It is thevalue at which the compensator will be calibrated. Ifthe two distinct extinctions are measured, thenEqs. (8a) and (9a), written in terms of polarizer scalereadings using Eq. (16a), can be solved for
(C-p')+ =Ps8 6 ,+ap cos(Psel+fBp) -7r=FTr
/1AI -2CI
+ (18a)T
or
(C-p )+=Pse2 +ap COs(Ps,2+j#P)i2Tr
+11CR T-1AI, (18b)
where the subscript + denotes that Psel and PS 2 aredetermined for the specific compensator setting Cs+.If 1
31AI, 02CI, and ,31cR are neglected, then these equa-
tions permit (C-p')+ to be determined. Once thequantity (C-p')+ is determined, it can be used as acalibration constant, rather than c". That is, the valueof C for any value of Cs is given by
C=p'+ (C-p')++ (Cs-Cs+)
+ac[cos(Cs+c) - cos(Cs++00cf (19)
rather than by use of Eq. (16b).For a given C and a given zone, there are actually
four different possibilities for Ps and A s; e.g., (Ps, A s),(Ps, As+7r), (Ps+7r, As), and (Ps+7r, As+ir). Forbest accuracy, (C-p')+ in Eq. (18a) should be deter-mined for all four possible Ps,, As1 measurements and
and
9I
aP
August 1973 953
HI
WILLIAM RALPH HUNTER
(C-p'), in Eq. (18b) should be determined for all fourpossible PS2 , AS2 measurements. Any discrepancybetween the two average values of (C-p')± can becaused by fSAI, Oscl, and o1cR. Although averaging thetwo values of (C-p')+ eliminates any error caused by131Al, there is no assurance that the 31cR and 02Cr termscancel. Finally, as long as only two-zone measurementsare made with Cs-Cs+, then ac and jOc do not need tobe determined. However, if the four-zone averagingmethod of Azzam and Basharal 6 is to be used, then itis essential that C (and not just Cs) is changed by Xr.
In other words, it is important to determine either acandflc, or (C-p')- for CS-=Cs+_i27r.
The residual polarizer ellipticity X can be determinedby evaluating the left-hand sides of Eqs. (8b) and (9b)from the extinction data by use of Eqs. (16 c) and (19).If the left-hand sides of Eqs. (8b) and (9b) have oppo-site signs, and approximately magnitudes > 0.03°, theyare probably reasonably good measures of X. Otherwise,the possible effects of 132CR and ,ici and experimentalerror obscure the situation. If the left-hand sides areboth equal and very small, it probably means that X,32c0R, and f#,ci can be considered negligible.
We turn our attention now to measurements basedon the P-C-A near-extinction relations that can bemade to determine independently the compensatorparameters T and d'. This is especially important whenone-zone measurements are made, or one-zone calcula-tions are compared to two-zone calculations.
We first discuss the use of Eq. (lOb) to obtain T.Data can be taken in one of two ways, (i) P is held con-stant at Pel and the values of C at minimum irradiancefor each of several values of A near Ael are measured,or (ii) A is held constant at A.,, and the values of C atminimum irradiance for each of several values of Pnear Pi are measured. In case (i), the data fit a straightline of slope
/ aCs\ 1
UA A81/) 1 +T2
whereas in case (ii), the data have the slope
(20a)
S1=A , A=Aci. (20b)
Similar equations derived from Eq. (lib) can bewritten. Any appreciable deviation of T from its idealvalue of 1 will be measurable and a least-squares esti-mate of the standard deviation for the measured slopeis useful as a measure of error. Strictly speaking, thequantities on the left-hand sides of Eqs. (2 0a) and(20b) should be corrected for the effects of ap, Op, ac,1C, UA, and PA, but the deviations from extinction valuesare small and hence the corrections are negligible.
Equations (lOc) and (1 ic) suggest that a similarprocedure can be used to obtain the slopes T6' and5'/T. However, because 5' is so small, large deviations
of p1 and P2 would be required to get an appreciablechange of vi or V2. Numerical calculations (not pre-sented here) have shown that such large deviations ofAl and U2 cause Eqs. (10c) and (11c) to be no longersatisfied unless X is identically equal to zero.
The parameter 6' can still be determined by use ofEqs. (la) and (11a), where a nonzero X has negligibleeffect even if the deviations of Pi and P2 are large. Theonly way S' can be obtained from Eqs. (1Oa) or (11a)with adequate accuracy is to hold C fixed and measurethe value of P at minimum irradiance for each of severalvalues of A. However, experimental complications inapplying this procedure require an explanation, whichsuggests an unambiguous procedure.
Assume that for a given C, the extinction angles A.and Pe, are measured. If the values of P, call them P>and P<, at minimum irradiance for A C+AA andA,-AA, respectively, are measured, then Eqs. (1Oa)or (11a) predict that
P> Pc=Pe-P<. (21)
In practice, Eq. (21) is not satisfied when AA is madelarge enough for (P>-P,) or (Pc,-P<) to be measura-ble. The problem appears to be due to polarizer-beamdeviation that occurs for the large variations of A and Prequired to determine P> and P<. This beam deviationcauses areas of different sensitivity on the photomulti-plier cathode to be struck by the beam."5 A first-orderTaylor-series expansion for the detector sensitivity andbeam irradiance shows that the four angles measured forthe same photomultiplier response (one for each sideof each minimum) are of the form
(22a)
(22b)
(22c)
(22d)
P>+ = Pa+ (-E+ A-P) (1-ay),
P>- = P.+(-.E-AP) (I+'y),
P<+ = P.+ (,e+ AP) (I1- -),
P..-_= P. (1e- AP) (1 +-),
where (i) eL-•AA6/T or e-ŽAATb' for Eqs. (1Oa) and(11a), respectively, and (ii) AP=2(P+-P). If thereis no detector-sensitivity variation due to polarizer-prism beam deviation, then By is zero. Then,
P>=' (P>++P>i) = P.- -AP`Y,
P<=2 4(P<++P<-) =Pc+E+APY.
(23a)
(23b)
Note that P> and P< in Eqs. (23) do not satisfyEq. (21), unless y=0. However, even if Py O as aresultof beam deviation and detector sensitivity,
P>-P<=-26E.
Therefore, we may write
W1It P<-P>
T 2AA
(24)
(25a)
Vol. 63
EFFECTS OF ELLIPSOMETER IMPERFECTIONS
TABLE I. Determination 6' from P-C-A near-extinction data.a
Aij-0
viO As, Ps. Ps> Ps< Ps>-Ps, Ps,-Ps< Ps<-Ps>i (deg) (deg) (deg) b (deg) b (deg) (deg) (deg)
1 45.73 46.52 46.595 46.40 0.075 0.12 -0.1951 45.73 226.495 226.63 226.46 0.135 0.035 -0.171 225.71 46.525 46.58 46.40 0.055 0.125 -0.181 225.71 226.495 226.61 226.465 0.115 0.030 -0.145
-0.172oeave
2 135.72 136.50 136.42 136.245 -0.08 0.255 -0.1752 135.72 316.545 316.72 316.58 0.175 -0.035 -0.142 315.725 136.50 136.46 136.29 -0.04 0.21 -0.172 315.725 316.545 316.735 316.585 0.190 -0.04 -0.15
-0.1591dave
& For all measurements, C-45'. Also, the instrument used obeyed A =-As+a"+p' and P= -90-Ps+p', with a" and p' small.The third decimal place in the table comes from an average.
b Experimental conditions were AA = 5I and (P+- P-)-210 .I For i= 1 data, with x=O, T=0.93, i1= 1.11 (calculated), and all fl's=0, the calculated value of 6' is -0.27°.d For i= 2 data, with x=O, T=0.93, 6 = 1.09 (calculated), and all #'s=0, the calculated value of 6' is -0.30°.
and
=T'= , ' A2 0, V2-02AA
(25b)
where 4, and t2 are correction factors close to 1 thatcan be calculated to account for the small change ofslope for large AA (use of this correction is justifiedonly if the experimental error is small enough). Experi-mentally, the right-hand sides of Eqs. (25a) and (25b)were evaluated for all eight possible Pse, Ase combina-tions with AA = 150 and P+-P-22 10. All eight orien-tations gave approximately the same value. That thisprocedure gives a valid measurement of 6' is evidencedby the experimental observation that (P<-P>) inEqs. (25a) and (25b) is directly proportional to AA.These experimental results are summarized in TablesII and III.
C. P-S-A Measurements
This subsection discusses the practical applicationof the P-S-A extinction relations, Eqs. (12) and (13),and the P-S-A near-extinction relations, Eqs. (14)and (15).
There are distinct advantages to using a samplehaving PiŽr and A!:7r4-r, i.e., ,3-Ž-4j. First, if X isnonzero, its effect on both Pei and Pe2 becomes negli-gible. Another reason is related to minimizing errors inT and 8' determined from a P-C-S-A two-zone measure-ment (see below). Three possible choices for a samplewith ifr_45' and Az-900 at 5461 A are (i) thermallyoxidized polished-silicon substrate having an oxidethickness of 900 A used at 67.750 angle of incidence.(ii) an aluminum film coated with about 400 A ofsputtered SiO2 used a 700 angle of incidence, or (iii) an
aluminum film coated with about 600 A of sputteredSiO2 used at a 680 angle of incidence.
We now restrict the discussion to a surface withfi=+j (however, similar results are obtained in themore-general case). Then, substituting Eq. (16a) atextinction into Eqs. (12a) and (13a) and solving for p',we obtain
P' = i-4-r- [Pse1+ap cos(Psi+13P)]-fl2SR /
31A I
and
= [t2'r EPse2+ap Cos(Pse2+3p)]
+1lSr+fl1AI,
(26a)
(26b)
which permit p' to be determined if the f3's are neglected.Comments similar to the first two following Eq. (19)apply here also. In addition, we might in general expectthe off-diagonal elements of Ts to be greater than forTe, because of roughness of the surface of the sample;hence, a greater difference should be expected betweenthe two values of p' obtained from Eqs. (26a) and (26b)than between the two values of (C-p')+ obtained fromEqs. (18a) and (18b).
Substituting Eq. (16c) into Eqs. (12b) and (13b) andevaluating the left-hand sides of both equations, we
TABLE III. Variation of (Ps<-Ps>) with AA for determinationof 6' from P-C-A near-extinction data.a
As. Ps. Ps> Ps< Ps>-Ps. Ps,-Ps< Ps<-Ps>A (deg) (deg) (deg)b (deg)b (deg) (deg) (deg)
7.5 45.72 46.52 46.60 46.53 0.08 -0.01 -0.0711.25 45.72 46.52 46.64 46.53 0.12 -0.01 -0.1115 45.72 46.52 46.67 46.52 0.15 0 -0.15
a For all measurements, CL-45'. Also, the instrument used obeyedA = -As+a" +p' and P = -90 -Ps +1', with a" and p' small.
b Experimental condition was P+-P- kept approximately constant at25.5%k4l.5'.
August 1973 955
WILLIAM RALPH HUNTER
obtain measures of (X+-2,s1) and (X+-I-SR), respec-tively. These two quantities should be small; any mag-nitude in excess of 0.05° probably indicates a poor de-termination of a" or p'.
A second calibration procedure due to McCrackinet al.2' 3 is based on the simultaneous solution of Eq. (7)with the near-extinction relations Eqs. (14a) or (15a).But McCrackin et al. did not consider off-diagonalelements, which prevent Eqs. (14a) or (15a) from beinguseful, in the same way that X$-O prevents Eq. (14b)or (15b), with Eq. (7), from being useful even if theoff-diagonal elements are zero. Rather than residualpolarizer ellipticity, nonzero off-diagonal elements wereprobably responsible for the discrepancies theyobserved.
D. P-C-S-A Measurement Check
As a final check of all instrumental parameters, thesame sample used in P-S-A measurements should bemeasured with the compensator inserted, without re-orienting the sample, preferably for all four possibilitiesin each of the two zones. Because the off-diagonal ele-ments are not actually measured in the calibration pro-cedures, P-C-S-A data must be reduced by assumingthat all 3's=O. Both one- and two-zone values for itand A can be calculated and checked for consistencyby use of Eqs. (7) and (8) of Ref. 4, respectively. Also,T and 5' can be calculated from the two-zone mea-surements by use of the analog of Eq. (8) in Ref. 4 forTe-i2 , which is derived in the Appendix, and comparedwith the independent measurements made in theP-C-A configuration. The errors of T and 6' so obtainedare mostly minimized' by the choice p = i1 j. However,two-zone P-C-S-A measurements of T and 5' generallyshow greater fluctuations, especially in 6', from the in-dependently measured P-C-A values than are expectedfrom reasonable errors of the calibration constants ormeasured angles; the effects of component depolariza-tion, broad-bandwidth operation, or especially beamdeviation may be responsible.
E. Relation to Previous Literature
Most authors2' 3' 6 combine c" and p' in Eq. (16b) intoa single constant (e.g., c"+p'=3C) and similarly for(a"+p') in Eq. (16c). There are, however, distinctadvantages of the separation used here. After all,fundamentally only c" [or (C-p')+] is measured inP-C-A extinction measurements, and not c"+p' (or C).Also, only a" and not (a"+p) is measured in P-A ex-tinction measurements. Second, by including p' inEqs. (16a), (16c), and (19), angle-of-tilt errors arisingfrom sample misalignment' are equivalent to errors ofthe calibration constant p'. Finally, if the results fortwo-zone measurements derived by Azzam andBashara6 are recast in terms of the calibration constantsused here, it can be seen that only errors in the calibra-
tion constant (C-p')+ affect the two-zone calculationof A.17
Azzam and Bashara6 also show that X is the onlydiagonal-element parameter that must be accuratelydetermined if two-zone calculations of ,6 are to be freeof error. Thus, if two-zone measurements are made,only P-C-A extinction measurements [which give(C-p')% and the best estimate of X] must be done withgreat precision.
III. SUMMARY AND CONCLUSIONS
Ideally, the analysis of ellipsometric data in thepresence of nonzero off-diagonal elements should betreated rigorously, as Azzam and Basharal have done.The application of their method in practice is ham-pered by eccentricity of the mounting of the compensa-tor divided circle. Furthermore, instrument imperfec-tions that cannot, for the most part, be systematicallyaccounted for (e.g., collimation errors, broad-band-width operation, and polarizer-prism deviation, etc.)probably introduce errors at least as great as thoseintroduced by off-diagonal elements. The proceduresuggested here is to continue to use the theories withall O's= 0 in P-C-S-A data reduction (e.g., Ref. 3 whenX=O or Ref. 4 when XW 0), which for two-zone calcula-tions are essentially identical to the two-zone averagingscheme of Azzam and Bashara,".6' 7 to use two-zonemeasurements and calculations (since this eliminatesthe effects of error in the majority of the instrumentconstants" 6' 7) and to measure those calibration con-stants that can affect two-zone measurement withgreat accuracy, namely, (C-p')+ and X. Althoughaccurate determination of the other parameters T, 5',p', and a" defined in this paper is theoretically notnecessary, a careful attempt to carry out the calibrationprocedures for all parameters will not only educate theserious user on the nonideality of ellipsometer behavior,but also give an indication of the uncertainty caused byoff-diagonal elements in the critical parameters (C-p')+and X. That is to say, measurements can be taken thatconsistently disagree with the mathematical models onwhich analysis is based by amounts greater than theexperimental error.
The important conclusions of this paper are (i) off-diagonal elements do not affect the ability to measureaccurately the analyzer azimuthal-offset parameter a"in P-A measurements. (ii) Eccentricity of the mountingof the polarizer and analyzer divided circles can besystematically measured and accounted for; i.e., ap,1P3 , aA, and ,8A can be determined from P-Ameasurements. (iii) Off-diagonal elements do affect theability to measure accurately the compensator azi-muthal-offset parameter c" [or the alternative pa-rameter (C-p')+] in P-C-A measurements. (iv) Thevalue of X can be inferred from analyzer extinctionreadings in P-C-A measurements. (v) The value of Tcan be obtained from near-extinction measurements in
956 Vol. 63
EFFECTS OF ELLIPSOMETER IMPERFECTIONS
P-C-A measurements. (vi) The suggested procedurefor determining 6' from near-extinction P-C-A measure-ments eliminates the effects of polarizer-prism beamdeviation. (vii) Off-diagonal elements affect the abilityto measure accurately the polarizer-azimuthal-offsetparameter p' in P-S-A measurements. (viii) The valueof X can be inferred from P-S-A measurements. How-ever, it is probably less accurate than the value in-ferred from P-C-A measurements, owing to samplesurface roughness. (ix) The method of McCrackin et al.2
for determining p' is influenced by off-diagonal ele-ments. The use of extinction readings described here iseasier because it requires fewer readings. (x) Use of asample having 4t1r and Ac-r- 4- during P-S-Ameasurements and P-C-S-A check is advantageous.(xi) Polarizer-prism beam deviation and its associatedaffect on photomultiplier response has tentatively beenidentified as a source of error in ellipsometrymeasurements.
APPENDIX: DERIVATION OF TWO-ZONECALCULATION FOR T AND d'
Eliminating p in Eq. (6) of Ref. 4 with i= 1, usingEq. (7) of Ref. 4 with i= 2, gives a quadratic equation in
er-Te-A = -Te-j 6 ,
Zff 2+ba+c= 0where
d=rz1z2 tanC(Y 2 - Y1), (A2a)
b = (Z1Z4 (Y2 + tan2CYi) -Z2Z3 (Yi + tan2CY2 ) ), (A2b)
C= -Z 3 Z4 tanC (Y 2 -Yl), (A2c)
withXi-j tanx,
1+jXi-2 tanx,
i=1,2
i = 3,4.
In Eqs. (A2a)-(A3),
Xi =tan (C-Pi), i =1,2 (A4a)
Yi =-1/tanAi, i= 1,2 (A4b)
where P1,A 1 are one-zone measurements for a given Cand P2,A2 are the other zone for a given C. The solutionof Eq. (Al) constitutes a two-zone calculation of Tand 8'.
REFERENCES
'R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am.61, 1236 (1971).
2F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L.Steinberg, J. Res. NatI. Bur. Stand. (U.S.) A 67, 363(1963).
3F. L. McCrackin, A Fortran Program for Analysis ofEllipsometer Measurements, Natl. Bur. Std. (U.S.) Tech.Note No. 479 (U.S. Government Printing Office,Washington, D.C., 1969).
4W. R. Hunter, D. H. Eaton, and C. T. Sah, Surf. Sci. 20, 355(1970).
5F. L. McCrackin, J. Opt. Soc. Am. 60, 57 (1970).6R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am.
61, 600 (1971).7R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am.
61, 773 (1971).8R. H. Muller, Surf. Sci. 16, 14 (1969).9Here the non-essential normalization (Ep ,O)x = 1 is made.
Furthermore, a small real part to (Ep o), need not beconsidered, since it can be made zero by redefining theprincipal frame of the polarizer by rotation through a smallangle.
'°The term extinction is used here even though the intensity atthe minimum is not zero but rather some small valueresulting from nonzero X and ,B's.
"The combination A + /3 A R occurs in all extinction relationsused in the calibration measurements described later. Azzamand Bashara (Ref. 1) showed that the same holds true forP-C-S-A measurements. By redefining the principal frameof the analyzer, by rotation through a small angle, /3 4 R canbe made zero in the new principal frame.
'2 It is assumed here that the components rotatecounterclockwise when looking from the detector toward thelight source. Two primes are used in the notation c " and a"because they essentially account for the offset of thecompensator and analyzer relative to the polarizer, whereasp' accounts for the offset of the polarizer relative to theplane of incidence. The subscript S applied to the angles P,C, and A denotes scale reading (it is not to be confusedwith the subscript S applied to /3, which denotes sample).
3 A 0.05-mm misplacement of the axis of rotation on a 75-mmdivided circle will cause an error of 0.04' in certainorientations of the circle. In this example I a I = 0.05/75 rad= 0.04'.
14 One instrument in use by the author has ap = 0.025, which
is readily detectable."Variation of T and 8' in the presence of beam deviation is
neglected. See, e.g., W. G. Oldham, J. Opt. Soc. Am.57, 617 (1967).
'Numerical calculations have shown that when p = ±j, then(i) 0.03 ' errors in X, p ',(C - p ')+, and a " have no effecton T; (ii) 0.03 ' errors in X and (C - p ')+ have no effect on8'; and (iii) 0.03 ' errors in p' and a ' produce 0.06 ' error in8'.
17Assuming that the a's and /3's are negligible or are known
accurately enough.
August 1973 957
