(

JOURNAL OF RESEARCH of the National Bureau of Standards-A. Physics and Chemistry Vol. 67A, No. 4, July- August 1963 Measurement of the Thickness and Refractive Index of Very Thin Films and the Optical Properties of Surfaces by Ellipsometryl

) Frank 1. McCrackin, Elio Passaglia, Robert R. Stromberg, and Harold 1. Steinberg

(April 11, 1963)

The use of t he ellipsometer for t he measurpment of the thickness a nd refractive index of very thin film s is reviewed . The P oin care sphere representation of t he state of pola riza­ t ion of light is de\'eloped a nd used to describe t he reflection process. Details of the operation of t he ellipsometer a rc examined criticall y. A computational mC'thod is presented by which the thickness of a film of known refractive index on a re fl ecting substrate of known optical co nstants may be calculated direct ly from the ellipsometer r ea din g~. A met hod for co mput­ in g both the refractivc index a nd thickness of an unknown film is also developed. These methods have been applied to the determination of the thickness of a n adsorbed water layer on chromium ferrotyp e p lates and on gold surfaces. In thc former case the t hi ckness ,ras 23 to 27 A, a nd in the lat,tcr was 3 to 5 A. The measurement of the thickness and refractive index of barium fiu ori de film s c\'aporated 0 11 ch romium fC'ITotyp e surfaces is used as a n illust ration of t he simultaneous deterlllina tion of these t wo qua ntities . 1. Introduction erally appli cable computational method is described. The complex and lengthy calculations required to Ellipso metry is a co nvenient and accurate tech­ solve Lhc equaLions involved have bee n programmed nique for the measurement of Lllicknesses and for an electroni c computer. Ivre asuremen ts carried r efractive ind exes of very thin film s on solid surfaces out on chrom e and gold surfaces wi th and without and for th e measurement of optical co nstants of filill s will also be described. refl ecting surfaces. The lower limit of film thick­ 1n addition , the Poincar e sphere representation of n esses lhat can be stud ied by ellipso metry is at the state of pol arizR,tion of ligh t will be developed least an order of magn i t ud e smaller than can be sin ce this is the most useful representation for the studied by other means such as interferometry. co nsid eration of the eIl'ects OCCUlTing on reilection . Artifacts such as those caused by vacuum ill the case or electron microscopy arc not enco untered. 2. Representations of Elliptically Polarized Neither interferometry nor electron mi croscopy are adaptable, as is ellipso metry, Lo the study of film s Light under liquids, and only elli pso meLry will give the index or refraction or film s of unknown thicknc s. The state of polarization of ellipLicall y polarized li ght nuty be described in many ways. For this The technique of ellipsometry is concerned wi th purpose, consider a ligh t wave Lraveling along the the measurement of changes in the state of polariza­ Z axis of a coordina te sysLem. The electric vector tion or li ght upon reflection from a surface. For of the wave is then given by: a clean refl ecting surface the optical constants of the surface and the reflection coefficients of the E X= al cos (T+ 8, ) system may be calculated from these changes. A (1) thin transparent film on the refl ecting surface causes E y = a2 cos (T+ 82) j additional changes from which the thickness and rel'ractive index o/' the film may be determined. where T = W wand v are the angular frequ;J1cy The principle of the ellipsometer has been de­ (t- ~ } scribed by several authors [e.g., 1- 7]2 and a bibli­ and linear velocity of the light, respectively, and th e ography o/' the theoretical contribu tions of many total amplitude is the vector sum of a, and a2. The authors, star ting with the original equations of polarization can be described by the amplitudes, Drude [1] is given by Winterbottom [2]. This paper al and a2, and the phase difference, 8= 82 - 8" or the will describe cer tai n i1l.easurement techniques and components. As is well known, in a stationary plane an application of the exact solution of the algebraic whose normal is parallel to the Z ax is the locus of the equations of Drude to the measurement of optical end of the electric vector is the ellipse shown in eonstanLs of surfaces and thickness and refractive figure 1. index of thin films covering the surfaces. A gen- The ellipse, however, may also be described in relation to coordinates X ' and Y' along the axes of the ellipse. Thus, the inclination cp of these cOOl·di­ I The wo rk reported here was supported in part by the Army Researcb Office nates and the semiaxes a and b of the ellipse also (Durham) anel in part by tbe Il ureau of Naval II·eapons. , Figures in brackets indicate tbe literature references at tbe cnd of th is paper. describe the polarization of the light. 363 \

-{\ y Therefore, the state of polarization may be repre­ sented by a point on a sphere of radius 8 0 by the spherical coordinates 8 0, 2cp, 2x. This sphere is called the Poincare sphere. Alternatively, the Stokes parameters 8 1, 8 2, and 83 may be used as Cartesian coordinates to describe the polarization. This representation is illustrated in figure 2. Since the intensity of the light is of secondary importance for use of the ellipsometer, the projection of the point on a sphere of unit radius is convenient. The coordinates are then given by

FIGURE 1. The locus of the electric vector fm' elliptically 80 = 1 polarized light in a plane normal to the direction of propa­ gation. I-tan2 a cos 2x cos 2cp The coordinate axes XY and X ' Y ' are both convenient lor the description of 2 the ellipse. 1+ tan a \( 2 tan a cos 0 82 cos 2x sin 2cp (6) It will be convenient to introduce auxiliary angles 1+ tan2 a a and X defined by 2 tan a sin 0 8:l sin 2x. 1+ tan2 a (2) Light of a given staLe of polarization is therefore b represented by a point on the sphere, with the polar (3) tan x=±-a' angles 2 cp and 2 X representing, respectively, twice the azimuth and twice the ellipticity of the light. For example, for plane polarized light, the ellipticity The numerical value of tan X represents the ratio of X is zero, and 83=0. Therefore, plane polarized light the minor to major axes of the ellipse and the sign of is represented by points on the equator of the X distinguishes the two senses in which the ellipse Poincare sphere. Also, the ellipticity of circularly may be described. The angles X and cp are called the polarized light is 45°, whence 81=82=0, 2x= ± 7T-j2; ellipticity and azimuth of the light, respectively, and thus, circularly polarized light is represented by the the representation may be made either in terms of poles of the sphere. aI, a2, and 0; a, 0, and the total amplitude; or Cp , X, The Poincare sphere may be used to represent and the total amplitude. elliptically polarized light with respect to any Another representation of the state of polarization, physical axes, say X" and Y" at an angle cp" with and one which leads quite naturally to the Poincare the XY axis. The ellipticity X is not dependent on sphere is by parameters which all have the same the choice of axes and the azimuth of the light is t physical dimensions, called Stokes parameters [8] : changed by cp". Thus the 8 1 and 8 2 axes must be rotated an angle 2 cp" around the 8 3 axis.

80 = ai + a ~ 81 = ai - a ~ (4) 8 2= 2ala2 cos 0 8 3= 2ala2 sin o.

The parameter 80 is proportional to the intensity of the wave and is related to the other parameters by the identity 8~ = 8f + 8~+ 8i , so that the Stokes parameters are not all independent. It has been shown [9 , 10] that the Stokes parameters are also given by

SI

8 1= 8 0 cos 2x cos 2cp FIGURE 2. Representation of polarized light on the Poincare 8 2 = 8 0 cos 2x sin 2cp (5) sphere. 'l'he point P represents ligh t with ellipticity x and azimuth "'. The car tesian 8 3= 8 0 sin 2x. coordinates are Stokes parameters (after Born and Wolf [9]).

364 L_ f

tive absorption of the two components. Thus, the ratio of the refl ec tion coeffi cien t for light polarized in the plane of in cidence to tha t for ligh t polarized in the plane of the surface is given (3) by

(7) r I rp where p is the ratio of the refl ec tion coeffi cients 1' p and 1's, and if; and Ll are functions of the op tical constants of the surface, the wavelength of the light used, the angle of incidence, and, for a film covered sm-face, the thickness and refractive index of the film. (See also eqs 40 to 46 .) For consideration 0[' the refl ection process on the Poincare sphere, rep res en t the light with respect to Cartesian coordina tes with the X and Yaxes in and normal to the plane of incidence and the Z axis in the direction of propagation . Then tan if; = as!a p • F I GU RE 3. The intersection 0/ the Poincare sphere and a plane The Stokes parameters of the ligh t before r efl ec tion nonnal to the 81 axis at S l = (1 - tan201. ) / (1 + /an201. ) for the con­ siderati on of the effect of a doubly refracting plate with I"e­ are given by eq (6) a nd after reflection ar e tar dation 0 on light with ellipticity represented by P . The plate rotates P to L . The fast axis of the plate is along S,. I - tan 2 acot2 if; 1+ tan 2 acot2 if; 2 tan a cot if; cos (O+ Ll ) The Poincar e sphere is con veni en t for considera­ s ~' 2 2 (8) tion of the eff ec ts of doubly refracting plates and l + tan a cot if; reflection on the state of polarization of a ligh t beam. " 2 tan a cot if; sin (O+ Ll ) For example, let the polarized wave represen ted in I. figure 1 pass through a doubly refracting plate of 8 3 = 1+ tan2 a cot2 if; relative phase retardation o. Choose the X and Y ? axes in figure 1 along the slow and fas t axes of the The refi ection is represented on a Poincar e sphere plate Sin ce tbe azimuth of the fast axis of th e in figure 4. The point I p represents the intersection plate is zero, it is represen ted by the posi ti ve 8 1 axis. In this case there will be no ch ange in the s, amplit udes al and a2 of the componen ts 0[' the ligh t, and therefore no change in a or in 8 1 as shown by eq (6), the only eff ec t bein g to change the relative phase of the componen ts by 8. Thus, upon passin g through tbe plftte the point represen ting the polariza­ tion will remain on the curve representin g con stan t 81, or on the in tersection of the sphere and the plane 81 = constant. The intersection of the Poincare sphere by a plan e perpendicular to the SI axis at 81 , . fi 3· . I f d· 2 tan a o. sh own 111 gure , IS a clrc e 0 ra lUS 1 + tan2a . J The ligh t incident on th e plate is represented by w the point P , and after passing through the plate is represen ted by th e poin t L. The eff ect of the doubly L' refracting plate is seen to turn points on the sphere about th e SI axis by an angle o. If the plate had been oriented with its fast axis at an azimuth cpl with Ipl'------YL- ...:::'l.I I s respect to th e X Y axes, the rotation by 8 would h ave been about the line from the center of the sphere and the point 2 cp ' on the equator . > R eflection at ft metftl surfftce may be represen ted similarly . For refl ection, the inciden t li ght wave is b resolved in to componen ts in th e plane of incidence and normal to the plane of incidence (the plane of F I GURE 4. Representation of metallic reflecti on on the Ponicare th e surface). The process of refl ection introduces ft sphere. /' T he azimuth ofthe plane of incidence is represented by J. and the plane of the phase differ ence Ll between t hese two componen ts, surface by J• . In part a t he view is along the S , axis. T he point Pis rotated to and changes tbe ratio of their amplitudes by a L aronnd the S , axi s by the retardation 6. of the surface. Relative absorption of the components in and normal to the plane of in cidence causes translation of factor tan if;; that is, tan if; is a measure of the rela- L to L' along the great circle Jp-L. The view in b is normal to this great circle. 365 \

of the axis 8 1 with the sphere and has the Cartesian and the chord I pL' may be calculated as coordinate 8[= 1,82=0, 83= 0. By eq (6) this cor­ ~ responds to zero ellipticity, x, and azimuth, 'P , so it 2 represen ts the orientation of the X axis, or the plane (13) I of incidence. Likewise, th e point I s has Cartesian coordinates (-1, 0, 0), so it corresponds to zero Moreo ver, by eqs (11) and (12), we have ellip ticity and an azimuth 7r/2. Therefore, the {' point I s' represents the orientation of the Yaxis or plane of the surface. The poin t P representing the (14) { polarization of the inciden t li ght is rotated around .' the 8[ axis through an angle .1 , the phase change produced by refl ection , to the point L. The Stokes and, in general, the ratio of the chords between a parameters of L are poin t repr esen tin g the state of polarization of a given ligh t wave and any two diam etrically opposed points on the spher e is equal to the ratio of the amplitudes of the componen ts when the electric vector of th e light is resolved along axes represen ted by these two opposed poin ts. (9) The Poincare sphere represen tation of the refi ec­ tion of light from a surface with known complex 2 tan a sin (iJ +:~ ) refl ec tion coeffi cient tan if;eiil may now be sum­ l + tltn 2 a marized. The point P , representin g the state of polarization of the incid en t ligh t is moved by the .J. , In addi tion, the relative absorption of the com­ refl ec tion process through an angle .1 on the spher e ponents translates the point L to the point L' , the in a plan e normal to the 8 ] axis to the poin t L. The Stokes parameters for which are given by eqs (8) . chords I pL and I sL are m easured and tan a calculated From eqs (8) and (9) we hfwe 8'{ /8'; = 8; / 8~ = t an (lJ+ .1). by eq (ll). Finally the angle I pl sL ' is calculated Points L and L' will, therefore, lie on the great circle from eq (12), or the chord I ])L' is calculated from o'iven by the locus of points on the sphere with eq (1 3), thus determining the state of polarization ~3 / 82 = tan (8+ .1 ). This locus is shown in figures 4a of the refi ected light. We shall show later how this and 4b and is given by the intersection of the sphere represen tation m ay be used con venien tly for the cal­ with a plane passing through the point L and normal culation of the eff ects occurring on refl ec tion . Firs t, J to the axis 8 1, Thus the poin t L must be moved however, we give a brief description of the instru­ 1 along th e great circle defin ed above to poin t L ' to ment. r epresent the polarization of the refl ected light. The 2.1. Instrument position of L' on this great circle will now be derived. The Car tesian coordinates of I p and of I s are (1, 0,0) Figure 5 shows the various components of an and (-1, 0, 0), respectively, and the coordinates of L ellipsometer. Collimated monochromatic light, usu- ar e given by egs (9). Therefore, the length of the ally the mercury green line (5 460.73A), is used. chords I pL and I sL are 1;::;; The polarizer, a Glan-Thompson or a Nicol prism I 2 tan a mounted in a graduated circle, serves to pola.rize the light emitted by the source. The compensator, also mounted in n. gr aduated circle, is a birefri ngent pla. ' e (10) usually of quarter-wave t hickn ess; it is used to C011 - 2

so ~~gu~~ e 8 Collimotlng Lenses I pL . Photocell I sL = tan a. (ll) \ I ~ Monochroma tic CoII ,mo te d ~'i!. / Fill" The angle I vIls is a righ t angle since it is inscribed ~o~~~~~~~~~~ ~ Plone liqhl in a semicircle. Therefore, the angle I 1J sL is equal I t o a, and it is clear by eq (2) that the tangen t of this Monochr omatic ./ Lig ht , angle is equal to the ratio of the amplitudes of the .-v , componen ts of the incident ligh t. The position of L ' is determined by t he angle I pl sL '. The tangen t of this angle again gives the ., r atio of the amplitudes of the componen ts of the r efl ected ligh t, which is now equal to tan a/tan 1/; . Therefore,

Pho tometE r I pL' . (I I I') .1. I sL,= tan p s ~ = tan a /ta n 'I' (12) FIGURE 5. The component parts of an ellipsometer. 366 L_ j

vert the lin early polarized light into elliptically mlllHYlUm in th e photometer reading is sough t at polarized ligh t. The lig ht inciden t upon the sample which the scale r eadings differ by 90°. The planes h as a n azimuthal angle and an ellipticity predictable of transmission of the two prisms should th en be in from th e settin gs of the polarizer and compensator. the plane of incidence a nd normal to it . The prisms In gen er al, the ligh t will h,wc its ellipticity a nd azi­ may now be rotated in their holders until th e scale muth ch anged by r efl ection from the ample. r eadings are 0 ± 180°, and 90 ± 180°, ~Lnd the aline­ The "aper ature" is an adju table opening allowing ment is complete. a variation in the area of surface examined and the In practice, however, cer tain sub tle effects occur. amount of light r eaching the photo tube. The After having fu·st crossed the prisms wh en reflecting a nalyzer, a second prism in a gr aduated circle, from the surface, a minimum may be ,w hi eved either can be rotated until a minimum intensity is achieved, by fixing P and adj usting A , or vice versa. If as indicated by the photometer. If som e ellipticity the light issuing from t he polarizer is accurately exists in the light reaching the analyzer, extin ction plane polarized , then the minimum achieved by of the light cannot be obtained by rotation of only either of these two means, . and with the P a nd A the analyzer. The polarizer and analy zer ar e t hen scales crossed , occurs at a single value of P a nd A ' ~ adjusted alternately to remove this ellipticity and independent of the procedure followed . However, I, obtain extinction. W·hen this occurs, the light r e­ if the light issuing from the polarizer h as some > flected from the sample is plane polarized and can sligh t ellipticity due to some imperfection in the b e thereby extinguished by the analyzer. The polarizer, then there ar e two positions at which tbe photo tube and photometer permit determination of prisms are crossed and minimum photometer readings the null point with a high degree of sensitivity. are ob tained, depending upon whether t he minimum For more accurate null point determinations graph­ is achieved by fixing P and adjusting A or vice ical plots of the in tensity of the transmitted beam versa. Moreover, the deepest minimum or most I • versus angle of polarizer and analyzer may be used complete extinction of the reflected lig bt n ow [11]. occurs at a t hird point, at which the scales are not crossed . Fortunately, with cer tain assumptions, 2.2. Alinement even in t his case alinemen t may be achi eved quite accurately. We shall first give a theoretical explana­ Alinement of the ellipsometer is not too critical t ion of the phenomewL observed , a nd then give a for the measurement of t hickness a nd refractive procedure for tLlining t he ellipsometer. index of thin films. H er e the impOl'tan t quantity We shall assume that the polarizer produces lig ht is the change in readings of the polarizer and ana­ of a co nstant ellipticity X whose azimuth changes as lyzer as compar ed to t he values for the bar e substrate. tlte polarizer is rotated. The analyzer is taken to However, for ob taining accurate values of the optical be perfect. It is assumed that the P a nd A prisms constan ts of surfaces, alinement is cl'it icttl. More­ h ave been adj usted without r efl ection so that when over, during alinement certain co nfusing phenomena minimum transmission is achieved t he scale readings may occur, and it is worthwhile h ere to develop the differ by 90 ° . The compensator is removed frorn theory of the ,dinement process. the system. The process will be depicted on t he The procedure for alining the ellipsometer is typi­ Poincare sphere. W e consider only ellipticity a nd cal of that gener ally used in optical spectrom eters azimuth much smaller than unity. In t hi s case the except for the adjustment of the analyzer (A) a nd Stokes parameters given byeq (6) ar e approximated polarizer (P ) scales. When alined , the scales for the by p olarizer and analyzer read zero when t he planes of s[= 1 tr ansm ission of the prisms are parallel to th e plane &2= 20' cos8 = 2cp (15) r of in cidence. This is accomplish ed by flrst adjustin g s3 = 20'sin8 = 2x· I~ th e polarizer and analyzer prisms in their scales so f that these scales differ in setting by 90° when th e H ence t he state of polarization Ill ay now be repre­ '1- prisms ar e crossed . These scales, imagined as a unit, setIted by tbe plane polar coordinates 20' a nd 8, must then be rotated so that when the P and A scale and this corresponds to r epr esenting as a plane r ead 0 and 90° r espectively, the planes of transmission surface the portion of the sphere on t he equator of th e two prism s are r espectively in the plane of around the azimuth represent ing the plane of incidence and normal to the plane of incidence. That incidence, i.e., around I p. is, th e coordin ate system defined by th e 0 and 90° U nder the above assumptions, the states of polar­ settings of both P and A mLi st be made to correspond ization of t he light issuing from the polarizer all lie to th e coordinate system defined by th e plane of on tbe line incidence and th e plane of t he surface. In principle, the adj ustment is r elatively simple. 20' sin 8= 2x (16) ·With th e compensator removed a nd the polarizer and a na,l yzer arms in the 'straight-through ' position, p = _ X_ . (17) th e polarizer and t),nalyzer prisms are adjusted tan 8 in their respective sc,Lles un til exti nction is achieved with thc scale r eadings differing by 90°. The arms An.\T point P? witb the polar coordinates 20', 8, on this are th en set for reflection from a metal surface. A line is rotated and translated by the process of re- 367 \ flection to the point P' with coordinates 20",0', with S3 .. the relations

J" 2O" =~ tan ~ (18) R 0'= 0+ 60. p. I , I ' This process is illustrated in figure 6, where 1 - ]' 8 " represents the locus of points representing the inci­ '~ dent light and R - R' the reflected light. Se The state of polarization of the light after reflec­ tion, represented by the point P', may be resolved into components along the plane of transmission of the analyzer and normal to it. The latter direction is represented in the figure by the point 2L. Under these conditions of very small azimuth and ellip­ ticity, and by eqs (11) and (2) , the amplitude of the light transmitted by the analyzer is proportional to R' the distance D between P' and 2L. By the triangle P', 2L, I p , the intensity of the FIGU RE 6. The Poincare re prese ntation of metallic reflection transmitted light is proportional to for light of very small azimuth and ellipticity caused by an imperf ect polmizer. 8Lx t. . The paint a.lon g [ - [' such as p ,a is translated to a point on R - R ' snch as P' (COS-- -slnuA) (19) by the process of reflection . This is passed through as analyzer with plane of tan ~ tan 0 transmission at an azimuth L+,,/2. The amplitucie of tbe light issuing from the analyzer is proportional to D . where 2L is to be considered an algebraic quantity, negative in figure 6. The process of taking readings may be accom­ p plished in two ways. First, the azimuth of P may be set and A adjusted for minimum transmission. This is equivalent to fL'Cing 0 and adjusting L, I.e. , ABSOLUTE MINIMUM O(D 2) = 0 oL ' t whence

L tan ~ = x(cos 60 cos o- sin 60 ) or (A+~) tan f=Pcos 6o -x sin 60 (20) which is the equation of the locus of points obtained in this manner. Equation (20) is plotted in figure 7. The polarizer FIGU RE 7. The lows of points P and A re presenting ext1:nction and analyzer will be crossed for only one point on of reflected light for an impelject polarizer and pelject analyzer. this curve. For this point, P = A+~, and The npper line is ach ieved hy fixing A and adjusting P for each settin g, and the lower by fixing P and adjusting A for each setting. Note that the upper line passes th rough zero azimn th. x sin 60 P (21) cos t.- tan ~ and and this crossed position is in general not at zero (A+~) tan f cos t.=P (22) azimuth but is determined by the constflnts (o =~} which is different from eq (20) . The point for which 60 and ~ of the surface. P and A are crossed is given by However, the scales may also be adjusted by first setting A and moving P. In this case, the condition P (l - tan f cos 60 )= 0. (23) for a minimum in transmitted light is Since the q uan ti ty in paren theses is no t in general O(D 2) = 0 zero, L tan ~ tan 0 cos 6o =x 00 ' P=o. (24) 368 Thus it seems clear that the crossed position of the - x P = -­ (26) 7 prisms achieved by setting A and adjusting P tan Ll for a minimum will always be the true azimuth under these circumstances, i. e. , an imperfect polarizer, but By eqs (17) and (26) perfect analyzer. This is represented in figure 7, tan Ll = - tan 0 where we show both the li ne achieved by fixing P or and adjusting A and the line achieved by fixing A (27) and adjusting P. These curves wer e calculated from eqs (4) and (6) for Ll = 135 deg and tan f = ?~ , and P so that the deepest mlHlmum occurs when the re­ and A are in the uni ts of x. It will be noticed that the fl ected light is linearly polarized, as would be ex­ curve obtained by first fixing A passes through the pected. This minimum is shown in figures 7 and zero azimuth. by the intersection of the two lines. The absolute For comparison, a curve obtained experimentally minimum determined by alternate adjustment of P is shown in figure 8, where the numbers indicate and A is the same as this point of intersecLion within meter readings and hence are proportional to the experimental error. It will be noticed that this point transmission. Since the axes of figure 8 are inverted with respect to figure 7, the ellipticity of the polarizer is not on the line P = A+~' However, if x= O, then must be negative. However, the similarity is strik­ the deepest minimum occurs at zero azimuth for ing. The value of x computed from thi s figure and both P and L , and thus either method of alinement eq (2 1) is - 0.1 8 deg. 111 ay be used if the li ght is perfectly plane polarized. It will be clear from a consideration of eqs (2 0), Thus, there appears to be only one unambiguous (22), and (12), that it is best to use a polarization way of ali n ing the ellipsometer when the light from azimuth near the plane of incidencemther than near the polarizer has ellipticity, namely, to seek thftt the plane of the surface, since tan f is generally less than uni t~T , and this has the effect of magnifying the point ftt which both P = A ± ~ and minimum trans­ angular difference between P and A ± n-j2 by an Illission is obtained when A is set and P adjusted. amount l /tan f . If the polarizer were set near the plane of the surface the angular spread between P 2.3. Alinement Procedure and A+ 7['/2 would be decreased by an amoun t tan if; , thus decreasing experimental precision. A step-by-step proccdure for alinement of the The condition for the deepest minimum or best ellipsometer is as follows: extinction is First, remove the quarter-wave plate. Lower the a( D 2) = a( D 2) = 0 arms to the "straigh t through" position. Set the P aL '00 scale to reftd zero or any co nvenient value. Adjust A until minimum transmission is achieved. The A and is thus given b\' the simultaneous solution of scale will now in general read (P ± 90 )+ E. Rotate eqs (20) and (22). This gives the A prism in its holdcr until Eis zero. The P and A cales are now crossed when the prisms are crossed. The scales should track within ± 0.02 deg throughout (25) one revolution. With a metal surfrtce set for reflection, raise the arms to an angle neal' the principal angle. Set A so that the plane of transm ission of the rtnftl?zer is approximately in the plane of the su rf rtce. Adj ust P until minimum transmission is achi mred, and note the readings of the P and A scales. Move A by 0.1 deg and again adjust P and note the values. If the meter reftding is higher than for the previous set of values, ad just A b? 0.1 deg in the otiler direction. If it is lower, continu e in the same direction. Take a series of readings in thi s mrtnnel' , changin g A by 0.1 deg and rtdj usting P for minimum trftnsmission at each setting of A, making sure that the settings encompass the lowest meter reading. Plo t the values of P and A and determine the readings of the P and A scales at whicll P = A ± 90°. At this point the plane of transmission of' the P prism is in the plane of incidence and the plane 92 93 94 of transmission of the A prism is in the plane of the A,deg surface. However, the scale readings will in general FIGU RE 8. The experimental countel'pal't of fig1l1'e 7. not be some multiple of 90 °. Let the values of P The P and A coordinates of the intersection of the line P = A -"./2 and the and A at this poin t be, for example, 0° + E' and 90° + E' " fIX in g A " line represe nt the azimuth of the plane of incidence an d til e pl ane of the surface, respec tivf'l y. respectively. The quantity E' is now the amount by 369 i ~ \ which both scales are displaced from the true zero readings in all. However, since both analyzer and azimuth. Set A at the value obtained from the polarizer may be rotated by 7r without affecting the graph. Adjust the P prism in its holder until ~' is results, there are 16 polarizer and analyzer settings zero by turning the prism a small amount in its falling in to four independen t zones. Since the holder and again adjusting the P scale for minimum compensator may also be rotated by 'If without transmission. This will require a series of adjust­ affecting the results, there are 32 possible sets of ments. (It is imperative that the A scale and prism readings on the ellipsometer. not be moved at this stage and that the adjustment. Rather than calculatin g Ll and f directly from the be done in this way. If, for example, the P scale P and A values, it is useful to calculate three other < were h eld and A adjusted, the ellipsometer would be quantities, p, ap , and as from the P and A values, p being related to the P readings, ap related to the alined to the value of P=A±§' obtained by fixing P A readings in zo nes 1 and 4, and as related to the and adjusting A, which has been demonstrated to be A readings in zones 2 and 3. For a perfect quarter­ incorrect.) Having adj usted ~' to be zero, remove wave plate these are related to Ll and f by the eq (2) the reflecting surface and lower the arms to the 1 "straight through" p08ition. Set the P scale at (28) zero and adjust the A scale for minimum. The A Ll = 2"7r + 2p scale will now read 90 + ~'. Turn the A prism in its f = ap = a s' (29) holder nn til ~' is zero. The ellipsometer P and A scales are now alined. If the compensator is not a perfect quarter-wave This procedure eliminates alinement errors due to plate, the relationships are ellipticity produced by the polarizer. The effect of ellipticity in the analyzer on alinement has not been -tan Ll = sin 0 cot 2p (30) determined but is expected to be small since the tan2 f = tan a" tan as (31 ) analyzer is used for extinction of the light. Thus, any ellipticity by the analyzer after extinction of the where 0 is the relative retardation of the com­ light will have no eff ect on the photometer reading. pensator. If the retardation 0 is near 7r/2, then, to 2.4. Determination of Ll and f From Ellipsometer first order, eqs (28) and (30) are the same. However, it is now necessary to distinguish between ap and as. Scale Readings The relationship of p and ap and as to the P and A By eq (7) the reflection of light from a surface is readings observed on the ellipsometer is best derived characterized by the complex reflection coefficient by a consideration of the process as represented on tan f ei l1 , and hence by the two quantities f and Ll. the Poincare sphere, and since this has been It will later be shown how the refractive index of a adequately treated by Winterbottom [2], we shall surface and the thickness and index of films on a merely list the results. The meanings of p, ap , surface are calculated from values of Ll and f . and as in the four zones are as follows: However, the determination of Ll and f from P and Zone 1. The fast axis of compensator is at -7r/4. A readings merits some discussion, for this is not The polarizer plane or transmission makes an angle always a perfectly direct matter. of +p with the plane of incidence. The analyzer For any given surface there is a multiplicity of plane of transmission makes an angle of +ap with polarizer, analyzer, and compensator scale settings the plane of incidence. that produce extinction by the analyzer of the Zone 2. The fast axis of the compensator set reflected light from the surface, and it becomes at + 7r/4. The polarizer plane of transmission somewhat of a problem to determine the values of makes an angle of -p with the surface (an angle 1l and f from these various readings. In order to of 7r/2 - P with the plane of incidence). The explain how th ese numerous readings arise and how analyzer plane of transmission makes an angle of 1l and f may be computed from them it is well to +as with the plane of incidence. keep two facts in mind: (a) all azimuthal angles are Zone 3. The fast axis of compensator is at - 7r /4. measured positive counter-clockwise from the plane The polarizer plane of transmission makes an angle of incidence when looking into the light beam, and of +p with the surface (an angle of p + 7r/2 with (b) the compensator, which may be set at any azi­ the plane of incidence). The analyzer plane of muth, is generally set so that its fast axis is in an transmission makes an angle of -as with the plane azimuth of ± 7r/4. The present discussion is for an of incidence. instrument such as that shown in figure 5 with the Zone 4. The fast axis of compensator is at + 7r/4. compensator before reflection. If the compensator The polarizer plane of transmission makes an angle is placed after reflection, suitable corrections will of - p with the plane of incidence. The analyzer have to be made. plane of transmission makes an angle of - ap with The various readings fall into four sets called the plane of incidence. zones, two with the fast axis of the compensator set The meanings of p, ap , and as are now clear. The at 7r/4, IlUlnbered 2 and 4, and two with it set at first is the angle between the polarizer plane of trans­ j - 7r/4, numbered 1 and 3. In each zone there is mission and either the plane of incidence or the l one independent set of polarizer and analyzer read­ plane of the surface, while a p and as represent the ings, making foul' independent sets of P and A angle between the analyzer plane of transmission 370 --, I

and the plane of incidence, being labeled either a p When working with a completely unknown surface, l' or as according to whether p is the angle between and particularly if multiple refl ections are used, it is the polarizer plane of transmission and the plane bes t to take a complete set of 16 readings for the of incidence or the plane of the surface, respectively, two settings of the compensator. Then, wi th the With these relationships, it is a simple matter to relations (33) and (3 5), and the further co ndition that compute the relationship of P and A to p and a p the values of p, ap , and as calculated from these or as, and hence to ~ and 1/;, by means of eqs (28 ) readings must be approximately the same, it is a and (29), or (30) and (3 1). These r elationships are simple matter to identify each of the cale r eadings :> summarized in table 1. with one of the possibilities outlined in the table. To con tinue to do this when the surface is known } TABLE 1. Relation of P and A readings to p , a p , and a. would be redundant and one reading in each zone is sufficient. W e have found that the average of one Zone Compensator P A r eading in each of two zones with the same compen­ sator setting gives approximately the same result as I -,,/4 l' (/p the average of one reading in each of all four zones. 1'+" all l' CL p +7r p+". up +7r 2 5. Differences Among the Zones ) 3 -,,/4 p+,,/2 7r-aa p+3.-/2 7r-a" p+"./2 211'"-a . In practice, the values of p, ap , and as found from 1'+3"./2 27r-a, the P and A readings in the various zones ar e no t 2 +,,/4 ,,/2- 1) a. identical. A typical set of values is shown in table 2. 3"/2- ,, a. ,,/2-1' (1, .+71" Differences of as much as 3 deg occur in the values of 3,,/2- p (1 3+11'" p and somewhat less in the valu es of a 1) a nd as. On 4 +"./4 7r - P 7r - U p t he other hand, it will be noticed that the average of 2 -rr-J) 7r-U 11 7r - /J 27r-a p the observed values o[ p in zon es 1 and 3 is the same 27r-p :l7r - a /1 as the average o[ the value obtained in zones 2 and4. This strongly implies the following eq uations PI = P+ O P4 = P+ O' ( ) For the purposes of this table, it has been assumed P3=P- 0 P2 = P - O' 36 that the P , A, and compensator scales h ave been adjusted to read angles with respect to the plane of wh ere 0 and 0' are error terms, These error terms are incidence as previously defin ed. .I [ the in strument not co nstant from run to run, but are usually 1 to is not arranged in this manner, it is necessary to ] .5 deg as in table 2. (In this particular set, 0 and adjust the readings before usin g table l. 0' are approximately the same. This is no t true in The table gives t he 16 possible readings for the general). They are no t caused by misalinem ent or two settings o[ the compensator. When working any other r e~L dil y detectable cause, and their source with a completely unknown surface it is still not a remains obscure. However, ft lltOn g very many simple matter to identify the P and A readings with experiments, no t a single one has been found where one o[ the entries on this table and hence to com­ the averages from zones 1 and 3 did no t check the > aver age from zones 2 and4 within experim en tal error. pute p, a p , or as. As a first step in this identification it is worthwhile to consider the ranges for ~ and1/; T ABLE 2. "Values of P and A lor a typical chrome slide in ail' a nd hence [or p, a p , or as. For all surfaces, including in all lour zones multiple reflections,

Zo ne P A l' (t _ (t. (32) --- I 19. 71 3 1. 04 19. il 31. 04 2 167. 17 31. 73 22.83 31. 73 3 Jl3.00 147.75 23.00 32.25 I In fact, in the large majority of cases 19. 80 31. 48 r ~ 4 160.20 148.52

A\'cragc of fo ur zon cs~ ______21. 34 31. 63 (33) A verngc o f zo nes 1 & 3 ______. __ 21. 36 31. 65 Average of zones 2 & L ______21. 32 31. 61

aud from eq (29) the range of a p and as is the same. ). For the special case of r efl ection from a dielectric The situation for a p and as is Hot quite so clear-cut. surface n,t an angle of incid ence smaller than the If the compensator is not a . per fect quarter-wave Brewster angle, ~ = - 71". For all other cases plate, then differences in ap and as are to be cxpect~cL In fact, it rnay be shown that [or a compensator WIth (3 4) retardation 0 near 71" /2,

which gives as a range for p, a p - as = (0 - 71" /2) cos 2p sin 21/; (3 7) 71" 371" (35) where, for the purposes or this calculation, we l?av.e - -4-

cates that the average of a p and as from zones 1 and 3 now being hydrophilic. However, in less than 1 hI' is equal to the average of a p and as from zones 2 and the surface became sufficiently contaminated to 4 which again strongly implies become hydrophobic, even when enclosed in a covered container. These precleaned slides were passed al = ap + o a4 = av+o' through a flame immediately before use to remove (38) this contamination; this flaming of the slides restored a3 = a S -o a2=as-8' the hydrophilic char acter to the surface. For studies where the a's with the nwnerical subscript are ob­ under liquids, the slides were placed under the liquid served values in the various zones. Many experi­ while still warm from the flame. The cleaning of ments indicate that these equations seem to be surfaces by ihming has been described by Pfttrick followed quite faithfully. The values of 8 and 8' in [12] and Bartell and Betts [13] . eqs (38) are, in general, different from those in eqs (36). 2.7. Cells It will be observed that the equations are not independent, and they cannot be solved for a p and Cells have been used for ellipsometer meftsure­ a s directly from the observed values of a. Fortu­ ments in vacuum or gaseous environments [2, 14, 15] nately, for a small value of ap-as, and for ~ not and under liquids [2 ]. The cell shown in fi gure 9 has too close to zero it may be shown that been used in our studies for measurements under liquids. The specimen is situated on the base of the (tan a p tan as)1/2 = tan (ap~as) (39) cell which is filled with liquid of known refractive index. The light beam enters and leaves the cell through optically flat windows. These windows ftre to first order in ap-as. Thus rather than using eq inclined at the angle of incidence if> with respect to the (31) for computing ~ , the above equation is used. base of the cell so that the light passes through them This requires a compensator not too different from at normal incidence. Therefore, reflection of light quarter-wave, and independent measurements indi­ at the surface of the cell windows will be independent cate that the compensator used in this work has a of its direction of polarization and the polarization of retardation which differs from 7r/2 by about 1 deg. the light will not be changed. There should be no Because the averages from zones 1 and 3 check so stresses in the glass sufficien t to cause detectable closely the averages from zones 2 and 4 for both p birefringence; moreover, the inner and outer sides of and a, measurements ar e usually taken in zones 1 each window should be parallel. They may be sealed and 3 only and averaged. to the cell body either by fusion or by epoxy resin; the epoxy seal has been used more successfully for 2.6. Surfaces our cells. The differences in polarizer and analyzer The substrates most easy to work with are those r eadings in air for a metallic reflecting surface out­ with high reflectance, such as metals. Both smooth­ side and inside the cell were no larger than 0.10 for ness and flatness are factors that must be considered our cells. in the selection of a substrate. Irregularities that Difficulties with photometer fluctuations have oc­ are small compared to the dimensions of the light CUlTed as a result of convection currents caused by beam, which is commonly of the order of 1 mm2, are evaporation of liquid. A small area of liquid in con­ averaged and do not affect the results [4]. Long tact with the air, a tightly fitting cover, and solu­ ran~e regularity (flatness) is desirable if more than tions filtered through fritted glass disks minimize one location on a specimen is to be studied. An indi­ this effect. cation of the regularity is obtained by determination The angle of incidence should generally be chosen of p and a at several points on the specimen. Meas­ to give the maximum sensitivity for the measure­ urements on I-in. long steel gage blocks of 0.09 fJ. in. ment of film thickness. For this purpose, sensitivity tolerance varied only about 0.10 in the pohtrizer readings may be defined as the change of P reading or A from top to bottom, and less in analyzer readings. reading with film thickness, t, i.e., dP/dt and dA/dt. Acceptable slides prepared by shearing small rec­ One is then faced with the problem of selecting an tangles from a large ferrotype plate varied about 0.20 angle of incidence such that these two quantities for similar lengths and these make convenient and are a maximum. No general rules can be laid out readily available surfaces for study. for this selection, and the choice of angle of incidence Various techniques have been used for the prepara­ will depend upon the particular substrate, film and tion and cleaning of the substrate surface. The surrounding medium. However, for the common case chrome ferrotype plate slides used in these investiga­ of an organic film on a chromium surface in an tions were washed in warm distilled organic solvents organic medium, the sensitivity for both P and A to remove organic contamination. Since the result­ has been calculated for various lJngles of incidence ing surface was hydrophobic, the slides were further and film thicknesses up to 1000 A. The results are cleaned with warm chromic acid cleaning solution shown in figures 10 and 11. followed by several washings in warm distilled water. The maximum sensitivity for A occurs at angles This trefttment should not appreciably affect the of incidence between 75 and 80 deg, and decreases character of the chromium-chr omium oxide surface, with film thickness. For P, the maximum sensitiv­ but it did remove organic contamination, the surface ity (the absolute value of dP/dt) occurs at fin angle

372 ! of incidence of 70°, and decreases with increasiQg film thickness up to a thickness of about 700 A. At this thickness, the scnsitivity in P is very small, making the thickness determination entirely depend­ ent on A, which is relatively iosensitive in this region . Thus, under these conditions, the accurate determi­ nation of film thickness is diffi cult for fums 600 to 800 A thiclc For fums thicker than 800 A, the sen­ sitivity in P again becomes usable, but the maximum FIGURE 9. A cell used for studies of sUljaces u nde?' li quids. occurs at angles less than 70 deg. For any film thickness, therefore, the best angle of incidence is always a compromise between the tr 0.8 most sensitive angle for P and that for A. The selection will, in par t, depend on the relative impor­

I 0.7 tance of P or A to the determination of the thickness. These sensitivities apply only to the given specific conditions but the sensitivities for other conditions 0.6 wi th a nonabsorbing film are expected to hfl,ve similar behavior although specific values will be different , 0.5 100

<( 0 2.8. Methods of Computation 0 0 .4 300 "g' " A typical system for study by the ellipsometer :;1:0 0.3 consists of a film of index n2 and thickness cl on a 700 1000 refl ectin g substrate of index n3 imm ersed in a 0.2 medium of iodex nl, as shown in fi gure 12. Let all media be iso tropic an d nl represen t a real index 700 0.1 600 of r efr action, while n2 and na may be complex. 3 00 ri ' 1000 Consider ligh t incid en t at the boundary between O ~~IO~O~ __~ __~ __~ __~ __~ __-L __ -L __ -L __ ~ the immersion m edium and film . The cosine of 60 65

FIGU RE 10. The sensitivity, dA/dt , of A for meawring thick­ n f cos 'P2= [ 1-( sin 'PI )2J1/2 • (40) F' ness t , plotted as a f uncti on of angle of inddence. i The parameters arc those for a t ypical organic film on chrome; n, = 1.4268, m=1.4700, n,=3.l480Q-4,142i9j. I The parallel and normal reflection coefficients [or light incident at this boundary are:

0.7 T I - (41 ) 05 and (42)

1000

700 60 0~ I 300 Immersion 100 NORMAL Medium 0 )' l I

- I

45 50 55 60 65 70 75 80 85 90 <1>, deg

FIGURE 11. The sensitivity, dP(dt of P for measuring thickness t, for the same conditions as figure 10. FIGURE 12. R efl ection fro m a film-covered surfa ce. Note the very low sensitivity at a fil m thickness of iOO A, The substr ate is a refl ecting metal surface.

373 r \ respectively. The reflection coefficients, T~3 and r~3 at the boundary between the film and substrate are 500 given by similar expressions. 600 The total reflection coefficients, Rv and R s that 400 include the contributions of reflections from lower boundaries are given [2] by: 700 12 8 Tf2 -1- r~3 exp D (43) 1 +Tf2r~3 exp D and 132 R s= Tf2 + r ~3 exp D I (44) " 1 + 1'f2r~3 exp D ::,J where cos 'P3 values needed for these reflection 136 coefficients are given by an expression similar to eq (40) and D represen ts the quan ti ty

(45) 14 0 where A is the wavelength of the light used , ill vacuum, and j ="/- l. The ratio of the parallel and normal total reflection coefficien ts IS defined 144 [~ as p: \ (46) n2 'L60 34 36 38 40 44 ~ This may be expressed in terms of the relative

p= tan if; exp ULl) (47) where 0 1, O2 , and 0 3 are complex functions of t he refractive indexes, angles of incidence, Ll and if; . as in eq (8) . Thus p is determined from ellipsom­ For a given value of the coefficients eq (49) gives two eter readings. solutions for exp D and a value of d m.ay be calculated The value of the complex index of a reflectirw from each. Since the coefficients are complex, the surface can be calculated from the equation b film thicknesses calculated from this equation would also be e.ll.'Pected to be complex, However, the cor­ 4p sin 2 'PjJI IZ rect film thickness, d, must be a real number as it n3=n1 tan 'PI [ 1- ( p+ 1)2 (48) represents a real quan ti ty. Therefore, the solution of the quadratic that yields a real film thickness is where 'PI is the angle of incidence and p is d etermined the correct solution. In practice, various experi­ from ellipsometry measurements on the base sub­ mental errors will r esult in both solutions yielding strate. complex values for d. The thickness with the small­ Several methods of determining the thicknesses of est imaginary component is selected as the correct films on reflecting .substrates from ellipsometry meas­ solu tion; the re~tl portion is taken as d and the imagi­ +- 1 urements are avaIlable [2- 5]. When the indexes of nary part, el }, is taken as a relative measure of errol'. the substrate and film are known, tables or graphs of The real portion of d is then used to compute Ll and Ll and if; may be computed from given values of cZ. if; by eqs (43) to (47) . As the imaginary component This is accomplished by calculating values of p from of d has been dropped, these values will differ from eqs (40) to (46) and values of Ll and if; from eq (47). the experimental angles by amounts /iLl and oif;, and FIgure 13 shows curves com.puted for three different d}, oLl , and oif; are all measures of the experimental values of film index nz. The numbers along each error. However, /iLl and oif; must be within the limits curve are thickness values of the film, d2 • Experi­ of experimen tal error of if; and Ll for the results to be mental values of Ll and if; are t hen interpolated in valid. This is a more direct determination of the such tables or related graphs to determine unknown validity of an experiment than the magnitude of cl}. thicknesses. If both the thickness and index of refraction of the However, it is usually more efficien t to solve the film, n2, are not known, the equations cannot be equations directly frOllt the thickness of a film. Sub­ solved for cl and nz in closed form. For this case, a stituting eqs (43), (44), and (46) in eq (47) and series of refractive indexes are assumed and a thick­ rearranging gives a quadratic of the form: ness is calculated from the experimental measure­ " ments. These calculations will result in error terms, described above, of different magnitudes. The ~I 374 ;!

r ange of refractive indexes and r ela ted thicknesses T A BI,E 3. j\![easured refmctive index, n = N - jK, of chrome a Ii within the experimen tal enol'S M and Of are then and gold a in ait· and in water ) chosen. The magni tud e of the possible ranges of Chrome Gold I refractive indexes and thicknesses will depend upon rm mersion mediu 111 the magnitude or the eITor and on several parameters 7 N J( N I< . such as n l, n2, n3, etc. Some of the principles described above are Air ._. ______2. 954 4.224 0.446 2. 134 l illustrated in figure 13, which shows theoretical 6. and \Vater ______3.231 4.354 . 528 2. 165 >/I values for a range of given film indexes and thick­ Air . ______3.005 4.27'1 .427 2. 11 9 nesses. An experimentally determined point (Ll e, \ Vater ______. 3.218 4.302 .5'16 2.100 }\ir ______. ______2.950 4.258 >/Ie) for a film of unknown n2 falls neal' one of these WaLer. __ . ______. " curves. For an assumed film index of 1. 5, a thick­ 3.285 4. 426

ness will be calculated corresponding to the point a Slides had been fl amed prior to examinaLio n. (Ll e, >/Ie) . The errol' terms, M and Of are also illus­ tr ated. The error terms are usually much smaller " than indicated on this figure. water are probably caused by the presence o[ an The calculations discussed above can be com­ adsorbed film of water on the slides when measured ( puted m anually, but the equations are complicated in air. Such a film would not be observed, of course, ~ and the computation or any appreciable amount or when the meas uremen ts were m ade under wa ter, data would be impractical. Several "Fortr an " assuming the refractive index of the film to be the programs have been written in this laboratory for sam.e as tb at or water. It is presumed, therefore, IBM 704 and 70 90 compu ters to enable the com­ that the measurements m ade under water yielded putation or data from ellipsometer measurements. the correct indexes [or these slides. The programs are generaJ enough to permit their use The measurements described above were also in most situations encountered. The followin g is a repeated for immersion m edia of differen t refractive brief description of problems that can be solved with indexes. The op tical constants of the m etal calcu­ tbese progr ams. lated under these conditions are given in table 4. The refrac tive index of the substrate, n 3, may be r computed directly from experimen taJ r eadings on the >. bar e substrate. Immersion media of diA'el'ent re­ TABLE 4. R efmctive indexes, N- jIC, calculated for clu·ome fractive indexes may be used fo r this detennina, tion . slides immersed in various liquids immediately aftm· flaming ,Vhen a" and as are known , the correct r etHrdaLion Refractive index a L of the compensator call be caJcul ated. This retarda­ [ Illillcrs ion IlIcd iUIIl " n r-' tion may then be used in subsequent calculation s. N J( Thus, the ellipsometer data m ay be used even i[ the compensator is no t a quarter-wave plate. De(lree 60 lllcLh all ol . ______I. 329 3.212 4. 335 Curves and tables of 6. >/I , and refl ection coeffi cients (i0 water ______. I. 337 3.41(j 4.358 can be compu ted for a series of given film thicknesses 70 water ______I. 3J7 3.278 4.3iL GO acetone ------I. 3GO 3.275 4.3G3 and refractive indexes, n2, as sbown in fig ure 13. iO acetonc ------I. 360 3.296 4.398 iO cyclohc,ane ______I. 426 3.254 '1.396 For experimental values 0 [' Ll and >/I , a thickn ess, d, 70 tOlucne ______I. 498 3.344 4.437 and error terms, 0>/1 and 06., are computed for a given 60 air h ------1.00 2.899 4. ]70 n2. If n2 is unknown, a series o[ n2 can be assumed 70 air h ------I. 00 2.918 4. 198 and corresponding film thichness fmd error terms computed. The selection o[ the correct n2 a,n d :J. All refractivc indexes arC averages of two or 1110re se ts of measuremcnts , ex­ cep t for me thanol. thickness has been discussed and is illustrated in b Slides JIlcHsured in ail' had been exposed to ai r from 1 to 24 hr. the next section. All the a,bove calculations h ave been extended for multiple films and fo r multiple refl ections. The slides measured under liquids were fl amed and immediately immersed in the liquid before a film 3. Applications could form on them from tl'le air. These slides are therefo re not expected to have any adsorbed film on Inasmuch as surfaces in air will usually have an them , so tha.t their true refr active indexes are adsorbed film of water or other contaminan ts, in measured. If some film did remain on them, its order to determine the index of a substrate it is refractive index would proba bly be near that of the necessary either to measure the surface in a vacuum ill1.J1l.ersion liquid, causing only a small error j 11 the or in a medium with a refractive index identical to measured refractive index o f" the slide. Hence t he that of t he adsorbed fi lm (nl = n 2). rerfactive indexes of the slides u nder the various In the work reported here chrome slid es wi th and liq uids are all approximately th e same, and inde­ > without a vacuum deposit of gold were fl amed to penden t of the liquid and the angle of incidence. ~ remove adsorbed gases and immediately immersed The variation is due m a.inly to vari ations among the I in n, liquid. T able 3 gives the refractive indexes of individual slides used fo r the different measurements. ~ slides n1.easured in air and measured under water. The refractive index measured in air shows a The differences in the refractive index of the m etals difference due to neglect of the films that is larger calculated from m easuremen ts made in air and under than the variations among the individual slides . 375 This film is expec ted to be adsorbed water and, TABLE 6. 'Thickness of evaporated barium fluoride jilm on perhaps, other gases and is expected to .have. an chrome slide" b index near th at of water. By first m easunng shdes .- in air and th en measuring them under water, it is aLl f ·f n, d therefore possible to cal culate a thickness of the '" Degree Degr ee Degree Degree A adsorbed layer in air, assuming the refractive index 133.96 3.64 36.30 0.60 1. 420 753 or the adsorbed film to be the same as that of liquid 2.64 . 40 I. 430 726 1. 58 .22 1. 440 701 water. Such results are shown in table 5, where I. 08 .14 1. 445 688 0.58 . 07 1. 450 6i6 M and Of are the differences obtained as described .28 .03 I. 453 668 above. The reproducibility of the results is seen to . 18 . 02 I. 454 665 be quite good, although the actual value of the . 10 . 01 I. 455 662 . 00 . 00 1. 456 660 thickness may be somewhat m error due to th e . 10 . 01 1. 457 657 ass umption of the r efractive index value. .20 . 02 I. 458 654 .28 . 03 1. 459 651 .38 .04 1. 460 647 TABLE 5. T hickness of ad SOl bed layer a on slides in air .70 .38 1. 475 453

a. IJlllllCrsed in acetone, nl =1.359, at an angle of incidence 0(60°. rrhe refractive Slide Ll aLl

table 6. "'The reproducibility of the readings is 134.0 ± O.l o for Ll and ± 0.05° for1/; and the maximum ~ experimen tal error is expected to be about ± 0. 2° I for Ll and ± O.l o for 1/; . It will be observed from table 6 t hat the index 1.456 gives zero error terms M and Of , and is, therefo re, the best fi t with the 134.8 experimen tal rea dir~ g. However, n: ,:alues .fI' ~m 500 1. 454 to 1.458 also gIve error terms \ Vl thm the 111111 ts of experimen tal error, and are the range of possible 1352 indexes of the filrn . The correspondin g thicknesses are 654 and 665 A. 135.6 L_...L_---1._-::-L,-_...L_d:--_L----;~----1.----' The actual range of possible thicknesses is greater th an this range as is illustrated in figure 14. This F I GURE 14. An enlargement of jig lll'e 13, showing how error fio-ure is an enlargement of the center portion of in 11 , and,p, leads to an unceTtainty in the calculated refractive rigure 13. Cun ·· es for a::i ditional refracti ve indexes index and thickness of a jilm. 376 >. 4. Summary index. If the refractive index chosen is not the I exact index of the film, and if the measurements are ~ The use of the ellipsometer for the meflSUl'ement not sufficiently precise, the calculated thickness of I of the thickness and index of refraction of very thin the film is complex. The imaginary part of the films is described and illustrated by eXflmples. The computed complex thickness is taken as a relative representation of elliptically polarized lil,?ht by means measure of error eitber of the asswned refractive of the Poincare sphere is developed in detail since it index or of the original measuremen t. The film is necessary for a complete understanding of the reJractive index with its corresponding thickness changes in the polarization of light in the ellipsometer. that yields the smallest error terms is taken as the Some unusual effects in alinement have been best fit. This calculation is illustrated 1'01' a barium observed and explained as due to the liaht from the fluoride film. polarizer having a slight ellipticity. A method of a~inement that is not affected by the ellipticity is 5. References gIven. Chrome ferro type plates have been found to be a [1] P . Drude, Ann. Physik 272, 532 (1889); ibid. 272, 865 (1889) ; ibid. 275, 481 (1890). co n venient substrate for study of thin films. Means [2] A. B. Winterbottom, Optical Studies of Me tal Surfaces, of prepflring and cleaning the plates are discussed. The Royal Norwegian Scientific Socicty R eport No.1, Also, cells used to study films under liquids are J 955, published by F . Bruns, Trondhcim, Iorway; described. Trans. Faraday Soc. 42, 487 (1946). [3] L. Tronstad, Z. Physik . Chem. AH2, 241 (1929); Trans. In order to study a film by ellipsometry, the Faraday Soc. 31, 1151 (1935); The Royal Norwegian reflection coefficient of a bare substrate is first Scicntific Socicty R eport No. 1 (1931 ). measured and the complex refractive index of tbe [4] F. A. Lucy, J. Chcm. Phys. 16, 167 (1948). substrate computed from the reflection coefficient. [5] A. Rothen, Annals N cw York Academy Sci. 53, Art. 5, 1054 (1951); Rev. Sci. Instr. 16, 26 (1945); and M . A flim is then deposited on the substrate and the Hanson, ibid. 19, 39 (1948), 20, 66 (1949) ; ibid. 28. reflection coefficient of tbe combination is measured. 283 (1957) . If the index of the film is known, the tbickness of [6] J. B. Bateman and M . W . Harris, Annals N cw York the film may be computed. In one method, the Acadcmy Sci. 53, Art. 5, 1064 (1951 ) . [7] J. A. Faucher, O. M. McManus, and H . J. Trurnit, reflection coefficient of the substrate with a film is J. Opt. Soc. Am crica 48, 51 (1958) . computed for many thicknesses o~ the film. Then [8] O. G. Stokes, Trans. Cambro Phil. Soc. 9, 399 (1852). the measured reflection coefficient is interpolated in [9] M. Born and E. Wolf, Principlcs of Optics (P ergamon the calculated results to determine the thickness Prcss, 1959). [10] W. A. Shurcliff, Polarizcd Light (Harvard Univcrsity of the film. However, in this paper the required Press, (J 962). equations are solved to give the thickness of the [ll] R. L. Patrick, Alpha Rand D Inc., Bluc Island, Ill. • film directly in terms of the measured reflection (private communication) . coefficients so that interpolation is not required. [12] R. L. Patrick and O . O. Paync, Jr., J . Colloid Sci. , 16, 93 (1961). These measurements and calculations are applied [1 3 ] L. S. Bartell ancI J . F . Betts, J. Phys. Chem. 64, 1075 to determining the thickness of an adsorbed film on (1960) . ferrotype plates. The index of refraction of this [1 4] F . B. Bowden and W . R. Throsscll, Proc. Royal Soc. [ film was assumed to be that of water. A209, 297 (1951) . [15] J. Kruger and W. J. Ambs, J. Optical Soc. Amcrica 49. > Both the index of refraction and thickness of a 11 95 (1959) . ? film may be calculated from the complex reflection coeffi cient of the film-substrate combination. The ,/' procedure is to assume a series of refractive indexes I and compute a corresponding thickness for each (Paper 67 A4- 227)

I ~

t ..• 377 Iy I