Surface Soence 62 (1977) 1-20 © North-Holland Pubhshmg Company
RELATIVE S E N S I T M T Y FACTORS F O R Q U A N T I T A T I V E A U G E R ANALYSIS OF BINARY ALLOYS P.M. HALL and J.M. MORABITO
Bell Telephone Laboratories, Incorporated, AIlentown, Pennsylvama 18103, USA and D.K. CONLEY
lqestern Elecmc Company, Allentown, Pennsylvania 18103, USA Received 23 August 1976, manuscript received in final form 18 October 1976
Quantitative analysis by Auger electron spectroscopy is commonly performed by applying elemental sensltivaty factors (Prel) to Auger peak height data taken while sputtenng. The accuracy of this approach depends on the statistical reproducibility of Auger data and vaxmUons m Prep Recent work has identified the parameters of the Auger emission process which determine Prel Some of these parameters are a function of concentration e.g. backscattenng and escape depth. Vanations m the escape depth and backscattermg factor with concentration, however, are often found to affect major peaks proportionally, leaving Prel independent of composition in binary alloys. If sputtenng is involved (as it usually is) varmtions m sputterIng ylelds with concentration can also be slgmficant, and a method to correct for this has also been developed. These methods have been apphed to a number of binary systems of technologacal Importance, vlz Cu-Au, NI-Au, N1-Cr, Fe-Cr, Ta-S1, TI-O, and TI-N. The accuracy of quantitatwe analysis is found to be <30% using peak heights and relative sensltlVaty factors. Better accuracy Is possible using standards m the concentration range of the unknown which are analyzed under the same experimental conditions. The relative sensitivity factor method Is, however, more convement for routine quantitative analysis and of suffloent accuracy for most apphcations m the thin film and silicon technologies.
1. Introduction Quantitative Auger electron spectroscopy (AES) analysis based on relative elemental sensitivity factors [ 1 - 3 ] is limited by statistical reproduclbdity , spectral interference effects [ 5 - 7 ] , and problems caused by differential sputtering [ 4 - 8 ] . In addition, the relative elemental sensitivity factors can vary with concentration  and the chemical state o f the specimen under analysis. Work to date indicates that the accuracy is best in the absence o f sputtering , using non-differentiated spectra [ 9 - 1 1 ] rather than the more commonly used differentiated spectra Sputtenng is routinely used dunng AES analysis to remove surface oxides and hydrocarbons which can attenuate signals o f interest , but sputtering itself lntro-
P M Hall et al / Relative sensitivity factors
duces the possibility of composmonal changes due to preferential removal [ 4 - 8 ] , redeposmon of sputtered material , knock-on effects  etc Sputtenng can also change the surface morphology, which can affect the Auger signal strength The effects of sputtering on surface analysis methods including AES are &scussed by Wehner [ 13]. If the energy distribution of the secondary electrons is represented by N(E), the response for a deflection type analyzer (e.g. a CMA) is proportional to E - N(E). Electronic differentiation of this signal (d(EN)/dE) removes background due to backscattered primary electrons and melastmally scattered Auger electrons, but the use of &fferentlated spectra for quantltatwe analysis from first pnnciples assumes that all Auger peaks are of the same general shape. This assumption is not necessary with non-&fferentlated spectra (E • N(E)) data wtuch also provade a more accurate measure of the Auger current present m a given peak and a better signal-to-noise  ratio. The peak height (PA, PB ..... Pz) of elements m the differentiated spectra can be shown, however, to be directly proportional to Auger current for gaussmn peaks with slnusoldal modulation  and the method based on the use of relative senSmvlty factors (Prel) to convert these data to quantitative AES analysis has become common practme [5,15] This paper shows that many, but not all, binary (AB) systems have constant relative sensitivity factors (Prel) even m the presence of sputtering For those binary systems studied where vanatmns in Prel were observed, the vanatmns can be explained by sputtering efficmncles which vary with concentration. Variations in escape depth (de) and backscattenng factors (r) whxch, m part, determine Prel can be treated by introducing a hnear term m the relatwe sensitwlty factor expressmn A method  to obtain the sputtering correctmn factor (R) and xts vanatxon wath concentratmn is also discussed. This method is limited to cases where the concentratlon independence of Prei has been estabhshed by hneanty m PsA versus pS plots , where the superscript in&cates measurements after sputtenng. The product RPre1 defines a sputtering relatwe sensitivity factor, PrSl The accuracy of quantitative AES based on sputtermg relative sensmwty factors ~s shown to be =<30% Better accuracy is possible by the use of calibration curves obtained on standards [8,16].
2. Basic equations 2.1. Relationship between Auger peak heights, Auger current, and composition for binary alloys
The height of the peaks m the &fferentlated spectrum associated with the elements A and B m a binary (AB) alloy after sputtenng can be represented by
P M. Hall et al /Relative sensmvtty [actors
where p S and pS are peak-to-peak signals in the differentiated spectrum, the superscript "0" refers to the pure element, C s and C s are the atomic concentration of element A and B on the surface after sputtenng (assumed constant within the escape depth region -10A), f(CSA) and f(CSB) are functions undetermined except that they are greater than zero, and are unity for pure A and B (i.e.,f(1) = 1). Since f(CSa) and f(C s) are not strong functions of composition , they can be expanded (for a binary alloy) about C s = 0, and CSA= 0 giving
f(C s)= l +aC s +a2(Cs) 2 + ., f(C s) = 1 + bCSA + b2(CS) 2 +
If terms of order C2 and higher are neglected,
(pSA/p°A) + (pSBIp°) = c s ( 1 + aC s) + cS(1 + bCS) .
Using C s + C s = I, this becomes
(pS /pO) + (pS/pO) = 1 + (a + b)CSACs
Experimentally, we find that a plot of pSA/~A versus PsB//~B is often a straight line (see section 3). Llnearlty In plots of PSA//~A versus PSB//~n not only justifies the neglecting of higher order terms in eq. (2) but lmphes that the last term in eq. (4) is negligible, which means that b ~- - a (unless they are both negligible). This is equivalent to saying that both f(CSA) and f(CSB) are linear functions of C s and they both have the same slope. Parameters which determine f(C s) and f(C s) are (1) the escape depth, de, (2) backscattering coefficient, r, (3) peak width, 8, (4) peak shape, and (5) spectral overlap. If there is no spectral overlap (which is usually true) and if the peak shape happens to be independent of composition, the other three dependences can be expressed as
for a given experimental configuration, primary electron beam current, primary electron energy, modulation voltage, and electron multiplier voltage. The (80/8) 2 factor arises because p S is really proportional to H, the second derivative of the total integrated Auger current. For example, if the non-differentiated peak is gaussian) it is shown in the appendix that H = 1 94ISt/82 ,
where I ts is the total Integrated Auger current associated with the peak under consideratlon, and 8 is the energy separation between the maximum and minimum of the peak in the differentiated spectrum. The quantity 8 is linearly related to the peak width In the non-differentiated spectrum, so that H = 2 68 IStiF2 ,
P M Hall et a l / Relatwe sensttwtty factors
where F is the full width at half maximum of the peak m the non-differentiated spectrum. For a non-guasslan shape, the constants are &fferent, but the 5 2 dependence is the same, where 5 is any measure of the width of the peak, provided that the shape is independent of composltmn This assumes that the modulated signal indeed does represent the true denvatwe. If the modulatmn amphtude, Vm, is not neghglble compared to 5, then (at least to second order in V2m) the peak-to-peak hmght PSA IS reduced and 5 2 is increased (see Appen&x) by the factor 1+(1/4)(Vm/Sn) 2 where 8 n is the "natural" line width This leaves the product PSA52 unchanged. The appendix also shows that if the hne wadth changes as a functmn of concentratmn, then PsA is proportional to ISt/5 2
2.2. Relattve sensttwtty factor, Prel For a two-component system (AB), It xs possible to express the surface concentration of A or B m terms of the measured peaks heights (PsA, PB) s by
C s= 1 +-~pAPrCl
where PreJ Is a factor whmh corrects for the different Auger yields of A and B and is known as the elemental relatwe sensmvlty factor  From the fact that CSA + Csa = 1, it follows that c s = [1 + ( ~ / ~ ) 1 - 1
It follows from eqs (1), (8) and (9) that
P,e, = ~ f ( c S ) / e ~ f ( c s)
Thus, from eq. (2) for b = - a
From eq. (2) at xs clear that the absolute value of a must be less than umty for the peak heights to be posture everywhere Thus to first order m a (espemally for small C s ) this gwes
and S S (PA/PB) 0 O (l + a)] - I CSA = I1 + (PB/PA)
It is possible m prmmple to measure Prd by observmgPA/P B on the unsputtcrcd surface of alloys of known composlhon Unfortunately, however, thc first few monolayers may have a &ffercnt composition from the bulk, duc to surface segrc-
P M liar et a l / Relattve senstttvtty factors
gatlon . Even worse is the practical difficulty of obtaining surfaces free of spurious surface contaminants such as organics that form on all surfaces in the atmosphere, producing measurable peaks for C, O, C1, S and others. Whereas these contaminants may not cause spectral overlaps, they normally attenuate the signals of interest , which usually increase mmally as the contamination is sputtered away. Special methods of obtammg clean surfaces such as vacuum cleaving and scribing have limitations [ 18], with the result that much more is known about AES analysis of surfaces that have been sputtered than of clean non-sputtered surfaces.
2.3. Calculation o f concentratton before sputtenng from p S and p S All of the above refers to calculations of surface concentrations after sputtering from Auger spectra which are generated after sputtering. It is often described, however, to refer the concentrations at a given depth (C A and CB) before sputtering from spectra generated after sputtenng to that depth. This is handled by introducing a factor,R, defined as
R = CSACB/CSCA
Shimlzu et al  have identified this factor with the ratio of sputtering efficlenCleS R = SB/SA,
where S A and S B are expressed in terms of atoms/ion/at%. If the ratio of sputtering efficlencles is independent of concentration, then R is simply a constant. Otherwise, R can vary with concentration. Even without this Identification,however, we can proceed using eq. (14) as a definition. Since C A + C B = 1, it follows [analogously to eq. (9)] that CA = [1 + CB/CA] -1 ,
which, when combined with eqs (8), (9) and (14) yields CA = [1 +RPreI(PS/[~S)] - ' ,
and this suggests a new factor, prSel, defined by prSel = RPrel,
which is the calibration factor required to find what the concentration was before sputtenng if P sA and pS are measured after sputtering. From eqs. (16), (17) and (18) we find
PSe, = (pSA/pS) (Cs/CA) ,
and this quantity is measurable. Given a series of bulk, binary AB alloys of known composition C B and C A, it is possible to construct a log-log plot of PSA/pSs versus S CA/C B which provides a measure ofPre ~.
P M Hall et al / Relative senstttvl~ Jactors
0 0 0 0
0 ~ 0 0 0 0 0 0 0 0 0 0 0 0 0
PM Hallet al / Relative sensttwtty factors
3. Experimental Most of the Auger peak to peak measurements (pS pS) were taken with a primary electron beam of 5 kV and 50 taA beam current. See table 1. Under these conditions the focused beam size was shghtly larger than 100 tam. Ion (A + and Xe +) sputtering was performed at 1 kV and 2 kV using 5 to 30 mA emission current which corresponds to a sputter current density of approydmately 40 taA/cm 2. Peakto-peak measurements were recorded (for the bulk samples) after steady state was obtained (less than 5 min). Several measurements were taken and there was no change m the equilibrium peak heights after the first few minutes of sputtering. Some allows were sputtered and analyzed for over an hour without any noticeable change. Several different areas were examined on each alloy and concentration variations due to lateral surface segregation effects were small. The copper-gold and lron-chrommm alloy systems were chosen since they exhibit a continuous series of solid solutmns A series of C u - A u and Ee-Cr alloys were prepared by quadruple arc melting, inverting between each melt and chill casting. Excellent correlatmn was found between electron microprobe and wet chemical analysis of these alloys [19,20], showing that the region within 5000 A of the surface had a composition indistinguishable from that of the bulk. The N1-Cr alloy series was obtained from the Materials Research Corporation [21 ]. In addition to the bulk alloys, several systems were studied in thin film form. Among these was C u - A u , which was prepared by evaporating a very thin titanium adherence layer onto glass substrates followed by copper and gold evaporation. The films were then alloyed by a series of heat treatments . The N1-Au and P d - A u alloy series  were prepared in a similar way. Reactive sputtering was used to prepare Ta-S1 , TI-N , and T1-O  films of varying composition.
4 1 Llnearity in pSA/I~A versus pS/pO plots Examples of the hnearlty of pS~pO versus pS/pg plots for the binary alloy systems C u - A u , P d - A u , T I - A u , NI- Au, N1-Cr, and F e - C r are shown m fig. 1 The four lowest curves (Au-alloys) were generated by sputtering through partially lnterdlffused bi-layer Films to expose alloys of different composition. The composition is not accurately known for any given sputter time, but this is not necessary in order to show that these plots are straight lines (the correlation coefficients are >0.999) The normalization factors pO and pO are obtained by extrapolating the straight lines to the pure materials The two sets of Cr alloys (top two curves) were obtained from a series of bulk samples whose composition before sputtenng was known by Independent means [19,23]. They were, however, also sputtered to arnve at steady state values of pS and pS, to eliminate any surface contamination,
P.M Hall et al /Relattve senstttvtty factors
~o~~ \ \
Fig 1. Plots of P~A/P~ versus P~]P~ for alloys of NI-Au, C u - A u , P d - A u , TI-Au, N1-Cr and F e - C r . The Cr alloys were bulk samples. The Au alloys were analyzed by sputtering through diffusion couples.
P M Hall et al / Relattve senstttvtty factors
and to avoid surface segregation effects-(except those that might occur at room temperature). The bulk alloys on fig. 1 show more scatter (correlation coefficients of 0.989 and 0.993) than the films but there is stdl no mdicatlon of significant departure from linearity. The increased scatter m the bulk alloys is attributed to difficulties m reproducing ~dentxcal conditions of alignment, focus, etc. since for the bulk alloys, the sample holder had to be rotated between measurements. One Indication of the reproduclbdlty is given by the two sohd points in the middle of the top (Fe-Cr) graph on fig. 1 which were taken in two different spots on the same sample. Lmeanty m these plots lmphes that Prel is independent of concentration over a wide range.
4 2 Peak width (6) as a function of concentration An attempt was made to measure the change m 8 of the 69 eV Au, the 920 eV Cu and the 510 eV oxygen peaks while sputtering through a bi-layer C u - A u film which had been heated m air untd some of the Cu came to the surface and oxadized. Since the C u . O peak ratio was qmte constant, the specmaen may be consldered a mixture of Au with copper oxide. The Au concentration went from 38 to 96% as the surface was sputtered into. During this time, the width of the 920 eV Cu peak decreased very slightly (by 8 + 6%). The 69 eV Au increased 6 -+ 5% and the 510 eV oxygen peak increased by 4 + 4%. None of these is considered statistically different from being constant (the in&cated errors are at the 95% confidence level). These small changes in ~ had no effect on the hnearity of the plots ofpSA/P°A versus p S /p °. The peaks of the different elements, however, did &splay different widths. The 920 eV Cu peak was 8.0 eV wide (+-0.5 eV at the 95% confidence level), the 69 eV Au peak was 5.9 eV wide -+0.4 eV, and the 510 oxygen peak was 5.7 eV wide -+0.3 eV. These peak widths can be corrected (to first order) using eq. (A.15) with Vm = 2 5 eV This produces the "natural" line widths (Sn) of 7.2, 4.8 and 4.5 eV for the Cu, Au and oxygen peaks respectively
4 3 Log-logplots ofPSA/p s versus CA/C B The results discussed below were obtained m three different Auger mstruments, system M, system C, and the system used m ref. . Figs. 2--8 are examples of log-log plots of PA/PBS S versus CA/C B for the systems Fe-Cr, Ag-Cu, Ta-S1, A u - C u , N-T1, and O-T1. For all of these cases except C u - A u the data are fitted to a straight line of slope one, and PrsI is given m table 1, along with some of the experimental details. One pomt on fig. 2 and one on fig. 7 are considered "outhers", and have been deleted from the calculations of PrSel of table 1. They are both for very dilute (~1 at%) alloys where the accuracy is expected to be very poor. The different AES systems at different operating condinons (e.g. ton energy,
Pr=el = 0 91, SYSTEM C ' ~
Prel =0 66, SYSTEM M
Fig 2. Calibration plot for bulk F e - C r alloys for system M and C.
P;.~=o 6 z - ~ SYST ES~prsel
(N~,Cr) =0 4B
Fig. 3. Cahbration plot for bulk N1-Cr alloys for systems M and C Samples were MRC-VP Grade, analyzed by Mass Spec. and Electron Microprobe, with good agreement. The values of CCr and CN, used here are from the microprobe.
P M Hall et al / Relattve senszttvtty factors
/ 01 01
Fig. 4. Cahbratlon plot for bulk Ag-Cu alloys data taken trom Braun and Faxber  for sputtered surfaces
primary electron energy, and incident angles) produced different cahbrahon factors. Note that PrSl for both N]-Cr and F e - C r Is higher m system C at 1 kV 1on energy than in system M at 2 kV. Results for sputtering with xenon ions in system C (not shown here) also show good hnearlty, but with somewhat &fferent values of prSel . Some of these &fferences are no doubt attributable to the dependence of R on sputter parameters. For all these material systems except Au-Cu, the independence of prSel on composition should be true for all Auger systems. For these cases, it ts concluded that the higher order terms m eq. (2) are indeed neghglble, and R and a are both mdependent of concentration, since erSe! = (eO/eO)R(1 + a)
West  has also found PrSl independent of composmon for a series of alloys of Cr-Pd and N b - U , and Ho et al.  have come to the same conclusion for NI-Cu alloys. Mathleu and Landolt  have shown for Ag-Pd and N1-Pd alloys that Prel IS independent of composlhon, and the factor a is neghglble compared to 1, so that PA IS proportional to CA. They also find R values almost independent of composition. Thus PrSel is independent of composmon for these alloys. In the case of Au-nch alloys of Cr-Au, Holloway  finds PrSel varying with
P M Hall et al / Relative senstttvttv factors
PrIlJ I T o , S I I ~ !
C$1 C Ta
Fig. 5 Cahbratlon plot for S1-Ta bulk alloys for system M CS~ and CTa determined by electron microprobe
concentration For the case of C u - A u and some others * our data also seem to deviate from slope one, and Pr~el(Au,Cu) vanes from 0 25 to 0 11 as CAu goes from 1 to 99 at%. Nonlinearity m the C u - A u system has been noticed before . Fig. 9 is a plot of PrSel for this system, fitted to a straight line (least mean squares) Since this system displayed linearlty m fig. 1, the dependence ofprSel on C is attributed to a variation m R. 5 Estimate of accuracies Once Prel or prSel IS d e t e r m i n e d for a gaven s y s t e m , t h e c o n c e n t r a t i o n is r e a d d y f o u n d f r o m eq. ( 8 ) , or eqs. ( 1 7 ) a n d ( 1 8 ) , d e p e n d i n g o n w h e t h e r It is required to k n o w the c o n c e n t r a t i o n b e f o r e or a f t e r s p u t t e r i n g . * T a - O , T a - N and T a - C  do not show hneanty, but are not included here
P M Hall et a l / Relative senstttvtty factors tO
~., (~..c.)-o 25 J o
PrS., (A.. C.). ttt
I t O&u Ccu
Fig 6. Cahbratlon plot for bulk C u - A u alloys for system M.
I I= 1 20
cTI Fig. 7. Cahbratlon plot for reactively sputtered TI-N "alloys for system M. Concentrations obtained from electron microprobe.
P M Hall et al /Relattve senstttvzty factors
Prel (OXY,TO=1 53
I I I III Oi
I I III iO
Fig 8. Cahbratlon plot for reactively sputtered T1-O thin f'dms for system M. The accuracies involved m Prel can be estimated from the repeatablhty of Prel" From a series of thirty-seven runs of C u - A u bl-layer samples with varying amounts of mterdlffuslon, taken during several different pump-downs on several different days, it was found that Prel(AU,Cu) was 1.13 + 0 14, where one standard deviation is indicated as the error. The effect of such an error is indicated on fig 10, as well as the effect of an error of two standard deviations. In the region of ddute alloys, the concentration of the minor constituent is off by a factor of 0.88 If a one-sigma error is assumed, or a factor of 0 75 If a two-sigma error is used. For example, for a concentration of 10 at%, the calculated concentration would be 8.8 at% (1o) or 7 5 at% (2o). If the alloy IS non-dilute the error factor is closer to umty as shown m fig. 10, which gwes the error factors for both Cu and Au as a function of Ccu At the 50 at% concentration, the error factors are 0.93 and 0.86 for one and two sigma errors respectively, corresponding to 46 5 at% (1 o) and 43 at% (20). The accuracxes involved in prSel can be esttmated from the data shown m figs 2 - 8 . Each point shown was used to determine a value of Prel, and these values were averaged for each pair of metals Table 1 gwes these averages and their standard devlatlons. In the case of C u - A u , the standard deviation is high because of the trend of decreasing PSel with increasing Au concentration, mentioned before (fig. 9) This dependence of PrSel could not be attributed to variations in a, de, or r pS /po because that would have changed Pre] as well, and the plot of Au/" Au versus P cSu / P cou (fig. 1) would not have been linear. Thus, it is concluded that this variation is due to variation m R, the ratio of sputter efficlencles. For the other cases,
P M Hall et a l / Relattve sensmvtty factors
Fig 9. Dependence of PSel(AU,Cu) on CAu The slope ot the hne is -(0 115 +- 0.044) to 95% confidence. where no systematic dependence of PrSel on concentration was observed, it was concluded that the ratio o f sputter efficlencles is essentlaUy constant, within the hmlts o f the data o f table 1. The accuracies involved could be summanzed with the statement that Prel for any one run has a standard devmtlon amounting to less than a few %, but from run-to-run it is about 12%. The standard deviations for/~rel are somewhat larger, as much as 30%, because they also Include the variations in R This value is comparable to the uncertainty o f 27% found by West  for/~rel
6. Conclusions It was found for all six binary alloy systems stu&ed that p S versus p S is strictly hnear. F r o m this It is concluded that the elemental relative sensitivity factor Prel is essenhally independent of composition. Ttus does not say that the escape depth, de, and the backscattenng factor, r, are independent of compos~tlon. It does imply, however, that if r increases as a function of composition, it increases the major peaks more-or-less proportionally. Dkewlse, if de(A ) increases with composltton,
P M Hall et al /Relattve senstttvtty factors 16
CAutCALC CAu,A C T U A L \
1 4 --
ERROROF ? S I G M A ~ ~"~ ~
~J I ~
ERROROF 1 SIGMA~
Fig. 10. Error factors caused by error m Prelso does de(B ) m a proportional manner. The variation in Prel as a function o f composition for any cahbratlon run is much less than the variation from run-to-run, which causes Prel to have a standard deviation of-+12%. It is also concluded that for many element pairs, PrSel (the elemental sensltlwty factor relating peak heights after sputtering to composition before sputtering) is independent o f composition, at least for the sputtering voltages used here. This implies a constant ratio of sputter efficlencles but does not ~mply that the two sputter efficlencies are equal. F o r these pairs, one or two points can determine the calibration factors, which can be expected to be accurate to -+30%. For other matehal pairs, the variation of sputter efficiency ratio causes considerable variation in s Prel, and a more extenswe cahbration curve must be generated m order to obtain the same accuracy. This level of accuracy is sufficient for most apphcations m the thm fdm and slhcon technologies.
P M Hall et al /Relattve senstttvtty factors
Acknowledgment It is a pleasure to acknowledge R.H. Mills and C.T. Hartwig who operated the AES system M to obtain much of the data shown here. D.F. Lesher assisted m operating system C. The reproducibility of the data is in no small measure attributable to their skill, care and pataence.
Appendix A This section is a calculation of the relation between the total (integrated) Auger current associated with a particular gaussxan peak and the peak-to-peak height in the &fferentiated spectrum. Since only the region very near a single peak is being considered, we need consider only N(E), ignoring the fact that the response of the analyzer IS actually EN(E). Let the total Auger current be I, such that I =:
Then lfN(E) is gausslan, centered around the energy E o,
N(E) = ~
exp L-~~ - - - ~ ]
where S t s a measure of the width of the peak. It is common to express S m terms of the full-width-at-half-maximum, F which Is F = 2Sx/~-In 2 = 2.355 S
The derlvatwe of N(E) is given by
• V(E) dE
I(E - Eo)
V , [E - Eo~ 51
This function has a maximum whose value is [dN(E)/dE]max = 1/$22X/~-~7 = 0.242 I/S 2 ,
which occurs at E = Eo - S.
Slmdarly, there is a minimum whose value is -0.2421/S 2 wluch occurs at E = E o + S. Thus, the peak-to-peak height of dN[dE is given by
H = 2I/$2
2~-~ = 0.4841/S 2 .
P M Hall et al /Relattve senstttvtty Jactors
The energy separation between the maximum and mlnmaum In the differentiated spectrum is 8 = 2S
Thus H = 81/629xF--_Tre= 1 936 I/82
( a 9)
And in terms o f F , this becomes H = (16 In 2/2X/2-~)(I/F 2) = 2 684 I / F 2
Now let the energy of the analyzer, E, be smusoldally modulated about energy E1 E=E I+V msmwt,
where Vm is 1 the peak-to-peak modulation energy. The lock-m amphfier is intended to measure the derwatwe (dN[dE) It does this by measuring the amphtude of the second harmonic, which is shown by Taylor  to be
P = Vm
+--~-- dE----5 + - - + 192dE s
Substituting N ( E ) from eq (A 2), thas function can be dlfferentaated to find the energy of its maximum and minimum, gwlng E I ( m ~ n ) = E o + S 1 +~-S-g 3 8 ~ +
Thus the energy separaUon (8) between the maximum and minimum m the differentiated spectrum increases as Vm is increased (notwithstanding a comment by Taylor  to the contrary) 6 = 2S
This equation can be used to correct measured values of S by solving to second order, giving S=~
The height of the peak (even when normahzed by Vm) also depends on Vm
Vm 82 2 N ~
P M Hall et al / Relative sensmvtty factors
The p r o d u c t p 8 2 / V m can n o w be shown to be i n d e p e n d e n t o f b o t h Vm and S (to second order m V2m/S2),
Vm = ~
( A 17)
Thus, if a peak broadens as a f u n c t i o n o f c o n c e n t r a t i o n , the peak height decreases by the same factor that 62 increases, and this can be corrected for by measuring and using a c o r r e c t e d P Pcorr = Pmeas (8/80) 2 , where 8 o is the value o f 8 for the pure material.
[ 1 ] P.W. Palmberg, Anal. Chem. 45 (1973) 549 A, P.W. Palmberg, G.E. Rlach, R.E. Weber and N.C MacDonald, Handbook of Auger Electron Spectroscopy (Physical Electronics, Edma, MN, 1972).  J.M Morablto, Thin Sohd Films 19 (1973) 21  C.C Chang, Surface Scl. 48 (1975) 9.  L.A West, J. Vacuum Scl Technol. 13 (1976) 198  J M Morablto and P.M. Hall, Scanning Electron Microscopy/1976 (Part I (liT Research Institute, Chicago, Ii1., 1976), pp. 221-230  M L Tarng and G K Wehner, J Appl Phys 42 (1971) 2449, 43 (1972) 2268  J T Grant,M.P. Hooker and T.W. Haas, Surface Scl 51 (1975) 318.  H Shlmlzu, M Ono and K Nakayama, Surface Scl 36 (1973) 817 [91 J T. Grant, T.W. Haas and J.E. Houston, Phys Letters 45A (1973) 309.  J E. Houston, Appl. Phys Letters 24 (1974) 42. I 11 ] J.T Grant, T.W Haas and J.E. Houston, Surface Scl 42 (1974) 1  B Blanchard, N. Hilleret and J.B. Quotrm, m Proc. Pittsburgh Analyt. Chem Appl. Spectrosc. Cont. 171 (1973) p. 131.  G K Wehner, m Methods of Surface Analysis (Elsewer, Amsterdam, 1975) pp 5 - 3 8  N J Taylor, Rev Scl Instp 40 (1969) 792.  A. Joshl, L E. Davis and P W Palmberg, in Methods of Surface Analysis (Elsevaer, Amsterdam, 1975) pp 159-222 [ 16 ] J.M. Morablto, Anal. Chem. 46 (1974) 189  J.M McDavld and S.C. Fain, Jr., Surface Scl. 52 (1975) 161.  P. Braun and W. Farber, Surface Scl. 47 (1975) 57.  J W Colby and D K Conley, OptNue des Rayons X et Mlcroanalyse (X-ray Optics and Mlcroanalysls) (Hermann, Pans, 1965) pp 263-276  J W Colby, W.N. Wise and D.K. Conley, In Advances in X-ray Analysis, Vol. 10 (Plenum, New York, 1967) pp. 447-461. [21 ] VP Grade Nlcralloy.  P.M. Hall, J.M. Morabtto and N T. Panousls, Thin Solid Films, In press.  P.M. Hall, J.M. MorabRo and J.M. Poate, Thin Solid Films 33 (1976) 107.  N. Schwartz and F.G Peters, m Proc. 1973 Electronic Components Conf., p. 251.  J.M Morablto, unpubhshed results.
20     
P M Hall et al /Relattve senstttvtty factors
P.S. Ho, J.E. Lewis, H S. Wfldman and J K. Howard, Surface Sca. 57 (1976) 393. H.J Mathleu and D. Landolt, Surface Scl. 53 (1975) 228. P.H Holloway, to be pubhshed. J.M. Morablto, Surface Scl 49 (1975) 318. D.K. Conley and D.F. Lesher, m Proc. 1974 Conf. Mlcrobeam Analysis Soc. (1974), paper 44.
Notes added in proof (1) If the number o f atoms per unit volume, n, is a function o f concentration, the right hand side of eq. (5) should be multlphed by n/n ° This has the effect of multiplying Prel by the factor nB/nA. o o Thus Prel is still mdependent of concentratxon, but this is one more reason why P~l is not simply/~A//~a, and a is not neghglble compared to one (2) It might be argued that l m e a n t y of the type seen xn fig 1 could occur if the surface exposed by sputtering the bl-layer films consisted o f two phases, one of pure A and one o f pure B Ten as the area fraction of A vaned from 0 to 1, llnearlty would result from the area-averaging process even lfPret were a functxon o f composition L m e a n t y was also observed, however, m the plots for the two series of homogeneous bulk alloys o f F e - C r and N a - C r (fig. 1). In addition, data for the C u - A u bulk alloys (used in fig. 6 to evaluate PS~l) are consistent with h n e a n t y in plots of this type The avadable compositions were mostly rather dilute alloys, however, so they are not included in fig. 1 Also, m the thin film bl-layers, some conslderable lnterdlffusaon must have taken place on an atomac basis, since thear mdepth composatlon profiles  show large changes without &sturbmg the hnearlty in pScu versus/~Au, and diffusion &stances as small as a few lattice spacings could affect the lmearlty ff Prel were a function of composition. Thus we can stall conclude that Pre~ is independent of composat~on