Scanning Electron Microscopy
Volume 1982 Number 1 1982 Article 5
1982
Analytical Models in Electron Backscattering
Heinz Niedrig Technische Universität Berlin
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ANALYTICAL MODELS IN ELECTRON BACKSCATTERING
HEINZ NIEDRIG Optisches lnstitut, Technische Universitat Berlin D-1000 Berlin 12, Germany (West)
Phone: (030) 314-2735
ABSTRACT l. INTRODUCTION
The different aspects of electron backscattering from solid The knowledge about the backscattering of medium ener films are calculated in terms of simple analytical models for gy electrons from solids is of great importance for its applica thin films and bulk targets, for normal and oblique inci tion in scanning electron microscopy, in electron beam dence. Well-known models regard only one scattering aspect , microlithography, in film thickness determination, etc. For e.g.: single Rutherford scat tering by Everhart's model, dif this reason there is much practical interest in a theory which fusion from a point source by Archard's model, diffusion describes backscattering as thoroughly as possible . from sources continuously distributed over the depth of the In most cases electron backscattering cannot be described target by Thiimmel's model. A few analytical models have by single scattering processes solely. Existing multiple scat been developed recently regarding single scattering and con tering theories (see, e.g. Bothe 1933, or Thiimmel 1974), tinuous diffusion, e.g. by Werner and by the author. These however, normally describe certain limiting cases, or they are model s allow us to calculate analytically the backscattering difficult to evaluate. Calculations which most properly fit the coefficient for bulk targets versus atomic number, for thin experimental values are those using the Boltzmann transport films versus thickness and incident electron energy, the equation (e.g., Fathers and Rez 1979, Schmoranzer and angular inten sity distribution, th e tot al and angular energy Hoffmann 1980, Hoffmann and Schmoranzer 1982) or those distributions, and the surface den sity distribution of back using the Monte-Carlo method (e.g. Berger 1963, Shimizu sca tter ed electrons. The numerical result s of the se models are and Murata 1971, Krefting and Reimer 1973, Kyser 1981). By compared with experimental results thu s revealing the limit s these methods numerical results can be obtained only with of th e pre sent models. expensive computer calculations (except some special cases, e.g. Lehmann 1980). Therefore a strong interest exists in simplifying theoretical models which can be treated analytic ally and can be discussed more easily. In the following section we will discuss analytical models regarding only one scattering concept, e.g ., only single Rutherford scattering, or only diffusion. Then we will treat Dedicated to Prof. Han s Boersch on the occasion of his analytical models combining single scattering and diffusion. 75th birthday . 2. MODELS REGARDING ONLY ONE SCATTERING CONCEPT
2.1. Everhart's single scattering approach Everhart (1960) calculated the electron backscattering from single scattering processes into angles i/>90° (Fig. I). He used the differential Rutherford scattering cross-section
Keywords: Electron backscattering, single scattering, elec da tron diffusion, Rutherford scattering cross-section, bulk tar gets, thin films, normal incidence, oblique incidence, angular d!1 intensity distribution, energy distribution, single-scattering model, point-source-diffusion model, diffusion depth dis where Z is the atomic number, e is the electron charge, € is tribution, continuous-diffusion-depth model. 0 the dielectric constant, and E is the electron energy. Addi tionally a continuous energy loss of the electrons travelling within the target is assumed according to the Thomson Whiddington law (Thomson 1906, Whiddington 1912, 1914):
51 Heinz Niedrig
LIST OF SYMBOLS Reduced electron exit energy 1 Velocity of incident electrons [ms - ) z Atomic number Velocity of electrons within the target 1 M Relative molecular or atomic ma ss [ms - ) 5.05 x 1033 m6 kg - 1s - 4 : Terrill' s con [g,mol - 1) stant Electron charge [As) Single scattering coefficient (Everhart, Dielectric constant [As V - 1 m - 1J 1 Werner) Loschmidt/ Avogadro number [mo1- ] Everhart's adjusted coefficient Electron mass [kg) a; Density of target material (g cm - 3; a~ Diffusion coefficient (Niedrig) kg m - 3) a::eZ/5 Diffusion coefficient (Werner) X Target depth normal to surface [m; aN a +a~ nm] 5 N ' N ' R Range of electrons in the target [m; ad , a Defined by equations 21a and 23a w nm] aw a s+ad y=x / R Reduced target depth Defined by equation 30 Ym Maximum reduced target depth allow Backscattering coefficient ing electron exit Backscattering coefficient of film of Depth of complete diffusion (diffu thickness D sion depth) [m; nm) Backscattering coefficient of bulk tar Reduced diffusion depth get y/yd 'P' (y),'P, (u) Differential diffu sion probabilitie s Path length of electron within the tar P'(u),P'(t),P'(T) get [m; nm] EATNW Upper indices , indicating that the cor Mean path length of electrons [m; nm] responding quantity is calculated from Reduced path length the models of Everhart, Archard, Maximum reduced path length for Thiimmel, Niedrig, or Werner backscatter direction iJ s ' d Lower indice s, indicating the single T= - fn(l-t) Quantity measuring the reduced path scattering or the diffu sion fraction of length the corresponding quantity Quantity corresponding to tm b=0.6 Value of u, where 'PN'(u) has its max Film thickness [m; nm] imum 0 Scattering angle [ ; rad], measured w= - cos8 Substituted quantity (for integration from the forward direction purpo ses) 8=1r-li Scattering angle [0; rad], mea sured from the backward direction Opening angle of exit cone related to 0 reduced target depth y ( ; rad] 2 0 <{) Azimuth angle [ ; rad], measured around the direction of incidence where vis the electron velocity at depth x, v0 is the velocity of , , 33 6 1 4 e ,'P Angles measured from the target nor incident electrons, c T = 5.05 x 10 m kg - s - is Terrill's 0 mal [ ; rad] constant (Terrill 1923) and Q is the density of the target mate O' Incidence angle of primary beam mea rial. By this continuous slowing down all the inelastic small 0 sured from the target normal [ ; rad] angle scattering processes are represented, neglecting the f(8' , 52 Analytical Models in Electron Backscattering y j incident f lo(y) J --dy 4 f beam 0 1- y 1 and lo(y) are the inten sities of the primary beam at depths 0 exit cone y = 0 and y = y, respectively. The angular integration ha s for depth y e- backscattered been done in such a way that the primary beam lose s all elec electron trons scattered into angles (} ::: 90°, whereas all electrons with a sca tterin g angle (} < 90° are treated as if they were not scattere d at all. I The factor a s (Everhart's single sca ttering coefficient) is an I I abbreviation for I I 7rz2e4N L z 2 I a s = ------0.024 - 5 \ I (4H 0)2ml cTM M \ I \ I where NL is Loschmidts's or Avogadro's number and M is \ \ I the relative molecular or atomic ma ss ("atomic weight") mea /< range s phere y= 1 '--. sured in g/ mol. Since M "'2 Z g/ mol we have " _,,,/ for depth y (x=R) ''-- ..__ 6 ------Fig. 1. Geometry of the Everhart backscattering model. Equation 4 can easily be solved yielding the flux of primary electrons 7 " 9 diffusion t heo ry which decrease s with increa sing depth. ,.,r I Archard I For the backscattering coefficient of a layer at the reduced depth y scatterin g into the solid angle drl we then obtain I I ._ Schonlond 100 keV (1923) I. 0 Pol lue l l6 ke V (19i, 7) 8 ! • Ho llld oy,Sterngloss 20 keV (1957) 0,3 - 'Q' Kanter 10 keV (1957) • Mu lvey.Doe 23 keV (196 0) a Bishop 10/30 k eV (1965) • Cos.s!ett _ Thomas 10/20/45 keV (1965) • A He1nr1ch 10/49 keV (1965) 1 Sieber 20 / 53 keV (1969) The index 's' denotes 'single' scatte rin g, and for 8 7r - (} = Everhart's l Or e sch er , Re1mer, S eide l see Fig. I. In calcu lating the back scatter ing coefficient of 0,2 the ory dj usted / 9.3 / 102 ,.v (19701 / YRe, mo,.lollkomp 20ke V (1980) such a layer by angular integration we have to consider, that according to the model only those electron s can escape sin gle+ dou ble scatte ring through the incidence surface (and thus contribute to the I Bod y I total backscattering coefficient) which have travelled less 0,1 large ang le than the range within the target, or, which are scatte red into single elastic scattering theory IE verhar t ) the 'exit cone' with the opening half ang le 8 111(y). According to Fig . I em is determined by W Pt AuPb81 U 70 80 90 y ----z y + 9 cos8m(y) Fig. 2. Backscattering coefficients of bulk solids measured by various authors together with the results of some Integration of equation 8 over the azimuth angle (}, over the theoretical models. scattering angle 8 with respect to equation 9, and up to the maximum reduced depth y = 0.5 (x = R/2 ) from which an Fig. 2 shows the backscattering coefficients for bulk targets electron just can escape gives Everhart's formula for the calculated from equation 10 compared with experimental values for electron energies E 10 to 100 keV from various ba cksca ttering from a bulk target at normal incidence : 0 = authors which are taken from earlier reviews (Niedrig I 978, 1981 a,b). From Fig. 2 we can see, that the values calculated E 10 from equation 10 are smaller than the experimental values by 1/5 00 a factor of about 3. This is not surprising since multiple scat tering and diffusion are taken into account only by the 53 Heinz Niedrig energy losses connected with these processes in using the space for the contributions of single and plural scattering Thomson-Whiddington law (equation 2). Everhart therefore processes. For low atomic numbers (Z < 13) the model breaks down because then the diffusion depth becomes adjusted his coefficient a 5 (equation 6) as follows greater than half the range (yd > 0.5) and no electron can 11 escape. Tomlin (1963) gave a different calculation of the dif fusion depth but the principal disadvantages of the model re to fit the experimental values. With this adjusted coefficient main the same. equation 10 describes the observed backscattering coeffici A more sophisticated model has been used by Kanaya and ents fairly well up to atomic numbers Z ==40 (Fig. 2). Okayama (1972) and by Kanaya and Ono (1976, 1978) who The Everhart model has been extended to thin films by assume the center of the diffusion sphere to be located at the Nakhodkin et al. (1962a), Cosslett and Thomas (1965), and most probable energy dissipation depth, which is smaller Iafrate et al. (1976). Here we find the same discrepancy by a than the diffusion depth (Kanaya 1982). Radzimski (1978) factor of about 3 between model calculation and experiment extended the diffusion model to oblique incidence of the (Niedrig and Sieber 1971, Hohn and Niedrig 1972) as before primary beam. if we do not use the adjusted coefficient from equation 11. The above treatment of the point-source-diffusion model Nakhodkin et al. (1962 a,b, 1964 a,b) and Shimizu and means that the real diffusion process is replaced by a single Shinoda (1963) also extended the model to oblique incidence scattering process with isotropic scattering cross-section in of the primary beam. By this extension they were able to ex the discrete diffusion depth. This has also the consequence plain qualitatively the increase of the backscattering coeffi that the model is not applicable to thin films with thicknesses cient with increasing deviation from the normal incidence as x < xd because a thickness dependence cannot be introduced. has been observed experimentally by, e.g., Kanter (1957). 2.3. Thiimmel's continuous-diffusion-depth model 2.2. Archard's point source diffusion model To avoid the disadvantages of the point-source-diffusion In the diffusion model an isotropic propagation of the model Thtimmel (1974) assumed that the transition of the in electrons within the target is assumed. For the incident elec cident electrons into isotropic diffusion does not occur at the tron beam this state of diffusion is reached in the 'depth of discrete diffusion depth, but is distributed over the path of complete diffusion' (Lenard 1918, Bothe 1933, Bethe et al. the incident electrons in such a way as is given, e.g., by the 1938) as the final state of the preceding multiple scattering differential turning-back probability P' (u) (Fig. 4, u = processes. y/yd) as calculated by Bothe (1933, see also Thtimmel 1974, Archard (1961) simplified this concept in his point-source p. 186). For reasons of analytical integration Thtimmel used diffusion model. He assumed the incident electrons to travel an elementary parabolic function r,oT' (u) (Fig. 4) and ob straight into the target up to the depth of complete diffusion tained a more realistic relation for the backdiffusion versus (later on called 'diffusion depth' xd = yd• R, where yd is the atomic number (Fig. 5). reduced diffusion depth), after which they diffuse isotropic The author (Niedrig 1981 a,b) used a different elementary ally in all directions in such a way that their overall paths distribution function equal the full range R (Fig. 3), thus forming a point source of electrons at the diffusion depth. Then the backdiffusion coefficient for bulk targets is sim 15 ply equal to the solid angle of the exit cone divided by the full solid angle 41r, yielding with b = 0.6, which approximates the Bothe function more closely (Fig. 4) and which avoids cut-off effects at u == 0,2 I - 2yd (see P' (u), Fig. 4) and at u = 2 (see r,oT' (u), Fig. 4), r,oN' (u) A T/ctoo 12 is chosen in such a way- corresponding to the other distribu 2 1 - Yct tion functions - that where the index 'd' denotes 'diffusion'. The diffusion depth 0, yd has been calculated by Archard ( 1961) in a low-energy J r,o' (u) du = 16 approximation to be 0 Yct == 40/ (7Z) 13 which means that finally all incident electrons are in the state of diffusion. (see also Thtimmel 1974, p. 164 ff., and 1981). Introducing A layer in the reduced depth y then contributes yd into equation 12 we get Archard's formula for the back diffusion in bulk targets for normal incidence I 1 - 2y d71d= r,o' (y) • - • -- dy 17 2 1 - y 7Z - 80 A T/ct0, 14 14Z - 80 (see equation 12) to the total backdiffusion. Regarding equa tion 15 we get by integration over y = 0 to 0.5 the following which is also plotted in Fig. 2. For atomic numbers Z > 40 backdiffusion coefficient of bulk targets for normal inci this relation gives the right order of magnitude but leaves no dence: 54 Analytical Models in Electron Backscattering incident beam e- rediffused electron 0.5 Thummel y=0.5 I (x=R/2)~/ / this work t., //range / sphere \___,,.,. 2 3 u=y/yd Fig. 3. Geometry of the Archard point-source-diffusion ------model. Fig. 4. Differential diffusion depth distributions used in dif --- ferent backdiffusion models. 2byd I +5b 2y~ + 2b 4 y~ l ,,,TN ·•doo ----- [------arc tg -- ir(l + b2y~) 2 2 by ct 2by ct ✓ l + 4b2. ·2 l b2.·2 - Pn -yd - + 2 -yd ] 18 byd (b = 0.6). With Archard's diffusion depth (equation 13) we obtain from equation 18 the backdiffusion curve in Fig. 5 0,4 which shows distinctly smaller values than the experimental ones thus leaving space for single and plural scattering con ♦ Schonlond 100keV (1923) tributions. Contrary to the point-source model this model 0 Polluel 16 keV (1947) gives backdiffu sion values also for low Z and for thin films. • Hollidoy ,S t«ngloss 20 keV (1957 ) 0,3 v Kanter 10 keV (1957) • Mulvey , Doe 23keV (1960} o Bishop 10/30 keV {1965) 3. COMBINED MODELS • Cosslett . Thomas 10/20/1.5 keV {1965) •6Heinrich 10/.t.9 keV (1965) I Siobe, 20/ 53 ••V (1969) A good combined model describing backscattering of t Ore-sch er , Re-imer, Seidel 0,2 9,3/ 102 keV (1970) solids should include single scattering and diffusion but also •Reimer , Tollkomp 20 keV (1980) the contributions of double and multiple scattering proces ses. However there are no simple models for double and mul tiple scattering. An attempt for considering double scattering has been made by Archard (1961), and Body (1962) has im 0,1 proved this approach. The result is plotted in Fig. 2. Unfor tunately we found it not very suitable for a combined model treatment. FeCuGe MoPdAg W PtAuPbBi U The only simple models combining at least two scattering 0 ~0~ ...,1'0~ ~20--' - 3',-oL.....~, o--'--"LLs~o::..__6~0 - 1~0--'----'-'e.;:,o.,:__:_~go:;_ concepts in a satisfactory way are those regarding single scat - --z tering processes on the one hand and diffusion on the other hand as the two limiting concepts . Such models have been Fig. 5. Backdiffusion coefficients of bulk targets according developed by Werner (1978, Werner et al. 1982), Dudek to Archard's point-source-diffusion model, to Thiim (1979, 1980, 1982 a,b), and Niedrig (1981 a,b) . mel's continuous-diffusion-depth model, and to the Dudek 's model is not treated analytically throughout and modified continuous-diffusion-depth model pro will be therefore not discussed in detail here but it gives inter posed by the author, compared with the experimental esting results for the angular distributions (1982 a,b) and for backscattering values taken from Fig. 2. In all cases the energy distribution (1980) of backscattered electrons. Archard's diffusion depth yd = 40 I (7 Z) has been The most sophisticated model is that of Werner, whereas used. 55 Heinz Niedrig the author's model (Niedrig 1981 a) is slightly simpler and condition equation 9) the following relation for the backscat will be discussed first. Then the Werner model will be treated tering coefficient of bulk targets for normal incidence of the in a slightly different way compared to the original version electron beam: (Werner 1978, Werner et al. 1982), and will be extended to the calculation of angular distributions, to allow finally an aN-2+0.5aN - f overall comparison of both models. + a~------ 22 aN(aN- 1) 3.1. The author's approach for a combined model where In a combination of single Rutherford scattering and dif fusion one has to consider that both processes influence each 23 other because the flux of incident electrons within the target is reduced by both scatte ring processes. We therefore have to For thin films of thicknesses less than half the electron range make an assumption on the differential scattering from a we obtain thin layer within the target regarding single scattering and diffusion, and then calculate the flux of incident electrons versus depth by a balance consideration. Using the result of thi s balance consideration we are finally able to calculate the backscattering coefficient. 3.1.1. Introduction of diffusion into the Everhart model The general treatment described before is just the treat ment of Everhart (1960) . We only have to add a diffusion 24 term to the expression for the scatte ring of a layer in the reduced depth y into the solid angle ctn (Niedrig 1981 a,b): If we neglect diffusion (a~ = 0: aN = aJ equation 22 be comes precisely the Everhart formula equation 10. In equa a lo(y)dy ctn 5 tions 22 and 24 the first term is the contribution of single d 2I(y,0,,p) = ------Rutherford-scattering, and the second term gives the contri 4 27r 1 - y 2cos ~ bution of the diffusion. For a quantitative evaluation, how - 2 ever, a~ has to be determined first. (Remark: The angular integrations lead ing to equation 20 a~ I0(y)dy + ----- ctn 19 have been done -as befor e in section 2.1.-in such a way, 21r 1 - y that the primary beam loses all electrons scatte red into angles 0 '.S90 °, whereas all electrons with a scattering angle 0 > 90° The first term describes sing le Ruth erfo rd scattering with the are treated as if they were not scattere d at all. This implies typical angular dependence cos - 4 0 / 2 according to Ever that electrons scatte red into 0 > 90°, Rutherford scattered or hart's treatment, whereas the seco nd term describes isotropic diffused, remain in the prif'lary beam and can suffer further scatter ing according to the diffusion model with a coefficient scatte ring events of these types. This was Everhart's ap a~ to be determined later. ,p is the azimuth angle. The deno proach. A different treatment would be to assume all dif fused electrons to be lost by the primary beam as was done by minator (I - y) in the second term has been added mainly for Werner in his model, see section 3.2. An application of this reasons of integration . As in the first term, where the deno treatment in the above model would give for the flux of pri minator (I - y) comes from the energy dependence of the mary electrons scattering cross section (equation I) together with the Thom son-Whiddington law (equation 2), this factor increases the 21a diffusion probability with decreasing energy of the incident electro ns travelling into the target. For the backscattering coefficient equations 22 and 24 can be Following Everhart's treatment as in section 2.1. we obtain from a balancing consideration after angular integration used further, if a~ is replaced by ar and aN is replaced by 23a y ( Io(y) J -- dy 20 Because of the procedure to determine a~, or now a~ ', to be 0 1- y described in the following section, the results for the back scattering coefficient presumably will not differ very much which is a transport equation of the same type as in equation from each other. However, this has to be proved numerical 4, now with the solution ly). 21 3.1.2. Determination of the diffusion coefficient a~ and results for normal incidence instead of equation 7. Introducing this into equation 19 we obtain by integration over ,p, 0, and y (observing the exit The model described in the preceding section is a continu- 56 Analytical Models in Electron Backscattering ous-diffusion-depth model as concerns the diffu sion frac 1.5 tion. Compared to the Thummel model s de scr ibed in section 2.3. it ha s a different distribution of the diffusion probability alon g the path of the primary beam, being proportional to a~(l - y)aN - 1. Now, we will determine a~ in such a way, that diffusion for vanishing sing le sca ttering (a s - 0) equation 22 gives the same va lues for the backdiffusion coefficie nt as the pure dif constant fusion model equation 18 ( iedri g 1981 a,b). This condition O N d is ad(Z)- 2 + 0.5 ~(Z) - I 25 a~(Z) - 1 and allows us to determine a~ as a function of the atomic Everhart's number Z by regarding Yct = 40 / (7 Z) (equation 13). The constant= num er ical so lution of equation 25 is plotted in Fig. 6. 0.5 2 Now we can eva luate equations 22 and 24 for the backscat Os= 0.024 · z IM tering coefficients. Fig. 7 shows the calculated curves for the a5~0.012 ·Z backscattering from bulk targets: the cur ve for the total backscattering lies close to the experimental va lue s. The dif fusion fraction proves to be about twice as large as the single scattering fraction, thus explaining the discrepancy between Ever har t's result and the experiment. The eva luation of eq uation 24 for thin films is plotted in 0 -11<----'---L---'---'----'---'----'------'----'------'-- Fig. 8 show ing good agreement with the experimental values, o 50 100 regarding that no adjustment between theory and experiment ------►- z has been made. For film thicknesses D ~ R/2 (y ~ 0.5) Fig. 6. Everhart's single scattering coefficient as from Eqs. equation 24 simplifie s to 5 + 6 and the diffusion coefficient a~ calculated from Eq. 25 vs. atomic number Z. a s + a~ 1t(D) ~ --- D 26 R exhibiting a linear slope as was observed also experimenta lly (Fig. 8). However, for very thin films one would expect a reduced slope, because then the diffusion fraction should +/ tend to zero compared with the sing le sca ttering fraction. In very careful expe riment s this has been conf irmed for very 0,1. 3 ~""""''"' thin films D ~ 10- R of elements of low atomic number 'Sch ontond 100keV (1923 ) v, o Polluel 16keV (19G.7) (Muller and Schroder 1977, Schmoranzer and Grabe 1976). • Hollldoy ,S tf'!"ngloss 20 k" \t (195 7) 0 0 V Therefore for this thickness region equation 26 has to be re C I v Konll!'r 10 kl!'V (1 957) 0,3 placed by • Mulvl!'y , 001!' 23k l!'V (1960) o Bis hop 10/ 30 k p\J (1965) ,. CoSSll!'tt . Tho ma s 10/20/l.5 keV (1965 ) • A He1nr,ch 10/ l.9 kl!'V (1965) I 51e,bpr 20 / 53 kl!'V (1969 ) 27 l Orl!'SChl!'r , R1!'1mf!r , 5 1!'1d l!'I 9,) / 102 keV (1970) 0,2 • Reimer , Tollkomp 20 k e V (1980) It is a disadvantage of this model that it doe s not produce this result automatically. It is caused by the diffu sion probability used in thi s model , as has been remarked at the beginning of 0,1 - this sectio n , which does not vanish for y - 0 (Niedrig 1981 single scattering fr act,on a,b). This discrepancy ha s been avoided in Werner's model (Werner 1978) . For practical purposes thi s critical thickness Fe Cu Ge Mo Pd Ag W Pt AuPb81 U is so small (Muller and Schroder 1977), that it hardly can be 10 20 30 1.0 50 60 70 80 z 90 represented in plots like Fig. 8. In the thickness range 0.2 R/2 to 0.4 R/ 2 equation 26 seems to be a useful approximation. Fig. 7. Backscattering of bulk targets calculated from the --- combined single scattering and diffusion model due 3.1.3. Extension to oblique incidence to the author (Eq. 22), compared to the experimental In calculating the backscattering coefficient for oblique in- values from Fig. 2. 57 Heinz Niedrig rt (DJ 0.2 ri (D) O.L. Ag 20 keV(o) Al 0 0 0 • • 0 l 0.3 l 0 0.1 0.2 0.1 0 0 0 1000 2000 3000 4000nm 0 100 200 300 400 500 600nm ---o ---o 0.3 Cu 20 keV(0 ,o) 0 0 0 0 0.5 • 0 • 0 Au • 0 'v 20 keV (• .o,o) r 0 • 0 0 0 0.2 'v 0.4 0 30keV(,;;,) 'v r 'v 0.3 V (•,") D 0.1 0.2 0 0 --- D ·53 keV ( □) 0.1 0 0 0 200 400 600 800nm 0 100 200 300 400nm --o ... D Fig. 8. Backscattering coefficients of thin films from the --- combined model due to the author (Eq. 24) com cidence of the primary beam on the target surfac e we can pared to experimental values (full symbols: Seidel follow the treatment of Nakhodkin et al. (1962a) who made 1972, half-filled symbols: Cosslett and Thomas 1965, the corresponding calculation using the Everhart model. We open symbols: Niedrig and Sieber 1971). have only to add the diffusion term as in equation 19, and y has to be replaced by y/ cosa, where a is the angle of inci e- backscattered dence measured to the surface normal (see Fig. 9). 1f we electron transform the coordinates 8 and dfl 4dfl ' 28a I 4 I cos 8 / 2 \ \ y with cos a. =1 ',...._- ___ ,...,"';' range sphere 1 for depth y f(rp' , 8', a)= I - sin8'cos 58 Analytical Models in Electron Backscattering (or 0 increases) with the result that in addition to (i) the 29 Rutherford sca ttering fraction strongly increases. The calculation of the backscattering from films leads to somewhat longer analytical expressions (Niedrig 1981 a). For with the abbreviation thin films (D ~ R/2) we obtain the following approximation 30 33 The further treatment corresponds to that of Everhart (1960) and yields for bulk target s and oblique incidence for the which directly exhibits the two tilting effects (i) and (ii) des single Rutherford fraction cribed before . 3.1.4. Angular distribution of the backscattered intensity Inserting the flux of primary electrons (equation 29) into equation 19, regarding the transformation 28 an d integrating 31a over the depth gives the differential backscattering coeffi cient or the angular back scatter ing distribution of bulk tar and for the isotropic diffu sion fraction of the back scattering gets: coefficie nt Li)JN(0' ,!{)' ,O') aN COSC\' LiD' 1J~ (a)= ___5!{ I - -- [1 - y~ - I] } 31b 00 q q - I Y111(0 ' ) )q] • [ I - ( I - - - -- 34 where cosa with f from equation 28b. COSC\' By the term in square brackets the di str ibution tends to 32 Ym = I + COSC\' zero for grazing observation (0 ' - 90°). The first term (with a 5 in round bracket s) describes the Rutherford scatter ing is the maximum red uced scat tering depth to allow backscat fraction, the second term (with a;;) the isotropic fraction. The tered electrons to reach the su rface again with respect to their maximum depth y 111(0 ' ) from which an electron can escape into the direction 0 ' (Fig. 9) is given by range. In Fig. 10 !J~(a) = !J~00 (a) + 1J~00 (a) is plotted versus a for aluminum, copper, silver, and gold, together y (0 ' ) Y (0 ' ) with the two fractions and with experimental values of differ 111 111 + ent au th ors. COSC\' cos0 ' The increase of the backscattering coefficient with increas COSC\'• COS0' ing inclination of the incident beam obviously can be ex y (0') = 111 35 plained by two effects: (i) with increa sing a the range spher es cosa + cos0 ' shift towards the surface and therefore the exit cone angles incr ease (Fig. 9). This effect appears for both scatteri ng frac The comparison with experimental curve s from Seidel (1972) tions. (ii) With increa sing a the sca ttering angle {) decreases in Fig. 11 shows that her e we have considerable differences I - 1 - '100 1 'loo Al t Ag t Au 0.5 0.5 0.5 'l_doo 0 +-- -'--'--'--'----'--" ----' - LC>...J 0 t-- -'--'--'- -'-- --'--'---'~ ~ 0 +-- ..J.._-'-_J__L____L_c----1_ L.:,,.J 0 20 l0 60 80 0 20 l0 60 80 90 20 l0 60 80 ---a -a --ex -a Fig. 10. Backscattering coefficients !J~ (a ) of bulk targets hodkin et al. (1962a, crosses), and Drescher et al. for oblique incidence (Eq. 31), compared with ex (1970, circles). ))~00 and !J~00 are the single scattering perimental values of Kanter (1957, points), Nak- and diffusion fractions, respectively. 59 Heinz Niedrig Al CX=0° / ~ experime nt (30 keV) .,Kl \ ( Seidel 1972) I I I theory I '1 ""(total) \ \ d 1ffus1on / . fra ct\on \ single scattering L fra ct1an 001. 0.02 0 002 0.01. A.Q. ----6 0 e Au ct=0° 0.2 experiment ( 30keV ) Au cx=60° --, ~ ( Se idel 1972) \ \ I H7i~;,~ J d1ffus 1on \ \ f ract1on \ _ singl e scattering fraction 0.2 0.1 01 ~ 02 0 0.2 0.1. 0.6 08 1.0 L:J 1 ---- 60. --ETI Fig. 11. Angular distributions of the differential backscat model due to Niedrig 1981a,b), broken curve: ex tering coefficient of bulk targets for normal and periment (Seidel 1972). oblique incidence . Full curves: Eq. 34 (combined between calculated and experimental distr ibution curves backscattering inten sitie s, as ha s been investigated experi alth ough the total back sca ttering coefficients (int egrated mentally by Wells (1972, 1974) by introducing an energy bar over 0 ' and \0' ) agree rather well. The theoretical model rier for the detected electrons . The treatment clo sely follows gives by far too flat distribution s. that of Nakhodkin et al. (1962 a,b), McAfee (1976), and There are two possible reasons for that result: (i) double Iafrate et al. (1976) . and multiple scatter ing have not been regarded . Transport Regarding only those backscattered electrons which at equation calcu lations give better results in this respect maximum have lost only a fraction E of their initi al energy (Fathers and Rez 1979). (ii) The Rutherford scattering cross E0, so that their exit ene rgy E ~ EE0 (E = 0 to 1), we ob tain sect ion shou ld be replaced by the Mott scatter ing cross for the maximum reduced electron path s within the target section (see, e.g ., Ladding and Reimer 1981). The Mott cross-sec tion s show a distinct peak arou nd 0 = 0° (t'J = 36 180°), increasing with Z . Their use in this model would shift the theoretical distribution curves nearer to the observed which is the reduced form of the Thomson-Widdington law curves . The problem for doing so is to find a simple analytic (equation 2). Then it follows from Fig. 9. expression for the Mott cross-sec tion allowing for the integ rations neces sary in thi s model. + 37 3.1.5. Surface density distributions of backscattered cosa cos0' electrons and the maximum depth from which an electron can escape For the calculation of proximity effects in electron beam with an energy E = E E0 is lithography (e.g. Chang 1975) we need the knowledge of the surface density distribution . Monte Carlo calculations on cosa • cos0' 2 this problem have been reviewed by Kyser (1981). For results y (8 ' ) (I - t: ) ------ 111 = 38 of calculations from the described model see Niedrig (1981 b). cosa + ca se' 3. 1.6. Energy spectra of backscattered electrons Introducing this into eq uation 34 we get (Niedrig 1981a) The model can easily be extended to calculate low-lo ss 60 Analytical Models in Electron Backscattering P l (78) t.,t(e ' , I Ml ' I I l"..'.1 I d e I 2 I / Au (79) • [i - (co sa+t cos8' )q] I Eo=20- L.OkeV 39 I ~ cosa + cos8 ' I 'i lt.7) t: ~ 29) Especially the single scattering fraction of this result (a~=O) ;; shows shape s of the angular intensity distribution (Niedrig 0 Al (13) 1981 a,b) which agree very well with experimental results of Wells (1972, 1974, 1975, and 1980) for low loss measure ~ 02 Ol ~0 608 • 1 0 ment s at oblique incidence. -- e.=EIE 0 For normal incidence (ex = 0, 8 ' = 8) we obtain from equation 38 the depth from which an electron escapes with Fig. 12. (a) Energy distributions of electrons backscattered the relative energy t into the direction 8: from bulk targets of different atomic numbers. Full curves: author's combined model (Eqs. 43a + 43b). cos8 Broken curves: Everhart's model (Eq. 43a with 2 Ym(8) = (l-t )-- 40 aN =a, - a~= 0.045 Z). I + cos8 (b) Energy distribution s of electrons backscattered from bulk targets of different atomic numbers, and further measured by Kulenkampff and Spyra (1954). case 41 dy = - 2t--- dt w I+ cos8 • --- dw 43b l - w By introducing dy into equat ion 19 and regarding the ex pression 21 for the primary electron flux and the relation For vanishing diffu sion (a~ = 0, aN = aJ equation 43a has 2 2 2 been derived by Nakhodkin et al. (1962a) and by McAfee 2cos (8 /2) = l + cos8 we obta in with d 11=d 1/10 and integ ratio n over To get the energy distribution of the total back scattered in 3.2. Werner's approach for a combined model tensity we have to add equations 42 a and b and to integrate Werner (1978, Werner et al. 1982) in principle used the over 8 = 90° to 0°. By substituting the term in square bra ck same concept of combining single Rutherford scatteri ng and ets the single scatte ring fraction equation 42a can easily be in diffusion as described before . However , there are several dif tegrated yielding ferent feature s. He distinguishes between the effective path length s of electrons within the target and the depth x below a 2 - 0.5aN(l+t 2)aN[aN(l - t 2)+2 ] the target surface. We will therefore use the following re 2t ---5 ------43a aN (aN+ I) (l -t 2)2 duced quantities: whereas the diffusion fraction equation 42b must be integ (i) rated numerically. With the substitution cos8 = - w we 44 then get - 1 2 where s is the mean path length for an electron coming to 2ta~ J [ l + (1 - t ) l: w ] aN- 1 0 rest at E = 0, and 0 61 Heinz Niedrig (ii) which is drawn for different atomic numbers in Fig. 13. Con X trary to the previous models the total diffusion probability y =- 45 lies within the mean electron path length . Fort > I the diffu R sion probability ha s no real value which makes physical sense. as the red uced depth as in the previous chapters. Especially The electron intensity d 2Id diffused from a layer of reduced for the diffu sed electrons s is larger than x by a factor 1.37 (Werner 1978, p. 50). The reduced quantities y and t are thickness dt = (I - t) d Tinto the so lid angle ctn is now related to each other by 52 40+Z 40+Z y=----t=---t 46 20(1 + ,/3) 54,64 A balance consideration as described m previous sections gives the following transport equation For Z '.".15 it follows that y > t although x < s. This is due to the fact that y is related to the range R, and t is related to 1 21r 7f the mean path length s . Both quantities s and R differ by a 0 0 53 factor increasing with the atomic number Z (Thlimmel 1974, p. 202): s0 Z if we assume that the incident beam is reduced by all elec - =I + 47 R 40 trons which are converted to diffusion. By solving equation 53 we obtain the flux of primary electrons in the target to be Most of the following considerations are made regarding the reduced path length t and not the reduced depth y. This 54 makes no difference in the results as far as we deal with bulk targets . For thin films, however, equation 46 has to be re Inserting 10 exp( - aJT) from equation 54 into equation 52 garded. and regarding the diffu sion probability (Eq. 49) yie lds 3.2.1. Werner's diffusion model 55 Werner introduces an auxiliary quantity T measuring the reduced electron path length t: The factor aJTI (I + a~vT) has the effect, that the diffusion I T c:c - fn(l - t) 48 For t /4 I we have T ==t, whereas for t - I: T - oo. By thi s 6 tran sformation the range of values fort (0 to I) is expanded Au (79) to the range Oto oo for T. Using this variab le T Werner make s the following approach for the differential probability dis w,5 tribution of the electrons to become diffused: p 49 with a diffusion coefficient a~; which Werner estimates to be / Ag (47) 3 z at==- = 0.2 Z 50 C ( 6) 5 2 Similar to the author's diffusion distribution function (equa "' tion 15) Werner's function (equation 49) is close to the Bothe distribution P ' (section 2.3) if drawn in a sca le correspond ing to u in Fig. 4 (Werner 1978, p. 29) . pw , is normalized corresponding to equation 16, that is, the total diffusion probability is I. 0--,L,:=------===""----,--:=---=---~- Tran sferred to the reduced path length t equation 49 be o come s 0.5 - t = ~o Fig. 13. Diffusion probability distribution pw · for different 51 elements vs. reduced path length t s/ s of the in = 0 cident electrons (Werner 1978). 62 Analytical Models in Electron Backscattering vanishes for very small depths as to be expected, whereas for tain for the differential sca ttering coefficient of a layer within larger depths this factor tends to I. the target ( iJ :". 90°) We do not carry out the further calculation but include this diffusion model into a combined model to be described in the following section. 3.2.2. Combination of single scattering and diffusion 62 For the single scattering contribution Werner also uses the Everhart model. Regarding dt = (1 - t)dT we obtain which after angular and depth integration and observing the 56 exit condition for the maximum depth for the backscatter direction iJ cosiJ Werner estimates the single scattering coefficient a to be t =--- - or Tm= fn( l - cosiJ) 63a,b 5 m cosiJ - I z results in the backscattering coefficient for bulk targets a ,=-=0.0125Z 57 5 80 which agrees well with Everhart's value (equation 6). Equations 55 and 56 can be combined to give the inten sity sca ttered from a layer of reduced thickness dt by both pro cesses arT dQ + a~t---- 10(T)d7 -- 58 l + arT 4 1r where 1 ( T) now is the total f1ux of primary electrons, when 0 both Rutherford single scatter ing (first term) and diffu sio n (second term) are present. To calculate 10(7) we make again The numeri cal evaluation is drawn in Fig. 14 together with a balancing consideration which leads to the following tran s the result of the combined model described in the section 3.1. port equation: Fig. 14 show s that both model s reflect the experimental situa tion sat isfacto rily . If one compares the ratios of the diffu sion to the single scatte ring fractions in both models it turns out, 59 that in Werner's model this ratio is considerably higher than in the author's model. So in Werner's model the single scat tering is only a correction of about 20% (except for very thin The preceding angular integration has been extended over the films). For thin films of reduced thickness y the relation 46 back half sphere for the single scattering term (as in the Ever between the reduced path t and y has to be regarded. Cor hart model) and over the full sphere for the diffusion term rectly this relation has to be applied only to the diffusion (as in Werner's diffusion model) . The solution of equation 59 fraction (Werner 1978, p. 50). However, since in Werner's gives the f1ux of the incident electrons within the target model the single scattering fraction is only about the fifth part of the diffusion fraction (except for very thin films) we 1 (7) = 1 (1 +a~ 'T)e - awr do not make a large error in applying relation 46 to the total 0 0 60a backscattering . The calculation of backscattered inten sity for thin films or with T = - fn(l - t) yields where 61 Introducing the f1ux (equation 60a) into equation 58 we ob- 63 Heinz Niedrig 3.2.3. Angular intensity distribution Curves due to From the differential scattering coefficient of a layer with models of, in the target (Eq. 62) we obtain the differential backscatter Werner (1978) ing coefficient (or the angular intensity distribution) of a Niedrig (1981lv bulk target by integration over T and regarding the exit condi- tion (Eq. 63b): Experimental values due to , ♦ Schontanc, )(}() keV (1923) o Polluel 16keV ( 19'.7) • Hot11doy . Stl!'l'"ngloss 20 keV (1957 i tin 9 Kanter 10 keV (1957) •Mulvey . Doe 23keV (1960) o Bi sh o p 10/ 30 keV (1965) ,. Coss lett . Thomas 10/20 / 45 keV (1965 . ., £J. He1nr, c h 10/ t.9 keV (1965) 1 Sieber 20 / 53 keV (1969) l Dre sc her, Reimer , ~; ; ~;; keV (lg?Qj 9 • {1 - (1 + fn(l + cos8)aw)(1 + cos8) - aw}] •Reimer , Toll komp 20 keV (1980) C Al 51 Fe Cu Ge Mo Pd Ag W Pt AuPbBiU 67 10 20 30 4 0 so 60 70 80 90 -z Fig. 14. Backscattering curves for bulk targets due to the In Fig. 16 the angular intensity distributions according to combined models of Werner (1978) and Niedrig equation 67 are drawn for aluminum and gold targets. The (198la,b) in comparison with experimental results. curve for aluminum (total) approximately shows a cos8- shape, as is observed experimentally (Fig. 11). The curve for gold shows a very flat shape contrary to the experimental observation. This shape results from the very flat single scat tering distribution added to the half sphere shape of the dif fusion fraction, which is to be expected from a diffu sion source ju st below the surface. As discussed in section 3. 1.4. the use of the Mott scattering cro ss-section could improve the angular shape for high atomic numbers. 3.2.4. Energy spectra aw I + [{ --- (aw - l)) fn-+ Corresponding to the treatment in section 3.1.6. the path 2(1 - t) 1- t length tm from which an electron escapes with the relative energy E into the direction 8 is 65 cos8 tn,(8) =(l -E 2) -- - 68 1 + cose For t = 0.5 equation 65 transforms into the result for bulk targets (equation 64). Fig. 15 shows a plot of the backscat and tered intensity versus film thickness for aluminum, copper, silver and gold due to Werner's model (broken curves) drawn 69 into the curves of Fig. 8. Although the general agreement of model calculation and experiment is of similar quality than Hence for the model described before (the agreement for high atomic numbers and higher energies seems to be not so good), the Werner model has the advantage that for very thin cos8 dT = - 2Ee - ' ---- dE 70 films the diffusion fraction tends to zero compared with the I+ cos8 single scattering fraction, as one has to expect physically: For t ~ I we obtain from equation 65 and by inserting dT into equation 62 we obtain the spectrum of electrons backscattered into the solid angle 21r sin8d8 66 ( = - 2 1r sin!? di?): 2 The model described in section 3 .1. could not show this fea d 71~ (8 ,E) ( w l)r COS8 sin0 4a E(I + awT )e - a - 111 - ---- 71a ture. = 5 dE de ct m (I + cos0) 3 64 Analytical Models in Electron Backscattering ri (D) ---- 02 · ri (D ) OL. Ag ~ 20keV( 0 ) Al ,/ / 0 0 /:-. -- =--- / 0 " • r / 0.3 / ~ I .. I I /4 0.1 0.2 0.1 -- □) 0 0 0 1000 2000 3000 4000nm 0 100 200 300 400 500 600nm ---o --o ------./ / ,/ / 0.3 f"l(D) Cu ,-: -20 keV (0 ,o) ~ 0 '/ 0 0.5 • I 0 • • ~ Au I I • I 0 --- 20keV (•,H • / I o ,/ 0 .:, 0 0.2 'v 0.4 / 0 30 keV ( 'v) r / c '7/ / eV / 0.3 0.1 0.2 ./ -- ~v ( □ r-- - -- 0.1 o-~+- -.!....__.,__ _.,___.,___...L_ _ _L_ _ _L_ __ 0 --'-r.-~ ..+-- --'---'----'---~---'---'--- 0 200 400 600 800nm 0 100 200 300 400nm .., D .., D Fig. 15. Backscattering coefficients of thin films due to the combined models of Werner (1978, broken curves) and of Niedrig (1981a,b, full curves) compared to experimental values (see Fig. 8). Al ex=0° Au ex=0° 0.02 0.04 0.1 0.05 0 0.05 t:;.11 - !:;.11 Ml' --- !:;.(l' Fig. 16. Angular distributions of the differential backscat tering coefficient of bulk targets due to Werner's combined model, Eq. 67. 65 Heinz Niedrig 2 more realistic energy distribution than the previous models d 17~v (8, t) 2 -(aW - J)7 cos8sin8 ---- = 2aw E T e "'----- 71b using a scattering probability increasing monotonically with dt d8 d m (I +cos8) the depth . For yielding the energy distribution of the total backscattered intensity we now have to integrate over 8 = 1r I 2 to 0. With 4. CONCLUSIONS It has been shown, that it is possible to combine at least e - (a w - I) r111 [ - (I _ t2) ___cos8 _ ] a w - I = 1 72 single scattering and diffu sion in analytical backscattering I+ cos 8 models. Without adjustment to the experiment these models give good result s for the total backscattering of bulk targets and with the substitution cos8 w we get = - and of thin films, also for oblique incidence. Comparing the two combined models the Werner model exhibits some fea tures which are phy sically more justified, e.g., the reduction of the backscattering to single scattering for very thin films and the more realistic energy distribution . The author's model is slightly simpler, an advantage which especia lly be w wdw • [1 + (1 -t 2>-- ] aW- [ __ _ 73a comes obvious for oblique incidence. Both models fail in 1- w (1 - w)3 reproducing the experimenta l angular distributions . In this respect th e mode l calculations have to be improved either by - I 2 a suitable approach for plural scattering contributions, or by - 2af t J fn [I + (I - t )- -w- ] • replacin g the Rutherford cross-section by a suitab le analytic 1-w 0 al form of the Mott cross-section, or by doing both . w ] w wdw REFERENCES • [ l + (l - t2)-- a - I __ 73b 1- w 1- w Archard GD. (1961). Back scattering of electrons . J. Appl. which mu st be evaluated numericall y. The result is plotted in Phys . 32, 1505-1509. 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