An analysis of Rutherford backscattered spectra to determine dechanneling in single crystals

Nuclear Instruments and Methods 174 (1980) 577-584 © North-Holland Publishing Company

AN ANALYSIS OF RUTHERFORD BACKSCATTERED SPECTRA TO DETERMINE DECHANNELING IN SINGLE CRYSTALS John Alan MOORE Physics Department, Brock University, St. Catharines, Ontario, Canada Received 31 October 1979 and in revised form 11 March 1980

A numerical method has been formulated to analyse aligned and random backscattered spectra to determine the random fraction x(z) of aligned MeV ion beams in single crystals. The analysis takes account of the difference in stopping power between channeled and random trajectories, the variations of stopping power and scattering cross sections with energy and a deviation of backscattered spectra from single scattering theory. The results of calculations for 1 MeV fl+ spectra from Si and Au are presented and compared to those obtained from the commonly used analysis which takes the ratio of the aligned and random spectra. The results show that the detailed numerical method is required in the analysis of the spectra at backscattered energies less than approximately 0.7k2Eo, where k 2 is the usual kinematic scattering factor, if a reasonable precision (<10%) is to be maintained in x(z) at these larger depths.

tively. The depth function z(Ea) is calculated by assuming a channeled stopping power, Sc, along the ingoing path and a random stopping power, St, along the outgoing path. There is some uncertainty in the best choice of Er corresponding to Ea. However, usually, E a and Er are chosen to correspond to the same backscattering depth. Other more detailed computations [4,8,9] have been described, but not generally used, which take account of the consequence of the general expectation [10] that S e < S r ; namery, there is no unique depth scale z(Ea) for the aligned spectra. Jack [8] effectively determined X(z) by fitting calculations for Na, based on various assumed trial functions for the dechanneling rate dx/dz, to the measured aligned spectra. The results were applied to the case o f 150 keV I-I+ along (110) Cu. The analysis described by Grasso [4] does not fully take into account the significant effects produced by the variation of stopping power with energy. Nashiyama [9] developed a general analysis to include the effects o f dechanneling by defects and this he applied to the determination of proton induced damage profiles in Si. All analyses make the following assumptions; (1) So(E) ~ aSr(E) where a < 1 ; (a) a channeled particle does not backscatter directly from on-site lattice atoms; (3) the backscattered particle experiences only one "change in direction" collision; we shall call this a single scattering theory. However, random backscattered spectra are not

1. Introduction The dechanneling of MeV particles (e.g. I f , He*) in good quality single crystals has been studied quite extensively during the past ten years [ 1 - 7 ] . The theoretical description has been gradually refined and good agreement (10-20%) between theoretical calculations and experimental results are now reported [6,7], at least for those crystals (e.g. Si) in which defect dechanneling is confidently estimated to be negligible. The dechanneling is normally measured by calculating X(z), the variation o f the random fraction with depth, from an analysis of the aligned and random backscattered spectra. There has been rather less detailed attention given to the analysis of the spectra than to the theoretical calculation. However, in view of the reasonable agreement between experiment and theory that now exists in some cases, it seems worthwhile to reexamine the significance of the approximations that have normally been introduced into the measurement of X(z). Perhaps the most commonly used approximation [6] gives ×(z)

Na(Ea) -

Nr(Er) '

where Na(Ea) and Nr(Er) are the aligned and random backscattered yields per unit energy interval respec577

J A Moore /Dechannehng in single crystals

578

always adequately described by the single backscattering theory. Indeed, for 1 MeV H + backscattering from high Z targets the difference between theory and experiment becomes 2 0 - 3 0 % at the lower energy end of the spectrum. The purpose of tiffs article is to present a detailed analysis of the aligned and random spectra from good quahty crystals, in which defect scattering is negligible, in order to determine ×(z). The analysis takes into account the consequences of Sc < Sr and the observed deviation of the random spectrum from that calculated by single scattering theory. The first section descrtbes the single scatterang formulation followed by the introduction of a modification to take account o f the fraction of backscattered particles which experience more than one scattering in the target. The final section compares the results of this detailed analysis with various ones reported previously and is applied to H* dechanneling in Au and Si.

2. Analytical formulation

2.1. Single scattering The basac assumption as that a channeled particle does not backscatter. This means that we consader scattering from defect free crystals or crystals containing types of defects which do not directly backscatter a channeled particle. Initially, we make the further assumption that a dechanneled and backscattered particle has experienced only one large angle scattering into the detector. Referring to fig. 1, z' and z are the depths at which the particle is dechanneled and backscattered respectively, Ea(z' ,z) as the energy o f a backscattered particle leaving the surface of an aligned crystal, Ea(z', z) is the energy of the dechanneled particle at z. Let dx(z') be the increase in × in the depth interval dz at z' and dEa(Z' , z) be the range m values o f Ea for partacles dechanneled at z' and backscattered in the interval dz at z. We may then write the aligned backscattered yield, Na(Ea) as: Na(Ea) =Kip ~

dx" E~ "

'

(1)

where Na(Ea) is the number of events recorded per unit energy interval, K is a proportional constant,

I 1s the incident particle fluence, p is the number of crystal atoms per umt volume. The sum is over all pairs (z', z) whlch correspond to the same energy E a and includes all values of z' m the range 0 ~< z' ~< b where b xs the scattermg depth for a particle that backscatters Immediately after dechanneling and then emerges from the crystal with energy Ea. Similarly, the random backscattered yield per unit energy, Nr(Er) , is written 1 dz Nr(Er) = Kip E~ dE r '

(2)

where Er(z ) is the energy of a backscattered particle leaving the surface of a non-ahgned crystal, El(z) is the energy of the incident particle at depth z, dEr(z ) is the range in values of Er for particles backscattered in the interval dz at z. The problem is to determine X(z) from an analysas of the observed data, N a and Nr, using eqs. (1) and (2). Let So(E), Sr(E) be the effective channeled and random stopping powers respectively. In general, we expect So(E) ~- ~ S ~ ( E ) ,

where a is a proportionality constant. If a < 1 then the value of the product E~ • (tEa varies from term to term m the sum f o r N a given by eq. (1). For example, in the case of 1 MeV H + -+ Au with ¢x = 0.5, the differences in value range up to 30%. To deduce an approximate expressaon for X(z) in terms of the ratio of the ahgned to random backscattered yields, we rewrite eq. (1) as 1

=Kto

×(z)

e,

(la)

where the depth function z(Ea) is calculated by assuming a channeled stopping power, Sc, along the lngoing path and a random stopping power, Sr, along the outgoing path. The function f(Ea) represents a sort of weighted average for E~ dE a. Combining eqs. (la) and (2) we have

x(z) -

Na(E.)

f(E.)

Nr(/~r) E~ dLr

The commonly used analysis described briefly by

J A Moore / Dechannehng in single crystals SURFACE

Pederson [61 gwes

X(z) - Na(Ea) Nr(Er)

(3) "

K i p = Nr,o" E:o [Sr,o] ,

(4)

where Eo is the incident energy and [St,o] is the energy loss factor for scattering o f a random beam from the surface. k: 1 cos01 S r ( E ° ) + c o s 0 2

Z!

Eo

However, in eq. (3) the correct value o f Er corresponding to Ea can only be determined by evaluating f(Ea) numerically and putting f(Ea)/(E~dEr) = 1. In the absence of such a calculation there is, of course, uncertainty associated with any choice in value of Er. The best choxce is not evident and is not always clearly stated [6]. In some analyses [7] Ea and Er are chosen to correspond to backscattering from the same depth, and so Er < Ea- In the approximate calculations for ×(z) presented in section 3 of this paper, and referred to as the Pederson analysis, we have used eq. (3) and put E r = E a. In order to avoid the uncertainty in the choice of E r m eq. (3), we have carried out a numerical solution of eqs. (1) and (2) to determine X(Z). Of course, X(z) is still subject to errors resulting from the uncertainty in the assumed value for the effective channeled stopping power, Sc(E). We use eq. (2) to express the product KID in terms of the random yield, Nr,o due to scattering from the surface. Using eq. (2) we write

[Sr'°]

579

Sr(k2E°)'

(5)

Z~Ed

Ea

Fig. 1. z' and z are the depths below the surface at which the incident beam particle, energy E0, dechannels and backscatters respectively. E d and E a are the energies of the dechanneled particle at z and the backscattered particle leaving the surface of the aligned crystal, respectively.

(Eai ]) = l(Ea, 1 + Eai, l _ l ) , AX~ = Xi - X~-1 ,

AEal I = Eal,l_ 1 - Eal,! In terms o f this discrete depth interval notation we write eq. (6) as Na((Ea) ) =Nr,o" Eo2" [Sr,o]

× ,/ (<_~d,j>2 . ,~Ea~/ ~z

(7)

,SXi •

All terms in eq. (7), except AXz , are measured directly or m a ~ be calculated using St(E) and the assumed value for Sc(E). The sum is over all pairs (if) corre-

where k 2 is the kinematic scattering factor and 01,02, are the directions of the incident and backscattered particles with respect to the surface normal [11]. Combining eqs. (1) and (4) we have

SURFACE !

Na(Ea) =Nr, o " Eg" [Sr,o] ~

X" E'--~ •

.

(6)

Recall that the sum is over all pairs (z', z) which result in the backscattered particle leaving the crystal with energy E a. The target is divided into a set o f equal depth intervals, thickness z2~z(fig. 2) with z' = i~z and z =/z3z where i, 1 = 1 , 2 , 3... (j > i). ×~ is the random fraction at z', Edt/ the energy o f the dechanneled particle at z, and Easy is the energy o f the backscatter particle leaving the surface o f the ahgned crystal. Let
Z~ X i Eo

ZpEdq

I

Eaq Fig. 2. Az is the depth mterval employed in the numerical calculation of Ea~] and ×(z). The trajectory shown corresponds to a particle which dechannels and backscatters in the intervals i and 1 respectively and then emerges from the aligned crystal with energy Ea#.

J.A. Moore /Dechanneling in single crystals

580

sponding to the same value for (Ea). The stopping power is taken to be constant in the depth interval and is evaluated at the mean energy for the interval, as determined by iteration. The calculation is performed in two stages. First, we determine (Edi/)2. AEa,I and (Ea0) using a program designed to compute the value of E=I corresponding to backscattering from successive depth intervals ]Q"> i). We then use ttus computed data, together with the measured aligned spectrum Na(Ea) , in the solution of eq. (7) to compute, in succession, the values of AX/ (i = 1,2, 3...) and hence X(Z) = ~ AX/ where the sum is from i = I t o z/Az.

..I-

1 MeV H-~Si

surface

1000

Nr \ \

1.10

\

\

2.2. Double scattering

\

D1 1.05

The preceding analysis assumes a single large angle scattering for the backscattered particle leaving the crystal surface. This is the normal assumption made in all surfaces analyses using Rutherford backscattering [11]. Figs. 3 and 4 show the random backscattered energy spectrum for 1 MeV H÷ scattering from Au and Si respectively. The full curve represents the

u

÷

200 I

,

|

i

200

400

Er

!

i

600

800

1.0(

KeV

F i g . 4 . A s i n fig. 3 b u t f o r 1 M e V H + ~ SL

theoretical spectrum shape, normalized to the surface edge, assuming a single Rutherford scattering collision, and is given by eq. (2), namely,

,

1MeV H ~ A u qll

3000

\

1

o

1

Nr(Er)
.

surface

2000

Nr 130 N

100(;

\

120

\ N

DI

\ \

110

\

I

,

o

I

200

400

600

Er

800

100

1000

KeV

Fig. 3. A comparison of the single scattering theoretical (full line) and experimental (full ckcles) backscattered energy spectrum for 1 MeV H + ~ Au, normalized at the surface. The scattering angle is 150 ° and the the stopping power data are from Andersen and Ziegler [12]. The broken line gives the ratio o f the experimental to theoreUcal spectra and is denoted by D l in the text.

The theoretical curve was computed by determining Er for equal depth intervals. We used the stopping power data St(E) given in the compilations by Ziegler and Anderson [12]. Similar differences between measured and theoretical random spectra have been observed to varying degrees in all cases we examined, e.g. I-I*-* Si, Cu, W, Au and has been noted briefly in previous publications [6,13]. In the case of Au we have observed the same difference in single crystal targets and polycrystalline Films. It appears that the difference is not a consequence of target texture effects or the use of incorrect stopping power data. Evidently, single scattering theory is not quite adequate to give a good description of the backscattered spectrum at the lower energy end. Multiple small angle scattering causes angular divergence of the incident and backscattered beams and the range of backscattering angles for particles recorded by the detector Increases with scattermg depth. Rutherford scattering is inversely proportional to sin40/2, consequently the correction to single

581

J A. Moore /Dechanneling in single crystals surface

Eo

y

II

Er

first term in eq. (7) by a factor DI((Ea) ) which is equal to the ratio of the measured random yield to the theoretical yield calculated by assuming single scattering. However, the last term :n the sum, i.e. i = b / A z , corresponds to particles which backscatter immediately after dechanneling. For this group there can be no double scattenng contribution, so, the corresponding multiplying factor is equal to one. We therefore, introduce the factors D, into eq. (7) which then becomes Na((Ea)) = Nr,o' Eo2" [Sr,o] × ~

0

Fig. 5.1 and 2 are single and double scattering trajectories for particles which leave the crystal surface in the same direction and with the same energy, E r.

scattering theory yield due to multiple scattering increases with the decrease in backscattered energy, Er. Using the results of the Moli~re multiple scattering theory [14,15], we have calculated that the correction to the 1 MeV H ÷ ~ A u yield increases to approximately 3.5% at Er = 400 keV. This is insufficient to account for the observed deviation from single scattering theory which, for 1 MeV H ÷ ~ Au, increases to approximately 20% (fig. 3). We now consider the effects of more than one large angle scattering. Fig. 5 shows, schematically, large angle single and double scattering trajectories for a random particle which give the same backscattered energy Er. Computations which sum over all double scattenng events contributing to the backscattered spectrum yield at Er, indicate that large angle double scattering within the target may account for the difference between measured and theoretical random spectra [16]. We therefore introduce the following modification to eq. (7) which takes an approximate account of the observed deviation of random spectra from single scattering theory. In eq. (7), the sum for Na((Ea)) extends from i = 1 to b / A z where, recall, b is a backscattering depth corresponding to (Ea) and is calculated by assuming channeled and random stopping powers along the ingolng and outgoing paths, respectively. The first term represents the contribution of AX1, the random beam component created m the first depth interval, and is calculated according to single scattering theory. The actual contribution of AX1 to the measured aligned yield at (Ea) is therefore given by multiplying the

X,

(Earl)2

"

AE d#

"D,

.

(8)

How do the factors D i vary with index number i? Consider the following simple argument in which we neglect the variation of stopping power and scattering cross section with energy. With reference to fig. 1, the probability that a particle which dechannels at z', suffers a first scattering in the angular range <-~90° and then is subsequently scattered into the detector with energy Ea, is proportional to the random path length z'z for the corresponding single scattering trajectory. Because of the presence of the target surface, this proportxonality does not hold true for double scattermg events in winch the first scattering angle is ::>90, when z' -+ 0. Nevertheless, the total probability that a particle which dechannels at z' will double scatter into the detector with energy E a will decrease as the depth z' increases and the random path length z'z decreases. Consequently D i decreases monotonically with increase in index number i. This conclusion is supported by some preliminary results of numerical calculations winch also take into account the variation of stopping power and scattering cross section with energy. For the numerical calculation of X(z), however, it appears to be sufficient to make the simple approximation that Di decreases linearly with index number 1 from the empirical value Dl((Ea)) at i = 1 to 1 at i = b[Az. (See the results of calculations presented m fig. 8.) Therefore, the numerical calculations using eq. (8) are performed as outlined m the single scattering theory to compute successive values of AX/ and hence X(z), using a linear approximation for the factors D i.

3. Results of calculations

Fig. 6 is a family of computed curves giving E~ • dE a as a function o f E a for 1 MeV H ÷ ~ Au. For each

J.A Moore /Dechanneling m szngle crystals

582

2.5 +

MeV

H ~ Au +

I MeV H~Si

3.0

~25

hi

U.l c~ '-¢:i l.U

C~'~ I,I

~

2

0

~6

z/

2.5

2.0

x 107 xlO

7

2.0 i

300

i

~

L

i

500

700

Ea

i

0

i

900

KeV

Fig. 6. The curves are derwed from a numerical computat]on of the backscattered energy Eatl; each curve corresponds to a constant value for z' (fig. 2). The interval ~z = 1000 A and z' is in units of 1000 A. The stopping power data are from Andersen and Ziegler [12] and a value a = 0.5 lS assumed. The scattering angle is 150 ° .

curve, the dechannel depth z' is constant and the backscatter d e p t h z is variable. The variation in the value o f (Ea) 2 • A E a over the terms in eq. (8) is given by reading along a vertical line drawn through the energy axis at the point (Ea). This variation increases 1 2 as ( E a) decreases and reaches a value o f 30% at 3k Eo. Fig. 7 shows the corresponding data for 1 MeV H ÷ Si. Figs. 3 and 4 show the comparison b e t w e e n the e x p e r i m e n t a l l y measured r a n d o m s p e c t r u m and the single scattering calculation, for 1 MeV H ÷ scattered from Au and Si, respectively. The correction function D I ( E ) , which takes account o f the deviation o f r a n d o m spectra f r o m single scattering t h e o r y , is given in these figures. The full line curve in fig. 8 is the r a n d o m fraction X(z) for 1 MeV H * - + ( l l 0 ) Au, calculated by using eq. (8) and the data illustrated in figs. 3 and 6. The broken lines give the results o f calculations using e x t r e m e values for the factors D~, n a m e l y D i = 1 and D i = DI((Ea) ). This dlustrates that, for the calculation o f X(z), the hnear a p p r o x i m a t i o n a d o p t e d for D i is adequate. In each case the stopping power ratio, a = 0.5. S h o w n for comparison are the results o f an analysis using X ( z ) = N a ( E a ) / N r ( E r ) w i t h E r = Ea and

1.5

' 300

'

5'0

0

' 700

'

' 900

'

Ea

KeV

Fig. 7. As in fig. 6 but for 1 MeV H÷ ~ SL In this case, the interval, Az = 2000 A and z' is in units of 10 000 A.

3O

1MeV

%

4-

H..~Au

20

r,X" •

////

V

•

X 10

i

i 1

i Z

i 2

~

i 3 /6~.m

Fig. 8.The full line gives X(Z) for 1 MeV H÷--, (110) Au calculated using eq. (8) and a = 0.5. T/us zs compared wzth the results of analyses according to Pederson et al. [6] (triangles) with a = 0.5 and the Aarhus Convention [15] (ctrcles). The broken lines represent the calculated x(z) using D z = 1 and D i = D 1((Ea>). The scattering angle is 150 ° .

J.A Moore/Dechannelingin single crystals

583

+ I

MeV H--*Au

I MeV

H~<110>Si

3O

0.5

IJ3 O.8

%

1,0 3C

%

05 20

2C

X

X 10

.4. i

i

i

~

I

I

Z

2

3

/,~m

F~g. 9. The vaxmtlon of ×(z) with a for 1 MeV H+ ~ <] 10> Au calculated using eq. (8). The results of analysis according to Pederson et al. [6] with a = 0.5 (triangles) and the Aarhus Convention [15] (ctrcles) axe shown again for comparison. The scattering angle is 150 ° .

z(Ea) given b y assuming (a) a random stopping along both the ingoing and outgoing paths (i.e. the Aarhus convention [17]) and (b) a stopping power along the ingoing path equal to 0.5St (i.e. the analysis described by Pederson et al. [6]). For shallow depths (<15 000 A) the present analysis [eq. (8)] gives results midway between those calculated using the Aarhus convention and the Pederson analysis. As the depth increases b e y o n d 15 000 A our results diverge from those of the Pederson analysis; the difference is 28% at 25 000 A,. The depths 15 000 3, and 25 000 A correspond to backscattered energies of 0.7k2Eo and 0.4k2Eo, respectively. The full curves m fig. 9 show the calculated [eq. (8)] vanation o f X with a for 1 MeV H* -~ <110> Au. We note two features here. First, the variation o f X with a is approxamately one half o f that given by the Pederson analysis. Second, for a = 1, the present analysis gives a value for X which exceeds that given by the Aarhus convention. This is a consequence o f the slgmficant deviation o f the spectra from single scattering theory for 1 MeV H ÷ -~ Au. Fig. 10 shows the results for 1 MeV If"-~ <110> Si. Again, the full curves corresponding to a = 0.5 and 1 are the results o f our calculations using eq. (8). For Si, the deviation o f the spectra from single scattering

7 Fig. 10 AS in fig. 9 but for 1 MeV H+ ~ Si. theory is relatxvely small (fig. 4). Consequently, our results for a = 1 must be close to those calculated directly using the Aarhus convention; this is seen to be so in fig. 10. Indeed, this case of Si with a = 1, prowdes a check on the validity o f the numerical technique for the solution o f eq. (8) to determine ×(z). For a = 0.5 our results are in rather good agreement with those given by using the Pederson analysis for depths up to 35 000 3,. Beyond this depth the results o f the two analyses diverge and reach a difference o f 22% at 60 000 A. The depths 35 000 A and 60 000 A correspond approximately to backscattered energies o f 0.7k2E0 and 0.4k2Eo. For the two cases which have been chosen to illustrate the problem, H + ~ $1 and Au, the commonly used analysis described by Pederson et al. [6] does provide a useful and simple method for calculating the random fraction up to depths which correspond approximately to backscattered energies of 0.7k2E0. However, if use is to be made of the lbwer energy part o f the spectrum (<0.7k2Eo) it appears to be desirable to use a more detailed calculation [eq. (8)] in order to determine the random fraction to a reasonable precision, say 10%. Furthermore, the results o f the detailed calculations are less sensitive to the value o f the stopping power ratio, a. For example, m the case of Au it appears that even if the precise value o f a in the range 0.5 < a < 1 is un-

584

ZA. Moore /Dechanneling m smgle crystals

known one could still calculate X(Z) to a precision of approximately 10% by taking the mean value between ct = 0.5 and 1. Finally, the deviatxon o f backscattered spectra from single scattering theory is significant for heavy elements m the lower energy part of the spectrum. Assuming that tins difference is due to double scattering within the target, then we may correct for the effect in the detailed calculation of X(z). In the case o f H + -~ Au, the correction amounts to approximately 10% for a depth o f 2.5 /~m and backscattered energy o f 0.4k2Eo . I would hke to thank Dr. L.M. Howe (Chalk River Nuclear Laboratories) for providing the 1 MeV H ÷ S1 experimental data which has been used here to illustrate the analysis and for useful discussions on this problem.

References [1] J. Lmdhard, Dan. Vid. Selsk., Mat. Fys. Medal. 34 (1965) 14. [2] G. Foti, F. Grasso, R. Quattrocchl and E. Riminl, Phys. Rev. B3 (1971) 2169. [3] E. Bonderup, H. Esbensen, J.U. Andersen and H.E. Sch~btt, Rad. Eft. 12 (1972) 261.

[4] F. Grasso, Channeling (ed D.V. Morgan, Wiley, N.Y, 1973). [5] H.E Sch4tt, E. Bonderup, J.U. Andersen and H. Esbensen, Atomic Collisions in Solids, vol. 5 (ed S. Datz, Plenum Press, N.Y., 1975) p, 843. [6] M.J. Pedersen, J.U. Andersen, D.J. Elliot and E. Laesgaard, 1bid., p. 863. [7] N. Matsunaml and L.M. Howe, Rad. Eft, to be published. [8] H.E. Jack, Jr., Rad. Eft. 13 (1972) 101. [9] I Nashtyama, Phys. Rev. B 17 (1978) 17. [10] G. Dearnaley, Channeling (ed. D.V. Morgan, of Wiley, N.Y, 1973). [ 11 ] W.K. Chu, J.W. Mayer and M.A. Nlcolet, Backscattermg spectrometry (Academic Press N.Y., 1978) ch. 3. [12] H.H. Andersen and J.F. Ziegler, H stopping powers and ranges in all elements (Pergamon, N.Y., 1977). [13] W.K. Chu and J.F. Zlegler, J. Appl. Phys. 46 (1975) 2768 [14] H.A. Bethe, Phys. Rev. 89 (1953) 1256. [15] L Meyer, Phys. Stat. Sol(b) 44 (1971) 253. [16] Prehmlnary numerical calculations (J.A. Moore) for 180 ° H back scattered from Au indicate that double scattering in the target may be sufficiently significant to account for the difference. [17] This procedure of usmg the random stopping power for calculating x(z) was recommended at a meeting m Aarhus in 1970 to factlitate comparison of experimental data from different laboratories.