- Email: [email protected]
Surface analysis by RBS and NRA
Vacuum/volume 34lnumber Prmted in Great Britain
Surface M Braun,
12f pages 1045 to 1052f 1984
analysis
0042-207X84$3.00+ .OO Pergamon Press Ltd
by RBS and NRA
Research institute of Physics, S- 104 05 Stockholm, Sweden
The use of Rutherford backscattering spectroscopy (RBS) and nuclear reaction analysis (NRA) for surface analysis is discussed. For the RBS technique, emphasis is laid on cases which are not discussed in existing review articles of the subject. The present work intends to describe a calculation procedure with the aid of which it is possible to obtain the depth distribution of a high concentration and non-homogeneously binary compound sample. This complicates the determination of the stopping and scattering cross-sections of the incoming particles at a certain depth below the surface. In addition, a method is described by which the thickness and composition of a two-element film, deposited on a single-element substrate, can be determined by RBS. One advantage with the method presented here is that it is not necessary to detect any signals from the lighter component of the film. in order to determine the composition. This improves the RBS technique to study light elements in connection with thin layers. Finally, the NRA method to measure concentration distributions of deuterium beneath a surface is presented. In the case discussed here, the analysis is done by the D(=He, H)4He nuclear reaction.
Introduction
Rutherford backscattering spectrometry (RBS) has become a standard technique for surface analysis. It is routinely used to determine (a) composition and thickness of thin films, (b) surface impurities and (c) depth concentration distributions in bulk materials. An excellent review of the technique is given in ref 1, which covers most aspects on the subject. However, for some special cases there exists no survey in the literature of how to use the RBS technique for surface analysis. This yields, for example, the case in which impurities of high concentrations are distributed non-homogeneously below the surface of a bulk material or binary compound thin films in which the concentration of one of the elements is too low to be detected by the RBS method. The intention with this work is to give a comprehensive presentation of general analytical expressions, with the aid of which it is possible to use the RBS technique to analyse samples of the kind mentioned above. It should be emphasized that the present study does not contain any new basic principles of the RBS technique, but aims to give the concepts of the calculation procedures necessary for the analyses. In addition, the nuclear reaction analysis (NRA) technique is discussed by the help of which depth distributions of high concentration deuterium can be obtained by the D(3He, H)4He reaction. Some of the material presented here, has been discussed in detail in earlier articles2-4.
Nowhomogeneous
high-concentration
depth distributions
Normally, depth distributions are determined by the RBS technique in situations where the impurity concentrations are low or homogeneously distributed in depth. In such cases the analysis is relatively straightforward because the scattering and stopping cross-sections of the analysing beam particles, necessary for the
depth calculation, are not influenced by the impurity concentration variation in depth. However, when the impurity concentration is high and non-homogeneous, the scattering and stopping cross-sections vary as a function of depth due to the energy loss of the penetrating particles which in turn depends on the bulk composition at a specific depth. As the composition variation is not known a priori, this complicates the analysis. The problem mentioned above, has been discussed in several articles ‘-‘. The method described here, which is applicable to the case ofa binary compound, makes use of a simple and fast iterative procedure to find the concentration distribution and requires knowledge of the backscattering signals from one of the components only. The calculation procedure takes into account the variation of the stopping and scattering cross-sections due to both the energy losses of the analysing beam particles and the change of the bulk composition as a function of depth. Let A and B denote the elements in a bulk material and AB, the composition in molecular form at a depth x,. Figure 1 illustrates the backscattering events from the front surface as well as from a slab with the thickness 6xi at a certain depth x, as a result of an incoming ion beam with energy E,. The angles of incidence and ejection are 8, and 02, respectively. Collisions with atoms A are not considered in the figure. In the forthcoming, subscripts ‘i’ refer to incoming particle energies and superscripts ‘i’ are used in the energy notation for outgoing particles. As shown in Figure 1, particles backscattered from the front surface obtain the energy k,E,, where k,, is the kinematic factor for elastic collisions between the beam particle and B atoms. E,.; and k,,& are the energies of particles just before scattering at the front surface of the ith slab and immediately after the collision, respectively. Particles scattered at the backside of the ith slab emerge from the front surface of the slab with an energy k,E,,, - &JZ;, where the last term is the energy loss of the particles 1045
M &am:
Surface analysis by RBS and NRA
Figure 1. Illustration of backscattering B are shown in the figure.
events taking place at the surface and at the front and back sides of a slab at the depth x,. Only collisions
through that slab. Particles leaving the surface, and emerging from the back and front side of the ith slab have energies J$- 6E and EA, respectively. Here 6E corresponds to the channel width (keV/channel) in a backscattering spectrum. As only collisions with element B are considered, the subscript B is left out in the energy notation in the following. It should, however, be noted that all expressions given below also can be derived for collisions with element A. The principal idea to calculate the depth distribution from a backscattering spectrum is as follows. First, determine the surface concentration where the scattering cross-section is known and the correct stopping value is obtained by iteration. Then, the composition of the first slab is calculated by the knowledge of the surface composition and iteration. In this way the concentration distribution below the surface is determined successively by an iterative procedure for each slab.
Figure 2 schematically shows an example of a RBS spectrum obtained from a sample containing a concentration distribution of a light element B in a substrate of a heavier element A. As the backscattering signals from element B are superimposed on the ones from element A, these background signals have to be subtracted to obtain the contribution of element B only. This is indicated in Figure 2 by the dotted curve. If element B is heavier than element A, the backscattering signals from element B are obtained directly in the RBS spectrum. Let HA.,, and H,,, denote the spectrum heights corresponding to scattering events close to the surface, and H,(E') the height at a depth x, in the ith slab (see Figure 2). Then, the height of a backscattering spectrum at the detecting energy E’ is given by
passing
H(E’)=R . Q
Hs.o
II
HJE’)
+I A ,<’
scattering Figure 2. Idealized
backscattering
spectrum
of a compound
O,,
I
(1) Q the integrated
particle
’ I
,
__-- -
7
,;-’
’ 1 1
EF EL energy
K&o
1
sample (M, > M,,). The lower dotted curve corresponds
only, obtained after subtraction of the signals from element A. 1046
N . BxJcos
I
I
-
. a(&).
where R is the solid angle of detection,
; -
with atoms
to scattering
signals from element
B
M Braun; Surface analysis by RBS and NRA
dose, a(,$) the differential scattering cross-section at the energy Ei and N the atomic density of the substrate. The elastic scattering cross-section for element B is:
Consequently, the surface composition, i.e. the number of B atoms expressed in terms of A atoms, can be written as no =
4({1 -[(M,/M,)sin
8]2~“2+cos
sin 48{1 -[(M,/M,)sin
_. 0]2)1’2
0)2 (2)
Here Z, and Z, are the atomic numbers of the incoming beam particle and element B, respectively. The corresponding masses are M, and M,. The electronic charge is denoted e. The corresponding expression for the scattering cross-section of element A is valid. For nuclear resonance scattering, expression (2) is no longer true. In this case the scattering cross-section might be much higher. As mentioned above, the first step in the calculation procedure is to determine the surface composition 11~.This is done in the following manner, by designating N,ARas the molecular (AB,) density and N:; and N&: as the atomic densities of atoms A and B in the compound close to the surface. Then the signal heights of element A and B (see Figure 2) are H4,0=R
* Q . CT~(&) . N$,
* G.~,~,/cos 8,
Furthermore,
. N;P, . G.~,~,/cos 6,.
the surface energy approximation
H A.0
(11) ’ oB(EO)
This expression is used to calculate the surface composition by first taking the ratio [.s(E,,, r~~)]~~/[e(&,, no)]:” equal to one. This first value of n, is then used to determine a zeroth-order approximation of the stopping cross-section ratio and so on. The iteration is stopped when the difference between two successive rt,-values have converged to less than, for example, 1 “,. The next step of the calculative procedure is to determine the composition in the first slab by using the known n,-value for a first-order approximation of n,. After iteration the composition is calculated in the second slab etc. until the whole depth distribution concentration is obtained successively. We now derive the expressions necessary for these calculations, bearing in mind that only collisions with element B are considered. The energy loss of particles passing through the ith slab (see Figure 1) is SE,! = [E(E;, ni)]tB . NAB . 6X’i
(4)
(13)
(5)
in which
C&, n;)lfi" = . 6s,~,.
+
C&A(Eo) + ~~o~B@o)l 1
&
&
[cA(Ej)+ nie”(Ei)] 1
(6)
The subscript ‘O’refers to surface conditions in all expressions. Let E* and cR denote the stopping cross-sections of element A and B, respectively. Then, by applying Bragg’s rule of linear additivity, the stopping cross-sections in expressions (5) and (6) are
$
[~~tkAEo)+ ~~o~B(~AEo)l
(7)
+ ---&-
[eA(k&)
El-AE
[cAB(E, n)]-’
dE =
. 1W:,
+ &
CEA(EO)+ noe”(Eo)l
2
C~A(h&o)+
no~R(Go)l.
H
‘*O= [E(&, no)];‘.
6E cos 8,
. 1~,,E_ rf;,,[EAB(E, n)l - ’ dE. , ’
Since SE’< k,,E, and SE 6 E’ one obtains
(8)
Now, Nty,= N,Aa and N~$=n,N~R which, by substituting equations (5) and (6) into (3) and (4) leads to R . Q .a,(&).
04)
However, expression (13) above cannot be used directly to calculate the concentration 11,as the factor 6El is unknown. In order to derive an expression for 6E,!, consider the outward path length of particles scattered from (a) the front surface of the ith slab and (b) the rear surface of the ith slab with the starting point at thefront surfaceat theslab. As these twopathlengthsareequal, one has (see Figure 1) E’
&
i- n;a’(k&;)].
2
and
C.+%,, no)ltB=
(12)
where N”H is the AB,, molecular density. Then, the spectrum height at’ the depth si becomes
and
Ch% noXB=
nO)1tB.
gives
SE= [a(EO, no)];” . N,AB. c%Y,~,
6E = [c(EO, no)];’
CE@09
(3)
and HB.O=R . Q . a,(E,)
H R.0
(9)
bE,f = 6E
cAB(k,Ei, ni) cAR(Ei, no)
(15)
where eAB(k,Ej, n,)= eA(kHEj)+ n@(k,E,)
(16)
and cAB(Ei, ~,)=E~(E’)+~~~~(E~).
(17)
and H
R. Q . a,(E,)
.6E . no
‘,O = [E(&, no)]tB . COs 8, .
(10)
From relation (15) it is evident that for surface conditions 6EI=6E, whereas 6Ei#6E beneath the surface where the difference increases with the analysing depth. Inserting (15) into 1047
M Braun: Surface analysis by RBS and NRA (13) gives the molecular slab ni =
concentration
(ni) of element
the difficulty to determine the Q. Q factor experimentally, where the detection geometry and the total analysing ion beam dose have to be measured accurately. In cases where the molecular density N,AHin equation (19) is not known, one has to express the slab thickness in pg cme2, i.e.
B at the ith
H,(E’) [&(E,, rti)-gB EAB(Ei,n,) . cos 8, C2. Q .6E . uB(Ei) . E*‘(~,E,,n;)
(18)
In this expression H,(F), cA”(E’, n,), cos 6,. Cl. Q and 6E can be derived directly from the experimental data. For the other quantities one needs to know the energy E,, i.e. the energy just before the scattering event at depth x,. By combining equations (12) and (15) the thickness of the ith slab reads
&xi =
6di=1.6606x
t”A +
6E . &*‘(k,Ei, ni)
Then, by first considering the energy loss 6E” felt by particles they pass inwards through the slab thickness 6xi
~E”=E*‘(E;,
n,) . I’/;’
. ik;/cos 8,,
6E
(20)
The corresponding
E'=k,E,-i
txAR(Ei.n,) . ~*‘(lc,,E;, ni)
[c(Ei, n;)];” . ~*‘(l?, n,) energy relation
(21)
cos 0,
for the detected
energy is
(22)
.6E.
b% cm-‘1.
ni"B)
(23)
To illustrate the effect of taking a highly varying depth concentration into account in the calculation, an RBS experiment was carried out with a silicon-gold sample. This sample was prepared by evaporating a Au film of a few hundred Angstrom on top of a semiconductor-grade Si wafer. The sample was then annealed to allow the gold material to diffuse into the bulk silicon substrate. A RBS spectrum was obtained by the use of a 2 MeV He+ ion beam from a Van de Graaff accelerator and conventional multichannel analysis techniques. Figure 3 shows the recorded backscattering spectrum. From the figure it is clear that the gold material had diffused into the silicon substrate, giving rise to a steep concentration gradient of gold atoms into the substrate. As the front edge of the silicon signal in the spectrum (indicated in the figure) corresponds to the energy of particles scattered from silicon sited right on top of the surface it can be concluded that the surface of the sample had become silicon enriched due to the heat treatment. One also observes a depletion zone of silicon near the surface region as a result of the gold concentration gradient. By using the above discussed calculation procedures, the diffusion profile of the gold material in the silicon substrate was determined by using the data shown in Figure 3. The result of this
as
and by inserting (19) into (20) the energy just before scattering can be written as
Ei+, =E, -
[E&, n,)];” . E*‘(E’, no)
Experiment
(19)
[E(&, ni)]$*. .cAB(Ei,no). NAB’
6E . ~*~(lc,E,, ni)
10-lS.
With help of equations (18, (19), (21) and (22) the depth concentration distribution now can be calculated in a successive order. The total depth is obtained by summing different slab thicknesses 6r,. Stopping cross-sections for protons and helium particles are obtained by tabulated coefficients8.9. The factor R Q, appearing in equation (18) can be derived separately from expression (9) which gives n. Q= H,Js(E,. q,]:” cos O,/dE o,(E,). In this way one overcomes
AU
l...,
Hly.0 -
*IO’
I
l
.
. . . . . ‘***.o.*~*
Si
l*..
.
..
I 00.. .*:
.
*0.3
. . . .
i
. . I
I
1.0
1
1.1
.
.
.
.I
backscattering Figure 3.
1.5
energy
.* 1
I.5
1
I
1.1
l.8
d l.9
(M&l
RBS spectrum from a sihcon -gold sample. In the example shown. gold atoms have diffused into the silicon substrate.
concentration 1048
..A.._.
.*
gradient
in depth of the gold material.
giving rise to a steep
M Braun: Surface analysis by
RBS
and NRA
depth
t pg/cm*)
Figure 4. Calculated depth distribution of gold in silicon from the data shown in Figure 3. Filled circles show the result from the full calculation discussed in the text. and open circles correspond to the case when the calculation is done without respect to the gold content in the material.
calculation is presented in Figure 4 by the filled circles. It is observed that the atomic concentration of gold at the surface is about 30”,, and decreases smoothly to nearly zero at a depth corresponding to about 180pg cm-‘. As the mass density variation of the silicon -gold system is not known accurately, the depth scale cannot easily be converted to a length scale. However, if one assumes a constant density of 3.8 g cm- 2 in the whole bulk material, 180 pg cm - ’ corresponds to a depth of about 5000 A. Note, that the slab thickness 6d, decreases as a function of depth, the reason being that the stopping cross-section increases for lower energies of the penetrating particles. The upper curve in Figure 4 (open circles) shows the result of the depth distribution calculation when the influence of the gold content in the silicon substrate on the stopping cross-sections is neglected (ni is zero in equations (14), (16) and (17)). The n,-value in equation (23), however, is not changed so that the depth scales of the two curves in Figure 4 are the same fig cm-* of the silicon-gold material. Evidently, a much deeper distribution is obtained when the gold content is neglected as compared with the casein which the varying gold concentration is taken into account in the calculations. This example clearly shows the importance of including the composition change in the depth distribution calculations, as this will alter the stopping cross-sections and thereby the analysing particle energy.
Composition
of compound thin films
RBS is a widely used technique to investigate thin films deposited on a substrate material. However, in the special case in which the composition of a compound thin film has to be analysed, the RBS technique suffers from the fact that the sensitivity decreases with the atomic number Z of the substrate atoms as the scattering cross-section is proportional to Z2 (see equation (2)). Thus, in many cases of practical interest, the backscattering signals from the lighter element in the film are superimposed on the substrate signals. Ifthe concentration of the lighter element is low, this may complicate or make an analysis impossible. The intention here is
SUBSTRATE
as
FILH
k&o
\
a)
7
HA
\
“il
/ 'HA.O
-
HS \
Es
A
S
ES
.
BACKSCATTERING
I
ENERGY
Figure 5. (a) illustration of backscattermg events taking place al the surface and ar the interface region between a compound thin film deposited on top ofa substrate. (h) The resulring idealized backscattering spectrum.
to show that the RBS technique still can be used to analyse thin films (>, 1000 A) of the type mentioned above. This is done by extracting all necessary information from signals originating from the heavier component in the film and the substrate only. This improves the RBS technique, as the analysis can be done without respect to the mostly poor detectable lighter element. Only binary compound thin films are considered here. Some examples of such 1049
M Braun: Surface analysis by RBS and NRA
thin films are boride, carbide, nitride or oxide coatings on different substrate materials. Consider a binary compound film with the atomic composition A=B, --z and the atomic density N, and N, of the element A and B, respectively. If N,, denotes the total atomic density of the film, the relative atomic composition is given by z=-.
NA
method
H = z R. Q . a,(E) .hE . cAB(kAE, 2) A [E(E, z)]:” . eAB(EA* z)
(25)
and
H = i-2. Q .o,(E). 6E. eAB(ksE, z) s [E(E)];. EAB(&, z) . cos 0, where the stopping factors are
cross-sections
E~~(E, z)=z~~(E)+(l
-z)cB(E)
C4E,z)l;'
-&
=
1
(26)
and stopping
cross-section
(27)
cAB(E, z) + &
2
eAB(k,AE,2)
(28)
and
C@)l: =
&
1
E’(E)+ &
By combining equations can be written as
HA. a,(E). ’ =
H, .a,(E).
HA . G.(E) 4. Q,(E)
(25) and (26), the atomic
E = E, - (k,E, - E,)/2
(k,E
+
+p
cos
1 -1
1
8,
~~~~~~~~~~~ Z)
(36)
where t,,, the film thickness expressed in a length scale, can be obtained if the atomic density of the film is known. The analysing method described above has been tested with several series of different kinds of films and substrate materials. One example is shown in Figure 6, which is a RBS spectrum from a Ti,N, __ film (z 1000 A thick) deposited on a pure Si substrate. The nitrogen concentration in the film was determined to be 5.5 at 0. Figure 6 demonstrates the advantage of using only the signals from the heavier element (Ti) and the substrate (Si) in the analysis. In the example shown the nitrogen signals are suppressed by the higher background signals originating from the silicon substrate, making an analysis, in which the backscattering events from nitrogen are necessary, impossible. The results from the analyses of several combinations of film -substrate materials have shown that the atomic concentration of a film can be determined with an accuracy of 2”,, in the most favorable cases and with about 59,) in more general cases.
cAB(Es, 2). .zAB(kAE,-_)
(30)
VI
E300-
a 0 “0 ,200-
(32)
Y>;..
‘+...,,,.
Ti
i
. “‘:..,. _ SI :..-+....,_ ..“.G&,, .‘%..+.&, 1 ‘:.‘.$
.+. . :
1
. .
E z loo-
. .
(31)
EA)/~.
EAB(Ein, Z)
N,d,,=(k,Eo-E,)
0 =
(35)
are used for starting values in the calculation. Then equations (30) to (33) are iterated to find better values of z and E. Once good values of the energy E and composition z have been found, the film thickness (expressed in atoms cm-‘) can be calculated according to
400 -
Now, the energy E can be derived with help of the energy loss 1050
and
composition
and
E A,,,“,
(34)
E~(~,E).
In this expression one has to know the energy E just before the scattering event at the interface. This is done by first considering the inward and outward energies of the analysing particles which are given by the ‘mean energy approximation’ (see ref 1, Section 3.2.2). Ei” = (E, + E)/2
(33)
2
[E(E, z)tB . cAB(ksE, z) . eAB(EA,z) [c(E)];.
. COS 8,
k, . cAB(Einrz)cos 8, + E~~(E~.~“,,z)cos 8,
The composition of a compound thin film can now be calculated in the following way. First, the zero-order approximations of the Z- and E-values z=
Figure S(a) illustrates the backscattering events taking place at the surface of a compound thin film A$_, and at the interface between the film and an elemental substrate S. Only collisions with atoms A are considered here. E,, 0,, e2 and the indexed k-factors have the same meaning as the ones used in the previous section (see Figure 1). E is the energy of the incoming particles reaching the interface and k,E and k,E the energies just after scattering from atoms A and S at the interface. The corresponding energies of particles emerging the film are E, and Es. The resulting RBS spectrum is shown in Figure 5(b) for the case M, > MS > M,. The signal heights from collisions with atoms A and S at the interface are H, and MS, respectively. H,,,is the height of the surface signal from element A. Applying similar arguments which was used above to derive equation (18), the signal heights H,Aand Hs can be expressed as
(EA + E,) . EAB(Ein,Z)
E=
(24)
N AB
(see ref 1, Section 3.3.1) to yield
0
Figure 6. RBS spectrum from a Ti;N, ~_ sample deposited on top of a silicon substrate. The nitrogen content in the film is 5.5 at”;,.
M Braun: Surface analysis by
RBS and NRA
Depth distribution of deuterium Hydrogen and its isotopes are difficult to analyse with conventional analysing techniques. One method to determine deuterium concentrations in bulk materials is by the D(3He, H)4He nuclear reaction. This technique, which has been described by different authors’0-12, makes use of a 3He+ ion beam impinging on a deuterium-rich sample. Following the maximum of the nuclear reaction cross-section ’ 3,the energy of the incoming 3He+ ion beam usually is around 650 keV or above. As a result of the nuclear reaction, ‘He or H particles are ejected, the former having energies in the region l-3 MeV and the latter 12-15 MeV, depending on the beam-target geometry. Thus, either the emitted protons or 4He particles can be detected for the analysis of deuterium. In this work we will only concentrate on proton detection bearing in mind that 4He detection can be used equally well. The principles of the NRA technique are similar to the RBS analysing method, the only differences being that nuclear reaction cross-sections have to be used and that the incoming and outgoing analysing particles are not identical, i.e. one has to account for a mass change at the point where the nuclear reaction takes place. Consider Figure 7, which illustrates the situation where the nuclear reactions take place at the surface and at a depth xi perpendicular to the sample surface of an element A. Henceforth, the different particle elements involved in the analysis are denoted in such a way that subscript 1 refers to H particles, 2 to D particles, 3 to 3He particles and 4 to 4He particles. The deuterium concentration in the ith slab at a depth xi is given by II,( cos 8, nD(xi) = (37)
$j
i2.Q. [
6xi
E3,i)
‘El -6E Figure 7. Illustration ofnuclear reaction events taking place at the surface and at the front and back sides of a slab at the depth x,.
The energy E3,i can be determined by the recursion formula ,Z&+,
6E2iE3.i)
i_
1
1 ‘a&i g
(42)
A EsH,(&,i) &
+ G(& ,i) &
and the energy El.; in MeV of the protons just after the nuclear reaction is given byI
El,i=
1L
=E3
M4
E3,i
. 18.352 +
._f W,+M,)*
M,+M,
where the sub- and superscript ‘i’ is used in the same way as defined in the previous section describing depth profiling by RBS. H,(E’,) is the signal height of the detected protons leaving the surface with energy E; and [da/dR(E3.i)], is the differential crosssection of the nuclear reaction in the laboratory frame. Now, the slab thickness can be written as
The differential nuclear cross-section in the laboratory frame, appearing in equation (37) is obtained through the expression of the nuclear cross-section
6x, =
in the centre of mass system expressed in mb/sr”
6E . Ei(EI.i) N&E’)
1 6E, i A L E+JE~,J SE,.;
(38) &-
[
+ EAH(E, .i) Lcos 8, 1
g 1 g 1 (E3.i)
[
CM
(E3.i)cM =
475 . (E3,i)3 1-26.2(E3,i)3.43+ 36.5(E3.i)3.9*
together with the expression
where the relation 6E,! =SE &El ,i)/ci(E’) has been used and c&e and E; are the stopping cross-sections for 3He and H particles in the material A, respectively. The kinematic factor (6E, .,)/SE,,,) is14 f dJ% i Z=(M,+M,)~-
M,
M3 ’ M4
M,+M,
18.352 ._. E~,~
(44)
(45) where
g
-I’* cos 8 (39)
where
sin 19~
dQ-=d%
sin8
1 (46)
l+~cos&-
and
f=2~M,~M,*cos28+(M,+M4)(M4-M,) MI
+ 2 cos 0 g”*
(40)
*M3
E3.i l/2
(47)
cos 8, = cos 8( 1 - y2 sin 2f9)‘i2 - 1’sin *0.
(48)
'=M,
and g=M:~M~~cos2~+M1~M3~(M1+M~)(M~-MM3) + M, . MS . MJM,
+ Mb). 18.352/E3,i.
and (41)
1051
M Braun:
Surface analysis by RBS and NRA
Now, equations (37) and (38) can be used to calculate the concentration of deuterium as a function of depth. As the deuterium atoms, present in the substrate material, normally do not contribute to any significant degree to the stopping crosssection. the deuterium content can be neglected in the calculations in most cases. For very high concentrations of deuterium, however, the influence on the stopping cross-section can be taken into account by applying similar calculation procedures as described in the section above yielding depth distributions by the RBS technique. One example where a high concentration of deuterium is possible is in carbon substrates. Here. up to 70 at’,, of deuterium can be trapped in the bulk lattice and for accurate depth profiling the deuterium content should be considered in the calculations.
Conclusions
It has been demonstrated that depth distributions, where the concentration is high and non-homogeneous in depth, can be determined accurately by the RBS technique. The importance of taking the varying stopping cross-section into account in such cases has been illustrated. Determination of compound thin film composition and thickness by RBS has been discussed. With the special technique presented here. it was shown that for low concentrations of the lighter element in the film, it might be advantageous if the RBS signals from that element are not used in the analysis. This improves the RBS technique to analyse light materials. where the scattering cross-sections are low and the signals of which usually are superimposed on a heavier element. Finally, the NRA technique to determine deuterium depth
1052
profiles was presented, for the calculations.
including
all necessary expressions
needed
Acknowledgements
The author contribution thin films.
acknowledges the help of Dr I Petrov for his to the technique of calculating the composition of
References
I W K Chu. J W Mayer and M-A Nicolet. Backscarrerirtg Specrrometr~. Academic Press. New York (1978). a M Braun. R Brewer. H Stuessi and S Vepiek. Appl Pltys, AZ&25 (1982). 3 T Fried and M Braun. N~tcl Ittsrrwn Merh, 198, 523 (1982). ’ I Petrov. M Braun. T Fried and H E Satherblom, J Appl Ph.rs. 54, 1358 (19831. s D K Brice. Thin Solid Film, 19, 121 (1973). ’ J F Ziegler. R F Lever and J K Hirvonen. loft Beam Swfhce Larer Am/~si.t (Edited by 0 Meyer, G Linker and F Kappeler), Plenum Press. New York (1975). _ P BorSesen. R Behrisch and B M U Scherzer, Appl P~J.\. A27, 183 (1982). ’ H H Andersen and J F Ziegler, Hyd,oyerl Stopping Potters ad Ratyes ill All Ehems. Pergamon Press, New York (1977). 9 J F Ziegler. Helium Stopping Porvm ad Ranges irl A// Elemems, PerSamon Press. New York. 1977. ” P P Pronko. P/IF.\ Ret.. 9, 2870 (1973). I’ M Hufschmidt V Heintze. W Moller and D Kamke, Ntrcl Irtarrtrru Merh. 124, 573 (1975). ” D Dieume8ard. D Dubreuil and G Amsel. Nacl Insrrtrrn Mrrh. 166,431 (1979). I3 J L Yarnell. R H Lo\ berg and W R Stratton. P11.v.tRet.. 90.292 (1953). Ii W Eckstein R Behrisch and J Roth, Iort Beartt Srrt:face La!w Analysis, Vol 2, p 821. Plenum Press. New York (1976). If, C J Alstetter. R Behrisch. J Bottiger. F Pohl and B M U Scherzer. fv’ttcl lmrwm .Mrrh. 149. 59 (19781.
Surface M Braun,
12f pages 1045 to 1052f 1984
analysis
0042-207X84$3.00+ .OO Pergamon Press Ltd
by RBS and NRA
Research institute of Physics, S- 104 05 Stockholm, Sweden
The use of Rutherford backscattering spectroscopy (RBS) and nuclear reaction analysis (NRA) for surface analysis is discussed. For the RBS technique, emphasis is laid on cases which are not discussed in existing review articles of the subject. The present work intends to describe a calculation procedure with the aid of which it is possible to obtain the depth distribution of a high concentration and non-homogeneously binary compound sample. This complicates the determination of the stopping and scattering cross-sections of the incoming particles at a certain depth below the surface. In addition, a method is described by which the thickness and composition of a two-element film, deposited on a single-element substrate, can be determined by RBS. One advantage with the method presented here is that it is not necessary to detect any signals from the lighter component of the film. in order to determine the composition. This improves the RBS technique to study light elements in connection with thin layers. Finally, the NRA method to measure concentration distributions of deuterium beneath a surface is presented. In the case discussed here, the analysis is done by the D(=He, H)4He nuclear reaction.
Introduction
Rutherford backscattering spectrometry (RBS) has become a standard technique for surface analysis. It is routinely used to determine (a) composition and thickness of thin films, (b) surface impurities and (c) depth concentration distributions in bulk materials. An excellent review of the technique is given in ref 1, which covers most aspects on the subject. However, for some special cases there exists no survey in the literature of how to use the RBS technique for surface analysis. This yields, for example, the case in which impurities of high concentrations are distributed non-homogeneously below the surface of a bulk material or binary compound thin films in which the concentration of one of the elements is too low to be detected by the RBS method. The intention with this work is to give a comprehensive presentation of general analytical expressions, with the aid of which it is possible to use the RBS technique to analyse samples of the kind mentioned above. It should be emphasized that the present study does not contain any new basic principles of the RBS technique, but aims to give the concepts of the calculation procedures necessary for the analyses. In addition, the nuclear reaction analysis (NRA) technique is discussed by the help of which depth distributions of high concentration deuterium can be obtained by the D(3He, H)4He reaction. Some of the material presented here, has been discussed in detail in earlier articles2-4.
Nowhomogeneous
high-concentration
depth distributions
Normally, depth distributions are determined by the RBS technique in situations where the impurity concentrations are low or homogeneously distributed in depth. In such cases the analysis is relatively straightforward because the scattering and stopping cross-sections of the analysing beam particles, necessary for the
depth calculation, are not influenced by the impurity concentration variation in depth. However, when the impurity concentration is high and non-homogeneous, the scattering and stopping cross-sections vary as a function of depth due to the energy loss of the penetrating particles which in turn depends on the bulk composition at a specific depth. As the composition variation is not known a priori, this complicates the analysis. The problem mentioned above, has been discussed in several articles ‘-‘. The method described here, which is applicable to the case ofa binary compound, makes use of a simple and fast iterative procedure to find the concentration distribution and requires knowledge of the backscattering signals from one of the components only. The calculation procedure takes into account the variation of the stopping and scattering cross-sections due to both the energy losses of the analysing beam particles and the change of the bulk composition as a function of depth. Let A and B denote the elements in a bulk material and AB, the composition in molecular form at a depth x,. Figure 1 illustrates the backscattering events from the front surface as well as from a slab with the thickness 6xi at a certain depth x, as a result of an incoming ion beam with energy E,. The angles of incidence and ejection are 8, and 02, respectively. Collisions with atoms A are not considered in the figure. In the forthcoming, subscripts ‘i’ refer to incoming particle energies and superscripts ‘i’ are used in the energy notation for outgoing particles. As shown in Figure 1, particles backscattered from the front surface obtain the energy k,E,, where k,, is the kinematic factor for elastic collisions between the beam particle and B atoms. E,.; and k,,& are the energies of particles just before scattering at the front surface of the ith slab and immediately after the collision, respectively. Particles scattered at the backside of the ith slab emerge from the front surface of the slab with an energy k,E,,, - &JZ;, where the last term is the energy loss of the particles 1045
M &am:
Surface analysis by RBS and NRA
Figure 1. Illustration of backscattering B are shown in the figure.
events taking place at the surface and at the front and back sides of a slab at the depth x,. Only collisions
through that slab. Particles leaving the surface, and emerging from the back and front side of the ith slab have energies J$- 6E and EA, respectively. Here 6E corresponds to the channel width (keV/channel) in a backscattering spectrum. As only collisions with element B are considered, the subscript B is left out in the energy notation in the following. It should, however, be noted that all expressions given below also can be derived for collisions with element A. The principal idea to calculate the depth distribution from a backscattering spectrum is as follows. First, determine the surface concentration where the scattering cross-section is known and the correct stopping value is obtained by iteration. Then, the composition of the first slab is calculated by the knowledge of the surface composition and iteration. In this way the concentration distribution below the surface is determined successively by an iterative procedure for each slab.
Figure 2 schematically shows an example of a RBS spectrum obtained from a sample containing a concentration distribution of a light element B in a substrate of a heavier element A. As the backscattering signals from element B are superimposed on the ones from element A, these background signals have to be subtracted to obtain the contribution of element B only. This is indicated in Figure 2 by the dotted curve. If element B is heavier than element A, the backscattering signals from element B are obtained directly in the RBS spectrum. Let HA.,, and H,,, denote the spectrum heights corresponding to scattering events close to the surface, and H,(E') the height at a depth x, in the ith slab (see Figure 2). Then, the height of a backscattering spectrum at the detecting energy E’ is given by
passing
H(E’)=R . Q
Hs.o
II
HJE’)
+I A ,<’
scattering Figure 2. Idealized
backscattering
spectrum
of a compound
O,,
I
(1) Q the integrated
particle
’ I
,
__-- -
7
,;-’
’ 1 1
EF EL energy
K&o
1
sample (M, > M,,). The lower dotted curve corresponds
only, obtained after subtraction of the signals from element A. 1046
N . BxJcos
I
I
-
. a(&).
where R is the solid angle of detection,
; -
with atoms
to scattering
signals from element
B
M Braun; Surface analysis by RBS and NRA
dose, a(,$) the differential scattering cross-section at the energy Ei and N the atomic density of the substrate. The elastic scattering cross-section for element B is:
Consequently, the surface composition, i.e. the number of B atoms expressed in terms of A atoms, can be written as no =
4({1 -[(M,/M,)sin
8]2~“2+cos
sin 48{1 -[(M,/M,)sin
_. 0]2)1’2
0)2 (2)
Here Z, and Z, are the atomic numbers of the incoming beam particle and element B, respectively. The corresponding masses are M, and M,. The electronic charge is denoted e. The corresponding expression for the scattering cross-section of element A is valid. For nuclear resonance scattering, expression (2) is no longer true. In this case the scattering cross-section might be much higher. As mentioned above, the first step in the calculation procedure is to determine the surface composition 11~.This is done in the following manner, by designating N,ARas the molecular (AB,) density and N:; and N&: as the atomic densities of atoms A and B in the compound close to the surface. Then the signal heights of element A and B (see Figure 2) are H4,0=R
* Q . CT~(&) . N$,
* G.~,~,/cos 8,
Furthermore,
. N;P, . G.~,~,/cos 6,.
the surface energy approximation
H A.0
(11) ’ oB(EO)
This expression is used to calculate the surface composition by first taking the ratio [.s(E,,, r~~)]~~/[e(&,, no)]:” equal to one. This first value of n, is then used to determine a zeroth-order approximation of the stopping cross-section ratio and so on. The iteration is stopped when the difference between two successive rt,-values have converged to less than, for example, 1 “,. The next step of the calculative procedure is to determine the composition in the first slab by using the known n,-value for a first-order approximation of n,. After iteration the composition is calculated in the second slab etc. until the whole depth distribution concentration is obtained successively. We now derive the expressions necessary for these calculations, bearing in mind that only collisions with element B are considered. The energy loss of particles passing through the ith slab (see Figure 1) is SE,! = [E(E;, ni)]tB . NAB . 6X’i
(4)
(13)
(5)
in which
C&, n;)lfi" = . 6s,~,.
+
C&A(Eo) + ~~o~B@o)l 1
&
&
[cA(Ej)+ nie”(Ei)] 1
(6)
The subscript ‘O’refers to surface conditions in all expressions. Let E* and cR denote the stopping cross-sections of element A and B, respectively. Then, by applying Bragg’s rule of linear additivity, the stopping cross-sections in expressions (5) and (6) are
$
[~~tkAEo)+ ~~o~B(~AEo)l
(7)
+ ---&-
[eA(k&)
El-AE
[cAB(E, n)]-’
dE =
. 1W:,
+ &
CEA(EO)+ noe”(Eo)l
2
C~A(h&o)+
no~R(Go)l.
H
‘*O= [E(&, no)];‘.
6E cos 8,
. 1~,,E_ rf;,,[EAB(E, n)l - ’ dE. , ’
Since SE’< k,,E, and SE 6 E’ one obtains
(8)
Now, Nty,= N,Aa and N~$=n,N~R which, by substituting equations (5) and (6) into (3) and (4) leads to R . Q .a,(&).
04)
However, expression (13) above cannot be used directly to calculate the concentration 11,as the factor 6El is unknown. In order to derive an expression for 6E,!, consider the outward path length of particles scattered from (a) the front surface of the ith slab and (b) the rear surface of the ith slab with the starting point at thefront surfaceat theslab. As these twopathlengthsareequal, one has (see Figure 1) E’
&
i- n;a’(k&;)].
2
and
C.+%,, no)ltB=
(12)
where N”H is the AB,, molecular density. Then, the spectrum height at’ the depth si becomes
and
Ch% noXB=
nO)1tB.
gives
SE= [a(EO, no)];” . N,AB. c%Y,~,
6E = [c(EO, no)];’
CE@09
(3)
and HB.O=R . Q . a,(E,)
H R.0
(9)
bE,f = 6E
cAB(k,Ei, ni) cAR(Ei, no)
(15)
where eAB(k,Ej, n,)= eA(kHEj)+ n@(k,E,)
(16)
and cAB(Ei, ~,)=E~(E’)+~~~~(E~).
(17)
and H
R. Q . a,(E,)
.6E . no
‘,O = [E(&, no)]tB . COs 8, .
(10)
From relation (15) it is evident that for surface conditions 6EI=6E, whereas 6Ei#6E beneath the surface where the difference increases with the analysing depth. Inserting (15) into 1047
M Braun: Surface analysis by RBS and NRA (13) gives the molecular slab ni =
concentration
(ni) of element
the difficulty to determine the Q. Q factor experimentally, where the detection geometry and the total analysing ion beam dose have to be measured accurately. In cases where the molecular density N,AHin equation (19) is not known, one has to express the slab thickness in pg cme2, i.e.
B at the ith
H,(E’) [&(E,, rti)-gB EAB(Ei,n,) . cos 8, C2. Q .6E . uB(Ei) . E*‘(~,E,,n;)
(18)
In this expression H,(F), cA”(E’, n,), cos 6,. Cl. Q and 6E can be derived directly from the experimental data. For the other quantities one needs to know the energy E,, i.e. the energy just before the scattering event at depth x,. By combining equations (12) and (15) the thickness of the ith slab reads
&xi =
6di=1.6606x
t”A +
6E . &*‘(k,Ei, ni)
Then, by first considering the energy loss 6E” felt by particles they pass inwards through the slab thickness 6xi
~E”=E*‘(E;,
n,) . I’/;’
. ik;/cos 8,,
6E
(20)
The corresponding
E'=k,E,-i
txAR(Ei.n,) . ~*‘(lc,,E;, ni)
[c(Ei, n;)];” . ~*‘(l?, n,) energy relation
(21)
cos 0,
for the detected
energy is
(22)
.6E.
b% cm-‘1.
ni"B)
(23)
To illustrate the effect of taking a highly varying depth concentration into account in the calculation, an RBS experiment was carried out with a silicon-gold sample. This sample was prepared by evaporating a Au film of a few hundred Angstrom on top of a semiconductor-grade Si wafer. The sample was then annealed to allow the gold material to diffuse into the bulk silicon substrate. A RBS spectrum was obtained by the use of a 2 MeV He+ ion beam from a Van de Graaff accelerator and conventional multichannel analysis techniques. Figure 3 shows the recorded backscattering spectrum. From the figure it is clear that the gold material had diffused into the silicon substrate, giving rise to a steep concentration gradient of gold atoms into the substrate. As the front edge of the silicon signal in the spectrum (indicated in the figure) corresponds to the energy of particles scattered from silicon sited right on top of the surface it can be concluded that the surface of the sample had become silicon enriched due to the heat treatment. One also observes a depletion zone of silicon near the surface region as a result of the gold concentration gradient. By using the above discussed calculation procedures, the diffusion profile of the gold material in the silicon substrate was determined by using the data shown in Figure 3. The result of this
as
and by inserting (19) into (20) the energy just before scattering can be written as
Ei+, =E, -
[E&, n,)];” . E*‘(E’, no)
Experiment
(19)
[E(&, ni)]$*. .cAB(Ei,no). NAB’
6E . ~*~(lc,E,, ni)
10-lS.
With help of equations (18, (19), (21) and (22) the depth concentration distribution now can be calculated in a successive order. The total depth is obtained by summing different slab thicknesses 6r,. Stopping cross-sections for protons and helium particles are obtained by tabulated coefficients8.9. The factor R Q, appearing in equation (18) can be derived separately from expression (9) which gives n. Q= H,Js(E,. q,]:” cos O,/dE o,(E,). In this way one overcomes
AU
l...,
Hly.0 -
*IO’
I
l
.
. . . . . ‘***.o.*~*
Si
l*..
.
..
I 00.. .*:
.
*0.3
. . . .
i
. . I
I
1.0
1
1.1
.
.
.
.I
backscattering Figure 3.
1.5
energy
.* 1
I.5
1
I
1.1
l.8
d l.9
(M&l
RBS spectrum from a sihcon -gold sample. In the example shown. gold atoms have diffused into the silicon substrate.
concentration 1048
..A.._.
.*
gradient
in depth of the gold material.
giving rise to a steep
M Braun: Surface analysis by
RBS
and NRA
depth
t pg/cm*)
Figure 4. Calculated depth distribution of gold in silicon from the data shown in Figure 3. Filled circles show the result from the full calculation discussed in the text. and open circles correspond to the case when the calculation is done without respect to the gold content in the material.
calculation is presented in Figure 4 by the filled circles. It is observed that the atomic concentration of gold at the surface is about 30”,, and decreases smoothly to nearly zero at a depth corresponding to about 180pg cm-‘. As the mass density variation of the silicon -gold system is not known accurately, the depth scale cannot easily be converted to a length scale. However, if one assumes a constant density of 3.8 g cm- 2 in the whole bulk material, 180 pg cm - ’ corresponds to a depth of about 5000 A. Note, that the slab thickness 6d, decreases as a function of depth, the reason being that the stopping cross-section increases for lower energies of the penetrating particles. The upper curve in Figure 4 (open circles) shows the result of the depth distribution calculation when the influence of the gold content in the silicon substrate on the stopping cross-sections is neglected (ni is zero in equations (14), (16) and (17)). The n,-value in equation (23), however, is not changed so that the depth scales of the two curves in Figure 4 are the same fig cm-* of the silicon-gold material. Evidently, a much deeper distribution is obtained when the gold content is neglected as compared with the casein which the varying gold concentration is taken into account in the calculations. This example clearly shows the importance of including the composition change in the depth distribution calculations, as this will alter the stopping cross-sections and thereby the analysing particle energy.
Composition
of compound thin films
RBS is a widely used technique to investigate thin films deposited on a substrate material. However, in the special case in which the composition of a compound thin film has to be analysed, the RBS technique suffers from the fact that the sensitivity decreases with the atomic number Z of the substrate atoms as the scattering cross-section is proportional to Z2 (see equation (2)). Thus, in many cases of practical interest, the backscattering signals from the lighter element in the film are superimposed on the substrate signals. Ifthe concentration of the lighter element is low, this may complicate or make an analysis impossible. The intention here is
SUBSTRATE
as
FILH
k&o
\
a)
7
HA
\
“il
/ 'HA.O
-
HS \
Es
A
S
ES
.
BACKSCATTERING
I
ENERGY
Figure 5. (a) illustration of backscattermg events taking place al the surface and ar the interface region between a compound thin film deposited on top ofa substrate. (h) The resulring idealized backscattering spectrum.
to show that the RBS technique still can be used to analyse thin films (>, 1000 A) of the type mentioned above. This is done by extracting all necessary information from signals originating from the heavier component in the film and the substrate only. This improves the RBS technique, as the analysis can be done without respect to the mostly poor detectable lighter element. Only binary compound thin films are considered here. Some examples of such 1049
M Braun: Surface analysis by RBS and NRA
thin films are boride, carbide, nitride or oxide coatings on different substrate materials. Consider a binary compound film with the atomic composition A=B, --z and the atomic density N, and N, of the element A and B, respectively. If N,, denotes the total atomic density of the film, the relative atomic composition is given by z=-.
NA
method
H = z R. Q . a,(E) .hE . cAB(kAE, 2) A [E(E, z)]:” . eAB(EA* z)
(25)
and
H = i-2. Q .o,(E). 6E. eAB(ksE, z) s [E(E)];. EAB(&, z) . cos 0, where the stopping factors are
cross-sections
E~~(E, z)=z~~(E)+(l
-z)cB(E)
C4E,z)l;'
-&
=
1
(26)
and stopping
cross-section
(27)
cAB(E, z) + &
2
eAB(k,AE,2)
(28)
and
C@)l: =
&
1
E’(E)+ &
By combining equations can be written as
HA. a,(E). ’ =
H, .a,(E).
HA . G.(E) 4. Q,(E)
(25) and (26), the atomic
E = E, - (k,E, - E,)/2
(k,E
+
+p
cos
1 -1
1
8,
~~~~~~~~~~~ Z)
(36)
where t,,, the film thickness expressed in a length scale, can be obtained if the atomic density of the film is known. The analysing method described above has been tested with several series of different kinds of films and substrate materials. One example is shown in Figure 6, which is a RBS spectrum from a Ti,N, __ film (z 1000 A thick) deposited on a pure Si substrate. The nitrogen concentration in the film was determined to be 5.5 at 0. Figure 6 demonstrates the advantage of using only the signals from the heavier element (Ti) and the substrate (Si) in the analysis. In the example shown the nitrogen signals are suppressed by the higher background signals originating from the silicon substrate, making an analysis, in which the backscattering events from nitrogen are necessary, impossible. The results from the analyses of several combinations of film -substrate materials have shown that the atomic concentration of a film can be determined with an accuracy of 2”,, in the most favorable cases and with about 59,) in more general cases.
cAB(Es, 2). .zAB(kAE,-_)
(30)
VI
E300-
a 0 “0 ,200-
(32)
Y>;..
‘+...,,,.
Ti
i
. “‘:..,. _ SI :..-+....,_ ..“.G&,, .‘%..+.&, 1 ‘:.‘.$
.+. . :
1
. .
E z loo-
. .
(31)
EA)/~.
EAB(Ein, Z)
N,d,,=(k,Eo-E,)
0 =
(35)
are used for starting values in the calculation. Then equations (30) to (33) are iterated to find better values of z and E. Once good values of the energy E and composition z have been found, the film thickness (expressed in atoms cm-‘) can be calculated according to
400 -
Now, the energy E can be derived with help of the energy loss 1050
and
composition
and
E A,,,“,
(34)
E~(~,E).
In this expression one has to know the energy E just before the scattering event at the interface. This is done by first considering the inward and outward energies of the analysing particles which are given by the ‘mean energy approximation’ (see ref 1, Section 3.2.2). Ei” = (E, + E)/2
(33)
2
[E(E, z)tB . cAB(ksE, z) . eAB(EA,z) [c(E)];.
. COS 8,
k, . cAB(Einrz)cos 8, + E~~(E~.~“,,z)cos 8,
The composition of a compound thin film can now be calculated in the following way. First, the zero-order approximations of the Z- and E-values z=
Figure S(a) illustrates the backscattering events taking place at the surface of a compound thin film A$_, and at the interface between the film and an elemental substrate S. Only collisions with atoms A are considered here. E,, 0,, e2 and the indexed k-factors have the same meaning as the ones used in the previous section (see Figure 1). E is the energy of the incoming particles reaching the interface and k,E and k,E the energies just after scattering from atoms A and S at the interface. The corresponding energies of particles emerging the film are E, and Es. The resulting RBS spectrum is shown in Figure 5(b) for the case M, > MS > M,. The signal heights from collisions with atoms A and S at the interface are H, and MS, respectively. H,,,is the height of the surface signal from element A. Applying similar arguments which was used above to derive equation (18), the signal heights H,Aand Hs can be expressed as
(EA + E,) . EAB(Ein,Z)
E=
(24)
N AB
(see ref 1, Section 3.3.1) to yield
0
Figure 6. RBS spectrum from a Ti;N, ~_ sample deposited on top of a silicon substrate. The nitrogen content in the film is 5.5 at”;,.
M Braun: Surface analysis by
RBS and NRA
Depth distribution of deuterium Hydrogen and its isotopes are difficult to analyse with conventional analysing techniques. One method to determine deuterium concentrations in bulk materials is by the D(3He, H)4He nuclear reaction. This technique, which has been described by different authors’0-12, makes use of a 3He+ ion beam impinging on a deuterium-rich sample. Following the maximum of the nuclear reaction cross-section ’ 3,the energy of the incoming 3He+ ion beam usually is around 650 keV or above. As a result of the nuclear reaction, ‘He or H particles are ejected, the former having energies in the region l-3 MeV and the latter 12-15 MeV, depending on the beam-target geometry. Thus, either the emitted protons or 4He particles can be detected for the analysis of deuterium. In this work we will only concentrate on proton detection bearing in mind that 4He detection can be used equally well. The principles of the NRA technique are similar to the RBS analysing method, the only differences being that nuclear reaction cross-sections have to be used and that the incoming and outgoing analysing particles are not identical, i.e. one has to account for a mass change at the point where the nuclear reaction takes place. Consider Figure 7, which illustrates the situation where the nuclear reactions take place at the surface and at a depth xi perpendicular to the sample surface of an element A. Henceforth, the different particle elements involved in the analysis are denoted in such a way that subscript 1 refers to H particles, 2 to D particles, 3 to 3He particles and 4 to 4He particles. The deuterium concentration in the ith slab at a depth xi is given by II,( cos 8, nD(xi) = (37)
$j
i2.Q. [
6xi
E3,i)
‘El -6E Figure 7. Illustration ofnuclear reaction events taking place at the surface and at the front and back sides of a slab at the depth x,.
The energy E3,i can be determined by the recursion formula ,Z&+,
6E2iE3.i)
i_
1
1 ‘a&i g
(42)
A EsH,(&,i) &
+ G(& ,i) &
and the energy El.; in MeV of the protons just after the nuclear reaction is given byI
El,i=
1L
=E3
M4
E3,i
. 18.352 +
._f W,+M,)*
M,+M,
where the sub- and superscript ‘i’ is used in the same way as defined in the previous section describing depth profiling by RBS. H,(E’,) is the signal height of the detected protons leaving the surface with energy E; and [da/dR(E3.i)], is the differential crosssection of the nuclear reaction in the laboratory frame. Now, the slab thickness can be written as
The differential nuclear cross-section in the laboratory frame, appearing in equation (37) is obtained through the expression of the nuclear cross-section
6x, =
in the centre of mass system expressed in mb/sr”
6E . Ei(EI.i) N&E’)
1 6E, i A L E+JE~,J SE,.;
(38) &-
[
+ EAH(E, .i) Lcos 8, 1
g 1 g 1 (E3.i)
[
CM
(E3.i)cM =
475 . (E3,i)3 1-26.2(E3,i)3.43+ 36.5(E3.i)3.9*
together with the expression
where the relation 6E,! =SE &El ,i)/ci(E’) has been used and c&e and E; are the stopping cross-sections for 3He and H particles in the material A, respectively. The kinematic factor (6E, .,)/SE,,,) is14 f dJ% i Z=(M,+M,)~-
M,
M3 ’ M4
M,+M,
18.352 ._. E~,~
(44)
(45) where
g
-I’* cos 8 (39)
where
sin 19~
dQ-=d%
sin8
1 (46)
l+~cos&-
and
f=2~M,~M,*cos28+(M,+M4)(M4-M,) MI
+ 2 cos 0 g”*
(40)
*M3
E3.i l/2
(47)
cos 8, = cos 8( 1 - y2 sin 2f9)‘i2 - 1’sin *0.
(48)
'=M,
and g=M:~M~~cos2~+M1~M3~(M1+M~)(M~-MM3) + M, . MS . MJM,
+ Mb). 18.352/E3,i.
and (41)
1051
M Braun:
Surface analysis by RBS and NRA
Now, equations (37) and (38) can be used to calculate the concentration of deuterium as a function of depth. As the deuterium atoms, present in the substrate material, normally do not contribute to any significant degree to the stopping crosssection. the deuterium content can be neglected in the calculations in most cases. For very high concentrations of deuterium, however, the influence on the stopping cross-section can be taken into account by applying similar calculation procedures as described in the section above yielding depth distributions by the RBS technique. One example where a high concentration of deuterium is possible is in carbon substrates. Here. up to 70 at’,, of deuterium can be trapped in the bulk lattice and for accurate depth profiling the deuterium content should be considered in the calculations.
Conclusions
It has been demonstrated that depth distributions, where the concentration is high and non-homogeneous in depth, can be determined accurately by the RBS technique. The importance of taking the varying stopping cross-section into account in such cases has been illustrated. Determination of compound thin film composition and thickness by RBS has been discussed. With the special technique presented here. it was shown that for low concentrations of the lighter element in the film, it might be advantageous if the RBS signals from that element are not used in the analysis. This improves the RBS technique to analyse light materials. where the scattering cross-sections are low and the signals of which usually are superimposed on a heavier element. Finally, the NRA technique to determine deuterium depth
1052
profiles was presented, for the calculations.
including
all necessary expressions
needed
Acknowledgements
The author contribution thin films.
acknowledges the help of Dr I Petrov for his to the technique of calculating the composition of
References
I W K Chu. J W Mayer and M-A Nicolet. Backscarrerirtg Specrrometr~. Academic Press. New York (1978). a M Braun. R Brewer. H Stuessi and S Vepiek. Appl Pltys, AZ&25 (1982). 3 T Fried and M Braun. N~tcl Ittsrrwn Merh, 198, 523 (1982). ’ I Petrov. M Braun. T Fried and H E Satherblom, J Appl Ph.rs. 54, 1358 (19831. s D K Brice. Thin Solid Film, 19, 121 (1973). ’ J F Ziegler. R F Lever and J K Hirvonen. loft Beam Swfhce Larer Am/~si.t (Edited by 0 Meyer, G Linker and F Kappeler), Plenum Press. New York (1975). _ P BorSesen. R Behrisch and B M U Scherzer, Appl P~J.\. A27, 183 (1982). ’ H H Andersen and J F Ziegler, Hyd,oyerl Stopping Potters ad Ratyes ill All Ehems. Pergamon Press, New York (1977). 9 J F Ziegler. Helium Stopping Porvm ad Ranges irl A// Elemems, PerSamon Press. New York. 1977. ” P P Pronko. P/IF.\ Ret.. 9, 2870 (1973). I’ M Hufschmidt V Heintze. W Moller and D Kamke, Ntrcl Irtarrtrru Merh. 124, 573 (1975). ” D Dieume8ard. D Dubreuil and G Amsel. Nacl Insrrtrrn Mrrh. 166,431 (1979). I3 J L Yarnell. R H Lo\ berg and W R Stratton. P11.v.tRet.. 90.292 (1953). Ii W Eckstein R Behrisch and J Roth, Iort Beartt Srrt:face La!w Analysis, Vol 2, p 821. Plenum Press. New York (1976). If, C J Alstetter. R Behrisch. J Bottiger. F Pohl and B M U Scherzer. fv’ttcl lmrwm .Mrrh. 149. 59 (19781.