- Email: [email protected]
Calculation of high fluence ion implantation depth profiles
Nuclear Instruments and Methods in Physics Research B18 (1987) 121-130 North-Holland, Amsterdam
121
CALCULATION OF HIGH FLUENCE ION IMPLANTATION DEPTH PROFILES * David K. B R I C E Sandia National Laboratories, A Ibuquerque. New Mexico 87185. USA
Received 1 April 1986 and in revised form 4 August 1986
The method of equivalent atomic stopping (MEAS) is used to develop the equations governing the implanted ion depth profiles for high fluence implantations. Simultaneous multiple ion implantations are considered and sputtering is included in the formalism. Analytical steady state solutions are provided for the equations. The approximations inherent in MEAS are shown to be valid in particular at low energies 0 _
I. Introduction The transport equation normally used to obtain range and energy distributions for ions implanted into random media [1] is applicable only to uniform target materials. As a result, solutions to the equation are restricted to low fluence implantations such that the implanted ions do not interact to any significant extent with previously implanted ions. Winterbon [2] has generalized the standard transport equation to apply to inhomogeneous target materials, but his development does not include an explicit treatment of the time (fluence) dependence of the target composition. As a result, without further development, his treatment is also restricted to low fluence implantations into inhomogeneous targets. High fluence implantations are of particular interest in fusion energy studies [3] in which laboratory experiments simulate the effects on various materials of long term exposures to an energetic hydrogen plasma which may also contain impurities. In a recent paper [4], hereafter referred to as I, we have developed an approximate analytical technique for calculating ion implantation depth profiles in inhomogeneous target materials. This technique, labeled the Method of Equivalent Atomic Stopping, or MEAS, is a scaling technique whereby target atoms of type k are replaced by a number of "equivalent standard atoms", e~. A depth scale chosen proportional to the number of "equivalent standard atoms"/cm 2 renders the material "homogeneous" and the techniques applicable to such targets [1] then become relevant. In the present paper * This work performed at Sandia National Laboratories supported by the US Department of Energy under contract number DE-AC04-76DP00789. 0168-583X/87/$03.50 9 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
we derive the time (fluence) dependent equations governing the evolution of the depth profile within the approximations of MEAS. Since sputtering usually is of significance in high fluence implants the effects of sputtering are also included in our development. Not included explicitly in the development are diffusion, atomic mixing, and trapping. Since our procedure amounts essentially to a change of variables these effects are easly added to the equations which we develop here. We briefly discuss the form these added terms will take. In section 2 we review the essential features of MEAS and derive the appropriate time (fluence) dependent equations governing the depth profiles. In section 3 we show, by comparison with calculations using the TRIM particle transport code [5], the particular validity of the basic approximations of MEAS for ion energies in the range 0 _< E _< 10 keV with ion and target atom atomic numbers in the range 1 _< [Z l, Z2] _< 32. These combinations of energy and atomic number have relevance for magnetic confinement fusion energy studies. Finally, in section 4 we reanalyze experimental data for high fluence 1.5 keV and 3 keV oxygen implantation into graphite targets. Conclusions and discussion appear in section 5.
2. Evolution of high fluence profiles 2.1. Method of equivalent atomic stopping (MEA S)
As energetic projectiles penetrate a target material they are scattered by, and lose energy to, the various atomic species which they encounter along their trajectories. In MEAS [4] the central notion is that to some degree of approximation the effect of the t3resence of an
122
D.K. Brice / High fluence ion implantation depth profiles
atom of a given atomic species,,k, on the projectile trajectory can be represented as the same as that expected from c k atoms of a "standard" atomic species at that same location. Specifically, let Nk(x, t) be the atomic number density of atoms of type k in the target material at depth x and time t. The time dependence of N~ can be due to implantation, diffusion, surface erosion, etc. Now, define a new depth scale z by
z = (1/N~,~)f ~ E , ~ N ~ ( x ' , t) dx', Jx9 o( t}
(1)
tion in the precise choice of { ck } is not important. We remark, however, that the { c k } can be chosen as simple functions of range parameters for uniform targets [4]. For example we may choose
ck = NstdR pstd//Nk o R pk,
(4)
where R pk is the average projected range of an ion in a uniform target of k-atoms with density Nk0, while the quantities subscripted with std are similarly defined for a uniform target of standard atoms.
k
where N~td is some chosen standard atomic density. The depth scale z is a depth scale in a target which is uniform in "equivalent standard atoms" with an atomic number density Nstd- The lower limit, x0(t), represents the location of the target surface which may be eroded by sputtering or other processes; thus the time dependence. We imagine now that atoms of type j are implanted into the target with flux density q~j(t) (at./cm 2 s). The probability at time t that an implanted atom of type j comes to rest in (x, d x ) is given by Pj(x, t) dx. The time dependence of Pj arises from the time dependence of the surface location, Xo(t ), as well as from the time varying composition of the t~get material, {Nk(X, t)}. The central approximation in MEAS is the assumption that constant c k may be defined such that with z given by eq. (1) one may write
Pj(x, t) dx = O j ( z ) dz.
(2)
Thus, given eq. (1) and (2) one obtains
Q j ( z ) = Ns,dPj(x, t ) / Y ' . , k N k ( x , t).
(3)
k
The important result in eqs. (2) and (3) is that the coordinate transformation of eq. (1) changes the source function ~ ( x , t) into a source function Qj(z) which is independent of time and target composition. The source function Qi can thus be calculated for a uniform target of standard atoms. As indicated in I, there is some latitude in the choice of the standard target and in the choice of the {c k }. The best choice of standard target "atom" is one which represents the "average" atom in the target of interest, although this particular choice is not essential. The approximation errors are reduced for the "average" case by a cancellation effect [4], thus providing more accuracy in the procedure. In order that the method be easily adapted to cases of simultaneous multiple ion implantations it is also necessary that the { c k } be independent (or nearly independent) of incident ion type. We assume here that this is so, but the assumption must be verified for each application of MEAS to simultaneous implantations. An example of such verification will be provided in section 3. For the developments in the remainder of this sec-
2.2. Time dependent equation for Nk(x, 0 In this subsection we derive the time dependent equations for the evolution of the Nk(x, t). More accurately, we transform the set of equations
dNk(x, t)/dt=r
t) --Oo(t)Nk(x, t ) ~ [ X - X o ( t ) ] ,
(5)
by changing the depth scale according to eq. (1). In eq. (5) the first term on the fight represents the change in Nk due to implantation, while the second represents loss due to surface erosion. The Dirac delta function is given by 8(x), while Vo(t ) = d x o / d t is the velocity of surface recession due to erosion. We note that eq. (5) assumes a spatially uniform erosion rate. The succeeding development is therefore not strictly applicable when the sputter formation of various surface structures (cones, pits, etc.) is significant. As remarked earlier, terms involving diffusion, atomic mixing, trapping, etc., may be added to the fight hand side for a more complete description. We ignore those terms in the present development, since our attention is focused here on the effects of high fluence implantation on the depth profiles. Our procedure involves only a change of variables which can, in principle, be easily extended to these other effects. Before transforming to the new depth scale let us also introduce a new dependent variable, namely n k ( z, t ), defined by
nk(z, t)Ns, d dz = Nk(x, t) dx.
(6)
Also, let us define M(x, t) by
M( x, t ) ~ d z / d x = Y~,kNk ( x, t )/Nst a.
(7)
k
From eqs. (6) and (7) we note that
nk(z, t ) = Nk(x, t ) / [ M ( x , t ) . Ns,d] ,
(8)
and that the n k are not independent since
~~.,kn~(z, t) = 1.
(9)
k
From eqs. (5) and (7) we also note that
d M ( x , t ) / d t = G ( x , t) -Vo(t)M(x, t)8(X-Xo(t)),
(10)
D.K. Brice / High fluence ion implantation depth profiles
where the source term G(x, t) is given by
G(x, t ) = ~_,,ktkk(t)Pk(x, t)/N~ta.
(11)
k
Utilizing eqs. (5) through (10), the transformation to the z depth scale, eq. (1) yields ~ ank
Onk
8--~ + S ( z , t ) " ~z- + nkH(z, t ) = Hk(z, t).
(12a)
and aM~t + J(z, t)__~_ z~M= M ( z , t ) H ( z , t) -Vo(t)M2(z, t)8(z).
(12b)
In eq. (12)
Hk(z, t ) = ~ k ( t ) Q k ( z ) / N s t d and
(13a)
H ( z , t) = ~_,c~Hk(z, t),
(13b)
k
while
J ( z , t)=foZH(z ', t ) d z ' - v o ( t ) M o ( t ) .
(13c)
123
greater depths (AJ > 0). For case 1, described below, z a represents the depth at which these opposing effects just balance (i.e., J(z a, t ) = 0). When the net implantation rate of "equivalent standard atoms" is smaller than the surface erosion rate (case 2 below) then J(z, t ) < 0 at all depths.
2.3. Relaxation As the implanted ion concentration increases the target material may undergo relaxation due to a buildup of stresses in the material. For an assumed planar geometry such relaxation will occur parallel to the depth axis, in our case parallel to x. Normally, such relaxation effects are included in a mathematical treatment such as ours by a change of depth scale which, for example, may maintain a constant total target atomic density. Alternatively, the relaxed depth scale may provide a known response to the induced stresses, but the net result is still simply a change of depth variable. All such relaxation processes are implicitly included in the depth scale of eq. (1). Thus, relaxation does not explicitly enter in to the time development of the n k nor of M when the latter is expressed as a function of z.
By M(z, t) we mean M[x(z, t), t], and M 0 ( t ) =
M[xo(t), t] = M(z = O, t). Eq. (12), with the auxiliary eq. (13), provide the basic general equations for the evolution of the Nk, within the procedures of MEAS. We note that these equations, eq. (12), are coupled and nonlinear, since J(z, t) depends on the surface value of M(x, t), i.e., on Mo(t). Time dependent solutions to the equations will thus normally require numerical methods. Inclusion of other transport effects, such as diffusion, will further complicate the coupling, but the essential features of the treatment remain the same. Note in eq. (13c) that vo(t)Mo(t) is the normalized rate of erosion of "equivalent standard atoms" from the target surface, while the integral on the right hand side represents the normalized rate of implantation of "equivalent standard atoms" on the interval (0, z). Note also that J(z, t) is a monotonic increasing function of z, with J(O, t)< O. Thus, if J(oo, t) is positive then there is a net accumulation of material in the target, i.e., the total implantation rate is greater than the rate of loss by sputtering. Similarly, J(0o, t) < 0 implies a net loss of material, i.e., the implantation rate is less than the sputter rate. The function J(z, t) represents a local streaming velocity in the z-coordinate. If one follows with time the location of the volume element, Va, containing a particular implanted atom, J represents the net velocity of Va resulting from surface erosion and implantation/ redeposition. Surface erosion causes Va to move toward the surface (A J < 0) while implantation between the location of Va and the surface causes Va to move to
2.4. Steady state solutions Although the general equations, eq. (12), normally require numerical methods for their time dependent solution, analytical steady state solutions to these equations are readily accessible. Thus, we can provide analytical formulas for the implantation profiles at fluences sufficiently high that the steady state is approached. We assume that for sufficiently large time the erosion rates approach constant values, so that
Vo(t)Nk(xo, t)/Ns, a ~ R~,
(14a)
and
Vo( t ) Mo( t ) --* R = )'~r
(14b)
k
We also assume ckk(t)~ ~kk = constant and note from eq. (13c) that
dJ/dz = H(z).
(15)
Setting the time derivatives in eq. (12) to zero, and utilizing eqs. (14) and (15), we obtain for the steady state for z > 0 d ( J . nk ) / d z = J dnk/dz + nk d J / d z = Ilk(z) (16a)
and J dM/dz = M dJ/dz.
(16b)
Solution to eq. (16) is formally straightforward. Some care must be exercised in applying the boundary conditions, however, depending on whether J ( ~ ) is positive
D.K. Brwe / High fluence ion implantation depth profiles
124
or negative. We thus define two cases. For case 1, J(oo) is positive, and for case 2, J(oo) is negative. The special case J ( 0 ) = 0 can be obtained as a limit from case 1. (See also the discussion at the end of section 2.2) For case 1, J(0) is negative and J(oo) is positive. Thus there exists a positive value of z, say z a, such that J(za) = 0 whj'ch implies that
f0=iH(z') dz'-- R. JCz)n,(z) =S(0)n,(0) + s n,(z') dz'. -J(O) =
(17)
For z < z~ we may integrate eq. (16a) to yield
(18)
Now, from eq. (18) we have at z = z~ nk
(0) = -- (1/J(O)) foZ'H~ (z i) dz'. = (1/R)foZ'H,(z ") dz'.
(19)
Substituting Eq. (19) into (18), and rearranging terms yields o
n~,(z) = - ( 1 / J ( z ) ) ~ ' " H , ( z ' )
dz'.
(20)
Note also that even though J(za)= O, n,(z,,) exists as a limit to eq. (20), and
, , ( z~) = H, ( z . ) / H ( z.).
(21)
Now, for z > z a the solution to eq. (16a) is
J(z)n,(z)=
fz]Hk(z') dz',
auxiliary conditions the value of z~ is defined by eq. (17), and the solutions follow as indicated. A second reason for placing the Vo on the left hand side is that both vo and N, are quantities defined on the unrelaxed x depth scale. Since the net implantation rate exceeds the net erosion rate for the present case (case 1) we know that in the steady state Oo---' 0 and Nk --* oo, while their product remains finite. Thus, it is improper in case 1 to separate the factors in these products. Finally, before leaving case 1, there is some temptation to replace R , by ~Tq~jSj,/Nsta where the $1~ are "high fluence sputtering coefficients". As we have noted, however, at this point this is simply replacing unknown Rs with unknown Ss. In the applications section we will deal with a case in which the Sj, are known, but that application will be a case 2 application. For case 2 the boundary values are the values of nk as z --* oo, which we take to be the large z values of n , in the original, unimplanted target material. Thus, for example, for an initial elemental target we would take no(oO ) = n o = constant = 1/Co, and n,(oo) = 0, for k 0. In general
n, (oo) = N, (o0,0)/[ m(oo, 0). Us,d],
(25)
where the right hand side is evaluated at x = oo, and t = 0. Then one has as solution to eq. (16)
nk(z)=(X/J(z))[J(oo)n,(oo)-(22a)
~~176
) dz'], (26a)
and or
M ( z ) = M(oo)J(z)/J(oo). nk ( z ) = (1/S( z ) ) fz]H, (z') dz'.
(26b)
(22b)
2.5. Other considerations The form given in eq. (22b) is identical to eq. (20), so that the single equation holds for all values of z. Care must be exercised however, in the vicinity of z = z~ where the limit of eq. (21) becomes applicable. A similar treatment of eq. (16b) yields
~
= I J ( z ) l-
(23)
The concentrations Nk(z) can now be obtained from eqs. (22b) and (23) using eq. (8). Thus
t,oN,(z) = f f ' H , ( z ' ) dz"
Us,d .
(24)
Eqs. (23) and (24) are each written with the multiplier v0 on the left hand side for two reasons. First, ooNk(O) represents the steady state sputtering flux density of k-type atoms and a knowledge of this rate from experiment, or some other theoretical treatment is required before the solution to the equations is complete. That is, other input determining the values of R and R~. is required which is beyond the scope of the present treatment. Once R = voMo is determined from such
2.5.1. Saturation It is often observed that implanted atoms of a given species can only be present in a target material up to some saturation level, say Nk=nskNstd, for atomic species k, where n~, = constant. That is, further implantation of species k into a region where the atomic density is already n~, N~td only provides a highly mobile form of k atoms which rapidly diffuse away from the region in question. In many cases these mobile atoms are lost from the target material [3]. The treatment given above for steady state high fluence implantations is easily extended to the case of saturable implants. Saturation can be shown to be accounted for by simply calculating the n , as if saturation were not a factor and then truncating n k in those regions for which nk/no> ns,. The truncation must also be a accompanied by a "renormalization" to maintain the relationship of eq. (9) and a proper calculation of J in the saturated region. This is equivalent to replacing H k by n~kH o in these regions.
D.K. Brice / Highfluence ion implantation depthprofiles 2.5.2. Diffusion and atomic mixing Diffusion can be added to the above treatment by including a term Tk on the right hand side of eq. (5) where
125
may, under certain circumstances, all be satisfied. At any rate, solutions in such cases are correct up to the time at which saturation effects begin to occur [7]. The general time dependent solution for J(z, t)--*J(z) is given by
nk(z,t)J(z)=J(u)nk(u,O)-fUH,(z')dz', (29) When transformed into the z-depth scale and the n, M variables, eq. (27) becomes T,. = (N, td- M ) ~ z [ M D ~ z ( n ~ M ) ] .
where u(z, t) is defined by
f Zdz'/J(z') = t.
(30)
(28)
Eqs.' (12) are altered to include diffusion by adding a term TA/[M. N~td] to the right hand side of eq. (12a) and a term ~.kckTk/Nstd to the right hand side of eq.
(12b). The inclusion of diffusion into eq. (12) clearly increases the coupling between the various dependent variables as indicated in eqs. (27) and (28). General solutions to the equations must normally be sought by numerical means, however, so that the additional coupling does not significantly add to the complexity of solving the equations. Explicit expressions for additional terms in eq. (5) to account for atomic mixing will not be given here. Inclusion of such terms is formally straightforward however, and consists of adding a term to the right hand side of eq. (5) which is proportional to the right hand side of Littmark's [6] eq. (1) (note that Littmark uses Pk to stand for the concentration which we call Nk). We note that Littmark uses Pk to stand for the concentration which we call Nk). We note two simplifications which will occur in the treatment of atomic mixing by MEAS in place of the standard treatment. First, with MEAS it will not be necessary to define a time (fluence) varying depth scale which keeps up with various relaxation processes. As already discussed, such relaxation is implicit in our depth scale, z. Second, the relocation function Uk(x, x') as defined by Littmark, when transformed into z, z' variables, will become independent of target composition and will, furthermore, be applicable to high fluence cases. As a result, the atomic mixing terms in the MEAS format will be significantly simplified over the corresponding terms in the usual treatment [6].
2.6. Time dependent solutions In general, time dependent solutions to eq. (12) must be obtained by numerical methods. Explicit expressions can be given however for the general solution to the equations when J(z, t) is independent of time (or is a very slowly varying function of time). This occurs when the fluxes, {~k(t)}, are all constants (and thus the H k are independent of t), the sputtering rate is constant, and saturation effects are not present. These conditions
In eq. (29) nk(u, 0) is simply the initial (t = 0) value of nk(z, t) while the other functions have already been defined. We note that if J(z) has a zero on 0 < z < oo, say at z = z,, then from eq. (30) either u and z are both greater than Za, or u and z are both less than z a and u --* z, as t --* oo. Thus eq. (29) limits to the steady state solutions for case 1. If J(z) has no zero on the positive z-axis then u(z, t) --* oo as t ---, oo and the steady state solution for case 2 is recovered. For M(z, t) we obtain
M(z, t) = S ( z ) M ( u , O)/J(u), (31) with the caveat that voM(z, t) remains finite for a case 1 solution.
3.
Applications
In this section we wish to demonstrate hat the approximations involved in MEAS are valid over a sufficient range of projectile types, target atom types, and incident energies to provide an adequate treatment of a variety of high fluence implantation problems of interest in fusion energy studies. In particular we wish to show that values for the (ok } can be obtained for a given target material which are essentially constant, i.e., independent of projectile type and energy over some useful range of projectile masses and energies. We then wish to show that with these given {( k }, the Qi(z), for several composite targets are well represented by the Q,(z) for appropriately chosen standard target materials.
3.1. Values of {(k} The TRIM particle transport code [5] has been used to calculate projected range profiles for 1 and 10 keV projectiles with 1 < Z~ < 32 incident on a variety of elemental and alloy targets with homogeneous spatial composition. The constituents of these targets were also chosen such tht 1 < Z2 < 32. This range of values for the atomic number Z~ and Z2 and the range for projectile energies are those of interest in plasma materials interactions for fusion energy applications. Table 1 contains values of Nk0Rpk obtained from the TRIM calculations with elemental target materials.
126
D.K. Brice / High fluence ion implantation depth profiles
Table 1 Average projected range NkoRpk from TRIM calculations with 1000 histories; units are 10 Is atoms/cm 2 Projectile
E(keV)
Target D
Ti
Ge
724.83 225.22 132.72 114.57 161.43 273.95
327.78 98.30 54.72 35.98 34.93 37.81
205.12 61.96 34.44 22.52 21.13 21.87
134.06 42.83 23.91 14.11 12.47 12.21
116.07 40.92 21.29 12.09 9.70 8.90
101.98 36.75 20.43 10.76 8.39 7.47
4738.63 2596.99 1437.51 891.59 981.13 1323.45
2565.56 1095.94 532.98 262.02 208.43 207.33
1782.01 643.64 311.24 147.68 115.48 105.46
1069.16 366.10 184.24 85.72 63.36 54.42
839.39 284.88 141.98 64.29 48.15 40.08
661.69 240.49 125.30 57.62 41.09 33.37
1.0
D Li C Si Ti Ge D Li C Si Ti Ge
10
Li
C
After selecting a standard target material values of { c k } can be calculated for each of the materials and energies listed in table 1 by using eq. (4). We note that the standard target material should be chosen to represent the average composition of some target of interest. Fig. 1 shows typical results obtained for ~ck) versus Z2(k ) where ( ) represents averaging over the various incident projectile types and energies. The error bars show the standard deviation of the calculated ok-values about the average, while the curve is a fitted polynomial to the < c k )-valuesF r o m fig. 1 it is seen that the ck are reasonably constant over a broad range of incident projectile types and energies. The variation of ck at the extreme target types amounts to - 15% for Ge targets and - 50% for the deuterium targets. While the standard deviation for
I
I
I
I
I
I
I 25
I 30
vs Z 2
2
V 1
0 0
I 5
I 10
I 15
I 20
35
Z2 Fig. 1. Average values of c calculated from the data in table 1 using eq. (4). Standard deviation of c-values about the average shown as error bars. Solid line is fitted cubic polynomial.
Si
D targets would seem to be unacceptable, we note that " the ~k-values for D are quite small, i.e., - 0.1. Thus, the presence of D in a target of silicon-like atoms will have only a small effect on the development of the range profiles and a large uncertainty in the c A is thus acceptable. For targets in which D constitutes a significant fraction of the composition a different standard target is appropriate, and the resultant values of ~k for D will have a smaller range of variation. This is easily verified from the data of table 1.
3.2. Accuracy of MEAS Qi(z) In this subsection we wish to examine the accuracy with which the Q,(z) for a chosen elemental standard target material represent the corresponding profiles for composite targets of various types. Targets of equal amounts of C and D (i.e., C D targets), equal amounts of Ge and D (GeD) and equal amounts of D, C, Si, and Ti (TiSiCD) are considered for these comparisons. Fig. 2 shows a comparison of profiles Qi(z) calculated by M E A S (solid curves), using Ti as the standard atom with corresponding profiles calculated by T R I M for G e D targets (dashed curves) for 1 keV and 10 keV D, C, and Si projectiles. Comparisons for 1 keV are shown in fig. 2a and those for 10 keV in fig. 2b. The comparison for the two heavier projectiles is seen to be excellent at both energies while the M E A S profiles for D projectiles are somewhat less accurate, particularly at 10 keV. Similar accuracy is expected for implantations of these ions into inhomogeneous targets of Ge and D for which the D concentration does not exceed - 50%. We note that agreement between M E A S and T R I M calculations for D projectiles can be improved somewhat by choosing a different standard material, but the improvement will be at the expense of the profiles for the other projectiles. Choice of standard material thus
D.K. Brice / High fluence ion implantation depth profiles I
'N' I
I
I
l
I
I
I
I
I
(a)
(a) O(z) vs z
Olzl vs z
II I
I
~176 I
I
I
I
I
,.i
. I
I
z (arb. units) I
I
I
127
I
I
I
I
z (arb. units) I
I
I
I
]
I
1
I
I
I
I
I
1
(b)
Olz) Vl Z
~I
(b)
-.
O(z) vs z
9
"[i I
I|
\",.] I
!
I
I
I
I
J
\1
z (arb. units)
~
r l l//,
D
oo \',., \,
z (arb. units)
Fig. 2. Comparison of TRIM profiles for GeD (Dashed curves) and MEAS profiles (solid curves) obtained from the data of table 1 using Ti as standard target. Depth scales may be obtained by reference to Ti column in table 1. Curves shown for D, C, and Si projectiles. Tic marks on ordinate indicate decades, a) 1 keV. b) 10 keV.
Fig. 3. Comparison of TRIM profiles for CD (dashed curves) and MEAS profiles (solid curves) obtained from the data of table 1 using Li as standard target. Depth scales may be obtained by reference to Li column in table 1. Curves shown for D, C, and Si projectiles. Tic marks on ordinate indicate decades, a) 1 keV. b) 10 keV.
depends to some extent on the projectile of interest in a given application. Nonuniformity of accuracy in the Q, results from the fact that electronic stopping does not scale with projectile atomic number similarly to the scaling of elastic scattering. Fig. 3 shows results similar to those in fig. 2, but for C D targets using Li as standard atoms, and similar results for TiSiCD targets (Ti = standard atom) are shown in fig. 4. F r o m figs. 2 - 4 we see that the M E A S procedure can be expected to give reasonably accurate profiles Q~(z) for use in the time dependent equations of the previous section for 1 < [ Z t, Z2] < 32 with projectile energies in the range 0 < E < 10 keV.
3.3. An example calculation In a recent publication Wampler and Brice [8] used a preliminary version of M E A S to analyze experimental results for high fluence oxygen implantation into graphite. That data is reanalyzed here to present an example of the application of the equations developed here, and to provide an updated discussion of the data. In the experiment reported in ref. [8] oxygen ions were implanted at 1.5 and 3 keV into graphite at room temperature, and the amount of oxygen retained was monitored by 2 MeV 4He backscattering analysis. The retained oxygen was found to reach a steady state value
128
D.K. Brice / High fluence ion implantation depth profiles I
I
I
/Si
1
1
I ~'
I
I
1
(J)
O(z) vs z
/e
using carbon as standard target and fitting the resultant six data points to a polynominal in Z2. Since only oxygen is implanted, the carbon density in the x-depth scale remains constant at Nc. Further, from the measured sputtering coefficient we have voNc(0) = y o n c = ~ S ,
(32a)
and thus for the oxygen erosion rate we have v o N O (0) = ~ S N o ( O ) / N c ,
g,
(32b)
so that
_1
voM(O) Uc = tbS(1 + e o N o ( O ) / U c ) . It
I
lJ
I
~
1
1
I
I
Using eqs. (8) and (9), eq. (32c) becomes voM(O) Uc = ~ S / n c (0),
z (arb. units)
(32c)
(33)
so that eq. (14b) yields
'~:~~ C
Q(z)v$z
(b)
R = dpS/( n c (0) Nc ).
(34)
This is easily verified [10] to be a case 2 example so we have from eq. (26) nc(z) =J(oe)/S(z),
/
Ti Si C D I
I
I
I
k I
I
I
z (arb. units) Fig. 4. Comparison of TRIM profiles for TiSiCD (dashed curves) and MEAS profiles (solid curves) obtained from the data of table 1 using Ti as standard target. Depth scales may be obtained by reference to Ti column in table 1. Curves shown for D, C, and Si projectiles. Tic marks on ordinate indicate decades, a) 1 keV. b) 10 keV.
since nc(OO)= 1. Noting that J depends on nc(0) through R, eqs. (34) and (13c), one can set z to zero in eq. (35) and solve for n~(0). Doing that and substituting the value into the expression for R and for J one can then verify that J ( o e ) < 0, and that the example is indeed case 2. The resultant expressions for n c and n o from eq. (26a) then become n c ( z ) = 1/[1 + c o q ( z ) ] ,
(36a)
no(z) = q(:)/[1
(36b)
+,oq(z)],
and no(z)/nc(z
) = No(z)/N c
= q(z),
(36c)
where q(z)
of 1.6 • 1016 O / c m z and 2.7 x 1016 O/cm-' for the 1.5 and 3 keV implants, respectively. Surface erosion is significant in this system and measured values of the carbon loss rate at both 1.5 and 3 keV yield 1.1 _+ 0.1 carbon atoms removed per incident oxygen at the high fluence limit [9]. TRIM calculations yield oxygen depth profiles in graphite which are nearly Gaussian for the two different energy implants with projected range and straggling of 37.4 x 1015 C / c m 2 and 16.5 x 1015 C / c m 2, respectively at 1.5 keV and 72.6 x 1015 C / c m z and 30.8 • 1015 C / c m 2, respectively at 3 keV. Taking carbon as the standard target for this analysis, a calculation similar to that of fig. 1 using the data of table 1 yields an equivalency factor for oxygen of c o = 1.2. This value is obtained by calculating ( ( ( Z z ) ) for the data of table 1
(35)
= (I/S) f~~
z') d z'.
(36d)
In eq. (36) Q ( z ) is the low fluence depth profile for oxygen implantation into carbon we obtained above from the TRIM calculations. We also note that q ( z ) is a monotonically decreasing function of z. We have avoided using the normalization of Q(z) in the above expressions. As a result, the expressions also hold when saturation effects are important. In fact, if n~ represents the saturation value of the ratio N o / N c, then eq. (36) predicts that the steady state oxygen profile is saturated for all z less than z., where z, is defined by ,, = q(z~).
(37)
The retained oxygen can be calculated from eq. (36b) and is given by Io, where !o =
Ucf ~ n o ( z ) dz. Jo
(38)
D.K. Brice / High fluence ion implantation depth profiles Wampler and Brice [8] have argued that saturation effects are important in the present case. The saturation level can be obtained by adjusting ns, i.e., z, such that the experimental results are reproduced by eq. (38). The resultant value for n~ is 0.43. This compares with the value 0.38 obtained in the earlier analysis [8] using a preliminary version of MEAS in which oxygen and carbon atoms were considered as precisely equivalent, i.e., c0 ---, 1. Fig. 5 shows steady state depth profiles for 3 keV oxygen implants into carbon which illustrate the effects of projectile interactions with previously implanted projectiies along with the effects of a saturation limit. The solid curves show the MEAS predicted profiles when oxygen-oxygen interactions are taken into account while the dashed curves show profiles in the absence of such interactions. The profiles with flat tops correspond to the saturation limit just obtained from the comparison of I0, eq. (38), with the experimental results of ref. [8]. From the two solid curves one sees immediately the effects of both saturation and the oxygen-oxygen interactions. The saturation limit reduces the oxygen concentration in the near surface region giving the flat topped profile; the reduced near surface oxygen concentration for the saturated case allows deeper penetration of those oxygen projectiles which ultimately come to rest in the tail of the distribution. In contrast, the tails of the saturated and nonsaturated profiles for the noninteracting case are coincident. The large difference between the solid and dashed curves indicates the importance of including the oxygen-oxygen interactions in the profile calculations as is done here through MEAS.
'
3keY
I
!
I
O+-.,.-C
0
0
0.5
, 0
I 40
,
The method of equivalent atomic stopping (MEAS) has been used to derive the equations governing the evolution of high fluence implantation depth profiles. The equations include the influence of sputtering by the implanted species .and can be used in cases of multiple simultaneous implantations. It is indicated in the development how the equations can be extended to include the effects of diffusion, atomic mixing and trapping, which are straightforward exercises since MEAS involves only a change of variables. It is also noted that relaxation effects are implicitly included in the development. Steady state solutions to the equations in the absence of diffusion and atomic mixing are obtained. It is demonstrated that the approximations involved in MEAS are valid ( - 15%) in particular for projectile energies in the range 0 < E < 10 keV for cases in which the atomic number of projectile and target atoms are in the range 1 < [Z~, Z2] < 32. These sets of values are those which are of interest for plasma materials interactions in fusion reactors. An example showing the analysis of experimental data for high fluence oxygen implantations into carbon is presented to demonstrate the utility of the steady state solutions. The expressions given here provide, for the first time, a set of simple analytical equations for obtaining the depth profiles for high fluence ion implantation. While certain approximations are inherent in the MEAS procedures, the validity of the method can be verified by simple comparisons with calculations for spatially homogeneous targets as indicated here in section 3. The equations developed here are of broad applicability, and should be useful not only in fusion energy studies, but also in implantation metallurgy, high fluence damage studies, and other areas involving high fluence implantation. References
Z
0
4. Summary
1
1.0
Z
129
"I ----~--80
I~,
~
120
I 160
!00
x (angstroms) Fig. 5. Steady state depth profiles for 3 keV oxygen implantation into carbon illustrating saturation and projectile-projectile interaction effects.
[1] See, for example, J. Lindhard, V. Nielsen, M. Scharff and P.V. Thompson, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 33 (1963) no. 10; J. Lindhard, M. Scharff and H.E. Schiatt, ibid. 33 (1963) no. 14; J. Lindhard, V. Nielsen and M. Scharff, ibid. 36 (1968) no. 10; K.B. Winterbon, P. Sigmund and J.B. Sanders, ibid. 37 (1970) no. 14. [2] K.B. Winterbon, Appl. Phys. Lett. 31 (1977) 649. [3] See, for example, B.L. Doyle, W.R. Wampler, D.K. Brice and S.T. Picraux, J. Nucl. Mater. 93/94 (1980) 551; K.L. Wilson and A.E. Pontau, ibid. 93/94 (1980) 569; D.K. Brice, B.L. Doyle, W.R. Wampler, S.T. Picraux and L.G. Haggmark, ibid. 114 (1983) 277. [4] D.K. Brice, submitted to Nucl. Instr. and Meth. B17 (1986) 289.
130
D.K. Brice / High fluence ion implantation depth profiles
[5] J.P. Biersack and L.G. Haggmark,'Nucl. Instr. and Meth. 174 (1980) 257. [6] U. Littmark, Nucl. Instr. and Meth. B7/8 (1985) 684. [7] When only two atomic species are present in the target the truncation procedure described for the steady state solutions in a saturated region can also be used for the time dependent solution. This procedure has not been shown to be valid however when more than two atomic species are present.
[8] W.R. Wampler and D.K. Brice, J. Vac. Sci. Technol. A4 (1986) 1186. [9] E. Hechtl, J. Bohdansky and J. Roth, J. Nucl. Mater. 103/104 (1981) 333. [10] One can easily show from eq. (13) that when the initial target material is not present as one of the implanted atom fluxes the steady state value for J(0r is negative. Such cases will thus always be examples of case 2.
121
CALCULATION OF HIGH FLUENCE ION IMPLANTATION DEPTH PROFILES * David K. B R I C E Sandia National Laboratories, A Ibuquerque. New Mexico 87185. USA
Received 1 April 1986 and in revised form 4 August 1986
The method of equivalent atomic stopping (MEAS) is used to develop the equations governing the implanted ion depth profiles for high fluence implantations. Simultaneous multiple ion implantations are considered and sputtering is included in the formalism. Analytical steady state solutions are provided for the equations. The approximations inherent in MEAS are shown to be valid in particular at low energies 0 _
I. Introduction The transport equation normally used to obtain range and energy distributions for ions implanted into random media [1] is applicable only to uniform target materials. As a result, solutions to the equation are restricted to low fluence implantations such that the implanted ions do not interact to any significant extent with previously implanted ions. Winterbon [2] has generalized the standard transport equation to apply to inhomogeneous target materials, but his development does not include an explicit treatment of the time (fluence) dependence of the target composition. As a result, without further development, his treatment is also restricted to low fluence implantations into inhomogeneous targets. High fluence implantations are of particular interest in fusion energy studies [3] in which laboratory experiments simulate the effects on various materials of long term exposures to an energetic hydrogen plasma which may also contain impurities. In a recent paper [4], hereafter referred to as I, we have developed an approximate analytical technique for calculating ion implantation depth profiles in inhomogeneous target materials. This technique, labeled the Method of Equivalent Atomic Stopping, or MEAS, is a scaling technique whereby target atoms of type k are replaced by a number of "equivalent standard atoms", e~. A depth scale chosen proportional to the number of "equivalent standard atoms"/cm 2 renders the material "homogeneous" and the techniques applicable to such targets [1] then become relevant. In the present paper * This work performed at Sandia National Laboratories supported by the US Department of Energy under contract number DE-AC04-76DP00789. 0168-583X/87/$03.50 9 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
we derive the time (fluence) dependent equations governing the evolution of the depth profile within the approximations of MEAS. Since sputtering usually is of significance in high fluence implants the effects of sputtering are also included in our development. Not included explicitly in the development are diffusion, atomic mixing, and trapping. Since our procedure amounts essentially to a change of variables these effects are easly added to the equations which we develop here. We briefly discuss the form these added terms will take. In section 2 we review the essential features of MEAS and derive the appropriate time (fluence) dependent equations governing the depth profiles. In section 3 we show, by comparison with calculations using the TRIM particle transport code [5], the particular validity of the basic approximations of MEAS for ion energies in the range 0 _< E _< 10 keV with ion and target atom atomic numbers in the range 1 _< [Z l, Z2] _< 32. These combinations of energy and atomic number have relevance for magnetic confinement fusion energy studies. Finally, in section 4 we reanalyze experimental data for high fluence 1.5 keV and 3 keV oxygen implantation into graphite targets. Conclusions and discussion appear in section 5.
2. Evolution of high fluence profiles 2.1. Method of equivalent atomic stopping (MEA S)
As energetic projectiles penetrate a target material they are scattered by, and lose energy to, the various atomic species which they encounter along their trajectories. In MEAS [4] the central notion is that to some degree of approximation the effect of the t3resence of an
122
D.K. Brice / High fluence ion implantation depth profiles
atom of a given atomic species,,k, on the projectile trajectory can be represented as the same as that expected from c k atoms of a "standard" atomic species at that same location. Specifically, let Nk(x, t) be the atomic number density of atoms of type k in the target material at depth x and time t. The time dependence of N~ can be due to implantation, diffusion, surface erosion, etc. Now, define a new depth scale z by
z = (1/N~,~)f ~ E , ~ N ~ ( x ' , t) dx', Jx9 o( t}
(1)
tion in the precise choice of { ck } is not important. We remark, however, that the { c k } can be chosen as simple functions of range parameters for uniform targets [4]. For example we may choose
ck = NstdR pstd//Nk o R pk,
(4)
where R pk is the average projected range of an ion in a uniform target of k-atoms with density Nk0, while the quantities subscripted with std are similarly defined for a uniform target of standard atoms.
k
where N~td is some chosen standard atomic density. The depth scale z is a depth scale in a target which is uniform in "equivalent standard atoms" with an atomic number density Nstd- The lower limit, x0(t), represents the location of the target surface which may be eroded by sputtering or other processes; thus the time dependence. We imagine now that atoms of type j are implanted into the target with flux density q~j(t) (at./cm 2 s). The probability at time t that an implanted atom of type j comes to rest in (x, d x ) is given by Pj(x, t) dx. The time dependence of Pj arises from the time dependence of the surface location, Xo(t ), as well as from the time varying composition of the t~get material, {Nk(X, t)}. The central approximation in MEAS is the assumption that constant c k may be defined such that with z given by eq. (1) one may write
Pj(x, t) dx = O j ( z ) dz.
(2)
Thus, given eq. (1) and (2) one obtains
Q j ( z ) = Ns,dPj(x, t ) / Y ' . , k N k ( x , t).
(3)
k
The important result in eqs. (2) and (3) is that the coordinate transformation of eq. (1) changes the source function ~ ( x , t) into a source function Qj(z) which is independent of time and target composition. The source function Qi can thus be calculated for a uniform target of standard atoms. As indicated in I, there is some latitude in the choice of the standard target and in the choice of the {c k }. The best choice of standard target "atom" is one which represents the "average" atom in the target of interest, although this particular choice is not essential. The approximation errors are reduced for the "average" case by a cancellation effect [4], thus providing more accuracy in the procedure. In order that the method be easily adapted to cases of simultaneous multiple ion implantations it is also necessary that the { c k } be independent (or nearly independent) of incident ion type. We assume here that this is so, but the assumption must be verified for each application of MEAS to simultaneous implantations. An example of such verification will be provided in section 3. For the developments in the remainder of this sec-
2.2. Time dependent equation for Nk(x, 0 In this subsection we derive the time dependent equations for the evolution of the Nk(x, t). More accurately, we transform the set of equations
dNk(x, t)/dt=r
t) --Oo(t)Nk(x, t ) ~ [ X - X o ( t ) ] ,
(5)
by changing the depth scale according to eq. (1). In eq. (5) the first term on the fight represents the change in Nk due to implantation, while the second represents loss due to surface erosion. The Dirac delta function is given by 8(x), while Vo(t ) = d x o / d t is the velocity of surface recession due to erosion. We note that eq. (5) assumes a spatially uniform erosion rate. The succeeding development is therefore not strictly applicable when the sputter formation of various surface structures (cones, pits, etc.) is significant. As remarked earlier, terms involving diffusion, atomic mixing, trapping, etc., may be added to the fight hand side for a more complete description. We ignore those terms in the present development, since our attention is focused here on the effects of high fluence implantation on the depth profiles. Our procedure involves only a change of variables which can, in principle, be easily extended to these other effects. Before transforming to the new depth scale let us also introduce a new dependent variable, namely n k ( z, t ), defined by
nk(z, t)Ns, d dz = Nk(x, t) dx.
(6)
Also, let us define M(x, t) by
M( x, t ) ~ d z / d x = Y~,kNk ( x, t )/Nst a.
(7)
k
From eqs. (6) and (7) we note that
nk(z, t ) = Nk(x, t ) / [ M ( x , t ) . Ns,d] ,
(8)
and that the n k are not independent since
~~.,kn~(z, t) = 1.
(9)
k
From eqs. (5) and (7) we also note that
d M ( x , t ) / d t = G ( x , t) -Vo(t)M(x, t)8(X-Xo(t)),
(10)
D.K. Brice / High fluence ion implantation depth profiles
where the source term G(x, t) is given by
G(x, t ) = ~_,,ktkk(t)Pk(x, t)/N~ta.
(11)
k
Utilizing eqs. (5) through (10), the transformation to the z depth scale, eq. (1) yields ~ ank
Onk
8--~ + S ( z , t ) " ~z- + nkH(z, t ) = Hk(z, t).
(12a)
and aM~t + J(z, t)__~_ z~M= M ( z , t ) H ( z , t) -Vo(t)M2(z, t)8(z).
(12b)
In eq. (12)
Hk(z, t ) = ~ k ( t ) Q k ( z ) / N s t d and
(13a)
H ( z , t) = ~_,c~Hk(z, t),
(13b)
k
while
J ( z , t)=foZH(z ', t ) d z ' - v o ( t ) M o ( t ) .
(13c)
123
greater depths (AJ > 0). For case 1, described below, z a represents the depth at which these opposing effects just balance (i.e., J(z a, t ) = 0). When the net implantation rate of "equivalent standard atoms" is smaller than the surface erosion rate (case 2 below) then J(z, t ) < 0 at all depths.
2.3. Relaxation As the implanted ion concentration increases the target material may undergo relaxation due to a buildup of stresses in the material. For an assumed planar geometry such relaxation will occur parallel to the depth axis, in our case parallel to x. Normally, such relaxation effects are included in a mathematical treatment such as ours by a change of depth scale which, for example, may maintain a constant total target atomic density. Alternatively, the relaxed depth scale may provide a known response to the induced stresses, but the net result is still simply a change of depth variable. All such relaxation processes are implicitly included in the depth scale of eq. (1). Thus, relaxation does not explicitly enter in to the time development of the n k nor of M when the latter is expressed as a function of z.
By M(z, t) we mean M[x(z, t), t], and M 0 ( t ) =
M[xo(t), t] = M(z = O, t). Eq. (12), with the auxiliary eq. (13), provide the basic general equations for the evolution of the Nk, within the procedures of MEAS. We note that these equations, eq. (12), are coupled and nonlinear, since J(z, t) depends on the surface value of M(x, t), i.e., on Mo(t). Time dependent solutions to the equations will thus normally require numerical methods. Inclusion of other transport effects, such as diffusion, will further complicate the coupling, but the essential features of the treatment remain the same. Note in eq. (13c) that vo(t)Mo(t) is the normalized rate of erosion of "equivalent standard atoms" from the target surface, while the integral on the right hand side represents the normalized rate of implantation of "equivalent standard atoms" on the interval (0, z). Note also that J(z, t) is a monotonic increasing function of z, with J(O, t)< O. Thus, if J(oo, t) is positive then there is a net accumulation of material in the target, i.e., the total implantation rate is greater than the rate of loss by sputtering. Similarly, J(0o, t) < 0 implies a net loss of material, i.e., the implantation rate is less than the sputter rate. The function J(z, t) represents a local streaming velocity in the z-coordinate. If one follows with time the location of the volume element, Va, containing a particular implanted atom, J represents the net velocity of Va resulting from surface erosion and implantation/ redeposition. Surface erosion causes Va to move toward the surface (A J < 0) while implantation between the location of Va and the surface causes Va to move to
2.4. Steady state solutions Although the general equations, eq. (12), normally require numerical methods for their time dependent solution, analytical steady state solutions to these equations are readily accessible. Thus, we can provide analytical formulas for the implantation profiles at fluences sufficiently high that the steady state is approached. We assume that for sufficiently large time the erosion rates approach constant values, so that
Vo(t)Nk(xo, t)/Ns, a ~ R~,
(14a)
and
Vo( t ) Mo( t ) --* R = )'~r
(14b)
k
We also assume ckk(t)~ ~kk = constant and note from eq. (13c) that
dJ/dz = H(z).
(15)
Setting the time derivatives in eq. (12) to zero, and utilizing eqs. (14) and (15), we obtain for the steady state for z > 0 d ( J . nk ) / d z = J dnk/dz + nk d J / d z = Ilk(z) (16a)
and J dM/dz = M dJ/dz.
(16b)
Solution to eq. (16) is formally straightforward. Some care must be exercised in applying the boundary conditions, however, depending on whether J ( ~ ) is positive
D.K. Brwe / High fluence ion implantation depth profiles
124
or negative. We thus define two cases. For case 1, J(oo) is positive, and for case 2, J(oo) is negative. The special case J ( 0 ) = 0 can be obtained as a limit from case 1. (See also the discussion at the end of section 2.2) For case 1, J(0) is negative and J(oo) is positive. Thus there exists a positive value of z, say z a, such that J(za) = 0 whj'ch implies that
f0=iH(z') dz'-- R. JCz)n,(z) =S(0)n,(0) + s n,(z') dz'. -J(O) =
(17)
For z < z~ we may integrate eq. (16a) to yield
(18)
Now, from eq. (18) we have at z = z~ nk
(0) = -- (1/J(O)) foZ'H~ (z i) dz'. = (1/R)foZ'H,(z ") dz'.
(19)
Substituting Eq. (19) into (18), and rearranging terms yields o
n~,(z) = - ( 1 / J ( z ) ) ~ ' " H , ( z ' )
dz'.
(20)
Note also that even though J(za)= O, n,(z,,) exists as a limit to eq. (20), and
, , ( z~) = H, ( z . ) / H ( z.).
(21)
Now, for z > z a the solution to eq. (16a) is
J(z)n,(z)=
fz]Hk(z') dz',
auxiliary conditions the value of z~ is defined by eq. (17), and the solutions follow as indicated. A second reason for placing the Vo on the left hand side is that both vo and N, are quantities defined on the unrelaxed x depth scale. Since the net implantation rate exceeds the net erosion rate for the present case (case 1) we know that in the steady state Oo---' 0 and Nk --* oo, while their product remains finite. Thus, it is improper in case 1 to separate the factors in these products. Finally, before leaving case 1, there is some temptation to replace R , by ~Tq~jSj,/Nsta where the $1~ are "high fluence sputtering coefficients". As we have noted, however, at this point this is simply replacing unknown Rs with unknown Ss. In the applications section we will deal with a case in which the Sj, are known, but that application will be a case 2 application. For case 2 the boundary values are the values of nk as z --* oo, which we take to be the large z values of n , in the original, unimplanted target material. Thus, for example, for an initial elemental target we would take no(oO ) = n o = constant = 1/Co, and n,(oo) = 0, for k 0. In general
n, (oo) = N, (o0,0)/[ m(oo, 0). Us,d],
(25)
where the right hand side is evaluated at x = oo, and t = 0. Then one has as solution to eq. (16)
nk(z)=(X/J(z))[J(oo)n,(oo)-(22a)
~~176
) dz'], (26a)
and or
M ( z ) = M(oo)J(z)/J(oo). nk ( z ) = (1/S( z ) ) fz]H, (z') dz'.
(26b)
(22b)
2.5. Other considerations The form given in eq. (22b) is identical to eq. (20), so that the single equation holds for all values of z. Care must be exercised however, in the vicinity of z = z~ where the limit of eq. (21) becomes applicable. A similar treatment of eq. (16b) yields
~
= I J ( z ) l-
(23)
The concentrations Nk(z) can now be obtained from eqs. (22b) and (23) using eq. (8). Thus
t,oN,(z) = f f ' H , ( z ' ) dz"
Us,d .
(24)
Eqs. (23) and (24) are each written with the multiplier v0 on the left hand side for two reasons. First, ooNk(O) represents the steady state sputtering flux density of k-type atoms and a knowledge of this rate from experiment, or some other theoretical treatment is required before the solution to the equations is complete. That is, other input determining the values of R and R~. is required which is beyond the scope of the present treatment. Once R = voMo is determined from such
2.5.1. Saturation It is often observed that implanted atoms of a given species can only be present in a target material up to some saturation level, say Nk=nskNstd, for atomic species k, where n~, = constant. That is, further implantation of species k into a region where the atomic density is already n~, N~td only provides a highly mobile form of k atoms which rapidly diffuse away from the region in question. In many cases these mobile atoms are lost from the target material [3]. The treatment given above for steady state high fluence implantations is easily extended to the case of saturable implants. Saturation can be shown to be accounted for by simply calculating the n , as if saturation were not a factor and then truncating n k in those regions for which nk/no> ns,. The truncation must also be a accompanied by a "renormalization" to maintain the relationship of eq. (9) and a proper calculation of J in the saturated region. This is equivalent to replacing H k by n~kH o in these regions.
D.K. Brice / Highfluence ion implantation depthprofiles 2.5.2. Diffusion and atomic mixing Diffusion can be added to the above treatment by including a term Tk on the right hand side of eq. (5) where
125
may, under certain circumstances, all be satisfied. At any rate, solutions in such cases are correct up to the time at which saturation effects begin to occur [7]. The general time dependent solution for J(z, t)--*J(z) is given by
nk(z,t)J(z)=J(u)nk(u,O)-fUH,(z')dz', (29) When transformed into the z-depth scale and the n, M variables, eq. (27) becomes T,. = (N, td- M ) ~ z [ M D ~ z ( n ~ M ) ] .
where u(z, t) is defined by
f Zdz'/J(z') = t.
(30)
(28)
Eqs.' (12) are altered to include diffusion by adding a term TA/[M. N~td] to the right hand side of eq. (12a) and a term ~.kckTk/Nstd to the right hand side of eq.
(12b). The inclusion of diffusion into eq. (12) clearly increases the coupling between the various dependent variables as indicated in eqs. (27) and (28). General solutions to the equations must normally be sought by numerical means, however, so that the additional coupling does not significantly add to the complexity of solving the equations. Explicit expressions for additional terms in eq. (5) to account for atomic mixing will not be given here. Inclusion of such terms is formally straightforward however, and consists of adding a term to the right hand side of eq. (5) which is proportional to the right hand side of Littmark's [6] eq. (1) (note that Littmark uses Pk to stand for the concentration which we call Nk). We note that Littmark uses Pk to stand for the concentration which we call Nk). We note two simplifications which will occur in the treatment of atomic mixing by MEAS in place of the standard treatment. First, with MEAS it will not be necessary to define a time (fluence) varying depth scale which keeps up with various relaxation processes. As already discussed, such relaxation is implicit in our depth scale, z. Second, the relocation function Uk(x, x') as defined by Littmark, when transformed into z, z' variables, will become independent of target composition and will, furthermore, be applicable to high fluence cases. As a result, the atomic mixing terms in the MEAS format will be significantly simplified over the corresponding terms in the usual treatment [6].
2.6. Time dependent solutions In general, time dependent solutions to eq. (12) must be obtained by numerical methods. Explicit expressions can be given however for the general solution to the equations when J(z, t) is independent of time (or is a very slowly varying function of time). This occurs when the fluxes, {~k(t)}, are all constants (and thus the H k are independent of t), the sputtering rate is constant, and saturation effects are not present. These conditions
In eq. (29) nk(u, 0) is simply the initial (t = 0) value of nk(z, t) while the other functions have already been defined. We note that if J(z) has a zero on 0 < z < oo, say at z = z,, then from eq. (30) either u and z are both greater than Za, or u and z are both less than z a and u --* z, as t --* oo. Thus eq. (29) limits to the steady state solutions for case 1. If J(z) has no zero on the positive z-axis then u(z, t) --* oo as t ---, oo and the steady state solution for case 2 is recovered. For M(z, t) we obtain
M(z, t) = S ( z ) M ( u , O)/J(u), (31) with the caveat that voM(z, t) remains finite for a case 1 solution.
3.
Applications
In this section we wish to demonstrate hat the approximations involved in MEAS are valid over a sufficient range of projectile types, target atom types, and incident energies to provide an adequate treatment of a variety of high fluence implantation problems of interest in fusion energy studies. In particular we wish to show that values for the (ok } can be obtained for a given target material which are essentially constant, i.e., independent of projectile type and energy over some useful range of projectile masses and energies. We then wish to show that with these given {( k }, the Qi(z), for several composite targets are well represented by the Q,(z) for appropriately chosen standard target materials.
3.1. Values of {(k} The TRIM particle transport code [5] has been used to calculate projected range profiles for 1 and 10 keV projectiles with 1 < Z~ < 32 incident on a variety of elemental and alloy targets with homogeneous spatial composition. The constituents of these targets were also chosen such tht 1 < Z2 < 32. This range of values for the atomic number Z~ and Z2 and the range for projectile energies are those of interest in plasma materials interactions for fusion energy applications. Table 1 contains values of Nk0Rpk obtained from the TRIM calculations with elemental target materials.
126
D.K. Brice / High fluence ion implantation depth profiles
Table 1 Average projected range NkoRpk from TRIM calculations with 1000 histories; units are 10 Is atoms/cm 2 Projectile
E(keV)
Target D
Ti
Ge
724.83 225.22 132.72 114.57 161.43 273.95
327.78 98.30 54.72 35.98 34.93 37.81
205.12 61.96 34.44 22.52 21.13 21.87
134.06 42.83 23.91 14.11 12.47 12.21
116.07 40.92 21.29 12.09 9.70 8.90
101.98 36.75 20.43 10.76 8.39 7.47
4738.63 2596.99 1437.51 891.59 981.13 1323.45
2565.56 1095.94 532.98 262.02 208.43 207.33
1782.01 643.64 311.24 147.68 115.48 105.46
1069.16 366.10 184.24 85.72 63.36 54.42
839.39 284.88 141.98 64.29 48.15 40.08
661.69 240.49 125.30 57.62 41.09 33.37
1.0
D Li C Si Ti Ge D Li C Si Ti Ge
10
Li
C
After selecting a standard target material values of { c k } can be calculated for each of the materials and energies listed in table 1 by using eq. (4). We note that the standard target material should be chosen to represent the average composition of some target of interest. Fig. 1 shows typical results obtained for ~ck) versus Z2(k ) where ( ) represents averaging over the various incident projectile types and energies. The error bars show the standard deviation of the calculated ok-values about the average, while the curve is a fitted polynomial to the < c k )-valuesF r o m fig. 1 it is seen that the ck are reasonably constant over a broad range of incident projectile types and energies. The variation of ck at the extreme target types amounts to - 15% for Ge targets and - 50% for the deuterium targets. While the standard deviation for
I
I
I
I
I
I
I 25
I 30
2
V 1
0 0
I 5
I 10
I 15
I 20
35
Z2 Fig. 1. Average values of c calculated from the data in table 1 using eq. (4). Standard deviation of c-values about the average shown as error bars. Solid line is fitted cubic polynomial.
Si
D targets would seem to be unacceptable, we note that " the ~k-values for D are quite small, i.e., - 0.1. Thus, the presence of D in a target of silicon-like atoms will have only a small effect on the development of the range profiles and a large uncertainty in the c A is thus acceptable. For targets in which D constitutes a significant fraction of the composition a different standard target is appropriate, and the resultant values of ~k for D will have a smaller range of variation. This is easily verified from the data of table 1.
3.2. Accuracy of MEAS Qi(z) In this subsection we wish to examine the accuracy with which the Q,(z) for a chosen elemental standard target material represent the corresponding profiles for composite targets of various types. Targets of equal amounts of C and D (i.e., C D targets), equal amounts of Ge and D (GeD) and equal amounts of D, C, Si, and Ti (TiSiCD) are considered for these comparisons. Fig. 2 shows a comparison of profiles Qi(z) calculated by M E A S (solid curves), using Ti as the standard atom with corresponding profiles calculated by T R I M for G e D targets (dashed curves) for 1 keV and 10 keV D, C, and Si projectiles. Comparisons for 1 keV are shown in fig. 2a and those for 10 keV in fig. 2b. The comparison for the two heavier projectiles is seen to be excellent at both energies while the M E A S profiles for D projectiles are somewhat less accurate, particularly at 10 keV. Similar accuracy is expected for implantations of these ions into inhomogeneous targets of Ge and D for which the D concentration does not exceed - 50%. We note that agreement between M E A S and T R I M calculations for D projectiles can be improved somewhat by choosing a different standard material, but the improvement will be at the expense of the profiles for the other projectiles. Choice of standard material thus
D.K. Brice / High fluence ion implantation depth profiles I
'N' I
I
I
l
I
I
I
I
I
(a)
(a) O(z) vs z
Olzl vs z
II I
I
~176 I
I
I
I
I
,.i
. I
I
z (arb. units) I
I
I
127
I
I
I
I
z (arb. units) I
I
I
I
]
I
1
I
I
I
I
I
1
(b)
Olz) Vl Z
~I
(b)
-.
O(z) vs z
9
"[i I
I|
\",.] I
!
I
I
I
I
J
\1
z (arb. units)
~
r l l//,
D
oo \',., \,
z (arb. units)
Fig. 2. Comparison of TRIM profiles for GeD (Dashed curves) and MEAS profiles (solid curves) obtained from the data of table 1 using Ti as standard target. Depth scales may be obtained by reference to Ti column in table 1. Curves shown for D, C, and Si projectiles. Tic marks on ordinate indicate decades, a) 1 keV. b) 10 keV.
Fig. 3. Comparison of TRIM profiles for CD (dashed curves) and MEAS profiles (solid curves) obtained from the data of table 1 using Li as standard target. Depth scales may be obtained by reference to Li column in table 1. Curves shown for D, C, and Si projectiles. Tic marks on ordinate indicate decades, a) 1 keV. b) 10 keV.
depends to some extent on the projectile of interest in a given application. Nonuniformity of accuracy in the Q, results from the fact that electronic stopping does not scale with projectile atomic number similarly to the scaling of elastic scattering. Fig. 3 shows results similar to those in fig. 2, but for C D targets using Li as standard atoms, and similar results for TiSiCD targets (Ti = standard atom) are shown in fig. 4. F r o m figs. 2 - 4 we see that the M E A S procedure can be expected to give reasonably accurate profiles Q~(z) for use in the time dependent equations of the previous section for 1 < [ Z t, Z2] < 32 with projectile energies in the range 0 < E < 10 keV.
3.3. An example calculation In a recent publication Wampler and Brice [8] used a preliminary version of M E A S to analyze experimental results for high fluence oxygen implantation into graphite. That data is reanalyzed here to present an example of the application of the equations developed here, and to provide an updated discussion of the data. In the experiment reported in ref. [8] oxygen ions were implanted at 1.5 and 3 keV into graphite at room temperature, and the amount of oxygen retained was monitored by 2 MeV 4He backscattering analysis. The retained oxygen was found to reach a steady state value
128
D.K. Brice / High fluence ion implantation depth profiles I
I
I
/Si
1
1
I ~'
I
I
1
(J)
O(z) vs z
/e
using carbon as standard target and fitting the resultant six data points to a polynominal in Z2. Since only oxygen is implanted, the carbon density in the x-depth scale remains constant at Nc. Further, from the measured sputtering coefficient we have voNc(0) = y o n c = ~ S ,
(32a)
and thus for the oxygen erosion rate we have v o N O (0) = ~ S N o ( O ) / N c ,
g,
(32b)
so that
_1
voM(O) Uc = tbS(1 + e o N o ( O ) / U c ) . It
I
lJ
I
~
1
1
I
I
Using eqs. (8) and (9), eq. (32c) becomes voM(O) Uc = ~ S / n c (0),
z (arb. units)
(32c)
(33)
so that eq. (14b) yields
'~:~~ C
Q(z)v$z
(b)
R = dpS/( n c (0) Nc ).
(34)
This is easily verified [10] to be a case 2 example so we have from eq. (26) nc(z) =J(oe)/S(z),
/
Ti Si C D I
I
I
I
k I
I
I
z (arb. units) Fig. 4. Comparison of TRIM profiles for TiSiCD (dashed curves) and MEAS profiles (solid curves) obtained from the data of table 1 using Ti as standard target. Depth scales may be obtained by reference to Ti column in table 1. Curves shown for D, C, and Si projectiles. Tic marks on ordinate indicate decades, a) 1 keV. b) 10 keV.
since nc(OO)= 1. Noting that J depends on nc(0) through R, eqs. (34) and (13c), one can set z to zero in eq. (35) and solve for n~(0). Doing that and substituting the value into the expression for R and for J one can then verify that J ( o e ) < 0, and that the example is indeed case 2. The resultant expressions for n c and n o from eq. (26a) then become n c ( z ) = 1/[1 + c o q ( z ) ] ,
(36a)
no(z) = q(:)/[1
(36b)
+,oq(z)],
and no(z)/nc(z
) = No(z)/N c
= q(z),
(36c)
where q(z)
of 1.6 • 1016 O / c m z and 2.7 x 1016 O/cm-' for the 1.5 and 3 keV implants, respectively. Surface erosion is significant in this system and measured values of the carbon loss rate at both 1.5 and 3 keV yield 1.1 _+ 0.1 carbon atoms removed per incident oxygen at the high fluence limit [9]. TRIM calculations yield oxygen depth profiles in graphite which are nearly Gaussian for the two different energy implants with projected range and straggling of 37.4 x 1015 C / c m 2 and 16.5 x 1015 C / c m 2, respectively at 1.5 keV and 72.6 x 1015 C / c m z and 30.8 • 1015 C / c m 2, respectively at 3 keV. Taking carbon as the standard target for this analysis, a calculation similar to that of fig. 1 using the data of table 1 yields an equivalency factor for oxygen of c o = 1.2. This value is obtained by calculating ( ( ( Z z ) ) for the data of table 1
(35)
= (I/S) f~~
z') d z'.
(36d)
In eq. (36) Q ( z ) is the low fluence depth profile for oxygen implantation into carbon we obtained above from the TRIM calculations. We also note that q ( z ) is a monotonically decreasing function of z. We have avoided using the normalization of Q(z) in the above expressions. As a result, the expressions also hold when saturation effects are important. In fact, if n~ represents the saturation value of the ratio N o / N c, then eq. (36) predicts that the steady state oxygen profile is saturated for all z less than z., where z, is defined by ,, = q(z~).
(37)
The retained oxygen can be calculated from eq. (36b) and is given by Io, where !o =
Ucf ~ n o ( z ) dz. Jo
(38)
D.K. Brice / High fluence ion implantation depth profiles Wampler and Brice [8] have argued that saturation effects are important in the present case. The saturation level can be obtained by adjusting ns, i.e., z, such that the experimental results are reproduced by eq. (38). The resultant value for n~ is 0.43. This compares with the value 0.38 obtained in the earlier analysis [8] using a preliminary version of MEAS in which oxygen and carbon atoms were considered as precisely equivalent, i.e., c0 ---, 1. Fig. 5 shows steady state depth profiles for 3 keV oxygen implants into carbon which illustrate the effects of projectile interactions with previously implanted projectiies along with the effects of a saturation limit. The solid curves show the MEAS predicted profiles when oxygen-oxygen interactions are taken into account while the dashed curves show profiles in the absence of such interactions. The profiles with flat tops correspond to the saturation limit just obtained from the comparison of I0, eq. (38), with the experimental results of ref. [8]. From the two solid curves one sees immediately the effects of both saturation and the oxygen-oxygen interactions. The saturation limit reduces the oxygen concentration in the near surface region giving the flat topped profile; the reduced near surface oxygen concentration for the saturated case allows deeper penetration of those oxygen projectiles which ultimately come to rest in the tail of the distribution. In contrast, the tails of the saturated and nonsaturated profiles for the noninteracting case are coincident. The large difference between the solid and dashed curves indicates the importance of including the oxygen-oxygen interactions in the profile calculations as is done here through MEAS.
'
3keY
I
!
I
O+-.,.-C
0
0
0.5
, 0
I 40
,
The method of equivalent atomic stopping (MEAS) has been used to derive the equations governing the evolution of high fluence implantation depth profiles. The equations include the influence of sputtering by the implanted species .and can be used in cases of multiple simultaneous implantations. It is indicated in the development how the equations can be extended to include the effects of diffusion, atomic mixing and trapping, which are straightforward exercises since MEAS involves only a change of variables. It is also noted that relaxation effects are implicitly included in the development. Steady state solutions to the equations in the absence of diffusion and atomic mixing are obtained. It is demonstrated that the approximations involved in MEAS are valid ( - 15%) in particular for projectile energies in the range 0 < E < 10 keV for cases in which the atomic number of projectile and target atoms are in the range 1 < [Z~, Z2] < 32. These sets of values are those which are of interest for plasma materials interactions in fusion reactors. An example showing the analysis of experimental data for high fluence oxygen implantations into carbon is presented to demonstrate the utility of the steady state solutions. The expressions given here provide, for the first time, a set of simple analytical equations for obtaining the depth profiles for high fluence ion implantation. While certain approximations are inherent in the MEAS procedures, the validity of the method can be verified by simple comparisons with calculations for spatially homogeneous targets as indicated here in section 3. The equations developed here are of broad applicability, and should be useful not only in fusion energy studies, but also in implantation metallurgy, high fluence damage studies, and other areas involving high fluence implantation. References
Z
0
4. Summary
1
1.0
Z
129
"I ----~--80
I~,
~
120
I 160
!00
x (angstroms) Fig. 5. Steady state depth profiles for 3 keV oxygen implantation into carbon illustrating saturation and projectile-projectile interaction effects.
[1] See, for example, J. Lindhard, V. Nielsen, M. Scharff and P.V. Thompson, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 33 (1963) no. 10; J. Lindhard, M. Scharff and H.E. Schiatt, ibid. 33 (1963) no. 14; J. Lindhard, V. Nielsen and M. Scharff, ibid. 36 (1968) no. 10; K.B. Winterbon, P. Sigmund and J.B. Sanders, ibid. 37 (1970) no. 14. [2] K.B. Winterbon, Appl. Phys. Lett. 31 (1977) 649. [3] See, for example, B.L. Doyle, W.R. Wampler, D.K. Brice and S.T. Picraux, J. Nucl. Mater. 93/94 (1980) 551; K.L. Wilson and A.E. Pontau, ibid. 93/94 (1980) 569; D.K. Brice, B.L. Doyle, W.R. Wampler, S.T. Picraux and L.G. Haggmark, ibid. 114 (1983) 277. [4] D.K. Brice, submitted to Nucl. Instr. and Meth. B17 (1986) 289.
130
D.K. Brice / High fluence ion implantation depth profiles
[5] J.P. Biersack and L.G. Haggmark,'Nucl. Instr. and Meth. 174 (1980) 257. [6] U. Littmark, Nucl. Instr. and Meth. B7/8 (1985) 684. [7] When only two atomic species are present in the target the truncation procedure described for the steady state solutions in a saturated region can also be used for the time dependent solution. This procedure has not been shown to be valid however when more than two atomic species are present.
[8] W.R. Wampler and D.K. Brice, J. Vac. Sci. Technol. A4 (1986) 1186. [9] E. Hechtl, J. Bohdansky and J. Roth, J. Nucl. Mater. 103/104 (1981) 333. [10] One can easily show from eq. (13) that when the initial target material is not present as one of the implanted atom fluxes the steady state value for J(0r is negative. Such cases will thus always be examples of case 2.