Theory of improved resolution in depth profiling with sample rotation R. Mark Bradleya) Department of Physics, Colorado State University, Fort Collins, Colorado 80523
Eun-Hee Cirlin Hughes Research Laboratories, 3011 Malibu Canyon Road, Malibu, California 90265
~Received 3 January 1996; accepted for publication 19 April 1996! We advance a theory that explains why sample rotation during depth profiling leads to a dramatic improvement in depth resolution. When the sample is rotated, the smoothing effects of viscous flow and surface self-diffusion can prevail over the roughening effect of the curvature-dependent sputter yield and generate a smooth surface. If the sample is not rotated initially and the depth resolution declines, we predict that subsequent rotation leads to improved resolution. This phenomenon has already been observed experimentally. © 1996 American Institute of Physics. @S0003-6951~96!00126-X#
Off-normal incidence ion bombardment often produces periodic height modulations on solid surfaces.1–4 For incidence angles u less than a critical angle u c from the normal, the wave vector of the modulations is parallel to the component of the ion beam in the surface plane. The wave vector is perpendicular to this component for incidence angles close to grazing. It is now well-established that if the incident ions do not react chemically with the solid, these surface modulations form as a result of the curvature dependence of the sputter yield.1–3,5–8 Formation of surface ripples is problematic in a variety of applications, including secondary ion mass spectroscopy ~SIMS!, Auger electron spectroscopy ~AES!, and ion milling. SIMS is one of the most widely used techniques for dopant profiling of semiconductors, while AES is an important tool in the structural characterization of multilayers. In a typical SIMS or AES apparatus, the primary ions are incident at an angle uÞ0. Thus, as sputtering proceeds, ripples can be formed, and this leads to rapid degradation of the depth resolution. This is particularly problematic when SIMS or AES is used in conjunction with ion sputtering for depth profiling of modern thin-film materials and devices.9 Zalar first demonstrated that this problem can be overcome by rotating the sample about its surface normal as the depth profiling proceeds.10 Zalar rotation has subsequently been used by many other groups, who found that in many cases, the surface remains remarkably smooth as the solid is eroded.11 Moreover, Cirlin and co-workers have observed that if the sample is at first stationary and ripples are formed, subsequent rotation during erosion can lead to the production of a smooth surface.12 Sample rotation does not always suppress surface roughening, however. In some instances, the sample roughens while it is simultaneously eroded and rotated, albeit at a slower rate than it does when it is eroded without being rotated.12,13 There is presently no real understanding of these interesting and useful observations. In this letter, we will advance a theory that explains why sample rotation reduces or eliminates surface roughening during depth profiling of an amorphous solid. Our theory also accounts for the observations of Cirlin et al. mentioned a!
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above. Our theory uses the theory of surface ripple formation in the absence of sample rotation as a point of departure. We will close by suggesting a simple experimental test of our theory. Let us begin by considering a flat surface of an amorphous solid that is subjected to off-normal incidence ion bombardment. We choose stationary coordinate axes (x,y,z) with the unit vector zˆ normal to the surface. Let the unit vector along the ion beam direction of incidence be 2eˆ , and the polar and azimuthal angles of eˆ be u and f, respectively. ~The angle of incidence u is nonzero.! Finally, let h(x,y,t) denote the height of the interface above the point (x,y) in the x – y plane at time t. Naturally, h(x,y,t)5h 0 2 v 0 t, where h 0 is the initial height of the interface and v 0 is the speed of the surface while the solid is being eroded. A real surface is not perfectly flat initially, and this has a profound effect on the time evolution of the surface. For simplicity, we take h 0 50, so that h(x,y,0) is small for all x and y. Next, we introduce the Fourier-transformed height h(k,t)5 * dr exp(ik–r)h(r,t), where r[(x,y) and k [(k x ,k y ). In general, each Fourier component of the initial height h(x,y,0) will have a small, nonzero amplitude. Finally, let f k be the angle k makes with the x axis and k 5 u ku . At small enough times, each of the Fourier components h(k,t) evolves independently in time. For each kÞ0, we have1,7,8
] u h ~ k,t ! u 5 $ 2Fk2A @ G 1 cos2 ~ f k 2 f ! 1G 2 sin2 ]t ~ f k 2 f !# k 2 2Bk 4 % u h ~ k,t ! u .
~1!
Here, F5 g /(2 m ), where g is the surface tension and m is the viscosity; A5a v 0 sec u, where a is the ion range; G 1 and G 2 are dimensionless parameters that describe the curvature dependence of the sputter yield;1 and B is the surface selfdiffusivity. In Eq. ~1!, the first term on the right-hand side describes the smoothing effect of viscous flow,7,8,14 the second term arises because the sputter yield depends on the surface’s curvature,1 and the third term accounts for the effect of surface self-diffusion.15 For the sake of simplicity, we will neglect any variations in the material parameters with the depth.
3722 Appl. Phys. Lett. 68 (26), 24 June 1996 0003-6951/96/68(26)/3722/3/$10.00 © 1996 American Institute of Physics Downloaded¬07¬Jun¬2006¬to¬129.82.140.58.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://apl.aip.org/apl/copyright.jsp
FIG. 1. Plot of R n (k) ~solid curve! and R rot(k) ~dashed curve! vs k for F c,2,F,F c,1 and u , u c .
According to Eq. ~1!, u h(k,t) u 5 u h(k,0) u expuR(k)t u , where R ~ k! 52Fk2A @ G 1 cos2 ~ f k 2 f ! 1G 2 sin2 ~ f k 2 f !# k 2 2Bk 4 .
~2!
Since G 1 and G 2 are always negative,1 there may be k’s with positive R(k). If R(k).0 for a particular k, the amplitude of this mode grows exponentially in time, while this amplitude decays to 0 if R~k!,0. The surface is therefore unstable if R~k!.0 for any k. If R~k! is positive for more than one k, the amplitude of the mode with the largest value of R rapidly surpasses the amplitudes of the other unstable modes. Therefore, the wave vector k with the largest positive value of R is the experimentally observed ripple wave vector. In the absence of viscous flow ~i.e., for F50!, ripples always begin to develop.1 For u less than the critical angle u c , the ripple wave vector is parallel to the projection of eˆ onto the x – y plane. These conclusions still hold true if F is nonzero but is smaller than F c,1[2B(A u G 1 u /3B) 3/2. 7,8 The dependence of R n (k)[R(k, f k 5 f )52Fk2AG 1 k 2 2Bk 4 on k for F,F c,1 and u , u c is shown in Fig. 1. The observed ripple wave vector is k * . On the other hand, there are no values of k with positive R~k! if F.F c,1 . No ripples develop in this case; instead, the surface becomes smoother during ion bombardment. This type of behavior has been observed experimentally.7,8 We now turn to the effect of rotating the sample about the z axis during bombardment. This is completely equivalent to rotating the ion beam and holding the sample fixed. We shall adopt the latter viewpoint because it makes the analysis more transparent. Equation ~1! still holds, but with f 5 v t1 f 0 , where f 0 is the initial value of f and v is the angular velocity. For simplicity, we put f 0 50. The solution to Eq. ~1! in this case is u h ~ k,t ! u 5 u h ~ k,0! u exp@ R rot~ k ! t #
3exp
S
D
A ~ G 2 2G 1 ! 2 k sin@ 2 ~ v t2 f k !# , 4v
~3!
where R rot(k)[2Fk2 21A(G 1 1G 2 )k 2 2Bk 4 . Note that u h(k,t) u either grows or decays exponentially in time, depending on the sign of R rot(k). The second exponential factor in Eq. ~4! simply superimposes a periodic oscillation upon this overall exponential increase or decrease. Let us suppose that F,F c,1 , so that ripples are formed in the absence of sample rotation. We also assume
that u , u c for the remainder of the letter. In this case, G 1 ,G 2 ,0.1 Since R n (k)2R rot(k)5 21A(G 2 2G 1 )k 2 .0, we have R rot,R n for all values of k. Let F c,2 [2B(A u G a v u /3B) 3/2, where G a v [ 21(G 1 1G 2 ). Note that F c,2 ,F c,1 since u G a v u , u G 1 u . If F.F c,2 , then R rot(k) is negative for all k ~Fig. 1!. This means that even though ripples would form in the absence of sample rotation, when the sample is rotated, there is no unstable mode. Thus, sample rotation suppresses ripple formation if F c,2,F ,F c,1 , and ultimately produces a smooth surface. On the other hand, if the sample is initially bombarded without being rotated, ripples are formed. If we subsequently rotate the sample while continuing to bombard it, the surface will become smooth since the ripple amplitude decays exponentially in time. Both of these conclusions are in complete agreement with the experimental results of Cirlin et al.9,12,13 If F,F c,2 , there are k values with R rot(k).0, and therefore the surface roughens. This roughening proceeds more slowly than it would if the sample were not rotated, since R rot(k),R n (k) for all k. This is in accord with the experimental observations of Cirlin et al.,9,12,13 who in some instances find that roughening is slowed but not prevented by sample rotation. Note that a periodic surface modulation is not formed when the sample is rotated and F,F c,2 , since there is rotational symmetry about the z axis. The nature of the surface roughening in this case is influenced by nonlinear corrections to Eq. ~1!,16 and will be discussed in detail elsewhere.17 From a heuristic standpoint, why does Zalar rotation suppress surface roughening when F c,2,F,F c,1? Suppose for the sake of argument that the surface initially has the sinusoidal height variation given by h(r,0)5A cos kx, where A is a constant. Suppose further, that the azimuthal angle of the ion beam f50 and the sample is rotated with angular velocity v, so that the ripple wave vector k 5(cos vt,sin vt,0). Whenever k lies along the x axis, the amplitude A increases. However, when k is close to being perpendicular to the x axis, A decreases because the smoothing effects of viscous flow and surface self-diffusion prevail over the roughening effect of the curvature-dependent sputter yield. This smoothing more than compensates for the roughening that occurs when k is nearly aligned with the x axis, and as a result, A decays to zero as the solid is sputtered. Our work would be of little more than academic interest if the effects of ion bombardment induced viscous flow were always negligible or if F c,2 and F c,1 were very nearly equal. Let us discuss these two possible objections to our work in turn. Chason and co-workers have shown that the effects of viscous flow can be important during ion bombardment of SiO2 and Ge surfaces.7,8 In fact, their experiments demonstrate that viscous flow can suppress the formation of surface ripples altogether, and so cases are known in which F is greater than F c,1 . Now let us consider the relative magnitudes of F c,2 and F c,1 . Unfortunately, G 1 and G 2 have not yet been measured for any material. As a result, we cannot directly test our prediction that ripples are suppressed when the sample is rotated and F c,2,F,F c,1 . However, Bradley and Harper1 give theoretical expressions for G 1 and G 2 which may be used to get an estimate of F c,2 /F c,1
Appl. Phys. Lett., Vol. 68, No. 26, 24 June 1996 R. M. Bradley and E.-H. Cirlin 3723 Downloaded¬07¬Jun¬2006¬to¬129.82.140.58.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://apl.aip.org/apl/copyright.jsp
5 u G a v /G 1 u 3/2. Bradley and Harper’s theory is based upon the Sigmund theory of sputtering,18 in which the energy deposited by an incident ion is taken to have a Gaussian spatial distribution. This distribution is characterized by three parameters: a, the average depth of energy deposition, and a and b, the widths of the distribution parallel and perpendicular to the beam direction, respectively. For example, suppose that a 5a/2, b 5 a /4, and u560°. Using Bradley and Harper’s expressions for G 1 and G 2 , we find that F c,2 /F c,1 50.485... . This shows that F c,2 be substantially smaller than F c,1 . There is a simple experimental test of our theory. Suppose that for a particular choice of crystalline semiconducting sample, ion beam, and temperature, the surface remains smooth as it is simultaneously rotated and sputtered. This means that F.F c,2 . In this case, the ion bombardment produces an amorphous surface layer where viscous flow takes place. As we have shown, it is the combined effect of surface self-diffusion and viscous flow that prevents roughening. If the sample temperature is elevated sufficiently, however, the sample will remain crystalline.7 As a result, viscous flow is suppressed and F becomes negligibly small—F will certainly be less than F c,2 . The surface will therefore roughen. Thus,
our theory predicts a transition from a smooth to a rough interface as the sample temperature is increased. 1
R. M. Bradley and J. M. E. Harper, J. Vac. Sci. Technol. A 6, 2390 ~1988!. 2 K. Elst and W. Wandervorst, J. Vac. Sci. Technol. A 12, 3205 ~1994!. 3 S. W. MacLaren, J. E. Baker, N. L. Finnegan, and C. M. Loxton, J. Vac. Sci. Technol. A 10, 468 ~1992!. 4 See also Refs. 1–7 cited in Ref. 1 and Refs. 1–8 in Ref. 2. 5 R. M. Bradley and J. M. E. Harper, Def. Diffus. Forum 61, 55 ~1988!. 6 G. Carter, I. V. Katardjiev, and M. J. Nobes, Def. Diffus. Forum 61, 60 ~1988!. 7 E. Chason, T. M. Mayer, B. K. Kellerman, D. T. McIlroy, and A. J. Howard, Phys. Rev. Lett. 72, 3040 ~1994!. 8 T. M. Mayer, E. Chason, and A. J. Howard, J. Appl. Phys. 76, 1633 ~1994!. 9 E.-H. Cirlin, Thin Solid Films 220, 197 ~1992!. 10 A. Zalar, Thin Solid Films 124, 223 ~1985!. 11 For a recent review, see Ref. 9. 12 E.-H. Cirlin, J. J. Vajo, R. E. Doty, and T. C. Hasenberg, J. Vac. Sci. Technol. A 9, 1395 ~1991!. 13 E.-H. Cirlin, J. J. Vajo, and T. C. Hasenberg, J. Vac. Sci. Technol. B 12, 269 ~1994!. 14 W. W. Mullins, J. Appl. Phys. 30, 77 ~1959!. 15 W. W. Mullins, J. Appl. Phys. 28, 333 ~1957!. 16 R. Cuerno and A.-L. Barabasi, Phys. Rev. Lett. 74, 4746 ~1995!. 17 R. M. Bradley ~unpublished!. 18 P. Sigmund, J. Mater. Sci. 8, 1545 ~1973!.
3724 Appl. Phys. Lett., Vol. 68, No. 26, 24 June 1996 R. M. Bradley and E.-H. Cirlin Downloaded¬07¬Jun¬2006¬to¬129.82.140.58.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://apl.aip.org/apl/copyright.jsp
