Sputter Roughening Instability on the Ge(001) Surface: Energy and Flux Dependence
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Mat. Res. Soc. Symp. Proc. Vol. 396 ©1996 Materials Research Society
where
d Ih(q,t)12 /dt = Rq Ih(q,t)12 + aC
(2a)
Rq = - Fq + Sq2 - Dq 4
(2b)
The constant term, c, which is independent of spatial frequency describes the stochastic process of random atom removal. The positive Sq2 term in Rq is derived from the Sigmund theory and indicates the dependence of the sputter yield on the curvature. It has the following dependence on the sputtering parameters [I]: S = - fa/p Y0 (0) (1 1 cos 2 (Qp) + 1F2 sin 2 (Qp))
(3)
where f is the ion flux, a is the range of the ion in the solid, p is the atom density, 0 is the angle of incidence and Y 0 (0) is the sputter yield on a flat surface. (pis the azimuthal angle between the incident ion direction and the surface wavevector, q. F1 and F2 are dimensionless coefficients that describe the curvature dependence of the sputtering rate and depend on the range and lateral extent of energy deposition and on the angle of incidence. The negative terms in Rq lead to smoothing of the surface by the processes of viscous flow (F) and surface diffusion (D). The rates of smoothing, driven by surface energy minimization, have been calculated by Mullins [6] and Herring [7]: F = y/Yj
(4)
where y is the surface energy and i" is the coefficient of viscosity, and D = 2Dsyfl2 v/kT
(5)
where Ds is the surface diffusion coefficient, fl is the atomic volume, and v is the number of molecules per unit area of surface. Integrating this equation yields the time dependent behavior of h(q,t): jh(q,t)12 = ho(q) 2 exp(Rq t) + (ct/Rq ) (exp(Rq t) - 1)
(6)
where ho(q) is the initial roughness spectrum. The sign of Rq in eq. 6 determines whether a particular spatial frequency will grow or decay during sputtering. Because the surface remains crystalline during sputtering at the temperatures investigated in this work, we can ignore viscous flow smoothing and set F=O. In this case, Ro is positive for all q < (S/D) 1/2, with a maximum value of R* = S/4D when q* (S/2D)'/½. The existence of a critical wavevector (q*) that grow faster than all others is consistent with the development of a pronounced ripple morphology on the surface which has been observed experimentally [2,3,4,8]. EXPERIMENT We use X-ray reflectivity to measure the evolution of the surface roughness during sputtering. This technique allows us to measure the roughness in real time during the sputter bombardment with a resolution of 0.05 nm; for details of the measurement technique, see refs. 9 and 10. All the results reported in this work are for Ge (001) surfaces and Xe ions. The sputtering is performed in a UHV vacuum chamber mounted on the X-ray system. The sputter 144
source is a Kaufmann style gun and the chamber is backfilled during sputtering to a pressure of 10-5 torr. The ions are incident on the sample at an angle of 550 from normal and along a azimuth. The surface roughness was obtained by fitting the XRR spectra using an optical multilayer theory [9]. The roughness was assumed to have a gaussian form. The measured mean square
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