Angular dependence of sputtering yield of amorphous and polycrystalline materials

An analytical formula is developed for the evolution of angular dependence of sputtering yields by extending the theory of sputtering yield proposed by Sigmund. We demonstrate that the peak of sputtering yield at oblique incidence can be attributed to a balance between the increased energy deposited on the surface by incident ion which enhances the sputtering yield and the decreased depth travelled by recoil atom which reduces the sputtering yield. The predicted dependence of sputtering yield on the incident angle is in good agreement with experimental observations. (Some figures in this article are in colour only in the electronic version) Ion-induced sputtering is a subject of constant research by many scientists over the last few decades due to its wide application in the semiconductor industry, in surface analysis and deposition. The understanding of this phenomenon lies in the framework of Sigmund’s theory [1]. This theory was derived on the basis of the linear Boltzmann transport equation under the assumption of random slowing down in an infinite medium. For amorphous and polycrystalline targets, Sigmund revealed that the sputtering yield is proportional to the energy accumulated by ions on the surface. It was shown that this theory can be used successfully to predict energy-dependent sputtering yields for a wide range of energies and a variety of ion–target combinations [2–6]. Many surface features induced by ion bombardment, including ripple and nanodot formation are based on this theory [7–15]. However, one challenging problem associated with this process is the angle-dependent sputtering yield. According to Sigmund’s theory, the evolution of sputtering yield with ion energy E and incident angle θ measured from the surface normal is given by Y (E, η) = F(E, η), (1) where η = cos θ , = 0.042/(NU0), N is atomic density, U0 is surface binding energy, F(E, η) is energy distribution. This equation can be understood as the production of sputtered atom density (in unit of atoms per length) per bombarding ion and depth from which sputtered atoms come [1, 16]. By solving the linear Boltzmann’s equation under the assumption of an infinite medium using Thomas–Fermi cross section dσ = CmE−mT −1−m dT with m = 0 and C0 = 1 2πλ0a, where λ0 = 24 and a = 0.219, Sigmund obtained incidence dependent sputtered atom density F(E, η)/(πU0) and incidence independent depth 3/(4NC0) [1, 16]. The production of these two terms determines the sputtering yield (equation (1)). Assuming a Gaussian distribution of deposited energy distribution F(E, η), from equation (1) the normalized sputtering yield can be approximated as Y (E, η) Y (E, η = 1) = (cos θ) −fs , (2) where the exponent fs ≈ 1 ∼ 2, depending on the mass of ion and atom [1,5]. This means that sputtering yield increases with the incidence angle and goes to infinity for grazing incidence. 0022-3727/08/172002+04$30.00 1 © 2008 IOP Publishing Ltd Printed in the UK J. Phys. D: Appl. Phys. 41 (2008) 172002 Fast Track Communication It is well known from experiment that the sputtering yield reaches a maximum at an oblique incidence of about 70◦ and then approaches zero at θ = 90◦. Sigmund pointed out that this maximum sputtering yield at a certain glancing angle cannot be explained on the basis of the assumption of an infinite medium [1]. Although this subject is mostly of applied interest and has been intensively investigated over several decades [2, 17–20], angular dependence of sputtering yield is still not well understood. In this letter, starting with the recoil atom density [1, 16, 21], we show that the sputtered atom depth is proportional to the cosine of incident angle. The peak of sputtering yield can be attributed to a balance between two competitive effects: one is the deposited energy F(E, η), which increases with the incident angle and thus enhances the sputtering yield, and another is the sputtered atom depth, which decreases with the incident angle and thus reduces the sputtering yield. According to Sigmund’s theory [16, 21], the average number of recoil atoms passing through the surface plane with energy (E1, dE1) in the solid angle (Ω1, d 1) per incident ion is given by [21] Y = ∫ ∫ J (E1,Ω1) dE1 d 1, (3) where J (E1,Ω1) is the number of recoil atoms per unit energy and unit solid angle. Equation (3) gives the sputtering yield if we integrate over E1 cos2 θ1 > U , where θ1 is the angle between Ω1 and the outward surface normal, U/ cos2 θ1 is the surface binding energy. Following the approach suggested by Falcone and Sigmund [21], using power cross section with m = 0, J (E1,Ω1) is given by J (E1,Ω1) = 3F(E, η) 2π3 × ∫ ∞