Royal Society Publishing


Here, we uncover three new nanoplasticity mechanisms, operating in highly stressed interstitial-rich regions in face-centred-cubic (FCC) metals, which are particularly important in understanding evolution of surface stress and morphology of a FCC metal under low-energy noble-gas ion bombardments. The first mechanism is the configurational motion of self-interstitials in subsonic scattering during ion bombardments. We have derived a stability criterion of self-interstitial scattering during ion embedding, which consistently predicts the possibility of vacancy- and interstitial-rich double-layer formation for various ion bombardments. The second mechanism is the growth by gliding of prismatic dislocation loops (PDLs) in a highly stressed interstitial-rich zone. This mechanism allows certain prismatic dislocations with their Burgers vectors parallel to the surface to grow in subway-glide mode (SGM) during ion bombardment. The SGM growth creates a large population of nanometre-sized prismatic dislocations beneath the surface. The third mechanism is the Burgers vector switching of a PDL that leads to unstable eruption of adatom islands during certain ion bombardments of FCC metals. We have also derived the driving force and kinetics for the growth by gliding of prismatic dislocations in an interstitial-rich environment as well as the criterion for Burgers vector switching, which consistently clarifies previously unexplainable experimental observations.

1. Introduction

In recent years, the evolution of surface stresses in various solids caused by bombardment of low-energy (0.5–2 keV) noble-gas ions has been actively investigated, aiming at applications of stress engineering at the nanoscale (Wirth 2007). For such applications, ion bombardments are often used to generate surface stresses of a few newtons per metre, which in turn control the stability of self-assembled surface nanostructures (Chan et al. 2007; Ahmed et al. 2010). When ion bombardment is carried out on an ultra-thin metallic film of approximately 10 nm thickness attached to a soft stretchable substrate, the induced stresses produce localized wrinkles of the film; these wrinkled patterns have potential applications in flexible electronics, making nanoparticle filters and variable optical gratings and adhesion, wetting and friction control of soft and/or bio material surfaces (Genzer & Groenewold 2006; Rahmawan et al. 2010; Sun et al. 2011). For the study of surface stresses, advances in nanoscale measurement and observation tools such as multi-beam optical stress sensor (MOSS; Stoney 1909; Floro & Chason 2001) and scanning tunnelling microscope (STM; Binnig & Rohrer 1985) have enabled accurate measurement of surface stress evolution and direct imaging of corresponding surface morphology generated by various ion bombardments. In addition, these experimental tools together with recent advances in high performance computing for molecular dynamics (MD) simulations have allowed us to carry out hybrid analyses of the nanoscale mechanisms of surface stress evolution under ion bombardments (Kalyanasundaram et al. 2006; Chew et al. 2011).

Employing a wafer curvature technique, Dahmen et al. (2003) found that the bombardment of single crystal copper with noble-gas ions of 0.8–2.2 keV produced compressive surface stresses that initially increase linearly with fluence until they gradually saturate. The saturated value of the compressive stresses depended on the ion mass and ion energy; the lighter the ion mass or the higher the ion energy, the larger the saturated value of the compressive stress. Chan et al. (2007) also observed that the 1 keV Ar+ bombardment of a 200 μm polycrystalline copper foil of 10 μm grain size caused the build-up of compressive stresses. When the ion beam irradiation was turned off, however, the compressive stress subsequently relaxed to a tensile stress state at room temperature. Chan et al. (2007) speculated that this unusual transition from compression to tension was caused by diffusion of vacancies and self-interstitial atoms (SIAs) towards the free surface with different mobilities. In contrast to the build-up of compressive stress during the 1 keV Ar+ bombardment on a copper surface, Chan et al. (2008) observed that the surface stress induced by 1 keV Ar+ bombardment on a platinum surface was tensile. However, the surface stress was still observed to become increasingly more compressive (or less tensile) when the surface was bombarded by lighter noble-gas ions. Here we have three distinct questions. (Qn1) Why does ion bombardment of a lighter noble-gas ion of same energy produce more compressive surface stress? (Qn2) Why is the sign of the platinum surface stress opposite to that of copper under the same Ar+ bombardment condition? (Qn3) Why does the relaxation of the copper surface stress induced by Ar+ bombardment make transitions from compression to tension, not to the pre-bombardment stress-free state?

Using an STM, Girard et al. (1994) discovered that the bombardment of 0.6 keV Ar+ ions at room temperature resulted in the formation of atomic monolayer adatom islands on the Cu(100) surface. The size distribution of the adatom islands indicated that islands of 2–4 nm initial size grew to larger size islands through movement of atomic steps caused primarily by surface atomic diffusion. Later, Busse et al. (2001) discovered that a single bombardment of a 1 keV heavy noble ion (Ne+, Ar+ and Xe+) on Al(111) caused the subsequent eruption of three adatom islands, each of 2–4 nm diameter at low temperature (e.g. 100 K); under continued bombardment, some of these islands merged together to form larger islands. However, the bombardment of light He+ bombardment on Al(111) created small vacancy islands instead of adatom islands (Busse et al. 2000). By contrast, for Pt(111), the 1 keV Xe+ bombardment generated only vacancy islands on the surface even though the ion was heavy. Petersen et al. (2003) also observed similar vacancy islands on Ir(111) under 0.5 keV Xe+ bombardment. To explain the adatom islands on Al(111), Busse et al. (2000) suggested a conceptual melt outflow model of ion bombardment, and tested the model later with MD simulations; these simulations, however, could not explain the excessive atoms in adatom island eruption (AIE) on Al(111) under 1 keV Xe+ bombardment. The eruptions of approximately 200 atoms per single-ion impact, as a cluster of three atomic monolayer islands, on Al(111) under 1 keV Xe+ bombardments were distinctly different from adatom yields of several atoms per ion impact on Pt(111) under relatively high-energy (4.5 and 6.5 keV) Xe+ bombardment (Morgenstern et al. 1999). The small adatom yields on Pt(111) resembled the MD simulation results of melt outflow model (Busse et al. 2001). Petersen et al. (2003) also observed melt overflow adatoms on Ir(111) under relatively high-energy (4, 10 and 15 keV) Xe+ bombardments, which subsequently grew to adatom islands surrounding the vacancy islands by surface diffusion at high temperature (880 K). These observations lead us to a second set of questions. (Qn4) Why do atomic monolayer islands of 2–4 nm diameter erupt under the bombardments of 0.6 keV Xe+ on Cu(100) and 1 keV Xe+ on Al(111), but not under 1 keV Xe+ on Pt(111) and 0.5 keV Xe+ on Ir(111)? (Qn5) How is the eruption of adatom islands controlled by the ion size and mass for energetically similar bombardments? The work of Bullough et al. (1991) offers some hints to these open questions; TEM observations showed that 10 keV Xe+ bombardment created a double layer composed of segregated vacancy- and SIA-rich sublayers below the ion-bombarded surface of Cu(100). The SIA layer underneath the vacancy-rich layer was highly populated by prismatic dislocation loops (PDLs) of mostly 2–5 nm diameters. (Qn6) Then, how is the double layer formed? (Qn7) Considering that the nominal surface stress observed by MOSS is the result of the subsurface stress distributions in the vacancy- and SIA-rich sublayers, what mechanisms in the double layer contribute to the surface stress evolution?

Three generic models supported by MD simulations are introduced in the following sections to answer the aforementioned questions: (i) double-layer formation (DLF) model, (ii) subway-glide mode (SGM) growth model of PDLs, and (iii) AIE model for the evolution of surface stresses under ion bombardment. For quantitative analyses of these models, we have carried out time-accelerated simulations of cooperative lattice distortions and defect motions under multiple ion bombardments, in contrast to conventional MD simulations of an event of a single-ion bombardment (Busse et al. 2001) or of displacement cascades under primary knock-on atom condition (Calder et al. 2010). For covalently bonded materials such as carbon or silicon, the stress generation and relaxation mechanisms during ion bombardment have been well investigated by MD simulations (Zhang et al. 2003; Kalyanasundaram et al. 2006, 2008). Kalyanasundaram et al. (2006) made MD simulations on the development of surface stresses on Si(100) under multiple 0.5 and 0.7 keV Ar+ bombardments in which the covalent-bonding crystalline silicon transforms to an amorphous phase. Such transformations typically generate volume expansion, and compressive stress builds up proportionally to the embedded ion density. By contrast, face-centred-cubic (FCC) metals under relatively low-energy-flux ion bombardments maintain the crystal structure by recrystalization of the melt zone due to high mobility of metallic-bonding atoms. The maintained crystal structure under such ion bombardments provides rich hierarchical mechanisms of subsurface atomic transport with various configurational motions of defects in the crystal structure.

2. Overview of the double-layer formation, subway-glide mode growth and adatom island eruption models

As introduced in §1, there are two distinct phenomena (DLF and AIE) in low-energy heavy ion bombardments on a number of FCC metal surfaces. The experiments of Bullough et al. (1991) showed that 10 keV heavy ion bombardments made extensive DLF near a surface of Cu(100), unlike the formation of monotonically compressive amorphous layer on Si(100) (Kalyanasundaram et al. 2006). Moreover, the experiments of Girard et al. (1994), Busse et al. (2000) and Petersen et al. (2003) showed AIEs on Cu(100) and Al(111), but not on Pt(111) and Ir(111), under 0.5–1 keV Xe+ bombardments.

Figure 1 shows schematics of DLF and AIE processes. Figure 1a exhibits a schematic of a single low-energy ion bombardment process at a FCC metal surface. The speed of 0.5–2 keV noble-gas ions, except He+, ranges from several to tens of the slowest shear wave speeds in FCC metals. Ion entry in this speed range creates sputtering, melting and recrystalizing processes along the hyper and supersonic trajectory (Calder et al. 2010) of the ion near the surfaces of FCC metals within approximately 10 ps (Nordlund et al. 1998). MD simulations (Kim et al. 2011) and STM observations (Morgenstern et al. 1999) show that the sputtering of a low-energy single-ion impact on a metallic substrate erodes the surface (A1 in figure 1a) and leaves a small number of adatoms, while the adatoms diffuse along the surface (A2 in figure 1a). Along the hyper and supersonic trajectory of the ion (B in figure 1a), the process of melting and recrystalization produces primarily vacancies with a small number of SIAs (Nordlund et al. 1998). By contrast, under multiple ion bombardments, collection of multiple hyper and supersonic ion-entry trajectories creates a vacancy-rich tensile sublayer, L(v) in figure 1b, while collection of SIAs scattered during the final subsonic entry and bouncing processes of the ion often constructs a segregated SIA-rich compressive sublayer, L(i) in figure 1b. These hyper and supersonic, and subsonic entry processes are considered to build up the double layer. The compressive stress in L(i) then drives multiple cooperative configurational motions of subsurface defects and triggers AIE (C in figure 1a). Regarding the atomic scattering and migration processes, SIAs migrate mostly through hierarchical configurational motions of defects such as consecutive atomic replacement sequences, condensation of SIAs to form PDLs and configurational glide of the dislocation loops (Arakawa et al. 2007; Matsukawa & Zinkle 2007) in the subsonic transport regime. By contrast, in the hyper and supersonic regime atoms undergo ballistic (or Lagrangian) displacement cascades creating a melt zone (Calder et al. 2010).

Figure 1.

DLF, SGM growth of PDLs and AIE during noble-gas ion bombardments. (a) A schematic of the process of low-energy single-ion bombardment and its by-products: A1, sputtered atoms; A2, adatoms on the surface; B, ion penetration with hyper and super-sonic speed; and C, subsonic SIA transport. (b) C1, a schematic of the DLF mechanism that separates into the vacancy-rich (L(v)) and the interstitial-rich (L(i)) region (top). C2, accumulation of compressive stress in L(i); C3, formation of an interstitial PDL (Q1), growth by gliding (Q2) and emission of PDL towards the surface to make AIE (Q3). (c) A STM topograph of single-ion impact on Al(111) at 100 K with 1 keV Xe+ with a fluence of 0.0075 ions nm−2 (courtesy of Busse et al. 2001).

The subsonic configurational transport processes depicted by C in figure 1a, can be further decomposed into three sequential processes C1, C2 and C3 as shown in figure 1b. The schematics in C1 illustrate generation of SIAs caused by the final-stage subsonic bouncing process of a single low-energy ion after the ion transitions from supersonic to subsonic. In the bouncing process, the ion often rebounds out of the substrate before the melting zone is recrystallized, while it is occasionally captured to be embedded during the closing process of the ion-entry path. The rebounding ions creates several SIAs each, and the long-range interactions between the ions and the SIAs often drive the configurational motions of the SIAs to build up a segregated layer of interstitial-rich region L(i), inducing compressive stress σ(i) in the layer as illustrated in C2. Then, the SIAs often condense on the lowest atom-insertion energy planes (111) to form a seed of an interstitial PDL at stage Q1 in C3. The dislocation loops grow by gliding a relatively long distance compared with their sizes along their Burgers vector directions. While they grow, the PDL seeds with their Burger vectors not parallel to the surface prematurely move out of the surface. On the other hand, those with their Burgers vectors parallel to the surface can glide a long distance to grow; this mode of gliding is termed SGM. Among the condensed PDL seeds, primarily SGM–PDLs grow and lead to AIE. Unlike PDLs in perfect crystals, we have found that interstitial prismatic loops in a SIA-rich environment can be driven by a uniform applied stress, i.e. without a stress gradient. In the following sections, we describe how the FCC crystallographic structure makes it possible for the SGM–PDL to have a peculiar gliding mobility under the driving force of the uniform compressive stress σ(i) that enables the loop to glide and grow in a SIA-rich region. The growth-coupled gliding rate of the dislocation loop is several orders of magnitude faster than their thermal diffusive motions (Arakawa et al. 2007; Matsukawa & Zinkle 2007). Once the loop grows to a critical size at Q2, the motion of the loop switches its direction (Wolfer et al. 2004), changing its Burgers vector by a partial cross-slip, and shoots to the surface, causing AIE at Q3. The AIE is therefore considered to be an effective mechanism of relaxing the sublayer compressive stress σ(i) by removing the subsurface SIAs.

Figure 1c shows AIE in a STM topograph of sparse 1 keV Xe+ impacts on Al(111) at 100 K with a fluence of 5.3×10−4 monolayer. Adatom islands composed of roughly 70 atoms each as well as several adsorbates (small white dots) can be distinguished (courtesy of Busse et al. 2001). The circle Embedded Image marks a possible grouping of three islands to represent one impact event. The specimen was preconditioned by cyclic cleaning with 1 keV Ar+ bombardments at 527 K in an ultra-high vacuum–STM to remove the oxide layer, followed by annealing at 800 K. Because only the PDLs created by the preconditioning on Embedded Image, Embedded Image and Embedded Image, for {(plane);[admissible Burgers vectors]}, can move parallel to the surface in L(i), such dislocation loops are hardly removed by annealing, and a large population of such dislocation loops are considered to be sitting in L(i). Then, subsequent Xe+ bombardments after the preconditioning are believed to activate motions of the PDLs to cause AIE (from Q2 to Q3 in C3 of figure 1b). Experimental results (Busse et al. 2001) show that the initial adatom yield of approximately 200 atoms per ion impact reduced to a few per impact as the fluence increased, indicating the fading of the preconditioning effect as the steady-state AIE by the Xe+ bombardments was approached.

In the following two sections, we verify the DLF, SGM growth and AIE models by investigating the multiple cooperative processes depicted by C1, C2 and C3 in figure 1b with MD simulations. We use the classical MD simulation package lammps (Plimpton 1995), where the metal–metal atomic interactions are governed by the embedded atom method potential (Foiles et al. 1986), while the Ziegler–Biersack–Littmark potential (Ziegler et al. 1985) is used to describe both the ion–metal and ion–ion interactions.

3. Evolutions of surface stresses and double-layer formation in super-cell ion bombardments

In this section, we carry out time-accelerated MD simulations of noble-gas ion bombardments on the (001) surfaces of approximately 6×6×7 nm3 FCC metal super cells to analyse cooperative motions of defects under multiple ion beam bombardments. We align the (001) free surface of the super-cell substrate to be normal (in the x3-direction) to the incoming ion, and enforce periodic boundary conditions in the in-plane (x1 and x2) directions. The substrate is divided into four regions to account for various effects arising from the finite thickness of our simulation model. The bottom layer of atoms is held rigidly, above which a five monolayer thick viscous damping layer is included to absorb the stress waves caused by the ion impact. Above the damping layer is a 10 atomic layer thick thermal bath maintained at the target temperature of 300 K using a Berendsen thermostat (Berendsen et al. 1984). The thermal bath equilibrates the temperature of the atoms above it.

Because the MD simulations require time scaling, we choose a time step of 0.1 fs for 104 iterations (1 ps) to resolve the details of initial impact dynamics during the initial stage of each ion bombardment. Thereafter, the system is allowed to freely equilibrate with a time step of 1 fs for 3×104 iterations (30 ps). Finally, we quench the entire system to the target temperature of 300 K for 3×104 iterations (30 ps) by subjecting the topmost layer to a Berendsen thermostat, acting in conjunction with the thermal bath layer. The bombardment cycle is subsequently repeated for the next ion impact. The in-plane stresses induced by ion implantation are calculated using the Virial theorem. The reported stress is taken over a 5-nm-thick region from the free surface of the substrate, and is time-averaged over the final 15 ps of the post-relaxation process. As our interest is in the motion of defects activated by the transient event of each bombardment not by the thermal fluctuations of the equilibrium state between the bombardment events, the choice of the 61 ps bombardment cycle period can accelerate the simulation time by a factor of 109–1011 compared with the experiments at relatively low temperature, 100–300 K (Busse et al. 2001).

(a) Saw-tooth profiles in the surface stress evolution

Our simulation results of 0.5 keV Ar+ bombardment on several FCC super-cell substrates shown in figure 2a reveal two distinct types of behaviour. For the bombardments of Ar+ on Pd, Pt and Au, our MD results show no significant compressive stress build-up in the substrates. By contrast, for the bombardments of Ar+ on Ni, Cu and Ag substrates, a saw-tooth-shaped stress profile is observed, consisting of a gradual build-up of compressive stress followed by an abrupt release of this stress. The abrupt stress release occurs within a single-ion impact, and is associated with the AIE of approximately 3.0-nm diameter area on the surface, as shown in figure 1c. Similarly, for the bombardments of Ne+, Kr+ and Xe+ on metal substrates, we find that the stress evolution patterns fall into one of these two types of behaviour, which leads us to two major questions: (Qn8) What is the origin of the two distinct behaviours? (Qn9) How can we develop a unifying physical picture based on fundamental material parameters to explain the stress evolution in the two cases? In what follows, we investigate the origin of the mechanism responsible for the stress evolution of the saw-tooth-shape profile, with an aim of providing answers to these basic questions.

Figure 2.

(a) Evolution of effective surface stresses of a MD super cell of 6×6× 7 nm3 during multiple 0.5 keV Ar+ bombardments on six FCC metals. Atomic configurations denoted by arrows show surface morphology of the Cu substrate just before (left) and right after (right) the abrupt release of compressive stress. The newly created AIE is enclosed by dashed lines (right). (b) STM topographs after ion bombardment of Al(111) at 300 K with a fluence of 0.423 ions nm−2 of 1 keV (i) He+, (ii) Ne+, (iii) Ar+ and (iv) Xe+ ions (courtesy of Busse et al. 2000).

Figure 2b shows dependence of AIE on the ions of He+, Ne+, Ar+ and Xe+ ions bombarded with 1 keV on Al(111) in a set of photos provided by Busse et al. (2000). While figure 2a exhibits dependence of surface stress evolution on substrate properties for a given ion Ar+, figure 2b illustrates ion-dependent behaviour of surface morphology for a fixed target substrate, Al(111). In §3a, we will consider primarily a size effect of the ion relative to the substrate lattice on the behaviour of ion bombardment—in particular, DLF.

(b) The criterion of double-layer formation

We have traced the development of the two distinct stress patterns to the distribution of interstitials and vacancies in the substrate. Figure 3a shows four frames of vacancy and interstitial distributions along the depth from the free surface, according to the labels (i)–(iv) on the stress profile in the inset of the frame (i). As shown in figure 3a for the case of Cu, the bombardments of Ar+ create a meta-stable vacancy-rich layer of randomly distributed vacancies and SIAs near the surface, and a segregated subsurface layer of SIAs. Within the SIA-rich layer, the number of SIAs including those condensed in PDLs increases with the build-up of compressive stress from (i) to (iii); the abrupt release of compressive stress in (iv), however, occurs concurrently with the removal of interstitial defects, thus leaving behind only the vacancy-rich layer. By contrast, no such segregation of interstitial accumulation, and hence negligible compressive stress build-up is seen during the bombardments of Ar+ on Pt(001), as shown in the inset of frame (iv) of figure 3a. Our simulations on other ion/metal configurations have further confirmed that the formation of well-delineated vacancy-rich and interstitial-rich layers ultimately lead to the development of the saw-tooth stress profile.

Figure 3.

(a) Distribution of ions, vacancies and interstitials along the distance (Z) from the surface for the bombardments of Ar+ on Cu at a fluence of (i) 8.97, (ii) 11.35, (iii) 13.45 and (iv) 14.05 ions nm−2. Negligible interstitial accumulation occurs for the bombardment of Ar+ on Pt, as shown by the representative plot in the inset in (iv). (b) Two possible ideal configurations (A) and after (B) of SIA formation during subsonic ion embedding processes. I1–I4 depict the SIAs scattered in the L(i) region where the compressive stress is accrued. Black bars denote ions; white bars denote vacancies; grey bars denote interstitials.

To understand the mechanisms responsible for the phenomena discussed earlier, we focus on the implantation process of the ion that impacts the substrate at hyper and supersonic speeds with respect to the slowest shear wave in the substrate. Typical ion speed for low-energy bombardment ranges from 27 (0.5 keV Xe+) to 139 (2.0 keV Ne+) km s−1; for example, the Ar+ speed for 0.5-keV bombardment is 49 km s−1, whereas the shear wave speed in metals of our interest spans from 1.2 km s−1 (Au) to 2.3 km s−1 (Cu). Once the velocity of the travelling ion reduces to subsonic, our MD simulations show that the ion moves into a tetrahedral site of the perfect metal lattice and displaces four lattice atoms in the process. Then, the scattering instability condition of the displaced atoms to become dispersed SIAs is determined by the configurational energetics of the interaction between the embedded ion and the scattered SIAs in the lattice.

The scattering instability of the SIAs generated by the embedding (or bouncing) ion can be determined by comparison between two simple static configurations, A and B, shown in figure 3b. If the total energy of the configuration A, EA, is greater than that of B, EB, i.e. EAEB>0, then the displaced atoms become unstable and scatter to be dispersed SIAs. Because EAEB=EIE(i), the scattering instability condition is then expressed as Embedded Image3.1where EI is the static embedding energy of the ion at the atom-removed tetrahedral sites, E(i) is the formation energy of n interstitials next to the embedded ion, K is the bulk modulus of the substrate, n=4 the number of displaced interstitial atoms, ΔVI the effective volume expansion caused by embedding a single ion at a tetrahedral site where four lattice atoms are removed, ΔV (i) is the effective volume expansion of embedding a SIA in the substrate and Embedded Image is the effective volume of the ion. In (3.1), the interstitial formation energy E(i) is denoted by the energy required to create the additional dilatation nΔV (i), overcoming the radial stress σrr=−KΔVI/VI at the ion/matrix boundary. The values of EI, K, ΔVI, ΔV (i) and r0 are evaluated by molecular statics simulations. EI is calculated by the difference between the total energies of the ion-embedded and the four-atom-vacant configurations of a molecular static super cell. The values of ΔVI and ΔV (i) are assessed by comparing the molecular-static lattice displacement fields with the linear elastic field of a dilatation centre. We approximate r0 by the distance between the embedded ion and the first nearest neighbour atom. The logic behind the formulation of the scattering stability criterion is based on two fundamental tenets of solid mechanics. One is the Kelvin theorem (Thomson 1883) that the elastic field generated by a sudden point load in an infinite medium, within the distance of shear wave propagation from the load point, is identical to the static elastic field caused by the point load. The other is the cavity expansion model of deep indentation by Bishop & Mott (1945), in which the front half elasto-plastic filed ahead of the deep-indenter tip is close to the field of a corresponding cavity expansion.

As shown in (3.1), whether SIAs are dispersed depends critically on the sign of ΔVI. If embedding of the ion produces a compressive stress field, i.e. ΔVI>0, then (3.1) shows that interstitials are always generated. By contrast, if the stress field is tensile, i.e. ΔVI<0, then interstitials may or may not be generated depending on the comparison between the ion embedding energy and the positive formation energy of the four SIAs. We have compared the values of EI and E(i) for the bombardment of several different ions on six FCC metal substrates in table 1. As shown in table 1, the correlation between E(i) and EI in (3.1) agrees with the occurrence or non-occurrence of saw-tooth-shape profiles in the stress evolution in every case of the simulations. Indeed, the saw-tooth-shaped stress profile appears, implying accumulation of SIAs to form the double layer, in every case of E(i)<0 as shown in table 1. Moreover, as predicted by (3.1), ion bombardments of Kr+ on Pd(001), Ne+ on Ag(001) and Xe+ on Pt(001) generate saw-tooth-shaped profiles of the stress evolution, despite the fact that both EI and E(i) are positive but EI>E(i), as shown in table 1. On the other hand, the ion bombardments of Ar+ on Pd(001), Pt(001) and Au(001), for which both EI and E(i) are positive and EI<E(i), do not produce saw-tooth-shaped stress profiles as predicted by (3.1), and shown in figure 2a and table 1.

View this table:
Table 1.

Summary of the parameters obtained from MD simulations for the bombardment of several different ions on six FCC metal substrates.

4. Subway-glide mode growth of interstitial prismatic dislocation loops and adatom island eruption during ion bombardment

In this section, we explain how SIAs in the compressive SIA sublayer condense to form an interstitial PDL and how the SGM–PDL grows until it reaches the stage of AIE. Figure 4a shows a cross-sectional view of Cu Embedded Image plane at the four instances (i)–(iv) shown on the stress–fluence plot in figure 3a. At stage (i), no significant structural distortion is observed in the crystal, indicating lack of interstitial or vacancy defects. While interstitial-vacancy pairs are continuously created at this stage by energetic ion bombardments in the sublayer L(v) (figure 1b), some of them annihilate each other and some of these newly formed interstitials in L(v) diffuse to the free surface. However, once the density of segregated SIAs in the sublayer L(i) increases at stage (ii), the compressive surface stress increases and the SIAs begin to cluster together resulting in nucleation of an interstitial PDL. The compressive stress level increases as the SGM–PDL grows. Beyond the critical size of the PDL at stage (iii), the loop suddenly shoots to the surface and the stress-free configuration of crystalline lattice is recovered at stage (iv).

Figure 4.

(a) Cross-sectional views of atomic configurations and atomic virial–stress distributions at four instances around in-plane stress builds up and release for bombardments of Ar+ on Cu(001). The moments are corresponding to four instances in figure 3a. (b) (i) A schematic of a SGM–PDL absorbing SIAs within s on (111); side-view schematics of the SGM–PDL sitting in a SIA sublayer (ii-a) before and (ii-b) after a virtual displacement of dx.

(a) Driving force and growing kinetics of an interstitial prismatic dislocation loop in a self-interstitial atom-rich zone

In this section, we derive the driving force applied on an interstitial PDL in a SIA-rich region as well as its growth and migration rates in the region. For the derivation, figure 4b(i) shows a schematic of a typical interstitial SGM–PDL sitting on a (111) plane and undergoing growth-coupled subway glide in 〈110〉. The schematic of the PDL sitting on the SIA-soaking PDL plane of normal n, i.e. 〈111〉, and its Burger vector b (green colour), i.e. 〈110〉, are also shown in figure 5b. The segment length of the PDL is denoted by l. We assume that the loop expands by absorbing SIAs within the distance s on the PDL plane (111), whereas it glides along its Burgers vector direction 〈110〉.

Figure 5.

(a) Upper row: three MD frames of formation and growth of an interstitial PDL at fluence of 9.35, 10.85 and 13.45 ions nm−2 (left to right); lower row: three MD frames of a dislocation cross-slip process in a Burgers vector-switching event of a PDL, observed within a single Ar+ bombardment; corresponding Thompson tetrahedron is included as an inset. (b) A schematic of the Burgers vector-switching process from the green arrow of PDL IJKL to the black arrow of PDL IJLM. The process requires migration of atoms in JKL to the triangular region ILM. (c) TEM images (20×20 nm2) of the PDL observed in Fe (i) before and (ii) after the Burgers vector switching (courtesy of Arakawa et al. 2006).

Shown in figure 4b(ii) is a schematic view, in Embedded Image direction, of the PDL undergoing growth-coupled subway glide in an environment of SIA density Embedded Image. The schematic in (ii-a) is the current configuration and that in (ii-b) is the configuration after the PDL makes a virtual motion dx. In the growth-coupled subway glide of PDL, the configuration not only makes the translational variation from (ii-a) to (ii-b), but also the loop grows to the final configuration by absorbing SIAs. Considering the energy variation from (ii-a) to (ii-b), as analogous to the J integral in fracture mechanics (Rice 1968), the energy difference between the volumes (A) and (B) in figure 4b(ii-a), Embedded Image, is released by the virtual motion dx. The released energy then activates the atomic rearrangements at the near field of the PDL for glide and growth. Therefore, the average driving force (or translational configurational force) per unit length of the dislocation line caused by the uniform stress in the Burgers vector direction is derived as Embedded Image4.1It is worth noting that the driving force is linearly proportional to σkk, Embedded Image and s. In ion bombardments, the stress and the SIA density are proportional to the fluence, and the driving force is proportional to the square of the fluence, which causes the loop to glide and grow in one direction faster and faster as fluence increases, as shown in the supplementary movie, S1.

For the PDL kinetics in ion bombardments at low temperature, we assume that the loop motion is mostly activated by the event of every ion impact, not by thermal fluctuations of the equilibrium states between the events of ion impact. Because the ion bombardments are random, we further assume linear kinetics as Embedded Image4.2where B denotes the PDL drag coefficient and ξ the fluence of the ion bombardment. Then, (4.1) and (4.2) provide the linear kinetics of a SGM–PDL in the SIA sublayer expressed as Embedded Image4.3where Embedded Image, Embedded Image and Embedded Image are normalized SIA density, in-plane biaxial pressure and random-bombardment kinetic constant which is independent of fluence ξ, where K is the bulk modulus. Linear elastic approximation of the substrate gives Embedded Image with Embedded Image and ν the effective Poisson ratio of the substrate. The normalized SIA density of the SIA sublayer can be expressed as Embedded Image, in terms of the normalized fluence Embedded Image, the SIA formation yield of the ion bombardment η, and the normalized thickness of the SIA sublayer Embedded Image.

In (4.3), as Embedded Image and Embedded Image are linear functions of Embedded Image, integration of (4.3) with s assumed to be constant provides the migration distance of the PDL as Embedded Image4.4Then, employing the interstitial absorption compatibility Embedded Image with a0 the lattice constant, integration of (4.3) after being multiplied by dl/dx gives the PDL size as Embedded Image4.5where Embedded Image, Embedded Image, Embedded Image and Embedded Image are normalized variables of the segment length of the PDL, SIA absorption distance, glide distance of the dislocation loop and threshold fluence at which the loop begins to glide and grow, respectively.

(b) Transition of the interstitial prismatic dislocation loop from subway-glide mode growth to adatom island eruption

Figure 5a shows Embedded Image view of resolved atomic configurations of a PDL in a SIA sublayer of Ar+ bombardment on Cu(001), revealed by MD simulations and filtered for visualization based on the centro-symmetry parameter (Kelchner et al. 1998). Burgers vector analyses of the distorted lattices reveal that the distorted region is a PDL. Ion bombardment first creates dispersed SIAs within the segregated interstitial-rich sublayer in the Cu crystal (frame 1) which, after repeated bombardments, condense to form a PDL (frame 2). The PDL consists of two parallel Shockley partial stacking faults on Embedded Image slip planes interconnected by an additional two parallel Shockley partial stacking faults on Embedded Image slip planes. This PDL is highly mobile along the direction of its common [110] Burgers vector. The PDL glides and grows. Once the dislocation loop reaches a critical size (frame 3), the loop suddenly moves rapidly in the [011] direction towards the free surface, resulting in a coherent shift of an atomic layer on the surface, i.e. AIE (see electronic supplementary material, movies S1 and S2). This final step requires a switch in the Burgers vector of the PDL from the [110] to [011] direction.

(i) Burgers vector-switching mechanism of the interstitial prismatic dislocation loop

Following the notation of the Thompson tetrahedron composed of Embedded Image, Embedded Image, Embedded Image and δ:(111) slip systems, we note that the dislocation segment IJ (or LK) in figure 5a has its Burgers vector in DC:[110] and is dissociated into two partials Dα:[121] and Embedded Image on the Embedded Image plane of the Thompson tetrahedron. Similarly, the segment JK (or IL) also has its Burgers vector in DC:[110], but is dissociated into Dβ:[211] and Embedded Image on the Embedded Image plane. Although the PDL IJKL of prismatic slip is composed of dislocation segments sitting on different slip planes, they have a common Burger vector orientated at an off-angle to the normal of the loop. When the stress in the substrate builds up to a critical value corresponding to a critical loop size, the high in-plane compressive stress triggers cross-slip of a segment of the loop to switch the Burgers vector from [110] to [011] direction. This cross-slip mechanism initiates with the formation of two Shockley partials Embedded Image and Embedded Image on the α-plane of the dislocation segment IJ (frame 4, figure 5a). The Shockley partial αB slips across the α-plane of the segment IJ and combines with Dα to form DB:[011], whereas Bα recombines with αC to form a full dislocation Embedded Image. The recombined full dislocation Embedded Image is free to glide on the loop cross-slipped plane δ:(111) (frame 5) and eventually slips across the complete loop plane (frame 6). This process transforms the slip planes of the IL and JK segments from the Embedded Image plane to the Embedded Image plane, and switches the Burgers vector of the dislocation segments IJKL from [110] to [011], as shown by the schematic in figure 5b. The entire process from frame 1 to frame 3 occurs within 0.7 ps. Upon its completion, the high in-plane compressive stress, stress gradient and image force of the dislocation loop act to shoot the loop in the [011] direction towards the surface. Once the shooting of the loop commences, the image force becomes increasingly dominant and the loop moves unstably towards the surface at speeds of 4.7 km s−1, which is comparable to the bulk wave speed of the material. Similar mechanisms have been identified in the Ar+ bombardment of Ni and Ag in figure 2. Recently, Arakawa et al. (2006, 2007) observed a similar Burger vector-switching event of a PDL in Fe, a BCC crystal, with TEM as shown in figure 5c.

(ii) Critical size of the interstitial prismatic dislocation loop for adatom island eruption

Next, we consider the resolved shear stress τres required to cause cross-slip of a segment of the PDL along its (111) plane (Orowan 1948) Embedded Image4.6where ec is the partial dislocation core energy, γSF is the stacking fault energy and bp is the Burgers vector magnitude of the partial dislocation. For cross-slip of the partial to occur, the ‘bow-out’ radius r must exceed lc/κ for a segment length lc of the cross-slipping partial, where κ is the loop shape factor. Note that κ=2 in the absence of surrounding edge dislocations of loop segments IM, JL and LM (bold-dashed lines). Hence, the critical loop size to initiate cross-slipping is obtained from (4.6) as Embedded Image4.7where Embedded Image with m3 and Embedded Image the x3 components of the normal of the PDL and the partial Burgers vector of the cross-slip. Equation (4.5) shows that the PDL size l grows with ξ4, neglecting ξ0. However, energetics for cross-slipping in (4.7) shows that the loop can only reach a critical size lc at a critical fluence ξc, beyond which its Burgers vector switches direction and the loop shoots to the surface. By equating (4.5) and (4.7), we obtain the critical size of the PDL lc and the critical fluence ξc at which the dislocation loop switches its Burger vector by the cross-slip. In figure 6, the PDL size l is plotted as a function of ξ with a solid curve for its growth (4.7) and dashed curve for its cross-slip limit (4.7), respectively, for Ar+ bombardments on Cu(001) and on Ag(001). In our super cell MD simulations, however, we directly get lc, h, η, ec and ξc, and evaluate the parameters κ and Embedded Image from (4.5) and (4.7) as summarized in table 2. The simulations also give the effective SIA absorption distance s close to the lattice parameter. This value is much smaller than the absorption distance made by thermal activation, because the PDL dynamically jumps to glide approximately ten times the lattice parameter per fluence ξ (ion nm−2). Then, the critical travel distance of the PDL, Embedded Image, is obtained from (4.4) and (4.5).

Figure 6.

Plots for a Burgers vector-switching criterion for an interstitial PDL in a SIA sublayer under ion bombardment: Solid lines, from (4.5), denote the growth of a PDL with respect to fluence, whereas the dashed lines, plotting (4.7), show the PDL edge length with respect to fluence. The cross points of the solid and dashed lines represented by solid circles stand for the critical PDL edge lengths and corresponding critical fluences for bombardments of 0.5 keV Ar+ on Cu(001) and 0.5 keV Ar+ on Ag(001).

View this table:
Table 2.

Summary of the parameters in use for the cases of forming PDLs.

5. Discussion

As described in the previous sections, in low-energy ion bombardment of FCC metals, the substitutional-embedding incompatibility of the ion in a tetrahedral site of the substrate lattice, EIE(i) in (3.1), determines the scattering instability of the SIAs generated by the ion embedding or bouncing process. Scattering of the SIAs produces a segregated compressive sublayer of SIA-rich state underneath the vacancy-rich sublayer created by hyper and supersonic entry of multiple ions, leading to DLF. This answers the questions (Qn6) and (Qn7) posed in §1. Whether DLF occurs or not is determined by the sign of the incompatibility, EIE(i). The incompatibility depends on not only the ion size relative to the lattice parameter of the substrate but also the surface energy of a tetrahedral void in the substrate and the nominal interface energy of the inserted ion. When DLF occurs, the net surface stress results from the compressive stress in the SIA sublayer and the stress in the vacancy-rich sublayer, which depends on the densities of vacancies and embedded ions in the sublayer. Our MD simulations show that the stress in the vacancy-rich sublayer is typically tensile. Therefore, if the double layer is formed and the compressive stress in the SIA sublayer dominates the tensile stress of the vacancy-rich sublayer, the net surface stress is compressive (e.g. 1 keV Ar+ on a Cu surface; Chan et al. 2007). However, if the incompatibility does not allow the ion to scatter SIAs during the subsonic entry phase, then the SIA sublayer does not appear and the vacancy-rich sublayer, i.e. only a single layer, is formed to have the net surface stress tensile in general (e.g. 1 keV Ar+ on a Pt surface; Chan et al. 2008). This answers (Qn2). Regarding (Qn1), the lighter ions with the same energy have higher speed and momentum, and can penetrate the substrate deeper to be readily trapped. This is an ion mass effect. Once trapped, a light ion such as He+ becomes an interstitial to generate a compressive stress by itself, whereas trapped heavier ions occupy substitutional lattice sites causing either compressive or tensile stress depending on its compatibility. In short, both mass and size effects of the ion contribute to the evolution of the surface stress.

Once the density of SIA becomes high in the SIA-rich sublayer under a continuous ion bombardment, SIAs are readily condensed to form a perfect but tilted interstitial PDL on a preferential crystallographic plane, (111). While vacancy PDLs are generally known to be partial and easily transform to strongly immobile stacking fault tetrahedra, small interstitial PDLs are perfect dislocations and highly mobile in their Burgers vector directions in thermally driven diffusion (Arakawa et al. 2006). In addition, both types of PDLs are conventionally regarded as sessile under uniform applied stresses, because the signs of dislocation segments facing each other in a loop are always opposite (de Koning et al. 2003). The loop may be driven only by pressure-gradient (Seitz 1950), or by thermal fluctuations (Arakawa et al. 2007; Matsukawa & Zinkle 2007). However, we have shown for the first time in (4.1) that absorption of SIAs into a PDL in a high SIA density environment creates a growth-induced configurational force for the dislocation loop to glide and grow under a uniform applied stress. The non-conservative gliding rate of the PDL associated with such a mechanism is several orders of magnitude faster than their thermal diffusive motions (Arakawa et al. 2007; Matsukawa & Zinkle 2007). It is usually too fast to be detected by TEM imaging. Instead, MD simulations offer a viable tool to track the evolution and motion of these PDLs during ion bombardment. The driving force on a PDL for growth-coupled gliding moves the loop under the activation of ion-impact events far from equilibrium under low temperature ion bombardments. At high temperature, the thermal activation should be added to the ion-impact activation.

In §4a, it is found that the interstitial PDLs are driven by compressive stresses to soak up surrounding interstitials. Therefore, once DLF is achieved, PDLs in the SIA sublayer grow by gliding. However, only the PDLs with their Burgers vectors parallel to the surface can undergo SGM growth to survive in the SIA sublayer. The total number of atoms in the SGM–PDL is thus a small fraction of all the SIAs dispersed by the ion, and net η the formation yield of SIA is much less than four per ion impact. The crystallography of FCC allows SGM growth of PDLs only for ion bombardments on (11k) with k an arbitrary integer for Embedded Image Burgers vector, on (100) for [011] and Embedded Image Burgers vectors, and on (111) for Embedded Image, Embedded Image and Embedded Image Burgers vectors. For effective SGM growth of PDLs, the tilt angle tolerance of the above admissible surface normal is estimated 1–2° considering their critical gliding distance compared with their critical size. Therefore, SGM growth of PDLs can be operative only in a small fraction of the ion-bombarded surface area of polycrystalline FCC metals (Chan et al. 2007, 2008).

Once the SGM–PDLs reach a critical size during ion bombardment, their Burgers vectors switch directions and they shoot to the free surface, abruptly releasing the in-plane compressive stress. Such PDL motion creates adatom plateaus on the surface (AIE) causing stress-induced roughening that can explain the experimentally observed eruptions of adatom islands during the ion bombardment of Al (figure 2b; Busse et al. 2000). The newly discovered AIE mechanism of SGM–PDLs answers (Qn4) and (Qn5) introduced in §1. This mechanism together with the criterion of DLF explains why, how and when the surface stress evolution shows saw-tooth profile if applied to a periodic super cell in MD simulations, answering (Qn8) in §3a. However, unlike MD super cell results, random events of AIE in both time and space would cause the surface stress to linearly increase at the initial stage, and gradually saturate towards the critical stress for AIE. Furthermore, when a polycrystalline FCC metal such as copper is bombarded with relatively large noble-gas ions, the majority of PDLs in the compressive SIA sublayer are believed to be small immature non-SGM PDLs. Therefore, the non-SGM PDLs in the SIA sublayer are driven by the pressure gradient along their Burgers vector directions owing to the stress distribution in the double layer and the near-surface image force. The driving force would make the PDLs move unstably out to the surface, and the rate of PDL ejection should be governed by kinetics. This kinetics of PDL ejection would relax much of the compressive stress in the SIA sublayer, when the ion beam irradiation is turned off. In turn, overall surface stress of the double layer would relax from compression to tension, as observed is the experiment of Chan et al. (2007). This answers (Qn3) of §1. In summary, three major mechanisms of DLF, SGM growth and AIE explain the puzzles seen in experimental observations reported over the past two to three decades. This finally answers (Qn9) of §3a. Although Nordlund et al. (1999) and Peltola & Nordlund (2004) observed coherent displacement of atoms in their MD simulations of a much higher energy (approx. 50 keV) single self-atom bombardment on Cu, Al or Au, it is different from AIE induced by low-energy multiple ion bombardments. The atomic displacement caused by a single high-energy self-atom bombardment creates large amount of subsurface vacancy clusters to account for the atomic displacement, whereas the low-energy AIE is generated by eruption of a PDL driven by the cooperative compressive stress of the SIA-rich layer as discussed earlier.

Regarding changes in surface morphology, ion beam irradiation has been widely used for surface modification or to induce microstructural changes in the material, depending on the ion energy level over the past 30 years (Murty 2002). Under low-energy heavy ion bombardment in the range of 0.5–2.0 keV, periodic surface patterns of self-organized nanodots and ripples are often created (Chan & Chason 2007; Munoz-Garcia et al. 2008). Sigmund (1969) and Bradley & Harper (1988) modelled that the formation of these surface nanopatterns are largely attributed to coupled effects of erosion and surface diffusion. However, the model failed to quantify the characteristics of the bombarded surfaces in many cases of low-energy ion bombardments, suggesting the presence of other active mechanisms than erosion and surface diffusion (Munoz-Garcia et al. 2006; Chan & Chason 2007; Norris et al. 2011). In particular, the stress measurement with MOSS has suggested that subsurface stresses and the PDL motions in the double layer may significantly influence the formation of surface patterns (Chan et al. 2008; Medhekar et al. 2009). Medhekar et al. (2009) showed that the relaxation of defect-generated stresses by surface undulations can be a significant driving force for ripple formation during sputtering of crystalline surfaces. Up to now, the origin of these stresses, their relaxation mechanism and how they are coupled to surface morphology evolution in crystalline metals have been unknown. In this paper, we have provided some answers to these questions; in particular, DLF and SGM growth of PDLs leading to AIE play significant roles in formation of surface nanopatterns during low-energy noble-gas ion beam bombardments of FCC metals. However, further mathematical modelling that includes the newly discovered mechanisms is needed to understand and control surface patterning of FCC metals with ion bombardments. In addition, research is also necessary to understand mechanisms of surface stress evolution in metals of other types than FCC.

6. Conclusion

Employing time-accelerated MD simulations and nano and micromechanics modelling, we have discovered three new mechanisms in subsurface atomic transport processes, responsible for evolution of surface stresses and morphologies on FCC metal surfaces under low-energy noble-gas ion bombardment. The first mechanism is DLF of a vacancy-rich tensile sublayer made by melting and recrystallization processes in the zone of hypersonic and supersonic ion-entry displacement cascade, and a SIA-rich compressive sublayer by subsonic scattering processes of SIAs resulted from ion embedding incompatibility. The second mechanism is the SGM growth of PDLs in the SIA sublayer, which allows large population of a few nanometre size PDLs to glide and grow in the SIA sublayer. The third mechanism is the Burgers vector switching of PDLs at a critical size to make AIE. These three new mechanisms have enabled us to understand previously unexplainable experimental observations in evolution of surface stresses and nanoscale surface morphology during low-energy noble-gas ion bombardments on FCC metal surfaces.

The newly discovered mechanisms in subsurface atomic transport processes suggest that the stress distribution in the subsurface double layer plays a critical role in controlling surface morphology evolution during low-energy ion bombardment. The new finding is in contrast to conventional model predictions that attributed the morphology evolution to competition between roughening by ion erosion and smoothening by surface diffusion (Sigmund 1969; Bradley & Harper 1988). More importantly, the PDL-mediated subsurface atomic transport phenomenon represents a new nanoplasticity mechanism that has never been perceived before. While PDLs have been previously reported to be sessile under uniform applied stress, we have shown that their mobility scales with the long-range uniform applied stress and the SIA density in a highly stressed SIA-rich environment. In addition, these loops are able to switch their burgers vector directions by dislocation cross-slip once they reach a critical size. We believe that such PDL motion mechanism, which ultimately removes point defects near the loop's travel path in a material, plays an active role in localization and delocalization of radiation damage in nuclear materials (Bai et al. 2010).


K.S.K., S.P.K. and H.B.C. acknowledge the support of the KIST through the hybrid computation program and the NSF through grants no. CMMI-0855853 and DMR0079964, and DOE-BES through the computational materials science network. The work of E.C. and V.B.S. was supported by the US DOE, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under award no. DE-FG02-01ER45913.

  • Received January 26, 2012.
  • Accepted April 23, 2012.


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