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Surface nanopatterning mechanisms by keV ions: Linear instability models and beyond
Nuclear Instruments and Methods in Physics Research B 272 (2012) 178–182
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Surface nanopatterning mechanisms by keV ions: Linear instability models and beyond Eric Chason ⇑, Vivek Shenoy Brown University, Division of Engineering, Providence, RI 02912, USA
a r t i c l e
i n f o
Article history: Available online 1 February 2011 Keywords: Self-organization and patterning Surface structure Ion beam processing Sputtering
a b s t r a c t Sputtering solid surfaces with low energy ions is well-known to induce a wide array of nanoscale pattern forming behavior (sputter ripples). A simple continuum model originally developed by Bradley and Harper (BH) provides a useful framework that can qualitatively explain multiple types of patterning behavior seen experimentally and their relationship to the ion beam and material parameters. The basis of the model is a dynamic competition between roughening by ion bombardment and smoothing by surface transport that leads to the growth of roughness with a preferred periodicity on the surface. However, there are many experimental results that cannot be accounted for within this framework and additional physical mechanisms are discussed that may close the gap in our understanding. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Low energy ion bombardment of solid surfaces induces a surprisingly complex array of morphology evolution that depends on the ion beam, processing conditions and material being bombarded. A compendium of different observed behavior can be found in several recent reviews [1–4]. For different materials and conditions, the surface can spontaneously develop highly uniform waves (sputter ripples) or arrays of nanoscale quantum dots over large areas. The alignment of the pattern may be controlled by the direction of the ion beam or the crystallographic orientation of the surface. The surface roughness can grow exponentially or algebraically with time, or in some cases refuse to roughen at all. This wide variety of behavior has created intense interest in understanding its origin. Because it is self-organizing, ion-induced patterning holds promise as an inexpensive method for creating nanoscale structures over large areas, e.g., for magnetic storage [5] alignment of liquid crystals [6], optoelectronic materials [7] or enhanced catalysis [8]. Equally important, understanding the processes that control ripple formation provide a window into the non-equilibrium kinetic processes that occur under a combination of energetic particle bombardment and defect-mediated transport. These processes are critical for understanding the formation and stability of nanoscale structures and for the behavior of materials subjected to high flux environments such as in nuclear reactors. Much of our understanding of ripple formation has come from a linear instability model proposed by Bradley and Harper (BH) [9] ⇑ Corresponding author. Tel.: +1 401 863 2317. E-mail address: [email protected] (E. Chason). 0168-583X/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2011.01.060
that considers the surface evolution in terms of the kinetic balance between roughening and smoothing processes. This approach is at the core of many models of ion-induced pattern formation and it provides a useful and intuitive framework for considering different forms of pattern formation. Many aspects of ripple formation, such as the transitions between different types of behavior under different conditions, can be understood within this picture. However, systematic studies of ripple formation kinetics over recent years have exposed significant shortcomings of this approach, e.g., the growth rate of ripples has been found to be significantly faster in experiments than predicted by the theory. Furthermore, other aspects of ripple formation (such as amplitude saturation, the formation of quantum-dots and the angular dependence of the patterning behavior) fall outside the scope of the linear theory. This indicates that other mechanisms or approaches are needed beyond the simple linear instability model. In the current work, we review different types of ripple formation behavior that are seen on metal surfaces and use a kinetic phase diagram to show how the different behaviors are related to the regimes of flux and temperature where they are observed. The main features of the BH model are described in order to relate the wavelength and growth rate to the underlying surface kinetic parameters and the results are compared with experiments on Cu(0 0 1) and kinetic Monte Carlo simulations. Finally, we discuss some alternative mechanisms or approaches that may be able to explain observations that can’t be explained by the linear theory. 2. Types of patterning Ion-induced surface patterns have since been seen in a large number of materials systems including semiconductors, insulators
E. Chason, V. Shenoy / Nuclear Instruments and Methods in Physics Research B 272 (2012) 178–182
179
guished from the athermal BH regime by the fact that the ripple wavelength depends on the temperature in BH regime. 2.2. ES instability
Fig. 1. Kinetic phase diagram showing regimes of pattern formation observed on Cu and Ag (001) surfaces at different fluxes and temperatures.
and metals. Since the production of sputter ripples is a kinetic phenomenon (i.e., the pattern is not thermodynamically stable), the type of behavior can be modified by changing the processing conditions. To illustrate this, we present the results from a number of studies on Cu and Ag in a kinetic phase diagram in Fig. 1 The patterning is categorized into one of several characteristic types of behavior that we refer to as BH instability, ES instability, athermal BH/kinetic roughening and non-roughening. The different symbols represent data from different studies that are listed in Table 1 [2,10–16]. Although not all the measurements were done under the same conditions the boundaries roughly delineate regimes of temperature and flux in which the different types of behavior are observed The different types of behavior have different characteristic features that give insight into the physical mechanisms that are controlling their formation: 2.1. BH instability These types of ripples are named for the fact that they are explained by the original BH model. They have several distinct features [3]: the characteristic surface periodicity remains constant during the growth, the orientation of the pattern is determined by the direction of the ion beam and the amplitude grows exponentially (in the early stage before saturating). In addition, the orientation of the surface wavevector can be parallel or perpendicular to the ion beam direction, depending on the ion’s incident angle. The dependence of the ripple orientation on the ion beam direction indicates that their formation is dominated by the ion-surface interaction. Originally seen only on semiconductor and insulator surfaces, they have been observed on the Cu surface when the temperature and flux are sufficiently high, i.e., the upper right hand portion of the diagram in Fig. 1 The BH instability regime is distin-
Table 1 List of the experimental conditions and the source reference for the data represented by the different symbols in Fig. 1. Symbol s e
r D h
Surface
Sputtering conditions
References
Cu(0 0 1) Cu(0 0 1) Cu(0 0 1) Ag(0 0 1) Ag(0 0 1)
Ion Ar+ Ar+ Ar+ Ar+ Ne+
[10–12] [13] [14] [15,16] [2]
h 70o 45o 30o 0o 70o
Energy (eV) 800 400–3000 350 1000 1000
The label ES instability refers to the Ehrlich-Schwoebel [17,18] barrier to interlayer diffusion that is found on many metal surface and has been used in models to explain this type of patterning [2,19]. The ES instability region (also called the diffusive regime [2]) is distinguished by patterning that typically aligns with the crystallographic direction of the surface rather than the ion beam direction [20]. The importance of the surface crystallography indicates the behavior is dominated by effects of diffusion on the surface rather than the ion-surface interaction. These types of ripples are found within a limited range of temperatures where the diffusional effects are most important and transitions to all the other types of behavior have been observed by changing the temperature [16,11]. In contrast with the BH behavior, the amplitude often grows with a power law behavior and the wavelength is not constant but increases with the sputtering time. 2.3. Athermal BH/kinetic roughening At low temperature, surface diffusion decreases and therefore the ES-type patterning no longer dominates. In this regime, the surface morphology may be observed to kinetically roughen without developing any preferred periodicity. Alternatively, the ionsurface interaction can also induce a preferred periodicity that is independent of surface diffusion and hence temperature. This behavior (labeled athermal BH) is consistent with an extension to the BH theory by Makeev et al. [21] which is discussed below. 2.4. Non-roughening At high temperatures, if the flux is low than the smoothing due to surface diffusion can dominate over the ion-induced effects and prevent the nucleation of surface roughness. In this regime, the surface remains smooth even under prolonged sputtering. The transition to non-roughening behavior has been observed both by decreasing the flux from the regime where BH ripples form [10] and by increasing the temperature from the regime where ES instability patterns form [16]. The diagram shows how these patterning behaviors are related to different regimes of flux and temperature on the Cu(1 0 0) surface. High temperature and high flux are the best conditions for BH instability ripples since the effect of the ion beam is relatively large while the barriers to diffusion are relatively low. High temperature with lower flux can lead to non-roughening of the surface. The ES instability occurs at lower temperatures where the effect of diffusion barriers can dominate over ion-induced effects. At even lower temperature, limited mobility can lead to kinetic roughening or possible the formation of athermal BH patterns. 3. Linear instability model of Bradley and Harper (BH) To provide an understanding of how these observed patterning behaviors relates to underlying kinetic processes, we describe the BH model and refinements made to it to account for additional effects. Bradley and Harper (BH) [9] developed the first quantitative model for ripple formation by considering the combined effects of sputtering and surface diffusion. The approach was based on a mechanism proposed by Sigmund [22,23] to describe the sputter removal of surface atoms by incident ions. Sigmund’s model related the rate of atom removal to the energy deposited by the incident ion into the near surface region through a series of collisions.
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E. Chason, V. Shenoy / Nuclear Instruments and Methods in Physics Research B 272 (2012) 178–182
ion
Makeev et al. [1] extended the BH sputtering mechanism to include higher order effects of the ion-surface interaction within the context of the Sigmund mechanism. This manifests itself as additional terms of the form BI @ 4 h=@x4 due to the ion-surface interaction which added a temperature-independent term BI to the term B in eq. (1). With this addition, the characteristic wavelength on the surface is given by:
sputtered atom
Slower erosion at crests
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Bðf ; TÞ þ BI ðf Þ Ds ðTÞC s ðf ; TÞ þ AI k / / fT mmax ðf Þ
Faster erosion at troughs Fig. 2. Schematic of Sigmund mechanism leading to a curvature-dependent sputtering yield that erodes surface troughs faster than crests.
The resulting energy distribution was approximated by a Gaussian function around the incident ion track (shown schematically in Fig. 2) and the probability of sputtering an atom from the surface was taken to be proportional to the energy deposited at the surface site. BH extended this mechanism to consider the evolution of a surface profile and showed that it resulted in a sputter yield that is proportional to the curvature of the surface. For a surface with a sinusoidal profile, this means that the crests of the waves on the surface sputter slower than the troughs so that surfaces become rougher as they are sputtered. This roughening mechanism is balanced by surface relaxation which depends on the divergence of the surface curvature. Putting these together, they derived an equation for the evolution of the height h(x,y,t):
@h @2h @2h ¼ mx 2 þ my 2 Br2 r2 h @t @x @y
ð1Þ
where mx and my refer to the curvature dependence of the sputtering in directions parallel (x) and perpendicular (y) to the ion beam along the surface. The parameter B can describe smoothing due to classical surface diffusion [24,25],where B is proportional to the surface diffusivity (Ds) and concentration of mobile species (Cs). It has also been shown that a similar functional form for the smoothing can arise from ion-induced viscous flow in the surface layer [26]. This equation predicts that each Fourier component of the surface height (hk(t)) will grow (or shrink) exponentially with a rate that depends on the wavevector:
hk ðtÞ ¼ hk ð0Þerk t
ð2aÞ
where 2
2
2
2
r k ¼ ðmx kx þ my ky Þ Bðkx þ ky Þ2
ð2bÞ
The growth rate rk has a maximum value r⁄ at the wavevector k⁄:
m2
r ¼ max 4B ffi rffiffiffiffiffiffiffiffiffi
k ¼
mmax 2B
ð3Þ
where mmax is the larger of the two values for m. The surface develops a characteristic periodicity of k⁄ = 2p/k⁄ because the amplitude of this wavelength grows faster than all the others, i.e., it is maximally unstable. This model accounts for many of the features observed in ripples of the BH instability type. It predicts exponential growth of the ripple amplitude, a constant characteristic surface periodicity and a pattern orientation that is determined by the direction of the ion beam. In addition, depending on the ion’s incident angle, the pattern can be induced with the wavevector parallel or perpendicular to the ion beam direction as seen experimentally.
ð4Þ
where AI is a constant. The dependence of the parameters on flux (f) and temperature (T) is shown explicitly to indicate how the change in these parameters will affect the ripple wavelength. The addition of the temperature-independent term AI means that ripples can form even in the absence of thermally-activated diffusion and is consistent with the transition to athermal BH ripples at low temperature in some metals. It also sets a limit to the minimum wavelength that can be achieved by ion patterning. To explain the type of behavior seen in the ES instability regime, Valbusa [2] proposed incorporating an instability mechanism described by Villain [27,28] based on the roughening effect of ES barriers into the BH theory. By linearizing the theory for small slopes, they determined that the addition of an ES barrier can have a similar effect as the curvature dependence of the sputter yield but with the roughness aligned along crystallographic directions rather than direction of the incident ion beam. Although the conditions for linearization are not always met by the experimental; studies, this merging of the two mechanisms enables an understanding of how changing the temperature or incident ion conditions can induce changes from ES to BH ripples. 4. Comparison of model with experiments and simulations in the BH regime Measurements of ripple formation kinetics have been made in many systems [3] using STM/AFM, light scattering and X-ray scattering that qualitatively display the features predicted by the BH model. However, the model also provides specific predictions for the dependence of measureable experimental quantities ( k⁄, r⁄) on the underlying kinetic processes which can be probed experimentally (e.g., by changing f, T, ion angle). For example, systematic measurements of the ripple wavelength on Cu surfaces at different temperatures and fluxes [29] are shown in Fig. 3 The measurements show a complex non-Arrhenius temperature behavior and a flux dependence that is different at high T (kf-½) and low T (kf 0). To analyze these results, Chan et al. developed a model for the flux and temperature dependence of the surface defect concentration (details found in Ref. [29]) by considering different processes of defect creation and annihilation. At low temperature the defect concentration is determined primarily by the ion beam while at high temperature it is primarily determined by thermal generation. This concentration model was used to evaluate the ripple wavelength (eq. 4) and is able to explain both the temperature dependence and the changing flux dependence with temperature. In addition to the experimental studies, kinetic Monte Carlo (KMC) simulations have been developed that include the Sigmund mechanisms for sputtering as well as diffusion of defects [30–34]. In these KMC simulations, the interaction between the ion and surface is identical to the one in the Sigmund model, in contrast with the experiments in which we cannot be sure of the ion–solid interaction. Different simulation schemes utilize different mechanisms for modeling the defect kinetics. In the work of Chan et al. [30], the defect kinetics (including adatoms and vacancies) are implemented by allowing the individual atoms to hop around with transition rates that depend on the local atomic configuration so that
E. Chason, V. Shenoy / Nuclear Instruments and Methods in Physics Research B 272 (2012) 178–182
(a)
181
(b) T=481 K λ ∝ f -1/2
(c) T=409 K λ∝f0
Fig. 3. Measurements of the ripple wavelength on Cu(001) as a function of a) temperature and as a function of flux at b) T = 481 K and c) T = 409 K. The solid lines are a fit to a model based on the BH theory with a temperature- and flux-dependent defect concentration.
5.1. Ripple time evolution and multifield approach
Fig. 4. Results from KMC simulations. (a) Simulated ripple morphology and (b) simulated ripple wavelength as a function of flux and temperature. Solid lines are fits to BH theory using simulation parameters.
the time dependence of the surface evolution can be simulated. An image of the simulated surface morphology and results for the simulated wavelength at different temperatures and fluxes is shown in Fig. 4 The results show that the temperature and flux dependence of the wavelength and growth rate can be well explained by the BH theory (solid lines in the figure), indicating that the BH model is a good continuum approximation for the surface evolution when the ion–solid interaction is modeled by the Sigmund mechanism. 5. Limitations of the BH theory and alternative approaches The BH model and its extension provide a useful intuitive framework for understanding why ripples form and how they depend on the processing conditions. However, the large number of studies of ripple evolution point out several significant shortcomings of the linear stability approach and have identified features of ripple formation that do not fall within the linear instability model. In this section, we will discuss features of ripple formation that are not explained by the BH model and possible alternatives that are being explored.
By definition, the linear instability model is only appropriate in the early stages of roughening. As the roughness increases, the ripples cannot continue to grow exponentially and saturation of the ripple amplitude is seen in many experimental studies. This behavior points to the importance of non-linear terms in the surface evolution and attempts have been made to incorporate these into the BH formalism. However, more recently an alternative multifield approach has been used [35–37] which considers the coupled evolution of the surface height and the thin layer of mobile species on the surface. This is described as a ‘‘hydrodynamic’’ model, similar to that used to understand pattern formation in sand dunes. In particular, this approach is able to include the effect of redistribution of material on the surface during the evolution. A notable success of this model is that it can predict saturation and stationary surface features as well as the formation of quantum dot-like features seen on semiconductor surfaces. In the current formulation, it uses the same ion-roughening term as the BH theory, so in the early stages it is expected that the results are the same as for the BH theory. 5.2. Alternative ‘‘crater’’ functions and atom redistribution Another alternative approach is based on the recognition that the Sigmund mechanism for the ion-induced roughening only includes the effect of atom removal due to the incoming ion. Several groups have recognized that this does not agree with what is seen around ion impact craters [38] or with what is predicted by molecular dynamics [39,40] and that this can have a significant impact on the surface evolution [41–43]. Including the effect of atomic redistribution due to ion bombardment has enabled an understanding of the complicated dependence of the ripple formation rate on the ion incident angle seen in experiments on Si [44] have shown that changing the angle of incidence can prevent the formation of ripples which is in disagreement with the predictions of the BH theory. They show that the inclusion of a different crater function than the one used by Sigmund can account for this angular dependence. 5.3. Surface roughening rate and ion-induced stress From eq. 3, the BH theory predicts that there is a relationship between the growth rate and wavevector that depends only on
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10 -3
However, other features of ripple formation (saturation, angular dependence and rapid growth) cannot be explained. New mechanisms and better understanding of the coupling between ioninduced changes and surface transport hold out hope that a comprehensive predictive model of surface patterning induced by low energy ions can be achieved in the future.
10 -4 10 -5
r* ( /s)
10 -6 10 -7
BH + ATGS roughening
Acknowledgements
10 -8
The authors thank W. L. Chan, Y. Ishii and N. Medhekar for their contributions to this work. The research for this work was supported by the US DOE, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-FG0201ER45913.
ATGS roughening
10 -9 10 -10 10 -11
BH roughening 0.5
σ (GPa)
1.0
1.5
References
2.0
Fig. 5. Results of continuum model for effect of stress on ripple growth rate.
the ion-dependent parameter and not on the surface kinetics (r⁄ = ½ mmax k⁄2). When this is prediction is compared with measurements of rippling using Ar ions on Cu [3], the measured value of r⁄ is 240 times larger than the value predicted by the BH model, suggesting that there are other driving forces for ripple formation beyond sputtering. One potential mechanism is the effect of stress induced by the ion bombardment. Stress is known to lead to a surface instability similar to the BH mechanism (referred to as the Asaro-Tiller-Grinfeld-Srolovitz (ATGS) instability [45–47]) but with a rate of surface roughening that depends on @ 3 h=@x3 . Combining the stress induced roughening with the BH mechanism for curvature-dependent sputter yield to develop a continuum model for the surface evolution under the combined effects of sputter removal and ion-induced stress [48]. The combination leads to a linear instability model with a rate that depends on the wavevector as r ¼ Ajkj2 Bjkj4 þ Cjkj3 where C is a parameter that depends on the stress. The predicted wavevector and roughening rate are given by:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 1 kATG þ kBH2 k ¼ kATG þ 2 2 1 2 a Þ r ¼ Ak ð1 þ 2 ð4k 3aÞ
ð5aÞ ð5bÞ
where kBH = (A/(2B))1/2, kATG = 3a/4 and a = C/B. This theory predicts that the ripple will grow faster in the presence of stress than in the simple BH theory, with a rate that rises as the wavevector approaches the value predicted by the ATG theory. An example of the growth rate enhancement predicted by the theory is shown in Fig. 5 For low stresses, r⁄ is equal to the value from the BH theory but as the stress increases then r⁄ increases rapidly and approaches the ATGS rate. This may also explain why measurements on amorphous Si show better agreement with the BH model than on Cu because the ion-induced stress in amorphous Si is lower. Measurements are currently being performed to determine the significance of the stress in pattern formation 6. Conclusion In conclusion, many features of pattern formation by low energy ions can be understood within the context of the BH linear instability model. The effects of changing ion flux and temperature can be understood in terms of shifting the balance between roughening and smoothing processes occurring on the surface, leading to different characteristic behavior in different kinetic regimes.
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Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Surface nanopatterning mechanisms by keV ions: Linear instability models and beyond Eric Chason ⇑, Vivek Shenoy Brown University, Division of Engineering, Providence, RI 02912, USA
a r t i c l e
i n f o
Article history: Available online 1 February 2011 Keywords: Self-organization and patterning Surface structure Ion beam processing Sputtering
a b s t r a c t Sputtering solid surfaces with low energy ions is well-known to induce a wide array of nanoscale pattern forming behavior (sputter ripples). A simple continuum model originally developed by Bradley and Harper (BH) provides a useful framework that can qualitatively explain multiple types of patterning behavior seen experimentally and their relationship to the ion beam and material parameters. The basis of the model is a dynamic competition between roughening by ion bombardment and smoothing by surface transport that leads to the growth of roughness with a preferred periodicity on the surface. However, there are many experimental results that cannot be accounted for within this framework and additional physical mechanisms are discussed that may close the gap in our understanding. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Low energy ion bombardment of solid surfaces induces a surprisingly complex array of morphology evolution that depends on the ion beam, processing conditions and material being bombarded. A compendium of different observed behavior can be found in several recent reviews [1–4]. For different materials and conditions, the surface can spontaneously develop highly uniform waves (sputter ripples) or arrays of nanoscale quantum dots over large areas. The alignment of the pattern may be controlled by the direction of the ion beam or the crystallographic orientation of the surface. The surface roughness can grow exponentially or algebraically with time, or in some cases refuse to roughen at all. This wide variety of behavior has created intense interest in understanding its origin. Because it is self-organizing, ion-induced patterning holds promise as an inexpensive method for creating nanoscale structures over large areas, e.g., for magnetic storage [5] alignment of liquid crystals [6], optoelectronic materials [7] or enhanced catalysis [8]. Equally important, understanding the processes that control ripple formation provide a window into the non-equilibrium kinetic processes that occur under a combination of energetic particle bombardment and defect-mediated transport. These processes are critical for understanding the formation and stability of nanoscale structures and for the behavior of materials subjected to high flux environments such as in nuclear reactors. Much of our understanding of ripple formation has come from a linear instability model proposed by Bradley and Harper (BH) [9] ⇑ Corresponding author. Tel.: +1 401 863 2317. E-mail address: [email protected] (E. Chason). 0168-583X/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2011.01.060
that considers the surface evolution in terms of the kinetic balance between roughening and smoothing processes. This approach is at the core of many models of ion-induced pattern formation and it provides a useful and intuitive framework for considering different forms of pattern formation. Many aspects of ripple formation, such as the transitions between different types of behavior under different conditions, can be understood within this picture. However, systematic studies of ripple formation kinetics over recent years have exposed significant shortcomings of this approach, e.g., the growth rate of ripples has been found to be significantly faster in experiments than predicted by the theory. Furthermore, other aspects of ripple formation (such as amplitude saturation, the formation of quantum-dots and the angular dependence of the patterning behavior) fall outside the scope of the linear theory. This indicates that other mechanisms or approaches are needed beyond the simple linear instability model. In the current work, we review different types of ripple formation behavior that are seen on metal surfaces and use a kinetic phase diagram to show how the different behaviors are related to the regimes of flux and temperature where they are observed. The main features of the BH model are described in order to relate the wavelength and growth rate to the underlying surface kinetic parameters and the results are compared with experiments on Cu(0 0 1) and kinetic Monte Carlo simulations. Finally, we discuss some alternative mechanisms or approaches that may be able to explain observations that can’t be explained by the linear theory. 2. Types of patterning Ion-induced surface patterns have since been seen in a large number of materials systems including semiconductors, insulators
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guished from the athermal BH regime by the fact that the ripple wavelength depends on the temperature in BH regime. 2.2. ES instability
Fig. 1. Kinetic phase diagram showing regimes of pattern formation observed on Cu and Ag (001) surfaces at different fluxes and temperatures.
and metals. Since the production of sputter ripples is a kinetic phenomenon (i.e., the pattern is not thermodynamically stable), the type of behavior can be modified by changing the processing conditions. To illustrate this, we present the results from a number of studies on Cu and Ag in a kinetic phase diagram in Fig. 1 The patterning is categorized into one of several characteristic types of behavior that we refer to as BH instability, ES instability, athermal BH/kinetic roughening and non-roughening. The different symbols represent data from different studies that are listed in Table 1 [2,10–16]. Although not all the measurements were done under the same conditions the boundaries roughly delineate regimes of temperature and flux in which the different types of behavior are observed The different types of behavior have different characteristic features that give insight into the physical mechanisms that are controlling their formation: 2.1. BH instability These types of ripples are named for the fact that they are explained by the original BH model. They have several distinct features [3]: the characteristic surface periodicity remains constant during the growth, the orientation of the pattern is determined by the direction of the ion beam and the amplitude grows exponentially (in the early stage before saturating). In addition, the orientation of the surface wavevector can be parallel or perpendicular to the ion beam direction, depending on the ion’s incident angle. The dependence of the ripple orientation on the ion beam direction indicates that their formation is dominated by the ion-surface interaction. Originally seen only on semiconductor and insulator surfaces, they have been observed on the Cu surface when the temperature and flux are sufficiently high, i.e., the upper right hand portion of the diagram in Fig. 1 The BH instability regime is distin-
Table 1 List of the experimental conditions and the source reference for the data represented by the different symbols in Fig. 1. Symbol s e
r D h
Surface
Sputtering conditions
References
Cu(0 0 1) Cu(0 0 1) Cu(0 0 1) Ag(0 0 1) Ag(0 0 1)
Ion Ar+ Ar+ Ar+ Ar+ Ne+
[10–12] [13] [14] [15,16] [2]
h 70o 45o 30o 0o 70o
Energy (eV) 800 400–3000 350 1000 1000
The label ES instability refers to the Ehrlich-Schwoebel [17,18] barrier to interlayer diffusion that is found on many metal surface and has been used in models to explain this type of patterning [2,19]. The ES instability region (also called the diffusive regime [2]) is distinguished by patterning that typically aligns with the crystallographic direction of the surface rather than the ion beam direction [20]. The importance of the surface crystallography indicates the behavior is dominated by effects of diffusion on the surface rather than the ion-surface interaction. These types of ripples are found within a limited range of temperatures where the diffusional effects are most important and transitions to all the other types of behavior have been observed by changing the temperature [16,11]. In contrast with the BH behavior, the amplitude often grows with a power law behavior and the wavelength is not constant but increases with the sputtering time. 2.3. Athermal BH/kinetic roughening At low temperature, surface diffusion decreases and therefore the ES-type patterning no longer dominates. In this regime, the surface morphology may be observed to kinetically roughen without developing any preferred periodicity. Alternatively, the ionsurface interaction can also induce a preferred periodicity that is independent of surface diffusion and hence temperature. This behavior (labeled athermal BH) is consistent with an extension to the BH theory by Makeev et al. [21] which is discussed below. 2.4. Non-roughening At high temperatures, if the flux is low than the smoothing due to surface diffusion can dominate over the ion-induced effects and prevent the nucleation of surface roughness. In this regime, the surface remains smooth even under prolonged sputtering. The transition to non-roughening behavior has been observed both by decreasing the flux from the regime where BH ripples form [10] and by increasing the temperature from the regime where ES instability patterns form [16]. The diagram shows how these patterning behaviors are related to different regimes of flux and temperature on the Cu(1 0 0) surface. High temperature and high flux are the best conditions for BH instability ripples since the effect of the ion beam is relatively large while the barriers to diffusion are relatively low. High temperature with lower flux can lead to non-roughening of the surface. The ES instability occurs at lower temperatures where the effect of diffusion barriers can dominate over ion-induced effects. At even lower temperature, limited mobility can lead to kinetic roughening or possible the formation of athermal BH patterns. 3. Linear instability model of Bradley and Harper (BH) To provide an understanding of how these observed patterning behaviors relates to underlying kinetic processes, we describe the BH model and refinements made to it to account for additional effects. Bradley and Harper (BH) [9] developed the first quantitative model for ripple formation by considering the combined effects of sputtering and surface diffusion. The approach was based on a mechanism proposed by Sigmund [22,23] to describe the sputter removal of surface atoms by incident ions. Sigmund’s model related the rate of atom removal to the energy deposited by the incident ion into the near surface region through a series of collisions.
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ion
Makeev et al. [1] extended the BH sputtering mechanism to include higher order effects of the ion-surface interaction within the context of the Sigmund mechanism. This manifests itself as additional terms of the form BI @ 4 h=@x4 due to the ion-surface interaction which added a temperature-independent term BI to the term B in eq. (1). With this addition, the characteristic wavelength on the surface is given by:
sputtered atom
Slower erosion at crests
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Bðf ; TÞ þ BI ðf Þ Ds ðTÞC s ðf ; TÞ þ AI k / / fT mmax ðf Þ
Faster erosion at troughs Fig. 2. Schematic of Sigmund mechanism leading to a curvature-dependent sputtering yield that erodes surface troughs faster than crests.
The resulting energy distribution was approximated by a Gaussian function around the incident ion track (shown schematically in Fig. 2) and the probability of sputtering an atom from the surface was taken to be proportional to the energy deposited at the surface site. BH extended this mechanism to consider the evolution of a surface profile and showed that it resulted in a sputter yield that is proportional to the curvature of the surface. For a surface with a sinusoidal profile, this means that the crests of the waves on the surface sputter slower than the troughs so that surfaces become rougher as they are sputtered. This roughening mechanism is balanced by surface relaxation which depends on the divergence of the surface curvature. Putting these together, they derived an equation for the evolution of the height h(x,y,t):
@h @2h @2h ¼ mx 2 þ my 2 Br2 r2 h @t @x @y
ð1Þ
where mx and my refer to the curvature dependence of the sputtering in directions parallel (x) and perpendicular (y) to the ion beam along the surface. The parameter B can describe smoothing due to classical surface diffusion [24,25],where B is proportional to the surface diffusivity (Ds) and concentration of mobile species (Cs). It has also been shown that a similar functional form for the smoothing can arise from ion-induced viscous flow in the surface layer [26]. This equation predicts that each Fourier component of the surface height (hk(t)) will grow (or shrink) exponentially with a rate that depends on the wavevector:
hk ðtÞ ¼ hk ð0Þerk t
ð2aÞ
where 2
2
2
2
r k ¼ ðmx kx þ my ky Þ Bðkx þ ky Þ2
ð2bÞ
The growth rate rk has a maximum value r⁄ at the wavevector k⁄:
m2
r ¼ max 4B ffi rffiffiffiffiffiffiffiffiffi
k ¼
mmax 2B
ð3Þ
where mmax is the larger of the two values for m. The surface develops a characteristic periodicity of k⁄ = 2p/k⁄ because the amplitude of this wavelength grows faster than all the others, i.e., it is maximally unstable. This model accounts for many of the features observed in ripples of the BH instability type. It predicts exponential growth of the ripple amplitude, a constant characteristic surface periodicity and a pattern orientation that is determined by the direction of the ion beam. In addition, depending on the ion’s incident angle, the pattern can be induced with the wavevector parallel or perpendicular to the ion beam direction as seen experimentally.
ð4Þ
where AI is a constant. The dependence of the parameters on flux (f) and temperature (T) is shown explicitly to indicate how the change in these parameters will affect the ripple wavelength. The addition of the temperature-independent term AI means that ripples can form even in the absence of thermally-activated diffusion and is consistent with the transition to athermal BH ripples at low temperature in some metals. It also sets a limit to the minimum wavelength that can be achieved by ion patterning. To explain the type of behavior seen in the ES instability regime, Valbusa [2] proposed incorporating an instability mechanism described by Villain [27,28] based on the roughening effect of ES barriers into the BH theory. By linearizing the theory for small slopes, they determined that the addition of an ES barrier can have a similar effect as the curvature dependence of the sputter yield but with the roughness aligned along crystallographic directions rather than direction of the incident ion beam. Although the conditions for linearization are not always met by the experimental; studies, this merging of the two mechanisms enables an understanding of how changing the temperature or incident ion conditions can induce changes from ES to BH ripples. 4. Comparison of model with experiments and simulations in the BH regime Measurements of ripple formation kinetics have been made in many systems [3] using STM/AFM, light scattering and X-ray scattering that qualitatively display the features predicted by the BH model. However, the model also provides specific predictions for the dependence of measureable experimental quantities ( k⁄, r⁄) on the underlying kinetic processes which can be probed experimentally (e.g., by changing f, T, ion angle). For example, systematic measurements of the ripple wavelength on Cu surfaces at different temperatures and fluxes [29] are shown in Fig. 3 The measurements show a complex non-Arrhenius temperature behavior and a flux dependence that is different at high T (kf-½) and low T (kf 0). To analyze these results, Chan et al. developed a model for the flux and temperature dependence of the surface defect concentration (details found in Ref. [29]) by considering different processes of defect creation and annihilation. At low temperature the defect concentration is determined primarily by the ion beam while at high temperature it is primarily determined by thermal generation. This concentration model was used to evaluate the ripple wavelength (eq. 4) and is able to explain both the temperature dependence and the changing flux dependence with temperature. In addition to the experimental studies, kinetic Monte Carlo (KMC) simulations have been developed that include the Sigmund mechanisms for sputtering as well as diffusion of defects [30–34]. In these KMC simulations, the interaction between the ion and surface is identical to the one in the Sigmund model, in contrast with the experiments in which we cannot be sure of the ion–solid interaction. Different simulation schemes utilize different mechanisms for modeling the defect kinetics. In the work of Chan et al. [30], the defect kinetics (including adatoms and vacancies) are implemented by allowing the individual atoms to hop around with transition rates that depend on the local atomic configuration so that
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(a)
181
(b) T=481 K λ ∝ f -1/2
(c) T=409 K λ∝f0
Fig. 3. Measurements of the ripple wavelength on Cu(001) as a function of a) temperature and as a function of flux at b) T = 481 K and c) T = 409 K. The solid lines are a fit to a model based on the BH theory with a temperature- and flux-dependent defect concentration.
5.1. Ripple time evolution and multifield approach
Fig. 4. Results from KMC simulations. (a) Simulated ripple morphology and (b) simulated ripple wavelength as a function of flux and temperature. Solid lines are fits to BH theory using simulation parameters.
the time dependence of the surface evolution can be simulated. An image of the simulated surface morphology and results for the simulated wavelength at different temperatures and fluxes is shown in Fig. 4 The results show that the temperature and flux dependence of the wavelength and growth rate can be well explained by the BH theory (solid lines in the figure), indicating that the BH model is a good continuum approximation for the surface evolution when the ion–solid interaction is modeled by the Sigmund mechanism. 5. Limitations of the BH theory and alternative approaches The BH model and its extension provide a useful intuitive framework for understanding why ripples form and how they depend on the processing conditions. However, the large number of studies of ripple evolution point out several significant shortcomings of the linear stability approach and have identified features of ripple formation that do not fall within the linear instability model. In this section, we will discuss features of ripple formation that are not explained by the BH model and possible alternatives that are being explored.
By definition, the linear instability model is only appropriate in the early stages of roughening. As the roughness increases, the ripples cannot continue to grow exponentially and saturation of the ripple amplitude is seen in many experimental studies. This behavior points to the importance of non-linear terms in the surface evolution and attempts have been made to incorporate these into the BH formalism. However, more recently an alternative multifield approach has been used [35–37] which considers the coupled evolution of the surface height and the thin layer of mobile species on the surface. This is described as a ‘‘hydrodynamic’’ model, similar to that used to understand pattern formation in sand dunes. In particular, this approach is able to include the effect of redistribution of material on the surface during the evolution. A notable success of this model is that it can predict saturation and stationary surface features as well as the formation of quantum dot-like features seen on semiconductor surfaces. In the current formulation, it uses the same ion-roughening term as the BH theory, so in the early stages it is expected that the results are the same as for the BH theory. 5.2. Alternative ‘‘crater’’ functions and atom redistribution Another alternative approach is based on the recognition that the Sigmund mechanism for the ion-induced roughening only includes the effect of atom removal due to the incoming ion. Several groups have recognized that this does not agree with what is seen around ion impact craters [38] or with what is predicted by molecular dynamics [39,40] and that this can have a significant impact on the surface evolution [41–43]. Including the effect of atomic redistribution due to ion bombardment has enabled an understanding of the complicated dependence of the ripple formation rate on the ion incident angle seen in experiments on Si [44] have shown that changing the angle of incidence can prevent the formation of ripples which is in disagreement with the predictions of the BH theory. They show that the inclusion of a different crater function than the one used by Sigmund can account for this angular dependence. 5.3. Surface roughening rate and ion-induced stress From eq. 3, the BH theory predicts that there is a relationship between the growth rate and wavevector that depends only on
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10 -3
However, other features of ripple formation (saturation, angular dependence and rapid growth) cannot be explained. New mechanisms and better understanding of the coupling between ioninduced changes and surface transport hold out hope that a comprehensive predictive model of surface patterning induced by low energy ions can be achieved in the future.
10 -4 10 -5
r* ( /s)
10 -6 10 -7
BH + ATGS roughening
Acknowledgements
10 -8
The authors thank W. L. Chan, Y. Ishii and N. Medhekar for their contributions to this work. The research for this work was supported by the US DOE, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-FG0201ER45913.
ATGS roughening
10 -9 10 -10 10 -11
BH roughening 0.5
σ (GPa)
1.0
1.5
References
2.0
Fig. 5. Results of continuum model for effect of stress on ripple growth rate.
the ion-dependent parameter and not on the surface kinetics (r⁄ = ½ mmax k⁄2). When this is prediction is compared with measurements of rippling using Ar ions on Cu [3], the measured value of r⁄ is 240 times larger than the value predicted by the BH model, suggesting that there are other driving forces for ripple formation beyond sputtering. One potential mechanism is the effect of stress induced by the ion bombardment. Stress is known to lead to a surface instability similar to the BH mechanism (referred to as the Asaro-Tiller-Grinfeld-Srolovitz (ATGS) instability [45–47]) but with a rate of surface roughening that depends on @ 3 h=@x3 . Combining the stress induced roughening with the BH mechanism for curvature-dependent sputter yield to develop a continuum model for the surface evolution under the combined effects of sputter removal and ion-induced stress [48]. The combination leads to a linear instability model with a rate that depends on the wavevector as r ¼ Ajkj2 Bjkj4 þ Cjkj3 where C is a parameter that depends on the stress. The predicted wavevector and roughening rate are given by:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 1 kATG þ kBH2 k ¼ kATG þ 2 2 1 2 a Þ r ¼ Ak ð1 þ 2 ð4k 3aÞ
ð5aÞ ð5bÞ
where kBH = (A/(2B))1/2, kATG = 3a/4 and a = C/B. This theory predicts that the ripple will grow faster in the presence of stress than in the simple BH theory, with a rate that rises as the wavevector approaches the value predicted by the ATG theory. An example of the growth rate enhancement predicted by the theory is shown in Fig. 5 For low stresses, r⁄ is equal to the value from the BH theory but as the stress increases then r⁄ increases rapidly and approaches the ATGS rate. This may also explain why measurements on amorphous Si show better agreement with the BH model than on Cu because the ion-induced stress in amorphous Si is lower. Measurements are currently being performed to determine the significance of the stress in pattern formation 6. Conclusion In conclusion, many features of pattern formation by low energy ions can be understood within the context of the BH linear instability model. The effects of changing ion flux and temperature can be understood in terms of shifting the balance between roughening and smoothing processes occurring on the surface, leading to different characteristic behavior in different kinetic regimes.
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