You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We introduce two models for the action of impurities in epitaxial growth. In the first, the interaction between the diffusing adatoms and the impurities is “barrier”-like and, in the second, it is “trap”-like. For the barrier model, we find a symmetry breaking effect that leads to an overall down-hill current. As expected, such a current produces Edwards-Wilkinson scaling. For the trap model, no symmetry breaking occurs and the scaling behavior appears to be of the conserved-KPZ type.'
address: |
    $^{1}$ Theorie II, Institut für Festkörperforschung, Forschungszentrum Jülich, D-52425 Jülich, Germany\
    $^{2}$ Fachbereich Physik, Universität-GH Essen, D-45117 Essen, Germany 
author:
- 'Luís A. Nunes Amaral$^{1}$ and Joachim Krug$^{2}$'
date: 'September 9, 1996'
title: 'Impurity-induced diffusion bias in epitaxial growth'
---

[2]{}

The effect of impurities on growth rate and morphology is a classic topic of crystal growth theory [@cabrera]. The most thoroughly studied case is the step flow growth of a vicinal surface, when the (immobile) impurities pin the advancing steps and thus lead to step bunching [@hmk; @kandel]. These theories are [*mesoscopic*]{} rather than microscopic in nature, in the sense that they describe the interaction of preexisting steps with discrete impurities [@kandel] or an impurity concentration field [@hmk].

The advent of modern crystal growth techniques aimed at manufacturing layers of atomic scale thickness, notably molecular beam epitaxy (MBE) [@mbe], has lead to a renewed appreciation of the fact that small concentrations of impurity atoms on the growing surface can drastically influence the growth kinetics. A particularly striking aspect of these recent results is that the impurities may either lead to a deterioration of the growth quality – as would be expected according to the classic view [@cabrera] – or, conversely, they may play the role of [*surfactants*]{} in stabilizing smooth, layer-by-layer growth [@voigtlaender].

A clear example of the former type is the effect of hydrogen on the MBE of silicon [@eaglesham; @adams]. It was observed that the presence of H in the growth chamber during Si deposition on Si(001) leads to a decrease of the epitaxial height [@eaglesham], at which epitaxy breaks down and the growth becomes amorphous, proportional to the logarithm of the partial pressure of H [@adams]. The experiments ruled out the hypothesis that the breakdown of epitaxy might be due either to an increased coverage of H at the interface [@eaglesham] or to its incorporation into the bulk [@copel]. Rather, it was concluded [@adams] that the hydrogen greatly speeds up the development of surface roughness due to a reduction of the diffusion length of Si adatoms [@vasek].

The modification of the diffusion properties of the adatoms appears to be the most significant effect of the impurities also when they act as surfactants [@zhang; @kaxiras], though the nature of the modification — for example, whether the diffusion length is increased or decreased — depends on the chemical species in a complicated way [@voigtlaender]. This sensitivity to atomic details is rather unexpected, and calls for the development of models which are more microscopic than previous approaches [@hmk; @kandel].

In this Letter, we introduce two models for the action of impurities in epitaxial growth. Rather than attempting a detailed description of some particular material, our aim is to define a ‘minimal’ model in which the consequences of the impurity-adatom interaction on the large scale morphology of the surface can be clearly elucidated. The study of oversimplified models [@wolf; @dassarma] of ‘ideal MBE’ [@lai] has previously been very successful in clarifying the universality classes for kinetic roughening [@lai; @villain; @tang] and morphological instability [@villain; @siegert; @kps] in the absence of impurities [@reviews].

Our models reproduce the sensitive dependence on microscopic details mentioned above: Using two equally plausible microscopic interaction mechanisms – of [*barrier*]{}-type and of [*trap*]{}-type, respectively – we find that for the barrier model the impurities neutralize the destabilizing effect of step edge barriers [@villain; @kps] and thus lead to smoother growth [@markov], while for the trap model the asymptotic morphology remains unaffected. In terms of the coarse-grained continuum description of the surface [@villain; @reviews] we are able to trace the difference between the two models to the fact that the barriers modify the [*symmetry*]{} of the surface diffusion process, while the traps do not.

The models proposed in this Letter have three main ingredients (see Fig. \[f-models\]).

[*i) Deposition and Diffusion:*]{} For simplicity, we consider a one-dimensional discrete substrate. Material is randomly deposited at a rate $F$. The deposition occurs in a solid-on-solid (SOS) manner, i.e. deposition at a position $i$ implies that the surface height $h(i)$ is increased by one unit. Every atom which has only one occupied neighbor (namely, in the layer below) is considered a mobile [*adatom*]{} which diffuses at a rate $D$; atoms with more bonds are immobile. When an adatom diffuses to a step edge from above we implement an additional energy barrier [@schwoebel] by accepting a diffusion move down the step only with probability $p = \exp (- E_S
)$, where the barrier energy $E_S$ is measured in units of $k_B T$. In the absence of impurities such step edge barriers are know to lead to unstable growth [@villain; @kps; @reviews].

  =.9

[*ii) Impurities:*]{} Based on the experimental observations described previously, we assume that there is a flux $F'$ of impurities onto the growing surface. We restrict our study to the limit in which the diffusion rate of the impurities is much smaller than the diffusion rate of the adatoms, so that the impurities can be considered immobile. We also assume that impurities evaporate from the surface at a rate that keeps $\theta_I$, the impurity coverage, approximately constant. This is done by removing the impurity whenever a new atom is deposited on top of it. An important consequence of this rule is that the average lifetime of an impurity at a given site equals the monolayer deposition time.

[*iii) Interactions:*]{} Concerning the interactions between the impurities and the diffusing adatoms, we introduce two alternative models: the barrier and the trap model. In the [*barrier*]{} model, an adatom trying to diffuse onto a site occupied by an impurity will have its attempted move rejected. In the [*trap*]{} model, the same adatom will move on top of the impurity and then will swap positions. The end result is that it will no longer be able to diffuse because it has (at least) two “chemical bonds”: to the atom below and to the impurity.

  =.9

In our simulations we focus on the exponent $\beta$ describing the increase of the surface width [@reviews] $W(t,L) \equiv \langle (h
- \langle h \rangle)^2\rangle^{1/2} \sim t^\beta$ in the early time regime $t \ll L^z$; here $L$ denotes the system size and the dynamic exponent is $z = 1/(1-2\beta)$ for the class of (one-dimensional) models considered in this paper [@wolf]. To put the results into perspective, we will compare them to the predictions of the appropriate continuum equations for the coarse-grained height function $h(x,t)$; as usual, the average height $\langle h \rangle = Ft$ will be subtracted. In the present context the following equation suffices [@lai; @villain; @tang]: $$\label{geneq}
\frac{\partial h}{\partial t} = \nu \nabla^2 h - \frac{\lambda}{2}
\nabla^2 (\nabla h)^2 - \kappa (\nabla^2)^2 h + \eta,$$ where the stochastic force $\eta(x,t)$ models the shot noise in the beam, and can be taken to be Gaussian with zero mean and covariance $$\label{covariance} 
\langle \eta(x,t) \eta(x',t') \rangle =  F \delta(x - x') \delta(t-t').$$ The first term on the right hand side of (\[geneq\]) arises from the gradient expansion of an inclination-dependent, growth induced surface current [@villain; @kps]. When it is present, it dominates the large scale morphology: For $\nu > 0$ one obtains kinetic roughening of the Edwards-Wilkinson (EW) universality class [@reviews; @EW] with $\beta = 1/4$, while for $\nu < 0$ the growth is unstable and a mound morphology is expected to develop [@siegert; @reviews; @mounds]. If, for reasons of symmetry (see below) $\nu = 0$, the second, nonlinear term becomes important, and changes the roughening exponent to $\beta = 1/3$ [@villain; @lai; @tang] (the “conserved Kardar-Parisi-Zhang” universality class [@KPZ]). Finally, in many cases the growth-induced coefficients $\nu$ and $\lambda$ are small, and the early time behavior is dominated by the third term in (\[geneq\]), which arises from equilibrium surface diffusion [@mullins] and leads to a (transient) value $\beta = 3/8$ [@wolf; @dassarma].

We consider first the case of a nonzero step edge barrier in the absence of impurities; then $\nu < 0$ [@villain; @kps] and one expects asymptotically unstable growth. As shown in Fig. \[f-barrier\], the instability sets in after an initial power law transient which terminates at about 100 monolayers. Later on, the destabilizing effect of the step edge barrier leads to wavelength selection and mound formation with a very rapid growth of the surface width. Once large slopes have appeared on the surface, there is hardly any transfer of matter between the different mounds and the exponent $\beta$ reaches the limiting value $\beta = 1/2$ characteristic of random deposition [@lanczycki].

When we introduce impurities of the barrier-type a striking change occurs. As is visually apparent from Fig. \[f-barrier\], the presence even of small amounts of barrier impurities leads to a significative decrease of the interface width. The exponent takes the value $\beta = 0.24 \pm 0.03$, consistent with EW universality. The natural interpretation is that the impurities have caused the coefficient $\nu$ in Eq. (\[geneq\]) to change sign, from destabilizing ($\nu < 0$) to stabilizing ($\nu > 0$). The value of $\nu$ can be directly ascertained by measuring the average surface current for tilted substrates [@kps]. The results, shown in Figure \[f-current\](a), confirm our interpretation: Even a small flux of impurities ($F'/F = 0.05$) leads to a sizable positive value of $\nu$, both in the presence of a step edge barrier and for $E_S = 0$.

The reason for the change produced by the barrier impurities can be understood as follows. As is well known [@villain; @kps], the step edge barrier leads to an up-hill current because adatoms are rejected when trying to go down step edges and become integrated in the bulk when reaching an ascending step. This difference leads to an average current towards the up-step which destabilizes the surface, as described earlier. To visualize the effect of the impurities on this process, let us consider a step train moving from left to right; cf. Fig \[f-current\](b). An impurity can be deposited anywhere on a given terrace, so we can say that on average it is deposited in the [*middle*]{} between the two steps. However, as more material is deposited, the step edge to the left of the impurity advances towards it. On the other hand, the step edge to the right of the impurity moves away from it. Thus, the distance to the step to the left of the impurity is typically [*smaller*]{} than the distance to the right. Since the current away from the impurity on each side is proportional to the material deposited there (and thus to the length of that part of the terrace) we see that an average [*down-hill*]{} current is generated.

  =.9

  =.9

In contrast, the trap impurities do not seem to significantly change the dynamics of the growth process, apart from an increase of the prefactor of the width (Fig. \[f-trap\]); certainly they are not able to suppress the destabilizing effect of the step edge barriers. This is confirmed by a measurement of the surface current, which remains up-hill in the presence of traps, as shown in Fig. \[f-current\](a).

The traps do not induce a surface current because they cannot [*bias*]{} the diffusion of adatoms: The trap impurity makes itself felt only when the atom has already jumped onto it, and therefore no longer participates in the mass transport on the surface (when the impurity disappears due to the deposition of an additional atom, the trapped adatom remains immobile).

This point is brought out more clearly by considering the trap model without step edge barriers. For $E_S = 0$, the pure model ($F'=0$) has a symmetry which forces $\nu = 0$ in Eq. (\[geneq\]): For any local environment, the probability of a mobile adatom to jump to the right is equal to that for a jump to the left. Since this is true irrespective of the overall surface tilt, no growth-induced current can exist [@symmetry]. With $\nu = 0$ the behavior of (\[geneq\]) is dominated by the second, nonlinear term, and one expects $\beta = 1/3$. Our simulations lead to a value of $\beta =
0.33 \pm 0.03$, indicating that this symmetry is preserved by the trap impurities.

The increase in the prefactor of the width with increasing trap concentration can be interpreted as a decrease of the diffusion length $\ell_D$: Indeed, it can be shown [@brendel96] that the prefactor scales as $\ell_D^{-4d/(10+d)}$ for a $d$-dimensional surface. From the data shown in Fig. \[f-trap\] we therefore estimate that an impurity flux $F'/F = 0.1$ decreases the diffusion length by almost a factor of three.

  =.9

Finally, one may ask why none of our models shows an [ *impurity-induced*]{} growth instability of the kind considered in the classic theories of step bunching [@hmk; @kandel]. We believe that this is due to the fact that the lifetime of an impurity at a given position is, in our models, fixed to be of the order of the monolayer deposition time. In terms of the conventional step flow picture [@cabrera; @hmk] it is evident that impurities can effectively pin steps only if they remain at a position much longer than the time required for a step to pass over a terrace. Thus, it appears important to consider models with a variable lifetime for the impurities.

We acknowledge stimulating discussions with M. Rost, M. Schimschak and P. Šmilauer.

[10]{}

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