Dimensional crossover in surface growth on rectangular substrates

This paper extends the study of dimensional crossovers in interface growth from the Kardar-Parisi-Zhang class to the Edwards-Wilkinson, Mullins-Herring, and Villain-Lai Das Sarma universality classes, demonstrating that rectangular substrates with large aspect ratios induce transitions from two-dimensional to one-dimensional scaling behaviors in both growth and steady-state regimes, including nonuniversal roughness scaling for specific geometric constraints.

The Gist

Imagine you are watching a pile of sand grow on a table. In physics, we study how these piles (or "surfaces") get rough and bumpy over time. Usually, scientists study this on a square table or a long, thin strip. But what happens if you build your pile on a rectangular table that is much longer than it is wide?

This paper explores exactly that scenario. It asks: How does the shape of the table change the way the pile grows?

Here is the story of the paper, broken down into simple concepts and analogies.

1. The Setup: The "Long Hallway" Effect

Imagine you are painting a wall.

• Scenario A (Square Room): You paint a square wall. The paint drips and spreads out in all directions equally. The roughness grows in a "2D" way (like a puddle spreading).
• Scenario B (Long Hallway): Now, imagine painting a very long, narrow hallway. At first, the paint spreads out in all directions. But because the hallway is so narrow, the paint hits the side walls quickly. Once it hits the walls, it can't spread sideways anymore; it can only spread forward down the hall.

The authors found that surface growth behaves exactly like this. If you have a rectangular substrate (the base) where the length ($L_y$) is much bigger than the width ($L_x$), the growth starts as a 2D process but eventually gets "squeezed" into a 1D process (like a line).

2. The Race: Short Time vs. Long Time

The paper looks at two different "races" happening on this rectangular table:

• The Early Race (Short Time): When you first start depositing particles (like sand or atoms), the pile doesn't know the table is narrow yet. The "correlation length" (how far the bumps can "see" each other) is small. The surface grows like it's on a giant, infinite 2D plane. It gets rough quickly.
• The Late Race (Long Time): Eventually, the bumps grow large enough to touch the narrow edges of the table. Once they hit the edges, they stop growing sideways. Now, the surface only grows along the long direction. It behaves like a 1D line.

The "Crossover": The moment the surface realizes it's trapped by the narrow walls is called the crossover time. Before this time, it's a 2D party; after this time, it's a 1D line dance.

3. The Different "Personalities" of Growth

The authors didn't just look at one type of growth (like the famous KPZ model). They tested three different "universality classes," which are like different personalities of how surfaces grow:

1. The "Smooth" Ones (EW and MH): These are linear models. Think of them as gentle rain. The surface smooths itself out.
   • The Twist: For the "Edwards-Wilkinson" (EW) type, the math is a bit weird. Instead of growing at a steady speed, the roughness grows very slowly, like a logarithm (it takes forever to get rough). The authors had to invent a new math formula to describe this "slow grower" on a rectangular table.
2. The "Chaotic" One (VLDS): This is a non-linear model. Think of it as a chaotic wind blowing sand. It creates very bumpy, jagged surfaces.
   • The Twist: For this chaotic type, the shape of the height distribution changes. At first, the heights look like a specific 2D bell curve. Later, they morph into a 1D bell curve. It's like watching a crowd of people spread out in a square room, then realizing the room is a hallway, and everyone lines up in a single file.

4. The "Special" Rectangle (The $L_x = L_y^\delta$ Case)

This is the most mind-bending part of the paper. The authors asked: What if the width and length are related by a specific power law?

Imagine you have a magic rule: Width = Length to the power of $\delta$.

• If $\delta = 1$, it's a square.
• If $\delta = 0$, it's a line.
• If $\delta$ is somewhere in between (like 0.9), it's a very long rectangle.

They discovered a "Critical Threshold" ($\delta^*$).

• If $\delta$ is small: The rectangle is long enough that the surface has plenty of time to grow as 2D, hit the walls, and then switch to 1D. You see the crossover.
• If $\delta$ is too big (close to 1): The rectangle is almost a square. The surface gets "saturated" (fills up the whole table) before it ever has a chance to realize it's narrow. The 1D phase never happens! The crossover is "preempted."

It's like trying to run a marathon in a hallway. If the hallway is long enough, you run a marathon (2D to 1D). If the hallway is just a tiny bit longer than it is wide, you finish the race before you even realize you're in a hallway.

5. Why Does This Matter?

You might ask, "Who cares about sand on a rectangular table?"

The answer is Nanotechnology.
Scientists are currently building tiny electronic devices, nanowires, and nano-sheets. These are often rectangular. To build them correctly, engineers need to know exactly how the surface will roughen. If they assume the surface grows in 2D, but it actually switches to 1D halfway through, their device might fail.

Summary of the Big Takeaways

1. Shape Matters: A rectangular base forces a 2D growth process to eventually become a 1D process.
2. Timing is Everything: There is a specific time when this switch happens. It depends on how narrow the rectangle is.
3. Not All Growth is the Same: Different types of growth (smooth vs. chaotic) react to this shape change in different ways. Some switch smoothly; others get stuck in a "logarithmic" slow-growth mode.
4. The "Too Square" Problem: If the rectangle isn't long enough compared to its width, the surface fills up before it can switch modes. The "crossover" disappears.
5. New Math Needed: The authors found that for some of these shapes, the old math formulas don't work. They had to create new ones to predict how rough the surface will get.

In short, this paper tells us that geometry is destiny for growing surfaces. The shape of the container dictates the story of how the surface evolves, and sometimes, if the container is the wrong shape, the story changes entirely.

Technical Summary

1. Problem Statement

The paper investigates dimensional crossover phenomena in the kinetic roughening of interfaces growing on rectangular substrates with lateral dimensions $L_x \times L_y$ (where $L_y > L_x$). While previous work established that Kardar-Parisi-Zhang (KPZ) interfaces exhibit a crossover from two-dimensional (2D) to one-dimensional (1D) scaling behavior as time progresses, it remained unclear whether this phenomenon is a universal feature across other universality classes of interface growth.

The authors aim to determine if this 2D-to-1D crossover occurs in:

• The Edwards-Wilkinson (EW) class (linear, Gaussian fluctuations).
• The Mullins-Herring (MH) class (linear, higher-order curvature).
• The Villain-Lai-Das Sarma (VLDS) class (non-linear, conserved dynamics).

Additionally, the study explores the specific geometric condition where $L_x = L_y^\delta$ (with $0 < \delta < 1$) to understand how the aspect ratio affects the existence of the crossover and the scaling of saturation roughness.

2. Methodology

The authors employed extensive Monte Carlo simulations of discrete growth models on rectangular square lattices with periodic boundary conditions.

• Models Investigated:
  • KPZ: Restricted Solid-on-Solid (RSOS) and Etching models.
  • EW: Family model and RSOS with evaporation (RSOSev).
  • MH: Large Curvature (LC1 and LC2) models.
  • VLDS: Conserved RSOS (CRSOS) models.
• Simulation Parameters:
  • Substrate sizes varied with fixed $L_x$ and very large $L_y$ (up to 32,768) to avoid saturation during the growth regime analysis.
  • Sample sizes ($N$) were sufficient to ensure $N L_x L_y \gtrsim 10^8$.
• Quantities Analyzed:
  • Global Roughness ($W$): Standard deviation of the height distribution.
  • Line Roughness ($W_x, W_y$): Roughness calculated along specific directions to isolate anisotropic effects.
  • Height Distributions (HDs): Analyzed via probability density functions, skewness ($S$), and kurtosis ($K$).
  • Scaling Relations: Tested against Family's scaling hypothesis adapted for rectangular geometries.

3. Key Contributions and Results

A. Growth Regime: Temporal Crossover

The study confirms that for all investigated universality classes, the global roughness $W(t)$ exhibits a temporal crossover from 2D to 1D scaling when $L_y \gg L_x$.

• Scaling Behavior:
  • Short times ($t \ll t_c$): $W \sim t^{\beta_{2D}}$ (2D behavior).
  • Long times ($t \gg t_c$): $W \sim t^{\beta_{1D}}$ (1D behavior).
  • Crossover Time: $t_c \sim L_x^{z_{2D}}$ for KPZ, MH, and VLDS classes.
• The EW Exception:
  • In 2D, the EW class exhibits logarithmic growth ($W^2 \sim \ln t$) rather than power-law growth.
  • The authors derived a modified scaling relation for EW: $W(L_x, t) \sim (\ln L_x)^{1/2} F_{EW} [2\nu t / (L_x \ln L_x)^2]$.
  • Consequently, the crossover time for EW includes a multiplicative logarithmic correction: $t_c \sim (L_x \ln L_x)^{z_{2D}}$.
• Directional Analysis:
  • $W_x$ (roughness along the short axis) saturates at $t_c$ as correlations reach $L_x$.
  • $W_y$ (roughness along the long axis) continues to grow, driving the global roughness into the 1D regime.

B. Height Distributions (HDs)

• Linear Classes (EW & MH): The HDs are Gaussian in both 1D and 2D. Consequently, skewness and kurtosis remain zero, and no crossover is observed in these moment ratios, despite the roughness scaling changing.
• Non-Linear Classes (KPZ & VLDS):
  • KPZ: HDs change from flat 2D to flat 1D (or cylindrical to circular).
  • VLDS: The HDs are non-Gaussian. The authors observed a clear crossover in the temporal evolution of skewness ($S$) and kurtosis ($K$), transitioning from 2D characteristic values to 1D values.

C. Steady State Regime

In the saturation regime (where correlations span the entire substrate), the saturation roughness $W_s$ depends on the aspect ratio $R = L_y/L_x$.

• Scaling: $W_s^2 \sim L_x^{2\alpha_{2D}} H(R)$, where $H(R) \sim R^{2\alpha_{1D}}$ for large $R$.
• EW Adaptation: Similar to the growth regime, the EW class requires a logarithmic correction to the scaling prefactor: $W_s^2 \sim (\ln L_x) H_{EW}(L_y / (L_x \ln L_x))$.
• VLDS HDs: The skewness of the steady-state HDs interpolates continuously between 2D and 1D values as $R$ increases, confirming a continuous family of distributions.

D. The $L_x = L_y^\delta$ Case and Non-Universality

The authors analyzed the specific geometry where $L_x = L_y^\delta$.

• Critical Exponent $\delta^*$: A "special" exponent is identified: $\delta^* = z_{1D} / z_{2D}$.
  • If $\delta < \delta^*$, the system has enough time to exhibit the 2D-to-1D crossover before saturation.
  • If $\delta \ge \delta^*$, the saturation time $t_s$ becomes comparable to or smaller than the crossover time $t_c$. The system saturates before the 1D regime can fully develop, effectively suppressing the crossover.
  • Values: $\delta^*_{KPZ} \approx 0.931$ and $\delta^*_{VLDS} \approx 0.889$. For linear classes, $\delta^* = 1$.
• Non-Universal Scaling:
  • For $\delta < \delta^*$, the saturation roughness follows a non-universal power law: $W_s \sim L_y^\Lambda$.
  • The exponent $\Lambda$ is a weighted average of the 1D and 2D roughness exponents: $\Lambda = (1-\delta)\alpha_{1D} + \delta\alpha_{2D}$.
  • Significance: This demonstrates a genuine breakdown of universality in the steady state caused by the non-equilateral geometry, a phenomenon rarely observed in standard growth models.

4. Significance and Conclusion

• Universality of Crossover: The paper establishes that the 2D-to-1D dimensional crossover is a general feature of surface growth on rectangular substrates, applicable to linear, non-linear, and conserved dynamics, not just KPZ.
• Geometry-Induced Non-Universality: The study provides strong evidence that non-equilateral substrate geometries can lead to non-universal scaling exponents in the steady state ($W_s \sim L_y^\Lambda$), challenging the assumption that universality classes are solely defined by local dynamics.
• Experimental Relevance: The findings are highly relevant for the fabrication of nanostructures (nanowires, nanoribbons) where selective area growth creates rectangular geometries. The results suggest that the aspect ratio of these nanostructures dictates whether the growth dynamics will exhibit 1D or 2D characteristics and whether the system will reach a true 1D steady state.
• Theoretical Refinement: The work provides necessary corrections to scaling relations for the EW class, accounting for logarithmic behaviors that were previously overlooked in the context of dimensional crossover.

In summary, this work extends the understanding of kinetic roughening beyond the KPZ class, revealing that substrate geometry plays a fundamental role in determining both the temporal evolution and the final statistical properties of growing interfaces.