bookmarkrole Chapter 171 Coupling Terms
Chapter 171 Coupling Terms
2020 11 Month 22 Day.
Su Chen finally turned on the camera.
It wasn't because he wanted to watch the news, but because the lab's nitrogen supplier needed to contact him to confirm next week's delivery time.
The moment the phone was turned on, messages flooded in—238 WeChat messages, 17 missed calls, and 42 text messages.
Su Chen glanced at the message list and then went directly to Lin Wei's chat window.
Lin Wei's latest message was posted yesterday afternoon: "I'm handling the packaging. You focus on the 300mm."
Su Chen replied with a single word: "Okay."
He then found the nitrogen supplier's message and confirmed the delivery time.
Then he placed his phone on the lab table—this time he didn't turn it off, but he didn't intend to check any other messages.
Shen Zhiming walked over from the office next door, holding a freshly brewed cup of tea.
"You turned your computer on?"
"About nitrogen."
"Are you aware of what's going on outside? Bosch issued a statement yesterday denying what Lin Wei said."
"Um."
"You don't care?"
Su Chen looked up at Shen Zhiming, then pushed the draft paper in front of him over:
"Zhiming, look at this."
Shen Zhiming put down his teacup and looked down at the draft paper. The paper was densely covered with mathematical derivations, and on the last line, Su Chen had drawn a circle in red pen:
$T(r,theta) = T_0 + alpha cdot Delta T cdot f(r) + beta cdot (Delta T)^2 cdot g(r,theta) + gamma cdot (Delta T)^3 cdot h(r,theta,z)$
"What is this?" Shen Zhiming frowned.
"Temperature field distribution in a 300mm cavity." Su Chen pointed to the last item with the tip of his pen. "The first two items—the linear and second-order terms—are basically the same as in the 250mm cavity, only the parameters are different. But this third item…"
He pointed to $gamma cdot (Delta T)^3 cdot h(r,theta,z)$:
"This is a third-order nonlinear coupling term. On a 250mm scale, the magnitude of this term is approximately 10 to the power of negative 4, which can be ignored. But on a 300mm scale—"
Su Chen flipped to the next sheet of draft paper, on which was a contour map, hand-drawn, but extremely precise:
"在300mm尺度上，这一项的量级上升到了10的负2次方。特别是在腔体边缘区域——距离圆心120mm到150mm的环形区域——这个三阶耦合项会导致温度场出现一个局部非均匀扰动。
This perturbation directly affects the isotropic distribution of the SF₆ etching step in the Bosch process, ultimately manifesting in the sidewall angle.
"How big of an impact will it be?" Shen Zhiming's expression turned serious.
"Without addressing this coupling term, the prediction error for the sidewall angle of the 300mm cavity could widen to ±0.08 to ±0.12 degrees."
Shen Zhiming gasped, "Then the advantage of the equivalent thermoelastic approximation method is gone. With an accuracy of ±0.12 degrees, it's about the same as traditional simulation software."
"Yes." Su Chen nodded. "So this third-order coupling term is the core challenge of the 300mm method. The 250mm method succeeded because this term can be ignored. If this term isn't solved in the 300mm method, the entire approach is useless."
Do you have any ideas?
Su Chen remained silent for a few seconds.
Then he turned to the third draft paper.
The contents of this sheet of paper are completely different from the previous two. The first two were rigorous mathematical derivations, but this one is more like a sketch—Su Chen drew a cross-sectional diagram of a 300mm cavity, and then marked several arrows and question marks on the edge area.
In the upper right corner of the paper, he wrote a few lines of messy handwriting:
方法一：分区域标定——将300mm腔体分为中心区域（0-120mm）和边缘区域（120-150mm），对边缘区域单独建立三阶修正模型。
Problem: Regional calibration requires at least three trial etching experiments to obtain correction parameters for the edge regions. This is time-consuming.
Method 2: Equivalent order reduction - Find a second-order equivalent form to approximate the third-order coupling effect, thereby reducing the 300mm temperature field model back to second-order accuracy.
Question: Is it mathematically feasible? It is necessary to prove that the third-order term can be precisely approximated by the second-order equivalent form under specific boundary conditions.
Method 3: ? ? ?
Shen Zhiming finished reading the three draft papers and remained silent for a long time.
"Method one is too slow," he said. "More than three trial etchings—each requiring at least a week of preparation and execution time—that step alone takes more than a month. Add to that the subsequent parameter calibration and verification… six months might not even be enough."
"Yes. So method one is a last resort," Su Chen said.
What about method two?
"Method Two..." Su Chen's eyes narrowed slightly. "If Method Two succeeds, it would be the ideal path. It doesn't require additional etching experiments; it's purely a mathematical solution. But whether it's mathematically feasible—I'm still thinking about it."
"What about method three? You wrote three question marks."
Su Chen did not answer. He looked out the window, his gaze seemingly piercing through the glass, through the skyscrapers of Nanshan District in Shenzhen, and looking towards some distant place.
"Method three..." he said softly, "not yet."
Shen Zhiming looked at Su Chen's profile. Sunlight streamed in through the window, falling on the lab benches covered with draft papers. Three years ago, when he first met Su Chen, Su Chen was still in a rented room in an urban village, testing a development board with a multimeter.
Back then, Su Chen's eyes were the same as they are now—focused, calm, and carrying a kind of certainty that others couldn't understand.
"You will find it," Shen Zhiming said.
Su Chen glanced back at him, a slight smile playing on her lips, then lowered her head again and erased the three question marks after "Method Three" on the third draft paper.
A new line of text has been added:
Method 3: Starting from the physical essence of the temperature field—the Green's function expansion of the heat conduction equation in cylindrical coordinates. If the physical origin of the third-order coupling term is the discontinuity of heat flux in the edge region—then an analytical third-order correction term can be obtained by asymptotically expanding the Green's function.
He drew a line under this line and then wrote two words:
verification.
"I need to do some calculations," Su Chen said. "If this approach is correct—the third-order coupling term for 300mm can be handled directly using analytical methods, without needing to conduct etching experiments."
How long will it take?
"I don't know. Maybe a week. Maybe a month. Maybe longer."
Shen Zhiming nodded, picked up his teacup, and turned to walk towards the door. He paused at the doorway.
"By the way, you haven't had lunch yet."
"Um."
"I'll have someone bring it to you. What do you want to eat?"
"casual."
Shen Zhiming smiled and walked out of the laboratory.
……
At four o'clock in the afternoon, Su Chen wrote down the expansion of the Green's function in cylindrical coordinates on the fourth draft paper.
At 6 p.m., he completed the derivation of the first three terms of the expansion and found that the asymptotic form of the third term could indeed be used to describe the discontinuous heat flux effect in the edge region.
At 8 p.m., he began to try substituting the third-order expansion of the Green's function into the stress field model of the equivalent thermoelastic approximation method.
He stopped at 10 p.m.
Because he discovered a problem.
The third-order expansion of the Green's function is convergent under Dirichletian boundary conditions (fixed temperature), but convergence cannot be guaranteed under Neumannian boundary conditions (fixed heat flux).
During DRIE deep silicon etching, the actual boundary conditions at the cavity edge are von Neumann type.
In other words, the Green's function approach is physically correct, but mathematically it is still one step short—it needs to find a way to make the third-order expansion under the von Neumann boundary condition converge as well.
Su Chen put down his pen and rubbed his eyes.
The lab was quiet. Shen Zhiming had already gone home. The Shenzhen night view outside the window was still dazzling, but Su Chen had no interest in looking at it.
He stood up, walked to the sink in the corner of the laboratory, and washed his face with cold water.
Cool water droplets slid down his chin and landed on the lapel of the white coat.
Su Chen looked at himself in the mirror. Over the past three years, he had grown accustomed to this moment—the moment when a path had come to an end and a new breakthrough was needed.
At the 200mm size, the bottleneck he encountered was the nonlinear deviation of the coefficient of thermal expansion at larger dimensions. He spent two weeks developing an equivalent thermoelastic approximation method.
When working on the 250mm size, the bottleneck he encountered was parameter calibration—different sizes require different calibration methods. He spent three weeks finding a method to extrapolate the 250mm parameters based on the 200mm experimental data.
The current diameter is 300mm. The bottleneck is the convergence problem of the third-order nonlinear coupling term.
How much time does he need?
do not know.
But he knew one thing—he would find it.
Su Chen dried his face and returned to the lab table.
He didn't continue working out the design on the draft paper. Instead, he took out an old notebook from the drawer—not the 300mm design plan, but the notebook he had written in the urban village three years ago.
The first page reads:
2017年12月15日。退伍第87天。今天在华强北买了一块STM32开发板，280块。配了一块MPU6050六轴传感器，45块。开始学MEMS。
Su Chen flipped through a few pages and came across a page he hadn't looked at in a long time:
March 2, 2018. Today I saw a paper in the library—"Application of Green's Function Method in Heat Conduction Boundary Value Problems." The author is a man named Zhou Zhiyuan, from the Chinese Academy of Sciences. The paper is very well written, especially his improved treatment of the Green's function under the von Neumann boundary condition—using a regularization method.
I've written down the method. Maybe I'll need it later.
Su Chen stared at the notes he had written two and a half years ago and remained silent for a long time.
Zhou Zhiyuan.
He was the researcher from the Chinese Academy of Sciences who connected with him during the live broadcast and refuted Fang Jianhua's doubts on the spot.
He was the one who told Academician Wang Deming over the phone that "Su Chen's equivalent thermoelastic approximation method is a completely new theoretical framework."
Su Chen had read his paper two and a half years ago and learned his regularization method for handling von Neumann boundary conditions.
Now, the convergence problem of the third-order coupling term in 300mm requires a method to make the expansion of the Green function converge under the Neumann boundary condition.
Su Chen slowly closed the old notebook.
He picked up his pen again and wrote a few words on the blank space of the fourth draft paper:
Regularized Green's function – Neumann boundary correction
Then he began to deduce.
This time, the pen tip glided across the paper faster than before.
……
1:17 AM.
Su Chen put down his pen.
The draft paper in front of him had been replaced with the seventh sheet. Seven A3-sized sheets of paper covered the entire lab bench, each one filled with dense mathematical derivations.
On the last line of the last sheet of paper, he wrote an equation:
$Deltatheta_{300} = theta_0 + sum_{n=1}^{3} c_n cdot R_n(r,theta,z) + O(10^{-5})$
The remainder is on the order of 10 to the power of -5.
This means that, after processing the third-order coupling term using the regularized Green's function method, the prediction accuracy of the sidewall angle of a 300mm cavity can theoretically reach ±0.03 degrees.
±0.03 degrees.
It perfectly matches his target of 12.08 ± 0.03 degrees for 300mm.
Su Chen stared at the equation for a long time.
Then, below the equation, he slowly wrote four words:
The theory is feasible.
He wasn't excited. He didn't cheer. He didn't call anyone.
He simply and quietly numbered the seven sheets of draft paper in order, folded them neatly, and put them into the drawer on the left side of the lab bench—along with the previous six sheets of draft paper.
Thirteen sheets of draft paper. Three days have passed since the 250mm verification was successful.
Then he turned off the lights, locked the door, and walked out of the laboratory.
The air in Shenzhen was a bit chilly at one in the morning. Su Chen stood at the entrance of the laboratory building and looked up at the night sky.
There were no stars. The light pollution in Shenzhen is too severe.
But he knew the stars were there.
Just like he knows where the answer to 300mm is.
He opened his phone and sent Lin Wei a message:
"The theoretical framework for the 300mm model is now complete. The next step is parameter calibration. A trial etching experiment is needed to verify the corrected model."
Lin Wei replied two seconds after sending the message:
"You're still awake?"
"Go to sleep now. Aren't you asleep either?"
"I'm looking at the equipment quote for the packaging line."
"Thanks for your hard work."
"You've worked hard too."
Su Chen looked at the conversation on the screen, thought for a moment, and then typed another line:
"I solved the 300mm third-order coupling term. I used the regularized Green's function method. Guess who invented this method?"
"who?"
"Zhou Zhiyuan. He's the researcher from the Chinese Academy of Sciences who helped us connect during the live stream."
Lin Wei paused for a few seconds before sending a smiling emoji, followed by a sentence:
"fate."
Su Chen smiled as well.
"I'd like to schedule a meeting with him tomorrow. If his regularization method can be further optimized, parameter calibration for 300mm might only require one trial etching experiment, instead of three."
"Okay. I'll arrange it."
Thank you. Good night.
"Good night."
Su Chen turned off his phone screen and walked into the Shenzhen night at the end of November.
The lights in the laboratory building behind us were all off. But in a drawer of the DRIE laboratory on the third floor, thirteen sheets of draft paper quietly awaited their mission.
From the equivalent thermoelastic approximation method to the regularized Green's function correction.
From 200mm to 250mm to 300mm.
Every step was taken by Su Chen alone, with just a pen and a stack of draft paper.
But from this day forward—from the moment he decided to contact Zhou Zhiyuan—he was no longer alone on this path.
The story of 300mm has only just begun.