I started with a ship full of max-level veterans, and I'm just a complete noob.

"First point at the head, begin recording."

Fang Xudong reported the reading, and Lin Lanying pressed the current switch on the four-probe measuring station. The voltmeter pointer slowly stopped on the dial.

Jiang Ming stood on the other side of the measuring table, holding the single crystal growth record in his hand. He first filled in the total crystal length and melt holding temperature into the table, and then checked the probe spacing along the marking line.

After the first set of measuring points is completed, the fixture moves backward by two millimeters.

With the probe pressed to the second position, Fang Xudong recalibrated the current, and Lin Lanying read the voltage. The two data were written into the length and resistivity columns respectively.

The readings at the head of the single crystal are too high. As the resistivity moves towards the middle section, it gradually decreases. The decrease is significantly greater at the last few measurement points at the tail.

"Impurities are less concentrated at the head and begin to accumulate at the tail."

Jiang Ming marked each measuring point on a coordinate paper. The horizontal axis was the crystal length, and the vertical axis was the resistivity. The broken line descended from the high position at the head to the low position at the tail.

Lin Lanying took the coordinate paper and connected the measuring points with a ruler.

"The difference between the beginning and the end is about 30%."

Fang Xudong found the first regional purification curve and pasted it next to him. The direction of the change in resistivity of the single crystal along the length is consistent with the impurity distribution of the polycrystalline germanium.

Regional purification pushes impurities to the tail end, while single crystal growth redistributes impurities in the melt according to the segregation law. After the two processes are superimposed, the difference between the head and tail ends still exists.

Jiang Ming looked at the curve but didn't rush to draw conclusions.

Lin Lanying took out several foreign language documents from the filing cabinet, unfolded one of them, and the edges of the pages were already soft from being turned so many times.

"According to the segregation model in the literature, if the equilibrium segregation coefficient remains constant and the pulling speed is constant, the resistivity should change approximately exponentially along the length direction."

She used a pencil to draw the theoretical trend on the curve. The actual measured line was in the same direction as the index model, but the enrichment area at the tail was deviated further downward than the theoretical line.

"The trend is correct, but the absolute value is off."

"How much more is there at the tail?"

"According to the theoretical curve, the actual impurity enrichment is about eight percentage points higher."

Fang Xudong retrieved the crucible usage record.

"The first and second furnaces both used the same graphite crucible, and the melt holding times were 32 minutes and 40 minutes, respectively."

Jiang Ming wrote the two times on the blackboard and added the melt temperature and single crystal length.

"The crucible wall may have been involved in the distribution of impurities."

Lin Lanying looked up at him.

Does graphite release carbon into germanium?

"The longer the contact time between the high-temperature melt and the heat source, the more likely the release will increase. The segregation coefficient of carbon in germanium is different from that of phosphorus, arsenic, and antimony. After being mixed into the overall distribution, the theoretical curve will be deflected."

Fang Xudong picked up the notebook and added notes on the number of times the crucible was used, the melt holding temperature, and the growth time.

Jiang Ming asked him to include the weight of the charge for each furnace, as the contact area of ​​the crucible wall, the height of the melt, and the heating time could all affect the amount of impurities entering.

At 10:00 AM, the Institute of Physics called 502.

Wu Hanzhang answered the phone, listened for a few minutes, and then handed the receiver to Jiang Ming.

"Huang Kun is looking for you."

Jiang Ming took the microphone.

"Professor Jiang, Wang Shouwu from the semiconductor research group is scheduled to come to the Institute of Physics the day after tomorrow. We'll first look at the purification data from furnace No. 3, and then at the single crystal growth results."

"We will prepare the data and samples."

"Show your results to others, and see if they stand up to scrutiny. That's up to you."

After hanging up the phone, Wu Hanzhang took the cigarette from behind his ear and twirled it between his fingers twice.

"Wang Shouwu is a genuine semiconductor expert who returned from studying in the United States, and his team is also working on germanium. His purpose in coming to see your data is not just to visit."

"He would ask about the boundaries of temperature control and purification."

"They'll also ask you why you're making single crystals and what you plan to do with them."

Jiang Ming pointed to the curve on the blackboard.

"The resistivity of the first five millimeters of the head is relatively uniform, so it can be cut out for device testing. The remaining part still needs further process modifications."

Wu Hanzhang approached the blackboard and looked at the difference in height between the beginning and the end.

"If this line were shown to someone, the first question they would ask would be about the 30% difference."

"That's a good question, that's definitely the problem."

"Is there a way to suppress it?"

"The effective segregation coefficient must be calculated first; you can't just use the equilibrium value to fit the measured curve."

Fang Xudong brought over a blank coordinate sheet from the side, and Jiang Ming wrote the segregation parameters of common impurities in germanium on the left side, and then listed the pulling speed, melt temperature and rotation speed in three columns.

Lin Lanying found the equilibrium segregation coefficients of phosphorus, arsenic, and antimony in germanium melt from the literature brought by the physics department. Several sets of data were scattered in different articles, and the measurement conditions were also different.

Jiang Ming labeled the source, temperature, and type of impurities separately, and circled the data that could not be directly compared.

"The effective segregation coefficient is affected by melt flow. As the rotation speed increases, the concentration boundary layer near the liquid surface will become thinner, and the transport state of impurities to the solid-liquid interface will change."

Fang Xudong looked at the melt flow lines he had drawn.

"The seed crystal is rotating, and the melt moves along with it?"

"The area near the contact surface will be affected, and the concentration at the interface is determined by radial flow and axial replenishment."

"The rotation speed also needs to be included in the parameters for the next batch."

Jiang Ming nodded, and another column for rotation speed was added to the blackboard.

In the afternoon, the notification of Wang Shouwu's visit was delivered to the Institute of Physics. Huang Kun posted the visit schedule at the entrance of Furnace Room No. 3, and the temperature control records of Furnace No. 3, the six-stage purification curves, and the single crystal samples were all included in the exhibition items.

Lin Lanying was in charge of organizing foreign language documents, Fang Xudong bound the paper tapes in chronological order, and Lao Sun returned to 502 to continue inspecting the single crystal furnace transmission mechanism.

Jiang Ming put the single crystal sample back into the cork box, closed the lid, and then took the regional purification data into the secure compartment.

On the table were the curves of furnace No. 3, the resistivity curve of single crystal, the crucible usage record, and excerpts from foreign literature.

He first wrote out the equilibrium segregation model, then divided the measured curve into three segments: head, middle and tail, and finally left a blank space between the two sets of curves.

The effective segregation coefficient is not constant; the pulling speed and melt boundary layer will change the way impurities reach the solid-liquid interface.

Jiang Ming wrote down the boundary layer thickness in the blank space, and then drew the relative positions of the bottom of the seed crystal, the surface of the melt, and the crucible wall.

The rotating seed crystal drives the melt flow, the velocity boundary layer spreads close to the solid-liquid interface, and the concentration boundary layer controls the migration of impurities to the crystal surface.

He correlated the rotation speed of 12 revolutions per minute, the lifting speed of 0.8 millimeters per minute, and the 30% difference between the beginning and end, and tried to deduce the effective condensation coefficient from the actual measurement results.

Several inconsistent results appeared on the calculation paper.

Simply increasing the lifting speed can explain some of the tail enrichment, but it cannot explain why the measured curve is eight percentage points higher than the theoretical curve.

Even after adding the carbon impurities released from the crucible to the model, the tail offset was not completely eliminated.

Jiang Ming opened his kraft paper notebook and drew the velocity boundary layer and concentration boundary layer near the solid-liquid interface on a blank page. He extended the pencil lines outward along the liquid surface and then wrote the Greek letter δ next to the boundary layer thickness.

The thickness of this boundary layer is related to the laminar flow above the rotating disk.

The golden margins of the Tongtianlu unfolded deep within my consciousness, and the outlines of European academic traditions, previously obscured by the gray fog, gradually emerged, with the names of aerodynamics, applied mathematics, and precision fluid mechanics intertwined and connected.

Jiang Ming placed the pencil next to δ and looked down at the unfinished streamline.

Fang Xudong's footsteps could be heard outside the door, followed by a light knock on the door.

"Jiang, the data needs to be sent to the Institute of Physics tomorrow morning. Wang Shouwu will arrive the day after tomorrow."

"I know."

"The data room has already asked Liu to check the diffusion coefficient. There might be a set of measured values ​​for germanium melt in a Soviet journal."

"Have him test for phosphorus first, then arsenic and antimony."

Fang Xudong pushed open the door and came in. He saw the dense curves and formulas on the blackboard and walked over to the diffusion coefficient D.

Is this value mentioned in the literature?

"The Soviet Union had a set of measured values ​​for the diffusion coefficient of phosphorus in germanium melt. I asked Liu to check the journal in the archives."

Fang Xudong nodded, picked up a pencil and wrote a question mark next to D, then changed the question mark to a check mark.

"First, provide the source information; Wang Shouwu will ask questions."

Jiang Ming put the curved paper into the file bag, his gaze returning to the letter δ.

"Calculate this boundary layer before tomorrow."

Fang Xudong looked at the blackboard for a while, then closed the door again.

The footsteps faded away down the corridor, and the confidential cubicle fell silent again, the flame of the kerosene lamp maintaining a steady brightness inside the glass cover.

Jiang Ming opened his old hardcover memo book, found a few pages of Russian excerpts about rotating fluids, then returned to his kraft paper notebook and added the rotational angular velocity next to the boundary layer diagram. One set of numbers was still missing.

The relationship between rotation speed and boundary layer thickness determines whether the 30% difference can be reduced in the next crystal pulling operation.

He pressed down on the page, his gaze fixed on the unfinished formula. His consciousness sank deeper along the golden edge of the Heavenly Record.