Thermal Expansion of Silicon Between 293k and 873k Using a Capacitor Dilatometer

doi:10.21203/rs.3.rs-2992865/v1

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Thermal Expansion of Silicon Between 293k and 873k Using a Capacitor Dilatometer

https://doi.org/10.21203/rs.3.rs-2992865/v1

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published 24 Jul, 2023

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A fused silica dilatometer consisted of a rod and tube that had capacitance discs attached at one end. The cylindrical specimen of polycrystalline pure silicon 2.54 centimeters long and 0.635 centimeters in diameter was clamped between the other end of the rod and tube. These were enclosed in a fused silica furnace tube within the furnace. The furnace tube was sealed with helium that did not flow. The capacitance discs were sealed in a container outside of the furnace. Helium flowed within that container. These disks were used to measure the thermal expansion of pure silicon between 200 and 600 Celsius. The capacitance measurements were made by an Andean Hager ling capacitance bridge with a readout of ten digits and a sensitivity of 0.5 attofarads. The rate of heating and cooling was 10C per minute. The specimen temperature was measured using a calibrated platinum thermometer 3 millimeters away from the specimen. To correct for changes in disc expansion during a test, a silicon diode measured the temperature. Calibration of the capacitor extensometer was made using NIST fused silica and NIST stainless steel. The total thermal expansion of three silicon specimens measured between 200 and 600 Celsius was 2074.83 strain with a standard deviation of 17.43 strain. The hysteresis difference between the warming and cooling curves for the silicon was less than 7 strain.

Measurements were made of the thermal expansion of silicon between 293K and 873K using a capacitance extensometer. The extensometer consisted of a fused silica rod and tube with attached capacitance disks which were used to measure small extensions precisely. An Andeen-Hagerling capacitance bridge had 10 digits with a 0.5x10^-18 farad sensitivity. The temperature of the specimen was measured with a platinum resistance thermometer. The furnace was temperature controlled with both a heating and cooling rate of 1⁰C/minute. The specimen consisted of a cylindrical prism 2.54 centimeters in length and 0.635 centimeters in diameter. The measuring apparatus, designated as the Advanced Materials Laboratory Inc. (AML) dilatometer was designed by the author. Construction was both contracted and done by the author. Since this was unique equipment, details of the apparatus and measurement method are given. The author also personally made the measurements.

A very expensive part broke that consisted of silicon braised to silicon carbide at 600⁰C. Using the best published data for silicon expansion, Table 3, it was predicted that the parts should have survived braising. Fracture was predicted using the thermal expansion of silicon as 2,074.0x10^-6, this work, and the thermal expansion of silicon carbide as 2,277.5x10^-6. Details of the equipment, calibration, and measurement procedure is presented so that the reader can have confidence that the silicon thermal expansion results are close to the true thermal expansion of silicon.

Figure 1 shows the AML high temperature dilatometer used for high temperature thermal expansion tests. Round capacitor plates used for thermal expansion measurements were electrically connected via coaxial cables to an Andeen Hagerling bridge providing capacitance measurements in 10 digits with a 0.5x10^− 18 farad sensitivity. The furnace and high temperature dilatometer rested upon a vibration isolation table. This minimized vibration that otherwise would cause the tube capacitor plate to move relative to the tube and rod capacitor plate to move relative to the rod. Also, that minimized vibration of the specimen that might have caused undesired movement during expansion measurements.

Figure 2 showed a plastic external sealing container filled with helium that enclosed and protected the capacitor plates. The capacitor plates measured thermal expansion with a sensitivity of 2 nanometers.

Figure 3 shows the rod capacitor plate (F) that rested upon the CTE measuring rod (A) without any clamps. The tube capacitor plate (G) rested upon the CTE measuring tube (K). It was found that clamping these plates onto the CTE measuring tube (K) and the CTE measuring rod (A) respectively caused hysteresis loops during thermal expansion tests as well as non-reproducible expansion results. Since both capacitor plates, and the rod and tube are made of fused silica, this connection arrangement prevented errors in thermal expansion measurement of the specimen as the rod slid back and forth within the tube, Fig. 4.

The CTE measuring tube was housed within a fuse silica furnace tube (L) that was sealed against the aluminum sealing disk (H) outside of the furnace. The inside of the furnace had an alumina tube that surrounded the silica tube. Due to the relatively high thermal conductivity of the alumina tube compared to the silica tube, the temperature within the high temperature dilatometer region enclosing the specimen was fairly uniform. The aluminum cooling vanes (J) kept the capacitor plates close to room temperature as well and the associated apparatus enclosed within the external sealing container (Q). Flowing helium, 10 ml/min. within that container also cooled its contents.

The tube capacitor plate, Fig. 3, had a sputtered gold film approximately 1,700 Angstroms thick over its entire surface. The tube capacitor plate was connected to the high side of the capacitor bridge. The rod capacitor plate had a circular central active area, 5.6086 cm², connected to the low side of the capacitor bridge. Surrounding the active area was a grounded area separated from the active area by 0.0025 centimeters. Both of these areas had 1,700 Angstroms of sputtered gold. The circumferential edge of the tube capacitor plate did not have sputtered gold while the edge of the rod capacitor plate had sputtered gold. These areas were designed to minimize induced currents from other electrical apparatus.

There was a calibrated platinum thermometer (S) attached to the silica tube was 3 mm away from the specimen that accurately measured the temperature to less than a tenth of degree Celsius. The four wires of the thermometer had fused lead wires that were connected to a Lake Shore Cryotronics 805 temperature controller. This controller provided one milliamp current to the platinum thermometer which measured the voltage of both the platinum thermometer and a silicon diode thermometer (R) that was attached to the aluminum frame (P), Fig. 3. These voltages were then converted to temperatures which were automatically transferred to a computer. The platinum thermometer temperature T_A was used to measure the temperature of the specimen while the silicon diode temperature T_B was used to measure the temperature of the capacitor plates.

A brass plate was attached to the aluminum frame with two stainless steel screws. This plate was also attached to a bellows (D) of stainless steel. The other end of the bellows was attached to a brass ring that in turn clamped onto the CTE measuring rod via a nylon screw. In this way the bellows acted as a non-rotating spring in compression providing force upon the rod. A coil spring would cause rotation of the rod during thermal expansion tests resulting in measurement errors.

The tube had two constrictions that surrounded and supported the rod, Fig. 4, with a few thousandths of a centimeter clearance. The rod was coated with Teflon to minimize friction, coefficient of friction = 0.04, while sliding within the tube. The Teflon region was outside of the furnace so that it was not exposed to a temperature significantly different than that of room temperature.

The frame, bellows, and capacitor plates were housed inside an air tight external sealing container (Q) within which helium gas flowed via the inlet valve (N) at 10 ml/minute during the thermal expansion tests. Aluminum cooling vanes (J) were used to keep the contents within the external sealing container near room temperature while flowing helium gas also provided cooling during expansion tests. The container was attached at one side of an aluminum sealing disk (H) while the other side of the disk had the furnace silica tube attached. Neoprene “O” rings provided air tight seals. Helium gas was used to purge the contents within the silica tube prior to thermal expansion tests, but no helium flowed in that tube or the CTE measuring tube or the CTE measuring rod during testing. The reason for this is that gas flow would cause significant temperature differences between adjacent surfaces of the CTE measuring tube and CTE measuring rod, the specimen, and the platinum thermometer. Positive helium pressure caused by flowing helium within the air tight container prevented air from entering the silica tube contents.

Calibration and corrections

The high temperature dilatometer was calibrated with NIST fused silica 739 and NIST stainless steel SRM 738. The reason for calibrating with these two materials was that the pure silicon thermal expansion was 2074.0 µstrain which was between the two NIST specimens used for calibration. The total thermal expansion of pure silicon at 600⁰C was 2,107.2 µstrain using only the NIST fused silica 739 calibration data. The maximum difference between the NIST reported value for thermal expansion and measured thermal expansion between 293K and 873K for fused silica was 1.07 µstrain and for stainless steel it was 2.8 µstrain, see Tables 1 and 2.

Table 1

Comparison between NIST fused silica 739 and AML calibration curve
Temperature, ⁰K	Total thermal expansion, ΔL/L₂₉₃, microstrain
Temperature, ⁰K	NIST fused silica	AML calibration	Difference
293	0	0	0
320	13.5	13.42	-0.08
340	24.5	24.03	-0.47
380	36	35.63	-0.37
420	72	71.58	-0.20
460	97	96.67	-0.33
500	122	121.89	-0.11
560	159	159.97	0.97
640	206	206.88	0.88
720	249	247.93	-1.07
800	288	288.27	0.27
840	307	306.91	-0.09

Table 2

Comparison between NIST stainless steel SRM 738 and AML calibration curve
Temperature, ⁰K	Total thermal expansion, ΔL/L₂₉₃, microstrain
Temperature, ⁰K	NIST stainless steel	AML calibration	Difference
293	0	0.58	0.58
300	69	68.99	-0.01
340	466	465.5	-0.5
380	872	872.8	0.8
420	1288	1289.5	1.5
460	1714	1715.7	1.7
500	2149	2150	1.3
540	2593	2593.6	0.6
580	3048	3045.8	-2.2
620	3511	3508.2	-2.8
660	3984	3981.7	-2.3
700	4467	4468.2	1.2

The standard deviation for the difference between the AML calibration and the NIST fused silica is 0.5860 µstrain and 1.5904 µstrain for the NIST stainless steel.

The cylindrical specimens of NIST fused silica 739 and the NIST stainless steel SRM738 each had a length of 2.525 centimeters and a diameter of 0.635 centimeters. The rate of warming and cooling was controlled by the furnace controller at 1⁰C/minute and thermal expansion measurements were made every three degrees Celsius which was the minimum rate of the furnace controller. Below 200⁰C the rate of cooling was slower due to the heat capacity of the furnace and the low rate of heat transfer between the furnace and surrounding air. Two calibration runs were made for the fused silica and three for the stainless steel. The expansion results were generated by a computer program using the measured calibration data, see Tables 1 and 2. The reason for using both NIST fused silica and NIST stainless steel was to calibrate the dilatometer over a large range of thermal expansion that bracketed the thermal expansion for silicon.

Corrections were made for tilt angle between the capacitor plates and the tilt angle was kept constant for all test at 0.0049 radians by measuring the separation of the plates top and bottom with a microscope. The angular position of the capacitance plates was kept the same for each test. Corrections were made using the bellows force, with a spring force of 786 grams/mm with a spring constant of 3.82 grams/µm, and the elastic constants of the fused quartz rod and tube and specimen during thermal expansion changes. Changes in thermal expansion of the capacitor plates were also corrected using the temperature measurements made by the diode thermometer. The line capacitance was 0.001066 pF and the conductive loss was 0.0019 nanosiemens for a capacitance of 13pF. Fringing fields were considered negligible. There were 14 error tests performed with two NIST quartz calibration tests, two silicon tests, and ten silicon carbide tests. At around 320K the average closure error in thermal expansions was 6.086 microstrain with a standard deviation of 4.355 microstrain. With the error between tests of the same material was 10.87 microstrain with a standard deviation of 3.210 microstrain. The maximum difference between the warming and cooling curves was within 7 microstrain. This small hysteresis difference indicates that the frictional forces between the CTE measuring rod and tube were minimal as well as a small temperature difference between the specimen and platinum thermometer while heating compared to cooling. Calculations for the thermal expansion error due to the frictional force created by movement between the rod and tube was 7x10^− 6.

Experimental procedure

The dilatometer was taken out of the surrounding silica tube that remained in the furnace and the external sealing container (Q) was removed. It was found that the friction between the sides of the specimen and tube was high if the specimen (M) rested on the CTE measuring tube. Therefore, two Teflon shims, each 0.005 centimeters thick, were placed at the base of the opening of the tube and then the specimen was placed on top of the shims between the CTE measuring rod and the CTE measuring tube.

Two stainless screws were loosened so that a brass shim 0.0518 centimeters thick could be placed between the aluminum plate (B) and frame (P). Then the screws were tightened. The nylon screw in the brass ring (E) was loosened so that the rod could move freely in both axial and radial directions. The rod was rotated until a scribed mark on the rod aligned with a scribed mark on the rod capacitor plate. In this way the radial orientation remained the same for every test. The rod was then pressed against the specimen and the nylon screw was tightened. The stainless screws (B) were then loosened and the brass shim was removed. Then the screws tightened the aluminum plate against the frame providing a clamping force of 410 grams upon the specimen via the bellows and CTE measuring rod. The Teflon shims were then removed, bottom one first, leaving the specimen suspended between the round end of the rod (A) and the tube (K), thereby minimizing the chance of moving the specimen. This procedure eliminated hysteresis loops during thermal expansion tests. The rod capacitor plate was then gently moved until the capacitance was between 12 and 14 pf resulting in a plate separation between 0.041 and 0.035 cm.

The external sealing container was replaced and the entire contents within the container and furnace tube was purged with helium gas. Then the contents within the furnace tube was then sealed with helium under pressure and inserted into the furnace. This resulted in no flow of helium gas within the specimen chamber. It was discovered that the flow of helium gas surrounding the specimen caused significant errors in temperature measurements resulting in a large hysteresis between heating and cooling data. The furnace was increased and decreased in temperature 1⁰C/minute with the furnace controller. Thermal expansion measurements were made for every 3⁰C temperature change.

Figure 5 shows the thermal expansion of pure polycrystalline silicon from 293K to 873K. The hysteresis opening was less than ±7 µstrain which was about one third of the ASTM Standard E228. The average thermal expansion ΔL/L₂₉₃ at 600⁰C for three tests (2090.0, 2074.0, 2060.5 µstrain) was 2074.83x10^− 6. Using a range estimate for the standard deviation, the standard deviation was [(2090.0–2060.5)x0.5908 = 17.43x10^− 6, for a sample size of 3. From Eq. (1), ΔL/L₂₉₃ = 2074.8x10^− 6 at 600⁰C. Here ΔL/L₂₉₃ = 0 at 293K.

ΔL/L₂₉₃ = − 822.94 + 2.47562T + 1.03143x10^− 3T² − 7.44467x10^− 8T³ µstrain (1)

Equation (2) gives within 95% probability an estimate for the error [2 standard deviations (s)] for the total thermal expansion ΔL/L₂₉₃ which includes calibration and hysteresis errors.

2s = 2 x {[(17.43/600) x (T-293)]² + 0.5860² + 1.5904² + (7/2)²}^1/2 (2)

Greater accuracy in determining the Coefficient of Thermal Expansion (CTE) was obtained from equations (3) and (4). A minimum of 40 data points were required to differentiate a curve fit to determine correct values of CTE. The Coefficient of Thermal Expansion (CTE) at 600⁰C was 4.0792x10^− 6/⁰C, Eq. (3), where T was the absolute temperature, degrees Kelvin.

From 293K to 340K

CTE = -0.522764 + 1.16897x10^− 2T − 1.90583x10^− 5T² + 4.84252x10^− 8T³ (3)

From 340K to 873K

CTE = 1.72388 + 5.56566x10^− 3T − 5.34039x10^− 6T² + 2.35454x10^− 9T³ (4)

Table 3 presents the total thermal expansion of silicon between 293K and 873K from various references. Table 4 presents the Coefficient of Thermal Expansion (CTE) at 873K from various references. Each reference is described below.

From Watanabe et al. (Table 1 [1]), ΔL/L₂₉₃ = 2,136.5x10^− 6 at 600⁰C with a 95% probability, k = 2, expanded uncertainty of 5.6x10^− 6. His CTE = 4.19x10^− 6/⁰C at 600⁰C. They used a heterodyne laser Michelson Interferometer to make their measurements. The difference in total thermal expansion between Watanabe [1] and results in this report, Altshuler, was {[1](ΔL/L₂₉₃) - [Altshuler]ΔL/L₂₉₃)} = 2,136.5x10^− 6 – 2074.8x10^− 6 = 61.7x10^− 6. If the reported measurement errors were included, the difference between Altshuler and Watanabe should be less than 36.1x10^− 6 with a 95% probability. The larger difference might be attributed to the flow of helium within the testing chamber due leakage from the furnace, Watanabe [1]. This would cause a temperature difference between the thermocouples and the specimen. It was discovered that flow of helium did cause a significant temperature difference between sample and specimen. Therefore, as described in the experimental procedure section of this report, there was no flow of helium gas within the specimen chamber to give more accurate results. The difference in their CTE and Altshuler was {[1]CTE - [Altshuler]CTE)} = 4.19 x10^− 6- 4.08 x10^− 6 = 0.11 x10^− 6 at 600⁰C. This difference might be attributed to the requirement of 40 data points as presented in the result section to give a more accurate value of CTE.

Swenson [2] gave a summary of thermal expansivity of silicon which he considers the most precise data. He presents (Table 2 [2]), the total thermal expansivity at 600⁰C which was ΔL/L₂₉₃ = 2,142.2x10^− 6 and his CTE = 4.225x10^− 6. The difference in his summary and that of Altshuler, was {[2](ΔL/L₂₉₃) - [Altshuler]ΔL/L₂₉₃)} = 2,142.2x10^− 6 – 2074.x10^− 6 = 67.4x10^− 6. The difference in his CTE and Altshuler was {[2]CTE - [Altshuler]CTE)} = 4.24 x10^− 6- 4.08 x10^− 6 = 0.16 x10^− 6.

Roberts [3] presents results of silicon polycrystals using a polarized interferometer with a platinum resistance thermometer. Roberts calculated from his own data ΔL/L₂₉₃ = 2,142.2x10^− 6 using Swenson {Table 1 and Eq. (1)[2]}.

Okaji [4] measured the thermal expansion of single crystal silicon from 300K to 1200K. His reported results for total thermal expansion from his Eq. (2) are somewhat higher than those of Roberts [3], but they seem reasonable.

Mazur [5] measured single crystal silicon oriented in [111] and [100] directions using a push rod dilatometer up to 1100⁰C. His results are quite different from the other referenced results.

Thermal Expansivity of Silicon (ΔL/L₂₉₃) between 20⁰C and 600⁰C

Table 3

Total thermal expansion of silicon between 293K and 873K
Reference	Polycrystal	Single Crystal [110]	Single Crystal [111]	Dilatometer
Present work	2,074.0x10^− 6			Push rod capacitor discs
Watanabe [1]	2,136.5x10^− 6			Michelson interferometer
Roberts, Swenson [2]	2,142.2x10^− 6			Polarized interferometer
Okaji [4]			2,228.9x10^− 6	Interferometric dilatometer
Mazur [5]		3,718.8x10^− 6	3,362.5x10^− 6	Push rod dilatometer

Coefficient of Thermal Expansion of silicon (CTE) per degree Celsius at 600⁰C

Table 4

Coefficient of Thermal Expansion of silicon at 873K
Reference	Polycrystal	Single Crystal [110]	Single Crystal [111]	Dilatometer
Present work	4.08x10^− 6/⁰C			Push rod capacitor discs
Watanabe [1]	4.19x10^− 6/⁰C			Michelson interferometer
Roberts, Swenson [2]	4.24x10^− 6/⁰C			Polarized interferometer
Okaji [4]			4.15x10^− 6/⁰C	Interferometric dilatometer
Mazur [5]		4.35x10^− 6/⁰C	3.90x10^− 6/⁰C	Push rod dilatometer

For the above stated reasons, it is believed that the total thermal expansion and CTE results given in this report is closer to the true behavior of silicon than the referenced published results.

5.1 The total thermal expansion of pure silicon between 20 and 600 degrees Celsius was 2074 µstrain.

5.2 The Coefficient of Thermal Expansion (CTE) at 600 degrees Celsius was 4.08 µstrain/⁰C

The surface of the specimen must not touch the CTE measuring tube except at its end. The specimen should be suspended by clamping its ends between the CTE measuring tube and the CTE measuring rod.

5.3 Helium gas must not flow in the specimen chamber. Otherwise, there would be a difference between the temperature of the measuring thermometer and the specimen creating an error in thermal expansion determination.

5.4 The total expansion measurement hysteresis between the heating and cooling curves of the specimen is indicative of:

temperature differences between the specimen and measuring thermometer.
the effect of friction between the specimen and dilatometer

The thermal expansion of silicon is important in designing and building mirrors for use in space.

7.1 Financial Interests

No funding was received for conducting this study.

7.2 Conflict of Interest

The author certifies that the has no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.

H.Watanabe, N. Yamada, and M. Okaji, “Linear Thermal Expansion Coefficient from 293 to 1000 K, International Journal of Thermophysics, Vol. 25, No. 1, pp 221-234, January 2004.
C.A.Swenson, “Recommended Values for the Thermal Expansivity of Silicon from 0 to 100K”, Journal of Physical and Chemical Reference Data, Data 12, 179, pp 179-182, (1983), https://doi.org/10.1063/1.555681 Accessed 15 October 2009
Roberts, R. B., "Thermal expansion reference data: silicon 300-850 K." Journal of Physics D: Applied Physics 14, no. 10 (1981): L163.
Okaji, M., “Absolute thermal expansion measurement of single-crystal silicon in the range 300-1300K with an interferometric dilatometer”, International Journal of Thermophysics, 9, No. 6, pp. 1101-1109, (1988).
Mazur, A.V., Gasik, M.M., “Thermal Expansion of Silicon up to 1100^oC”, Journal of Materials Processing Technology, Vol. 209, Issue 2, pp 723-727, January 2009.

No competing interests reported.

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Thermal Expansion of Silicon Between 293k and 873k Using a Capacitor Dilatometer

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