Modified La-Fe-Si magnetocaloric alloys
20220238262 · 2022-07-28
Inventors
- Arjun K. Pathak (Willamsville, NY, US)
- Yaroslav Mudryk (Ames, IA, US)
- Oleksandr Dolotko (Karlsruhe, DE)
- Vitalij K. Pecharsky (Ames, IA)
Cpc classification
B33Y70/00
PERFORMING OPERATIONS; TRANSPORTING
B22F2999/00
PERFORMING OPERATIONS; TRANSPORTING
Y02B30/00
GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
C22C33/0257
CHEMISTRY; METALLURGY
B22F10/28
PERFORMING OPERATIONS; TRANSPORTING
C22C38/005
CHEMISTRY; METALLURGY
B33Y80/00
PERFORMING OPERATIONS; TRANSPORTING
B22F2999/00
PERFORMING OPERATIONS; TRANSPORTING
C22C33/0257
CHEMISTRY; METALLURGY
Y02P10/25
GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
International classification
B33Y70/00
PERFORMING OPERATIONS; TRANSPORTING
Abstract
A magnetocaloric material comprising a La—Fe—Si based alloy composition that is compositionally modified to include a small but effective amount of at least one of Al, Ga, and In to improve mechanical stability of the alloy (substantially reduce alloy brittleness), improve thermal conductivity, and preserve comparable or provide improved magnetocaloric effects. The alloy composition may be further modified by inclusion of at least one of Co, Mn, Cr, and V as well as interstitial hydrogen.
Claims
1. A magnetocaloric alloy comprising a magnetocaloric intermetallic compound phase having a crystal structure in which La, Fe and Si and at least one of Al, Ga, and In are present in solid solution in an amount of 0.07 atomic % to 3.5 atomic % of the alloy which amount is selected to reduce alloy brittleness as compared to an identical alloy devoid of Al, Ga, and In to increase the number of temperature/magnetic field cycles without alloy cracking or decrepitating.
2. (canceled)
3. The alloy of claim 1 wherein said at least one of Al, Ga, and In is/are substituted for an amount of the Si of the alloy.
4. The alloy of claim 1 also including at least one of Co, Mn, Cr, and V.
5. The alloy of claim 4 wherein said at least one of Co, Mn, Cr, and V is/are substituted for some of the Fe of the alloy.
6. The alloy of claim 1 including interstitial hydrogen.
7. The alloy of claim 8 which is solidified and then heat treated.
8. The alloy of claim 1 which is solidified from a bulk alloy melt.
9. A magnetocaloric alloy having reduced brittleness and one of the following compositions: LaFe.sub.13-zSi.sub.z-yX.sub.y, where X═Al, Ga, In with 1.1≤z≤1.9 and 0.01≤y≤0.5 La(Fe.sub.1-wM.sub.w).sub.13-zSi.sub.z-yX.sub.y, where X═Al, Ga, In, and M=Co, Mn, Cr, V with 0.05≤w≤0.1, and 1.1≤z≤1.9, and 0.01≤y≤0.5 LaFe.sub.13-zSi.sub.z-yX.sub.yH.sub.v, where X═Al, Ga, In with 1.1≤z≤1.9, 0.01≤y≤0.5, 0<v 2.3 La(Fe.sub.1-wM.sub.w).sub.13-zSi.sub.z-yX.sub.yH.sub.v, where X═Al, Ga, In, and M=Co, Mn, Cr, V with 0.05≤w≤0.1, and 1.1≤z≤1.9, and 0.01≤y≤0.5, and 0<v≤2.3.
10. The alloy of claim 9 exhibiting a magnetocaloric effect that is tunable between about 170 K and about 350 K.
11. The alloy of claim 9 exhibiting increased thermal conductivity as compared to the parent La—Fe—Si alloy devoid of Al, Ga, and In.
12. (canceled)
13. The alloy of claim 9 that is machinable.
14. A magnetic regenerator comprising the alloy of claim 1.
15. The regenerator of claim 14 which is in the form of particles or plates or packed spheres.
16. The regenerator of claim 14 which comprises packed irregularly shaped particles.
17. The regenerator of claim 14 comprising 3D-printed spherical particles comprising said alloy and/or 3D-printed irregularly shaped particles comprising said alloy.
18. The regenerator of claim 14 which comprises parallel plates.
19. A magnetic refrigerator using a regenerator made from material of claim 1.
20. A magnetic heat pump using a regenerator made from material of claim 1.
21. The alloy of claim 1 wherein said at least one of Al, Ga, and In is/are present in an amount of 0.6 atomic % to 3.5 atomic % of the alloy.
22. The alloy of claim 1 wherein said at least one of Al, Ga, and In is/are present in an amount of 0.7 atomic % to 3.5 atomic % of the alloy.
Description
BRIEF DESCRIPTION OF THE DRAWINGS
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DETAILED DESCRIPTION OF THE INVENTION
[0041] Embodiments of the present invention relate to certain magnetocaloric materials that exhibit the so-called magnetocaloric effect, which is a thermal response of the material when subject to an external applied magnetic field change. The magnetocaloric effect is more prominent closer to large magnetization changes in respect to the temperature, i.e., at magnetic transitions. Different magnetic transitions may present large magnetocaloric effect: ferromagnetic (FM) to paramagnetic (PM); antiferromagnetic (AFM) to FM; spin glass to FM; etc.
[0042] The present invention embodies new chemical compositions that lead to improvements of La—Fe—Si based magnetocaloric alloys, such as the first-order phase-transition LaFe.sub.13-zSi.sub.z system, where 1.1≤z≤1.9, which are achieved by inclusion of a small but effective amount of at least one of Group 13 elements; Al, Ga, and/or In, to improve mechanical stability of the solid alloys by significantly reducing their brittleness, enabling the solid alloys to be manually handled and serve as working bodies in magnetic regenerators without decrepitation (breaking apart) or disintegration from the handling or during operation. Moreover, this improvement is achieved while preserving the NaZn.sub.13-type crystal structure and preserving similar or providing even higher magnetocaloric effects as compared to other known and commercially available La—Fe—Si based magnetocaloric materials. Certain illustrative embodiments include at least one of Al, Ga, and/or In in the alloys in an amount of 0.07 atomic % to 3.5 atomic % to this end. These improvements are achieved in bulk arc-melt/drop-cast condition, and/or in heat treated condition, and/or in rapidly solidified condition (after melt spinning, atomization, or after any other rapid solidification technique). Moreover, the materials do not disintegrate or decrepitate after hydrogenation.
[0043] Certain illustrative embodiments of the present invention involve substituting at least one of Al, Ga, and/or In for at least some Si in the La—Fe—Si based alloy composition. Optionally, at least one of Co, Mn, Cr, and V can be substituted for Fe in the modified La—Fe—Si based alloy composition (i.e. modified with at least one of Al, Ga, In), and interstitial hydrogen optionally can be included in the modified La—Fe—Si based alloy composition. Another illustrative embodiment of the present invention involves the above first-order phase-transition magnetocaloric alloy that further includes M which is at least one of Co, Mn, Cr, and V substituted for Fe in the modified La—Fe—Si based alloy composition. Such alloy compositions are formulated by cheap and easily processed elements with large mechanical integrity.
[0044] The modified La—Fe—Si alloy compositions can be chemically tailored as just described in a manner to tune the magneto-structural transition temperatures of the materials; for example, to provide a large magnetocaloric effect that is tunable between about 170 K and about 350 K. Such alloys exhibit metamagnetic transitions from the paramagnetic states in zero or low magnetic fields (generally less than 0.1 Tesla) into ferromagnetic states when induced by the magnetic field of 0.5 Tesla or higher applied at or near the respective Curie temperature. The metamagnetic transitions proceed with phase volume change but without crystal symmetry change. A reverse metamagnetic transition occurs when the high magnetic field is reduced back to less than 0.1 Tesla. The magneto-structural transition temperatures of the modified alloy compositions also may be controlled by alloy chemistries and/or finely tuned by heat-treatments.
[0045] Illustrative embodiments of chemically modified alloy compositions pursuant to embodiments of the invention include, but are not limited to, the following compositions: [0046] a. LaFe.sub.13-zSi.sub.z-yX.sub.y, where X═Al, Ga, In with 1.1≤z≤1.9 and 0.01≤y≤0.5 [0047] b. La(Fe.sub.1-wM.sub.w).sub.13-zSi.sub.z-yX.sub.y, where X═Al, Ga, In, and M=Co, Mn, Cr, V with 0.05:5≤w≤0.1, and 1.1≤z≤1.9, and 0.01≤y≤0.5 [0048] c. LaFe.sub.13-zSi.sub.z-yX.sub.yH.sub.v, where X═Al, Ga, In with 1.1≤z≤1.9, 0.01≤y≤0.5, 0≤v≤2.3 [0049] d. La(Fe.sub.1-wM.sub.w).sub.13-zSi.sub.z-yX.sub.yH.sub.v, where X═Al, Ga, In, and M=Co, Mn, Cr, V with 0.05≤w≤0.1, and 1.1≤z≤1.9, and 0.01≤y≤0.5, and 0≤v≤2.3
and exhibiting large magnetocaloric effect tunable between about 170 K and about 350 K as well as having improved mechanical stability and even some machinability.
[0050] The magnetocaloric alloys can be made with improved properties both in bulk arc-melt/drop-cast condition with subsequent heat treatment and in rapidly solidified condition (melt spinning, any atomization method, e.g., gas atomization, or any other rapid solidification technique). An illustrative processing method involves arc-melting the elemental components of the alloy composition in the correct stoichiometry, or with an excess of 3.5 wt. % Mn, if used, to account for its loss due to evaporation. The solidified material is then (optional step) drop-cast from a high temperature molten state to form a more chemically homogeneous casting. The casting can be then re-melted and re-solidified, and subjected to a homogenizing heat treatment. Alternately, the casting can be rapidly solidified; for example, using melt-spinning, splat cooling or atomization. For example, the melt-spun ribbons are prepared using induction melting of ingots in a quartz crucible at ⅓ atmosphere pressure of high purity He gas at the temperature of approximately 1570 K, and then ejected at approximately 0.1 atmosphere overpressure of helium at onto a copper chill wheel rotating at a tangential speed of about 25 to 50 m/s, which are common parameters of the melt-spinning technique. The parent phase is easily formed and stable. Other methods of rapid solidification of molten alloy composition can be utilized including, but not limited to atomization, selective laser melting, and 3D printing. A regenerator can be custom designed and fabricated using spherical and/or irregular particles of the modified La—Fe—Si alloy pursuant to embodiments of the invention (without or with appropriate binder) using 3D printing technology using direct metal laser sintering, selective laser sintering, and/or powder bed and inkjet head 3D printing.
[0051] Following drop casting or rapid solidification, the solidified alloy material may then be heat-treated at a temperature and for a time to crystallize any possible remains of amorphous material, and/or to complete solid state reactions, and/or to release internal stress after the rapid quenching.
[0052] The rapidly solidified alloy material, or the heat treated alloy material from the preceding paragraphs, may be subjected to optional annealing heat treatment at different temperatures and times to closely control the magneto-structural transition temperature of the material and thus the operation temperature of magnetic refrigeration material.
[0053] The following examples are offered to further illustrate but not limit embodiments of the present invention:
EXAMPLES
Example 1
Alloy Preparation
[0054] Approximately 10 gm of a stoichiometric LaFe.sub.13-zSi.sub.z-yX.sub.y, where X═Al, Ga and/or In, and 1.1≤z≤1.9 and 0.01≤y≤0.5 alloys were prepared by arc melting process. Alloy compositions were prepared with high purity elements Fe, Si, and X being weighed stoichiometrically and arc melted three to five times under argon atmosphere followed by addition of high purity La and arc melting three to five times under high purity argon atmosphere. The arc melted buttons were then drop-cast into an about 1 cm diameter ingot for a finer grain size and homogeneous solidification structure throughout the ingot.
[0055] The alloy compositions so made are represented by: [0056] LaFe.sub.11.8Si.sub.1.1In.sub.0.1 [0057] LaFe.sub.11.8Si.sub.1.1Al.sub.0.1 [0058] LaFe.sub.11.8Si.sub.1.1Ga.sub.0.1 [0059] LaFe.sub.11.5Si.sub.1.4In.sub.0.1
[0060] The as-prepared, drop-cast alloys were heat treated at 1050′C for two weeks followed by slow cooling to room temperature. Room temperature x-ray diffraction (XRD) confirmed formation of the major phase with NaZn.sub.13-type crystal structure containing about 10-20% by volume of α-iron as an impurity phase, the presence of which did not deteriorate the value of the magnetic entropy change, yet the alloys demonstrated much improved mechanical stability.
[0061] In particular, improvement in mechanical stability of the heat treated solid alloys was evidenced by their being able to be manually handled and tested as described below without decrepitation, disintegrating, or shattering if dropped as a result of their substantially reduced alloy brittleness. In contrast, the parent heat treated solid alloys (i.e. the same alloys without inclusion of Al, Ga, and/or In) are excessively brittle and disintegrate in one's hand with the slightest of manipulation, and they also decrepitate on their own upon hydrogenation due to an intrinsic volume expansion.
Example 2
Hydrogenation
[0062] The Curie temperature of the alloy samples of Example 1 can be increased to and above room temperature by proper hydrogenation of the alloys, which preserves the strong metamagnetic transition thereby keeping the large magnetocaloric effect of the alloys. In order to bring the Curie temperature close to room temperature, several drop cast alloys were hydrogenated. Hydrogenation can be effected by annealing the heat treated solid alloy in a high purity hydrogen or hydrogen-containing atmosphere for a time sufficient for the alloy to absorb interstitial hydrogen. For example, the following procedure was used to introduce interstitial hydrogen into each of certain alloy samples, as illustrated below for LaFe.sub.13-zSi.sub.z-yX.sub.y, where X═In, z=1.2, and y=0.1, i.e., for the alloy with LaFe.sub.11.8Si.sub.1.1In.sub.0.1 composition. Each sample received multiple activations set forth below.
First Activation:
[0063] Each LaFe.sub.11.8Si.sub.1.1In.sub.0.1 sample was loaded in an autoclave. [0064] The sample was heated to 300° C. for about 6 h in vacuum. After this the sample was cooled to 230° C. and kept at this temperature overnight (about 17 hours) under vacuum. [0065] While keeping sample at 230° C., the reservoir was pressurized to about 5 bar hydrogen pressure (V.sub.(reservoir)=12.26 ml, V.sub.(autoclave)=16.704 ml) and kept at the hydrogen atmosphere for 24 hours. [0066] After that the sample was evacuated, and while keeping sample in vacuum, heated to 360° C., held at this temperature for 1 hour. After full dehydrogenation the sample was cooled down to room temperature with a cooling rate of 1° C./min.
Second Activation (FIG. 2) of the Sample:
[0067] At room temperature the same sample was pressurized to about 5 bar hydrogen pressure. When the measurement was started, the sample was heated to 360° C. at a rate 1° C./min. [0068] After the temperature reached 360° C., the sample was kept for 10 hours at this temperature in hydrogen atmosphere. [0069] Then the sample was cooled down to room temperature in a cooling rate of 1° C./min. [0070] After this the sample was evacuated at room temperature. [0071] Keeping the sample under vacuum the autoclave was heated to 360° C., kept at this temperature for 5 hours and cooled to room temperature.
Third Activation (FIG. 3):
[0072] At room temperature the same sample then was pressurized to about 5 bar hydrogen pressure. When kinetic measurement was started, the sample was heated to 200° C. at a rate 1° C./min. [0073] The temperature of 200° C. was kept for 5 hours. [0074] After this temperature of the sample was increased to 370° C. with a rate of 1° C./min and kept at this temperature for 10 hours [0075] Then the sample was cooled down to room temperature in a cooling rate of 1° C./min. [0076] After this the sample was evacuated at room temperature. [0077] Keeping the sample under vacuum the autoclave was heated from room temperature to 360° C., kept at this temperature for 2 hours and cooled to room temperature.
Fourth Activation (FIG. 4):
[0078] At room temperature the same sample then was pressurized to about 5 bar hydrogen pressure. When kinetic measurement was started, the sample was heated to 200° C. at a rate 1° C./min. [0079] The temperature of 200° C. was kept for 5 hours. [0080] After this temperature of the sample was increased to 370° C. with a rate of 1° C./min and kept at this temperature for 10 hours [0081] Then the sample was cooled down to room temperature in a cooling rate of 1° C./min. [0082] After this the sample was evacuated at room temperature. [0083] Keeping the sample under vacuum the autoclave was heated from room temperature to 360° C., kept at this temperature for 2 hours and cooled to room temperature.
Final Hydrogenation and Homogenization of the Same Sample (FIG. 5):
[0084] At room temperature the same sample then was pressurized to about 5 bar hydrogen pressure. When kinetic measurement was started, the sample was heated to 200° C. at a rate 1° C./min. [0085] The temperature of 200° C. was kept for 5 hours. [0086] After this temperature of the sample was increased to 370° C. in a rate of 1° C./min and kept at this temperature for 10 hours. [0087] Then the sample was cooled down to 100° C. with a cooling rate of 1° C./min and held at this temperature for 12 hours. This step ensures homogenous distribution of hydrogen in the sample. [0088] After the homogenization the sample was cooled down to room temperature with a cooling rate of 1° C./min. [0089] According to data presented in
Interstitial hydrides so made are represented by: [0091] LaFe.sub.11.8Si.sub.1.1In.sub.0.1H.sub.2.3 [0092] LaFe.sub.11.8Si.sub.1.1Al.sub.0.1H.sub.2.1
[0093] Room temperature x-ray diffraction,
TABLE-US-00001 TABLE 1 Crystallographic data of LaFen.sub.11.8Si.sub.1.1In.sub.0.1H.sub.2.3 (before hydrogenation) and LaFen.sub.11.8Si.sub.1.1In.sub.0.1H.sub.2.3 (after hydrogenation) obtained as a result of Rietveld refinement. Space group Fm-3c. La occupies the 8a (¼, ¼, ¼) position, Fe1 occupies the 8b position (0, 0, 0), and a mixture of (Fe.sub.10.8Si.sub.1.1In.sub.0.1) occupies the 96i position (0, y/b, z/c). Only y and z coordinates of this position are listed in the table with the numbers in parentheses representing estimatederrors in the last significant digit Coordinates of (Fe.sub.10.8Si.sub.1.1In.sub.0.1) in the unit cell Material a (Å) y/b z/c LaFe.sub.11.88Si.sub.1.1In.sub.0.1 11.4754(3), 0.1170(2) 0.1793(2) LaFe.sub.11.8Si.sub.1.1In.sub.0.1H.sub.2.3 11.6210(4) 0.1152(2) 0.1783(2)
Example 3
Curie Temperatures
[0094] Isofield magnetization (M) measurements were carried out on samples of Example 1 in a Quantum Design Physical Property Measurement System (QD-PPMS) with a vibrating sample magnetometer insert.
[0095] The Curie temperature, T.sub.C, of the LaFe.sub.11.8Si.sub.1.1In.sub.0.1 sample was determined from magnetization vs temperature data, M(T), measured in several constant magnetic field shown in
[0096] Similarly, the Curie temperature, T.sub.C, of the LaFe.sub.11.8Si.sub.1.1Al.sub.0.1 sample was determined to be 173K while heating, see
[0097] Similarly, the Curie temperature, T.sub.C, of the LaFe.sub.11.8Si.sub.1.1Ga.sub.0.1 sample was determined to be 187K while heating, see
[0098] Similarly, the Curie temperature, T.sub.C, of the LaFe.sub.11.5Si.sub.1.4In.sub.0.1 sample was determined to be 207 K while heating, see
[0099] Similarly, the Curie temperature, T.sub.C, of the hydrogenated LaFe.sub.11.8Si.sub.1.1In.sub.0.1H.sub.2.3 sample was determined to be 345 K while heating, see
[0100] Similarly, the Curie temperature, T.sub.C, of the hydrogenated LaFe.sub.11.8Si.sub.1.1Al.sub.0.1H.sub.2.1 sample was determined to be 355 K while heating, see
[0101] In all of these samples, the shift of the Curie temperature with magnetic field is greater than 4K/Tesla,
Example 4
Magnetocaloric Effect (Magnetic Entropy Change, ΔS.SUB.M.)
[0102] All materials described in Examples 1, 2, and 3 exhibit strong metamagnetic transitions in isothermal M(H) measurements. The calculations of the magnetic entropy change, ΔS.sub.m, were performed employing Maxwell equation using isofield magnetization measurements carried with 1 K-min-1 temperature sweeps. Isofield magnetization, M(T), measurements were performed in a Quantum Design Physical Property Measurement System (QD-PPMS) with a vibrating sample magnetometer.
[0103] The corresponding values of the magnetic entropy change for LaFe.sub.11.8Si.sub.1.1In.sub.0.1 are shown in
Example 5
Rapidly Solidified Materials
[0104] The rapidly solidified melt-spun ribbons were prepared using induction melting of drop-cast La(FeSiX).sub.13 alloys (where X is at least one of Al, Ga, and In as indicated above) in a quartz crucible at ⅓ atmosphere of high purity He gas and then ejected at 0.1 atmosphere overpressure of helium at the melt temperature of 1570 K onto a copper chill wheel rotating at a tangential speed of about 25 m/s. The rapidly solidified ribbons then were optionally heat-treated in order to crystallize possible remains of amorphous material and/or release internal stress after the rapid quenching. Moreover, annealing of the ribbons at different temperatures allows control of the transition temperature and thus operation temperature of magnetic refrigeration material.
Example 6
Heat Capacity and Thermal Conductivity of Drop-Cast/Heat-Treated Materials
[0105] The heat capacity of drop cast and heat treated LaFe.sub.11.8Si.sub.1.1In.sub.0.1 sample of Example 1 was measured in magnetic fields up to 5 Tesla (
[0106] Thermal conductivity of drop cast and heat treated LaFe.sub.11.8Si.sub.1.1In.sub.0.1 sample of Example 1 was found to be 14 W/K-m at 300 K (
[0107] Alloys pursuant to the present invention thus show much improved inherent mechanical stability, higher thermal conductivity, and preserve equivalent or even higher magnetocaloric effects compared to other known and commercially available La—Fe—Si based magnetocaloric materials. Moreover, the alloys contain no toxic or hazardous elements and do not contain expensive elements. Curie temperature (operating temperature) T.sub.C can be properly tuned to room temperature, with an extremely large magnetocaloric effect (here evaluated as entropy change, ΔS). The expected commercial applications include, but are not limited to, magnetic refrigeration and magnetic heat pumping.
[0108] Although the present invention is described above with respect to certain illustrative embodiments, the invention is not limited to these embodiments and changes and modifications can be made therein within the scope of the appended claims.
[0109] References which are incorporated herein by reference: [0110] 1. S. Fujieda, A. Fujita, and K. Fukamichi, Appl. Phys. Lett. 81, 1276 (2002). [0111] 2. N. H. Dung, Z. Q. Ou, L. Caron, L. Zhang, D. T. Cam Thanh, G. A. de Wijs, R. A. de Groot, K. H. Jirgen Buschow and E. BrUck, Adv. Ener. Mat., V-1, 1215 (2011). [0112] 3. H. Wada, and Y. Tanabe, Appl. Phys. Lett., 79, 3302 (2001). [0113] 4. Y. K. Fang, C. C. Yeh, C. W. Chang, W. C. Chang, M. G. Zhuband W. Li, Scripta Materialia 57 453 (2007). [0114] 5. S. Dankov, A. M. Tishin, V. K. Pecharsky, K. A. Gschneidner Jr., Phys. Rev. B, V-57, 3478 (1998). [0115] 6. K. A. Gschneidner, V. K. Pecharsky, A. O. Tsokol, Rep. Prog. Physics, V-68, 1479 (2005). [0116] 7. V. K. Pecharsky, K. A. Gschneidner Jr, Phys. Rev. Lett., V-78, 4494 (1997). [0117] 8. Chen et al, United States Patent: US 2014/0166159 A1 [0118] 9. Hu et al, U.S. Pat. No. 9,657,971 B2 [0119] 10. Kogure et al, U.S. Pat. No. 7,186,303 B2 [0120] 11. A. Fujita, S. Fujieda, Y. Hasegawa, and K. Fukamichi, Phys. Rev.B, V-67, 104416 (2003).