- Email: [email protected]
Pressure-induced superconductivity in Li and Fe
Physica C 408–410 (2004) 750–753 www.elsevier.com/locate/physc
Pressure-induced superconductivity in Li and Fe Katsuya Shimizu
a,b,*
, Daigoroh Takao b, Shigeyuki Furomoto b, Kiichi Amaya
b
a
b
Research Center for Materials Science at Extreme Conditions, Osaka University, 1-3, Machikaneyama, Toyonaka, Osaka 560-8531, Japan Graduate School of Engineering Science, Osaka University, 1-3, Machikaneyama, Toyonaka, Osaka 560-8531, Japan
Abstract We have developed complex extreme condition of very low temperature down to 30 mK and very high pressure exceeding 200 GPa by assembling compact diamond anvil cell (DAC) on a powerful 3 He/4 He dilution refrigerator. Using the apparatus and techniques, we have searched for pressure-induced superconductivity in various materials under pressures. In this paper, our experimental techniques and the examples of pressure-induced superconductivity in Li and Fe are reviewed. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Pressure; Superconductivity; Lithium; Iron; Electrical resistance
1. Introduction A compact DAC could be easily assembled on a powerful 3 He/4 He dilution refrigerator then this combined system enabled us to perform various experiments under the extreme conditions of very low-temperature of 30 mK and very high-pressure exceeding 200 GPa at the same time. We have employed good thermal conductors and non-magnetic materials for all parts of the pressure cell and performed electrical resistance measurements and/or sensitive magnetization measurements, and searched for pressure-induced superconductivity in simple systems. One of the most aggressive challenges under high pressure research is the search for metallic hydrogen under its extremely dense phase. The metallic hydrogen is predicted [1] to show superconductivity even at room temperature and have been the long fascinate the highpressure physicists, which exists in fact at interior of giant planet such as Jupiter and Saturn. The metallic
*
Corresponding author. Tel.: +81-66-850-6675; fax: +81-66850-6662. E-mail address: [email protected] (K. Shimizu).
hydrogen may form the same electronic structure of alkaline metals. Then we expected that highly compressed lithium can be treated as prototype of the metallic hydrogen and have studied the possibility of superconducting transition of lithium. Another challenge topic is superconductivity against magnetism. It is well known that a ferromagnetic metal does not show superconductivity and even a small amount of paramagnetic impurities could suppress the superconductivity. However even in the case of iron, the superconducting transition temperature (Tc ) was theoretically predicted to be 0.25 K by Wohlfarth [2]. We may expect the appearance of superconductivity in magnetic metals in its non-magnetic state under certain pressure and low temperature, and we have long searched superconductivity in the high-pressure non-magnetic phase of iron.
2. Experimental For the purpose of detecting superconducting transition, we employed sensitive ac-4 terminal method for electrical resistance measurement [3]. In the conventional configuration Au or Pt electrodes is set on the pressure surface of diamond anvils. In this configuration
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.03.077
K. Shimizu et al. / Physica C 408–410 (2004) 750–753
the sample is pressed directly by the diamond and consequently an axial stress is applied to the sample. To improve the quality of pressure, proper pressure media such as mixed alcohol or NaCl is used. In both cases, hard metallic gasket such as rhenium is covered with thin layer of aluminum oxide for electrical insulation between the gasket and the electrodes. The setting of the electrodes is made by hand and the typical size of the sample of the order of 108 cm3 (typically for 100 GPa) makes this procedure difficult. Magnetization measurement is performed by a SQUID magnetometer with a pair of pick up coils; the detecting coil is for the sample in DAC and another coil is set for the reference sample as well as for compensation against uniform external noise. This dc method is available at temperatures below several K, where the pick up coil (such as NbTi) is at superconducting state. In the present study we performed in situ pressuredetermination by the ruby method. Several small ruby chips on the sample are irradiated by argon laser and empirical relation between observed wave length of ruby fluorescent line. The pressure value is employed at room temperature as well as low temperature below 10 K.
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reaction with surrounding materials such as diamonds and sample chamber materials. We made a sample chamber (pit) on a surface of the diamond-anvil as shown in Fig. 1 and confined there lithium sample at low temperature. This procedure worked well and enabled us to perform measurements up to 85 GPa. We could observe the resistance drop showing the appearance of superconductive state of lithium under pressure [8]. As shown in Fig. 2, superconductivity was observed at pressures above 25 GPa and the Tc reached 20 K at about 50 GPa in Run#1. The observed Tc value is the highest in elements. Further pressure studies (Run#2) shows, however, that the curve of Tc versus pressure becomes almost flat above 40 GPa and change the slope at around 60 GPa. The deference in Tc curve comes from experimental error on pressure determination in the sample chamber. This pressure
3. Pressure-induced superconductivity in lithium Lithium is the third element on the periodic table and well know as the lightest metal in the periodic table. For long time, mono-valent metals like alkaline metals are believed to show no superconductivity even at very low temperature. However Cs metal was found to become superconductive under pressure [4] and there arises possibility of pressure-induced superconductivity also in other lighter alkaline metals. The first motive force of searching superconductivity in lithium is the above possibility and also the possibility of the high-Tc in lithium. According to the conventional BSC theory, Tc is inversely proportional to the square root of the mass. The lightest metal of lithium is expect to show the highest Tc in the element. Recent calculation [5] predicted the pressure dependence of Tc of lithium and give considerable high Tc values reaching up to 80 K at around 40 GPa. The second arises from a recent theoretical prediction [6] which showed the new structural transition of Li towards the formation of pairing of Li atoms under higher pressure. In fact, we reported experimental studies showed a slight increment of electrical resistance by increasing pressure up to 30 GPa [7], however those still did not explain the tendency towards the insulating phase. We started the search for superconductivity of lithium by the electrical resistance measurement under pressure, however there were difficulties on confinement of Li sample in our DAC because of the strong chemical
Fig. 1. Photograph of a pit of 50-micron in diameter and 7micron in depth on the 200-micron pressure surface of synthetic Ib diamond-anvil; scale bar ¼ 0.5 mm.
Fig. 2. Pressure dependence of Tc of lithium. Dotted lines indicate the structural phase boundary observed by X-ray diffraction studies [8] at temperature of 180 K.
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K. Shimizu et al. / Physica C 408–410 (2004) 750–753
dependence of Tc may reflect the structural changes under pressure [9]. We did not observe the sign of the metal-to-insulator transition that predicted at pressure exceeding 100 GPa. The further experimental investigation for higher-pressure behaviour of Li should help the total understanding of the fundamental property of metals and possible room-temperature superconductivity [10] in metallic hydrogen.
4. Pressure-induced superconductivity in iron It is well known that a ferromagnetic metal does not show superconductivity down to the lowest temperature. So long as we look at the periodic table of elements, the idea of absence of Tc of magnetic metals seems to be accepted quite in general. We expected the appearance of Tc of magnetic metals in its non-magnetic state under certain high pressure and at low temperature. In the case of iron, there appears non-magnetic state at pressures after the crystal phase transition from bcc to hcp phase at around 15 GPa. The non-magnetism of hcp iron (eFe) is confirmed by M€ ossbauer experiments [11,12]. There are also theoretical predictions [13] on Tc of iron in its non-magnetic state, giving Tc below 1 K. We have searched for superconductivity of iron under pressure in the past 10 years. After searching in the wide range of pressure up to 100 GPa and temperature down to 50 mK, we noticed that purification of the sample and application of quasi hydro-static pressure using proper pressure media should be the most important. In fact, pressurization of sample in pressure media such as NaCl was found to be very effective in the present case of iron, avoiding excess crystal distortion and uniaxial pressure distribution in the course of pressurization. The final arrangement of the sample and pressure chamber is shown in Fig. 3 with which the superconductivity of iron was obtained [14]. Sample thickness is 40 lm and the metal gasket (Re) is covered with thin Al2 O3 layer for electrical insulation. The gold wires of 10 lm in diameter attached to the sample crystal and connected to the platinum electrode at the out side the chamber to perform 4-terminal electrical resistance measurements. The chamber is filled with sodium chloride (NaCl) powder as the pressure-transmitting medium. We studied the pressure dependence of Tc and found that Tc appears at 15 GPa and disappears at 30 GPa, giving the maximum value of 2 K at 21 GPa with very small resistance drop and Meissner signal. These may suggest that the superconductivity is sensitive to both sample and pressure quality. The superconductivity seem appear in the non-magnetic state, however, a theoretical calculation was performed by Jarlborg [15] in which the spin fluctuation coupling, especially of ferro-
Fig. 3. Photograph of the sample and electrodes on the pressure surface of the diamond-anvil under the pressure of 23 GPa; scale bar ¼ 0.1 mm.
magnetic nature, is estimated to be sufficiently large to explain superconductivity of hcp iron. We tried to estimate the phase boundary of bcc-to-hcp structure even if the transition under pressure has a wide pressure-width and is sluggish. The bcc-to-hcp (a to e) transition can be detected by electrical resistance changes as shown in Fig. 4. The experiments are done with the same arrangement as Fig. 3 with which the superconductivity of iron was observed. Resistance curves shows the broad step at the structural boundary; in the case of increasing pressure at
Fig. 4. Resistance curves of iron under pressure across the phase boundary of a to e phase at temperature of 290 and 10 K. As shown in dotted line clear evidence was not observed in the curve with increasing pressure at low temperature of 10 K.
K. Shimizu et al. / Physica C 408–410 (2004) 750–753
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5. Summary Here we reviewed recent developments of complex extreme condition of low temperature and high pressure and recent results of pressure-induced superconductivity. Following the case of lithium or iron, a new type of superconductivity may be discovered elemental materials in high-pressure. Indeed, our final target is production of metallic hydrogen and detection of its possible high Tc superconductive state which are expected to be realized under ultra-high but almost attainable pressure of 400 GPa. Fig. 5. T –P phase diagram for iron. The area between two dotted lines indicates the coexistence phase of a þ e which obtained by electrical resistance measurements at 290 and 10 K.
290 K (open circles), the resistance starts to increase at the pressure of appearance of e phase (13 GPa) and starts to decrease at the finish of the transition (17 GPa). Then in the pressure range of the resistance slope (a þ e; 13–17 GPa) the two phases are coexist. The phase transition has a large pressure hysteresis, however we estimated the maximum allowance of coexistent phase (a þ e) and obtained the T –P phase diagram as shown in Fig. 5. Solid lines in Fig. 5 indicate the obtained structural boundary at room temperature (290 K) and low temperature (10 K). We estimate the intermediate boundary by dotted lines as shown in Fig. 5. In the higher pressure region than the coexistence phase of a þ e between two dotted lines, the transformation of a to e seems to be complete at P –T region for superconductivity. Recently the complete zero resistance in the superconducting transition and the nearly ferromagnetic character of normal state was reported [16]. Moreover superconductivity in the nearly (ferro) magnetic elementary metals is recently predicted [17]. We now have no explanation of the dome-shape of the superconducting phase. We need further and precise investigations of the magnetic property of the region of the boundary. We can expect analogously the similar behaviour in the case of ferromagnetic cobalt, nickel and antiferromagnetic manganese and chromium as well.
References [1] C.F. Richardson, N.W. Ashcroft, Phys. Rev. Lett. 78 (1997) 118. [2] E.P. Wohlfarth, Phys. Lett. A 75 (1979) 141. [3] K. Shimizu, M.I. Eremets, T.C. Kobayashi, K. Amaya, Science and technology of high pressure, in: Proceedings of AIRAPT-17, Hawaii, 1999, p. 77. [4] J. Wittig, Phys. Rev. Lett. 21 (1968) 1250. [5] N.E. Christensen, D.L. Novikov, Phys. Rev. Lett. 86 (2001) 1861. [6] J.B. Neaton, N.W. Ashcroft, Nature 400 (1999) 141. [7] K. Shimizu, H. Ishikawa, K. Amaya, J. Phys.: Condens. Matter 14 (2002) 10433. [8] K. Shimizu, H. Ishikawa, D. Takao, T. Yagi, K. Amaya, Nature 419 (2002) 597. [9] M. Hanfland, K. Syassen, N.E. Christensen, D.L. Novikov, Nature 408 (2000) 174. [10] N.W. Ashcroft, Phys. Rev. Lett. 78 (1997) 118. [11] G. Cort, R.D. Taylor, J. Willis, J. Appl. Phys. 53 (1982) 2064. [12] R.D. Taylor, M. Pasternak, R. Jeanloz, J. Appl. Phys. 69 (1991) 6126. [13] E.P. Wohlfarth, Phys. Lett. A 75 (1979) 141; A.J. Freeman, A. Continenza, S. Massida, J.C. Grossman, Physica C 166 (1990) 317; N. Suzuki, T. Souraku, I. Hamad, Proc. EHPRG ’39, 2001, Santander. [14] K. Shimizu, T. Kimura, S. Furomoto, K. Takeda, K. Kontani, Y. Onuki, K. Amaya, Nature 412 (2001) 316. [15] T. Jarlborg, Phys. Lett. A 300 (2002) 518. [16] D. Jaccard, A.T. Holmes, G. Behr, Y. Inada, Y. Onuki, Phys. Lett. A 299 (2002) 282. [17] T. Jarlborg, Physica C 385 (2003) 513.
Pressure-induced superconductivity in Li and Fe Katsuya Shimizu
a,b,*
, Daigoroh Takao b, Shigeyuki Furomoto b, Kiichi Amaya
b
a
b
Research Center for Materials Science at Extreme Conditions, Osaka University, 1-3, Machikaneyama, Toyonaka, Osaka 560-8531, Japan Graduate School of Engineering Science, Osaka University, 1-3, Machikaneyama, Toyonaka, Osaka 560-8531, Japan
Abstract We have developed complex extreme condition of very low temperature down to 30 mK and very high pressure exceeding 200 GPa by assembling compact diamond anvil cell (DAC) on a powerful 3 He/4 He dilution refrigerator. Using the apparatus and techniques, we have searched for pressure-induced superconductivity in various materials under pressures. In this paper, our experimental techniques and the examples of pressure-induced superconductivity in Li and Fe are reviewed. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Pressure; Superconductivity; Lithium; Iron; Electrical resistance
1. Introduction A compact DAC could be easily assembled on a powerful 3 He/4 He dilution refrigerator then this combined system enabled us to perform various experiments under the extreme conditions of very low-temperature of 30 mK and very high-pressure exceeding 200 GPa at the same time. We have employed good thermal conductors and non-magnetic materials for all parts of the pressure cell and performed electrical resistance measurements and/or sensitive magnetization measurements, and searched for pressure-induced superconductivity in simple systems. One of the most aggressive challenges under high pressure research is the search for metallic hydrogen under its extremely dense phase. The metallic hydrogen is predicted [1] to show superconductivity even at room temperature and have been the long fascinate the highpressure physicists, which exists in fact at interior of giant planet such as Jupiter and Saturn. The metallic
*
Corresponding author. Tel.: +81-66-850-6675; fax: +81-66850-6662. E-mail address: [email protected] (K. Shimizu).
hydrogen may form the same electronic structure of alkaline metals. Then we expected that highly compressed lithium can be treated as prototype of the metallic hydrogen and have studied the possibility of superconducting transition of lithium. Another challenge topic is superconductivity against magnetism. It is well known that a ferromagnetic metal does not show superconductivity and even a small amount of paramagnetic impurities could suppress the superconductivity. However even in the case of iron, the superconducting transition temperature (Tc ) was theoretically predicted to be 0.25 K by Wohlfarth [2]. We may expect the appearance of superconductivity in magnetic metals in its non-magnetic state under certain pressure and low temperature, and we have long searched superconductivity in the high-pressure non-magnetic phase of iron.
2. Experimental For the purpose of detecting superconducting transition, we employed sensitive ac-4 terminal method for electrical resistance measurement [3]. In the conventional configuration Au or Pt electrodes is set on the pressure surface of diamond anvils. In this configuration
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.03.077
K. Shimizu et al. / Physica C 408–410 (2004) 750–753
the sample is pressed directly by the diamond and consequently an axial stress is applied to the sample. To improve the quality of pressure, proper pressure media such as mixed alcohol or NaCl is used. In both cases, hard metallic gasket such as rhenium is covered with thin layer of aluminum oxide for electrical insulation between the gasket and the electrodes. The setting of the electrodes is made by hand and the typical size of the sample of the order of 108 cm3 (typically for 100 GPa) makes this procedure difficult. Magnetization measurement is performed by a SQUID magnetometer with a pair of pick up coils; the detecting coil is for the sample in DAC and another coil is set for the reference sample as well as for compensation against uniform external noise. This dc method is available at temperatures below several K, where the pick up coil (such as NbTi) is at superconducting state. In the present study we performed in situ pressuredetermination by the ruby method. Several small ruby chips on the sample are irradiated by argon laser and empirical relation between observed wave length of ruby fluorescent line. The pressure value is employed at room temperature as well as low temperature below 10 K.
751
reaction with surrounding materials such as diamonds and sample chamber materials. We made a sample chamber (pit) on a surface of the diamond-anvil as shown in Fig. 1 and confined there lithium sample at low temperature. This procedure worked well and enabled us to perform measurements up to 85 GPa. We could observe the resistance drop showing the appearance of superconductive state of lithium under pressure [8]. As shown in Fig. 2, superconductivity was observed at pressures above 25 GPa and the Tc reached 20 K at about 50 GPa in Run#1. The observed Tc value is the highest in elements. Further pressure studies (Run#2) shows, however, that the curve of Tc versus pressure becomes almost flat above 40 GPa and change the slope at around 60 GPa. The deference in Tc curve comes from experimental error on pressure determination in the sample chamber. This pressure
3. Pressure-induced superconductivity in lithium Lithium is the third element on the periodic table and well know as the lightest metal in the periodic table. For long time, mono-valent metals like alkaline metals are believed to show no superconductivity even at very low temperature. However Cs metal was found to become superconductive under pressure [4] and there arises possibility of pressure-induced superconductivity also in other lighter alkaline metals. The first motive force of searching superconductivity in lithium is the above possibility and also the possibility of the high-Tc in lithium. According to the conventional BSC theory, Tc is inversely proportional to the square root of the mass. The lightest metal of lithium is expect to show the highest Tc in the element. Recent calculation [5] predicted the pressure dependence of Tc of lithium and give considerable high Tc values reaching up to 80 K at around 40 GPa. The second arises from a recent theoretical prediction [6] which showed the new structural transition of Li towards the formation of pairing of Li atoms under higher pressure. In fact, we reported experimental studies showed a slight increment of electrical resistance by increasing pressure up to 30 GPa [7], however those still did not explain the tendency towards the insulating phase. We started the search for superconductivity of lithium by the electrical resistance measurement under pressure, however there were difficulties on confinement of Li sample in our DAC because of the strong chemical
Fig. 1. Photograph of a pit of 50-micron in diameter and 7micron in depth on the 200-micron pressure surface of synthetic Ib diamond-anvil; scale bar ¼ 0.5 mm.
Fig. 2. Pressure dependence of Tc of lithium. Dotted lines indicate the structural phase boundary observed by X-ray diffraction studies [8] at temperature of 180 K.
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K. Shimizu et al. / Physica C 408–410 (2004) 750–753
dependence of Tc may reflect the structural changes under pressure [9]. We did not observe the sign of the metal-to-insulator transition that predicted at pressure exceeding 100 GPa. The further experimental investigation for higher-pressure behaviour of Li should help the total understanding of the fundamental property of metals and possible room-temperature superconductivity [10] in metallic hydrogen.
4. Pressure-induced superconductivity in iron It is well known that a ferromagnetic metal does not show superconductivity down to the lowest temperature. So long as we look at the periodic table of elements, the idea of absence of Tc of magnetic metals seems to be accepted quite in general. We expected the appearance of Tc of magnetic metals in its non-magnetic state under certain high pressure and at low temperature. In the case of iron, there appears non-magnetic state at pressures after the crystal phase transition from bcc to hcp phase at around 15 GPa. The non-magnetism of hcp iron (eFe) is confirmed by M€ ossbauer experiments [11,12]. There are also theoretical predictions [13] on Tc of iron in its non-magnetic state, giving Tc below 1 K. We have searched for superconductivity of iron under pressure in the past 10 years. After searching in the wide range of pressure up to 100 GPa and temperature down to 50 mK, we noticed that purification of the sample and application of quasi hydro-static pressure using proper pressure media should be the most important. In fact, pressurization of sample in pressure media such as NaCl was found to be very effective in the present case of iron, avoiding excess crystal distortion and uniaxial pressure distribution in the course of pressurization. The final arrangement of the sample and pressure chamber is shown in Fig. 3 with which the superconductivity of iron was obtained [14]. Sample thickness is 40 lm and the metal gasket (Re) is covered with thin Al2 O3 layer for electrical insulation. The gold wires of 10 lm in diameter attached to the sample crystal and connected to the platinum electrode at the out side the chamber to perform 4-terminal electrical resistance measurements. The chamber is filled with sodium chloride (NaCl) powder as the pressure-transmitting medium. We studied the pressure dependence of Tc and found that Tc appears at 15 GPa and disappears at 30 GPa, giving the maximum value of 2 K at 21 GPa with very small resistance drop and Meissner signal. These may suggest that the superconductivity is sensitive to both sample and pressure quality. The superconductivity seem appear in the non-magnetic state, however, a theoretical calculation was performed by Jarlborg [15] in which the spin fluctuation coupling, especially of ferro-
Fig. 3. Photograph of the sample and electrodes on the pressure surface of the diamond-anvil under the pressure of 23 GPa; scale bar ¼ 0.1 mm.
magnetic nature, is estimated to be sufficiently large to explain superconductivity of hcp iron. We tried to estimate the phase boundary of bcc-to-hcp structure even if the transition under pressure has a wide pressure-width and is sluggish. The bcc-to-hcp (a to e) transition can be detected by electrical resistance changes as shown in Fig. 4. The experiments are done with the same arrangement as Fig. 3 with which the superconductivity of iron was observed. Resistance curves shows the broad step at the structural boundary; in the case of increasing pressure at
Fig. 4. Resistance curves of iron under pressure across the phase boundary of a to e phase at temperature of 290 and 10 K. As shown in dotted line clear evidence was not observed in the curve with increasing pressure at low temperature of 10 K.
K. Shimizu et al. / Physica C 408–410 (2004) 750–753
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5. Summary Here we reviewed recent developments of complex extreme condition of low temperature and high pressure and recent results of pressure-induced superconductivity. Following the case of lithium or iron, a new type of superconductivity may be discovered elemental materials in high-pressure. Indeed, our final target is production of metallic hydrogen and detection of its possible high Tc superconductive state which are expected to be realized under ultra-high but almost attainable pressure of 400 GPa. Fig. 5. T –P phase diagram for iron. The area between two dotted lines indicates the coexistence phase of a þ e which obtained by electrical resistance measurements at 290 and 10 K.
290 K (open circles), the resistance starts to increase at the pressure of appearance of e phase (13 GPa) and starts to decrease at the finish of the transition (17 GPa). Then in the pressure range of the resistance slope (a þ e; 13–17 GPa) the two phases are coexist. The phase transition has a large pressure hysteresis, however we estimated the maximum allowance of coexistent phase (a þ e) and obtained the T –P phase diagram as shown in Fig. 5. Solid lines in Fig. 5 indicate the obtained structural boundary at room temperature (290 K) and low temperature (10 K). We estimate the intermediate boundary by dotted lines as shown in Fig. 5. In the higher pressure region than the coexistence phase of a þ e between two dotted lines, the transformation of a to e seems to be complete at P –T region for superconductivity. Recently the complete zero resistance in the superconducting transition and the nearly ferromagnetic character of normal state was reported [16]. Moreover superconductivity in the nearly (ferro) magnetic elementary metals is recently predicted [17]. We now have no explanation of the dome-shape of the superconducting phase. We need further and precise investigations of the magnetic property of the region of the boundary. We can expect analogously the similar behaviour in the case of ferromagnetic cobalt, nickel and antiferromagnetic manganese and chromium as well.
References [1] C.F. Richardson, N.W. Ashcroft, Phys. Rev. Lett. 78 (1997) 118. [2] E.P. Wohlfarth, Phys. Lett. A 75 (1979) 141. [3] K. Shimizu, M.I. Eremets, T.C. Kobayashi, K. Amaya, Science and technology of high pressure, in: Proceedings of AIRAPT-17, Hawaii, 1999, p. 77. [4] J. Wittig, Phys. Rev. Lett. 21 (1968) 1250. [5] N.E. Christensen, D.L. Novikov, Phys. Rev. Lett. 86 (2001) 1861. [6] J.B. Neaton, N.W. Ashcroft, Nature 400 (1999) 141. [7] K. Shimizu, H. Ishikawa, K. Amaya, J. Phys.: Condens. Matter 14 (2002) 10433. [8] K. Shimizu, H. Ishikawa, D. Takao, T. Yagi, K. Amaya, Nature 419 (2002) 597. [9] M. Hanfland, K. Syassen, N.E. Christensen, D.L. Novikov, Nature 408 (2000) 174. [10] N.W. Ashcroft, Phys. Rev. Lett. 78 (1997) 118. [11] G. Cort, R.D. Taylor, J. Willis, J. Appl. Phys. 53 (1982) 2064. [12] R.D. Taylor, M. Pasternak, R. Jeanloz, J. Appl. Phys. 69 (1991) 6126. [13] E.P. Wohlfarth, Phys. Lett. A 75 (1979) 141; A.J. Freeman, A. Continenza, S. Massida, J.C. Grossman, Physica C 166 (1990) 317; N. Suzuki, T. Souraku, I. Hamad, Proc. EHPRG ’39, 2001, Santander. [14] K. Shimizu, T. Kimura, S. Furomoto, K. Takeda, K. Kontani, Y. Onuki, K. Amaya, Nature 412 (2001) 316. [15] T. Jarlborg, Phys. Lett. A 300 (2002) 518. [16] D. Jaccard, A.T. Holmes, G. Behr, Y. Inada, Y. Onuki, Phys. Lett. A 299 (2002) 282. [17] T. Jarlborg, Physica C 385 (2003) 513.