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Anomalous Hall effect in the hopping regime of the insulating Al70Pd22.5Re7.5 quasicrystal
Journal of Alloys and Compounds 342 (2002) 352–354
L
www.elsevier.com / locate / jallcom
Anomalous Hall effect in the hopping regime of the insulating Al 70 Pd 22.5 Re 7.5 quasicrystal Kun-Hsien Chen, Tzung-I Su, Hung-Cheng Fang, Shih-Chen Lee, Shui-Tien Lin* Department of Physics, National Cheng Kung University, Tainan, Taiwan, ROC
Abstract The Hall coefficient R H is seen to rise rapidly with decreasing temperature in the hopping regime of the insulating Al 70 Pd 22.5 Re 7.5 quasicrystal with a resistivity ratio r 5 r (4.2 K) /r (300 K) equal to 24 and to exhibit the variable-range hopping behavior i.e. R H 5 R 0sT 0H /Td 1 / 4 . The obtained value of T 0H is much larger than the Mott’s temperature T 0 extracted from the conductivity in the hopping regime. This is contrary to the prediction of Gruenewald et al. [Solid State Commum. 64 (1987) 11] for disordered systems. No existing theories can account for this anomalous effect. 2002 Elsevier Science B.V. All rights reserved. Keywords: Quasicrystal; Hall effect; Hopping transport
1. Introduction The Hall effect has been studied in the face-centered icosahedral Al–Cu–(Fe,Ru) quasicrystals (QCs) [1–3]. The general features observed in these QCs are: (1) the Hall coefficient R H can be either negative or positive and the sign of R H is related to the composition and the conductivity at 4.2 K, s (4.2 K); (2) the value of R H is strongly dependent on temperature and its sign can be reversed as the temperature is varied. Although the resistivity, r, of these series of QCs is high (1.3–30 mV cm), they remain in the metallic state and are believed to be still far away from the metal–insulator (M–I) transition [4]. Only recently, it was found that M–I transition can occur in Al–Pd–Re QCs [5–7]. Through a series of studies on the temperature-dependent conductivity of Al 70 Pd 22.5 Re 7.5 QCs with the resistivity ratio r5 r (4.2 K) /r (300 K) ranging from 2 to 100, we have found that the extrapolated zero conductivity of the samples in the metallic state obeys the scaling law [7]: s (0) 5 s0s1 2 r /r cd y with y ¯ 1.0, s0 5 25.663.8 (V cm)21 , and the critical resistivity ratio r c | 12.860.5. This suggests that samples with r , r c are metals, while samples with r . r c are insulators, and thus gives us a good opportunity to study the Hall effect in the hopping regime of QCs. *Corresponding author. Fax: 1886-6-275-7575. E-mail address: [email protected] (S.-T. Lin).
In 1961, Holstein [8] first demonstrated that to observe the non-vanishing Hall effect in the hopping regime of insulating doped-semiconductors, a minimum of three hopping sites is necessary to be considered, and showed the effect arising from the interference between the amplitude for a direct transition between two sites and the amplitude for an indirect transition, involving intermediate occupancy of a third site. Later Gruenewald et al. [9], employing the percolation approach to the three-site clusters, obtained the Hall mobility for disordered hopping systems to be ln mH ~23 / 8(T 0 /T ) in addition to confirming Mott’s result ln s (t) ~2(T 0 /T )1 / 4 . Using mH 5 R H s, one obtains
S D
T 0H R H 5 R 0 exp ] T
1/4
(1)
where T 0H equal to (5 / 8)4 T 0 is much smaller than T 0 . In 1987, Koon et al. [10] found that the temperature dependence of the Hall coefficient R H in the hopping regime of Si–As exhibits the variable-range hopping (VRH) behavior and follows Eq. (1). In this work, the Hall effect in the hopping regime of the insulating Al 70 Pd 22.5 Re 7.5 QC was studied. We found that the temperature dependence of R H also obeys the VRH behavior. But in contrast to the prediction of the theory, the value of T 0H is much larger than the value of T 0 .
0925-8388 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0925-8388( 02 )00252-9
K.-H. Chen et al. / Journal of Alloys and Compounds 342 (2002) 352 – 354
353
S D
(2)
2. Experimental procedures Al 70 Pd 22.5 Re 7.5 samples with different values of r are prepared. The detailed descriptions of the sample fabrication are given in Ref. [11]. The bar-shaped samples for Hall voltage measurements have dimensions |1.031.537 mm 3 . The Hall voltage was measured between 1.9 and 280 K with a DC current of 0.5–10 mA under a magnetic field H 5 3 T and 5 T. The temperature stability during each measurement is better than 0.1%. Misalignment voltage and thermal effects have been considered and corrected.
T0 s (T ) 5 s (0) 1 s0 exp ] T
1/4
] The conductivity s (T ) plotted against ŒT for Al 70 Pd 22.5 Re 7.5 QCs with r55.6, 11.6 and 24 is shown in Fig. 1a. It shows that among these samples, the sample with r524 is the most resistive one. The inset in Fig. 1b shows W 5 [d ln s (T )] / [d ln (T )] as a function of T for the sample with r524. W is seen to start to rise at around 4.2 K and to fall at around 0.9 K. The rise of W with decreasing temperature indicates that electrons propagate via a hopping process, while the fall of W at low temperature implies that there exists a finite zero temperature conductivity s (0) (see discussions in Ref. [7]). For the sample with r . r c , the conductivity s (T ) in the hopping regime has been found to obey a modified Mott’s law [7]
ln [s (T ) 2 s (0)] versus T 21 / 4 for the sample with r524 is also shown in Fig. 1b. The extracted value of T 0 is 76 K. The measured R H for the sample with r524 is positive between 1.9 and 280 K, it increases slightly with decreasing temperature, but rises steeply at low temperatures. As for metallic samples with r55.6 and 11.6, the magnitude of R H is much smaller and the sign of R H is negative at low temperature; and for the former sample, its R H changes sign at about 100 K. In the inset of Fig. 2, the R H at 4.2 K as a function of r is plotted. The magnitude of R H (4.2 K) is seen to increase with increasing r. The value of R H (4.2 K) for the sample with r516.4 is positive (8.63 10 23 cm 3 / C). This indicates that R H reverses its sign at the value of r between 11.6 and 16.4, which is close to the critical resistivity ratio r c | 12.860.5 for the metal–insulator transition. Previously, we have shown that below | r c , increasing r improves the quasicrystalline lattice and decreases the density of states at the Fermi level N(EF ), while above |r c the quasicrystalline lattice and N(EF ) almost remain unchanged [6,12]. These results seem to suggest that R H reverses its sign at the sample with its N(EF ) near the least value. Behaviors such as the sign reversal of R H with temperature and strong temperature dependence of R H are the general features observed also in other icosahedral Al–Cu(Fe,Ru) QCs.
] Fig. 1. (a) Conductivity s (T ) as a function of ŒT for Al 70 Pd 22.5 Re 7.5 QCs with r55.6, 11.6 and 24. (b) ln [s (T ) 2 s (0)] plotted against T 21 / 4 for the sample with r524; the inset: W 5 [d ln s (T )] / [d ln (T )] as a function of T.
Fig. 2. Hall coefficient R H as a function of T for Al 70 Pd 22.5 Re 7.5 QCs with r55.6, 11.6, 24. Inset: R H at 4.2 K as a function of r.
3. Results and discussion
354
K.-H. Chen et al. / Journal of Alloys and Compounds 342 (2002) 352 – 354
4. Conclusion The observed VRH behavior in insulating QC once again reveals that the importance of the structural and / or chemical disorder in the low-T electronic properties of QCs because it can only result from a system with disorder. Eq. (1) is not a universal one and therefore new theories are still needed to understand the Hall effect observed here. In our view, the temperature dependence of the Hall coefficient provides richer information than that of the conductivity about the electronic properties of QCs and is more suitable for testing stringently the validity of the electronic transport theories developed for QCs.
Acknowledgements Fig. 3. ln (R H ) plotted against T 21 / 4 for Al 70 Pd 22.5 Re 7.5 sample with r524 measured at H 5 3 T and 5 T.
Only the rapid increase in R H at low temperature observed in the sample with r524, so far as we know, is first reported. The anomalous Hall effect is seen to occur in the temperature range where electrons transport via the VRH process (see Fig. 1b). Therefore, it is connected with hopping conduction and not observed in metallic samples with r55.6 and 11.6. Fig. 3 shows the logarithm of R H (measured at the magnetic field H53 T and 5 T) versus T 21 / 4 for the sample with r524. It is obviously seen that the temperature dependence of R H exhibits the VRH behavior i.e. R H 5 R 0 exp sT 0H /Td 1 / 4 . Within the experimental errors, the slopes of ln (R H ) versus T 21 / 4 measured at H 5 3 T and 5 T are found to be about the same. T 0H is determined to be |5373 K which is much larger than the characteristic temperature T 0 |76 K obtained from the conductivity in the hopping regime. This means that the rate of change of R H with temperature is much larger than the rate of change of s with temperature, contrary to the prediction of Gruenewald et al. [9] for insulatingly disordered systems.
We are indebted to the National Science Council of the Republic of China for the financial support of this work under the grant no: NSC 89-2112-M-006-041.
References [1] P. Lindqvist, C. Berger, T. Klein, P. Lanco, F. Cyrot-Lackmann, Y. Calvayrac, Phys. Rev. B 48 (1993) 630. [2] R. Haberkern, G. Fritsch, in: C. Janot, R. Mosseri (Eds.), Proceedings of the 5th International Conference on Quasicrystals, World Scientific, 1995, p. 460. [3] R. Tamura, K. Kirihara, K. Kimura, H. Ino, in: S. Takeuchi, T. Fujiwara (Eds.), Proceedings of the 6th International Conference on Quasicrystals, World Scientific, 1997, p. 631. [4] C.R. Lin, S.L. Lou, S.T. Lin, J. Phys.: Condens. Matter 8 (1996) L725. [5] Q. Guo, F.S. Pierce, S.J. Poon, Phys. Rev. B 52 (1995) I. [6] C.R. Wang, H.S. Kuan, S.T. Lin, Y.Y. Chen, J. Phys. Soc. Jpn. 67 (1998) 2383. [7] C.R. Wang, S.T. Lin, J. Phys. Soc. Jpn. 68 (1999) 3988. [8] T. Holstein, Phys. Rev. 124 (1961) 1329. [9] M. Gruenewald, H. Mueller, P. Thomas, D. Wuertz, Solid State Commun. 38 (1981) 1011. [10] D.W. Koon, T.G. Castner, Solid State Commun. 64 (1987) 11. [11] C.R. Lin, S.T. Lin, C.R. Wang, S.L. Chou, H.E. Horng, J.M. Cheng, Y.D. Yao, S.C. Lai, J. Phys.: Condens. Matter 9 (1997) 1509. [12] C.R. Wang, T.I. Su, S.T. Lin, J. Phys. Soc. Jpn. 69 (2000) 3356.
L
www.elsevier.com / locate / jallcom
Anomalous Hall effect in the hopping regime of the insulating Al 70 Pd 22.5 Re 7.5 quasicrystal Kun-Hsien Chen, Tzung-I Su, Hung-Cheng Fang, Shih-Chen Lee, Shui-Tien Lin* Department of Physics, National Cheng Kung University, Tainan, Taiwan, ROC
Abstract The Hall coefficient R H is seen to rise rapidly with decreasing temperature in the hopping regime of the insulating Al 70 Pd 22.5 Re 7.5 quasicrystal with a resistivity ratio r 5 r (4.2 K) /r (300 K) equal to 24 and to exhibit the variable-range hopping behavior i.e. R H 5 R 0sT 0H /Td 1 / 4 . The obtained value of T 0H is much larger than the Mott’s temperature T 0 extracted from the conductivity in the hopping regime. This is contrary to the prediction of Gruenewald et al. [Solid State Commum. 64 (1987) 11] for disordered systems. No existing theories can account for this anomalous effect. 2002 Elsevier Science B.V. All rights reserved. Keywords: Quasicrystal; Hall effect; Hopping transport
1. Introduction The Hall effect has been studied in the face-centered icosahedral Al–Cu–(Fe,Ru) quasicrystals (QCs) [1–3]. The general features observed in these QCs are: (1) the Hall coefficient R H can be either negative or positive and the sign of R H is related to the composition and the conductivity at 4.2 K, s (4.2 K); (2) the value of R H is strongly dependent on temperature and its sign can be reversed as the temperature is varied. Although the resistivity, r, of these series of QCs is high (1.3–30 mV cm), they remain in the metallic state and are believed to be still far away from the metal–insulator (M–I) transition [4]. Only recently, it was found that M–I transition can occur in Al–Pd–Re QCs [5–7]. Through a series of studies on the temperature-dependent conductivity of Al 70 Pd 22.5 Re 7.5 QCs with the resistivity ratio r5 r (4.2 K) /r (300 K) ranging from 2 to 100, we have found that the extrapolated zero conductivity of the samples in the metallic state obeys the scaling law [7]: s (0) 5 s0s1 2 r /r cd y with y ¯ 1.0, s0 5 25.663.8 (V cm)21 , and the critical resistivity ratio r c | 12.860.5. This suggests that samples with r , r c are metals, while samples with r . r c are insulators, and thus gives us a good opportunity to study the Hall effect in the hopping regime of QCs. *Corresponding author. Fax: 1886-6-275-7575. E-mail address: [email protected] (S.-T. Lin).
In 1961, Holstein [8] first demonstrated that to observe the non-vanishing Hall effect in the hopping regime of insulating doped-semiconductors, a minimum of three hopping sites is necessary to be considered, and showed the effect arising from the interference between the amplitude for a direct transition between two sites and the amplitude for an indirect transition, involving intermediate occupancy of a third site. Later Gruenewald et al. [9], employing the percolation approach to the three-site clusters, obtained the Hall mobility for disordered hopping systems to be ln mH ~23 / 8(T 0 /T ) in addition to confirming Mott’s result ln s (t) ~2(T 0 /T )1 / 4 . Using mH 5 R H s, one obtains
S D
T 0H R H 5 R 0 exp ] T
1/4
(1)
where T 0H equal to (5 / 8)4 T 0 is much smaller than T 0 . In 1987, Koon et al. [10] found that the temperature dependence of the Hall coefficient R H in the hopping regime of Si–As exhibits the variable-range hopping (VRH) behavior and follows Eq. (1). In this work, the Hall effect in the hopping regime of the insulating Al 70 Pd 22.5 Re 7.5 QC was studied. We found that the temperature dependence of R H also obeys the VRH behavior. But in contrast to the prediction of the theory, the value of T 0H is much larger than the value of T 0 .
0925-8388 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0925-8388( 02 )00252-9
K.-H. Chen et al. / Journal of Alloys and Compounds 342 (2002) 352 – 354
353
S D
(2)
2. Experimental procedures Al 70 Pd 22.5 Re 7.5 samples with different values of r are prepared. The detailed descriptions of the sample fabrication are given in Ref. [11]. The bar-shaped samples for Hall voltage measurements have dimensions |1.031.537 mm 3 . The Hall voltage was measured between 1.9 and 280 K with a DC current of 0.5–10 mA under a magnetic field H 5 3 T and 5 T. The temperature stability during each measurement is better than 0.1%. Misalignment voltage and thermal effects have been considered and corrected.
T0 s (T ) 5 s (0) 1 s0 exp ] T
1/4
] The conductivity s (T ) plotted against ŒT for Al 70 Pd 22.5 Re 7.5 QCs with r55.6, 11.6 and 24 is shown in Fig. 1a. It shows that among these samples, the sample with r524 is the most resistive one. The inset in Fig. 1b shows W 5 [d ln s (T )] / [d ln (T )] as a function of T for the sample with r524. W is seen to start to rise at around 4.2 K and to fall at around 0.9 K. The rise of W with decreasing temperature indicates that electrons propagate via a hopping process, while the fall of W at low temperature implies that there exists a finite zero temperature conductivity s (0) (see discussions in Ref. [7]). For the sample with r . r c , the conductivity s (T ) in the hopping regime has been found to obey a modified Mott’s law [7]
ln [s (T ) 2 s (0)] versus T 21 / 4 for the sample with r524 is also shown in Fig. 1b. The extracted value of T 0 is 76 K. The measured R H for the sample with r524 is positive between 1.9 and 280 K, it increases slightly with decreasing temperature, but rises steeply at low temperatures. As for metallic samples with r55.6 and 11.6, the magnitude of R H is much smaller and the sign of R H is negative at low temperature; and for the former sample, its R H changes sign at about 100 K. In the inset of Fig. 2, the R H at 4.2 K as a function of r is plotted. The magnitude of R H (4.2 K) is seen to increase with increasing r. The value of R H (4.2 K) for the sample with r516.4 is positive (8.63 10 23 cm 3 / C). This indicates that R H reverses its sign at the value of r between 11.6 and 16.4, which is close to the critical resistivity ratio r c | 12.860.5 for the metal–insulator transition. Previously, we have shown that below | r c , increasing r improves the quasicrystalline lattice and decreases the density of states at the Fermi level N(EF ), while above |r c the quasicrystalline lattice and N(EF ) almost remain unchanged [6,12]. These results seem to suggest that R H reverses its sign at the sample with its N(EF ) near the least value. Behaviors such as the sign reversal of R H with temperature and strong temperature dependence of R H are the general features observed also in other icosahedral Al–Cu(Fe,Ru) QCs.
] Fig. 1. (a) Conductivity s (T ) as a function of ŒT for Al 70 Pd 22.5 Re 7.5 QCs with r55.6, 11.6 and 24. (b) ln [s (T ) 2 s (0)] plotted against T 21 / 4 for the sample with r524; the inset: W 5 [d ln s (T )] / [d ln (T )] as a function of T.
Fig. 2. Hall coefficient R H as a function of T for Al 70 Pd 22.5 Re 7.5 QCs with r55.6, 11.6, 24. Inset: R H at 4.2 K as a function of r.
3. Results and discussion
354
K.-H. Chen et al. / Journal of Alloys and Compounds 342 (2002) 352 – 354
4. Conclusion The observed VRH behavior in insulating QC once again reveals that the importance of the structural and / or chemical disorder in the low-T electronic properties of QCs because it can only result from a system with disorder. Eq. (1) is not a universal one and therefore new theories are still needed to understand the Hall effect observed here. In our view, the temperature dependence of the Hall coefficient provides richer information than that of the conductivity about the electronic properties of QCs and is more suitable for testing stringently the validity of the electronic transport theories developed for QCs.
Acknowledgements Fig. 3. ln (R H ) plotted against T 21 / 4 for Al 70 Pd 22.5 Re 7.5 sample with r524 measured at H 5 3 T and 5 T.
Only the rapid increase in R H at low temperature observed in the sample with r524, so far as we know, is first reported. The anomalous Hall effect is seen to occur in the temperature range where electrons transport via the VRH process (see Fig. 1b). Therefore, it is connected with hopping conduction and not observed in metallic samples with r55.6 and 11.6. Fig. 3 shows the logarithm of R H (measured at the magnetic field H53 T and 5 T) versus T 21 / 4 for the sample with r524. It is obviously seen that the temperature dependence of R H exhibits the VRH behavior i.e. R H 5 R 0 exp sT 0H /Td 1 / 4 . Within the experimental errors, the slopes of ln (R H ) versus T 21 / 4 measured at H 5 3 T and 5 T are found to be about the same. T 0H is determined to be |5373 K which is much larger than the characteristic temperature T 0 |76 K obtained from the conductivity in the hopping regime. This means that the rate of change of R H with temperature is much larger than the rate of change of s with temperature, contrary to the prediction of Gruenewald et al. [9] for insulatingly disordered systems.
We are indebted to the National Science Council of the Republic of China for the financial support of this work under the grant no: NSC 89-2112-M-006-041.
References [1] P. Lindqvist, C. Berger, T. Klein, P. Lanco, F. Cyrot-Lackmann, Y. Calvayrac, Phys. Rev. B 48 (1993) 630. [2] R. Haberkern, G. Fritsch, in: C. Janot, R. Mosseri (Eds.), Proceedings of the 5th International Conference on Quasicrystals, World Scientific, 1995, p. 460. [3] R. Tamura, K. Kirihara, K. Kimura, H. Ino, in: S. Takeuchi, T. Fujiwara (Eds.), Proceedings of the 6th International Conference on Quasicrystals, World Scientific, 1997, p. 631. [4] C.R. Lin, S.L. Lou, S.T. Lin, J. Phys.: Condens. Matter 8 (1996) L725. [5] Q. Guo, F.S. Pierce, S.J. Poon, Phys. Rev. B 52 (1995) I. [6] C.R. Wang, H.S. Kuan, S.T. Lin, Y.Y. Chen, J. Phys. Soc. Jpn. 67 (1998) 2383. [7] C.R. Wang, S.T. Lin, J. Phys. Soc. Jpn. 68 (1999) 3988. [8] T. Holstein, Phys. Rev. 124 (1961) 1329. [9] M. Gruenewald, H. Mueller, P. Thomas, D. Wuertz, Solid State Commun. 38 (1981) 1011. [10] D.W. Koon, T.G. Castner, Solid State Commun. 64 (1987) 11. [11] C.R. Lin, S.T. Lin, C.R. Wang, S.L. Chou, H.E. Horng, J.M. Cheng, Y.D. Yao, S.C. Lai, J. Phys.: Condens. Matter 9 (1997) 1509. [12] C.R. Wang, T.I. Su, S.T. Lin, J. Phys. Soc. Jpn. 69 (2000) 3356.