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Diamagnetism in quasicrystals
Solid State Communications 133 (2005) 473–475 www.elsevier.com/locate/ssc
Diamagnetism in quasicrystals Yu.Kh. Vekilova, E.I. Isaeva,*, B. Johanssonb,c a
Theoretical Physics Department, Moscow State Institute of Steel and Alloys (Technological University), 4, Leninskii prospect, 119049 Moscow, Russian Federation b Condensed Matter Theory Group, Uppsala University, SE-751 21 Uppsala, Sweden c Department of Materials Science and Engineering, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden Received 22 September 2004; received in revised form 29 November 2004; accepted 30 November 2004 by H. Eschrig Available online 8 December 2004
Abstract A reliable explanation of diamagnetism in quasicrystals is given. We show that the weak diamagnetism in perfect icosahedral quasicrystals is due to an atomic-like diamagnetic contribution of tightly bound conduction electrons in electron pockets of a multiconnected Fermi surface. The Landau–Peierls diamagnetic term is small due to large effective masses. At temperatures above the Debye temperature the intervalley electron–phonon scattering makes the electrons ‘free’, and the temperature dependence of the Pauli paramagnetism related to a pseudogap in the density of states at the Fermi level becomes important. q 2004 Elsevier Ltd. All rights reserved. PACS: 71.23.Ft; 75.20.Kg Keywords: A. Quasicrystals; D. Conductivity electrons; D. Diamagnetism
The icosahedral (i-) quasicrystals (QC) like i-Al–Cu–Fe, i-Al–Pd–Mn with low Mn concentration, and i-Al–Pd–Re exhibit a weak diamagnetism over a wide range of temperatures [1–5]. At very low temperatures their magnetic susceptibility c behaves as a Curie–Weiss like law with decreasing temperatures and its zero-temperature limit becomes positive (Al–Cu–Fe, Al–Pd–Mn) or nearly zero (Al–Pd–Re). At elevated temperatures c increases with increasing T, and the enhancement of c is accounted for by a temperature dependence of the Pauli paramagnetism. The diamagnetism is observed in high quality perfect samples. The poor quality samples are always paramagnetic. There is also a certain correlation between conductivity and diamagnetism in QC, because the perfect samples are better insulators and more diamagnetic. So far, no reliable explanation has been given to the
* Corresponding author. Tel.: C7 95 2304506; fax: C7 95 2362105. E-mail address: [email protected] (E.I. Isaev). 0038-1098/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2004.11.040
diamagnetism in QC. In the paper [6] it has been assumed that diamagnetism in QC is of the Landau–Peierls type for conduction electrons, and the very low electron masses in some directions for the electron pockets of multiconnected Fermi surface (FS) of QC are responsible for this. However, this assumption is not correct. Certainly, in QC one has to take into account a large number of effective Bragg reflections, which lead to small gaps on the FS due to its interaction with Bragg planes, as well as to small electron– hole pockets on it. Consequently, the group velocities of electrons at the Fermi level EF should be small, and correspondingly the effective masses should be large. The experimental data and electronic structure calculations also show that the bands near the Fermi level in QC are rather flat, and thus the FS should have many electron–hole pockets with large electron effective masses [7]. Diamagnetism of conduction electrons in QC is difficult to calculate, and its calculation is a difficult problem even for ordinary metals [8]. Kjeldaas and Kohn [9] using Kohn– Luttinger representation and perturbation theory, have obtained the following expression for the diamagnetic
474
Yu.Kh. Vekilov et al. / Solid State Communications 133 (2005) 473–475
susceptibility for Bloch electrons c ZK
e2 k0 m C c2 k02 C c4 k04 C . 2 2 m* 12p mc
(1)
where k0 is the radius of the Fermi sphere, the first term in this series is just the Landau–Peierls expression for the diamagnetic susceptibility of degenerate electrons (cLPZ K(e2k0)/(12p2m*c2)) and higher order terms in (1) are corrections due to band structure effects which are not included in the effective mass m*. The first two terms of (1) give the exact result in the following limiting cases: (1) number of conduction electrons goes to zero; (2) free electrons; (3) tightly bound electrons. The third case is more suitable for QCs, although the existence of Bloch states is not allowed for them. Certainly, QC could be considered as a structural limit of a sequence of periodic approximants with increasing lattice parameter, a/N. Then the use of the Harrison procedure [10] for FS construction becomes possible, and due to intersection with different Bragg planes FS of QC will be multiconnected with a large number of electron–hole pockets (‘fractional’ FS). In the limit of infinite lattice parameter, i.e. when m/m* vanishes exponentionally [1–5], and k0/0 like the reciprocal of the interatomic spacing, it is only the second term in formula (1) that survives, and the Landau–Peierls contribution in diamagnetic susceptibility of conduction electrons is negligible. Kjeldaas and Kohn [9] have shown that the second term in (1) reduces correctly to the conventional atomic diamagnetism c wK 6cn2 hr2 imm Accordingly, in the quasicrystalline case we have the atomic like diamagnetism of conduction electrons tightly bound in each electron pockets of the multiconnected FS. The total susceptibility will be given by a sum of atomic-like diamagnetic contributions of all electron pockets from different Bragg planes and it can exceed the paramagnetic contribution over a wide temperature interval. Let us consider the temperature dependence of the magnetic susceptibility. The experimental curves of c versus T are usually fitted by the equation c Z c0 C aT Ka C AT 2
(2)
Here c0 is a temperature independent susceptibility which may include the following three contributions: the diamagnetic susceptibility of ion core electrons ccore which is small for the above mentioned objects [1–5], the Pauli spin paramagnetic susceptibility (cP) and diamagnetic susceptibility of conduction electrons which was discussed above. The Curie-like term ccZaTKa with a%1 must originate from interaction of localized moments existing in the samples, and it prevails at very low temperatures [1–5]. There is a definite correlation between the value of cc and resistivity r at 4.2 K: a sample with larger r (4.2 K) has larger values of cc and in this sense there is some analogy with the behavior of objects which show metal-insulator transition. For example, at these temperatures the magnetic
properties are quite similar to those of the phosphorus doped Si (Si:P) semiconductor [11]. In Si:P the cZaTKa term occurs in samples with carriers near the metal-insulator transition either from the metallic or insulating side, and is attributed to the behavior of an assembly of localized spins, associated with P and distributed randomly in space. At high temperatures the third term in (2) becomes important, c increases with increasing T and an increase in c is accounted for by a temperature dependence of the Pauli paramagnetism [12], cKAT2, with a prefactor
AZ
2 m2B NðEF ÞðZkB Þ2 1 d NðxÞ 3 NðxÞ dx2 1 dNðxÞ 2 K NðxÞ dx jxZEF
(3)
where mB denotes the Bohr magneton, EF is the Fermi energy at TZ0 K, and N(x) is the electronic density of states (DOS) at the Fermi level in QC. The second derivative of the DOS, d2 NðxÞ=dx2 jxZEF must be positive, because the second term in (3) is always negative [1]. The temperature dependence of c can be qualitatively explained in the framework of multiconnected (‘fractional’) FS model. This model has been successfully used to explain the electronic transport and electron localization in QC [13– 15]. The temperature-dependent conductivity of QC can be described by the power law, that is Ds(T)wTb, where b is roughly equal to 1 for poor quality samples with higher conductivity, while b is different in both the low- and hightemperature regimes for the higher quality samples with lower conductivity (the very high quality i-Al–Pd–Re samples even reveal at T!4 K the variable range hopping conductivity). In defectless samples the FS has an infinite number of zero area pockets (valleys). In real QC the electronic states are smeared by both temperature and scattering which results in the existence of a FS with a finite number of valleys whose number depends on the strength of disorder and temperature. Because the characteristic size of a valley is connected to the uncertainty of the electron energy, the localization of electronic states occurs in the limit of ordered (defectless) state resulting in zero conductivity at zero temperature, and only deviation from ordering leads to the possibility of intervalley scattering at zero temperature which provides the nonzero, but small value of zerotemperature conductivity. At finite temperatures electron– phonon interaction is able to change the electron momentum only by a small amount, of the order of T/u, where u is the sound velocity and there exists a characteristic temperature T*wu/a (a is the average interatomic distance) below which the phonons are unable to scatter electrons from one pocket to another. When the temperature exceeds T* the scattering by phonons provides the effective intervalley scattering [13– 15]. At temperatures T* which is order of the Debye
Yu.Kh. Vekilov et al. / Solid State Communications 133 (2005) 473–475
temperature, electrons become ‘free’-like, and the Pauli paramagnetic contribution prevails. In conclusion, we have shown that the weak diamagnetism in icosahedral quasicrystals is connected to atomic-like diamagnetic contribution of conduction electrons in electron pockets of multiconnected FS of quasicrystal. The Landau– Peierls diamagnetic term is small due to large electron effective masses. At temperatures above the Debye temperature the Pauli paramagnetic contribution prevails due to the temperature dependence of the Pauli paramagnetism related to a pseudogap in the DOS at the Fermi level.
Acknowledgements We are grateful to S. Simak for valuable dicussions and remarks. Yu.Kh.V. and E.I.I. thank the Royal Swedish Academy of Sciencies (KVA), the Russian Foundation for Basic Researches (RFBR, grant #03-02-16970) for financial support.
References [1] Z.M. Stadnik (Ed.), Physical Properties of Quasicrystals, Springer, Berlin, 1999, p. 295.
475
[2] C.R. Lin, S.T. Lin, C.R. Wang, S.L. Chou, H.E. Horn, J.M. Cheng, Y.D. Yao, S.C. Lai, J. Phys.: Condens. Matter 9 (1997) 1509. [3] S. Matsuo, H. Nakano, T. Ishiyama, Y. Fukano, J. Phys.: Condens. Matter 1 (1989) 6893. [4] S. Matsuo, T. Ishiyama, H. Nakano, Y. Fukano, J. Phys. F: Met. Phys. 18 (1988) L175. [5] A. Kobayashi, S. Matsuo, T. Ishimasa, H. Nakano, J. Phys.: Condens. Matter 9 (1997) 3205. [6] F. Cyrot-Lackmann, Solid State Commun. 103 (1997) 123. [7] T. Fujiwara, Theory of electronic structure in quasicrystals in: Z.M. Stadnik (Ed.), Physical Properties of Quasicrystals (1999), p. 169. [8] A.A. Abrikosov, Fundamentals of the Theory of Metals, North-Holland, Amsterdam, 1988. [9] T.K. Kjeldaas, W. Kohn, Phys. Rev. 105 (1957) 806. [10] W.A. Harrison, Pseudopotentials in the Theory of Metals, W.A. Benjamin Inc., New York, 1966. [11] A. Roy, M. Turner, P. Sarachik, Phys. Rev. B 37 (1988) 5522. [12] A.M. Wilson, The Theory of Metals, Cambrige University Press, London, 1965. [13] S.E. Burkov, T. Timusk, A.W. Ashkroft, J. Phys.: Condens. Matter 4 (1992) 9447. [14] S.E. Burkov, A.A. Varlamov, D.V. Livanov, Phys. Rev. B 53 (1996) 11504. [15] Yu.Kh. Vekilov, E.I. Isaev, D.V. Livanov, J. Exp. Theor. Phys. 94 (2002) 172.
Diamagnetism in quasicrystals Yu.Kh. Vekilova, E.I. Isaeva,*, B. Johanssonb,c a
Theoretical Physics Department, Moscow State Institute of Steel and Alloys (Technological University), 4, Leninskii prospect, 119049 Moscow, Russian Federation b Condensed Matter Theory Group, Uppsala University, SE-751 21 Uppsala, Sweden c Department of Materials Science and Engineering, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden Received 22 September 2004; received in revised form 29 November 2004; accepted 30 November 2004 by H. Eschrig Available online 8 December 2004
Abstract A reliable explanation of diamagnetism in quasicrystals is given. We show that the weak diamagnetism in perfect icosahedral quasicrystals is due to an atomic-like diamagnetic contribution of tightly bound conduction electrons in electron pockets of a multiconnected Fermi surface. The Landau–Peierls diamagnetic term is small due to large effective masses. At temperatures above the Debye temperature the intervalley electron–phonon scattering makes the electrons ‘free’, and the temperature dependence of the Pauli paramagnetism related to a pseudogap in the density of states at the Fermi level becomes important. q 2004 Elsevier Ltd. All rights reserved. PACS: 71.23.Ft; 75.20.Kg Keywords: A. Quasicrystals; D. Conductivity electrons; D. Diamagnetism
The icosahedral (i-) quasicrystals (QC) like i-Al–Cu–Fe, i-Al–Pd–Mn with low Mn concentration, and i-Al–Pd–Re exhibit a weak diamagnetism over a wide range of temperatures [1–5]. At very low temperatures their magnetic susceptibility c behaves as a Curie–Weiss like law with decreasing temperatures and its zero-temperature limit becomes positive (Al–Cu–Fe, Al–Pd–Mn) or nearly zero (Al–Pd–Re). At elevated temperatures c increases with increasing T, and the enhancement of c is accounted for by a temperature dependence of the Pauli paramagnetism. The diamagnetism is observed in high quality perfect samples. The poor quality samples are always paramagnetic. There is also a certain correlation between conductivity and diamagnetism in QC, because the perfect samples are better insulators and more diamagnetic. So far, no reliable explanation has been given to the
* Corresponding author. Tel.: C7 95 2304506; fax: C7 95 2362105. E-mail address: [email protected] (E.I. Isaev). 0038-1098/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2004.11.040
diamagnetism in QC. In the paper [6] it has been assumed that diamagnetism in QC is of the Landau–Peierls type for conduction electrons, and the very low electron masses in some directions for the electron pockets of multiconnected Fermi surface (FS) of QC are responsible for this. However, this assumption is not correct. Certainly, in QC one has to take into account a large number of effective Bragg reflections, which lead to small gaps on the FS due to its interaction with Bragg planes, as well as to small electron– hole pockets on it. Consequently, the group velocities of electrons at the Fermi level EF should be small, and correspondingly the effective masses should be large. The experimental data and electronic structure calculations also show that the bands near the Fermi level in QC are rather flat, and thus the FS should have many electron–hole pockets with large electron effective masses [7]. Diamagnetism of conduction electrons in QC is difficult to calculate, and its calculation is a difficult problem even for ordinary metals [8]. Kjeldaas and Kohn [9] using Kohn– Luttinger representation and perturbation theory, have obtained the following expression for the diamagnetic
474
Yu.Kh. Vekilov et al. / Solid State Communications 133 (2005) 473–475
susceptibility for Bloch electrons c ZK
e2 k0 m C c2 k02 C c4 k04 C . 2 2 m* 12p mc
(1)
where k0 is the radius of the Fermi sphere, the first term in this series is just the Landau–Peierls expression for the diamagnetic susceptibility of degenerate electrons (cLPZ K(e2k0)/(12p2m*c2)) and higher order terms in (1) are corrections due to band structure effects which are not included in the effective mass m*. The first two terms of (1) give the exact result in the following limiting cases: (1) number of conduction electrons goes to zero; (2) free electrons; (3) tightly bound electrons. The third case is more suitable for QCs, although the existence of Bloch states is not allowed for them. Certainly, QC could be considered as a structural limit of a sequence of periodic approximants with increasing lattice parameter, a/N. Then the use of the Harrison procedure [10] for FS construction becomes possible, and due to intersection with different Bragg planes FS of QC will be multiconnected with a large number of electron–hole pockets (‘fractional’ FS). In the limit of infinite lattice parameter, i.e. when m/m* vanishes exponentionally [1–5], and k0/0 like the reciprocal of the interatomic spacing, it is only the second term in formula (1) that survives, and the Landau–Peierls contribution in diamagnetic susceptibility of conduction electrons is negligible. Kjeldaas and Kohn [9] have shown that the second term in (1) reduces correctly to the conventional atomic diamagnetism c wK 6cn2 hr2 imm Accordingly, in the quasicrystalline case we have the atomic like diamagnetism of conduction electrons tightly bound in each electron pockets of the multiconnected FS. The total susceptibility will be given by a sum of atomic-like diamagnetic contributions of all electron pockets from different Bragg planes and it can exceed the paramagnetic contribution over a wide temperature interval. Let us consider the temperature dependence of the magnetic susceptibility. The experimental curves of c versus T are usually fitted by the equation c Z c0 C aT Ka C AT 2
(2)
Here c0 is a temperature independent susceptibility which may include the following three contributions: the diamagnetic susceptibility of ion core electrons ccore which is small for the above mentioned objects [1–5], the Pauli spin paramagnetic susceptibility (cP) and diamagnetic susceptibility of conduction electrons which was discussed above. The Curie-like term ccZaTKa with a%1 must originate from interaction of localized moments existing in the samples, and it prevails at very low temperatures [1–5]. There is a definite correlation between the value of cc and resistivity r at 4.2 K: a sample with larger r (4.2 K) has larger values of cc and in this sense there is some analogy with the behavior of objects which show metal-insulator transition. For example, at these temperatures the magnetic
properties are quite similar to those of the phosphorus doped Si (Si:P) semiconductor [11]. In Si:P the cZaTKa term occurs in samples with carriers near the metal-insulator transition either from the metallic or insulating side, and is attributed to the behavior of an assembly of localized spins, associated with P and distributed randomly in space. At high temperatures the third term in (2) becomes important, c increases with increasing T and an increase in c is accounted for by a temperature dependence of the Pauli paramagnetism [12], cKAT2, with a prefactor
AZ
2 m2B NðEF ÞðZkB Þ2 1 d NðxÞ 3 NðxÞ dx2 1 dNðxÞ 2 K NðxÞ dx jxZEF
(3)
where mB denotes the Bohr magneton, EF is the Fermi energy at TZ0 K, and N(x) is the electronic density of states (DOS) at the Fermi level in QC. The second derivative of the DOS, d2 NðxÞ=dx2 jxZEF must be positive, because the second term in (3) is always negative [1]. The temperature dependence of c can be qualitatively explained in the framework of multiconnected (‘fractional’) FS model. This model has been successfully used to explain the electronic transport and electron localization in QC [13– 15]. The temperature-dependent conductivity of QC can be described by the power law, that is Ds(T)wTb, where b is roughly equal to 1 for poor quality samples with higher conductivity, while b is different in both the low- and hightemperature regimes for the higher quality samples with lower conductivity (the very high quality i-Al–Pd–Re samples even reveal at T!4 K the variable range hopping conductivity). In defectless samples the FS has an infinite number of zero area pockets (valleys). In real QC the electronic states are smeared by both temperature and scattering which results in the existence of a FS with a finite number of valleys whose number depends on the strength of disorder and temperature. Because the characteristic size of a valley is connected to the uncertainty of the electron energy, the localization of electronic states occurs in the limit of ordered (defectless) state resulting in zero conductivity at zero temperature, and only deviation from ordering leads to the possibility of intervalley scattering at zero temperature which provides the nonzero, but small value of zerotemperature conductivity. At finite temperatures electron– phonon interaction is able to change the electron momentum only by a small amount, of the order of T/u, where u is the sound velocity and there exists a characteristic temperature T*wu/a (a is the average interatomic distance) below which the phonons are unable to scatter electrons from one pocket to another. When the temperature exceeds T* the scattering by phonons provides the effective intervalley scattering [13– 15]. At temperatures T* which is order of the Debye
Yu.Kh. Vekilov et al. / Solid State Communications 133 (2005) 473–475
temperature, electrons become ‘free’-like, and the Pauli paramagnetic contribution prevails. In conclusion, we have shown that the weak diamagnetism in icosahedral quasicrystals is connected to atomic-like diamagnetic contribution of conduction electrons in electron pockets of multiconnected FS of quasicrystal. The Landau– Peierls diamagnetic term is small due to large electron effective masses. At temperatures above the Debye temperature the Pauli paramagnetic contribution prevails due to the temperature dependence of the Pauli paramagnetism related to a pseudogap in the DOS at the Fermi level.
Acknowledgements We are grateful to S. Simak for valuable dicussions and remarks. Yu.Kh.V. and E.I.I. thank the Royal Swedish Academy of Sciencies (KVA), the Russian Foundation for Basic Researches (RFBR, grant #03-02-16970) for financial support.
References [1] Z.M. Stadnik (Ed.), Physical Properties of Quasicrystals, Springer, Berlin, 1999, p. 295.
475
[2] C.R. Lin, S.T. Lin, C.R. Wang, S.L. Chou, H.E. Horn, J.M. Cheng, Y.D. Yao, S.C. Lai, J. Phys.: Condens. Matter 9 (1997) 1509. [3] S. Matsuo, H. Nakano, T. Ishiyama, Y. Fukano, J. Phys.: Condens. Matter 1 (1989) 6893. [4] S. Matsuo, T. Ishiyama, H. Nakano, Y. Fukano, J. Phys. F: Met. Phys. 18 (1988) L175. [5] A. Kobayashi, S. Matsuo, T. Ishimasa, H. Nakano, J. Phys.: Condens. Matter 9 (1997) 3205. [6] F. Cyrot-Lackmann, Solid State Commun. 103 (1997) 123. [7] T. Fujiwara, Theory of electronic structure in quasicrystals in: Z.M. Stadnik (Ed.), Physical Properties of Quasicrystals (1999), p. 169. [8] A.A. Abrikosov, Fundamentals of the Theory of Metals, North-Holland, Amsterdam, 1988. [9] T.K. Kjeldaas, W. Kohn, Phys. Rev. 105 (1957) 806. [10] W.A. Harrison, Pseudopotentials in the Theory of Metals, W.A. Benjamin Inc., New York, 1966. [11] A. Roy, M. Turner, P. Sarachik, Phys. Rev. B 37 (1988) 5522. [12] A.M. Wilson, The Theory of Metals, Cambrige University Press, London, 1965. [13] S.E. Burkov, T. Timusk, A.W. Ashkroft, J. Phys.: Condens. Matter 4 (1992) 9447. [14] S.E. Burkov, A.A. Varlamov, D.V. Livanov, Phys. Rev. B 53 (1996) 11504. [15] Yu.Kh. Vekilov, E.I. Isaev, D.V. Livanov, J. Exp. Theor. Phys. 94 (2002) 172.