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Fracton superconductivity in quasicrystals – a theoretical possibility suggested by experiment
4 October 1999
Physics Letters A 261 Ž1999. 119–124 www.elsevier.nlrlocaterphysleta
Fracton superconductivity in quasicrystals – a theoretical possibility suggested by experiment K. Moulopoulos a b
a,)
, F. Cyrot-Lackmann
b
UniÕersity of Cyprus, Department of Natural Sciences, P.O. Box 20537, 1678 Nicosia, Cyprus Laboratoire d’ Etudes des Proprietes ´ ´ Electronique des Solides, CNRS, 38042 Grenoble, France Received 23 July 1999; accepted 13 August 1999 Communicated by V.M. Agranovich
Abstract Scattering of electrons due to fractons can result in a resistivity that decreases with temperature. Such a behavior also appears in real quasicrystalline materials. If this is then attributed to effective fracton scattering, fracton-superconductivity would be theoretically possible. By fitting with the scattering result the experimental resistivity data for the AlPdMn quasicrystal, we estimate the corresponding Tc . This effective fracton interpretation, not unexpected for a self-similar system, is also found consistent with other experiments on thermal and acoustic properties of the AlPdMn quasicrystal. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 61.44.Br; 74.20.-z; 61.43.Hv Keywords: Quasicrystals; Fractal; Fractons; Eliashberg; Superconductivity; Resistivity
The quasicrystalline state w1x is a new and novel state of matter discovered during the past decade w2x, with anomalous physical properties that still pose a challenge to a fundamental theoretical understanding. Most concepts of conventional Solid State Physics are inappropriate for a rigorous theoretical modeling of ideal quasicrystalline systems. The basic reason is the lack of periodicity, that makes concepts from Bloch theory invalid, but also the lack of randomness, in its usual sense, that, rigorously speaking, prevents one from using concepts pertaining to disordered systems. In fact, from a structural point of
) Corresponding author. Tel.: q357-2-338671; fax: q357-2339060; e-mail: [email protected]
view, quasicrystals can be viewed as being in between periodic and purely disordered systems. Their seemingly irregular structure incorporates a complete Žaperiodic. long-range order. Nevertheless, there have been in the past decade important theoretical developments towards a complete structural characterization, especially based on the hyperspace construction or, alternatively, on matching rules in physical space. However, giÕen the structure of an ideal quasicrystal, almost nothing is known with certainty concerning its electronic properties. It is only in 1-d Žone-dimensional. quasiperiodic chains that rigorous results are known w3x, and this is basically a consequence of the trivial topology. Most of these results manifest a new type of electronic state that has been called ‘critical’, which can also be viewed as having a
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 5 8 4 - 8
120
K. Moulopoulos, F. Cyrot-Lackmannr Physics Letters A 261 (1999) 119–124
character that is between extended and localized states Žthe former being the generic electronic state in an ideally periodic system, the latter being the one in a 1-d disordered system.. What is fascinating Žalthough not entirely unexpected. is that these critical states have a multifractal nature, basically because of a type of self-similarity that characterizes a typical quasiperiodic structure and that is further discussed below. Experimentally there has been active research on structural aspects w4x with few questions still being unresolved, concerning especially the actual positions of atoms. In the past few years there has also been intense activity on electronic properties 1 as well. The latter demonstrate a number of anomalous transport properties that seem to be opposite to the typical behavior of metallic materials. The simplest of these, namely the temperature dependence of the resistivity, is the focus of this Letter, and will be given a new interpretation, related to the fractal nature of the quasicrystalline state. The type of self-similarity mentioned above, is the well-known scale-invariance of an ideal quasicrystal under appropriate inflationrdeflation transformations. It has also been suggested that in quasicrystalline phases there exist atomic clusters with hierarchical and quasiperiodic packings w5x, the existence of which is basically another manifestation of the same symmetry. It may therefore be expected that quasicrystals may effectively have properties that generically appear in fractal systems. In particular, the vibrational properties may be describable in terms of effective fracton spectra, a concept introduced w6x shortly before the discovery of quasicrystals and used in approximate theoretical modeling of certain disordered systems w7x. The term ‘fracton’ refers to quanta of special localized vibrational modes in fractal systems, in analogy with the term ‘phonon’ for the extended modes in periodic and in homogeneous systems. In a fractal lattice and at long length scales, the vibrational excitations are phonons with a linear dispersion law. But at length scales less than a crossover length, corresponding to frequencies greater than a crossover frequency Žcalled vc below.,
1
For a review see C. Berger, Ref. w4x.
the vibrational excitations are fractons with a nonstandard dispersion law, and with an exponent described by the ‘fracton dimension’ Ždenoted below by d˜ .. We demonstrate in this work the interesting fact that such an assumption of fractal vibrational behavior, and in particular the simulation of the vibrating quasicrystalline structure with a vibrating bond-percolation network w8x with the proper exponents, can actually account for the anomalous transport properties of this system. In more detail, a simple electronscattering calculation due to fractons associated with these vibrations, can lead to a resistivity that decreases with temperature, a behavior opposite to the normal electron-phonon scattering behavior manifested in metallic materials. On the other hand, it was already mentioned that quasicrystals also behave anomalously; the experimentally measured resistivity w9x, for example, always shows the same qualitative behavior, namely a resistivity lowering with temperature. The basic proposition in this Letter is to relate the two qualitatively similar results in a quantitative manner. Hence, by attributing the experimental behavior of the resistivity of real quasicrystalline materials to an effective fracton spectrum Žand to the eventual electron–fracton scattering., we determine the corresponding electron–fracton coupling. On the other hand, the mere existence of fractons and the scattering of electrons off them, will in principle lead to an attractive electron-electron interaction Žin a w10x elecway similar to the conventional Frochlich ¨ tron-phonon mechanism. and will in principle result in superconductivity due to the corresponding fracton-mediated electron-pairing. In this Letter: 1. we use the above mentioned electron–fracton scattering result to determine the electron–fracton coupling from the temperature-dependent resistivity of the AlPdMn quasicrystal; 2. we show evidence that this effective fracton interpretation is consistent with other experiments on the same material, namely, measurements of specific heat, thermal conductivity, and temperature-dependent velocity of sound; and 3. we use the above determined electron–fracton coupling within an approximate form of the Eliashberg theory w11x of strong-coupling superconductivity, to give an estimate of the range of the critical temperature Tc of the corresponding fracton-mediated superconductivity.
K. Moulopoulos, F. Cyrot-Lackmannr Physics Letters A 261 (1999) 119–124
In more detail, a calculation to second order in perturbation theory with respect to electron–fracton coupling, leads to a scattering time that is approximately given by 1
t Ž EF .
ž
s A
ne 2 m
/ž ž ln
"vF k B T y " vc
2
//
Ž 1.
with
ž
A
ne 2 m
/
2p s "
N Ž EF .
ž
˜
A simple scattering-time approximation for the resistivity Žequivalent to a neglect of the usual cosŽ k P k X . factor in the Boltzmann equation. is expected to be a good approximation in quasicrystals w14x because of the stochastically fragmented kspace. ŽIn an ideal quasicrystal, k-space is infinitely fragmented, with an eventual unpredictability in the positions of the allowed k-values.. As a result, the resistivity is approximately given by w13x
2
˜ i 4l20 dN ˜
2 rj 2v Fd vc2 d r D
/
Ž 2.
provided that one uses a fractal dimension D s 4 and a fracton dimension d˜s 4r3 Žwhat is known as the Alexander and Orbach ansatz w7x., which are the proper values for a bond-percolation network at the marginal dimensionality d s 6 where both mean-field exponents and hyperscaling are valid w12x. In Ž1. and Ž2. r is the bulk mass-density, Ni is the total number of atoms, l0 is the electron–fracton coupling constant, j is a characteristic length of fractal behavior Žsee below and Eq. Ž8.., v F is the fracton analog of the Debye frequency, vc is the phonon-fracton crossover frequency, and N Ž E F . is the electronic density of states per spin at the Fermi level. The above logarithmic behavior resulting from the special choice of a 6-d percolation network was actually noted in an earlier work w13x for a different system Ža metallic glass.. In that work, however, fractons should live in 3-d Žthey are actually treated together with phonons. and the choice D s 4 is not justified. Moreover, as a result of the competition between the two types of modes, a minimum in the resistivity was obtained, proposed to describe the Kondo effect. In our case, however, the physics is entirely different; we have in mind an effectiÕe fracton behavior that may summarize the entire vibrational physics, a picture that is not contradictory to the use of exponents pertaining to higher dimensionality and the omission of other modes Žphonons.. We will see below that such an effective fracton interpretation can reproduce the anomalous experimental behavior of the resistivity, is consistent with and offers an alternative Žand complementary. interpretation for other experiments, and can also lead to the prediction of a new ordered state, namely novel superconductivity.
121
žž
r s A ln
"vF k B T y " vc
2
//
Ž 3.
for " v c - k B T < " v F , and is seen to declinewith temperature Ž A is given by Ž2... We now discuss the possibility of superconductivity caused by the above electron–fracton scattering, which can in principle serve as a mechanism for electron-pairing. By using the corresponding fracton density of states ˜
NŽ v . s
˜ i v dy 1 dN
Ž 4.
v Fd˜
and fracton dispersion relation Dr d˜
j
v s vc
ž / 2
q
Ž 5.
in the framework of the Eliashberg theory of strongcoupling superconductivity w11x we arrive at an Eliashberg function
a 2F Ž v . s
˜ i 4l20 N Ž EF . dN v Fd˜ 2 rj 2 vc2 d˜r D
.
Ž 6.
The corresponding superconducting coupling paramemeter, defined in the standard way by
lŽ T s 0. s 2
vF
Hv
a 2F Ž v .
c
dv
v
,
is then found to be of the form
lŽ T s 0. s
˜ i 4l20 2 N Ž EF . dN v Fd˜ 2 rj 2
vc2 d˜r D
ln
vF
ž / vc
.
Ž 7.
We now need to estimate the electron–fracton coupling l20 from experimental measurements of resistivity by using Ž3. and combining with Ž2.. By
K. Moulopoulos, F. Cyrot-Lackmannr Physics Letters A 261 (1999) 119–124
122
fitting the resistivity data w9x on the quasicrystal Al 70.5 Pd 22 Mn 7.5 with the expression Ž3., for 100 F T F 150 K, we obtain " vc ; 0.4 K, " v F ; 19000 K, and A ; 300 mV cm. We should point out here that the very low value for the crossover frequency vc is actually expected due to the wide plateau measured recently w15x in the thermal conductivity of the AlPdMn system. This is related to a large characteristic length j below which we effectively have fractal behavior Žor, alternatively, above which the system looks homogeneous.. In the case of the AlPdMn system, and from the relation
vF vc
j
s
Dr d˜
ž /
Ž 8.
a0
we find j , 36 a0 , where a0 is the lowest atomic length scale, which we here take as the Bohr radius. We now show that the above is also consistent with other experiments on the same material: Measurements of low-temperature specific heat w16x can be rather well reproduced through a phonon-fracton crossover interpretation. Indeed, following the calculation of Avogadro et al. w17x we arrive at results in reasonable agreement with experimental points shown in Fig. 1 of Ref. w16x. ŽIn this temperature range, the electronic contribution is negligible compared to the vibrational terms, as also shown by the solid line of the same Figure.. In addition, the rise of the thermal conductivity Ž k . above the plateau w15x seems to be consistent with the linear decrease of the relative variation Ž d Õ srÕs . of the velocity of sound Ž Õs . with temperature w18x Žwhat could be called the ‘Bellessa effect’ w19x. if this rise is explained through a theory of phonon-assisted fracton-hopping proposed by Orbach and collaborators w20x. In this theory, the two slopes are related by
ž
d Õ srÕs T
/
0.1 sy kB
ž
j2 2
2p Õs
k
/ž / T
.
Ž 9.
The experimentally measured value w18x of Ž d ÕsrÕs . , y1.818 = 10y5rK Ždetermined from T the linear decrease in the temperature range 5 - T -
27 K and at the lowest measured frequency of v ; 28 MHz. then yields through Ž9. a value of Ž krT . , 8.889 = 10y5 WrK 2 m, which is in turn reasonably consistent with the experimental data above the plateau shown in Fig. 2 of Ref. w15x. Note that our fit of the resistivity data was carried out at temperatures aboÕe the plateau. In this temperature range we have excitations of energies higher than two-level systems w18x; it is actually such excitations that are responsible for the Bellessa effect as is well known w19x. The above experiments can therefore be viewed as probes of fracton hopping in higher temperatures Žalthough of course the presence of effective fractons will persist down to quite low temperatures, as is also shown by the very low value of the crossover frequency vc .. This fracton interpretation should also be viewed as complementary to other interpretations given earlier in this temperature range. In Ref. w15x, it was briefly suggested that this high-T behavior could be explained in the framework of the soft potential model, which is known to be a competing theory to the fracton model w7x. Also in other acoustic experiments w21x it is also briefly suggested that the diminuation of ultrasonic attenuation as the temperature is lowered, is in agreement with a picture of phason hopping. Again, we here propose the theory of phonon-assisted fracton-hopping w20x as an alternative interpretation. The above mentioned suggestions in the literature have just been tentatively made, they have not of course been established, and our alternative proposition actually complements the possibilities in a field that is still open. By taking, therefore, the above fracton interpretation at face value, we make use of the above parameters with an effective density rs ; 6 and an effective electron-to-atom ratio Z , 1.5, and together with some uncertainty associated with the effective electronic density of states Žwe use a N Ž EF . being between 101 and 15 of the free-electron density of states., we arrive at the following estimate for the range of l values 1.26 - l Ž T s 0 . - 1.79.
Ž 10 .
Finally, we use Kresin’s expression w22x for Tc Tc s
0.25² v 2 :1r2
(expŽ 2rl
eff
. y1
Ž 11 .
K. Moulopoulos, F. Cyrot-Lackmannr Physics Letters A 261 (1999) 119–124
which is supposed to be valid for any arbitrary strength l. The parameter leff is defined by
l y m) leff s
Ž 12 .
1 q 2 m ) q lm ) t Ž l .
and t Ž l . , 1.5exp Ž y0.28 l . .
Ž 13 .
The characteristic frequency appearing in Ž11. is defined by vF
²v2:s
Hv
a 2F Ž v . v dv
c
vF
Hv
2
a FŽ v.
c
Ž 14 .
dv
v
which, with the use of Ž6., leads to ² v 2 :1r2 , 4000 K. We also need an estimate for the Coulomb parameter m ) which we take from Thomas–Fermi screening theory. This parameter is defined by m m) s Ž 15 . vp 1 q m ln vF
ž /
with m given by a Fermi surface average, namely
m s N Ž EF . ²
4p e 2 V Ž k 2 q k 02 .
:FS ,
k 02
ž
ln 4 2
8kF
k F2 k 02
q1
/
Ž 16 . with the Thomas–Fermi screening wavevector k 0 given by k 02 k F2
16
s
ž / 3p 2
2r3
rs
Ž 17 .
and the plasma frequency given by
v p s 4.71 = 1.16 = 10 4rrs3r2 K.
Ž 18 .
For an effective rs ; 6 this yields m ) s 0.816, which signifies an especially strong electron-electron interaction, that is actually expected for quasicrystals from interpretation of transport data w23,24x. Substituting all this into Ž11. with an effective rs ; 6, we arrive at the following estimate of the range of critical temperatures 0.23 K - Tc - 17.2 K.
Ž 19 .
123
ŽNote that the critical temperature can actually be much higher, for smaller values of m ) , that can result, for example, from a model different from the Thomas–Fermi theory.. We have proposed a mechanism for electron-pairing and superconductivity that may be inherent to the properties of the generic quasicrystalline phase. With regard to real materials such a novel superconductivity has not so far been observed. Although this type of ordering is in principle possible, it is expected that there are other physical mechanisms, at least in AlPdMn, that might prohibit its appearance. The existence, for example, of magnetic moments, possible spin-orbit couplings etc. may have such a destructive effect and render this fracton-superconductivity unobservable. The question then of how to experimentally overcome these prohibiting factors, in this or some other quasicrystalline material, becomes of interest and would deserve further investigation.
References w1x C. Janot, Quasicrystals: A Primer, Oxford Science Publications, 2nd ed., 1994. w2x D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Phys. Rev. Lett. 53 Ž1984. 1951. w3x M. Kohmoto, L.P. Kadanoff, Ch. Tang, Phys. Rev. Lett. 50 Ž1983. 1870; S. Ostlund, R. Pandit, D. Rand, H.J. Schellnhuber, E.D. Siggia, Phys. Rev. Lett. 50 Ž1983. 1873; for rigorous theorems on more general quasiperiodic chains see J. Bellissard, A. Bovier, J.M. Ghez, Commun. Math. Phys. 135 Ž1991. 379; J.M. Luck, Phys. Rev. B 39 Ž1989. 5834. w4x F. Hippert, D. Gratias ŽEds.., For a summary of experimental activity in this field see Lectures On Quasicrystals, Les Editions de Physique Les Ulis, 1994. w5x C. Janot, M. de Boissieu, Phys. Rev. Lett. 72 Ž1994. 1674. w6x S. Alexander, R. Orbach, J. Phys. ŽParis. Lett. 43 Ž1982. L-625. w7x T. Nakayama, K. Yakubo, R. Orbach, Rev. Mod. Phys. 66 Ž1994. 381. w8x R. Orbach, Science 231 Ž1986. 814. w9x P. Lanco, T. Klein, C. Berger, F. Cyrot-Lackmann, G. Fourcaudot, A. Sulpice, Europhys. Lett. 18 Ž1992. 227. w10x H. Frohlich, Phys. Rev. 79 Ž1950. 845. ¨ w11x G.M. Eliashberg, Zh. Eksp. Teor. Fiz. 38 Ž1960. 966 ŽSov. Phys. JETP 11 Ž1960. 696.. w12x D. Stauffer, Phys. Rep. 54 Ž1979. 1. w13x D. Tian, Phys. Rev. B 45 Ž1992. 8116. w14x N.W. Ashcroft, personal communication. w15x M.A. Chernikov, Phys. Rev. B 51 Ž1995. 153. w16x M.A. Chernikov, Phys. Rev. B 48 Ž1993. 3058.
124
K. Moulopoulos, F. Cyrot-Lackmannr Physics Letters A 261 (1999) 119–124
w17x A. Avogadro, Phys. Rev. B 33 Ž1986. 5637. w18x N. Vernier, G. Bellessa, B. Perrin, A. Zarembowitch, M. De Boissieu, Europhys. Lett. 22 Ž1993. 187. w19x G. Bellessa, Phys. Rev. Lett. 40 Ž1978. 1456. w20x A. Jagannathan, R. Orbach, O. Entin-Wohlman, Phys. Rev. B 39 Ž1989. 13465.
w21x Y. Amazit, Europhys. Lett. 20 Ž1992. 703. w22x V.Z. Kresin, Phys. Lett. A 122 Ž1987. 434. w23x T. Klein, H. Rakoto, C. Berger, G. Fourcaudot, F. CyrotLackmann, Phys. Rev. B 45 Ž1992. 2046. w24x T. Klein, C. Berger, G. Fourcaudot, J.C. Grieco, P. Lanco, F. Cyrot-Lackmann, J. Non-Cryst. Sol. 156–158 Ž1993. 901.
Physics Letters A 261 Ž1999. 119–124 www.elsevier.nlrlocaterphysleta
Fracton superconductivity in quasicrystals – a theoretical possibility suggested by experiment K. Moulopoulos a b
a,)
, F. Cyrot-Lackmann
b
UniÕersity of Cyprus, Department of Natural Sciences, P.O. Box 20537, 1678 Nicosia, Cyprus Laboratoire d’ Etudes des Proprietes ´ ´ Electronique des Solides, CNRS, 38042 Grenoble, France Received 23 July 1999; accepted 13 August 1999 Communicated by V.M. Agranovich
Abstract Scattering of electrons due to fractons can result in a resistivity that decreases with temperature. Such a behavior also appears in real quasicrystalline materials. If this is then attributed to effective fracton scattering, fracton-superconductivity would be theoretically possible. By fitting with the scattering result the experimental resistivity data for the AlPdMn quasicrystal, we estimate the corresponding Tc . This effective fracton interpretation, not unexpected for a self-similar system, is also found consistent with other experiments on thermal and acoustic properties of the AlPdMn quasicrystal. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 61.44.Br; 74.20.-z; 61.43.Hv Keywords: Quasicrystals; Fractal; Fractons; Eliashberg; Superconductivity; Resistivity
The quasicrystalline state w1x is a new and novel state of matter discovered during the past decade w2x, with anomalous physical properties that still pose a challenge to a fundamental theoretical understanding. Most concepts of conventional Solid State Physics are inappropriate for a rigorous theoretical modeling of ideal quasicrystalline systems. The basic reason is the lack of periodicity, that makes concepts from Bloch theory invalid, but also the lack of randomness, in its usual sense, that, rigorously speaking, prevents one from using concepts pertaining to disordered systems. In fact, from a structural point of
) Corresponding author. Tel.: q357-2-338671; fax: q357-2339060; e-mail: [email protected]
view, quasicrystals can be viewed as being in between periodic and purely disordered systems. Their seemingly irregular structure incorporates a complete Žaperiodic. long-range order. Nevertheless, there have been in the past decade important theoretical developments towards a complete structural characterization, especially based on the hyperspace construction or, alternatively, on matching rules in physical space. However, giÕen the structure of an ideal quasicrystal, almost nothing is known with certainty concerning its electronic properties. It is only in 1-d Žone-dimensional. quasiperiodic chains that rigorous results are known w3x, and this is basically a consequence of the trivial topology. Most of these results manifest a new type of electronic state that has been called ‘critical’, which can also be viewed as having a
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 5 8 4 - 8
120
K. Moulopoulos, F. Cyrot-Lackmannr Physics Letters A 261 (1999) 119–124
character that is between extended and localized states Žthe former being the generic electronic state in an ideally periodic system, the latter being the one in a 1-d disordered system.. What is fascinating Žalthough not entirely unexpected. is that these critical states have a multifractal nature, basically because of a type of self-similarity that characterizes a typical quasiperiodic structure and that is further discussed below. Experimentally there has been active research on structural aspects w4x with few questions still being unresolved, concerning especially the actual positions of atoms. In the past few years there has also been intense activity on electronic properties 1 as well. The latter demonstrate a number of anomalous transport properties that seem to be opposite to the typical behavior of metallic materials. The simplest of these, namely the temperature dependence of the resistivity, is the focus of this Letter, and will be given a new interpretation, related to the fractal nature of the quasicrystalline state. The type of self-similarity mentioned above, is the well-known scale-invariance of an ideal quasicrystal under appropriate inflationrdeflation transformations. It has also been suggested that in quasicrystalline phases there exist atomic clusters with hierarchical and quasiperiodic packings w5x, the existence of which is basically another manifestation of the same symmetry. It may therefore be expected that quasicrystals may effectively have properties that generically appear in fractal systems. In particular, the vibrational properties may be describable in terms of effective fracton spectra, a concept introduced w6x shortly before the discovery of quasicrystals and used in approximate theoretical modeling of certain disordered systems w7x. The term ‘fracton’ refers to quanta of special localized vibrational modes in fractal systems, in analogy with the term ‘phonon’ for the extended modes in periodic and in homogeneous systems. In a fractal lattice and at long length scales, the vibrational excitations are phonons with a linear dispersion law. But at length scales less than a crossover length, corresponding to frequencies greater than a crossover frequency Žcalled vc below.,
1
For a review see C. Berger, Ref. w4x.
the vibrational excitations are fractons with a nonstandard dispersion law, and with an exponent described by the ‘fracton dimension’ Ždenoted below by d˜ .. We demonstrate in this work the interesting fact that such an assumption of fractal vibrational behavior, and in particular the simulation of the vibrating quasicrystalline structure with a vibrating bond-percolation network w8x with the proper exponents, can actually account for the anomalous transport properties of this system. In more detail, a simple electronscattering calculation due to fractons associated with these vibrations, can lead to a resistivity that decreases with temperature, a behavior opposite to the normal electron-phonon scattering behavior manifested in metallic materials. On the other hand, it was already mentioned that quasicrystals also behave anomalously; the experimentally measured resistivity w9x, for example, always shows the same qualitative behavior, namely a resistivity lowering with temperature. The basic proposition in this Letter is to relate the two qualitatively similar results in a quantitative manner. Hence, by attributing the experimental behavior of the resistivity of real quasicrystalline materials to an effective fracton spectrum Žand to the eventual electron–fracton scattering., we determine the corresponding electron–fracton coupling. On the other hand, the mere existence of fractons and the scattering of electrons off them, will in principle lead to an attractive electron-electron interaction Žin a w10x elecway similar to the conventional Frochlich ¨ tron-phonon mechanism. and will in principle result in superconductivity due to the corresponding fracton-mediated electron-pairing. In this Letter: 1. we use the above mentioned electron–fracton scattering result to determine the electron–fracton coupling from the temperature-dependent resistivity of the AlPdMn quasicrystal; 2. we show evidence that this effective fracton interpretation is consistent with other experiments on the same material, namely, measurements of specific heat, thermal conductivity, and temperature-dependent velocity of sound; and 3. we use the above determined electron–fracton coupling within an approximate form of the Eliashberg theory w11x of strong-coupling superconductivity, to give an estimate of the range of the critical temperature Tc of the corresponding fracton-mediated superconductivity.
K. Moulopoulos, F. Cyrot-Lackmannr Physics Letters A 261 (1999) 119–124
In more detail, a calculation to second order in perturbation theory with respect to electron–fracton coupling, leads to a scattering time that is approximately given by 1
t Ž EF .
ž
s A
ne 2 m
/ž ž ln
"vF k B T y " vc
2
//
Ž 1.
with
ž
A
ne 2 m
/
2p s "
N Ž EF .
ž
˜
A simple scattering-time approximation for the resistivity Žequivalent to a neglect of the usual cosŽ k P k X . factor in the Boltzmann equation. is expected to be a good approximation in quasicrystals w14x because of the stochastically fragmented kspace. ŽIn an ideal quasicrystal, k-space is infinitely fragmented, with an eventual unpredictability in the positions of the allowed k-values.. As a result, the resistivity is approximately given by w13x
2
˜ i 4l20 dN ˜
2 rj 2v Fd vc2 d r D
/
Ž 2.
provided that one uses a fractal dimension D s 4 and a fracton dimension d˜s 4r3 Žwhat is known as the Alexander and Orbach ansatz w7x., which are the proper values for a bond-percolation network at the marginal dimensionality d s 6 where both mean-field exponents and hyperscaling are valid w12x. In Ž1. and Ž2. r is the bulk mass-density, Ni is the total number of atoms, l0 is the electron–fracton coupling constant, j is a characteristic length of fractal behavior Žsee below and Eq. Ž8.., v F is the fracton analog of the Debye frequency, vc is the phonon-fracton crossover frequency, and N Ž E F . is the electronic density of states per spin at the Fermi level. The above logarithmic behavior resulting from the special choice of a 6-d percolation network was actually noted in an earlier work w13x for a different system Ža metallic glass.. In that work, however, fractons should live in 3-d Žthey are actually treated together with phonons. and the choice D s 4 is not justified. Moreover, as a result of the competition between the two types of modes, a minimum in the resistivity was obtained, proposed to describe the Kondo effect. In our case, however, the physics is entirely different; we have in mind an effectiÕe fracton behavior that may summarize the entire vibrational physics, a picture that is not contradictory to the use of exponents pertaining to higher dimensionality and the omission of other modes Žphonons.. We will see below that such an effective fracton interpretation can reproduce the anomalous experimental behavior of the resistivity, is consistent with and offers an alternative Žand complementary. interpretation for other experiments, and can also lead to the prediction of a new ordered state, namely novel superconductivity.
121
žž
r s A ln
"vF k B T y " vc
2
//
Ž 3.
for " v c - k B T < " v F , and is seen to declinewith temperature Ž A is given by Ž2... We now discuss the possibility of superconductivity caused by the above electron–fracton scattering, which can in principle serve as a mechanism for electron-pairing. By using the corresponding fracton density of states ˜
NŽ v . s
˜ i v dy 1 dN
Ž 4.
v Fd˜
and fracton dispersion relation Dr d˜
j
v s vc
ž / 2
q
Ž 5.
in the framework of the Eliashberg theory of strongcoupling superconductivity w11x we arrive at an Eliashberg function
a 2F Ž v . s
˜ i 4l20 N Ž EF . dN v Fd˜ 2 rj 2 vc2 d˜r D
.
Ž 6.
The corresponding superconducting coupling paramemeter, defined in the standard way by
lŽ T s 0. s 2
vF
Hv
a 2F Ž v .
c
dv
v
,
is then found to be of the form
lŽ T s 0. s
˜ i 4l20 2 N Ž EF . dN v Fd˜ 2 rj 2
vc2 d˜r D
ln
vF
ž / vc
.
Ž 7.
We now need to estimate the electron–fracton coupling l20 from experimental measurements of resistivity by using Ž3. and combining with Ž2.. By
K. Moulopoulos, F. Cyrot-Lackmannr Physics Letters A 261 (1999) 119–124
122
fitting the resistivity data w9x on the quasicrystal Al 70.5 Pd 22 Mn 7.5 with the expression Ž3., for 100 F T F 150 K, we obtain " vc ; 0.4 K, " v F ; 19000 K, and A ; 300 mV cm. We should point out here that the very low value for the crossover frequency vc is actually expected due to the wide plateau measured recently w15x in the thermal conductivity of the AlPdMn system. This is related to a large characteristic length j below which we effectively have fractal behavior Žor, alternatively, above which the system looks homogeneous.. In the case of the AlPdMn system, and from the relation
vF vc
j
s
Dr d˜
ž /
Ž 8.
a0
we find j , 36 a0 , where a0 is the lowest atomic length scale, which we here take as the Bohr radius. We now show that the above is also consistent with other experiments on the same material: Measurements of low-temperature specific heat w16x can be rather well reproduced through a phonon-fracton crossover interpretation. Indeed, following the calculation of Avogadro et al. w17x we arrive at results in reasonable agreement with experimental points shown in Fig. 1 of Ref. w16x. ŽIn this temperature range, the electronic contribution is negligible compared to the vibrational terms, as also shown by the solid line of the same Figure.. In addition, the rise of the thermal conductivity Ž k . above the plateau w15x seems to be consistent with the linear decrease of the relative variation Ž d Õ srÕs . of the velocity of sound Ž Õs . with temperature w18x Žwhat could be called the ‘Bellessa effect’ w19x. if this rise is explained through a theory of phonon-assisted fracton-hopping proposed by Orbach and collaborators w20x. In this theory, the two slopes are related by
ž
d Õ srÕs T
/
0.1 sy kB
ž
j2 2
2p Õs
k
/ž / T
.
Ž 9.
The experimentally measured value w18x of Ž d ÕsrÕs . , y1.818 = 10y5rK Ždetermined from T the linear decrease in the temperature range 5 - T -
27 K and at the lowest measured frequency of v ; 28 MHz. then yields through Ž9. a value of Ž krT . , 8.889 = 10y5 WrK 2 m, which is in turn reasonably consistent with the experimental data above the plateau shown in Fig. 2 of Ref. w15x. Note that our fit of the resistivity data was carried out at temperatures aboÕe the plateau. In this temperature range we have excitations of energies higher than two-level systems w18x; it is actually such excitations that are responsible for the Bellessa effect as is well known w19x. The above experiments can therefore be viewed as probes of fracton hopping in higher temperatures Žalthough of course the presence of effective fractons will persist down to quite low temperatures, as is also shown by the very low value of the crossover frequency vc .. This fracton interpretation should also be viewed as complementary to other interpretations given earlier in this temperature range. In Ref. w15x, it was briefly suggested that this high-T behavior could be explained in the framework of the soft potential model, which is known to be a competing theory to the fracton model w7x. Also in other acoustic experiments w21x it is also briefly suggested that the diminuation of ultrasonic attenuation as the temperature is lowered, is in agreement with a picture of phason hopping. Again, we here propose the theory of phonon-assisted fracton-hopping w20x as an alternative interpretation. The above mentioned suggestions in the literature have just been tentatively made, they have not of course been established, and our alternative proposition actually complements the possibilities in a field that is still open. By taking, therefore, the above fracton interpretation at face value, we make use of the above parameters with an effective density rs ; 6 and an effective electron-to-atom ratio Z , 1.5, and together with some uncertainty associated with the effective electronic density of states Žwe use a N Ž EF . being between 101 and 15 of the free-electron density of states., we arrive at the following estimate for the range of l values 1.26 - l Ž T s 0 . - 1.79.
Ž 10 .
Finally, we use Kresin’s expression w22x for Tc Tc s
0.25² v 2 :1r2
(expŽ 2rl
eff
. y1
Ž 11 .
K. Moulopoulos, F. Cyrot-Lackmannr Physics Letters A 261 (1999) 119–124
which is supposed to be valid for any arbitrary strength l. The parameter leff is defined by
l y m) leff s
Ž 12 .
1 q 2 m ) q lm ) t Ž l .
and t Ž l . , 1.5exp Ž y0.28 l . .
Ž 13 .
The characteristic frequency appearing in Ž11. is defined by vF
²v2:s
Hv
a 2F Ž v . v dv
c
vF
Hv
2
a FŽ v.
c
Ž 14 .
dv
v
which, with the use of Ž6., leads to ² v 2 :1r2 , 4000 K. We also need an estimate for the Coulomb parameter m ) which we take from Thomas–Fermi screening theory. This parameter is defined by m m) s Ž 15 . vp 1 q m ln vF
ž /
with m given by a Fermi surface average, namely
m s N Ž EF . ²
4p e 2 V Ž k 2 q k 02 .
:FS ,
k 02
ž
ln 4 2
8kF
k F2 k 02
q1
/
Ž 16 . with the Thomas–Fermi screening wavevector k 0 given by k 02 k F2
16
s
ž / 3p 2
2r3
rs
Ž 17 .
and the plasma frequency given by
v p s 4.71 = 1.16 = 10 4rrs3r2 K.
Ž 18 .
For an effective rs ; 6 this yields m ) s 0.816, which signifies an especially strong electron-electron interaction, that is actually expected for quasicrystals from interpretation of transport data w23,24x. Substituting all this into Ž11. with an effective rs ; 6, we arrive at the following estimate of the range of critical temperatures 0.23 K - Tc - 17.2 K.
Ž 19 .
123
ŽNote that the critical temperature can actually be much higher, for smaller values of m ) , that can result, for example, from a model different from the Thomas–Fermi theory.. We have proposed a mechanism for electron-pairing and superconductivity that may be inherent to the properties of the generic quasicrystalline phase. With regard to real materials such a novel superconductivity has not so far been observed. Although this type of ordering is in principle possible, it is expected that there are other physical mechanisms, at least in AlPdMn, that might prohibit its appearance. The existence, for example, of magnetic moments, possible spin-orbit couplings etc. may have such a destructive effect and render this fracton-superconductivity unobservable. The question then of how to experimentally overcome these prohibiting factors, in this or some other quasicrystalline material, becomes of interest and would deserve further investigation.
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w17x A. Avogadro, Phys. Rev. B 33 Ž1986. 5637. w18x N. Vernier, G. Bellessa, B. Perrin, A. Zarembowitch, M. De Boissieu, Europhys. Lett. 22 Ž1993. 187. w19x G. Bellessa, Phys. Rev. Lett. 40 Ž1978. 1456. w20x A. Jagannathan, R. Orbach, O. Entin-Wohlman, Phys. Rev. B 39 Ž1989. 13465.
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