Modeling the electronic transport properties of Al–Cu–Fe phases

doi:10.1016/S0921-4526(01)00237-X

Physica B: Condensed Matter

Volume 301, Issues 3–4, August 2001, Pages 267-275

https://doi.org/10.1016/S0921-4526(01)00237-X Get rights and content

Abstract

The temperature dependence of transport coefficients is theoretically examined for Al–Cu–Fe phases. We investigate the electronic conductivity, the thermoelectric power, the Hall coefficient, the thermalconductivity, and the Lorenz number. The spectral dependence of the resistivity is modeled by means of a wide and a narrow Lorentzian in accordance with results of ab initio calculations (LMTO-ASA, Kubo–Greenwood formula). This model extracts those properties from the ab initio results which are indispensables to consistently explain the transport coefficients mentioned above. Sub-diffusive transport is found in our ab initio results. The validities of both the Matthiessen rule and the Wiedemann–Franz law are analyzed. We generalize to the bulk quasicrystal where the narrow Lorentzian is found to become more pronounced than in the (1/1) approximant.

Introduction

Low electronic density of states at the Fermi energy has consequences for both phase stability and electronic transport [1], [2], [3], [4]. Such stabilizing pseudogaps on the $1 eV$ scale were also found in quasicrystals and related approximants [5], [6], [7]. Moreover, experiments have revealed spectral features on energy scales down to a few $10 meV$ [8], [9], [10]. The latter ones are believed to be closely related to the anomalous electronic transport properties of quasicrystals. Hence, growing interest is devoted to spectral fine structures and related topics [8], [9], [10], [11], [12], [13].

The most interesting quasicrystals are influenced by both strong electronic scattering and interference (e.g. i-AlPdRe [14], [15]). Transport parameters should thus be calculated from Kubo-type formulas rather than from a band-structure based Boltzmann approach [16], [17]. This confines reliable self-consistent calculations to small approximants which are not yet fully equivalent to the quasicrystals. However, one can try to make a model of the spectral conductivity of a small approximant which reasonably reproduces the ab initio results. This model must thus consider the important spectral features. In view of a related quasicrystal one can hope that, after fitting the model parameters to a few reliable experimental data, other experimental findings can be explained within this model.

Recently, we have proposed such a modeling procedure for i-AlCuFe [18]. The ab initio spectral resistivity of the (1/1) approximant has been fitted by a wide and a narrow Lorentzian. The first one represents what is common with amorphous phases whereas the latter one is specific for the approximant. The present work shows that the model thus obtained provides indeed the Hall coefficient in accordance with available experimental results [19], [20].

Section 2 presents the basic concepts employed in this work. 3 Ab initio results, 4 Resistivity model, 5 Scaling of the narrow resistivity peak, 6 Applications of the model deal with the approximant phase (Section 3—ab initio results, Section 4—Lorentzian resistivity model, Section 5—resistivity scaling with the level broadening, Section 6— transport properties calculated from the Lorentzian model). In Section 7, we propose a Lorentzian resistivity model for the bulk icosahedral quasicrystal and compare it with experiments. Conclusions are drawn in Section 8.

Section snippets

Basic concepts

We employ the Cockayne model [21] of a hypothetical i-AlCuFe (1/1) approximant. The atomic-sphere approximation of the linear muffin-tin orbital method (ASA-LMTO [22]) is applied to the 128-atom supercell with a special $k$ set of 20 non-equivalent points in the irreducible wedge [23].

Electronic transport parameters are obtained from the spectral curve of the conductivity, $σ ̂ (ε)$ , which is calculated by means of the Kubo–Greenwood formula [24], [25], $σ ̂ (ε)= he^{2} V ∑ ij 〈i|v| j〉〈 j|v|i〉δ(ε−ε_{i})δ(ε−ε_{j}).$ $V, v$ ,

Ab initio results

In the preceding paper [18] we have reported LMTO spectral curves of both the DOS and the conductivity (1). The icosahedral (1/1) approximant supplies a specific $100 meV$ conductivity pseudogap in addition to the $1 eV$ Hume–Rothery type feature. The latter one is due to electronic interference on the scale of the interatomic distance. Hence, significant electronic interference must act in the icosahedral phase on length scales beyond the interatomic distance. For proof we have shown [36] that the

Resistivity model

We believe that a model spectral resistivity of the icosahedral (1/1) approximant, suitably scaled, can account for the quasicrystal to certain extent, provided that the model includes significant spectral features. Recently, we have constructed such model resistivities by means of Lorentzians [18]. Each Lorentzian, $L(ε−(ε_{F}^{sc} +δ), γ)=(γ/π)/[(ε−(ε_{F}^{sc} +δ))^{2} +γ^{2}]$ , is characterized by its height, 1/(πγ), and its position, δ, with reference to ε_F^sc. The model for i-AlCuFe (1/1) requires two Lorentzians, $ρ$

Scaling of the narrow resistivity peak

In the present ab initio calculation of the spectral resistivity, $ρ ̂ (ε)= σ ̂^{−1} (ε)$ (1), the energy delta-functions are replaced by Lorentzians (half-width Γ). Fig. 3 shows spectral resistivity curves for a few selected Γ. Rising Γ causes the spectral fine structure to be smeared out. Note that the peak at $0.23 eV$ above ε_F^sc survives increasing Γ but its height and width are lowered and enlarged, respectively. Fig. 4 shows the inverse heights of the two prominent fine-structure peaks in Fig. 3, i.e.

Applications of the model

We deal with the temperature dependences of the electronic transport parameters listed in Section 2. However, we employ the spectral conductivity according to the Lorentzian resistivity model of the (1/1) approximant. The question is whether the well-known variety of experimentally observed temperature curves can be reproduced in passing in Fig. 1 from type A Fermi energies through type B ones, e.g. on adding Al. Note that this has been realized by different stoichiometries of Al–Cu–Fe phases

The i-AlCuFe quasicrystal

Here, we intend to obtain the parameters for our Lorentzians spectral resistivity model (10) which fits the corresponding experimental [48] conductivity and thermopower of the i-Al_62.5Cu₂₅Fe_12.5 bulk quasicrystal. We proceed as follows: (i) the Lorentzians are centered at just the same energies as in the (1/1) approximant (Table 1). (ii) $A, γ_{1}, α, γ_{2}, ε_{F}$ and ξ are fitting parameters. They are chosen as to reproduce reasonably experimental results [48] for both the resistivity at $4 K$ and the

Conclusions

There is strong indication that the peculiar electronic transport in quasicrystals and approximants is due to resistivity/conductivity spectral fine structure on the $10 meV$ scale which is missing in related amorphous and crystalline phases. The variety of temperature curves of the electronic transport coefficients can be derived from small shifts of the chemical potential in almost rigid spectral resistivity/conductivity curves. With rising temperature the physically significant spectral fine

Acknowledgements

We are grateful to R. Haberkern, P. Häussler, E. Maciá, and C. Madel for useful discussions and information. This work is supported by the “Deutsche Forschungsgemeinschaft”.

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Modeling the electronic transport properties of Al–Cu–Fe phases

Abstract

Introduction

Section snippets

Basic concepts

Ab initio results

Resistivity model

Scaling of the narrow resistivity peak

Applications of the model

The i-AlCuFe quasicrystal

Conclusions

Acknowledgements

Phys. Rep.

Mater. Sci. Eng. A

J. Non-Cryst. Solids

J. Non-Cryst. Solids

Physica B

Physica B

Mater. Sci. Eng. A

Mater. Sci. Eng. A

J. Non-Cryst. Solids

J. Inst. Met.

The Structure of Metals and Alloys

Phys. Rev. Lett.

J. Phys.: Cond. Matter

J. Phys.: Cond. Matter

Phys. Rev. Lett.

Phys. Rev. Lett.

Phys. Rev. Lett.

J. Phys.: Cond. Matter

Phys. Rev. Lett.

Int. J. Mod. Phys. B

Phys. Rev. B

Phys. Rev. Lett.

Phys. Rev. B

Phys. Rev. B