# Two-Level Electron Excitations and Distinctive Physical Properties of Al-Cu-Fe Quasicrystals

^{*}

## Abstract

**:**

## 1. Introduction

^{19–20}cm

^{−3}) concentration of conduction carriers in the low temperature limit; i.e., the average valence (electrons/atom) in the alloy at low temperatures is e/a ~ 10

^{−3}. As a consequence, the icosahedral phases have a low, but finite (~100 Ω

^{−1}·cm

^{−1}), metal-like conductivity, as well as low values of the Pauli susceptibility and electronic heat capacity. These differences are reflected in the hypothesis of the pseudogap. According to photoelectron spectroscopy experiments, the pseudogap has a width of ~1 eV [1].

## 2. Features of the Electronic Specific Heat Capacity

^{23}cm

^{–3}and room temperature, this contribution is no more than 0.5% of the lattice specific heat [19]. For quasicrystals, the corresponding contribution is even smaller and does not exceed 0.1%. This contribution was ignored, and the entire experimentally measured specific heat was considered to constitute the lattice specific heat.

_{63}Cu

_{25}Fe

_{12}and Al

_{62}Cu

_{25.5}Fe

_{12.5}nominal compositions. The purpose of these experiments was to verify whether the linear-in-temperature metal-like contribution γТ is the sole contribution to the electronic heat capacity of quasicrystals, or TICCs are capable of making their own contribution.

**C**

_{lat}and the metal-like heat capacity of ground state conduction electrons C

_{ml}= γT, that is,

_{exc}= C

_{total}−C

_{lat}−C

_{ml}

_{p}into C

_{v}, were performed in the temperature range from 1.5 to 400 K. It is a well-studied region, where a noticeable increase in the electrical conductivity (negative TCR) and an enhancement in the paramagnetism have been observed and where, consequently, a singular contribution can manifest itself. It was found that the inclusion of the lattice heat capacity C

_{lat}within the approximation of the Debye model leads to a very strong dependence of the final result on the model assumptions. To avoid this difficulty, we assessed a contribution of the lattice heat capacity C

_{lat}in the approximation of the empirical Gruneisen law, which implies a linear relation of the lattice heat capacity to the linear expansion coefficient. Details can be found in [20,21].

_{total}at high temperatures significantly exceeding the Dulong-Petit limit. We determined the excess heat capacity in this temperature range with respect to this limit. The asymptotic approximation to this limit may be assessed using the Debye model, exactly which we chose to perform. Details can be found in [8,18].

_{TLE}(T) curve is double peaked. It is unlikely that the heat capacity is a continuous function of the temperature. Rather, it consists of two functionally similar, but not interconnected parts; therefore their sum can be approximated by two contributions of the Schottky type [23] in the form of Equation (2) with excitation energies δE

_{1}= 0.02 eV and δE

_{2}= 0.25 eV, respectively.

## 3. A Two-Component Model of the Electronic Structure

_{F}. By suggesting that HL-localization (not the pseudogap) is the electron stabilizing factor of the quasilattice, we postulated also that the continual and discrete components are autonomic subsystems, i.e., the thermally activated carriers do not get into the conduction band. They propagate on the excited levels of the traps by hopping or tunnelling.

_{H}) in the specified temperature range [22]. With the attainment of the complete picture of the excess heat capacity and its approximation by two Schottky anomalies, there appeared a unique possibility to demonstrate this relation as a whole.

_{TLE}(T). The latter dependence was obtained by the numerical integration of the experimental curve C

_{TLE}(T). Well-coordinated behaviour of all the curves is evident. On the other hand, the dependence U(T) differs by obvious signs of alternating curvature, and the origin of this behaviour is clear enough. Since the singular heat capacity C

_{TLE}, according to Equation (2), consists of two functionally similar parts, the temperature dependence of the number of TLEs also consists of two functionally similar contributions in the form:

^{−1}coordinates, as shown in Figure 3a for the Al

_{63}Cu

_{25}Fe

_{12}icosahedral phase. A change in the activation regimes manifests itself in the fact that the curve has two linear portions. If the temperature dependent components of the electrical conductivity, the magnetic susceptibility, and the inverse Hall-effect, where these contributions are very difficult or impossible to separate, are represented in the same coordinates, then, as seen in Figure 3b–d, there appear patterns that are identical to that in Figure 3a.

_{ml}. Qualitatively, this implication is confirmed by the empirical relationship known as Inverse Matthiessen Rule [26]. According to this rule, the electronic transport in quasicrystals is determined by additive contributions to the electrical conductivity, rather than to the electrical resistivity as in metals, i.e.:

_{ml}+ σ

_{TICC}.

_{1}~ 0.02 eV and, as follows from the description of the high-temperature part of the excess heat capacity, it is δE

_{2}~ 0.25 eV. The pseudogap width, according to photoelectron spectroscopy, as was noted above, is equal to ~1 eV. The Fermi energy of residual metal like carriers in the free electron approximation and at a charge carrier concentration of 10

^{20}cm

^{–3}is E

_{F}~ 0.3 eV. It is easy to see that spatially localized states, if they exist, coincide in energy with the continuum of residual carriers. In the framework of homogeneous localization models, such a coincidence is impossible [27]. It may mean that in quasicrystals, we are dealing with a special sort of inhomogeneous electronic system with a dual manifestation of inhomogeneity, namely, in the form of several generations of electron traps and in the form of the coexistence of extended and spatially localized states with the same energies. In any case, it means definitely that the conjugated levels of the traps should be separated by real gaps, not pseudogaps. To verify how realistic this implication we turned to the analysis of the spectral characteristics of quasicrystals.

## 4. Correlations between TLE Heat Capacity and Tunnelling Spectra

^{2}), G(V) curves have an “averaged” shape with a sign-alternating curvature, but smoothly increase with V. In the case of a small area of the junction (<1 nm

^{2}), G(V) curves have a “local” oscillatory shape.

_{0}, which is due to remaining delocalized conduction electrons, and the field-dependent part G

_{v}, which is due to energetically and spatially well localized electrons:

_{0}+ G

_{v}

_{v}. It is obvious that, in view of the spatial localization of the features of G

_{v}, extremely narrow peaks of the density of states should be separated by real gaps rather than by pseudogaps. Consequently, multiple ZBA would be expected in the spectrum. However, this is not the case in experiments. A single ZBA is always observed near V = 0. This specificity of real spectra was the first reason for doubt that the fine structure of G(V) curves is the direct image of the electron density of states [36] and for the possibility of another mechanism.

_{64}Cu

_{23}Fe

_{13}in [32] and the C

_{TLE}(T) curve that we obtained for the i-phase of Al

_{62}Cu

_{25}Fe

_{12}and obtained the picture shown in Figure 4. All results in this figure are in situ, but all curves are represented in the same energy format. It is seen that the tunnel and thermal characteristics have visually similar maxima (they are indicated by arrows in Figure 4) at close energies.

_{e}) of these carriers slightly depends or does not depend on the field, it is easy to show that the conductance depends on V as:

_{v}should generally be sums of numerous elementary terms given by Equation (7), differing in the splitting energy of conjugate levels of traps. Under this assumption, we analysed in detail the local and averaged G

_{v}(V) curves using Equation (7) as a trial function [38].

_{i}corresponding to the best fit. The dotted line is a similar residual after the subtraction of lines 5, 16, and 80. This line has only one characteristic feature, ZBA, which is described by the dash-dotted line marked by the numeral 260. This residual is best approximated with the parameter δE

_{i}= 260 meV, which almost coincides with the second characteristic energy of two-level excitations from the specific heat δE

_{i}= 250 meV [25].

_{i}is visible. The remaining ZBAs become hidden.

_{i}= 400, 1400, and 5300 meV are detected. The procedure of their separation is similar to that described above. The only explanation that needs to be done relates to accounting the low-energy part of the spectrum in order to obtain a residual. The averaged curve in this part of the spectrum is strongly smoothed but is due seemingly to the same elementary terms as the above local curve. In view of this circumstance, to approximate the low-energy part of the spectrum at V < 70 meV, we used the entire set of “local” characteristic energies in the form of the LE contribution. This is the reason why the “high energy” residual shown by the dotted line in Figure 6 exhibits “dips” near V = 0. This does not affect the approximation of the high-energy residual of the averaged curve and this entire curve by the set of elementary terms (Equation (7)) with the characteristic energy gaps δE

_{i}= 5, 16, 80, 260, 400, 1400, and 5300 meV. The latter is shown by a thick solid line in Figure 6.

## 5. Two-Level Excitations and Optical Spectra

_{i}values as the real gaps. If so, a distinctive feature of the behaviour of optical conductivity σ (ω) in quasicrystals should be the summary effect of the same set of resonance absorption peaks.

_{63}Cu

_{25}Fe

_{12}at room temperature is shown in Figure 7 in accordance with experimental data [39]. It presents an intense, broad and practically featureless (except for maximum) band occupying virtually the all range of frequencies from 40 up to 25,000 cm

^{−1}. A number of authors have reported the existence of the similar bands in various icosahedral systems: AlPdMn [40], AlMnSi [41], AlPdRe [42], and AlCuFeB [43]. Accordingly, there exist three possible explanations for the band. It may be associated with strong scattering of electrons and then the maximum occurs at ω = 1/τ [44]; it may be associated with the direct gap in the zone structure and then σ(ω) = const x(ω − E

_{g})

^{1/2}/ω [42]; and, finally, it may be associated with resonance absorption [43] in the form of:

_{0i}, ω

_{i}, and γ

_{i}are the ith mode resonance frequency, damping, and mode strength of the ith harmonic oscillators, respectively.

_{0}= δE

_{i}directly. It was not so hard to see that the band maximum in Al

_{63}Cu

_{25}Fe

_{12}i-phase at about 1.4 eV coincides very well with δE

_{i}= 1400 meV from tunnelling experiment. We used this coincidence to deconvolute the broad band for a set of elementary optic terms using a Lorentzian peak (Equation (8)) as a trial function.

_{i}= 5300 meV. Unfortunately, the maximum of the peak, if it exists, takes place outside the experimental range. As to the peak with the smallest excitation energy at about 5 meV in tunnelling experiment, it could not be observed at room temperature. To achieve the best description, shown by a thick solid line in Figure 7a, we needed yet another peak at 90 meV shown by dash-dotted line in Figure 7b.

## 6. TLEs and Giant Thermally Induced Effects in Magnetic Susceptibility and dc Conductivity

_{t}(T) = ∑σ

_{i}and χ

_{t}(T) = ∑χ

_{i}, as well as that the elementary terms σ

_{i}and χ

_{i}are the linear functions of TLE density, i.e., σ

_{i}∞ N

_{TLEi}(see Equation (3)) and χ

_{i}∞ T

^{−1}N

_{TLEi}. These terms were used as trial functions. The final results of this analysis are shown in Figure 8 and Figure 9. An almost perfect curve fitting was achieved using the above-mentioned set of characteristic energies. It is surprising that such high-energy TLEs of up to 1400 meV were shown to be effective in a physical experiment.

_{g}/k

_{B}T) [45]. We argue that the reason for this is the multigap spectrum of threshold excitations.

## 7. Conclusions

## Acknowledgments

## Author contributions

## Conflicts of Interest

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**Figure 1.**Temperature dependence of the excess electronic heat capacity of the Al

_{63}Cu

_{25}Fe

_{12}quasicrystalline alloy with respect to the Sommerfeld metallic contribution. Solid lines show the approximation of the curve by two Schottky contributions with excitation energies of 0.02 and 0.25 eV.

**Figure 2.**Curves of temperature-induced increments in theelectrical conductivity Δσ (open circles), paramagneticsusceptibility Δχ (closed circles), and internal energy ΔU(solid line) of the Al

_{63}Cu

_{25}Fe

_{12}icosahedral phase in thetemperature range from 1.8 to 1000 K.

**Figure 3.**Change of regimes of thermal activation is a character feature of: two-level excitation density (

**a**); conductivity (

**b**); magnetic susceptibility (

**c**); and inverse Hall effect (

**d**) in the Al

_{63}Cu

_{25}Fe

_{12}phase.

**Figure 5.**Deconvolution of the local G(V) curve into elementary terms with the characteristic splitting energies of the levels δE

_{i}= 5, 16, 80, and 260 meV. The thin solid line is the experimental curve [32]. The dash-dotted lines are the elementary terms. The thick solid line is the sum of the terms. The dotted lines are residuals after subtraction of the terms.

**Figure 6.**The averaged experimental G(V) curve (thin solid line [32]) and its description (thick solid line) by the sum of the elementary terms with δE

_{i}= 5, 16, 80, 260, 400, 1400, and 5300 meV. The dash-dotted lines are the low-energy (LE) part of the spectrum and additional terms. The dotted lines are residuals after subtraction of the calculated curves.

**Figure 7.**(

**a**) Experimental optical conductivity of icosahedral Al

_{63}Cu

_{25}Fe

_{12}(thin solid line [39]). The dash-dotted line (1400) is a fit with the Lorentzian peak according to Equation (8). The dotted line is a residue after subtraction of the peak. Dash-dotted line (5300) is a fit with the Lorentzian peak according to Equation (8). Thick solid line is a fit with a sum of this Lorentzian peak and peaks in (

**b**). (

**b**) The residue shown in (

**a**) but on an enlarged scale (solid line). Dash-dotted lines (20, 90, 250, and 420) are the fit with the Lorentzian peaks according to Equation (8).

**Figure 8.**Temperature-dependent part of the magnetic susceptibility of the ordered Al

_{63}Cu

_{25}Fe

_{12}phase: (o), experimental results [2]; and solid line, description of the results by a sum of the magnetic elementary terms (dash-dotted lines) with δE = 5, 16, 80, 260, 400, and 1500 meV, respectively.

**Figure 9.**Temperature-dependent part of the dc conductivity of the ordered Al

_{63}Cu

_{25}Fe

_{12}phase: (o), experimental results [2]; and solid line, description of the results by a sum of the conduction elementary terms (dash-dotted lines) with δE = 5, 16, 80, 260, 400, and 1500 meV, respectively.

**Figure 10.**An updated version of the electronic structure model. Shown schematically is the superposition of two types of spectra: the continuum spectrum with a pseudogap and the discrete spectrum with seven types of two-level states. The Fermi level is fixed at the centre of the smallest gap separating the symmetric and antisymmetric states. For the sake of representativity, a nonlinear scale E

^{1/2}is used. To simplify, only the smallest (δE

_{1}) and the largest (δE

_{7}) gaps are indicated by arrows.

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**MDPI and ACS Style**

Prekul, A.; Shchegolikhina, N.
Two-Level Electron Excitations and Distinctive Physical Properties of Al-Cu-Fe Quasicrystals. *Crystals* **2016**, *6*, 119.
https://doi.org/10.3390/cryst6090119

**AMA Style**

Prekul A, Shchegolikhina N.
Two-Level Electron Excitations and Distinctive Physical Properties of Al-Cu-Fe Quasicrystals. *Crystals*. 2016; 6(9):119.
https://doi.org/10.3390/cryst6090119

**Chicago/Turabian Style**

Prekul, Alexandre, and Natalya Shchegolikhina.
2016. "Two-Level Electron Excitations and Distinctive Physical Properties of Al-Cu-Fe Quasicrystals" *Crystals* 6, no. 9: 119.
https://doi.org/10.3390/cryst6090119