The electrodynamic response of the icosahedral quasicrystal Al70Mn9Pd21

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Solid State Communications,Vol. 87, No. 8, PP. 721-726, 1993. Printed in Great Britain.

THE ELECTRODYNAMIC

0038-1098/93 $6.00+.O0 Pergamon Press Ltd

RESPONSE OF THE ICOSAHEDRAL QUASICRYSTAL AI7oMn9Pd21

L. Degiorgi, M.A. Chernikov, C. Beeli and H.R. Ott Laboratorium for Festkoerperphysik, ETH-Z0rich, CH-8093 Z0rich, Switzerland

(received June 1, 1993 by M. Cardona) We report an optical investigation of single-phase icosahedral AI70Mn9Pd21 in a very broad frequency range between 14 and 10s cm-1. The electrodynamic response of this material is characterized by a very low optical conductivity in the infrared and a prominent absorption in the visible frequency range. These unusual optical properties are compatible with the presence of pseudogaps in the electronic density of states of quasi-crystalline materials. The implication of the results of our ac-response experiments with respect to the thermodynamic and tic-transport results, obtained on the same specimen, are also discussed.

Since the discovery of icosahedral structural symmetry in rapidly quenched AI-Mn alloys [1] the consequences of quasiperiodicity for the physical properties of quasicrystals and possibly related peculiarities of their electronic structure have been studied intensively. It was, however, soon realized that AI-Mn and other metastable quasicrystals possess a high degree of intrinsic disorder that obscures the expected novel properties. New opportunities for experimental studies of the electronic properties of condensed matter with quasiperiodic structures are provided by the recent discovery of the thermodynamically stable quasicrystals AI-Cu-(Fe,Ru,Os) [2] and AI-(Mn,Re)Pd [3,4], which have face-centered icosahedrallyordered structures and structural coherence lengths up to 104 A [5] compatible with those of well ordered periodic crystals. Quasicrystals have been studied mostly with respect to their structural properties but also their transport and thermodynamic properties have been investigated. A remarkable feature of thermodynamically stable icosahedral quasicrystals is their surprisingly low electrical conductivity. While composed of good metals (AI,Cu,Fe etc.), they exhibit dc conductivities (adc), which can be thousands of times smaller than the conductivity of their constituent elements and, also, at least one order of magnitude smaller than in common amorphous metals. Furthermore, the temperature dependence of adc is quite unusual,

since Odc decreases significantly at low temperatures, without showing an activated behaviour [6]. Recent investigations of the electrodynamic response of the quasi-crystals AI63.5Cu24.5Fe12 [7] and AI75.5Mn20.sSi4 [8] from the far-infrared up to the visible frequency range reveal similarly peculiar behavior. In both cases the spectrum is rather different from that in many disordered metals, where a Drude model in the strong-scattering limit is applicable, or in semiconductors with a well developed conductivity gap. Starting from a very low dc value the optical conductivity increases with frequency, revealing a prominent peak in the visible frequency range (i.e., at about 104 cm -1) and implying a substantial transfer of spectral weight from low to higher frequencies. The appearance of this absorption is ascribed to excitations across a so-called pseudogap (see below) [7,8]. In this communication, we present our measurements of the complete electrodynamic response of a bulk single-phase sample of icosahedral AI70Mn9Pd21. The phenomenological analysis of the excitation spectrum, based on a classical dispersion model [9], allows us to evaluate several intrinsic parameters which are compared to related quantities arrived at by other experiments (i.e., from transport and thermodynamic properties [6]) performed on the same specimen. We also discuss our results in the light of recent theoretical models which suggest an electrodynamic response

721

722

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ICOSAHEDRAL QUASICRYSTAL AI70Mn9Pd2I

dominated by band-structure effects [10]. The specimen of (i)-AI70MngPd21 synthesized from 99.997% pure aluminium, 99.9% pure palladium and 99.94% pure manganese was arc melted several times to provide homogeneity, annealed for 2 days at 800 oc, and subsequently quenched into water directly from 800 oC. The sample composition is optimal for the formation of the thermodynamically stable icosahedral phase [11], and the absence of any inclusions of other phases was confirmed by surface analysis from back-scattered electron images. Also, selected-area electron-diffraction patterns showed a high degree of order and a low density of phason-defects. Figure 1 displays the dc-conductivity as a function of the temperature of AIToMngPd21 between 0.02 and 100 K [6]. The low residual conductivity o(0)=136.8 (Qcm) -1 and the temperature variation of the conductivity are comparable to values reported for single-grained quasicrystals in AI-Mn-Pd system with compositions close to ours and obtained by different methods (i.e., conventional solidification from the melt [12] and Czochralski method [13]) confirming the high quality of our sample. As extensively discussed elsewhere [6], the observed o(T) is compatible with a variation due to quantuminterference effects considering the influence of weak localization including strong spin-orbit scattering and the Coulomb interaction among electrons [6]. More details concerning the sample preparation, the structural analysis and its characterization are given elsewhere [11]. We have performed the optical investigations on the same specimen. The reflectivity R(v) has been measured within a broad frequency range between 14 cm "1 and 105 cm -1, using four spectrometers with overlapping frequency ranges. In the far infrared (FIR) we made use of a Bruker IFS113v Fourier interferometer with a Hg arc-light source and a He-cooled Ge-bolometer detector, while from the FIR up to the mid-IR range a fast scanning Bruker interferometer IFS48PC was used. In the visible spectral range a homemade spectrometer based on a Zeiss monochromator was employed and in the ultraviolet we used a McPherson spectrometer. From the mid-IR down to the FIR we have used the reflectivity of gold as reference. All measurements have been performed

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Fig. 1 :

The conductivity adc as a function of the temperature between0.02 and 100 K (note the logarithmic scale). The left-upper part shows a(T) below 1 K as a function of the square root of the temperature.

at 300 K and in the FIR also at 6 K. The specimen had two large rectangular shiny surfaces (-1.5x2 ram), on which the optical measurements have been performed. Subsequently, one of the two surfaces was polished and measured again, without a noticeable change of R(v), neither qualitatively nor quantitatively (besides an irrelevant increase of the overall reflectivity of a few percent). The lowtemperature results in the FIR also do not indicate substantial changes compared to 300 K. In Fig. 2a we show the complete R(v) spectrum at 300 K, indicating that the reflectivity decreases quickly with frequency to -60% at 800 cm-1 and is then essentially constant up to 3x104 cm -1, where it suddenly starts to drop, suggesting a plasma-edgelike behaviour. The measured reflectivity spectrum was extrapolated to lower frequencies (< 14 cm -1) assuming a metallic behaviour and thus using the Hagen-Rubens relation t-R(v)~~/v. Beyond the highest measurable frequency (105 cm-1), we first used the extrapolation R(v)~l/v 2, which simulates interband transitions, and at frequencies higher than 3x105 cm -1, we assumed R(v)~l/v 4, simulating the behaviour of free electrons. The reflectivity

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ICOSAHEDRAL QUASICRYSTAL Al70Mn9Pd21

Vol. 87, No. 8

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(a) The reflectivity spectrum R(v) at 300 K in the whole investigated frequency range (please observe the logarithmic frequency scale). (b) The optical conductivity Ol(V), obtained byKramersKronig transformation of the R(v)spectrum in (a). The line represents the phenomenological fit with the parameters: e**=1.087, Vph0=282 cm "1, 7ph=1244 cm "1, Vph=5828 cm "1, ~p=10915 cm 1 , FD=9929 cm "1, Ve0=9679 cm "1,Ye=19607 cm 1 and Ve=68557 cm-1.

spectrum over such a broad frequency range allows us to perform a reliable Kramers-Kronig transformation, from which we obtain the optical conductivity Gl(v) displayed in Fig. 2b. We identify several features in o1(v): (I) the low optical conductivity in the FIR with an almost complete and unusual absence of a Drude freecarrier response. Nevertheless, the v ~ 0 limit of O1(V) is in fair agreement with Odc (Fig. 1) [6]; (11)the

broad structure, split into two peaks at 250 and 300 cm -1 and ascribed to phonon-like modes [14], and finally (111)the overwhelming peak centered at about 104 cm -1. Moreover, in the ultraviolet range we observed a broad absorption at about 9x104 cm -1, which we assign to high-frequency electron interband transitions. There is qualitative agreement with the excitation spectrum reported for AI63.sCu24.5Fe12 [7]. The latter measurement was, however, performed on a mosaic of melt-spun ribbons, which had an irregular surface and were very brittle, making difficult any kind of polishing [7]. We also note that the investigated frequency range (i.e., from FIR up to the visible) was narrower than ours, so that the high-frequency roflectivity had to be numerically reconstructed from a set of Lorentzian oscillators. Our large bulk specimen and the broad frequency range covered by our experimental facilities allow an overall improvement of the available experimental data. If o1(v) is known on a very broad frequency range, we may estimate the total spectral weight, by applying the conductivity sum rule:

jo1((o)do) = ne2 (2~:Vp)2 mb = 8

(1)

where Vp is the plasma frequency. By integrating our experimental optical conductivity after having subtracted from o1(v) the contribution from the electronic interband transitions at about 9x104 cm -1, we obtain hvp=8.8 eV. We note that this value of hvp would correspond, in the case of rob=me (i.e., the free electron mass), to a free charge-carrier concentration ne=5.6xl022 cm °3. The meaning of these quantities is considered below. A rather simple but very helpful description of the complete excitation spectrum may be obtained by a phenomenological approach, based on the classical dispersion theory of the medium [9]. We model the dielectric function as follows: Vph2 ~p2 F.,(V)= £=,+ (Vph0)2. V2 " iVVph-V( v + iFD) re2 + (veO) 2 - v 2 - ivYe

(2)

724

ICOSAHEDRAL QUASICRYSTAL A170Mn9Pd2I

The first term (¢=,) in eq. (2) describes the highfrequency electronic interband transitions. The second contribution is due to the FIR absorption, which we ascribe to phonons [14], while the third and fourth ones are assigned to a Drude-like contribution, associated with the conductivity of the free charge-carriers responsible for the low Gdc, and to the broad excitation at about 104 cm -1, respectively. Vph0 and ve0 are the resonance

frequencies, "Ifph and 7e the dampings, and Vph and Ve the mode strengths of the two harmonic oscillators. The Drude parameters~p and F D are the plasma frequency and the free charge-carriers scattering relaxation rate, respectively. The solid line in Fig. 2 shows the best fit, obtained with the parameters given in the figure caption. We immediately realize that the effective plasma frequency Vp, associated with the low-frequency Drude contribution, corresponds to a very small fraction of the total spectral weight (Vp) evaluated before in eq. (1). Indeed, almost the entire spectral weight is shifted to the absorption at about 104 cm -1. Consequently, unless an effective free chargecarrier concentration neff smaller than ne is considered, this vanishingly small spectral weight would be suggestive of a substantial enhancement of the effective band mass (i.e., m*/me~40). However, we may combine our effective plasma frequency '~p with the low coefficient of the electronic specific heat 7=0.41 mJ/moleK 2 which was measured on the same sample [15], and calculate neff and m*. We find that the experimental values of ~'p and 7 are compatible with an effective mass m*=2.02me and an effective concentration of electrons neff=2.7xl021 cm -3. It is important to realize that nef f is the actual charge-carrier concentration contributing to adc, while ne is the total optical charge-carrier concentration involved in the complete excitation spectrum and associated to the total spectral weight (Vp). In other words, only 5% of the total charge-carrier concentration (i.e., ne/nelf -20) behaves as free electrons (i.e., Gdc=neffe2'c/m*), while almost all of it (i.e., ~ne) is involved in the broad absorption at 104 cm -1. Recently, it has been argued that it is possible to explain the experimentally observed dc and ac conductivity by introducing the so-called 'bandstructure hypothesis' [10], thus making it

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unnecessary to invoke a wavefunction localization or criticality because of the inapplicability of the Bloch theorem to quasi-crystals. Within this bandstructure approach the anomalously low conductivity is attributed to a particularly low density of states (DOS) at the Fermi level (as in semimetals) rather than to an anomalously strong scattering localization. The low DOS, in turn, is believed to be a consequence of a Hume-Rothery-type electronic stabilization mechanism, which implies a close contact of the Fermi surface with several Bragg planes, whose associated reciprocal lattice vectors G satisfy 2kF'IGI, where kF is the Fermi wave vector [10]. Interaction of the Fermi surface with the Bragg planes results in the opening of pseudogaps. In our case kF=(3~2ne)1/3=1.18 A -1, so that 2kF is indeed of the order of IG1~3 A-l, the positions of the two strongest peaks in the diffraction pattern of AI7oMn9Pd21 in reciprocal space [16]. Here, we calculate kF from he, since we are interested in the radius of the Fermi surface before the HumeRothery stabilization mechanism takes place. What makes these icosahedral quasi-crystals different from conventional metallic alloys is the high multiplicity (namely 42) of their reciprocal lattice vectors, which allows a better match between the Fermi sphere and the corresponding set of Bragg planes. This effect, besides the strong reduction of the dc conductivity through a significantly reduced effective charge-carrier concentration and a moderate enhancement of the effective mass, enhances the interband absorption across the pseudogaps. In our excitation spectrum the overwhelming absorption at 104 cm -1, which has been phenomenologically described by a harmonic oscillator [9], is associated with excitations across the pseudogap in the DOS. The energy of this absorption (veO) is also in accord with that inferred from electronic band structure calculations for crystalline approximants [17]. Finally, we observed that the scattering relaxation time ('~~I/FD) of the Drude component is rather short, compared to the value for pure AI. The complexity of these alloys and consequently their distinct chemical disorder are, besides electronphonon interactions, important sources for strong scattering. Nevertheless, it is instructive to use this value for '¢ in order to estimate the mean free path

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ICOSAHEDRAL QUASICRYSTAL AI70Mn9Pd21

I=VF¢. With neff=2.7xl021 cm -3 and m'=2.02me we calculate VF=0.25x108 cm/sec, which implies I=1.31 A. On the other hand ~FI~0.2 [6], which leads for ~'F=(3~2neff)1/3=0.43 A-1 to I-0.5 AI In contrast to the above evaluation of kF, here we are only interested in those electrons contributing to adc and we have thus considered neff . However, in the present case it is not obvious what the mean free path I=VF~ really means, since VF is far from being constant on the Fermi surface and even vanishes at some points. Moreover, such small values of the mean free path, as already recognized in other compounds [10], question the applicability of the free electron approximation. We also point out that mean free paths shorter than the average distance between atomic sites or'~'FI
725

dependence of the electrical conductivity can be well described by weak localization including important spin-orbit scattering and Coulomb interaction [6]. This is, however, quite surprising in view of the very low value of kFI, as determined from thermodynamic data. The analysis of our optical data in terms of the free electron approximation yields unreasonably small values for the mean free path, as well. Band-structure effects certainly influence the overall frequency dependence of the conductivity, while the weak-localization arguments seem to be more appropriate for the low-temperature dependence of the dc conductivity. It remains to be seen whether simply considering band-structure or localization effects is sufficient for a complete understanding of the peculiar properties of these quasicrystals. New theoretical approaches combining both schemes would be helpful.

Acknowledgements We thank R. Monnier for illuminating discussion and J. M011er for technical assistance. One of us (L.D.) is very much indebted to P. Wachter for important support. We are also very grateful to X. Wu, C.C. Homes, S.E. Burkov, T. Timusk, F.S. Pierce, S.J. Poon, S.L. Cooper and M.A. Karlow for having sent us preprints of their paper prior to publication. This work was in part supported by the Schweizerische National Fonds zur Foerderung der Wissenschaftliche Forschung.

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P.A. Bancel, in 'Ouasicrystals: The State of Art', edited by D.P.Divincenzo and P. Steinardt (World Scientific,1991), p. 17 M.A. Chernikov, A. Bemasconi, C. Beeli and H.R. Ott, Europhys. Lett. 21,767 (1993) C.C. Homes, T. Timusk, X. Wu, Z. Altounian, A. Salinoune and J.O. Stroem-Olsen, Phys. Rev. Lett. 67, 2694 (1991) X. Wu, C.C. Homes, S.E. Burkov, T. Timusk, F.S. Pierce, S.J. Poon, S.L. Cooper and M.A. I
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[13]

[14]

ICOSAHEDRAL QUASICRYSTAL Al70Mn9Pd21

F. Wooten, 'Optical Properties of Solids' (1972), Academic Press S.E. Burkov, T. Timusk and N.W. Ashcroft, J. Phys.:Condens. Matter 4, 9447 (1992) C. Beeli, Ph.D. Thesis, ETH-Z0dch (1992) S. Takeuchi, H. Akiyama, N. Naito, T. Shibuya, T. Hashimoto, K. Egagana and K. Kimura, J. of Non-Crystalline Solids 153-154, 353 (1992) Y. Yokoyama, T. Miura, A.P. Tsui, A. Inoue and T. Matsumoto, Mater Trans. JIM 33, 97 (1992) The two frequencies of 250 and 300 cm °1

[15] [16]

[17]

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are close to frequencies (rescaled by the concentration-weighted mean of the masses in (i)-AI70MngPd21) of the peaks in the phonon density of states of crystalline Ah G. Gilat and R.M. Nicklow, Phys. Rev. 143, 487 (1966) M.A. Chernikov, A. Bemasconi, A. Schilling, C. Beeli and H.R. Ott (to be published) M. Boudard, M. de Boissieu, C. Janot, G. Heger, C. Beeli, H.-U. Nissen, H. Vincent, R. Ibberson, M. Andrei and J.M. Dubois, J. Phys.: Condens. Matter 4, 10149 (1992) T. Fujiwara and T. Yokohawa, Phys. Rev. Lett. 66, 333 (1991)