Specific heat measurements on single-phased AlCuFe quasicrystals

Journal of Non-Crystalline Solids 153&154 (1993) 357-360 North-Holland

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Specific heat measurements on single-phased A1CuFe quasicrystals K. W a n g a, C. S c h e i d t a, p. G a r o c h e a a n d Y. C a l v a y r a c b "Laboratoire de Physique des Solides, Universit~ Paris-Sud, 91405 Orsay Cedex, France b C E C M / C N R S , 15 rue G. Urbain, 94407 I/itry Cedex, France

We have measured the low-temperature specific heat of samples of quasicrystalline AlCuFe alloys using an ac method. We observed weak electronic contributions in all the samples and excess vibrational contributions in phason-strained ones. The electronic contribution can account for the anomalously high resistivityusing a simple pseudogap model with a realistic electronic mean free path value. The evolution of the vibrational contribution suggests that phason strains induce a softening of low-energytransverse phonon modes by decreasing the elastic shear modulus. Some implications are discussed.

1. Introduction The production of nearly 'ideal' quasicrystals (QC) [1] has revealed some specific physical properties of these materials, among which are anomalous electronic transport [2,3] and structural instabilities related to phason strain [4]. In order to investigate the electronic structure and to study the influence of phason strain on atomic dynamics, we have measured the low t e m p e r a t u r e specific heat of quasicrystalline AICuFe samples of different chemical compositions and after different thermal treatments. It is well known that the electronic transport is closely related to the electronic structure near the Fermi level, whereas the low-temperature specific heat (and the corresponding Debye temperature) depends on the elastic properties.

2. Samples and experimental procedure The samples used in this study were p r e p a r e d at the C E C M / C N R S in Vitry. They were obtained by planar flow-casting and submitted to various thermal treatments. Four single flakes, Correspondence to: Dr K. Wang, Laboratoire de Physique des

Solides, Universit6 Paris-Sud, 91405 Orsay Cedex, France. Tel: +33-1 69415311. Telefax: +33-1 69416086.

each with a mass of about 1.5 mg, were investigated: a, A162Cu25.sFe12.5 as-quenched; b, A162 Cuzs.sFe12.5 annealed at 600°C; c, A162Cuzs.sFelz.5 annealed at 800°C; and d, A163Cuz4,sFe12.5 annealed to 800°C and then quenched. All compositions are nominal. The diffraction of the samples was measured by high resolution X-ray diffraction using synchrotron radiation [5]. Sample a contained mainly Q C phase, with a small proportion ( < 10%) of a foreign phase (cubic 13 phase). In sample b, the foreign phase was eliminated by the annealing process. This sample was single-phased and contained phason strain disorder, revealed by Bragg p e a k broadening. In sample c, phason strains were eliminated by further annealing at 800°C; this sample was single-phased. The X-ray diffraction p e a k width was of the order of the instrument resolution (Aq ~ 10 -4 ~k- l ) and the peak positions are exactly those of an ideal icosahedral quasilattice. It remained stable over the whole t e m p e r a t u r e range from 800°C to room temperature and was regarded as a 'perfect' sample. Sample d was also well ordered, but exhibited an evolution to a low t e m p e r a t u r e approximating crystalline phase around 650°C [5]. The experiments were performed using an a.c. calorimetric technique at a working frequency range of 2 - 6 Hz. This method is described in detail elsewhere [16].

0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

358

K. Wang et al. / Specific heat measurements on single-phased AICuFe quasicrystals 1.4

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4. Discussion

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T2 (K 2) Fig. i. Molar specific heat C of four quasicrystalline samples: (a) as-quenched A162Cuzs.sFe12.5; (b) A162Cu2s.sFe12.5 annealed to 600°C; (c) AI62Cu25.sFe12.5 annealed to 800°C; and (d) AI63Cu 24.5Fe12.5 quenched form 800°C.

3. Results

Figure 1 presents the molar specific heat of the four samples in a C / T versus T 2 diagram, in the temperature range 1-3 K. This figure shows that in this temperature range the specific heat for these samples can be described by the classical law C = 7T+/3T 3. The 7 and /3 values for these samples were obtained by a best fit of this customary plot (see table 1). The 7 values for samples b, c and d were in agreement with previous observations of Biggs et al. [2] and Klein et al. [3]. The /3 values for samples c and d were somewhat lower than those obtained by Biggs et al., but in agreement with the value given by Klein et al.

Table 1 3' and 13 values obtained from specific heat measurements and estimated electronic density of states D ( E F) and T = 0 Debye temperature @o0 for four quasicrystalline samples: (a) as-quenched A162Cu25.sFe12.5; (b) A162Cu25.5Fe12.5 annealed to 600°C; (c) A162CUE5.5Fe12.5 annealed to 800°C; and (d) A163Cu24.5Fe12.5 quenched from 800°C a 3, ( m J / K 2 mole) /3 ( m J / K 4 mole) D(E F) (state/eV spin at 10 -2 ) Oo0 (K)

b 0.67 0.09

10.8 290

c 0.33 0.11

5.3 273

d 0.3 0.05

4.8 350

0.31 0.07 5.0 315

The linear term TT can be unambiguously attributed to the electron contribution. The estimated electronic density of states (DOS) at the Fermi level D(E v) for each sample is listed in table 1. All these values are lower than those of the free-electron DOS, estimated as 0.16 s t a t e / eV spin at. with a conventional renormalization factor A ~ 0.3. This confirms the existence of a pseudogap at the Fermi level in all samples. The decrease in DOS values from sample a to samples b and c can be attributed to the disappearance of the/3 phase, which has a stronger 7 value, and to the different chemical composition of the quasicrystalline phase in sample a, the composition of the two phases there being different. This pseudogap picture can explain the extremely high room-temperature electrical resistivity [2,3] observed in these alloys using a simple model developped by Mott [6]. The conductivity o- associated with electronic states in a pseudogap is proportional to the square of the electronic DOS through a factor g 2 : Se2Lg 2

~r

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where L is the electronic mean free path, S = 4"rrkF2 (where k F is the Fermi vector), and g is the ratio between the reduced and the free DOS values, g =D(Ev)/D(EF)tree. The factor g 2 in eq. (1) comes simply from Fermi's golden rule. Assuming the Ioffe-Regel regime [7], so that the room-temperature electronic mean free path reaches its minimum value L = a (a being the interatomic distance, typically 3 ,& for metals), with k r ~ 1.5 ,&-i (this k F value can be obtained either from the mean electron concentration e/a -- 1.8 or from the H u m e - R o t h e r y condition k F ½kp with kp---3 ~ - 1 ) , and g = 0.3, we can estimate for samples b, c and d a room-temperature resistivity of the order of p = 2000 I ~ cm. This value is in satisfactory agreement with the observations of Biggs et al. [2] and Klein et al. [3] at room temperature. This is the first explanation of the anomalous transport properties observed in

K. Wang et al. / Specific heat measurements on single-phased AICuFe quasicrystals

stable Q C s with a realistic room temperature electronic mean free path value. This model can be generalized to other stable QC phases [8].

4.2. Excess vibrational specific heat and phason strains The T = 0 Debye temperatures OD0 calculated using /3 values are listed in table 1. These values show clearly an-excess of the low-temperature specific heat in the imperfect samples. The OD0 values for samples a (290 K), b (273 K) and d (315 K) are obviously lower than that for sample c (350 K). The highest Oo0 value is lower than that given in ref. [2], but is in agreement with the value of ref. [3]. We first consider the difference between samples b (phason-strained) and c (phason-free). The Oo0 difference indicates that the low-energy vibrational density of states is enhanced in sample b compared with c. This enhancement is clearly associated with the phason strains, which are eliminated by the further annealing at 800°C in sample c. This is indicated by the X-ray diffraction experiments [5]. Enhancement of the low-energy, vibrational density of states by the softening of the phonon transverse mode is often observed in metallic glasses. This leads to a T = 0 Debye temperature decrease in specific heat measurements [9]. Low-energy mode enhancements have also been observed by inelastic neutron scattering in the metastable QC PdSiU as compared to the crystalline phase [10]. However, our observation allows the first direct comparison between a phason-strained and a perfect QC phase. In the measuring temperature range this enhancement cannot be due to the phason 'hopping', because phason relaxations occur only at very high temperature ( ~ 650°C [5]). We show below that this anomaly can be attributed to a softening of the phonon transverse mode, implying a softening of the shear elastic modulus induced by phason strains. In fact, for an isotropic substance, the T = 0 Debye temperature can be written as [11]:

=

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359

where p,/x and B are the density, the shear and the bulk elastic moduli, respectively. For normal metal alloys, the relative change in the bulk modulus B due to the disorder can be expressed as A B / B = - ( A a / a ) ( a ' / F ( a ' ) ) (dF(a')/da') [12], where a is the mean interatomic distance and a' the distance between the defects and their nearest-neighbour atoms. F(a') has the asymptotic form ~ sin(2kFa')/a '3. According to X-ray diffraction experiments, the relative change in Aa/a for sample b can be estimated at about 0.1% relative to sample c [5]. This implies a finite F(a') [12]. The corresponding change in the bulk modulus A B / B cannot exceed 2%, if we take for the mean interatomic distance a ~ 3 .~ and k v = 1.5 ,~. Thus the decrease in OD0 in sample b is essentially due to the decrease in the elastic shear modulus/x. In fact, neglecting the vanishingly small density variation (0.3%), from eq. (2) we obtain AOD0/OD0 ~ A/x/2/x, and we can estimate for sample b a relative change in the elastic shear modulus A/x//x ~ - - 4 0 % as compared to sample c. The decrease in OD0 from sample a to b may be due to the elimination of the 13 phase, because sample b annealed at 600°C transforms to a homogeneous phason-strained QC phase. The OD, ~ value for sample d is lower than that for sample c, leading to a relative elastic modulus difference of A/x//x ~ - 2 0 % . Sample d can be considered as stabilized at high temperature by the entropy created by phason fluctuations, which can be frozen by the quenching process. Thus our results indicate that the shear modulus can also be reduced by frozen high-temperature phason fluctuations, which must involve p h a s o n - p h o n o n interactions. To our knowledge, these results give the first experimental evidence that phason strain disorder drastically reduces the QC elastic shear modulus. The possibility of a shear modulus decrease due to p h a s o n - p h o n o n coupling was considered several years ago by Jaric and Mohanty [13], based on a calculation for a hypothetical icosahedral cobalt. Furthermore, the onset of nonzero average phason strains is characteristic of lowtemperature phase transitions observed in some QC phases [14]. Our results support the proposi-

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K. Wang et al. / Specific heat measurements on single-phased AICuFe quasicrystals

tion that such transitions are due to an elastic instability driven by phason-phonon coupling [13,15].

5. Conclusions

We studied the electronic and vibrational properties of different single-phased quasicrystalline A1CuFe alloys by low-temperature specific heat measurements. The weak electronic specific heat contribution allows us to propose a simple model to explain the anomalous electronic transport with a realistic electronic mean free path value. The excess vibrational contribution in phason-strained samples indicates an enhancement of the low-energy vibrational density of states. This implies a softening of the phonon transverse modes caused by a decrease in the elastic shear modulus related to phason strain through phason-phonon coupling.

References [1] A. Tsai et al., Jpn. J. Appl. Phys. 26 (1987) L1505. [2] B.D. Biggs et al., Phys. Rev. B43 (1991) 8747. [3] T. Klein et al., Phys. Rev. Lett. 66 (1991) 2907.

[4] A.I. Goldman et al., in Proc. Anniversary'Adriatico Research Conference on Quasicrystals, eds. M.V. Jaric and S. Lundqvist (World Scientific, Singapore, 1989); P.A. Bancel, Phys. Rev. Lett. 63 (1989) 2741. [5] M. Bessi~re, et al., J. de Phys. I 1 (1991) 1823; Y. Calvayrac et al., J. de Phys. 51 (1990) 417. [6] N.F. Mott and E.A. Davis, Electronic Processes in NonCrystalline Materials (Clarendon Press, Oxford, 1979). [7] A.F. Ioffe and A.R. Regel, Progr. Semicond. 4 (1962) 237. [8] K. Wang, C. Scheidt and P. Garoche, to be published. [9] B. Golding et al., Phys. Rev. Lett. 29 (1972) 68. [10] J.-B. Suck et al., Phys. Rev. Lett. 59 (1987) 102. [11] For the relations between the sound velocities and the elastic constants in isotropic solids, see R.N. Thurston, in Physical Acoustics, Vol. 1, Part A, ed. W.P. Mason (Academic Press, New York, 1964). [12] A. Blandin, in Alloy Behaviour and Concentrated Solid Solutions, ed. T.B. Massalski (Gordon and Breach, New York, 1965). [13] M.V. Jaric and U. Mohanty, Phys. Rev. Lett. 58 (1987) 230; M.V. Jaric and U. Mohanty, Phys. Rev. B38 (1988) 9434. [14] M. Audier et al., Phil. Mag. B63 (1991) 1375. [15] M. Widom, in Proc. Anniversary Adriatico Research Conference on Quasicrystals, eds. M.V. Jaric and S. Lundqvist (World Scientific, Singapore, 1989); C. Henley, in Quasicrystals; The State of the Art, eds. D.P. Di Vincenzo and P. Steinhardt (World Scientific, Singapore, 1991). [16] P. Manuel and J.J. Veyssi6, Rev. G6n6rale de Thermique 111 (1976) 231.