Z. Phys. B 101, 235—245 (1996)
Elastic properties of several amorphous solids and disordered crystals below 100 K K.A. Topp, David G. Cahill* Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853, USA (Tel.: (607) 255-3943 Fax: (607)255-6428, e-mail:
[email protected]) Received: 5 January 1996
Abstract. We have measured the internal friction and speed of sound variation at temperatures between 60 mK and room temperature for amorphous CdGeAs , Polysty rene, and Stycast 2850FT epoxy, and the disordered crystals (ZrO ) (CaO) and (CaF ) (LaF ) . A comparison of our results with an extensive review of previously published data shows a remarkable similarity in the internal friction of disordered solids below &5 K. The low temperature elastic behavior of these solids is adequately described by the standard tunneling model, from which one finds a nearly universal density of tunneling states for glasses. Internal friction above &10 K for different materials, however, displays a wide range of magnitudes and temperature dependence that is far from universal. Attempts to directly link the tunneling states observed by internal friction at low temperatures to configurational states of localized oscillators existing at high temperatures must take into account this striking variation among disordered solids above 10 K. PACS: 62.40.#i; 63.20.!e; 63.50.#x
I. Introduction Low energy excitations appear to be common to all amorphous and certain disordered crystalline solids. At temperatures below a few degress Kelvin, these excitations dominate their thermal, elastic and dielectric behavior, and the dynamics of these excitations are well described by models based on quantum mechanical tunneling. A detailed understanding of the microscopic origin of these excitations remains elusive but a description has emerged based on localized oscillators with nearly degenerate ground state configurations. Localized oscillators have
*Present address: Department of Materials Science and Engineering, University of Illinois, IL 61801, USA
been identified in specific heat data at ¹&10 K and in inelastic neutron and light scattering experiments in a number of amorphous solids [1—4]. Connections between these localized vibrations and the tunneling states have been explored in amorphous solids [5—8] and in disordered crystals which exhibit similar vibrational anomalies [9]. In these disordered crystals, the lattice vibrations are often referred to as ‘‘glasslike’’ [10]. Measurements of elastic properties — for example, the elastic constants and internal friction — have proven to be a particularly versatile technique for the study of the tunneling states (TS) in amorphous and disordered crystalline solids. At intermediate temperatures, typically between a few Kelvin and 100 K, the elastic measurements also provide information about the distributions of potential barriers for thermally activated reorientations of atoms or molecules, and therefore yield additional information about the localized oscillator states described above. With this goal in mind, we have studied the elastic properties of several disordered solids below 100 K. These new data will be presented alongside an extensive review of earlier measurements on a wide variety of disordered solids. We will analyze the data at low temperatures, below &3 K, using the tunneling model (TM) [11, 12], and also inspect the data above that temperature to gain additional insight on the behavior of the oscillator states. We report new data for five materials: (a) CdGeAs . Each atom in this amorphous solid is fourfold coordinated. Therefore, the atomic bonding in CdGeAs might be expected to be over-constrained, greatly reducing the concentration of so-called ‘‘floppy modes’’ [13] which could give rise to tunneling states. Thermal measurements [14], however, showed evidence for tunneling states and so elastic measurements are needed to more fully characterize these states. (b) Polystyrene. The thermal and elastic properties of this polymer have been studied previously (Nittke et al. [15], and earlier references therein); however, elastic measurements at sufficiently low temperatures and high frequencies from which the coupling energies of the
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tunneling states to the phonons could be determined had not yet been made, and will be presented here. (c) Stycast 2850FT epoxy. This polymer-ceramic composite contains +40% by volume crystalline Al O in a heavily cross-linked polymer matrix. This adhesive is used in our laboratory for attaching samples to the acoustic transducers and therefore the internal friction of this material is needed to fully understand the background damping in our measurements. In addition to this practical motivation, this material represents an example of how the addition of crystalline filler to an amorphous matrix effects the elastic properties of the composite. (For measurements of the effect of partial crstallization on the specific heat of a crystallizable amorphous solid, polyethylene, see [16, 17].) (d) ZrO ) (CaO) , x"0.11, stabilized in the cubic \V V phase. This disordered crystalline solid was chosen since it is known that ZrO stabilized with Y O exhibits glass-like lattice vibrations, tentatively ascribed to the disordered arrangement of oxygen vacancies [18]. The generality of glass-like lattice vibrations in stabilized zironia is not known and therefore we examine whether the addition of CaO has the same effect as Y O . (e) (CaF ) (LaF ) , x"0.26. In this mixed fluoride \V V crystal, interstitial fluorine ions (created by substitution of the trivalent La ion for divalent Ca) appears to produce glassy behavior. The highly doped CaF system was chosen because its thermal conductivity is roughly a factor of two higher than that of the equally doped BaF system over a broad temperature range [18, 19], suggesting a smaller number of localized excitations, or alternatively a weaker coupling to the lattice, for the same amount of interstitial disorder. In our discussion of the results for these five materials, we will also make a comparison with earlier studies. We have therefore made an exhaustive search of earlier measurements of the internal friction for amorphous solids (including polymers) and glasslike crystals that we were able to find which covered the range between &3 K and 100 K. These investigations will be summarized, and representative data will be shown in the figures in which our measurements of internal friction are presented.
II. Experimental The materials measured here were obtained from a variety of sources: (a) The amorphous CdGeAs compound was produc ed from 99.9999%-pure Cd and As and 99.9995 %-pure Ge supplied by Alpha Chemicals. The method is described elsewhere [14]. (b) The clear and fully amorphous polystyrene (PS) sample was cut from commercially available extruded stock. (c) The exact composition of Stycast 2850FT epoxy is proprietary, but we know it to be a filled-polymer containing +80% crystalline Al O by weight [20]. Our sample (using Stycast catalyst 24LV) was mixed according to the directions of the manufacturer, and cast in a Teflon mold
after degassing the mixture in vacuum. To ensure that no significant amount of air had been trapped in the mold, we compared the measured density of our sample to the value quoted by the manufacturer. Both were 2.26 g/cm. (Using this density, with the known density of 3.97 g/cm for Al O , and assuming a typical value of +1 g/cm for the unknown polymer matrix, we calculate the proportion of Al O in the mixture to be roughly 42% by volume, or 74% by weight.) (d) Zirmat Corporation [21] supplied the single crystal of calcia-stabilized cubic zirconia (ZrO ) (CaO) . X-ray diffraction was used to confirm the cubic structure of the crystal and to measure the lattice constant of 5.135(6) As , as well as to orient the sample for cutting so that its torsion axis was along the crystallographic 100 direction. The CaO concentration was measured by X-ray fluorescence. (e) The (CaF ) (LaF ) crystal, purchased from Optovac [22], was grown from the melt under vacuum in a graphite crucible using the Stockbarger technique. The lanthanum concentration was measured by X-ray fluorescence, and the cubic lattice constant was determined by X-ray diffraction to be 5.634(5) As . The torsion axis of this sample was not chosen to be a specific crystallographic orientation, but later X-ray diffraction measurements showed it to be +15° from 1 0 0. We measure the internal friction using a composite torsional oscillator technique, described in detail in [23]. In this method, a 90 kHz quartz transducer and the sample form a composite torsion bar. The quartz end is attached to a thin Be-Cu pedestal [23] and the junctions (sample-quartz and quartz-pedestal) are each made with an approximately 25 mg drop of Stycast 2850FT epoxy. The sample length is tuned to be one half of a shear wavelength, so that the composite oscillator has a resonance frequency at room temperature within 1% of the bare quartz crystal resonance. This adjustment ensures that the epoxy joint between the quartz and sample has almost zero strain, and therefore contributes minimally to the observed internal friction. The two junctions produce a background contribution to the internal friction of roughly 10\ at low temperatures. The oscillator is driven by a set of electrodes which form a quadrupole configuration around the transducer and which simultaneously drive and detect its motion. The internal friction of the sample (Q\) is determined from the quality factor (Q ) Q? AMKN of the composite oscillator resonance by
I #(1#)I RP Q\ Q\" Q? (1) Q? AMKN I Q? where I and I are the moments-of-inertia of the transRP Q? ducer and sample respectively, and is a correction for the attachment of the transducer to the thin Be-Cu pedestal, +0.06 [23]. The measurements below 1.5 K were made in a dilution refrigerator, and those from 1.5 to 300 K in an insertable He cryostat [24]. The literature data summarized below were obtained using a variety of techniques — for example, longitudinal or flexural vibration, or ultrasound attenuation — and cover a wide range of measuring frequencies, 10 Hz to 450 MHz. Sample descriptions and measurement
237 Table 1. Materials for which internal friction has been measured from near 100 K down to the ‘‘plateau’’ region (( a few K). The relevant mass density and sound velocities at low temperatures (¹(5 K) are included from various sources. References are given in square brackets: they are placed after the mode of oscillation for the Measuring frequency Amorphous solids As S B O CdGeAs GeO Ge Se GeSe GeSe Ge S GeS GeS Pd Si Cu Se SiO Polymers PC PEMA PMMA PS 2850FT Epoxy
300 Hz 2.8 kHz 90 kHz 6.3 kHz 550 kHz 550 kHz 550 kHz 550 kHz 550 kHz 550 kHz 178 Hz 0.228, 35 kHz 66, 90 kHz 10 Hz 15 MHz 15—410 MHz 87 kHz 84 kHz
internal friction data reference, and after v for the density and speeds of sound reference, unless otherwise indicated. Values in parentheses are not measured, but derived in a manner indicated in the footnotes
Mode of oscillation
(g/cm)
v (km/s)
v (km/s)
flexural [25] flexural [27] torsional [ours] flexural [27] longitudinal [32] longitudinal [32] longitudinal [32] longitudinal [32] longitudinal [32] longitudinal [32] flexural [37] flex. [39], long [33] torsional [40, 23]
3.20 1.82 [28] 5.72 3.61 4.25 — — — 2.70 [35] 2.44 [35] 10.52 4.30 2.20
2.65 3.47 3.03 3.77 — — — — 2.73 [35] 2.48 [35] 4.60 2.00 5.80
1.44 [26] 1.88 [29] 1.86 [30] 2.36 [31] 1.44 [34] — — — 1.7 [36] 1.5 [36] 1.797 [38] 1.05 [34] 3.80 [41]
2.90 2. 955 [46] 3.150 2.8 [48] (3.65)
1.37 [44] — 1.570 [46] 1.34 [ours] 2.21 [ours]
torsional [42] long. atten. [45] ] long. atten. [46, 47] torsional [ours] torsional [ours]
1.19 1.12 1.18 1.06 2.26
Disordered crystals (BaF ) (LaF ) 90 kHz torsional [49] (5.55) (4.69) (2.40) [50] torsional [ours] 4.21 — 3.11 [ours] (CaF ) (LaF ) 87 kHz (KBr) (KCN) 500 Hz torsional [51] (1.90)
(3.82) (1.22) [52] (NaCl) (NaCN) 90 kHz torsional [9] 2.16 — 1.86 [9] (ZrO ) (CaO) 83 kHz torsional [ours] 5.71 — 3.43 [ours] B C 165 kHz torsional [53] 2.45 (13.0) 7.86 [53] GdB 205 kHz torsional [54] 2.813 — 7.77 [54] YB 164 kHz torsional [54] 2.57 — 8.54 [54] Ti Nb 66 kHz torsional [40] 6.00 (4.36) 2.64 [40] Ti V 86 kHz torsional [40] 4.73 (4.38) 2.66 [40] Room temperature value Data of Q\ also exist in [33] for this material in the MHz range, but they are presented with an arbitrary shift along the Q\ axis, and thus kcannot be used This is not a complete list of measured polymers. For other materials, see [42, 43] polycarbonate polyethylmethacrylate polymethylmethacrylate polystyrene Using the relation for amorphous solids v "1.65v from [41]. Note that this does not appear to be a good approximation for unfilled polymers. It can be used reasonably well for polycrystalline materials [40], but not for single crystals Extrapolated from values measured on other (BaF ) (LaF ) crystals [50] \V V
Interpolated from values measured on other (KBr) (KCN) crystals [52] \V V Calculated from interpolated values of elastic constants [52]
parameters are summarized in Table 1. Note that the studies of internal friction often did not report the speeds of sound; in these cases, sound velocity data were gathered from other sources. III. Tunneling model For the analysis of the low tempetarure data (¹(5 K), we use the phenomenological tunneling model (TM) [11, 12, 37] which assumes a constant spectral density
PM of tunneling states (TS); the coupling energies of the TS to longitudinal and transverse elastic strains are given by , respectively. J R The processes by which the tunneling systems dissipate elastic energy in electrical insulators at low temperatures, and the resulting temperature dependence of internal friction, can be examimined in two limiting cases. In the extreme low temperature limit, the maximum relaxation rate \ of the TS is small compared with the angular
frequency of the applied strain oscillations, \ ;, and
the occupations of the tunneling states are incapable of
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critically examine this assumption in our review of the data. The value of T is extracted by fitting our data AM below 1 K (from the plateau region through the low temperatrue T roll-off ) to a numerical evaluation of the TM prediction [55]; T is the only parameter used to adjust AM the fit along the temperature axis. The coupling constant, is then calculated from the T value [40]. R AM The TM predicts that the temperature dependence of the speed of sound is given by
Fig. 1. Internal friction of selected amorphous solids. SiO data (solid circles) below 1.5 K are from [40], and above 1.5 K, from [23]. Open circles are high temperature SiO data from [57], measured at 1.5 MHz. Diamond symbols are As S from [25]; other data exist for As S [58] but are omitted for clarity. (The measuring frequency of the latter is five orders of magnitude higher than that of the data shown here, but the two measurements show reasonable agreement. See Figs. 6 and 7.) Open triangles are B O , and squares are GeO , both from [27]; see also [59] for data on both materials above 4 K. CdGeAs data, this work, are shown as solid triangles. Plus symbols are Pd Si Cu from [37]. Five-point stars are Se data; from [33] above 3 K, and from [39] below
reaching equilibrium on the time scale of an oscillation. This leads to a T temperature dependence of internal friction (see for example a-SiO below 0.1 K in Fig. 1). At higher temperatures, where the TS relax quickly enough to reach equilibrium on the time scale of the oscillations, \
