Elastic properties of amorphous and crystalline B1−xCx and boron at low temperatures

Journal of Alloys and Compounds 270 (1998) 1–15

L

Elastic properties of amorphous and crystalline B 12x C x and boron at low temperatures 1

2

P.A. Medwick , B.E. White, Jr. , R.O. Pohl* Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York, NY 14853 -2501, USA Received 5 August 1997

Abstract We present measurements of the internal friction (Q 21 ) and speed of sound variation (dv /v0 ) of amorphous boron (a-B) and amorphous B 9 C (a-B 9 C). The elastic properties of these materials, which can only be produced as thin films, are consistent with those of other amorphous solids measured to date and exhibit good agreement with the tunneling model (TM) of amorphous solids. The TM parameter P¯ gt 2 /r vt 2 extracted from the elastic data has the same order of magnitude as that observed for all amorphous solids studied to date; a review will be presented. Using the results from the elastic measurements, we calculate the T 2 thermal conductivity L expected in the TM regime (T #1 K) for a-B. The predicted thermal conductivity falls within the expected range for amorphous solids and agrees with the thermal conductivity of the crystalline icosahedral boride MB 68-d (M5Y, Gd), which has been previously shown to exhibit glass-like excitations. We have also measured the internal friction and speed of sound variation of bulk polycrystalline c-B 12x C x at low temperatures (0.07 K,T ,10 K). The elastic properties evolve towards the behavior characteristics of amorphous solids for increasingly carbon-deficient (x,0.20) specimens. The magnitude of the internal friction for the most carbon-deficient crystalline c-B 12x C x sample (x50.1, c-B 9 C) is comparable to that for a-B and a-B 9 C, thereby confirming the inherent glass-like vibrational properties of carbon-deficient c-B 12x C x . Such behavior supports the glass-like character of carbon-deficient c-B 12x C x high temperature (T .50 K) thermal transport reported previously and provides the first experimental evidence for the presence of two-level systems (TLS) in these crystalline solids. However, discrepancies with the tunneling model are present; the data for c-B 12x C x bear some similarity to those for amorphous metals in which electronic relaxation channels are active, although details are still unclear. Previous studies have shown that the TM quantity C5Pgt 2 /r vt 2 (‘‘tunneling strength’’) is essentially independent of the material’s shear modulus G5 r vt 2 over a factor of |17. The elastic data presented in this work now extend the observed independence of the tunneling strength, C, over a factor of |70 in shear modulus.  1998 Elsevier Science S.A. Keywords: B 12x C x alloys; Elastic properties; Internal friction; Amorphous alloys

1. Introduction Boron-rich solids possess unusual physical properties that make them both intrinsically and technologically interesting. Their traditional applications as abrasives and light weight armor exploit the high melting points and extreme hardness of these materials [1]. Some borides, notably the crystalline boron carbides (c-B 12x C x ), are low mobility semiconductors with large Seebeck coefficients (S $200 mV K 21 at the highest temperatures measured to *Corresponding author. E-mail: [email protected] 1 Present Address: PPG Industries, Inc., Glass Technology Center, P.O. Box 11472, Pittsburgh, PA 15238-0472, USA. 2 Present Address: Advanced Materials Group, Materials Research and Strategic Technology, Motorola, 3501 Ed Bluestein Boulevard, Austin, TX 78721, USA. 0925-8388 / 98 / $19.00  1998 Elsevier Science S.A. All rights reserved. PII: S0925-8388( 98 )00119-4

date; |1200 K) [1–3]. This has made them attractive candidates for thermoelectric power conversion at high temperatures [4]. The conversion efficiency (figure of merit Z5 s S 2 /L) of such a device is a strong function of the thermal conductivity of the thermoelectric elements; where s is the electrical conductivity, S is the Seebeck coefficient and L is the thermal conductivity. An understanding of the thermal and mechanical properties of boron-rich solids is therefore critical for many applications. It is also of fundamental interest because the physical properties of these structurally complex icosahedral borides are not well-understood [5,6]. Previous work [7] demonstrated an evolution of the high temperature (T .50 K) thermal conductivity L from crystalline to glass-like behavior as the carbon concentration in c-B 12x C x is decreased below x50.20. However, one of the most definitive signatures of amorphous solids

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P. A. Medwick et al. / Journal of Alloys and Compounds 270 (1998) 1 – 15

is the common magnitude and temperature dependence of their thermal conductivities below 1 K: L |(3310 24 W cm 21 K 23 )T 2 ; the prefactor varies by only a factor of three either way for all amorphous solids measured to date. The magnitude and temperature variation of this thermal conductivity cannot be explained in terms of the Debye theory. This thermal conductivity has been attributed to low energy excitations, associated with two-level systems (TLS) or ‘‘tunneling states’’, which dominate the material’s thermal, elastic and dielectric properties at low temperatures [8,9]. Glass-like behavior has also previously been detected in certain disordered crystals [10]. However, detection of TLS excitations using thermal conductivity measurements can be thwarted by extrinsic factors, in particular, by microstructural features such as grain boundaries; this has previously been the case for c-B 12x C x and other borides [11]. As a result, there has been no direct evidence of TLS in c-B 12x C x from the existing low temperature thermal conductivity data. Similar problems exist for low temperature specific heat measurements, which are often affected by chemical impurities in the sample [11]. However, mechanical measurements of internal friction (Q 21 ) and sound speed variation (dv /vo ) are not as susceptible to such extrinsic factors [12] and allow extraction of TLS parameters. This work describes the first direct evidence of TLS in polycrystalline c-B 12x C x , obtained from such measurements at low temperatures (0.07 K,T ,10 K). It will be shown that the elastic properties evolve towards behavior characteristics of amorphous solids for increasingly carbon deficient (x,0.20) specimens. The data provide the first experimental evidence for TLS in these crystalline borides. However, the details of their relaxation dynamics appear to be more complicated than for TLS in amorphous dielectrics; rather, some similarity to amorphous metals is observed instead. For comparison, we have also measured the elastic properties of amorphous boron (a-B) and a-B 9 C. To date, it has not been possible to synthesize these solids in sufficient quantities to make such measurements using conventional techniques. However, one can produce amorphous borides as thin films. Using hyperpure silicon wafers with very little mechanical damping as substrates, we have studied the TLS in films of a-B and a-B 9 C and have thereby shown that the low energy excitations in carbondeficient c-B 12x C x are very similar to those of these amorphous solids. From these elastic data, one can predict the low temperature (T #1 K)L~T 2 thermal conductivity in the TLS dominated regime for bulk c-B 9 C (in the absence of grain boundary scattering) and a-B; these have been found to be in excellent agreement with those of all other amorphous solids. Finally, the reported universality (or near-universality) of the low energy excitations will be critically reviewed (and confirmed) by inspecting the low temperature internal friction and / or thermal conductivity of a large number of amorphous solids, including a-B and a-B 9 C, with shear moduli ranging over a factor of seventy.

2. Experimental

2.1. Preparation of bulk c-B12 x Cx The crystal structure of c-B 12x C x has been described previously [7,13–16]. The bulk c-B 13 C 2 (x50.133) and c-B 9 C (x50.1) samples were hot-pressed (2450 K, 6000 p.s.i.) from boron and graphite powders under an argon atmosphere in a graphite die lined with BN [17] by Dr. T.L. Aselage of Sandia National Laboratories. These cB 13 C 2 ( r 52.47 g cm 23 ) and c-B 9 C ( r 52.46 g cm 23 ) samples were similar to those prepared for the thermal conductivity measurements reported previously [7]; they were nearly fully dense (.98% of theoretical density). Samples prepared in this fashion have been examined via x-ray diffraction (XRD) and Raman spectroscopy to confirm good crystallinity, the absence of secondary phases, and the absence of carbon diffusion into the material from the graphite die [17,18]. Samples previously prepared using this procedure for the earlier study (ref. [7]) had grain sizes ranging from approximately 10–20 mm (for carbon-rich samples; x|0.20) to 50–60 mm (for carbon-deficient samples; x|0.10) [19]. The sample of hot-pressed c-B 4 C ( r 52.37 g cm 23 ; grain size, |3 mm) was obtained from F. Thevenot [20]; it was pressed from an ESK (Electroschmelzwerk Kempten) pure arc-melted boron powder and was |93% fully dense. A sample of polycrystalline b-B ( r 52.33 g cm 23 ) was obtained from Hughes Research Laboratories; details of its fabrication are unknown. Due to the extreme hardness of c-B 12x C x , it was not practical to cut the samples into the rectangular prismatic geometry required for the composite oscillator technique (described in Section 2.2). Electro-disintegration cutting, also known as spark cutting, was employed instead.

2.2. Composite oscillator technique The internal friction of bulk polycrystalline c-B 12x C x and b-B was measured using the composite torsional oscillator technique of Cahill and Van Cleve [21]; we briefly review the measurement technique here. The sample, which is fashioned into a rectangular prismatic geometry, is bonded to a cylindrical quartz piezoelectric transducer. Typical dimensions for rectangular samples are |0.330.332.54 cm; the quartz piezo typically has a diameter of 0.4 cm and a length of |1 cm. Stycast 2850FT [22], an epoxy widely used for cryogenic experiments [23], is used as the bonding agent. The composite oscillator is then secured with this epoxy resin to a specially designed BeCu pedestal mount [21]. Oscillations of the composite torsion bar are excited via a quadrupolar electrode assembly that surrounds the transducer; the torsion axis is along the long dimension of the oscillator. The lengths of transducer and sample are appropriately chosen such that the epoxy bond between the two corresponds to a displace-

P. A. Medwick et al. / Journal of Alloys and Compounds 270 (1998) 1 – 15

ment antinode (and, therefore, a stress node) for the fundamental torsion mode of the composite bar at the desired frequency (|160 kHz for this work). The internal friction and speed of sound variation of the sample are extracted from the resonant frequency, quality factor and known dimensions of the composite torsion bar using the following equations [21,24,25]:

F

G

It 21 21 Q s 5 s1 1 ad] 1 1 Q co I

S D F dv ] vo

s

s

It 5 s1 1 ad] 1 1 I s

(1)

GS D df ] fo

(2) co

where It and Is are the moments of inertia of the transducer and sample relative to the torsion axis along the composite oscillator’s length, respectively; Q 21 co is the internal friction of the composite bar, Q s21 is the internal friction of the sample alone, a is an empirical factor used to account for the finite thickness of the BeCu pedestal mount (a |0.06) [24] and (df /fo ) co is the relative variation of the resonant frequency of the composite oscillator. Here, fo and vo are resonant frequency and transverse sound speed values measured at an arbitrary reference temperature, To . All internal friction data reported in this paper, unless otherwise noted, are those of the sample with the subscript ‘‘s’’ being omitted for simplicity. Further details regarding the composite oscillator technique and its data analysis are found in references [21,24,25].

2.3. Preparation /characterization of a-B and a-B9 C films The amorphous boride films examined in this study, a-B and a-B 9 C, were deposited onto silicon double paddle oscillators (described below) by electron beam evaporation. Note that the boride films were deposited onto paddles that had already been secured to their Invar block bases and attached to their brass cryostat mounts (see Section 2.4 below); the temperature of the paddle oscillator was not controlled during the deposition. Granulated boron of 99.9995% purity was used as the source for the a-B film. A remnant of the hot-pressed c-B 9 C sample from Sandia was used as the source for the a-B 9 C film. Deposition parameters were base pressure of approximately 1310 27 torr, beam current 10 mA, beam voltage 18 kV, ˚ s 21 . Concerns about boron diffusion deposition rate 10 A into the silicon are addressed below. Analyses of film structure, composition and thickness were performed on test films that were deposited simultaneously on surplus pieces of silicon. When this was not possible, the analysis of the paddle’s film was done after making the low temperature mechanical measurements. Because a-B and a-B 12x C x films are known to efficiently getter oxygen and hydrogen [26], the coated paddles were immediately stored in a vacuum desiccator upon removal from the evaporator. After all preparations for cool-down of the cryostat were completed, the oscillator assembly was quickly transferred

3

to the cryostat and the vacuum can was sealed and pumped out. Direct exposure of the paddle to atmospheric contaminants was therefore limited to approximately 10 min before performing the low temperature measurements. Structural information was obtained via glancing angle x-ray scattering using a Seifert thin film diffractometer (CuK a radiation); a series of angular scans showed no sharp peaks. The presence of an amorphous halo was inconclusive; this was attributed to the weak x-ray scattering cross-section of low-Z elements. As a further check, long-time (2–12 h) glancing angle x-ray exposures using a Read camera were performed. Such long exposures should make any small amount of crystallinity clearly apparent. The developed Read camera films showed only an illdefined, diffuse and incomplete ring for exposures in excess of 8 h. This ring corresponds to a d-spacing of ˚ This is near the d-spacing approximately 2.28–2.30 A. ˚ of one of six diffuse halos in the electron (d52.33 A) diffraction pattern of a 0.1 mm a-B film evaporated onto an unheated substrate [27]. The reason for the absence of the other diffraction maxima for our films is not clear; this may be due to the weak x-ray scattering cross-section of boron relative to electron scattering, in addition to the comparatively small thickness of our films. Note that ˚ in x-ray Kobayashi [28] has reported maxima of |2.5 A and electron diffraction patterns for thick (|2 mm) a-B films deposited on silica via high-temperature (800– 8508C) chemical vapor deposition (CVD). Three other maxima, in good agreement with analyses for films similarly prepared by other workers, were also observed for Kobayashi’s thick CVD films. Further evidence for the amorphicity of our boride films is provided by their mass densities |1.7 g cm 23 (discussed below), which are similar to amorphous boride densities reported previously [26,29] for sputtered films on silicon; however, no diffraction data were reported for those low density films. The a-B films prepared by Kobayashi have densities |2.33 g cm 23 ; densification of the films as a result of the high-temperature CVD process is suspected in that case [28]. Furthermore, the low temperature elastic signatures of our films (see Section 3) are consistent with those expected for amorphous solids, thereby supporting our conclusion that the deposited films are indeed amorphous. Rutherford backscattering spectrometry (RBS; 2.17 and 2.72 MeV He 11 ) was used for compositional analysis of the deposited films. Hydrogen isotope content was examined using forward recoil elastic scattering (FRES), also known as elastic recoil detection (ERD). Increased detection sensitivity for light elements, such as oxygen and carbon, was achieved using nuclear resonance enhanced scattering. He 11 ion beam energies were 3.07 MeV for the oxygen resonance and 4.31 MeV for the carbon resonance. The a-B film showed the presence of only 2.8 at.% oxygen and less than 1 at.% hydrogen in the film; there was no carbon signal from the film. Ion beam analysis of the a-B 9 C film showed the carbon content to be between 9–10 at.%, which is very close to the composition of the target

P. A. Medwick et al. / Journal of Alloys and Compounds 270 (1998) 1 – 15

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material (10 at.% carbon). The oxygen content was 3.7 at.%; hydrogen impurity ,1 at.%. Despite the loss of some carbon during evaporation, this film is still referred to as ‘‘a-B 9 C’’ for convenience, to indicate the composition of the c-B 9 C evaporation source. Because the carbon-poor limit of the crystalline c-B 12x C x phase field lies somewhat above 9 at.% carbon, a comparison between the experimental results for this film and c-B 9 C will still be meaningful. Note that the above impurity contents represent upper limits. Due to some unavoidable exposure of the films to atmosphere immediately preceding the ion beam analysis, the impurity concentrations were likely to be even lower during the actual mechanical measurements at low temperature. The thickness t film of the deposited layers was measured ˚ for a-B and 1760 A ˚ by stylus profilometry to be 2510 A for a-B 9 C. Film densities, r film, were calculated using the areal density of atoms, s atom (via RBS), t film , average ¯ and Avogadro’s number, NA ; thus rfilm 5 molar mass, M, ¯ /(NA t film ). We obtained mass densities of 1.70 and satom M 1.68 g cm 23 for a-B and a-B 9 C, respectively. These are comparable to the mass densities of other amorphous borides reported previously [26,29]. For the purposes of this discussion, we define film porosity, p, as rfilm p 5 1 2 ]] (3) rth where rfilm is the film’s mass density and the ‘‘theoretical’’ density, rth , is taken to be the density of b-B (for a-B) and crystalline c-B 9 C (for a-B 9 C). We elect to define porosity in this fashion because the mass densities of bulk, fully dense a-B and a-B 9 C are undefined (i.e. bulk a-B and a-B 9 C have not yet been synthesized). The porosities are p50.32 and p50.33 for the a-B and a-B 9 C films, respectively. Using these values of rfilm and t film , the frequency shifts of the coated oscillators (discussed below) relative to the uncoated oscillator, and the known thickness (t sub 5300 mm) and shear modulus (Gsub 56.2310 11 dyne 22 cm ) of the silicon oscillator substrate, we calculate the shear moduli and transverse speeds of sound to be Gfilm 5

11.1310 11 and 7.4310 11 dyne cm 22 via Eq. (5) (discussed below) and vt 58.08310 5 and 6.64310 5 cm s 21 , for a-B and a-B 9 C, respectively. The calculated shear modulus of a-B is approximately 44% lower than that for crystalline b-B and may be due to the porosity of the a-B film as defined above. MacKenzie [30] has shown that the porosity, p, will reduce the Young’s modulus E according to the empirical relation: E 5 Eos1 2 1.9p 1 0.9p 2d

(4)

where Eo is the modulus of the fully dense phase. This relation assumes a Poisson ratio of 0.3 (typical for many solids) and porosities of less than 50%. The relative shift in the shear modulus should be similar to the Young’s modulus shift. For our a-B film, the predicted shift in the Young’s modulus is about 52%, based on Eq. (4), and agrees reasonably well with the measured shift (44%) in Gfilm for a-B (see Section 3). For our a-B 9 C film, the expected percentage deviation of the film’s shear modulus from that of c-B 9 C as a result of porosity is about 53%; this agrees reasonably well with the measured decrease of 51%. We therefore conclude that the reduced shear modulus of a-B and a-B 9 C is likely to be explained by the porosity as defined above. A summary of experimentally determined mass densities, shear moduli, and speeds of sound for our samples is given in Table 1.

2.4. Double paddle oscillator technique The mechanical properties of thin films can be measured using a sensitive mechanical resonator known as a double paddle oscillator (DPO) (Fig. 1) [24,31–37]. The double paddle oscillator is a quasi-two-dimensional mechanical resonator that is chemically etched from a hyperpure silicon (100) wafer. Non-compensated, microelectronicgrade silicon exhibits nearly perfect structural order and extreme chemical purity. The silicon DPO therefore pos21 28 sesses extremely low mechanical loss (Q |3–4310 at T ,1 K). Double paddle oscillators have been used for

Table 1 Mass densities r, shear moduli G, and speeds of sound for boride samples of this study. Data for single crystal YB 63 and GdB 62.5 included for comparison [51]. Transverse sound speed (vt ) calculated from resonant frequency of oscillator, shear modulus calculated from vt and measured mass density. Longitudinal sound speed vl calculated from vt using the approximation vl |1.65vt , as reported for a variety of amorphous solids by Berret and Meissner [49]. Debye speed of sound vD calculated from vl and vt via 1 /v 3D 5(1 / 3)[2 /v 2t 11 /v 3l ]. Measured sound speeds for c-B 12x C x agree (|6% difference) with those of Gieske and Aselage [56] Material

r (g cm 23 )

G (10 12 dynes cm 22 )

vt (10 5 cm s 21 )

vl |1.65vt (10 5 cm s 21 )

vD (10 5 cm s 21 )

c-B 4 C c-B 13 C 2 c-B 9 C a-B a-B 9 C b-B YB 63 GdB 62.5

2.37 2.47 2.46 1.70 1.68 2.33 2.57 2.81

1.78 1.84 1.52 1.11 0.74 2.00 1.87 1.7

8.67 8.64 7.86 8.08 6.64 9.27 8.54 7.77

14.3 14.3 13.0 13.3 11.0 15.3 14.1 12.8

9.58 9.55 8.69 8.93 7.34 10.2 9.44 8.59

P. A. Medwick et al. / Journal of Alloys and Compounds 270 (1998) 1 – 15

5

lators used in our experiments are approximately f 55.1– 5.3 kHz and Q52.5310 7 at T570 mK. Deposition of a film on the oscillator changes its resonant frequency and quality factor. For uniform coating, the shift in the oscillator’s resonant frequency, f, at a particular temperature, T, is given by the following equation [34–36]:

S D DfsTd ]] fsTd

fco (T ) 2 fsub (T ) ; ]]]]] fsub (T ) co

S

1 t film 3Gfilm (T ) rfilm 5 ] ]] ]]] 2 ]] 2 t sub Gsub (T ) rsub

Fig. 1. Silicon double paddle oscillator (scale indicates 1 cm). The oscillator shape is photolithographically patterned and chemically etched (using an anisotropic KOH wet chemical etch) out of a 300 mm thick hyperpure silicon wafer. For the present study, the device is operated in the antisymmetric torsion mode in which the head and wings twist 1808 out-of-phase with each other; the torsion axis is along the centerline of the structure parallel to the neck and tail. Mounting and excitation of the device are described in detail in refs. [24,31–37]. Boride films are deposited onto the oscillator; the resulting change in the oscillator’s quality factor and resonant frequency is attributed to the film. The internal friction of the film Q 21 film and its speed of sound variation (dv /v o ) film can be extracted from the quality factor and resonant frequency data for the coated oscillator (see text).

measurements of the internal friction of several disordered films [34–36]. The base of the oscillator (Fig. 1) is adhesively bonded, using Stycast 2850FT epoxy, to a block (3.132.931.0 cm) of Invar alloy; this block can then be clamped to a specially designed brass mount [34–36]. The resonator’s vibrations are capacitively driven and detected by two coaxial electrodes that are located very close to the paddle behind each wing. The electrodes are held in place by clearance holes cut in the brass mount. The entire assembly is then secured to the cold stage of a cryostat for low temperature measurements. For details, see references [24,31–37]. The paddle has several vibrational eigenmodes. The mode of interest for this work is the antisymmetric (AS) torsional mode, in which the head and wings oscillate 1808 out of phase, thereby twisting the paddle’s neck. Because the head’s moment of inertia (MOI) is much smaller than that of the wings, the angular deflection of the wings is much less than the deflection of the head. Therefore, the oscillator’s leg is deformed very little. This limits the transmission of mechanical energy down the leg and into the epoxy bond / Invar block base, where it can be dissipated, which reduces the background mechanical loss [34– 36]. The resonant frequency and quality factor of the oscil-

D

(5)

where the resonant frequencies of the bare paddle and composite (film / substrate) structure are fsub and fco , respectively. The frequency shift Df(T ) /f(T ) at temperature T is measured by subtracting the resonant frequencies of the coated and uncoated paddle at T and dividing by the uncoated paddle’s resonant frequency. The thicknesses of the substrate and film are t sub and t film ; their respective shear moduli and mass densities are Gsub (T ), Gfilm (T ) and rsub , rfilm [34–36]. Note that Df(T ) /f(T ) is a function of temperature because the shear moduli of the paddle and film both vary with T. The temperature-dependent shear modulus of the film Gfilm (T ) is the only unknown quantity (see Section 2.3 above) and can be calculated using the measured frequency shift via Eq. (5). We now derive an expression for the relative variation in the transverse speed of sound of the film (dv /vo ) film as a function of experimentally measurable quantities. The relative variation in the speed of sound of the film is defined as:

S D S dv ] vo

vsTd 2 vo ; ]]] vo film

D

(6) film

In this expression, v(T ) is the film’s sound speed at temperature T and vo is the film’s sound speed at an arbitrary reference temperature, To . We begin by solving Eq. (5) for the film’s temperature-dependent shear modulus Gfilm (T ):

F S D

1 2t sub Df(T ) Gfilm (T ) 5 ] ]] ]] 3 t film f(T )

G

rfilm 1 ]] Gsub (T ) rsub co

(7)

The transverse sound speed of the film is related to its shear modulus via: ]]] Gfilm (T ) v(T ) 5 ]]] (8) rfilm

œ

where the (unsubscripted) sound speed, v, is hereafter understood to refer to the transverse sound speed and is the transverse sound speed at the reference temperature (i.e. vo 5v(T o )) and the normalized sound speed variation of the film is related to its shear modulus variation via:

S

d v(T ) ]] vo

D

1 d Gfilm (T ) 5 ] ]]] 2 Gfilm (T ) film

(9)

P. A. Medwick et al. / Journal of Alloys and Compounds 270 (1998) 1 – 15

6

Analogous expressions hold for the silicon substrate. We take differentials of both sides of Eq. (7) and obtain: 2 t sub d Gfilm (T ) 5 ] ]] Gsub (T )d 3 t film

FS D G Df(T ) ]] f(T )

21

S D

co

d Gsub (T ).

(10)

In this expression, the differential in the first term is defined as:

d

FS D G S D S Df(T ) ]] f(T )

Df(T ) ; ]] f(T )

co

Df(T o ) 2 ]] f(T o ) co

D

(11) co

where T o is the reference temperature. Multiplying Eq. (10) by 1 / [2Gfilm (T )] and using the definition of (dv(T ) / vo ) film given by Eq. (9), we have:

S

d v(T ) ]] vo

D

Df(T ) ]] f(T )

F

co

1 1] 2

2 t sub Df(T ) ] ]] ]] 3 t film f(T )

1 rfilm 1 ] ]] 3 rsub

G

21

(17)

Note that it is first necessary to measure the background loss of the bare paddle oscillator Q 21 sub in a separate experiment prior to depositing a film such that this background may be subtracted out via Eq. (17) to give Q 21 film .

3.1. a-B and a-B9 C

co

d Gsub (T ) ]]] Gfilm (T )

1 Gsub (T ) t sub 21 21 21 Q film (T ) 5 ] ]]] ]] Q co (T ) 1 Q sub (T ). 3 Gfilm (T ) t film

3. Experimental results

FS D G S D

1 t sub Gsub (T ) 5 ] ]] ]]] d 3 t film Gfilm (T ) film

(16)

Q sub is the internal friction of the bare paddle and Q co is the loss of the composite film / substrate structure. The increase in measured Q 21 after film deposition is attributed to dissipative processes in the film. The internal friction of the film itself is extracted by rearranging Eq. (16):

co

1 rfilm 1 ] ]] d Gsub (T ) 3 rsub 2 t sub Df(T ) 1 ] ]] ]] 3 T film f(T )

3Gfilm (T ) t film 21 21 21 Q co (T ) 5 ]]] ]] Q film (T ) 1 Q sub (T ). Gsub (T ) t sub

We first consider the data for a-B and a-B 9 C (Figs. 2–5) because these results provide a framework within (12)

The bracketed factor in the second term of Eq. (12) can be rewritten via Eq. (5) as:

F

S D

2 t sub Df(T ) ] ]] ]] 3 t film f(T )

1 rfilm 1 ] ]] 3 rsub co

G

Gfilm (T ) ; ]]]. Gsub (T )

(13)

Substituting Eq. (13) into Eq. (12) yields:

S

d v(T ) ]] vo

D

1 t sub Gsub (T ) 5 ] ]] ]]] d 3 t film Gfilm (T ) film

FS D G Df(T ) ]] f(T )

co

1 d Gsub (T ) 1 ] ]]]. 2 Gsub (T )

(14)

The last term in Eq. (14) is equal to (dv(T ) /vo ) sub . We therefore obtain the final expression for (dv(T ) /vo ) film :

S

d v(T ) ]] vo

D

1 t sub Gsub (T ) 5 ] ]] ]]] d 3 t film Gfilm (T ) film

S

d v(T ) 1 ]] vo

D

.

FS D G Df(T ) ]] f(T )

co

(15)

sub

Thus, the normalized sound speed variation of the film can be obtained via Eq. (15). We now consider extraction of the internal friction of the film from the double paddle oscillator raw data. Deposition of a film on the resonator increases the mechanical loss relative to that of the bare oscillator. The internal friction of the composite film / substrate structure is given by [34–36]:

Fig. 2. Internal friction Q 21 of a-B. Open triangles, background dissipation of the bare silicon double paddle oscillator. Solid triangles, raw data ˚ a-B film. Solid circles, result for bulk a-B after for oscillator with 2510 A background subtraction and scaling (see text). The well-defined internal friction plateau within 10 24 ,Q 21 ,10 23 is a characteristic signature of solids with glass-like vibrational properties. Van Cleve’s data for a-SiO 2 (Suprasil W, open circles) are included for comparison [24].

P. A. Medwick et al. / Journal of Alloys and Compounds 270 (1998) 1 – 15

Fig. 3. Internal friction Q 21 of a-B 9 C. Open triangles, background dissipation of the bare silicon double paddle oscillator. Solid triangles, ˚ a-B 9 C film on paddle oscillator. Solid circles, result raw data for 1760 A for bulk a-B 9 C after background subtraction and scaling (see text). The well-defined internal friction plateau within 10 24 ,Q 21 ,10 23 is a characteristic signature of solids with glass-like vibrational properties. Van Cleve’s data for a-SiO 2 (Suprasil W, open circles) are included for comparison [24].

Fig. 4. Relative variation in resonant frequency df /fo for paddle oscillators ˚ a-B film (open squares) and 1760 A ˚ a-B 9 C film (solid carrying 2510 A squares). Data are normalized to resonant frequency at the lowest experimental temperature (|70 mK).

7

Fig. 5. Relative speed of sound variation dv /vo for a-B (solid circles) derived from df /fo data via Eq. 15. The relative change in the speed of sound dv /vo exhibits an increase and maximum at low temperatures. This behavior is a characteristic signature of amorphous solids and is not observed for crystalline materials. Van Cleve’s data for a-SiO 2 (Suprasil W, open circles) are included for comparison [24]. The solid line is a tunneling model fit, assuming phonon-mediated relaxations dominate the TLS dynamics (see text). Dashed line is the analogous curve for a-SiO 2 .

which to interpret the data for crystalline c-B 12x C x . The background loss (i.e. internal friction of the uncoated silicon oscillator; Fig. 2, open triangles) is |4310 28 at the lowest temperatures and is comparable to the baseline observed by White and Pohl [34–36]. Note that theoretical calculations place the internal friction of highly ordered, chemically pure crystals, such as silicon, at Q 21 ,10 212 [38]. The discrepancy between theory and experiment is attributed mainly to the effect of the oscillator mounting, the so-called ‘‘clamping losses’’ or ‘‘background losses’’. This is an unavoidable limitation when measuring the internal friction of low-loss solids. The key point is that the background loss is low (compared to the internal friction of amorphous solids, Q 21 |10 24 –10 23 ) and reproducible [34–36], hence, it can be subtracted out from measurements of the coated oscillator via Eq. (17). Because the baseline varies only very slowly above |0.8 K, the magnitude of the background is extrapolated to be constant for 2 K,T ,20 K (dashed line, Figs. 2 and 3). The internal friction of the paddle carrying a film of a-B is shown in Fig. 2 as the solid triangles. Deposition of the ˚ a-B film on the bare paddle increases the Q 21 by 2510 A about a factor of twenty above the baseline. The raw dissipation data for the coated oscillator are subsequently scaled according to Eq. (17), to obtain the internal friction of the film itself (solid circles in Fig. 2); it lies within the range (10 24 ,Q 21 ,10 23 ) observed for all amorphous solids. Van Cleve’s data for a-SiO 2 (Suprasil W) at a similar frequency (4.5 kHz) are included for comparison

8

P. A. Medwick et al. / Journal of Alloys and Compounds 270 (1998) 1 – 15

(open circles) [24]; the dissipation for a-B lies only about 20–30% lower than a-SiO 2 in the plateau region (|0.3, T , |10 K). The internal friction of the paddle carrying a ˚ film of a-B 9 C is shown in Fig. 3. The damping of 1760 A the coated oscillator (solid triangles) lies about a factor of ten–twelve above the baseline (open triangles). Subtraction of the background loss and appropriate scaling via Eq. (17) yields the internal friction of the a-B 9 C film itself (solid circles); it also lies within the glass-like range (10 24 , Q 21 ,10 23 ). The level of loss for a-B 9 C is nearly indistinguishable from that for a-SiO 2 in the plateau. It is difficult to say with certainty whether or not the temperature dependence of Q 21 for the two boride films is the same as for a-SiO 2 , because the boride data only extend to about 70 mK. Furthermore, Van Cleve suspected some self-heating of his a-SiO 2 paddle oscillators at the lowest temperatures T , |20 mK [24]. The magnitude of the internal friction plateau seen in the a-B and a-B 9 C data support the conclusion that these films are indeed amorphous. Furthermore, we dismiss the possibility that the increase in Q 21 is due to boron diffusion into the silicon during or after deposition. The order of magnitude and temperature dependence of the loss expected for silicon doped with boron [33] are inconsistent with the results for a-B and a-B 9 C reported here. Raw data showing the relative variation in resonant frequency of the oscillator coated with the a-B film are shown in Fig. 4 (open squares); the maximum variation is |7310 27 . From the df /fo data, we calculate the speed of sound variation for a-B via Eq. (15); dv /vo is plotted as solid circles in Fig. 5. dv /vo increases at low temperatures, reaches a maximum, and subsequently decreases at higher temperatures; data for a-SiO 2 are included for comparison. This behavior is a characteristic signature of amorphous solids; crystalline solids show a dv /vo whose slope approaches zero as T →0 (see the data for crystalline b-B, quartz c-SiO 2 and c-B 4 C, shown in Fig. 8). The solid and dashed lines in Fig. 5 are fits to the phenomenological TM for amorphous solids (to be discussed below). Note that the data for both a-SiO 2 and a-B deviate from the model above the sound speed maximum; this has been previously reported for other amorphous solids and suggests deficiencies in the standard TM [39]. Figs. 4 and 6 show the corresponding resonant frequency and sound speed variation for the a-B 9 C experiment. The maximum df /fo excursion (|1.3310 26 ) is larger than for a-B. A logarithmic temperature dependence and maximum in dv /vo are observed, as expected for an amorphous solid. The high sensitivity of double paddle oscillators for detecting changes in internal friction caused by thin films is well-established; films of a-SiO 2 as thin as 0.75 nm have been detected in this way [34–36]. The measurements shown in Figs. 4–6 are the first ones in which dv /vo has been measured on thin films. This was possible because of the large shear moduli of a-B and a-B 9 C. The observed magnitude and temperature dependence of

Fig. 6. Relative speed of sound variation dv /vo for a-B 9 C (solid circles) derived from df /fo data via Eq. 15. The relative change in the speed of sound dv /vo exhibits an increase and maximum at low temperatures. This behavior is a characteristic signature of amorphous solids and is not observed for crystalline materials. Van Cleve’s data for a-SiO 2 (Suprasil W, open circles) are included for comparison [24]. The solid line is a tunneling model fit, assuming phonon-mediated relaxations dominate the TLS dynamics (see text). Dashed line is the analogous curve for a-SiO 2 .

the Q 21 and dv /vo data for a-B and a-B 9 C are as expected for amorphous solids. These data will be compared to data for crystalline c-B 12x C x which are discussed next.

3.2. c-B12 x Cx The internal friction of bulk polycrystalline c-B 12x C x is shown in Fig. 7. The internal friction of a-SiO 2 [21,24] and a-Pd 78 Si 16 Cu 6 [40] (vibrating reed measurements), shown for comparison, exhibit a plateau with a magnitude in the range 10 24 ,Q 21 ,10 23 below |20 K, which is characteristic for all amorphous solids. In dielectric glasses (e.g. a-SiO 2 ) below |1 K (at the frequencies characteristic of this work), the internal friction decreases and asymptotically approaches a T 3 dependence at the lowest temperatures. This is ascribed to the freezing out of phononmediated TLS relaxations in dielectric glasses [40]. In metallic glasses (e.g. a-Pd 78 Si 16 Cu 6 ), in which electronic carriers also participate in TLS relaxations, the internal friction is expected to remain independent of temperature down to the lowest temperatures measured here. Raychaudhuri and Hunklinger’s internal friction data for a-Pd 78 Si 16 Cu 6 are essentially constant down to 0.02 K; they estimated that Q 21 for a-Pd 78 Si 16 Cu 6 should not begin to roll-off until temperatures of the order 10 mK (at

P. A. Medwick et al. / Journal of Alloys and Compounds 270 (1998) 1 – 15

Fig. 7. Internal friction Q 21 of bulk c-B 12x C x . Solid stars, composite 21 oscillator background (Q co ) as measured for a quartz transducer (c-SiO 2 , 165 kHz). Open stars, composite oscillator Q 21 co for a sample of b-B (166 kHz); these data are also dominated by clamping losses and have therefore not been converted to Q 21 , the internal friction of b-B. The remaining data in the figure are the internal friction of the respective samples obtained from the composite oscillator Q 21 co according to Eq. 1 [21,24]. Solid triangles, x50.20 (c-B 4 C, 153 kHz). Open circles, x5 0.133 (c-B 13 C 2 , 159 kHz). Solid circles, x50.10 (c-B 9 C, 165 kHz). Open triangles below 1.5 K, a-SiO 2 measured at 160 kHz [24]. Dashed line above 1.5 K, a-SiO 2 measured at 86 kHz [21]. Solid line, a-Pd 78 Si 16 Cu 6 amorphous metal (vibrating reed measurements at 1030 Hz) [40]. For increasingly carbon-deficient c-B 12x C x , an internal friction plateau develops with a magnitude that is characteristic of glass-like solids. The absence of a roll-off in Q 21 for c-B 13 C 2 and c-B 9 C at the lowest temperatures, as observed for amorphous dielectrics such as a-SiO 2 , may indicate the presence of alternative TLS relaxation channels (e.g. carriermediated relaxations) in addition to phonon-mediated TLS relaxations. Note that electron-mediated TLS relaxations are believed to maintain the approximately constant level of mechanical loss in a-Pd 78 Si 16 Cu 6 down to the lowest temperatures shown.

frequencies |1 kHz) are reached when carrier-mediated TLS relaxations begin freezing out [40]. A similar temperature independence has also been observed in the normal phase of a-Pd 30 Zr 70 , even at 720 MHz in measurements extending to 0.1 K [41], and also in a-Cu 30 Zr 70 in the kHz range [41,42]. The background (or ‘‘clamping’’) loss in Fig. 7 for the composite oscillator at 165 kHz is shown by the solid stars (data for the bare quartz transducer mounted on its BeCu pedestal, labeled as c-SiO 2 ). Clamping losses also dominate the a-SiO 2 data above |100 K [21]. Topp and Cahill [43] have shown that the rapid rise in Q 21 above |100 K is caused by the increased dissipation in the transducer / pedestal epoxy bond at T .100 K. The background losses at high and low temperatures are smoothly connected by the data points for polycrystalline b-B (open stars). This

9

agreement is taken as evidence that the b-B data are also dominated by clamping losses. The Q 21 for c-B 12x C x in Fig. 7 is the internal friction of 21 each sample after converting the composite oscillator Q co to that of the sample according to Eq. (1) (see Section 2). The background was not subtracted in this case; this has a negligible effect on the data in the temperature range of interest, T <100 K. The internal friction of the carbonsaturated specimen (x50.20, c-B 4 C, r 52.37 g cm 23 , vt 58.67310 5 cm s 21 ) is given by the solid triangles. It exceeds the background by about a factor of two, and may reflect the small amount of microscopic disorder which exists in the material at this composition [7]. The internal friction of c-B 13 C 2 (x50.133, open circles, r 52.47 g 23 5 21 cm , vt 58.64310 cm s ) is more than a factor of ten higher than the c-B 4 C data; it approaches the behavior expected for amorphous solids at low temperatures. However, there are some dissimilarities; specifically, the peak at |6 K and the lack of a roll-off at the lowest temperatures. The most carbon-deficient specimen, c-B 9 C (x5 0.10, solid circles, r 52.46 g cm 23 , vt 57.86310 5 cm s 21 ), has an even higher Q 21 . Although the dissipation peak in the c-B 13 C 2 data is not visible in the c-B 9 C data, the failure of Q 21 to roll-off below |1 K is even more pronounced for c-B 9 C. Nonetheless, the trend is clear: As the carbon content decreases, the magnitude of the internal friction increases to levels that are typical for all amorphous solids. This observation supports the previous suggestion of a transition to glass-like properties based on high temperature (30 K,T ,300 K) thermal conductivity data [7]. The temperature dependence of Q 21 for the metallic glass a-Pd 78 Si 16 Cu 6 is very similar to that for c-B 9 C, although the magnitude of the loss for the latter is about a factor of four higher than for a-Pd 78 Si 16 Cu 6 , which is close to the lower limit observed for amorphous solids. The relative variation in the speed of sound for crystalline c-B 12x C x is plotted in Fig. 8, together with that for a-SiO 2 [24], b-B and a-Pd 78 Si 16 Cu 6 [40]. For all data, the reference value vo is the speed of sound at 70 mK. The data for crystalline c-B 4 C (T .1 K) show a temperature dependence similar to that for crystalline b-B. The slope of dv /vo for c-B 4 C and b-B approaches zero at the lowest temperatures, as expected for an ordered crystalline solid such as quartz (c-SiO 2 , solid stars). This observation is consistent with the crystal-like internal friction signature of c-B 4 C discussed above. In contrast, the dv /vo data for c-B 13 C 2 and c-B 9 C more closely resemble, in magnitude and temperature dependence, those for a-Pd 78 Si 16 Cu 6 amorphous metal than those for the amorphous dielectric. In fact, the slope of dv /vo for c-B 13 C 2 is nearly the same as for a-Pd 78 Si 16 Cu 6 below the maximum. For a-SiO 2 , the maximum value of the dv /vo variation is |2.4310 24 , whereas carbon-deficient c-B 12x C x (x50.1, x50.133) shows a less pronounced variation of ,1310 24 . The dv /vo data for c-B 9 C (solid circles) show a maximum at

10

P. A. Medwick et al. / Journal of Alloys and Compounds 270 (1998) 1 – 15

Fig. 8. Relative speed of sound variation dv /vo for bulk c-B 12x C x . Solid circles, c-B 9 C (165 kHz). Open circles, c-B 13 C 2 (159 kHz). Solid triangles, c-B 4 C (153 kHz). Open stars, b-B (166 kHz). Solid stars, quartz transducer (c-SiO 2 , 165 kHz). Open triangles, a-SiO 2 (Suprasil W) at 160 kHz as reported by Van Cleve [24]. Solid line, a-Pd 78 Si 16 Cu 6 amorphous metal (1030 Hz) as reported by Raychaudhuri and Hunklinger [40]. The data for c-B 9 C and c-B 13 C 2 exhibit the characteristic signature of an amorphous solid (compare to a-SiO 2 and a-Pd 78 Si 16 Cu 6 ), whereas data for c-B 4 C retain the crystalline signature d[dv /vo ] /dT →0 as T →0 (compare to c-SiO 2 and b-B).

|0.7–0.8 K, followed by a softening of the sample for T ,0.7–0.8 K. The temperature dependence of the softening levels off below 0.2 K; the slope of dv /vo appears to be approaching zero in that region; such an effect has not been observed for amorphous dielectrics, except in a-SiO 2 , albeit at much lower temperatures [44]. Care was taken to ensure that this effect was not a result of self-heating of the sample induced by its oscillations. The combined message of the Q 21 and dv /vo data is clear: c-B 4 C possesses a crystalline elastic signature; the elastic properties of carbon deficient (x,0.20) c-B 12x C x are more similar to those of amorphous dielectrics, although there are clear differences which are reminiscent of amorphous metals.

4. Discussion

4.1. Summary of tunneling model predictions for Q 21 and dv /vo The low temperature thermal, dielectric and elastic properties of amorphous solids can be described by the TM [8,9]. This phenomenological model posits the existence of

tunneling states or TLS, intrinsic to the disordered solid. A comprehensive understanding of their microscopic nature has not yet been attained; TLS are assumed to be associated with atoms or small groups of atoms that quantummechanically tunnel between two nearby local energy minima with respect to some configurational coordinate (e.g. in a-SiO 2 , this coordinate may be an angle, corresponding to a slight rotation of an SiO 4 tetrahedron between two positions). The shape of the potential is assumed to be an asymmetric double-well. As a result of the quantum tunneling, the lowest energy states of a given double-well are two closely spaced levels, hence the name ‘‘two-level system’’. Disorder is believed to give rise to a broad distribution of double-well energy splittings and ¯ is asymmetries. As a result, the spectral density of TLS, P, a constant. If the solid is perturbed by an elastic wave, for example, the shape of a given double-well is perturbed and the tunneling entity must relax to an eigenstate of the perturbed potential. The TLS relaxations are assumed to occur via interaction with phonons or charge carriers; the broad distribution in double-well energy splittings and asymmetries leads to a broad distribution of relaxation times. For T #10 K, the TM predicts the internal friction (in our torsion mode) to be constant and given by its ‘‘plateau’’ value: 2 p P¯ g t 21 Q plat 5 ] ]] 2 2 rvt

(18)

independent of the relaxation mechanism. Here, P¯ is the spectral density of TLS, gt is the coupling energy between TLS and transverse phonons, r the mass density and vt is the transverse sound speed. From the plateau value of Q 21 , we define the tunneling strength, C: P¯ g 2t 2 21 C 5 ] Q plat 5 ]] p r v 2t

(19)

¯ 2 From the experimental value for Q 21 plat , [Pg t ] can be calculated using the known values of the mass density and sound speed: 2 2 [P¯ g t ] ; r v t C.

(20)

At very low temperatures, the TM predicts a decrease in Q 21 ; the temperature dependence of this decrease depends upon the details of the TLS relaxations. To date, only phonon- and electron-mediated relaxations have been considered [40,41]. For recent discussions, see refs. [45– 47]. The TM also predicts a maximum in the speed of sound at low temperatures; the temperature at which the maximum occurs is denoted as T max . When phonon-mediated relaxations dominate, which is expected in dielectric glasses, the model predicts the variation to be logarithmic in temperature:

P. A. Medwick et al. / Journal of Alloys and Compounds 270 (1998) 1 – 15

S D

d v v(T ) 2 v T ] ; ]]]o 5 C ln ] vo vo To

(for T , T max ).

(21)

where vo is the sound speed at some reference temperature, T o . When carrier-mediated relaxations dominate, as could be the case in amorphous metals:

S D

d v v(T ) 2 v 1 T ] ; ]]]o 5 ] C ln ] vo vo 2 To

(for T , T max )

(22)

In the general case, when the disordered solid possesses mobile charge carriers at low temperatures, both relaxation channels will operate in parallel [41,46]. To avoid this complication, discussions to date have been concentrated on amorphous dielectrics or superconductors below their transition temperatures. But even for the latter, complications have been noted [42]. For T 4T max , the TM predicts that the sound speed will decrease logarithmically for both dielectric and metallic glasses:

S D

d v v(T ) 2 v 1 T ] ; ]]]o 5 ] C ln ] vo vo 2 To

(for T 4 T max ).

(23)

For dielectrics, the behavior of dv /vo at intermediate temperatures has to be evaluated numerically.

5. a-B and a-B 9 C The a-B and a-B 9 C internal friction data in Figs. 2 and 3 show the temperature-independent Q 21 plateau (in the range 10 24 ,Q 21 ,10 23 ) expected for amorphous solids. CQ 21 and [P¯ g 2t ] Q 21 are evaluated using the value of Q 21 plat at approximately 1.5 K (where the subscript Q 21 is used to designate their origin); results are found in Table 2. They are similar to those found for the archetypal amorphous solid a-SiO 2 . Figs. 5 and 6 show fits to the dv /vo data for a-B and a-B 9 C, assuming phonon-mediated processes dominate the TLS relaxations. Note that there is no a priori justification for assuming the TLS relaxations are domi-

11

nated by phonons; both phonons and electronic carriers may contribute to the dynamics, in principle. Without a priori knowledge of their relative contributions, we assume phonons dominate and examine the consequences. Values 2 of Cd v / v o and [P¯ g t ] d v / v o from the fits are also listed in Table 2 (where the subscript indicates quantities derived from the dv /vo data). The corresponding fit to a-SiO 2 in Figs. 5 and 6 is shown as the dashed line. For a-SiO 2 , as well as for a-B and a-B 9 C, deviations between model and experiment are observed for T .T max ; this has been previously shown to be typical of TM fits [39,48] and is believed to illustrate unexplained shortcomings of the model. For a-B 9 C, Cd v / v o is larger than CQ 21 by |20%. The magnitude of this discrepancy is comparable to that previously reported for a-SiO 2 [24] and, thus, seems acceptable. Note also that Berret and Meissner [49] reported TM fits for a survey of eighteen amorphous dielectrics. They found discrepancies between values of CQ 21 for longitudinal and transverse elastic waves to be typically 3–27%, although these discrepancies were even larger (up to a factor of two) for a-Se and PMMA (polymethylmethacrylate). For a-B, the ratio CQ 21 /Cd v / v o is |2. It is not known whether this result is attributable to the typical discrepancies between the TM and experiment, or whether alternative relaxation channels (e.g. charge carrier-mediated processes) are at work in a-B. Note that Raychaudhuri and Hunklinger [40] obtained a ratio of CQ 21 /Cd v / v o |1.09 for the amorphous metal aPd 78 Si 16 Cu 6 , assuming carrier-mediated relaxations dominate (see Table 2). The discrepancy observed for a-B may be the result of phonon- and carrier-mediated relaxation channels operating in parallel [41]. These results, as well as those of Van Cleve et al. [48], Berret and Meissner [49] and Classen et al. [39] underscore the fact that there remain substantial discrepancies between model and experiment. Despite its successes, the TM is probably far from complete. What we can conclude is that the Q 21 and dv /vo signatures of a-B and a-B 9 C are clearly those of amorphous solids. The magnitudes of the tunneling model

Table 2 TLS parameters derived from Q 21 and dv /vo data. Sample densities and speeds of sound as listed in Table 1 or in indicated references Material

Q 21 plat (10 24 )

CQ 2 1 (10 24 )

[Pg 2t ] Q 21 (10 8 erg cm 23 )

Cdv / v o (10 24 )

[Pgt 2 ] dv / v o (10 8 erg cm 23 )

CQ 21 /Cd v / v o

c-B 4 C c-B 13 C 2 c-B 9 C a-B a-B 9 C a-SiO 2 [24] Pd 78 Si 16 Cu 6 [40] YB 63 [51]

0.163 a 1.417 b 4.413 b 3.260 a 4.804 a 5.430 a 1.313 b 3.252 a

0.104 0.902 2.809 2.225 3.040 3.460 0.836 2.070

0.207 1.709 3.957 2.469 2.250 1.076 0.284 3.872

(N /A) 0.734 c

(N /A) 1.391

(N /A) 1.23

d

d

d

1.056 e 3.672 e 3.150 0.766 c 1.400

1.172 2.717 0.980 0.261 2.622

2.11 0.83 1.10 1.09 1.48

b 21 Notes: a Q 21 Q plat value at |70 mK. c Slope measured below dv /vo maximum. d Slope not well-defined below dv /vo maximum. plat value at |1.5 K. e Cd v / v o determined from full tunneling model fit to dv /vo data, assuming only phonon-mediated TLS relaxations (see text). (N /A), not applicable; Cd v / v o not defined for this sample.

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P. A. Medwick et al. / Journal of Alloys and Compounds 270 (1998) 1 – 15

parameter C5P¯ g 2t /r v 2t lie in the range found for all amorphous solids studied to date for both a-B and a-B 9 C.

5.2. Thermal conductivity estimates for c-B9 C and a-B

5.1. c-B12 x Cx

At low temperatures, the thermal conductivity, L, of all amorphous dielectric or superconducting metallic solids exhibits a magnitude of approximately 1–9310 24 W cm 21 K 21 at 1 K and a T 2 variation with temperature. Within the TM framework, L is expressed as:

The internal friction data for c-B 12x C x in Fig. 7 evolve towards a glass-like signature as the carbon content is decreased. For carbon-rich c-B 4 C, the internal friction signature is crystalline; the fact that Q 21 is elevated above the background for this sample may reflect the small amount of disorder known to exist in this material [7]. Carbon deficient c-B 13 C 2 and c-B 9 C both possess a nearly temperature-independent internal friction plateau with a magnitude close to that which is characteristic of amorphous solids. Values of CQ 21 and [P¯ g 2t ] Q 21 for these solids are calculated from Q 21 plat near the lowest experimental temperature (|70 mK) and are listed in Table 2. It is only at the lowest temperatures that the Q 21 plateau becomes well-defined. The qualitative similarity in the temperature dependence of the Q 21 data for c-B 9 C and a-Pd 78 Si 16 Cu 6 amorphous metal suggests that non-phonon-mediated processes may also contribute to relaxations in c-B 9 C and c-B 13 C 2 . The dv /vo data for c-B 4 C (see Fig. 8) are those expected for a crystal; data for c-SiO 2 (quartz) and b-B at similar frequencies are included for comparison. The slope of the dv /vo data for these crystals approaches zero at the lowest temperatures. In contrast, dv /vo for c-B 13 C 2 goes through a maximum and then decreases at the lowest temperatures as for an amorphous solid; the well-defined logarithmic variation of dv /vo below |0.5 K gives Cd v / v o 50.7343 10 24 for this sample (see Table 2). Note that this slope is nearly identical to that for the amorphous metal aPd 78 Si 16 Cu 6 (solid line). The resulting CQ 21 /Cd v / v o ratio for c-B 13 C 2 is |1.23. The dv /vo for c-B 9 C also shows the maximum expected for an amorphous solid, but does not exhibit a well-defined logarithmic variation below the maximum. Therefore, we are not justified in calculating Cd v / v o for this sample. Note that the decrease in dv /vo for c-B 9 C diminishes below |0.2 K; a similar anomaly was observed for Pd 0.775 Si 0.165 Cu 0.06 metallic glass at 0.96 GHz by Golding et al. [50] and has been discussed by Coppersmith and Golding [46]. In summary, the elastic data for c-B 12x C x do not fit as neatly into a TM interpretation as data for dielectrics such as a-SiO 2 previously reported, although the data show similarities with those reported for amorphous metals. Nevertheless, the following conclusions are justified. Based upon Q 21 and dv /vo data, the elastic signature of c-B 4 C is that of a crystal. The magnitude of the Q 21 signature for carbon-deficient c-B 1 2 x C x evolves into that expected for an amorphous solid as the carbon content is decreased. The lack of a robust decrease in Q 21 at the lowest temperatures indicates that non-phonon-mediated processes may contribute to the dynamics in c-B 13 C 2 and c-B 9 C.

S

D

r k 3B 1 g 2t 2 ] L 5 ]]2 ]] 2v 1 v T t 4p " P¯ g t2 g l2 l

(24)

where all quantities have been previously defined. Using the elastic data from this work, the low temperature thermal conductivity of c-B 9 C and a-B, as predicted by the TM, can be calculated for the first time. In evaluating Eq. (24), we use the value of P¯ g 2t derived from the Q 21 plateau for each sample (i.e. [P¯ g t2 ] Q 21 ). For vt , we use the transverse sound velocity measured on the polycrystalline sample and on the film, respectively. By using the approximations vl ¯1.65vt and g l2 ¯2.5g t2 as empirically observed by Berret and Meissner [49] for a collection of amorphous solids, Eq. (24) can be written as: 3

2

k B T 2.66 1 L 5 ]]2 ]] ]. 6p " C vt

(25)

where C is the tunneling strength as previously defined by Eq. (19). In Fig. 9, the calculated thermal conductivities are compared to measurements on a single crystal MB 68-d (M5 Y, Gd); MB 68-d was previously shown to have glass-like lattice vibrations [51]. The calculated conductivities for the amorphous boron-rich solids and for c-B 9 C lie within the 24 21 23 2 glass-like range L |(3310 W cm K )T at T #1 K, as expected. The calculated conductivities agree well with low temperature measurements for the crystalline icosahedral boride MB 68-d , which, in turn, agree well with a similar calculation for YB 63 [51]. We therefore conclude that there is no difference in the low energy excitations of these icosahedral borides whose unit cells contain between |12 (for c-B 12x C x ) and |1600 (for MB 68-d ) atoms and of a-B, which is believed to consist of boron icosahedra that are randomly bonded, thereby precluding any long-range order in the structure [52–54]

5.3. On the universality of the low energy excitations In this investigation of what we call the glass-like lattice vibrations of the disordered crystalline c-B 9 C, it has been stated repeatedly that all amorphous solids share some characteristic low temperature properties, specifically, a temperature-independent internal friction plateau (Q 21 plat ) with a magnitude ranging from |10 23 to |10 24 . In the 21 TM, Q plat 5(p / 2)C, where the tunneling strength is given by: P¯ g 2 C 5 ]] . rv2

(26)

P. A. Medwick et al. / Journal of Alloys and Compounds 270 (1998) 1 – 15

Fig. 9. Tunneling model (TM) prediction for the low temperature thermal conductivity of c-B 9 C, a-B and YB 63 ; data points for the glass-like crystalline icosahedral boride MB 68-d (‘‘M’’5Y, Gd) and b-B (dashed line) [57] are included for comparison. T 2 curves below 2 K are derived 2 using the values of [P¯ g t ] Q 21 obtained from the value of the internal friction in the plateau (for c-B 9 C and a-B, this work) and from ref. [51] (for YB 63 ). Solid line, a-B. Dash-dot line, c-B 9 C. Dotted line, YB 63 . Note: The solid and dotted lines below 2 K are indistinguishable. As expected from the elastic measurements, the calculated thermal conductivities lie within the L |(3310 24 W cm 21 K 23 ) range characteristic of all amorphous solids below |1 K. In addition, the TM predictions are in excellent agreement with thermal conductivity data for MB 68-d . Solid circles, GdB 62.5 [51]; open circles, GdB 66 [58]; open squares, YB 61.7 [59]; open triangles and open diamonds, two different samples of YB 66 [57]. The single data point (solid star at 335 K) is for a-B as reported by Talley et al. [60]. The minimum thermal conductivity, Lmin , is computed as described in ref. [51] and fits the measurements well at high temperatures (T .200 K). This figure demonstrates that the vibrational properties of these icosahedral borides are inherently glass-like at all temperatures.

For transverse waves, which we have concentrated on in this investigation, the denominator equals the shear modulus of the solid: G 5 r v 2t .

(27)

Since the shear moduli of a-B and a-B 9 C are relatively large, our measurements on these solids enable us to perform a test of this claim of universality over a wider range of G than before. For this test, we will draw on a representative collection of measurements either of internal friction or ultrasonic attenuation, or of low temperature thermal conductivity, from which C was derived with the help of Eq. (25), using the simplifying assumptions mentioned there. The uncertainty of extracting C from the data is estimated to be less than a factor of two; sample dependence is generally negligible. As shown in Fig. 10, the tunneling strength C shows no sign of a dependence on the shear modulus, with the latter

13

Fig. 10. Transverse tunneling strength C5P¯ g 2t /r vt 2 obtained from low temperature elastic (internal friction and ultrasonic attenuation; open circles) and thermal conductivity (solid circles) measurements. Materials and sources of information are as follows. SiO 2 , GeO 2 , CdGeA 2 , Se, As 2 S 3 , B 2 O 3 , PMMA (polymethylmethacrylate), PS (polystyrene) are all from ref. [43]. ‘‘9606’’: Corning code 9606 (glass ceramic ‘‘Pyroceram’’ measured in its amorphous form) from refs. [61,12]. ‘‘K, Ca, NO 3 ’’: KCa(NO 3 ) 3 from ref. [59] using sound speed from ref. [62]. ‘‘Mn, PO 4 ’’: (MnF 2 ) 0.65 (BaF 2 ) 0.2 (NaPO 4 ) 0.15 from ref. [63]. ‘‘B, Li’’: (B 2 O 3 ) 0.5 (Li 2 O) 0.5 from refs. [59,64]. ‘‘Zr, Pd’’: Zr 0.7 Pd 0.3 from ref. [65] using sound speed from ref. [66]. ‘‘V52’’: (ZrF 4 ) 0.575 (BaF 2 ) 0.3375 (ThF 4 ) 0.0875 from ref. [63]. ‘‘LiCl’’: LiCl17H 2 O from ref. [67]. ‘‘FS’’: amorphous feldspar of composition (NaAlSi 3 O 3 ) 0.5 (CaAl 2 Si 2 O 8 ) 0.5 from ref. [68] using mass density and sound speed from ref. [69]. ‘‘7740’’: borosilicate glass (Corning code 7740) of composition (in wt.%) SiO 2 (87%), B 2 O 3 (13%) from ref. [70] using sound speed from ref. [71]. ‘‘9754’’: Aluminogermanate glass (Corning code 9754) of composition (in wt.%) GeO 2 (50%), Al 2 O 3 (25%), CaO (15%), BaO (5%), ZnO (5%) from ref. [70] and sound speed from ref. [72]. ‘‘SC5’’ and ‘‘SC8’’: Scotchcast 5 and Scotchcast 8 from ref. [73]. ‘‘PB’’: polybutadiene from ref. [74]. ‘‘PET’’: polyethyleneterephthalate from ref. [75] using mass density and sound speed from ref. [76].

varying by over a factor of close to 70. In a previous review of acoustic data [49], the range of G had spanned only a factor of seventeen. While the scatter of the data clearly demonstrates some materials dependence on C, the data further strengthen the earlier conclusion that the quantity P¯ g 2t , which describes properties of the tunneling defects (two-level systems) within the TM, is indeed proportional to the shear modulus, a quantity which is determined by the host lattice. Previous attempts to explain this observation have assumed strong interactions amongst the tunneling defects (as reviewed in ref. [55]), although recent work on very thin amorphous films have failed to produce evidence for such interactions [36]. In light of the additional evidence of the universality of C, presented here, more theoretical work is clearly needed.

6. Conclusion We have measured the low temperature elastic properties of films of a-B and a-B 9 C using a double paddle oscillator technique [34]. These materials possess glasslike elastic signatures that are comparable to all amorphous solids reported to date. The tunneling model parameters

14

P. A. Medwick et al. / Journal of Alloys and Compounds 270 (1998) 1 – 15

extracted from the experimental data are consistent with those of other glasses. The low temperature (0.07,T ,10 K) internal friction and sound speed variation of polycrystalline c-B 12x C x show an evolution from crystalline to glass-like behavior for carbon-deficient (x,0.20) specimens. This evolution towards a glass-like signature supports the previously reported [7] similarities between c-B 12x C x and amorphous boron (a-B) observed in their high temperature (T .50 K) thermal transport. The elastic data can be interpreted using the tunneling model of amorphous solids [8,9]; the parameter P¯ g 2t /r v 2t extracted from the data lies within the typical range observed for all amorphous solids. A resemblance to amorphous metals is observed for c-B 13 C 2 and c-B 9 C, which suggests the presence of non-phonon-mediated relaxations of TLS in these solids. Details of those relaxations are unclear. Using the TLS parameters extracted from the elastic data, we have calculated the T 2 thermal conductivity expected for bulk c-B 9 C and a-B in the TLS dominated regime (T ,1 K). The calculated thermal conductivities fall within the range observed for all amorphous solids. Previous measurements [11] on polycrystalline c-B 9 C in this temperature range differ from the calculated value for bulk single-crystal B 9 C due to the scattering of phonons by grain boundaries; note that experimental data for bulk single-crystal B 9 C are not available. In the case of a-B, the theoretical thermal conductivity is that expected if this material could be produced in bulk form; whereas, to date, it exists only as a thin film. In addition, the calculated thermal conductivities show excellent agreement with existing data for single-crystal MB 68-d , which has been previously shown to exhibit glass-like excitations, thereby demonstrating the inherent glass-like character of the lattice vibrations of the icosahedral borides a-B, a-B 9 C, c-B 12x C x and c-MB 68-d . Finally, our results support the claim that the tunneling strength parameter C5P¯ g 2t /r v 2t shows no dependence upon the material’s shear modulus G5 r vt 2 . In previous reviews of acoustic data, the range of G over which glass-like properties had been found only spanned a factor of seventeen. The present work expands this range to nearly a factor of 70 in shear modulus, without revealing any deviation from the universality of those properties.

Acknowledgements The authors wish to thank Dr. P. Wagner (Wacker Siltronic) for supplying the silicon wafers used in this investigation, Dr. T. Aselage (Sandia National Laboratories) for the samples of bulk boron carbides and stimulating discussions, Mr. G. Schmidt of the Materials Science Center (MSC) at Cornell for depositions of some boride films, Dr. M. Weathers (MSC) for x-ray analyses, Dr. P.

Revesz (MSC) for the RBS analyses, Mr. B. Addis (MSC) for electro-disintegration cutting of the bulk boride specimens, and Dr. D. Emin (Sandia) for helpful discussions. We also thank Dr. J.E. Van Cleve for permission to publish his data on the internal friction and speed of sound of a-SiO 2 . This work was supported by the United States National Science Foundation, Grant DMR-91-15961 and by the Cornell University Materials Science Center through the use of its central facilities. The use of the National Nanofabrication Facility at Cornell University is also acknowledged. Medwick acknowledges financial support from the International Business Machines Corporation as an IBM Fellow in the Physical Sciences, the U.S. Department of Education as a Graduate Fellow for National Needs in Materials Physics, and the Fannie and John Hertz Foundation. White acknowledges the U.S. Department of Education Graduate Fellowship Program in Materials Science, grant no. P200A10148.

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