Non-Debye excess heat capacity and boson peak of

Journal of Non-Crystalline Solids 357 (2011) 534–537

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Journal of Non-Crystalline Solids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j n o n c r y s o l

Non-Debye excess heat capacity and boson peak of binary lithium borate glasses Yu Matsuda a,⁎, Hitoshi Kawaji b, Tooru Atake b, Yasuhisa Yamamura a, Shuma Yasuzuka a, Kazuya Saito a, Seiji Kojima a a b

Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan Materials and Structures Laboratory, Tokyo Institute of Technology, Yokohama, Kanagawa 226-8503, Japan

a r t i c l e

i n f o

Keywords: Alkali borate glasses; Low-temperature heat capacities; Specific heat; Boson peak; Low-energy excitation; Vibrational density of states; Relaxation calorimetry; Non-Debye

a b s t r a c t The non-Debye excess heat capacities of binary lithium borate glasses with different Li2O compositions of x = 8, 14 and 22 (mol%) are investigated to understand origin of the boson peak. The low-temperature heat capacities are measured between 2 and 50 K by a relaxation calorimeter. The experimental non-Debye heat capacities with x = 14 is successfully reproduced using the excess vibrational density of states measured by inelastic neutron scattering. This finding indicates that the non-Debye heat capacities of lithium borate glasses originate from the excess vibrational density of states measureable by inelastic neutron scattering. Moreover, it is demonstrated that all of the excess heat capacity spectra lie on a single master curve by the scaling using boson peak temperature and intensity. © 2010 Elsevier B.V. All rights reserved.

1. Introduction The structures of glasses are random without any translational symmetry, and the lack of long-range order in the glassy network leads to physical properties markedly different from those of crystalline solids [1–3]. One of the significant differences is in the low-temperature heat capacity (Cp) [4]. Although Cp for ideal crystals at low-temperatures is well described by the Debye T 3 law as an elastic continuum approximation, that of glass presented in a Cp/T 3–T plot shows a broad maximum centered at approximately 5–10 K above the Debye expectation, known as the “non-Debye excess Cp”. The origin of such a high Cp for glasses has been discussed in connection with the low-energy excess vibrational density of states (VDoS), known as the “boson peak”, which is usually observed by inelastic neutron scattering (INS) or by Raman scattering. Despite considerable effort [1–4], the origins of the non-Debye excess Cp and boson peak still remain serious open questions in condensed-matter physics and materials science. From the experimental viewpoint, the non-Debye Cp of the alkalimetal borate glass family except for lithium borate glasses (LiB) has been studied. Results for pure B2O3 glass [5–7], sodium borate glasses (NaB) [8] and cesium borate glasses (CsB) [9] have been reported. Moreover, the alkali-metal dependence for a specific composition (x = 14) [10] has recently been published. In particular, in Ref. [9] the non-Debye Cp of CsB was discussed by comparison with that of NaB,

⁎ Corresponding author. Present address: Glass Research Center, Central Glass Co. Ltd., Matsusaka, Mie 515-0001, Japan. E-mail addresses: [email protected] (Y. Matsuda), [email protected] (S. Kojima). 0022-3093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2010.06.066

and different behaviors were revealed. The hump in the excess Cp of NaB shifted to a higher temperature, and the hump itself became much less intense with increasing Na2O. It was proposed that the excess Cp of NaB is related to the propagating nature of the correlated excess VDoS, which are acoustic-like or strongly coupled with the acoustic phonons of borate network structures. In contrast, it has been revealed that the position of the hump for CsB does not change with the composition, indicating a localized nature for the vibrations resulting in the excess Cp of CsB. The intensity of the hump of CsB exhibits an anomalous maximum in the Cs2O composition dependence. To obtain deeper insight into the origins of the non-Debye excess Cp and the boson peak of the alkali-metal borate glass family, it is highly desirable to first study the composition dependence of the nonDebye Cp of binary LiB as a fundamental system of the borate family. However, there are almost no reports of detailed studies on the lowtemperature Cp of LiB with reliable absolute values. In addition to the experimental viewpoint mentioned above, the LiB system itself is of considerable interest. It has been revealed that the physical properties of LiB such as density [11], sound velocity [11,12], stretched exponentiality [13], and Raman [14] and INS spectra [15] drastically vary with the composition, in contrast with CsB, and they show maxima or minima in their composition dependences owing to changes in the intermediate structure units. The Angell's fragility of LiB also markedly varies from strong to fragile [16], indicating that the LiB system can cover a wide variety of glass-forming materials from the viewpoint of strong-fragile classification. In this study, we investigate the non-Debye Cp at low-temperatures and the boson peak of the LiB system as a function of Li2O composition. The main purpose of this paper is to present the composition dependences of the low-temperature Cp of LiB as absolute

Y. Matsuda et al. / Journal of Non-Crystalline Solids 357 (2011) 534–537

0.7

Reduced Molar heat capacity Cp /T 3 (mJ K-4 mol-1)

values. Such reliable information is desirable for the development of a model for the boson peak and to verify its reasonability. In this paper, we first demonstrate that the non-Debye Cp of LiB can be reproduced by the excess VDoS obtained by INS. Secondly, we discuss the composition dependence of the non-Debye Cp in analogy with the boson peak. Finally, we present the scaling behavior of the spectral shape of the non-Debye Cp.

2. Experimental The chemical formula of LiB is denoted by xLi2O·(100-x)B2O3, where x is the Li2O composition in mol%. The LiB samples with x = 8, 14, 22 with high homogeneity were prepared by the solution method. A detailed description of the method of the sample preparation is given in Ref. [11]. B and Li in the samples were isotopically enriched by 11B (N99 atom%) and 7Li (N97 atom%), respectively, to avoid neutron absorption by 10B and 6Li. The samples were previously used for INS experiments by a direct geometry chopper-type ToF spectrometer, AGNES, belonging to the Institute for Solid State Physics, University of Tokyo [15]. The same samples were used for the heat capacity measurements to ensure experimental consistency. The heat capacities of all of the samples were measured by a relaxation calorimeter (Physical Properties Measurement System (PPMS)) with heat capacity option in the temperature range between 2 and 50 K [17,18]. The sample was bonded to the calorimeter by Apiezon N grease. The heat capacities of the selected sample with x = 14 were also measured by an adiabatic calorimeter developed by Saito laboratory, University of Tsukuba, in the temperature range between 14.5 and 50 K. A detailed description of the adiabatic calorimeter is given in Ref. [19].

3. Results

Molar heat capacity, Cp (J K-1 mol-1)

Fig. 1 shows the temperature dependences of the molar heat capacity Cp(T) for the different compositions, where the open symbols denote the values measured by the PPMS and the solid stars denote those of x = 14 measured by the adiabatic calorimeter. The error in the present data presented in Figs. 1 and 2 are smaller than the size of the data points. It should be emphasized that the absolute values observed by the present PPMS are in excellent agreement with those obtained by the adiabatic calorimeter. The results indicate the reliability of the absolute values determined in this work.

x = 8 (PPMS) x = 14 (PPMS) x = 22 (PPMS) x = 14 (Adiabatic)

8 7

x=8 x = 14 x = 22

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0

5

10

15

20

25

30

35

40

45

Temperature, T (K) Fig. 2. Temperature dependence of non-Debye excess Cp presented in a Cp/T 3–T plot. Solid symbols denote the Debye contributions at the zero-temperature limit.

To clarify the non-Debye contributions, the Cp(T) is illustrated in Cp/T 3–T plots in Fig. 2. The Debye contributions at the zerotemperature limit [20], calculated by 234NA kB CDebye ≈ m

T θ Debye

!3 ð J = g = KÞ;

ð1Þ

are also plotted as the solid symbols, where NA is the Avogadro constant, kB is the Boltzmann constant, m is the mean atomic mass in the chemical formula unit, and θDebye is the Debye temperature, which was calculated using mass density ρ [11] and the transverse VT and longitudinal VL sound velocities [11,12]. The values of ρ, VT, VL and θDebye used in the calculations are listed in Table 1. A broad asymmetric hump can be clearly observed for all the samples in Fig. 2. This is evidence of the non-Debye excess Cp(T) in comparison with the Debye contributions. The composition dependence of this hump will be discussed in the next section. At temperatures below approximately 3 K, linear contributions can be observed. These have been discussed on the basis of the Phillips and Anderson two-level tunneling system [4]. Although the microscopic origin of the tunneling objects remains an open question, this is anyway beyond the scope of the present paper. 4. Discussion We first demonstrate that the non-Debye excess Cp(T) of LiB can be reproduced by the excess VDoS measured by INS. In general, the vibrational Cp is given by the knowledge of the harmonic oscillator heat capacity and phonon VDoS, and hence it is important to determine whether or not the non-Debye excess Cp(T) originates from the boson peak observable in the INS experiments. Recently, we reported the raw dynamic structure factor S(Q ave, ω) of LiB with the different compositions [15], where Q ave is the mean momentum transfer (1.7 Å− 1) and ω is the energy transfer in cm− 1. S

10 9

535

6 5 4 3 2 1 0 0

5

10

15

20

25

30

35

40

45

Temperature, T (K)

Table 1 Physical properties of lithium borate glasses. The values of ρ, VL and VT are determined by interpolation of the reported data (see text). x (mol%)

Fig. 1. Temperature dependences of raw heat capacity of isotope-substituted lithium borate glasses, xLi2O·(100-x)B2O3. Open symbols denote the results measured by the relaxation calorimeter (PPMS) and solid stars denote the results obtained by the adiabatic calorimeter.

8 14 22

VL (m/s) 3

4.43 ⁎ 10 5.16 ⁎ 103 6.05 ⁎ 103

VT (m/s) 3

2.47 ⁎ 10 2.88 ⁎ 103 3.38 ⁎ 103

ρ (*106 g/m3)

θ Debye (K)

1.95 2.04 2.14

3.6 ⁎ 102 4.3 ⁎ 102 5.1 ⁎ 102


SðQ ave ; ωÞ ; g ðωÞ = α⋅ ω⋅nðω; T Þ

ð2Þ

where n(ω, T) is the Bose population factor for the energy-gain side, and α is the scaling constant relating S(Q ave, ω), measured in an arbitrary scale by INS, to the absolute values of the density of states per unit energy and per chemical formula unit. Then, Cp(T) at a given temperature can be calculated by ∞

CP ≅ CV = 3ZNA kB ∫ o

ℏω kB T

2

expðℏω = kB T Þ g ðωÞdω; ½expðℏω = kB T Þ−12

f ðωÞ =

2B A ⋅ ⋅ π 4ω + A2

ð4Þ

Then, the three unknown parameters (α, A, B) were determined by a fitting procedure, which was repeated until the reasonable values were obtained. Fig. 3 shows the final results of the calculations; in Fig. 3(a) the open circles denote the raw S(Q ave, ω) of x = 14 in an arbitrary unit, the solid line denotes the fitted Lorenz function and the solid circles denote the inelastic components; in Fig. 3(b) the open triangles denote the experimental results obtained by PPMS and the solid squares denote the calculated Cp(T). The general trend is well reproduced, and the values of Cp(T) calculated from S(Q ave, ω) are in good agreement with the experimental values. This result provides direct evidence for the relationship between the non-Debye excess Cp of LiB and the boson peak. In other words, the non-Debye Cp of LiB originates from the excess VDoS, which are measurable by INS. Such direct evidence was achieved by taking advantage of the availability of both inelastic neutron and heat capacity measurements. It should be noted that, to conclude that the non-Debye Cp of the entire alkali-metal borate glass family originates from the VDoS measurable by INS, further experiments may be desirable for the entire alkali-metal borate glasses, because it has been pointed out that the non-Debye Cp of CsB cannot be reproduced by the VDoS observed by INS [10]. Secondly, we discuss the compositional variation of the non-Debye Cp(T). As can be seen in Fig. 2, the peak temperatures increase the higher with increasing x, while the magnitudes of the excess Cp(T) become much less intense with increasing x. These general trends are very similar to the trends of the boson peaks obtained by Raman scattering [14,15] and INS [15], both of which relates to the excess VDoS, while the Raman intensity is weighted by the light-vibration coupling constant. Such similar behaviors are reasonable, because the non-Debye Cp(T) results from the excess VDoS, as has been shown above. A detailed discussion of the composition dependences of the boson peaks observed by INS for the glass structures is given in Ref. [15], which can also account for the present results of Cp(T)/T 3 as described below. Pure B2O3 glass is constructed from random networks of a sixmembered boroxol ring consisting of BØ3 triangular planar units, in which boron atoms are 3-coordinated with 3-bridging oxygens (“Ø” denotes a bridging oxygen atom) (see Ref. [13] and therein). It has been revealed that the BP of pure B2O3 glass is related to some

(a)

80

Raw S(Qave, E) Lorenz function Inelastic component A = 0.95 B = 0.00023

60 40 20 0

ð3Þ

where Z is the number of atoms per chemical formula unit [6,10,20]. Cp(T) for the selected composition (x = 14) as a representative example was calculated as follows. S(Q ave, ω) obtained by INS consists of the components of elastic, quasi-elastic and inelastic scattering, and thus, the inelastic component is masked by the tails of the elastic and quasi-elastic contributions. The sum of the elastic and quasi-elastic components was assumed to be fitted by the Lorentz function f(ω), which has two adjustable parameters A and B:

14Li2O 86B2O3

*10-6

Raw dynamic structure factor S(Qave, E) (arb. unit)

(Q ave, ω) can be converted into the generalized (neutron weighted) VDoS g(ω) on the basis of the incoherent approximation [21], by

0

4

8

12

16

20

Energy transfer, E (meV) 0.4

Reduced molar heat capacity Cp/T 3 (mJ/mol/K4)

536

(b)

Experiments Calculated values

0.3

Scaling factor = 1.68

0.2

0.1

0

10

20

30

40

Temperature, T (K) Fig. 3. (a) Dynamic structure factor measured by INS in AGNES, which is proportional to the VDoS. Open circles denote raw values. Solid circles denote the calculated inelastic component. The solid line denotes the sum of the elastic and quasi-elastic components represented by the Lorenz function. See text for details. (b) Comparison of the calculated and experimental Cp. Solid squares denote the values of Cp calculated using the inelastic components in the INS dynamic structure factor. Open triangles denote the experimental results obtained by PPMS.

types of the vibrational or librational motion of the planar boroxol rings [22,23]. The planar boroxol rings may contain a large void, which gives rise to an “excess” degree of freedom of motion. This explanation is consistent with the results for the permanently densified SiO2 [24] and of a recent MD simulation using two-parameter model, where the low-density defective structures are assigned to be the origin of the boson peak [25]. The addition of Li2O to B2O3 glass induces a change in the coordination number of the boron atom from 3 to 4, and thus some of the planar BØ3 units convert into three-dimensional tetrahedral BØ4 units. The further addition of Li2O causes further conversion from BØ3 to BØ4 units, at least in the composition range studied. A tetrahedral BØ4 unit has much more efficient packing in space than the boroxol ring, as has been shown in Ref. [13]. The dramatic increase in the network connectivity by the conversion of the coordination number of the boron atom with increasing x results in a decrease in the excess VDoS originating from the boroxol rings. Therefore, the non-Debye excess Cp(T) of LiB at low-temperatures becomes less intense with increasing x. The above explanation accounts for the composition dependences for LiB and NaB, and may not be applicable to the CsB, as has been pointed out by Crupi et al. [9], where it was suggested that the much larger ionic radius of the Cs cation than the Li cation contributes to the localized vibrations, which may be predominant in the boson peak of CsB. Finally, we discuss the universal spectral shape of the boson peak of LiB. The spectral shape of the boson peak shall be determined by the distribution function of the excess VDoS. An important open question with regard to the spectral shape is whether or not the universal distribution exits. Very recently, the spectral shape and the scaling behavior of the pressure or temperature dependence of the boson


Cp(T) / CT=0 (T) Debye

3.5

(a)

3.0

density of states (VDoS) measured by inelastic neutron scattering (INS). This finding indicates that the non-Debye Cp of lithium borate glasses originates from the excess VDoS measurable by INS. Moreover, it is revealed that the non-Debye Cp becomes much less intense with increasing x and that the raw spectral shape drastically changes. In contrast to the dramatic changes in the raw spectra, it is demonstrated that the non-Debye excess Cp spectra lie on a single master curve using the temperature and the maximum value of the Cp/T3–T curve, rather than the Debye contributions.

x=8 x = 14 x = 22

2.5 2.0 1.5 1.0 0.5 0.00

0.02

0.04

0.06

0.08

0.10

T / θ Debye

CpT -3 / (CpT -3)max

1.4

Acknowledgements We wish to thank Professor Masao Kodama (Department of Applied Chemistry, Sojo Univ.) for helpful discussion about the sample preparation and characterization, and Mr. Yu-ta Suzuki (Graduate School of Pure and Applied Sciences, Univ. of Tsukuba) for helpful support in the heat capacity measurement by the adiabatic calorimeter. YM thanks JSPS for support in the form of a Fellowship for Young Scientists (DC1, 19·574). This research was partially supported by Collaborative Research Projects of the Materials and Structures Laboratory, Tokyo Institute of Technology, 2008–2010.

(b)

1.2

537

1.0 0.8 0.6 0.4 0.2

References

0

1

2

3

4

T /T max Fig. 4. Scaling for non-Debye excess Cp. (a) Scaling by the Debye Cp at the zerotemperature limit and the Debye temperature. (b) Scaling by the maximum values of the Cp/T 3–T curve.

peak have been hotly debated and controversial results have been presented [26–28]. One of the issues is whether or not the spectral shape of the boson peak can be scaled by the Debye level. In Fig. 4, the two types of scaled non-Debye Cp(T) in the Cp/T 3–T plot of LiB are presented. In Fig. 4(a) each curve with different x was scaled by the Debye Cp at the zerotemperature limit and Debye temperature, both of which were calculated by Eq. (1), while in Fig. 4(b) the scaling was performed using each temperature and the maximum value of the Cp/T 3 peak. As demonstrated in both figures, the non-Debye excess Cp spectra of LiB were not able to be scaled by only the Debye levels, but they were able to be scaled by the maximum values. For the case shown in Fig. 4(b), it was found that despite the marked changes in the raw spectra, all scaled spectra remain surprisingly similar at all compositions and lie on a single master curve. This finding indicates that the mechanism by which excess VDoS are distributed around the boson peak is the same for LiB with different compositions. This universal scaling has also been found in the Raman and INS spectra for LiB [15]. Recently, it was demonstrated that the boson peak spectra of alkali silicate glasses also lie on a single master curve [29]. The results for both lithium borate glasses in the present paper and alkali silicate glasses [29] support the idea that the distribution of the excess VDoS governing the BP shall be explained by a universal physical scenario [30]. 5. Conclusion The non-Debye excess heat capacities Cp of binary lithium borate glasses (LiB) with different Li2O compositions of x = 8, 14 and 22 (mol%) have been measured by a relaxation calorimeter, PPMS. The absolute values of Cp for x = 14 are in good agreement with the values determined by an adiabatic calorimeter. Moreover, the non-Debye Cp for x = 14 is successfully reproduced using the excess vibrational

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