The Boson Peak and its Relation with Acoustic Attenuation in Glasses
aa r X i v : . [ c ond - m a t . d i s - nn ] N ov The Boson Peak and its Relation with Acoustic Attenuation in Glasses
B. Ruffl´e , D.A. Parshin , , E. Courtens , and R. Vacher Laboratoire des Collo¨ıdes, Verres et Nanomat´eriaux, UMR 5587 CNRSUniversit´e Montpellier II, F-34095 Montpellier Cedex 5, France Saint Petersburg State Technical University, 195251 Saint Petersburg, Russia (Dated: Received 11 july 2007; revised 31 august 2007)Experimental results on the density of states and on the acoustic modes of glasses in the THzregion are compared to the predictions of two categories of models. A recent one, solely based onan elastic instability, does not account for most observations. Good agreement without adjustableparameters is obtained with models including the existence of non-acoustic vibrational modes atTHz frequency, providing in many cases a comprehensive picture for a range of glass anomalies.
PACS numbers: 63.50.+x, 78.35.+c, 81.05.Kf
The boson peak is an excess in the vibrational densityof states (VDOS), g ( ω ), observed in many glasses at fre-quencies ω/ π of the order of one THz. It appears as ahump in g ( ω ) /ω vs. ω , above the acoustic Debye level g D ( ω ) /ω . It is typically located at Ω BP ∼ . × ω D ,where ω D is the Debye frequency. This excess producesthe well-known specific heat anomaly of glasses at tem-peratures T ≃ ~ Ω BP / k B ∼
10 K [1]. It is generallyagreed that the boson peak is a vibrational signature ofthe disordered structure of glasses beyond the nanometerscale. Its correct understanding is thus of considerableimportance. Two main categories of dynamical boson-peak models currently exist. The first, which we call harmonic random matrix (HRM) models, is based on theconcept that purely harmonic elastic disorder producesan excess of low frequency modes. The alternate pictureis that there exist in glasses additional –non-acoustic– quasi-local vibrations (QLVs) at low frequencies [2, 3].The purpose of the present Letter is to compare HRMand QLV models to actual experimental results. We findthat QLVs, using only independently determined param-eters, mostly provide a much better agreement betweentheory and experiment than HRMs can do.The HRM models postulate randomly fluctuatingspring constants K ij , as e.g. in [4, 5, 6, 7, 8]. Theseassume a distribution p ( K ), generally bounded betweentwo values K min < K max . A truncated gaussian distri-bution centered at K > K min < − K min is above a threshold. Just before this instability, alow frequency excess in the VDOS appears as precursor,which is interpreted as the boson peak [4]. In [5], a squaredistribution p ( K ) is used instead. A negative K min is notnecessary if p ( K ) ∝ /K over the interval { K min , K max } as shown in [6]. Such a distribution can be rationalizedby free-volume considerations. A similar distribution, in-cluding some negative force constants, was used in [7] andhandled with so-called Euclidean Random Matrices. Theshape and position of the predicted boson peaks are notuniversal as they depend on the selected K -distribution.While earlier treatments were based on microscopic har- monic models, more recently macroscopic tensorial elas-tic approaches were developed [8, 9]. In [9], the gaussiandisorder affects only the shear modulus which tends tozero at critical disorder. This now allows performing re-alistic comparisons with experimental results, and it willbe used below. In all these models, the boson peak isproduced by shifting the maximum in g ( ω ) /ω to low ω owing to a softening of the elastic constant distribution.On the other hand, inelastic neutron-scattering mea-surements of the dynamic structure factor of silicashowed unambiguously that its boson-peak modes arenot sound waves [10], providing an early justification forthe QLV hypothesis. That the boson peak of this net-work glass relates to librations of structural units hassince been confirmed by hyper-Raman scattering [11],and similarly for boron oxide [12]. Additional manifesta-tions of excess excitations are the thermal and acousticanomalies observed below liquid-He temperatures thatare described by two-level systems [13]. A theory en-compassing these and QLVs is the soft potential model[3, 14]. The latter predicts an onset in the excess VDOS, g V = g − g D , with g V /ω ∝ ω [14], evolving into aboson peak when the total number of QLVs is appro-priately limited [15]. This is now further understoodon the basis of a physical model that considers the me-chanical instability of an initially uniform distribution ofquasi-harmonic QLVs of density g ( ω ), interacting viatheir common strain field [16, 17]. The authors assumethat g ( ω ) has a high cut-off, at ω ∼ ω D . The inter-action being weak, it only destabilizes those oscillatorsof relatively low frequencies, below ω c ≪ ω [18]. Thedestabilized oscillators, restricted by their anharmonic-ity, form a renormalized density g c ( ω ) ∝ ω up to ω c [16].This result is independent from the initial g ( ω ) as wellas from the size of the anharmonicity. A last but cru-cial step is that, owing to anharmonicity, the displacedoscillators produce random static forces affecting othernearby oscillators. This creates a soft gap, g V ( ω ) ∝ ω ,up to ω b < ω c . The reader is referred to [16, 17] fordetails. This model thus leads to a peak in g V ( ω ) /ω ,with an onset in ω up to ≃ ω b , and a decay in ≈ ω − x g ( E ) / ( E ) ( m e V - ) v -SiO T = 51 K Ω BP / ω D = 0.10 ε = 0.011 Debye
Energy Transfer E (meV) Glycerol T = 120 K Ω BP / ω D = 0.15 ε = 0.052 Debye
FIG. 1: The measured boson peaks of v -SiO [21] and glycerol[22] compared to the Debye levels (dashed horizontal lines),to the best prediction of the HRM model [9] (dotted lines),and to the onset calculated using the soft-potential model,Eq. (1a) (thin solid lines) and Eqs. (1a+b) (thicker solidlines). (color online) between ω b and ω c . While the onset power is univer-sal, the decay above ω b depends on the separation be-tween ω b and ω c . For small separation, the decay is be-tween ω − and ω − [17], in good agreement with obser-vation [19]. Contrary to HRMs, the QLV boson peak is not a shifted down end of acoustic branches. The highfrequency VDOS is but slightly modified by the succes-sive interactions. Complemented with the soft-potentialmodel, the QLV model makes specific predictions con-cerning high-frequency acoustic modes and their Ioffe-Regel limit beyond which they cease existing as planewaves [17, 20]. Those will be checked below.First consider the VDOS in the boson peak region.High quality data are available on many glasses, in par-ticular from neutron scattering. For example, the exper-imental results, normalized to unity and divided by E ,where E = ~ ω , are shown in Fig. 1 for the strong networkglass silica, v -SiO [21], and for the intermediate molec-ular one glycerol [22]. The corresponding Debye levels, g D ( E ) /E , are shown by dashed lines. These are calcu-lated from the known velocities of longitudinal (LA) andtransverse (TA) acoustic modes, v L and v T respectively.The boson peak of v -SiO , at E BP = ~ Ω BP ≃ g V /g D ∼ ∼ .
2. The dottedlines in Fig. 1 illustrate for comparison the predictionsof the HRM model, Eq. (2) of [9]. These are determinediteratively using the experimental values of the sound ve-locities and of Ω BP /ω D to extract the appropriate barevelocities and separation parameter ε of [9]. It is obviousfrom Fig. 1 that the elastic instability model falls shortof reproducing the observed peak strength. This alreadysuggests that additional modes should be involved. Forfurther comparison, the prediction of soft potentials iscalculated below E BP , in the region where the growth of g V /E is in E , using [15, 19] g V ( E ) /E = P s
24 1 W (cid:18) EW (cid:19) . (1 a )Here P s is the density of additional modes around the eigenvalue zero, calculated per atom, and W isthe crossover energy between vibrational and tunnelingstates. These parameters are obtained from thermody-namical data around liquid-helium temperature ( T ), andare available, both for silica [23] and for glycerol [24]. Us-ing (1a), and adding the Debye contribution g D ( E ) /E ,one obtains the thin lines shown in Fig. 1. It is wellknown that there exist, in addition to the VDOS, quasi-elastic contributions to the scattering signal arising fromrelaxations in double-well potentials. This is describedwith the same soft potential parameters [19], g rel ( E, T ) /E ≈ P s EW (cid:18) k B TW (cid:19) / , (1 b )where k B T is the thermal energy. Adding g rel to g V , aquantitative agreement is obtained in the onset region be-low E BP , shown by thick lines in Fig. 1. This is not a triv-ial result, as the parameters P s and W entering Eqs.(1)are derived from independent very low T measurements.It emphasizes the self-consistency of QLVs and two-levelsystems embodied in the soft potential model. It is anadditional indication that the excess indeed arises fromQLVs. The solid lines in Fig. 1 do not account for thesaturation in the growth of g ( E ) /E near E BP . Thiscould be easily included, as shown e.g. in [19]. Accordingto [15, 17], it requires at least one additional parameterwhich is beyond the purpose of the present discussion.A second important information is the linewidth ofacoustic modes of very high frequency Ω / π , which nearthe THz range should increase as Γ ∝ Ω , both in theHRM and QLV models. In HRMs, the acoustic widtharises from elastic Rayleigh scattering by disorder. Withsoft potentials, it is the resonant absorption of sound byexcess modes that gives the leading contribution [23], ~ Γ = π C L E (cid:18) EW (cid:19) . (2)Here, Γ is the full width of LA modes relating to the energy mean free path ℓ = v L / Γ, E is the phonon en-ergy ~ Ω, and C L is the tunneling strength for LA modesthat can be obtained from the height of the acoustic at-tenuation plateau near liquid-He T . This strong growth,Γ ∝ E , leads to a rapid end of acoustic plane wavesas E increases. This occurs at a Ioffe-Regel crossover E IR = ~ Ω IR where ℓ decreases down to half the wave-length, i.e. for Γ = Γ IR = Ω IR /π [25]. This crossoverhappens to fall at frequencies and wavevectors well abovethose reached in optical or UV Brillouin scattering, butmost often near the lowest values reachable with inelasticx-ray scattering. The energy E IR can be determined ex-perimentally provided Γ( E ) is available over an adequaterange [25]. It can also be calculated from Eq. (2) and theknown soft-potential parameters, E IR = 2( π C L ) − / W .In the few cases where both can be done, their agreementis remarkable [26]. Ω BP Li O-2B O [24] T = 300 K Ω BP / ω D = 0.17 ε = 0.060 Ω BP Polybutadiene [30] T = 140 K Ω BP / ω D = 0.11 ε = 0.019 Ω BP v -SiO [26] T = 1050 K Ω BP / ω D = 0.11 ε = 0.016 2 10 Ω BP E (meV) d -SiO [32] T = 565 K Ω BP / ω D = 0.15 ε = 0.0400.1110 Ω BP h Γ ( m e V ) Selenium [29] T = 295 K Ω BP / ω D = 0.18 ε = 0.068 Ω BP B O [27] T = 543 K Ω BP / ω D = 0.13 ε = 0.037 Ω BP NiZr [33]T = 297 K Ω BP / ω D = 0.15 ε = 0.050 Ω BP Orthoterphenyl [31]T = 156 K Ω BP / ω D = 0.08 ε = 0.006 Ω BP Glycerol [28]T = 175 K Ω BP / ω D = 0.15 ε = 0.052 FIG. 2: The full widths ~ Γ of x-ray Brillouin spectra mea-sured on nine glasses in function of the mode energy E , bothderived from adjustments with damped harmonic oscillatorlineshapes [25, 27, 28, 29, 30, 31, 32, 33, 34]. The dottedcurves are calculated following [9] as explained in the text,leading to the parameter ε which is indicated. The lines ofslope 4 are either calculated using Eq. (2) (solid lines), ortraced through the experimental points (dashed lines), ter-minating then at the E IR values obtained from the data asexplained in [25]. Arrows indicate the approximate boson-peak positions taken from [35] where the experimental Ω IR values are also discussed. (color online) The current situation for nine glasses is summarized inFig. 2. The dots show the x-ray data [25, 27, 28, 29, 30,31, 32, 33, 34]. These are full widths of damped harmonicoscillator lineshapes adjusted to x-ray Brillouin spectraafter convolution with the instrumental response. TheHRM predictions (dotted lines) are calculated using Eq.(4) of [9]. Again the known values of v L , v T , and Ω BP /ω D are used to extract the separation parameter ε . It is clearthat in all but one case the best HRM predictions fall wellbelow the observed widths. Interestingly, the single ex-ception is a metallic glass [34] for which a real Ioffe-Regelcrossover was not observed [36] and might not exist. Forcomparison, the QLV predictions below the crossover aredrawn as solid lines for the three cases for which reli-able values of both W and C L are available [37]. Abovethe crossover, Eq. (2) ceases to apply, and the widthscease increasing in E . This presentation shows that theexperimental widths smoothly prolongate the QLV pre-diction, which again supports the quantitative validityof this model on the basis of independently determinedparameters. However, it also illustrates that the x-raydata unfortunately starts at somewhat too high energy -30 -20 -10 0 10 200204060 C oun t s p e r c h a nn e l i n s a) Q = 2.4 nm -1 -20 -10 0 10 20 300102030 Energy transfer E (meV) b) Q = 4.25 nm -1 FIG. 3: Two inelastic spectra obtained on lithium diborate[25] at wavevectors Q near and above the Ioffe-Regel crossoverat Q IR ≃ . − . The dotted curves are the best predic-tions of the HRM model [9]. The relative amplitudes are sig-nificant. The instrumental full widths at half maximum forthe Brillouin peaks, including the effect of the finite collectionangle, are indicated by horizontal bars. (color online) in these three glasses to properly investigate the onsetregion ~ Γ ∝ E . On the opposite, this onset falls at suf-ficiently high E to be well observed in lithium diborateLi O-2B O [25] and in densified silica glass of density 2.6g/cm , d -SiO [33]. In these cases, as well as for the fourother glasses in Fig. 2, either one or both soft-potentialparameters are unknown. For glycerol and polybutadi-ene W is known, but for C one only has an average valuederived from low T thermal conductivity. It turns outthat C L and C T (for TA modes) can differ by factors aslarge as ∼
3. Thus, the solid line in Fig. 2 cannot bedrawn for these two glasses. Returning to the HRM pre-dictions, it is also remarkable that the slope of the dottedlines tapers off gently in all cases, and that no single linecrosses the value Γ = Ω /π within the ranges shown. Thismeans that the HRM model does not predict Ioffe-Regelcrossovers at places where these are in fact observed [35].It is also of interest to compare spectral shapes withspecific HRM predictions of [9]. This is illustrated in Fig.3 for spectra observed on Li O-2B O [25]. In drawingFig. 3, the HRM lineshapes have been convoluted withthe instrumental response function, and their height wasadjusted to the corresponding experimental peak ampli-tude at low Q < Q IR . The instrumental response beingwider than the spectral broadening, the latter is hard tojudge just inspecting Fig. 3a. Only the numerical analy-sis of the data leads to the widths given in the correspond-ing panel in Fig. 2. However, as Ω increases beyond Ω IR ,the HRM predictions, including the peak position , startdeparting strongly from the observed spectra which alsobecome very wide. This presentation clearly shows thatthe best HRM prediction cannot provide an explanationfor the observed spectra. In the QLV model, one expectsthat in the crossover region the acoustic modes hybridizewith the boson-peak modes. In such a strong couplingcase, perturbation approaches become inadequate. Forthis reason, a reliable expression for the Brillouin line-shapes in the QLV model is not yet available.In conclusion, it clearly appears that acoustic modesalone are not able to account for observations on mostglasses in the THz range. The acoustic modes are merelya subset of all the low frequency vibrations in disorderedsystems. QLVs predicted by the soft potential model takethese into account using parameters that are determinedfully independently by two-level-system measurements[13, 14]. It is truly remarkable that the latter, performedat low frequencies and sub-liquid-He temperatures, agreeso well with hypersonic measurements at elevated tem-peratures. For network glasses, there exists separate ev-idence on the non-acoustic nature of the boson-peak ex-cess which relates to rigid librations of structural units[10, 11, 12]. For molecular glasses, the disagreement be-tween the observed widths and the HRM predictions inFig. 2 is also striking. It is intuitive that for these thelowest frequency non-acoustic modes could be rigid libra-tions as well, though there exists so far little experimentalevidence for this. Contrary to crystals in which modes areorthogonal, in disordered systems the QLVs can linearlycouple to strains and to planar acoustic modes which arenot true eigenmodes. For this reason the acoustic modesseen in Brillouin scattering are strongly affected as theirfrequency nears the boson peak. The observed relationbetween E IR and E BP [20, 25, 35] is also well explainedby this coupling, as shown in [17]. [1] R.C. Zeller and R.O. Pohl, Phys. Rev. B , 2029 (1971).[2] QLVs in crystal lattices were investigated by I.M. Lif-shitz, J. of Phys. (USSR) , 86 (1943); id. , 215 (1943);id. , 82 (1944). Lifshitz found that a single defect pro-duces a narrow peak in the VDOS, the finite width of thepeak showing that the vibration is not really local, henceits name. The concept was first applied to glasses in [3].[3] V.G. Karpov, M.I. Klinger, and F.N. Ignat’ev, Sov. Phys.JETP , 439 (1983).[4] W. Schirmacher, G. Diezemann, and C. Ganter, Phys.Rev. Lett. , 136 (1998).[5] S.N. Taraskin, Y.L. Loh, G. Natarajan, and S.R. Elliott,Phys. Rev. Lett. , 1255 (2001).[6] J.W. Kantelhardt, S. Russ, and A. Bunde, Phys. Rev. B , 064302 (2001).[7] T.S. Grigera, V. Martin-Mayor, G. Parisi, and P. Verroc-chio, J. Phys.: Condens. Matter , 2167 (2002).[8] V. Gurarie and A. Altland, Phys. Rev. Lett. , 245502(2005).[9] W. Schirmacher, G. Ruocco, and T. Scopigno, Phys. Rev. Lett. , 025501 (2007).[10] U. Buchenau, M. Prager, N. N¨ucker, A.J. Dianoux et al. ,Phys. Rev. B , 5665 (1986).[11] B. Hehlen, E. Courtens, R. Vacher, A. Yamanaka, M.Kataoka, and K. Inoue, Phys. Rev. Lett. , 5355 (2000).[12] G. Simon, B. Hehlen, E. Courtens, E. Longueteau, andR. Vacher, Phys. Rev. Lett. , 105502 (2006).[13] Amorphous Solids: Low-T Properties , edited by W.A.Phillips (Springer, Berlin, 1981), and refs. therein.[14] D.A. Parshin, Phys. Solid State , 991 (1994).[15] M.A. Ramos, L. Gil, A. Bringer, and U. Buchenau, phys.sta. sol. (a) , 477 (1993).[16] V.L. Gurevich, D.A. Parshin, and H.R. Schober, Phys.Rev. B , 094203 (2003).[17] D.A. Parshin, H.R. Schober, and V.L. Gurevich, Phys.Rev. B , 064206 (2007).[18] This differs from previous models postulating a stronginteraction renormalizing the full VDOS, as in E.R.Grannan et al. , Phys. Rev. B , 7799 (1990), or R. K¨uhnand U. Horstmann, Phys. Rev. Lett. , 4067 (1997).[19] U. Buchenau, A. Wischnewski, M. Ohl, and E. Fabiani,J. Phys.: Condens. Matter , 205106 (2007).[20] D.A. Parshin and C. Laermans, Phys. Rev. B , 132203(2001).[21] A. Wischnewski et al. , Phys. Rev. B , 2663 (1998).[22] J. Wuttke et al. , Phys. Rev. E , 4026 (1995).[23] M.A. Ramos and U. Buchenau, Phys. Rev. B , 5749(1997).[24] C. Tal´on et al. , Phys. Rev. B , 012203 (2001).[25] B. Ruffl´e, G. Guimbreti`ere, E. Courtens, R. Vacher, andG. Monaco, Phys. Rev. Lett. , 045502 (2006).[26] For v -SiO , B O , and Se, we find experimentally for E IR the values 5 . ± .
7, 3 . ± .
9, and 2 . ± . et al. , Phys. Rev. Lett. , 1236 (1998).[28] A. Matic et al. , Europhys. Lett. , 77 (2001).[29] F. Sette et al. , Science , 1550 (1998).[30] T. Scopigno et al. , Phys. Rev. Lett. , 025503 (2004).[31] D. Fioretto et al. , Phys. Rev. E , 4470 (1999).[32] G. Monaco et al. , Phys. Rev. Lett. , 2161 (1998).[33] B. Ruffl´e et al. , Phys. Rev. Lett. , 095502 (2003).[34] T. Scopigno et al. , Phys. Rev. Lett. , 135501 (2006).[35] B. Ruffl´e et al. , Phys. Rev. Lett. , 079602 (2007).[36] E. Courtens et al. , Phys. Rev. Lett. , 079603 (2007).[37] The values of W are taken from [23]. For 10 × C L we use2.8, 2.2, and 1.1, for v -SiO , B O , and Se, respectively.These are obtained from the tunneling value C TUNL , usingthe relation C L ≃ C TUNL / . v -SiO and Se, C TUNL is from J.F. Berret and M. Meissner, Z. Phys. B , 65 (1988), for B O it is from M. Devaud et al. , SolidState Ionics9&10