Acoustic attenuation in glasses and its relation with the boson peak

0.15 MB·English

by  W. Schirmacher

Acoustic attenuation in glasses and its relation with the boson peak
W. Schirmacher1, G. Ruocco2,3, T. Scopigno3
1 Physik-Department E13, Technische Universit¨at Mu¨nchen, D-85747, Garching, Germany
7 2 Dipartimento di Fisica, Universit´a di Roma “La Sapienza”, I-00185, Roma, Italy.
0 3 CRS SOFT-INFM-CNR c/o Universit´a di Roma “La Sapienza”, I-00185, Roma, Italy.
0
2
A theory for the vibrational dynamics in disordered solids [W. Schirmacher, Europhys. Lett.

n 73, 892 (2006)], based on the random spatial variation of the shear modulus, has been applied to
a determine the wavevector (k) dependence of the Brillouin peak position (Ω ) and width (Γ ), as
k k
J well as the density of vibrational states (g(ω)), in disordered systems. As a result, we give a firm
5 theoretical ground to the ubiquitousk2 dependenceof Γ observed in glasses. Moreover, we derive
k
a quantitative relation between the excess of the density of states (the boson peak) and Γ , two
k
] quantities that were not considered related before. The successful comparison of this relation with
i
c theoutcome of experimentsand numerical simulations gives furthersupport tothe theory.
s
- PACSnumbers: 65.60.+a
l
r
t
m
Themoststrikingdifferencesbetweenglassesandcrys- Inthis regime,however,the resonancefrequency Ω still
k
.
t tals, at a macroscopic level, concernthe thermal proper- exhibits a linear dispersion with the wavevector k, al-
a
ties. At few tens of Kelvin, the specific heat of glasses beit with an apparent sound velocity being somewhat
m
exhibits an excess over the Debye expectation, which, larger than the ultrasonic speed [16]. More interesting,
- in a C(T)/T3 vs. T plot, appears as a characteristic in this frequency range the width Γ of the excitations,
d k
n maximum. Similarly, at low-T the thermal conductivity a quantity proportional to the sound attenuation coeffi-
o increases as T2 with increasing T and, in the same tem- cient, shows a k2 dependence in most of the investigated
c peraturerangewhereC(T)/T3 hasamaximum,exhibits materials [6], whereas at lower frequencies the width ap-
[
a plateau. While the firstobservationcanbe ascribedto pearstohaveastrongerkdependence[9]. Thesefindings

1 the presence of an excess of states over the Debye den- are still awaiting a theoretical explanation.
v sity of states g (ω) (the ”boson peak”, i. e. the peak
D Inthis letter we exploita recently developedtheoryof
2 observed in g(ω)/ω2 vs. ω), the second one was until
vibrational excitations in disordered elastic media [29]
1 recently not understood at all. Both anomalies appear
1 and apply it to the calculation of the dynamic struc-
to be strongly affected by the characteristics of the nor-
1 ture factor. This theory is based on the model assump-
mal modes of vibrations in the THz frequency region.
0 tion that the disorder leads to microscopic random spa-
7 The nature of these modes has been the subject-matter tial fluctuations of the transverse elastic constant (shear
0 of a very intense and controversial debate in the liter-
modulus)[30]. As insimilar,moreschematicapproaches
/ ature since many decades. The issue has been recently
t [20,23,25]theexcessDOShasbeenshowntoarisefroma
a revitalized thanks to new neutron, X-ray, and other in-
bandofdisorder-inducedirregularvibrationalstates,the
m elasticscatteringexperiments[1,2,3,4,5,6,7,8,9,10],
onsetofwhichapproacheslowerfrequenciesasthedisor-
- computer simulations [11, 12, 13, 14, 15, 16, 17], and
der is increased. Within this framework it was possible
d
analytical theory [18, 19, 20, 21, 22, 23, 24, 25, 26, 27].
n to formulate a theory for the energy diffusivity, which
o The boson peak shows up in a frequency range where gave the first explanation of the plateau of the tempera-
c thebroadeningoftheacousticexcitationsbecomesofthe ture dependent thermal conductivity and its relation to
:
v order of magnitude of the resonance frequency, thus in- the excess DOS. When applied to the calculation of the
i dicating a possible mode localization. This observation shape of the Brillouin resonance, this theory explains i)
X lead different authors to hypothesize a relation between the ubiquitous k2 dependence of Γ and ii) the observed
k
r thepositionofthebosonpeakandtheexistenceoflocal- increase of the apparent sound velocity with frequency
a
ized vibrations. Acoustic waves that become Anderson- [31]. Moreover the theory predicts iii) that the excess of
localized, indeed, could produce the plateau in the ther- vibrationalstates,ifproperlynormalizedwiththeDebye
mal conductivity. Following this idea, investigations of DOS,takesalmostauniversalvalueandiv)theexistence
(Anderson-)localizationpropertiesofwavesindisordered ofaquantitativerelationshipbetweentheexcessoverthe
systems based on simulations [12], model calculations Debye DOS and the width of the Brillouinline. The lat-
[19, 20] and field-theoretical techniques [28] have shown ter, previously unexpected, relation is shown to agree
thatAnderson-localizedstatesindisorderedmediadoac- withthefindingsofexperimentalinelasticscatteringand
tuallyoccur,butinamuchhigherfrequencyrange(near computer simulation experiments.
the upper band edge) than the boson peak frequency.
We now shortly summarize the theory. We
Sothequestionisstillopen: whatistheverynatureof consider an elastic medium with a mass density
2
the states near andabovethe bosonpeak frequency? As m0, shear modulus G, bulk modulus K =λ+ 3G
these states are neither really propagating nor localized, (λ = longitudinal Lam´e constant). These elastic con-
Feldman and coworkers [12] suggested to call them “dif- stants are related to the longitudinal and transverse
fusons” (they behave like diffusing light in milky glass). local sound velocities as c2T=G/m0; c2L=(K+ 34G)/m0
2
2 a
1
ω) D
ω
(gD ω)/ ω2
ω)/ 1 Γ( 0.01
(g f .

d
n 0.0001
a 0.1
ω) 1 b ω) ε)/ωD
(D (g ω(
g 0.1 ∆ 0.01
(ω)/g 0.01 ω.D 0.0001 0ε. 0=01 1 − (γ/0γ.0c1) 0.1
∆ 0.01 0.1 1
ω/ω
0.01 0.1 1 D
ω/ω
D
FIG.2: ExcessDOSg(ω)−g (ω)forthesameǫvaluesasin
D
=FIG0..0010:1,a)0.R00e1d,uc0e.0d1DaOndS 0(g.1(ω()f/rogmD(ωle)ftfotroǫri=ght1)−aγnGd/fγoGcr lFoiugi.n1liannedwifdorth(cfL(/ccLT,)c2T=)Γ6((ωfu)/llωlDin=es)2ωtofg(ectLh,ecrTw)Σit′h′(ωth)/eωBDrcil2L-
(c /c )2=2(fulllines)and(c /c )2 =6(dots). b)Reduced (dotted lines.)
L T L T
excess DOS ∆g(ω)/g (ω) for thesame parameters [33]. Insert: Onset frequency (full line), boson peak frequency
D
(dashed line) and Ioffe-Regel frequency ω = c (ω)k/π
IR L
(dash-dottedline)asafunctionoftheparameterǫ=1−γ /γc.
G G
=(λ+2G)/m0. We now assume that the shear modu-
lus (and not λ, which is set to a constant value λ0) ex-
From Eq. (1), the DOS can be calculated as:
hibits a random spatial variation: G(r)=G0[1+∆G˜(r)].
The random function ∆G˜(r) is supposed to be Gaus- 1
g(ω)=(2ω/3π)Σ Im{χ (k,ω)+2χ (k,ω)} (2)
sian distributed around its average G0 with a variance k<kDk2 L T
∝γ . The parameter γ describes the “degree of disor-
G G As in other theories of a harmonic solid with quenched
der” of the system. We emphasize that this phenomeno-
disorder [20, 23, 26] the system becomes unstable if the
logical model may be adequate both for a topologically disorder exceeds a critical value γc, which — in the
disordered system (glass) or a disordered crystal. Us- G
present case — slightly depends on the ratio of sound
ing standard field-theoretic techniques [25, 28, 29, 32] velocities and ranges, for example, from γc=0.1666 for
the self-consistent Born approximation (SCBA) for the c2/c2=2 to γc=0.227 for c2/c2=6. It hasGbeen demon-
(complex)selfenergyfunctionΣ(ω)=Σ′(ω)+iΣ′′(ω)has L T G L T
strated [20, 23, 25, 26, 29], that the quenched disorder
been derived [29], resulting in the set of equations:
produces an excess over the Debye DOS, the offset of
whichapproacheslowerandlowerfrequenciesasγ →γc.
Σ(ω)=γ Σ [χ (k,ω)+χ (k,ω)] (1) G G
G k<kD L T In Fig. 1 we report the reduced DOS g(ω)/g (ω) (i. e.
2 2 2 2 −1 D
χ (k,ω)= k [−ω +k (c −2Σ(ω))]
L L,0 theusualbosonpeakrepresentation)andthereduced ex-
χT(k,ω)= k2[−ω2+k2(c2T,0−Σ(ω))]−1. cessDOS[g(ω)/gD(ω)]−1againstω/ωDforseveralvalues
of the separation parameter ǫ=1−γ /γc. It is remark-
G G
Here χ (χ ) are the longitudinal and transverse dy- able thatthe value ofthe normalizedexcess doesnot ex-
L T
namic susceptibilities, resp.,and cL,0, cU,0 are the sound ceed the value 1, which it takes for small enough values
velocities of a system without disorder. The sum over ofǫ. Itisalsoremarkablethattheexcessessentiallydoes
k indicates integration up to the Debye cutoff, k = not depend on the ratio of the sound velocities. More-
D
[6π2N/V]13, according to the identification Σk<kD → over, the excess is significantly different from zero only
(3/k3)RkDk2dk. The SCBA is the simplest form of an above a certain frequency threshold (onset frequency),
D 0
effective-medium theory for the disorder [19, 20, 23], in belowwhichthe DOSbasicallycoincides withthe Debye
which the disorder effects lead to a frequency dependent expectation. The excess DOS turns out to vanish as ω4
modification of the sound velocities (“complex acoustic for ω →0.
indices of refraction”) by the function Σ(ω). The real We now turn the attentionto the dynamical structure
part Σ′(ω) causes a renormalization and dispersion of factor, S(k,ω), which can be measured by inelastic neu-
the sound velocities accordingto c (ω)2 =c2 −2Σ′(ω) tron, X-ray and light scattering:
L L,0
and cT(ω)2 = cT,0 −Σ′(ω), whereas its imaginary part 1 1
S(k,ω)= (n(ω)+1)Im{χ (k,ω)}= (n(ω)+1)
describes the sound attenuation (see below). Apart L
π π
−1
ωfrom=tcheknat[u3r3a]lthleenrgetahreanodnlyfretqwuoenncoyn-stcriavleiaslkpDaraamned- ×2k4Σ′′(ω)(cid:8)[k2cL(ω)2−ω2]2+4k4Σ′′(ω)2(cid:9)−1
D D D
ters in this theory,namely the degreeof disorderγ and 1 k2 k2Σ′′(ω)/ω
G ≈ [n(ω)+1] (3)
the ratioofthe renormalizedsoundvelocitiescL/cT [33]. π 2ω[cL(ω)k−ω]2+[k2Σ′′(ω)/ω]2
3
Material V/N c c A B k c ω A ρ R R Err
L T D D D D Exp Th
˚A3 m/s m/s ps3 ps ˚A−1 m/s ps ps3 ps2 ps2 %
MD-LJ 39.5 1320 [16] 540 [34] 2.1 10−2 [34] 0.120 [16] 1.14 610 6.99 8.8 10−3 2.4 0.099 0.103 4
MD-SiO2 14.1 5940 [17] 3570 [17] 3.7 10−5 [17] 0.045 [17] 1.61 3950 63.7 1.2 10−5 3.1 0.00055 0.00060 8
SiO2 15.1 5960 [9] 3750 [9] 5.3 10−5 [35] 0.042 [4] 1.58 4130 65.1 1.1 10−5 4.9 0.00101 0.00053 64
GeO2 16.1 3650 [36] 2150 [36] 1.4 10−4 [37] 0.034 [37] 1.54 2380 36.8 6.0 10−5 2.3 0.00237 0.00187 24
Glycerol 8.2 3600 [38] 1870 [38] 9.9 10−5 [39] 0.031 [40] 1.93 2090 40.4 4.5 10−5 2.2 0.00177 0.00194 10
oTP 10.7 2940 [41] 1370 [41] 3.5 10−4 [41] 0.036 [40] 1.77 1540 27.3 1.5 10−4 2.4 0.00445 0.00523 16
Selenium 30.0 1800 [42] 895 [42] 2.2 10−3 [43] 0.070 [44] 1.25 1000 12.6 1.5 10−3 1.4 0.00935 0.02170 80
TABLE I: Data taken from the literature together with quantities derived from them: atomic volume V/N, longitudinal and
transversesoundvelocitiesc ;coefficientsA,A oftheω2dependenceofthemeasuredandDebyeDOS;coefficientBentering
L,T D
intothefrequencydependenceoftheBrillouin width,Γ(ω)=Bω2;Debyewavenumberk =[6π2N/V]1/3,Debyesoundvelocity
D
c [33]and frequencyω =c k ;ρ (=A/A ),R =(A−A )/B, R =f(c ,c )/ω2,Err=|R -R |/[(R +R )/2].
D D D D D exp D Th L T D Exp Th Exp Th
Fromthis equationwecanreadoffthe widthofthe Bril- ofLennard-JonesparticlesandforamodelofSilicaglass,
louin line (full width at half maximum) as: and for the experimental cases of Silica, Germania,Sele-
nium, Glycerol and o-Terphenyl.
2
2 ′′ ′′
Γ(ω)≈2k Σ (ω)/ω ≈ c (ω)2ωΣ (ω) (4) ThefirsttestisthatofFig. 1a)fromwhichweexpect
L
that the quantity ρ = [g(ω)/g (ω)] = A/A (where
D max D
As the “disorder function” Σ′′(ω) enters also into the A,AD arethe coefficients of the measuredandDebye ω2
DOSwefindfrom(2)forΣ′′(ω)<c (ω)2 thefollowing lawinthebosonpeakregime)isequalto2. Asystematic
L,T
approximate relation: variationupwardsisobserved,butthe overallsetofdata
is not far from this value [45]. Exceptions are the case
Γ(ω)
of silica, where the presence of specific localized modes
ω ∆g(ω)=ω [g(ω)−g (ω)]=f(c ,c ) (5)
D D D L T
ωD (tetrahedra rotation) are well documented, and that of
2 Selenium, where the ratio is low, most likely because of
2 4
with f(c ,c )= (c /c ) [1+(c /c ) ] (6)
L T D L L T the boson peak is not well developed, indicating a large
π
value of ǫ.
To check the validity of the approximation made in de-
The second test involves the quantity
riving Eq. (5), in Fig. 2 we compare the excess DOS
R =∆g(ω)/Γ(ω)=(A − A )/B, which is the ra-
with the Brillouin linewidth, multiplied with the factor exp D
tio of the excess DOS to the Brillouin width as
f(c ,c ). We see that within our theory the approxi-
L T
determined by the experiments, while R is the same
mate relationship given in Eq. (5) holds, particularly in Th
ratio as determined from the theoretical prediction, i.e.
the regime where both quantities follow approximately
an ω2 dependence. RTh=f(cL,cT)/ωD2. Thevaluesofthe twoquantitiesare
very similar in all the cases investigated, the maximum
Equation (5) represents the main result of the present
deviation, observed once more for the cases of silica
work. Ittells us that the Brillouinwidth Γ(ω) is propor-
tional to ω2 (and thus to k2 recalling the slightly chang- and selenium, being within a factor two. The column
Err = |R − R |/((R + R )/2) reports the
ingvalueofc (ω))inthewholefrequencyregioncovered Exp Th Exp Th
L
by the boson peak, and that it turns to a ω4 behav- percentage of deviation between the measured and pre-
dicted values. Overall, the agreement is very good, also
ior only below the onset frequency. The onset frequency,
in view of the approximation made in deriving Eq. (5)
thatmustnotbeconfusedwiththebosonpeakfrequency
and, more important, of the experimental uncertainties
ω (see insert of Fig. 2), lies well below ω , it is lo-
BP BP
associated with the determination of A and B.
cated at the low frequency edge of the boson peak (i. e.
atthatfrequency wherethe excessDOSvanishes)andit In conclusion, we applied a recent theory of vibra-
tends to zero as ǫ→0. Equation (5) also gives a quan- tional excitations in a disordered elastic medium [29] to
titative prediction: it establishes a relation between the the determination of the dynamic structure factor. This
absolute value of the excess DOS and the Brillouin peak theory, that successfully explained the plateau in the T-
width. The material dependent physical quantities en- dependenceofthethermalconductivityinglasses[29],is
tering into this relation are only macroscopicproperties, now found to explain one of the most intriguing features
as sound velocities and density, that can be determined in the collective high-frequency vibrational dynamics of
independently. disordered systems: the k2 dependence of the Brillouin
In order to test the theoretical prediction manifested linewidth, i. e. of the acoustic attenuation. The the-
in Equation (5) we have collected a number of experi- ory also predicts the existence of a relation between the
mental and numerical simulation data of the vibrational excessDOS(thebosonpeakintensity)andthesoundat-
DOSandtheBrillouinlinewidth,aswellassoundveloc- tenuation coefficient (the Brillouin peak width). A test
ity data (Table I). A complete set of data, to our knowl- ofthispredictionusingexistingexperimentalandsimula-
edge, only exists in the case of the numerical simulation tion data gives anexcellent outcome,thus giving further
4
support to the validity of the theory itself. Finally, the dots in the insert of 2) takes place at a frequency much
4
theory also indicates that the transition between the k lower than the boson peak frequency which, in turn, is
lawatverylowfrequencyandthek2 lawforΓ [9](open much smaller than the Ioffe-Regel limit frequency.
k
[1] F. Setteet al., Science280, 1550 (1998). Stat Sol. (c) 1, 17 (2004); E. Maurer, W. Schirmacher,
[2] P.Benassi et al.,Phys. Rev.Lett. 77, 3835 (1996). J. Low-Temp. Phys. 137, 453 (2004)
[3] M. Foret et al., Phys. Rev. Lett. 77, 3839 (1996); Phys. [22] W. G¨otze, M. R. Mayr, Phys.Rev. E 61, 587 (2000).
Rev.B. 66, 024204 (2002). [23] S. N. Taraskin et al., Phys. Rev. Lett. 86, 1255 (2001);
[4] G. Ruocco et al.,Phys. Rev.Lett. 83, 5583 (1999). Phys. Rev.B 65, 052201 (2002)
[5] D.Engbergetal.,Phys.Rev.B59,4053(1999);A.Matic [24] J.W.Kantelhardtetal.,Phys.Rev.B63,064302(2001).
et al.,Phys. Rev.Lett. 86, 3803 (2001). [25] W. Schirmacher et al., Phys. Stat Sol. (b) 230, (2002);
[6] G.RuoccoandF.Sette,J.Phys.Cond.Matter13,9141 ibid.(c),117(2004);E.Maurer,W.Schirmacher,J.Low-
(2001). Temp. Phys. 137, 453 (2004)
[7] B.Ruffl´eet al.,Phys.Rev.Lett.90,095502(2003);ibid. [26] T. S.Grigera et al.,Nature422, 289 (2003).
96, 04552 (2006). [27] M. Turlakov,Phys. Rev.Lett 93, 035501 (2004).
[8] A. I. Chumakov et al., Phys. Rev. Lett. 92, 245508 [28] S. John et al.,Phys. Rev.B 27, 5592 (1983).
(2004). [29] W. Schirmacher, Europhys.Lett. 73, 892 (2006).
[9] C. Masciovecchio et al., Phys. Rev. Lett. 97, 035501 [30] A relation between the excess DOS and elastic constant
(2006). fluctuations was already suggested by A. Sokolov, J.
[10] A. Monaco et al., Phys. Rev. Lett. 96, 205502 (2006); Phys. Condens. Matter 11, A213 (1999).
ibid97, 135501 (2006) [31] W. Schirmacher et al.,to be published.
[11] R.Dell’Anna et al.,Phys. Rev.Lett. 80, 1236 (1998). [32] D. Belitz and T. R. Kirkpatrick, Rev. Mod. Phys. 66,
[12] P. B. Allen and J. L. Feldman, Phys. Rev. B 48, 12581 261 (1994).
(1993): J. L. Feldman et al., Phys. Rev. B 48, 12589 [33] c =[(1/3)(c−3+2c−3)]−1/3, c =c (0) ,c =c (0).
(1993); ibid. 59, 3551 (1999). [34] UDnpublishedLdatafrTomthesimLulatioLninVT.MazTzacurati
[13] S. N. Taraskin and S. R. Elliott., Phys. Rev. B 56, et al.,Europhys.Lett. 34, 681 (1996).
8605 (1997); ibid61, 12017 (2000); S.I. Simdyankin et [35] U.Buchenau,N.Nu¨cker,andA.J.Dianoux,Phys.Rev.
al.,Phys.Rev. B 65, 104302 (2002). Lett 24, 2316 (1984).
[14] M.C.C.Ribeiroetal.,J.Chem.Phys.108,9027(1998); [36] L. E. Bove et al.,Europhys.Lett. 71, 563 (2005).
ibid.9859 (1998). [37] G. Baldi, PhD Thesis, Universita’ di Trento(2006).
[15] P.Jund and R.Jullien, Phys. Rev.B 59, 13707 (1999). [38] F. Scarponi et al.,Phys. Rev.B 70, 054203 (2004).
[16] G. Ruocco et al.,Phys. Rev.Lett. 84, 5788 (2000). [39] J. Wuttkeet al.,Phys. Rev.E 52, 4026 (1995).
[17] J. Horbach et al.,Eur. Phys.J. B 19, 531 (2001). [40] L. Comez et al., J. Non-Cryst.Solids 307, 148 (2002).
[18] V. G. Karpov et al., Sov. Phys. JETP 57, 439 (1983); [41] A. T¨olle et al., Rep. Prog. Phys. 64, 1473 (2001). In
U.Buchenau et al., Phys. Rev.B 43, 5039 (1991); ibid., this paper the reported DOS is normalized to have its
46,2798 (1992); V.L.Gurevichet al.,Phys.Rev.B48, frequency integral equal to 3.
16318 (1993); ibid.,67, 094203 (2003). [42] W. B. Payne et al.,cond-mat/0408425 (2004).
[19] W. Schirmacher, M. Wagener, in ”Dynamics of Dis- [43] W. A. Phillips et al., Phys. Rev. Lett 21, 2381 (1989);
ordered Materials”, Eds. D. Richter, A.J. Dianoux, U. Buchenau et al., cond-mat/0407136 (2004).
W. Petry, and J. Teixeira, Springer, Heidelberg, p.231 [44] T. Scopigno et al., Phys.Rev.Lett 92, 025503 (2004).
(1989); Sol. State Comm. 86, 597 (1993) [45] The values for ρ reported in V. N. Novikov et al., Phys.
[20] W. Schirmacher et al., Phys. Rev. Lett. 81, 136 (1998); Rev. E 71, 061501 (2005) differ somewhat from our val-
PhysicaB263-264, 160(1999); AnnPhys.(Leipzig), 8, ues. We shall discuss this in our forthcoming paper [31].
727 (1999); Physica B 284-288, 1147 (2000)
[21] R. Ku¨hn, U. Horstmann, Phys. Rev. Lett. 78, 4067
(1997).W.Schirmacher,E.Maurer,M.P¨ohlmann,Phys.