The boson peak in supercooled metallic liquids and glasses and relation to the glass transition

Journal of Non-Crystalline Solids 293±295 (2001) 615±619

www.elsevier.com/locate/jnoncrysol

The boson peak in supercooled metallic liquids and glasses and relation to the glass transition I. Kanazawa * Department of Physics, Tokyo Gakugei University, Koganei-shi, Tokyo 184-8501, Japan

Abstract We have presented a theory of three-dimensional supercooled metallic liquids and glasses, which is based on the gauge-invariant Lagrangian density with spontaneous breaking (Higgs mechanism), and compared it with the frustration-limited domain (FLD) theory. Using the present theoretical formulation, a qualitative picture of the boson peak is proposed. Ó 2001 Published by Elsevier Science B.V. PACS: 61.20.Gy; 61.43.Dq; 61.43.Fs; 64.70.Pf; 63.50.+x

1. Introduction Although there are many studies on the dynamics of supercooled liquids, the structural glass transition, and the boson peak, a satisfactory theory for these phenomena has not yet been developed. The theories of the glass transition may be classi®ed into three categories. Theories (the mode coupling theory) in the ®rst category [1,2] describe the glass transition as a purely dynamic phenomenon without any underlying thermodynamic phase transition. Theories (the entropy theory) in the second category [3±5] are based on the assumption that the behavior near the glass transition is attributed to an underlying thermodynamic phase transition at a temperature lower than the conventional glass transition temperature. Theories (the trapping hopping model) in the third category [6,7] are constructed by taking into

*

Tel.: +81-423 297 548; fax: +81-423 297 491. E-mail address: [email protected] (I. Kanazawa).

account jump motions with the trapping-type master equations. Recently Kivelson et al. [8,9] have presented an interesting theoretical approach, which is referred to as the frustration-limited domain (FLD) theory. This theory is described by a Coulombic function, and long-range interactions play an important role in the properties of supercooled liquids. In order to understand the glass transition, we must study the short-range structures of the supercooled liquids and amorphous solid. There exists an important development in studies on the structures of liquids and amorphous solids. Based on important ideas by Kleman and Sadoc [10], it has been proposed that the parameter q…r; u† in two-dimensional and three-dimensional metallic glasses is speci®ed by rigid-body rotation, which is related to gauge ®elds of SO(3) symmetry for S 2 and SO(4) symmetry for S 3 , respectively [11±13]. As the ®vefold co-ordinated disk (the pentagonal disk) in the two-dimensional system is favorable energetically in comparison to other excited solitons, the present author stresses that a pentagonal disk is one of the excited dominant soli-

0022-3093/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 7 5 9 - 1

616

I. Kanazawa / Journal of Non-Crystalline Solids 293±295 (2001) 615±619

tons in the two-dimensional system at high temperature, and has proposed a theoretical picture for these excited solitons, which is based on the gauge-invariant Lagrangian density with spontaneous breaking [14±16]. Then adopting ®vefold and sevenfold co-ordinated disks as the frozen hedgehog-like solitons, we have proposed the mean ®eld theory with the replica method on the two-dimensional metallic glass system [17±19]. In this study, we will present a theoretical formula based on the gauge-invariant Lagrangian with spontaneous breaking in the three-dimensional metallic glass system, and compare it with the FLD theory. Furthermore, on the basis of the present theoretical formula, a qualitative picture of low energy excitations (the boson peak) in threedimensional glasses and supercooled liquids will be proposed.

2. A model system First, we will investigate a two-dimensional glass by using frozen excited disks, such as the anomalous 5- and 7-co-ordinated disks. It has been shown that the curvature can be represented by using a component in the other axis than x or y. That is, it is preferable to think of the anomalous disk as the hedgehog-like soliton (defect), taking account of the curvature. We adopt the parameter ®eld q…r; u†  qa …a ˆ 1; 2; 3†, which is similar to that in the Sachdev and Nelson model [20]. It is furthermore assumed that the symmetry of the gauge ®elds Aal , which introduce the curvature of the hedgehog-like soliton, is extended from SO(3) to SU(2). It is thought that the SU(2) triplet ®elds, Aal , are spontaneously broken through the Higgs mechanism similar to the way in which the 6-coordinated symmetry in the triangular lattice is broken around the anomalous 5- and 7-co-ordinated disks. In other words, in order to introduce the cluster with some radius in this system in the gauge-invariant formula, we must use the Higgs mechanism. If the 5-co-ordinated disk is formed, we set the symmetry breaking of the triplet ®eld, h0jqa j0i, equal to …0; 0; v†. On the other hand, if the 7-co-ordinated disk is formed, we set symmetry

breaking, h0jqa j0i, equal to …0; 0; v†. Then we can introduce the e€ective Lagrangian density: Lˆ

1 2 …om Aal ol Aam ‡ g1 abc Abl Acm † 4 1 2 ‡ …ol qa gabc Abl qc † 2 i 1 h 2 2 ‡ m2 …A1l † ‡ …A2l † ‡ m‰A1l ol q2 2 n h i h ‡ gm q3 …A1l †2 ‡ …A2l †2 m22 3 2 …q † 2

m22 3 a 2 gq …q † 2m

A2l ol q1 Š io A3l q1 A1l ‡ q2 A2l

m22 g2 a a 2 …q q † ; …1† 8m2 p where jmj is vg and jm2 j is 2 2kv. The e€ective Lagrangian, Leff , represents two massive vector ®elds, A1l and A2l , and one massless vector ®eld, A3l . Because these masses are formed through the Higgs mechanism by introducing the 5- and 7-coordinated disks, the gauge ®elds A1l and A2l are only present around the disks. The inverse, 1=jmj, of the mass of A1l and A2l re¯ects the radius of the cluster. Since the U(1) gauge ®eld A3l is massless, it is thought that the gauge ®eld A3l mediates the longrange interaction between two excited disks (the hedgehog-like solitons). Now, we shall introduce a ®eld-theoretical model to treat three-dimensional liquids and glasses by using excited clusters (solitons). It has been proposed that the parameter q…t; r; u† ˆ q…t; x; y; z; u† in three-dimensional liquids and glasses is speci®ed by the rotation, which is related to the gauge ®elds of Aal for SO(4) symmetry for S3 [20]. It has been required that the curvature can be represented by using a component, u, in the otheraxis direction, when the three spatial dimensional axes are the x, y and z ones. It is preferable that we think of the anomalous density ¯uctuation in the three-dimensional liquids as the hedgehog-like ¯uctuation (cluster), taking account of the curvature. We adopt the parameter q…r; u†  qa …a ˆ 1; 2; 3; 4†, which is similar to that in the Sachdev and Nelson model [20]. The SO(4) quadruplet ®elds, Aal , are spontaneously broken through the Higgs mechanism similar to the way in which the ¯uid host is broken around the hedgehog-like ¯uctuation (icosahedral cluster) [21]. When the icosahedral cluster (soliton) is created,

I. Kanazawa / Journal of Non-Crystalline Solids 293±295 (2001) 615±619

we set the symmetry breaking of the quadruplet ®elds, h0jqj0i, equal to …0; 0; 0; v4 †. Now we introduce approximately the Lagrangian density as follows: 2 1 a om Al ol Aam ‡ g1 abc Abl Acm Lˆ 4 2 1 2 ‡ ol qb gbac Aal qc ‡ c2 qa qa k…qa qa † : 2 …2† Then we set the symmetry breaking as follows:
qa ! …0; 0; 0; v4 † ‡ q1 ; q2 ; q3 ; q4 : Thus we can introduce the e€ective Lagrangian density 2 1 a Leff ˆ om Al ol Aam ‡ g1 abc Abl Acm 4 2 1 b ‡ ol q gbac Aal qc 2   m2  2  2  2 ‡ 1 A1l ‡ A2l ‡ A3l 2   ‡ m1 A1l ol q2 A2l ol q1   ‡ m1 A2l ol q3 A3l ol q2   ‡ m1 A3l ol q1 A1l ol q3 (   2  2  2  ‡ gm1 q4 A1l : ‡ A2l ‡ A3l h i A4l q1 A1l ‡ q2 A2l ‡ q3 A3l

)

m22 4
2 q 2

m22 g2 a a 2 …q q † ; …3† 8m21 p where m1 is v4
g and m2 is 2 2k
v4 . The e€ective Lagrangian density, Leff , represents three massive vector ®elds A1l , A2l , and A3l , and one massless vector ®eld A4l . Because these masses are created through the Higgs mechanism by introducing the hedgehog-like clusters (solitons), the gauge ®elds A1l , A2l , and A3l are only present around the clusters. The inverse, 1=jmj, of the mass of A1l , A2l , and A3l reveals approximately the radius of the clusters. Since the gauge ®eld A4l is massless, it is thought that the gauge ®eld A4l mediates the long-range m2 g 4 a 2 q …q † 2m1

617

interaction between two excited clusters (the hedgehog-like solitons). The generation function Z‰J Š for Green's functions is shown as follows: Z Z‰J Š  DADBDqDCDC Z …4† exp i d4 x…Leff ‡ LGF‡FP ‡ J
U†: 1 LDF‡FP ˆ Ba ol Aal ‡ aBa Ba ‡ iCa ol Dl C a ; 2

…5†

where Ba and C a are the Nakanishi±Lautrup (NL) ®elds and Faddeev±Popov ®ctitious ®elds, respectively. J
U  J al Aal ‡ JBa Ba ‡ Jq
qa ‡ JCa
C a ‡ JCaCa : …6† BRS-quartet [22,23] in the present theoretical system are …q1 ; B1 ; C 1 ; C1 †, …q2 ; B2 ; C 2 ; C2 †, …q3 ; B3 ; C 3 ; C3 †, and …A4L;l ; B4 ; C 4 ; C4 †, where A4L;l is the longitudinal component of A4l . 3. The model for three-dimensional metallic glasses and the frustration-limited domain theory In order to study the supercooled liquids, Kivelson et al. [9] proposed an e€ective Hamiltonian with the local order variable, Oi , describing the extention of the locally preferred structure in a supercooled liquid X K X Oi
Oj : …7† Oi
Oj ‡ Hˆ J 2 i6ˆj jri rj j hiji The ®rst sum over hiji is taken only over nearest neighbors and the second is taken over all pairs. The ®rst term corresponds to short-range ordering interactions and the second one represents longrange frustration. The frustration-limited domain model by Kivelson et al., which is based on the Sethna±Sachdev±Nelson theoretical formula, supplies a signi®cant picture for supercooled liquids. But it is not clear how their model is related to the gauge formula. In the present theoretical formula, we can de®ne the topological number q for excited hedgehog-like solitons as follows [24]:

618

qˆ

I. Kanazawa / Journal of Non-Crystalline Solids 293±295 (2001) 615±619

1 24p2

Z S3

dSl elabc Tr…A4a A4b A4c †;

…8†

where S 3 is a sphere, whose radius is much larger than 1=jmj. If a sphere S 3 surrounds completely one hedgehog-like cluster (soliton) with some curvature, whose center position is ri , its topological number is represented as qi . When the hedgehog-like soliton (monopole-like cluster) is located at the position rj ˆ …xj ; yj ; zj † and jri rj j > 1=jmj is assumed, the gauge ®eld A4l …ri ; rj † at the position ri ˆ …xi ; yi ; zi † is represented as follows [25,26]: A4l …ri ; rj †

/

qj jri

rj j

:

Thus the interaction energy Vij between two hedgehog-like solitons at the position ri and rj is proportional to qi qj : jri rj j

p If gi in Eq. (3) is assumed to be equal to p=K , where K is the long-range interaction constant in the e€ective Hamiltonian, Eq. (7), by Kivelson et al. [9], it is found that the interaction energy between two hedgehog-like solitons in the present theoretical formula is attributed to the second term in the e€ective Hamiltonian, Eq. (7). When the system is cooled rapidly through the glass transition, many hedgehog-like solitons are frozen randomly. In this system we can introduce approximately the Hamiltonian as follows: X Hˆ Vij : …ij†

For the mean ®eld approximation, it is assumed that Vij describes n hedgehog-like soliton interaction, which is mediated by the massless A4l ®eld, in pair …i; j† via in®nite-range Gaussian-random interaction for simplifying the discussion ! Vij2 1 P …Vij † ˆ exp : …9† 1=2 2hVij2 i …2phVij2 i† Now we can evaluate the properties of the threedimensional metallic glasses from the analogy of

the Sherrington±Kirkpatrik (SK) formalism [27] by using the replica method [28]. Thus we can estimate the free energy bf for one frozen hedgehog-like soliton in the method of steepest descent and replica symmetry condition as follows: Z 1 ~ 2 1 2 2 p …bV †
…1 G† e …1=2†z bf ˆ 4 2p p  log z cosh bV~ Gz dz; …10† where 7G  hqai qbi i ˆ Qab ;

b ˆ 1=T andV~ ˆ

q N hVij2 i; …11†

a and b are the replica indices and hqai qbi i represents thePcanonical average with weight of exp… bV~ ab qai qbi Qab †. Then we can introduce G self-consistently from of =oG ˆ 0 as follows: Z 1  p  1 2 G ˆ p e …1=2†z tanh2 bV~ Gz dz: …12† 2p 1 In the temperature region below Tc ˆ V~ =kB , we can obtain the phase of G  hqai qbi i ˆ Qab 6ˆ 0 and hqai i ˆ 0. It is thought that this phase corresponds to the two-dimensional glass phase. More exactly we must treat this system in the replica symmetry breaking condition. We can de®ne the order paR ~  1 dx Q…x†, where Q…x† is the Parisi rameter G 0 order parameter and is derived from Qab , in Parisi's theoretical formula [29]. In the temperature region below V~ =kB , we obtained a phase of the ~ 6ˆ 0, and hqa i ˆ 0 which cororder parameter G i responds to the glass phase.

4. A qualitative picture of the boson peak In glasses and amorphous materials, one observes the thermal conductivity plateau at 10 K and the low energy broad peak observed in Raman and neutron scattering [30], the so-called boson peak. It is thought that vibrational states responsible for the boson peak also contribute to the

I. Kanazawa / Journal of Non-Crystalline Solids 293±295 (2001) 615±619

thermal conductivity plateau, because the energy range spanned by the plateau covers that for the boson peak spectra, indicating that acoustic excitations must cease to propagate when their wavelength k reaches the nm range [31]. That is, acoustic modes may cross over to strongly localized modes satisfying the Io€e±Regel condition. By a computer simulation of a soft sphere glass, it is found that there are (quasi-)localized modes with e€ective masses ranging from 10 atomic masses upwards, which are closely related to the boson peak [32]. In the present theoretical formula, the e€ective Lagrangian density Leff in Eq. (3) represents certainly three massive vector ®elds A1l , A2l , and A3l , which are localized within the radius, 1=jmj, around the hedgehog-like clusters. Thus it is suggested that the localized gauge ®elds A1l , A2l , and A3l around the hedgehog-like clusters (solitons) are much related to the (quasi-)localized modes of the boson peak. That is, the localized gauge ®elds A1l , A2l , and A3l introduce the localized strain tensor ukl  1 Cijkl eikl ejma ok om Aal …a ˆ 1; 2; 3† around the hedgehog-like clusters (solitons) [33], where Cijkl is the elastic tensor. It should be noted that localized modes around the hedgehog-like clusters (solitons) are required naturally through the gauge-invariant condition in the present theory.

619

On the basis of the present theoretical formula, a qualitative picture of the boson peak is proposed. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

5. Conclusion

[24]

We have introduced the theory of three-dimensional supercooled metallic liquids and glasses, which is based on the gauge-invariant Lagrangian density with spontaneous breaking, and compared it with the FLD model. The order ~ is introduced by using the topoparameter G…G† logical number of frozen hedgehog-like soliton in the three-dimensional glass system. In the mean ®eld theory with the replica method, the phase of ~ 6ˆ 0 andhqa i ˆ 0, which the order parameter G…G† i corresponds to the glass phase, is obtained in the temperature region below V~ =kB .

[25] [26] [27] [28] [29] [30] [31] [32] [33]

E. Lentheusser, Phys. Rev. A 29 (1984) 2765. W. G otze, Z. Phys. B 56 (1984) 139. G. Adam, J.H. Gibbs, J. Chem. Phys. 43 (1965) 139. U. Mohanty, Physica A 162 (1990) 362. U. Mohanty, I. Oppenhein, C.H. Taubes, Science 266 (1994) 425. T. Odagaki, Y. Hiwatari, Phys. Rev. A 41 (1990) 929. T. Odagaki, J. Matsui, Y. Hiwatari, Phys. Rev. E 49 (1994) 3150. D. Kivelson, S.A. Kivelson, X.L. Zhao, Z. Nussinov, G. Tarjus, Physica A 219 (1995) 27. D. Kivelson, G. Tarjus, S.A. Kivelson, Prog. Theor. Phys. Suppl 126 (1997) 289. M. Kleman, T.F. Sadoc, J. Phys. 40 (1979) L567. D.R. Nelson, Phys. Rev. Lett. 50 (1983) 982. J.P. Sethna, Phys. Rev. Lett. 51 (1983) 2198. S. Sachdev, D.R. Nelson, Phys. Rev. Lett. 53 (1984) 1947. I. Kanazawa, J. Non-Cryst. Solids 150 (1992) 271. I. Kanazawa, Phys. Lett. A 176 (1993) 246. I. Kanazawa, Physica C 235±240 (1994) 2645. I. Kanazawa, Trans. Mater. Res. Soc. Jpn. 20 (1996) 431. I. Kanazawa, Prog. Theor. Phys. Suppl. 126 (1997) 397. I. Kanazawa, Phys. Rev. E (to be published). S. Sachdev, D.R. Nelson, Phys. Rev. B 32 (1985) 1480. I. Kanazawa, Radiat. Phys. Chem. 58 (2000) 457. C. Becchi, A. Rouet, R. Stora, Commun. Math. Phys. 42 (1975) 127. T. Kugo, I. Ojima, Prog. Theor. Phys. Suppl. 66 (1979) 1. R. Rajaraman, Soliton and Instantons, Elsevier, Amsterdam, 1989. A.M. Polyakov, Gauge Fields and Strings, Harwood, New York, 1987. E. Fradkin, Field Theories of Condensed Matter System, Addison-Wesley, Reading, MA, 1991. D. Sherrington, S. Kirkpatrik, Phys. Rev. Lett. 35 (1975) 1792. S.F. Edwards, P.W. Anderson, J. Phys. F 5 (1975) 965. G. Parisi, Phys. Lett. A 73 (1979) 203. U. Buchenaa, H.M. Zhou, N. Nucker, K.S. Gilroy, W.A. Phillips, Phys. Rev. Lett. 60 (1988) 1368. J. Graebuer, B. Golding, L.C. Allen, Phys. Rev. B 34 (1985) 5969. B.B. Laird, H.R. Schober, Phys. Rev. Lett. 66 (1991) 636. H. Kleinert, Gauge Fields in Condensed Matter, World Scienti®c, Singapore, 1989.