Vibrational instability, two-level systems and Boson peak in glasses
aa r X i v : . [ c ond - m a t . d i s - nn ] J un Vibrational instability, two-level systems and Boson peak inglasses
D. A. Parshin
Saint Petersburg State Technical University,195251 Saint Petersburg, Russia andMax-Plank-Institut f¨ur Physik komplexer Systeme, D-01187 Dresden, Germany
H. R. Schober
Institut f¨ur Festk¨orperforschung, Forschungszentrum J¨ulich, D-52425, J¨ulich, Germany
V. L. Gurevich
A. F. Ioffe Institute, 194021 Saint Petersburg, Russia (Dated: November 12, 2018)We show that the same physical mechanism is fundamental for two seeminglydifferent phenomena such as the formation of two-level systems in glasses and theBoson peak in the reduced density of low-frequency vibrational states g ( ω ) /ω . Thismechanism is the vibrational instability of weakly interacting harmonic modes. Belowsome frequency ω c ≪ ω (where ω is of the order of Debye frequency) the instability,controlled by the anharmonicity, creates a new stable universal spectrum of harmonicvibrations with a Boson peak feature as well as double-well potentials with a widedistribution of barrier heights. Both are determined by the strength of the interaction I ∝ ω c between the oscillators. Our theory predicts in a natural way a small value forthe important dimensionless parameter C = P γ /ρv ≈ − for two-level systemsin glasses. We show that C ≈ ( W/ ~ ω c ) ∝ I − and decreases with increasing ofthe interaction strength I . The energy W is an important characteristic energy inglasses and is of the order of a few Kelvin. This formula relates the TLS’s parameter C with the width of the vibration instability region ω c which is typically larger or ofthe order of the Boson peak frequency ω b . Since ~ ω c & ~ ω b ≫ W the typical valueof C and therefore the number of active two-level systems is very small, less thanone per ten million of oscillators, in a good agreement with experiment. Within theunified approach developed in the present paper the density of the tunneling statesand the density of vibrational states at the Boson peak frequency are interrelated. PACS numbers: 61.43.Fs, 63.50+x, 78.30.Ly
I. INTRODUCTION
One of the most typical low-temperature properties of glasses is the existence of two-level systems (TLS’s) which determine at low temperatures, typically below a few Kelvin,and frequencies less than 1 GHz, phenomena such as specific heat, thermal conductivity,ultrasonic and microwave absorption and many others (such as the echo, etc) . Anotherremarkable universal property of almost all glasses is the Boson peak in the low frequencyinelastic scattering, proportional to the reduced density of the vibrational states g ( ω ) /ω .Compared to the TLS’s it is observed at much higher frequencies, between 0.5 and 2 THz,and persists to higher temperatures, sometimes up to the glass transition temperature T g . Usually these two important glassy features are considered separately and no definiteconnection between them has been established so far.The purpose of the present paper is to propose a physical picture in which these twoseemingly different phenomena are closely interrelated. We will show that the formationof the Boson peak in the reduced vibrational density of states (DOS) inevitably leadsto the creation of two-level systems (and vice versa). A mechanism of the Boson peakformation, implying also the formation of TLS’s was proposed in our recent papers .This mechanism was based on the phenomenon of vibrational instability of weakly in-teracting harmonic oscillators (HO’s). The instability takes place in the low frequencyregion, 0 ≤ ω . ω c , below some characteristic frequency ω c which is proportional tothe interaction strength I . Changing the interaction strength one can vary the widthof the instability region, the position of the Boson peak and the number of TLS’s. Theweakness of the interaction I implies that ω c ≪ ω , where ω is of the order of Debyefrequency. Thus the vibrational instability occurs far below the Debye frequency ω andhas little influence on the high frequency vibrations.The low-frequency harmonic oscillators we are speaking about are realized in glassesas quasi-local vibrations (QLV) which can be understood as local low frequency vibrationsbilinearly coupled to the sound waves . The existence of quasi-local vibrations inglasses has been confirmed in a number of papers (see e.g. the literature cited in Ref. ).The local low frequency vibrations are the cores of the QLV’s. The cores representcollective low frequency vibrations of small groups of atoms . If one plots the potentialenergy against the amplitude of one of these modes , one obtains a mode potential,as treated in the soft potential model . The vibrational instability results from theweak interaction I of these soft oscillators with high-frequency oscillators (with ω ≃ ω ).As a result of this harmonic instability and the anharmonicity of the glass the initialvibrational density of states g ( ω ) is reconstructed in the low frequency region 0 ≤ ω . ω c where instability takes place and shows the universal Boson peak feature.The microscopic origin of these high frequency oscillators, whose frequencies are muchhigher than the Boson peak one, is not important in this picture. It has been shownthat, in general, the frequency of the sound waves can pass through the Ioffe-Regel limitnear the Boson peak frequency . Therefore, the higher frequency modes mighthave a rather complex structure. They interact with the soft oscillators. This interactioncauses the vibrational instability and hence the Boson peak and TLS’s formation.The vibrational instability is a rather general phenomenon and occurs in any systemof bilinearly coupled harmonic oscillators. It can be considered in a purely harmonicapproximation. For example, a system of two oscillators with masses M , and frequen-cies ω , becomes unstable if the interaction I between the oscillators exceeds a criticalvalue I c = ω ω √ M M . If one of two frequencies is small then the critical interaction I c is also small. In our case we have such an instability due to the interaction of low andhigh frequency oscillators. Therefore, we can simplify the picture to treat it analytically.The physical reason for the instability in this case is the fact that the high frequencymodes adiabatically follow the motion of the low-frequency oscillator (a description ofthis fact can be given within the adiabatic approximation ). The squared frequency of thesoft oscillator is reduced by an amount proportional to the strength of the interactionsquared and, therefore, even can turn negative. For the example of two interacting os-cillators with frequencies ω ≪ ω the new frequency e ω of the low-frequency oscillatoris given by e ω = ω (cid:0) − I /I c (cid:1) . (1.1)It is important that it decreases to zero linearly with I c − I ( e ω ∝ I c − I , when I → I c ).This is the essence of the vibrational instability phenomenon in the general case. Asa result, switching on the interaction between the oscillators, we shift all low frequencymodes with ω . ω c towards the boundary point ω = 0 and some of them will crossthis point (so that the corresponding ω turns negative). Therefore, in such a case (ifthere is no hard gap around zero in the initial density of states, g ( ω )) we will have a constant distribution of renormalized ω around ω = 0. From that we immediately getthe right wing of the Boson peak. Indeed, the local anharmonicity does not change thisimportant property of uniform distribution of renormalized ω around zero. One canshow that it restores the mechanical stability of the system by simply reflecting all thenegative ω values back to the positive ω range (like in a mirror but with additionalstretching factor of 2, which is obviously not important, see Eq. (2.16)). The strengthof the anharmonicity itself plays no role in this mirror transformation . Now a constantdistribution of ω around zero (on the positive side) obviously leads to a universal linearlaw for the density of states, g ( ω ) ∝ ω in the interval 0 ≤ ω . ω c (independent of theinitial DOS g ( ω )). And in its turn this linear ω dependence of the reconstructed DOSjust gives us the right wing of the Boson peak, since g ( ω ) /ω ∝ /ω and this dependenceis also universal and independent of the initial DOS g ( ω ).We want to stress that this DOS transformation (due to the phenomenon of vibra-tional instability) is rather general and universal since any monotonous ”traffic” of ω from positive to negative values (due to interaction between the oscillators, or due tochanging the temperature or pressure, etc.) always gives a constant distribution of ω around zero. Therefore, the universal linear DOS, g ( ω ) ∝ ω , and the correspondinguniversal right wing of the Boson peak, g ( ω ) /ω ∝ /ω , inevitably emerge as a result ofthis instability.If the origin of the right wing of the Boson peak looks somehow natural, the left wingof the Boson peak appears as a result of the less obvious additional transformation of thelinear DOS at smaller frequencies. The point is that when the anharmonicity restoresthe mechanical stability of the system, single-well potentials describing the unstablesoft modes with negative ω are replaced by double-well ones. This means that theeffective potential energy of the glass in direction of the local soft mode has two minimaseparated by a rather low barrier. Thus, in this scheme, the two-level systems arecreated simultaneously with the Boson peak due to the same mechanism of vibrationalinstability. Besides their own high importance (the TLS’s physics in glasses and beyond)these double-well potentials play also an important role in building the left wing of theBoson peak. It can be explained as follows.Due to bilinear interaction between the oscillators, I ij x i x j , double-well potentials,with a particle vibrating in one of the wells and therefore having a non-zero averagedisplacement x i = 0, create random static forces f i ≈ I ij x j acting on other oscillators.In a purely harmonic case, these linear forces would not affect the frequencies (andlinear density of states g ( ω ) ∝ ω would not change). However, together with the localanharmonicity the static forces create a universal soft gap , g ( ω ) ∝ ω , in the lineardensity of states. This soft gap is a manifestation of the sea-gull singularity (see alsoRef. ) predicted in the framework of the soft potential model for glasses. Recently it wasshown that ω behavior of the DOS is indeed a universal feature in disordered systemsfor low frequency bosonic excitations which are not Goldstone modes .The physical reason for this gap is very transparent. Due to anharmonicity there isalways a blue shift of the soft oscillator frequency under the action of the static force f . For small ω this shift is proportional to | f | / and it is anomalously large for smallforces. The density of states in the gap then can be estimated as follows g ( ω ) ∝ ω Z ω dω ∞ Z −∞ δ (cid:0) ω − af / (cid:1) df ∝ ω . (1.2)As a result, the random static forces (together with anharmonicity) effectively ”push out”oscillators from the low frequency range to higher frequencies, creating the universal softgap, g ( ω ) ∝ ω . One can also see this gap in the context of mechanical stability of thesystem but of another kind. Due to local anharmonicity small frequencies cannot survivein the system in the presence of random static forces. In some sense they are not stableeven though random forces cannot transform single-well potentials into double-well ones.The width of the ω -gap is of the order of the Boson peak frequency ω b ∝ ( δf ) / ,where δf is the width of the random force distribution P ( f ). The Boson peak frequency ω b is typically smaller than the characteristic frequency ω c determining the width ofthe vibrational instability region ω b ≈ ω c (cid:20) g ( ω c ) g ( ω ) (cid:21) / . (1.3)The strong inequality ω b ≪ ω c occurs if g ( ω c ) ≪ g ( ω ) i.e. g ( ω ) is an increasingfunction of ω (since ω c ≪ ω ). As a result, g ( ω ) ∝ ω in the interval 0 ≤ ω . ω b .Frequencies higher than ω b are only weakly affected by the static forces. Therefore, thelinear DOS, g ( ω ) ∝ ω (and the right wing of the Boson peak) which was created in thecourse of the vibrational instability is conserved in the interval ω b . ω . ω c . Since atsmall frequencies ω . ω b we have the universal ω gap in the density of states, the leftwing of the Boson peak takes also the universal form, g ( ω ) /ω ∝ ω . As a result in theregion of the vibrational instability, 0 ≤ ω . ω c , we have a universal behavior of thedensity of states g ( ω ) ( g ( ω ) ∝ ω for 0 ≤ ω . ω b and g ( ω ) ∝ ω for ω b . ω . ω c ) withthe Boson peak feature, independent of the initial DOS g ( ω ). At higher frequencies,above ω c , we keep the initial DOS g ( ω ) almost undistorted.The Boson peak was the main topic of our previous papers . In the present work weshall concentrate on the two-level systems, i.e. the level splittings due to the tunnelingthrough the barriers separating the two minima of the two-well potentials. But thisconsideration is not independent of the Boson peak since we will see that the mainparameters of TLS’s will be strongly interrelated with the parameters of the Bosonpeak. Therefore, these two universal phenomena should be considered together.In the standard tunneling model the TLS’s are often characterized by the so-calleddimensionless tunneling strength C , Ref. C = P γ ρv (1.4)where P is the density of states of the TLS’s, γ the deformation potential, ρ the massdensity of the glass and v is the average sound velocity. The experimental value of C for different glasses is small and varies in a narrow band between 10 − and 10 − . In ourtheory such small numerical values for C will emerge in a natural way.Several authors proposed that the approximate universality and smallnessof C in glasses may be a consequence of the interaction between the TLS’s. Roughlyspeaking the idea was based on a mean-field approximation. The i th TLS produces ata distance r i a deformation ε i ≃ γ i ρv r i , (1.5)where γ i is the deformation potential of the i th TLS. As the deformation is inverselyproportional to r i the distribution function of the deformations in a glass is a Lorentzianwith width δε proportional to the total concentration N of the TLS’s: δε ≃ γNρv . (1.6)The energy E i (the interlevel spacing) of each TLS changes under the deformation ε as δE i = γ i ε (1.7)and from Eq. (1.6) one sees that the energies E i of the TLS’s are distributed in theinterval δE : δE ≃ γ Nρv . (1.8)For small energies, the density of states n ( E ) is independent of both the energy and theconcentration of TLS’s: n ( E ) ≃ NδE ≃ ρv γ . (1.9)This is the result of a purely classical approach. In this approach the dimensionlessparameter C cl C cl ≃ n ( E ) γ ρv ≃ − − − . This was the main difficultyof the theory outlined in Ref’s. . However, if one accounts for the quantumnature of tunneling, the situation is improved and the value of C is reduced strongly. Toexplain this on the qualitative level we remind that the energy E of a TLS consists of twocontributions, i. e. the classical asymmetry ∆ and the quantum tunneling amplitude∆ : E = p ∆ + ∆ . So far we have disregarded the latter.According to the standard tunneling model∆ = ~ ω exp( − λ ) , (1.11)where λ is the tunneling parameter, distributed uniformly in the interval λ min < λ <λ max . Usually λ min is taken to be about unity. Only TLS’s with λ ≃ λ max ≫
1, systems with λ ≃ λ max cannot tunnel and donot contribute to the observable properties. Therefore, the relative number of TLS’sparticipating in the tunneling is proportional to the small number 1 /λ max . As a result,we estimate the dimensionless parameter C in glasses as C ≃ C cl /λ max ≃ /λ max . (1.12)If for example λ max ≃ the dimensionless parameter C ≃ − . Thus the smallnessof the dimensionless parameter C in our theory is related to typically large values of thetunneling parameter λ max (and to typically high barriers in the system). We will discussthis point in Section IV.In the same Section we will show that two important parameters, namely C forTLS’s and the characteristic frequency ω c for HO’s, marking the onset of the vibrationalinstability are closely interrelated: C ≈ (cid:18) W ~ ω c (cid:19) . (1.13)Here W is an important characteristic energy in glasses . Typically it is of the orderof a few Kelvin. It determines for example the position of the minimum in the reducedspecific heat , C ( T ) /T ( W ≈ T min ) and some other properties of glasses above oneKelvin . For vitreous silica W ≈ I ∝ ω c between the oscillators the smaller is the TLS’sparameter C : C ∝ /I . And as we will see in Section IV the smaller will be also thedensity of tunneling states, P ∝ /I . It naturally explains the very old puzzle in thephysics of glasses, why the number of two-level systems is so small (one two-level systemfor a million of atoms).At a first glance this interesting result seems to be rather contradicting. The strongerthe interaction I between the oscillators, the larger is the width of the vibrational in-stability region ω c ∝ I and therefore the higher is the number of double-well potentialscreated in the course of stabilization of the system due to anharmonicity. The explana-tion of this seeming paradox is that majority of the double-well potentials created due tovibrational instability have so high barriers V that they cannot participate in tunnelingat all. As a result only a very small part of the double-well potentials contributes to thetunneling density of states, P .Since the experimental values of W and C are well known for many glasses one canestimate from Eq. (1.13) the important characteristic energy ~ ω c which gives the widthof the vibrational instability region in glasses ~ ω c ≈ W C − / . (1.14)For example for a-SiO W = 4 K and C = 3 · − giving ~ ω c ≈
60 K. This falls intothe Boson peak region ( ~ ω b ≈
70 K). As a result we see that indeed the Boson peak isplaced in the vibrational instability range.
II. VIBRATIONAL INSTABILITY
To illustrate the idea of a vibrational instability, we consider a cluster containing alow-frequency harmonic oscillator (HO) with frequency ω surrounded by a large number, s −
1, of HO’s with much higher frequencies ω j of the order of ω ≫ ω . Here ω is aorder of magnitude estimate of the high frequencies. In glasses it usually is of the orderof the Debye frequency. Let n be the total concentration of the HO’s in the cluster and g ( ω ) the normalized initial density of states (DOS), i. e. the DOS of the HO neglectingtheir interaction, g ( ω ) = 1 s s X i =1 δ ( ω − ω i ) . (2.1)Including the interaction between the HO’s, the total potential energy of the clusteris U tot ( x , x , ..., x s ) = X i k i x i − X i,j = i I ij x i x j + 14 X i A i x i , A i > . (2.2)Here x i are the generalized coordinates describing the vibrations of HO’s, k i > I ij determines the bilinear inter-action between the oscillators. To stabilize the system we have added in this equationthe anharmonic terms, A i x i (with A i > I ij = g ij J/r ij , J ≡ Λ /ρv (2.3)where g ij ≃ ± r ij is the distancebetween HO’s, ρ is the mass density of the glass and v is the sound velocity.The interaction between the HO’s is due to the coupling between a single HO andthe surrounding elastic medium (the glass). This HO-phonon coupling has the form H int = Λ xε, (2.4)where Λ is the coupling constant and ε is the strain. Introducing the masses of oscillators M i we have for the bare frequencies (neglecting the bilinear interaction) as usual ω i = p k i /M i . (2.5)These bare frequencies enter Eq. (2.1) for the initial DOS g ( ω ). We will assume thecharacteristic strength of the bilinear interaction I ≃ J n between the oscillators to be much smaller than the typical quasielastic constants, so that | I | ≪ M ω ≡ k ≃ k j , ( j = 1) , (2.6)where M is the typical mass of the HO’s.The equation of motion for the low-frequency oscillator is M ¨ x = − k x + X j =1 I j x j − A x (2.7)and for the high frequency ones M j ¨ x j = − k j x j + X i = j I ji x i − A j x j , j = 1 . (2.8)For a slow motion one can set the acceleration term M j ¨ x j = 0 in Eq. (2.8). For I ≪ M ω ,we have x j ≪ x (see below). Therefore we can neglect also the anharmonicity force term − A j x j and the interaction terms I ji x i ( i = 1) between the high frequency oscillators andget from Eq. (2.8) x j = ( I j /k j ) x , j = 1 . (2.9)According to Eq. (2.6) we see that x j ≃ ( I/M ω ) x ≪ x . Inserting this value of x j intoEq. (2.7) we finally get a reduced equation of motion of the low-frequency oscillator M ¨ x = − ( k − κ ) x − A x = − dU eff ( x ) dx (2.10)where κ = X j =1 I j k j ≃ I M ω (2.11)and the effective potential energy for the slow motion is U eff ( x ) = 12 ( k − κ ) x + 14 A x . (2.12)The physical origin for this reduction to a one-mode approximation is the adiabatic ap-proximation where the high frequency modes adiabatically follow the slow low-frequencymotion . As a result the interaction between the low and high frequency oscillatorsrenormalizes the quasielastic constant k for the low frequency motion to the new effec-tive value k = k − κ. (2.13)For k > κ the potential (2.12) is a one-well potential and the cluster of oscillators isstable, the equilibrium positions of all oscillators x i = 0. For k < κ the renormalizedquasielastic constant k is negative and the cluster is unstable. The effective potential(2.12) in this case is a symmetric double-well potential. This is what we call the vibra-tional instability . As a result of the instability the low-frequency oscillator is displacedto one of the two new minima x = ± p ( κ − k ) /A = ± p | k | /A , (2.14)while the displacements of the high frequency ones are x j = ( I j /k j ) x ≪ x ( j = 1)and are much smaller. The barrier height between the minima is V = ( κ − k ) A = k A . (2.15)As follows from Eq. (2.12) the new lowest frequency of the system of s coupledoscillators is given by ω = (cid:26) ( k − κ ) /M = k/M , κ < k , κ − k ) /M = 2 | k | /M , κ > k . (2.16)The first case κ < k corresponds to a vibration in the minimum of a one-well potential(2.12) while the second case κ > k corresponds to a vibration in either of the two wellsof a double-well potential (2.12). It is remarkable that for weak interaction I ≪ M ω the strength of the anharmonicity A does not enter the renormalized frequency (2.16).Using the Holtsmark method , we derived in our previous paper the normalizeddistribution function of κ ρ ( κ ) = 1 √ π Bκ / exp (cid:18) − B κ (cid:19) (2.17)where B = π r π J n √ M (cid:28) ω (cid:29) ≡ ω c √ M . (2.18)Here
J n ≃ I , and h /ω i ≃ /ω is the ω − moment of the normalized initial DOS, g ( ω ). This formula can serve as a definition of the important characteristic quantities ω c ≃ I/M ω , and k c ≡ M ω c = B . (2.19)The physical meaning of these quantities is that the typical clusters with frequencies ω . ω c become unstable due to the interaction between the soft oscillator and thesurrounding high frequency ones. Thus the characteristic frequency ω c indicates theonset of the mechanical instability region . We will see below that the creation of TLS’sand the formation of the Boson peak occur in this region.However, in the present paper this particular form of the function ρ ( κ ) is not suitable.The reason is the long-range power tail of this function, ρ ( κ ) ∝ /κ / for κ ≫ k c . Thistail leads to divergent integrals for large κ -values when calculating averages of the type h κ ν i for ν ≥ /
2. As follows from Eqs. (2.11) and (2.3) the long-range tail of thedistribution is related to close pairs with small distances between the low and highfrequency oscillators, r ij ≪ n − / . However, usually the distance between the HO’s ina glass can not be arbitrarily small and, therefore, the function ρ ( κ ) drops faster andapproaches zero as κ − ( n +3) / (for g ( ω ) ∝ ω n , with n > κ with a characteristic scale κ ≃ k c ≃ I /k . In Fig. 1 this function is shown for different interaction strengths J ( n = M = ω = 1, see Ref. for details). Knowing the function ρ ( κ ), we can calculate FIG. 1: Distribution function ρ ( κ ), calculated as ensemble average by exact diagonalization ofsystems of N = 2197 oscillators with g ( ω ) = 3 ω and J = 0 . , .
10 and 0 .
15 (from left toright). the distribution function of the renormalized quasielastic constants, Φ( k ). Let F ( k ) bea normalized distribution function of quasielastic constants k i in Eq. (2.2). In the caseof equal masses of the oscillators, M i = M it is related to the normalized initial DOS g ( ω ) as follows F ( k ) = g ( ω )2 M ω , where k = M ω (2.20)and the normalized distribution function Φ( k ) is given asΦ( k ) = h δ ( k − k + κ ) i k ,κ = ∞ Z dk F ( k ) ∞ Z dκρ ( κ ) δ ( k − k + κ ) . (2.21)Integrating over the delta-function, it is convenient to present the expression for Φ( k )for positive and negative k separately. We have from Eq. (2.21)Φ( k ) = ∞ Z dk F ( k ) ρ ( k + | k | ) , for k < , (2.22)0and Φ( k ) = ∞ Z dκρ ( κ ) F ( k + κ ) , for k > . (2.23)Since the distribution function ρ ( κ ) is nonvanishing only for κ . k c and rapidly dropsto zero for κ & k c we conclude from Eqs. (2.22) and (2.23) that the function Φ( k ) for | k | ≪ k c is approximately a constantΦ( k ) ≈ Φ(0) = ∞ Z dκρ ( κ ) F ( κ ) ≈ F ( k c ) , for | k | ≪ k c . (2.24)For negative k and | k | & k c the function Φ( k ) rapidly drops. For positive k & k c ,Φ( k ) ≈ F ( k ). In Fig. 2 this function is shown for different interaction strength J ( n = M = ω = 1, see Ref. for details). FIG. 2: Distribution function of the renormalized quasielastic constants, Φ( k ), calculated asensemble average by exact diagonalization of a systems of N = 2197 oscillators with g ( ω ) =3 ω ( F ( k ) = 3 √ k/
2) and J = 0 . , . , .
15 (solid curves, from right to left). Dotted curve:Result of convolution, Eq. (2.21), for J = 0 . III. INTERACTION BETWEEN LOW-FREQUENCY OSCILLATORS
In the previous section we have considered the effect of the interaction between alow-frequency oscillator and the surrounding high frequency oscillators in a cluster. Asa result of this interaction the quasielastic constant k has been renormalized to a neweffective value k = k − κ . Negative values of k indicate a vibrational instability of1the cluster. So far we have neglected the interaction between low-frequency oscillatorsbelonging to different clusters. This interaction is much weaker and cannot producea new instability. However, it causes internal random static forces acting on the low-frequency oscillators. As we have seen in the previous section, in the case of instabilitythe interaction between low frequency and high frequency oscillators shifts the positionsof their minima (static displacements). These shifts in turn act as forces if we takethe interaction between unstable low-frequency HO’s into account. As was shown inour previous paper (see also Section V) these forces are responsible for the universal g ( ω ) ∝ ω (see also Ref. ) dependence of the excess vibrational density of states for lowfrequencies.One can insert these forces into Eq. (2.12) as a linear term − f x where f is theinternal random force created by the other unstable low-frequency oscillators. Theeffective potential energy then reads U eff ( x ) = − f x + 12 kx + 14 Ax . (3.1)Here and henceforth the index 1 will be omitted.The distribution function of the random forces P ( f ) has been obtained in our previouspaper P ( f ) = 1 π δff + ( δf ) , (3.2)where δf is the width of the distribution. The Lorentzian form of the distributionis related to the fact that the forces between harmonic oscillators decay as r − ij [seeEq. (2.3)].One can estimate the width, δf , of the distribution as follows. The static force f i exerted on the i th oscillator by the j th one is f i = I ij x j . (3.3)Its characteristic value is given by displaced harmonic oscillators with frequencies oforder of ω c . For these oscillators we have I ( c ) ij ≈ J n c , J ≈ I/n ≈ M ω c ω /n ≈ M ω c /n g ( ω ) . (3.4)Here n c ≈ n g ( ω c ) ω c is the concentration of these unstable harmonic oscillators (double-well potentials) while n is the total concentration of HO’s. Due to the normalizationcondition ω g ( ω ) ≈
1. From Eq. (2.14) it follows that the characteristic static displace-ment of these unstable oscillators is x j ≈ p k c /A = ω c p M/A . As a result, one getsfrom Eq. (3.3) (see Ref. ) δf ≈ J n c ω c r MA ≈ M r MA ω c g ( ω c ) g ( ω ) . (3.5)As mentioned already in the beginning of this section, these internal random forcesdo not produce a new vibrational instability. However, they can transform some double-well potentials into single-well ones. For k > k < | f | < f ⋆k where f ⋆k = 2 | k | / / √ A. (3.6)2For | f | > f ⋆k (and k <
0) (3.1) is a one-well potential. For | f | = f ⋆k ( k <
0) the potential(3.1) is one-well with a bending point.It is interesting to compare the width of the distribution δf with the characteristicvalue of f ⋆k . They become equal for | k | = k ⋆ ≡ M ( ω ⋆ ) where ω ⋆ ≈ ω c [ g ( ω c ) /g ( ω )] / ≪ ω c . (3.7)The strong inequality ω ∗ ≪ ω c occurs if both ω c ≪ ω and g ( ω c ) ≪ g ( ω ). We shallsee in Section V that the frequency ω ⋆ plays a role of the Boson peak frequency. IV. TWO-LEVEL SYSTEMS
As follows from Eq. (3.1) negative values of k for f = 0 correspond to symmetricdouble-well potentials. In a purely classical treatment the oscillator will vibrate in eitherof the wells. Taking quantum mechanics into account, there will be a finite probability ofpenetration through the barrier separating the two wells, i. e. there is a finite tunnelingprobability. This causes a splitting of the vibrational levels. We are interested in thelowest pair of levels. This constitutes a two-level system (TLS). These systems areubiquitous in glasses and determine their low temperature properties .Tunneling systems can be described effectively in terms of a tunnel splitting ∆ andan asymmetry ∆. We will derive expressions for these quantities and their distributions.Neglecting the linear force term in Eq. (3.1), the tunnel splitting is given in the WKBapproximation as∆ ≈ W exp ( − S/ ~ ) , S = x Z − x | p | dx = 2 x Z p M [ U eff ( x ) + V ] dx. (4.1)Here we approximated the dependence of the prefactor on the vibrational frequency byan order of magnitude estimate W (see also Ref. ) W = ~ (cid:18) ~ A M (cid:19) / (4.2)that is of the order of the interlevel spacing in a purely quartic potential V ( x ) = Ax / , one finds values W fordifferent glasses of the order of a few Kelvin, e. g. for vitreous silica W ≈ U eff ( x ) (for f = 0) aredenoted by ± x = ± p | k | /A and V = k / A is the barrier hight [see Eqs. (2.14) and(2.15)].Evaluating the integral in Eq. (4.1) we get S = 2 √ | k | / M / A (4.3)and ∆ = W exp − √ | k | / M / ~ A ! = W exp − √ ~ | k | / M / W ! . (4.4)3The second quantity characterizing the TLS is the asymmetry of the two-well config-uration. For | f | ≪ f ⋆k we have from Eq. (3.1)∆ = 2 f x = 2 f p | k | /A. (4.5)We are interested in the two quantum states with the lowest energies. These statesbelong to both wells. TLS are often described by the interlevel distance E and thedimensionless tunneling parameter p : E = q ∆ + ∆ , p = (∆ /E ) . (4.6)The Jacobian of the transformation from the variables | k | and f to E and p is J = ∂ ( | k | , f ) ∂ ( E, p ) = (cid:18) (cid:19) / / M / W / ~ L − / p √ − p , where L = ln WE √ p . (4.7)In the new variables the distribution function reads F ( E, p ) = n P (0)Φ(0) |J | = (cid:18) (cid:19) / / n P (0)Φ(0) M / W / ~ L − / p √ − p . (4.8)Here we have replaced P ( f ) and Φ( k ) by P (0) and Φ(0). This can be justified byestimating the relevant ranges of f and | k | given by Eq. (4.11). Using Eqs. (4.5), (4.4)and (3.5) we express f in terms of ∆ and ∆ fδf ≈ (cid:18) (cid:19) / L − / E √ − pW (cid:18) W ~ ω c (cid:19) g ( ω ) g ( ω c ) . (4.9)Taking rough estimates L = 10, E = 1 K , W = 4 K , ~ ω c = 100 K , ω /ω c = 3, and g ( ω ) ∝ ω one finds a typical value f /δf ≈ · − . To estimate | k | we derive fromEq. (4.4) | k | /k c = 4 (9 / / L / ( W/ ~ ω c ) . (4.10)This set of parameters gives the typical value of | k | /k c ≈ .
05. Therefore, in the rangeof parameters where the notion of TLS’s is applicable, the characteristic values of f and | k | satisfy the conditions f ≪ δf, and | k | ≪ k c . (4.11)Our result can be compared with the standard tunneling model where the distribu-tion functions are P (∆ , ∆ ) = P ∆ , F ( E, p ) = 12 Pp √ − p (4.12)with a constant density of tunneling states P . Comparing these distributions withEq. (4.8), one gets P = (cid:18) (cid:19) / / n P (0)Φ(0) M / W / ~ L − / . (4.13)Both distributions, Eqs. (4.8) and (4.12), coincide regarding their dependencies on E and p , apart from the factor L − / describing a weak logarithmic dependence on E and p . The same factor is found in the soft potential model (see Refs. ).4To compare the tunneling strengths of the TLS’s with experiment we study theirinteraction with strain, described by the deformation potential γ . According to Eq. (2.4),the variation of asymmetry ∆ ε due to a strain ε is∆ ε = 2Λ x ε = 2Λ ε p | k | /A. (4.14)The deformation potential is defined as γ = 12 ∂ ∆ ε ∂ε (4.15)and from Eqs. (4.14), (4.4) and (4.2) follows γ = Λ r | k | A = 3 / / ~ Λ √ M W L / . (4.16)In the standard tunneling model the TLS’s are often characterized by the dimen-sionless tunneling strength C , given by Eq. (1.4). For different glasses its value variesbetween 10 − and 10 − . Using Eqs. (4.13) and (4.16) one gets C = 2 √ n P (0)Φ(0) Λ ρv W / √ M ~ . (4.17)This value is independent of E and p as in the standard tunneling model.The different factors entering the expression for C can be estimated from our modelas Λ /ρv = J ≈ I/n , I ≈ M ω c ω , Φ(0) ≈ F ( k c ) = g ( ω c ) / M ω c (4.18)and from Eqs. (3.2), (3.5) and (4.2) P (0) = 1 πδf ≈ π W / ~ √ M g ( ω ) ω c g ( ω c ) . (4.19)As a result we arrive at the important estimate C ≈ √ π (cid:18) W ~ ω c (cid:19) (4.20)that is independent of the initial DOS of HO’s g ( ω ) and in this sense is universal . Itonly depends on the characteristic energy W and the frequency ω c that is proportional tothe interaction I . The larger the interaction between the original oscillators the smalleris constant C , C ∝ I − .In the discussion of the Boson peak (Section V) we will see that ω c is two or threetimes larger than the Boson peak frequency ω b (or ω ⋆ ) which slightly depends on initialDOS g ( ω c ) (see Eqs. (5.10), (5.11) and Fig. 3). Using values appropriate to SiO , W = 4 K and ~ ω c = 100 K we get C ≈ − . (4.21)Thus the unified approach of this paper gives a value of the tunneling strength C in goodagreement with experiment. Since values of C and W are well known from experiment5for many glasses , Eq. (4.20) can be used to estimate the important characteristic energy ~ ω c giving onset of the vibrational instability in glasses ~ ω c ≈ W C − / . (4.22)It was demonstrated in Ref. that taking into account the experimental data for C and W this energy is indeed correlated with position of the Boson peak in glasses, ω b ≃ ω c .The two factors entering C , Eq. (1.4), can be estimated separately as P ≈ n ~ ω (cid:18) W ~ ω c (cid:19) L − / , γ ρv ≈ ~ ω n ~ ω c W L / . (4.23)The first of these quantities is ∝ ~ / and the second ∝ ~ / and thereby C ∝ ~ . Since C ≪ C = ~ / e S where e S ≫ ~ is some classical action. From Eqs. (4.20), (4.3) and (4.2) we get C ≈ π ~ S c ≪ , S c = 2 √ | k c | / M / A , (4.24)i.e. e S ≈ πS c . The classical action S c corresponds to a typical double-well potential(3.1) with | k | = k c (and f = 0).Using the estimates ω c /ω ≈ / W = 4 K and ~ ω c = 100 K we estimate theconcentration of tunneling systems with energies in the range 0 < E < W and tunnelingparameter p ≃ n TLS ≃ P W ≃ n ω c ω (cid:18) W ~ ω c (cid:19) L − / ≈ · − n . (4.25)The number of active TLS’s is thus less than one for ten million of oscillators . This ex-plains why the concentration of observed TLS’s in glasses is so small. Since according toEq. (4.25) n TLS ∝ I − the number of TLS’s decreases rapidly with increasing interactionstrength I .It would be instructive to derive a dimensionless parameter C cl by a classical proce-dure (neglecting the tunneling probability ∆ ). We take the width of the force distribu-tion from Eq. (3.5) and estimate the typical asymmetry ∆ c from Eq. (4.5) δf ≈ I ( c ) x ( c )0 ≈ J n c p k c /A, ∆ c ≈ δf p k c /A ≈ J n c · ( k c /A ) . (4.26)With n c the concentration of double-well potentials we get the classical estimate fortheir density of states P c ≈ n c ∆ c ≈ AJ k c ≈ n ~ ω (cid:18) W ~ ω c (cid:19) (4.27)that is independent of ~ .For the deformation potential γ c we have from Eqs. (4.14) and (4.15) the estimate γ c ≈ Λ p k c /A, and γ c ρv ≈ J k c A ≈ ~ ω n (cid:18) ~ ω c W (cid:19) , (4.28)also independent of ~ . Finally C cl = P c γ c ρv ≈ , (4.29)6i.e. the dimensionless parameter C cl in this classical approach is of the order of unitywhich is a consequence of the 1 /r interaction between the TLS’s . We wish toemphasize that C cl (unlike C ) does not determine any physical property of glasses.The reason for the difference between the two approaches (quantum and classical)is the following. In the classical approach we take all the double-well potentials intoaccount. They have typically | k | ≃ k c . Their concentration, n c , is unimportant sinceit is canceled in Eq. (4.27) for P c . In the quantum approach only the small portionof TLS’s which are able to tunnel (they have | k | << k c ) contribute to the observablequantities. For all other TLS’s the high barriers V and asymmetries ∆ ∼ ∆ c preventtunneling (but they contribute to the internal random static force δf ).To further clarify this point let us consider TLS’s with a quasielastic constant k inthe interval ∆ k ≃ k where | k | ≪ k ⋆ ≪ k c [see Eq. (3.7)]. Their concentration n k ,asymmetry ∆ k and the deformation potential γ k are given by n k ≈ n | k | Φ(0)( f ⋆k /δf ) , ∆ k ≈ f ⋆k p | k | /A, γ k = Λ p | k | /A. (4.30)In this expression for n k we took into account, that for | k | ≪ k ⋆ , only the small fractionof all potentials, where f ⋆k /δf ≪
1, is of double-well type. Keeping in mind that Φ(0) ≈ F ( k c ) we get the density of states P k ≈ n k ∆ k ≈ n | k | F ( k c ) δf p | k | /A (4.31)and the parameter C k C k = P k γ k ρv ≈ n F ( k c ) J | k | / δf √ A ≈ (cid:18) | k | k c (cid:19) / ≪ . (4.32)Here we have used Eq. (4.26) for δf and the estimate n c ≃ n k c F ( k c ) ≃ n ω c g ( ω c ).If we now fix | k | by the condition that the exponent in Eq. (4.4) is of order of unity,i.e. | k | ≃ M W / ~ , we reproduce our quantum result, Eq. (4.20), C k ≈ ( W/ ~ ω c ) . Theclassical result, C cl , Eq. (4.29) would be recovered for k ≈ k c when C k ≈
1. We concludethat the physical reason for smallness of the parameter C for TLS’s in glasses is thescarcity of those TLS’s that are able to tunnel compared to their total number .Again we can compare our results with the standard tunneling model . In this modelthe tunneling amplitude ∆ = ~ ω exp( − λ ) and the dimensionless parameter λ is uni-formly distributed in the interval λ min < λ < λ max . The lowest value λ min ≃
1. Accordingto Eq. (4.1) λ = S/ ~ , and therefore the maximal value λ max ≈ S c / ~ . Taking into accountEq. (4.24) we get λ max ≈ S c / ~ ≈ / πC. (4.33)Thus λ max is related to the small parameter C . For SiO , C = 3 · − and λ max ≈ V. THE BOSON PEAK
In this section we relate the results obtained for the TLS’s to the Boson peak prop-erties — see Ref. . For this we calculate the vibrational density of states (DOS) g ( ω ).We start from the case f = 0 (i.e. neglecting the interaction between the clusters).There are two types of harmonic vibrations. In the one-well case, k >
0, according to7Eq. (2.24) the distribution function of k for k ≪ k c is constant, Φ( k ) ≈ Φ(0). Therefore,since according to Eq. (2.16) (top) k = M ω , the renormalized DOS for ω ≪ ω c is e g I ( ω ) = 2 n M Φ(0) ω. (5.1)For harmonic vibrations in either well of a symmetric double-well potential [Eq. (3.1), k < f = 0] | k | = M ω / k ) = Φ(0) for | k | ≪ k c ( ω ≪ ω c ) the DOS is e g II ( ω ) = n M Φ(0) ω. (5.2)It is half of the one-well contribution. The total DOS for f = 0 and ω ≪ ω c is the sumof the two contributions e g tot ( ω ) = e g I ( ω ) + e g II ( ω ) = 3 n M Φ(0) ω. (5.3)It is a linear function of ω independent of the form of the initial DOS g ( ω ). This linearbehavior follows from the finite value of Φ(0).If the low-frequency HO’s were isolated their density of states would be determinedby Eq. (5.3). As we have shown in the Section III there is, however, an interactionbetween these oscillators which we have to take into account. According to Eq. (3.3)the low-frequency harmonic oscillators, displaced from their equilibrium positions (andforming the double-well potentials), create long-range random static forces f actingon other oscillators. In a purely harmonic case, these linear forces would not affectthe frequencies. Anharmonicity, however, renormalizes the low frequency part of thespectrum, a manifestation of the so-called sea-gull singularity treated in detail in Ref. (see also Ref. ).We begin with the case k >
0. It corresponds to one-well potentials. Consideran anharmonic oscillator under the action of a random static force f . The effectivepotential is given by Eq. (3.1) where p k/M is the oscillator frequency in the harmonicapproximation for f = 0. The force f shifts the equilibrium position from x = 0 to x = 0, given by Ax + kx − f = 0 , (5.4)where the oscillator has a new (harmonic) frequency M ω = k + 3 Ax . (5.5)With Φ( k ) as the distribution function of k [see Eq. (2.23)] and P ( f ) as the distributionof random forces f [see Eq. (3.2)] the renormalized DOS is given by g I ( ω ) = n ∞ Z Φ( k ) dk ∞ Z −∞ df P ( f ) δ ( ω − ω new ) . (5.6)Assuming ω ≪ ω c and integrating Eq. (5.6) with Φ( k ) = Φ(0) we get the integral g I ( ω ) = 2 n Φ(0) M ω √ A Mω Z dk P [ f ( k )] √ M ω − k (5.7)where according to Eqs. (5.4) and (5.5) f ( k ) = Ax + kx = 13 r M ω − k A (2 k + M ω ) . (5.8)8Taking the Lorentzian distribution, Eq. (3.2), for P ( f ) and introducing a new variable t = p − k/M ω we finally get g I ( ω ) = 12 π n M Φ(0) ω ω ⋆ (cid:16) ωω ⋆ (cid:17) Z dt ω/ω ⋆ ) t (3 − t ) (5.9)with ω ⋆ = √ A / ( δf ) / / √ M . (5.10)The function g I ( ω ) depends on a single parameter, ω ⋆ characterizing, as we will seebelow, the position of the Boson peak ω b ( ω b ≈ ω ∗ ). The frequency ω ⋆ is determinedby the characteristic value of the random static force δf acting on an HO with thecharacteristic frequency ω c . As a result, taking into account Eq. (3.5), we get theestimate ω ⋆ ≈ ω c (cid:20) g ( ω c ) g ( ω ) (cid:21) / , ω ⋆ ≪ ω c . (5.11)Again, as in Eq. (2.16) in lowest order the anharmonicity A does not enter this formula.This equation for ω ⋆ coincides with Eq. (3.7) obtained from the condition f ⋆k ≃ δf .According to Eq. (5.11), for weak interactions I ( ω c ≪ ω ), the frequency of theBoson peak ω ∗ ≪ ω c only in the case when the initial DOS, g ( ω ), is monotonically (andrapidly) decreasing (to zero) function of ω . For example we can take g ( ω ) ∝ ω n with n >
0. Then for n = 2 and ω c = ω / ω ∗ ≈ ω c /
2. For thesame n and smaller interaction, ω c = ω /
5, we get ω ∗ ≈ ω c /
3. In the opposite case ifthe initial DOS drops to zero too slowly or remains nearly constant, g ( ω ) ≃ const, wehave from Eq. (5.11) that ω ∗ ≃ ω c . In this case the Boson peak frequency ω ∗ is of thesame order as the characteristic frequency ω c .For small frequencies, ω ≪ ω ⋆ , only small forces f ≪ δf contribute to the integral inEq. (5.7). In this case the distribution function P ( f ) can be approximated by a constantvalue, P (0), and we get from Eqs. (5.7) and (5.9) g I ( ω ) = 4 n Φ(0) P (0) M / ω √ A = 12 π n M Φ(0) ω (cid:16) ωω ⋆ (cid:17) ∝ ω . (5.12)As a result, at low frequencies the renormalized excess DOS, g I ( ω ) ∝ ω , Ref. . Forsufficiently large frequencies, ω ≫ ω ⋆ (but still smaller than ω c ) the action of randomstatic forces on the HO spectrum can be discarded. In this case the integral in Eq. (5.9) isequal to ( π/ ω ⋆ /ω ) . We recover the linear DOS, Eq. (5.1), g I ( ω ) = 2 n M Φ(0) ω ∝ ω .For k < f is givenby Eq. (3.1). The simple analysis in Section III shows that for sufficiently small force, | f | < f ⋆k , where f ⋆k is given by Eq. (3.6), the potential (3.1) has two minima (double-wellpotential). For a large force | f | > f ⋆k the potential (3.1) has only one minimum (one-wellpotential) while for f = f ⋆k the potential is a one-well potential with a bending point.The calculation of the DOS for the lower minimum in the two-well case can beconsidered together with the one-well case ( k < , f > f ⋆ ). The position x of theminimum can be found from the equation Ax − | k | x = f. (5.13)For f > f < M ω = −| k | + 3 Ax . (5.14)9The density of states can be calculated from the Eq. (5.6). For ω ≪ ω c g II ( ω ) = 2 n Φ(0) M ω √ A Mω / Z d | k | P [ f ( | k | )] p M ω + | k | (5.15)where according to Eqs. (5.13) and (5.14) f ( k ) = Ax − | k | x = 13 r M ω + | k | A ( M ω − | k | ) . (5.16)Comparing f ( k ) with f ⋆k given by Eq. (3.6) one can see that the region of integrationin Eq. (5.15), 0 < | k | < M ω / f ( k ) > f ⋆k ) and region M ω / < | k | < M ω / f ( k ) < f ⋆k ). Taking into account Eq. (3.2) for P ( f ) andintroducing a new variable t = p | k | /M ω we finally get from Eq. (5.15) g II ( ω ) = 12 π n M Φ(0) ω ω ⋆ (cid:16) ωω ⋆ (cid:17) √ / Z dt ω/ω ⋆ ) t (3 − t ) . (5.17)As follows from this equation at small frequencies, ω ≪ ω ⋆ , g II ( ω ) ∝ ω and formoderately high frequencies satisfying the inequality ω ⋆ ≪ ω ≪ ω c , the integral inEq. (5.17) is equal to ( π/ ω ⋆ /ω ) . Therefore in this case g II ( ω ) = n M Φ(0) ω whatcoincides with Eq. (5.2).Combining results (5.9) and (5.17) we get for the total DOS at T = 0 g tot ( ω ) = g I ( ω ) + g II ( ω ) = 12 π n M Φ(0) ω ω ⋆ (cid:16) ωω ⋆ (cid:17) √ / Z dt ω/ω ⋆ ) t (3 − t ) . (5.18)We want to mention that Eq.(5.18) differs from the corresponding equation (22) of Ref. not only by the prefactor and the upper limit but also by the power of (3 − t ) inthe integrand. Thus we are correcting our error in Ref. . Fortunately this does onlymarginally alter the plots of the quoted paper, where the analytical theory is comparedto the results of simulation and experiment, since the plots of the present function g tot ( ω )and the one given in Ref. differ only slightly.For ω ≪ ω ⋆ the integral in Eq. (5.18) is equal to p / g tot ( ω ) = 12 π r n M Φ(0) ω (cid:16) ωω ⋆ (cid:17) ∝ ω , ω ≪ ω ⋆ . (5.19)Taking into account Eq. (4.2) one can present the density of states as function of energy E = ~ ω (for E ≪ E ⋆ ≡ ~ ω ⋆ ) in the following way n ( E ) = g tot ( ω ) / ~ = 1 √ n Φ(0) P (0) M / W / ~ (cid:18) EW (cid:19) . (5.20)This result for n ( E ) can be compared with Eq. (4.8) giving the density of states forTLS’s. It is clear from this comparison that for E ≫ W the density of states of HO’s0is much bigger than the density of states for TLS’s. For large frequencies ω ⋆ ≪ ω ≪ ω c the integral in Eq. (5.18) is equal to ( π/ · ( ω ⋆ /ω ) and we have g tot ( ω ) = 3 n M Φ(0) ω, ω ⋆ ≪ ω ≪ ω c , (5.21)which coincides with Eq. (5.3).Since at low frequencies ω ≪ ω ⋆ , the total DOS g tot ( ω ) ∝ ω and at high frequencies ω ∗ ≪ ω ≪ ω c , g tot ( ω ) ∝ ω we have a peak in the reduced density of states g tot ( ω ) /ω at ω ≈ ω ⋆ , the Boson peak. In the Fig. 3 we plot the function g tot ( ω ) /ω . We see fromthis figure that ω b ≈ ω ⋆ . This figure is valid only for the case ω ∗ ≪ ω c . If ω ∗ ≃ ω c FIG. 3: The Boson peak: the reduced density of states g tot ( ω ) /ω given by Eq. (5.18). ( g ( ω ) ≈ const) then ω b ≃ ω c and at ω ≫ ω c , according to our previous results (seeRef. , Eq. (27)), g tot ( ω ) ≈ g ( ω ) ≈ const. In this case the right wing of the Boson peakis determined by the initial density of states g ( ω ) and g tot ( ω ) /ω ∝ /ω (instead of g tot /ω ∝ /ω in the previous case).The DOS for the higher minimum in the double-well potential (3.1) is different.Though the thermal occupation number of this minimum is smaller than the one of thelower minimum it can contribute to the total DOS at finite temperatures. Starting fromEq.(5.6) the position of the higher minimum can be obtained as the smallest negativeroot or largest positive root of Eq. (5.13) for f > f <
0, respectively. The resultingDOS for ω ≪ ω c is g III ( ω ) = 2 n Φ(0) M ω √ A ∞ Z Mω / d | k | P [ f ( | k | )] p M ω + | k | (5.22)with f ( k ) = Ax − | k | x = 13 r M ω + | k | A (2 | k | − M ω ) . (5.23)Inserting the Lorentzian distribution for P ( f ), Eq. (3.2), and introducing a new variable t = p | k | /M ω we get for the DOS g III ( ω ) = 12 π n M Φ(0) ω ω ⋆ (cid:16) ωω ⋆ (cid:17) ∞ Z √ / dt ω/ω ⋆ ) t (3 − t ) . (5.24)1At low frequencies, ω ≪ ω ⋆ , the integral in Eq. (5.24) is equal to ( π/ · / )( ω ⋆ /ω )and g III ( ω ) = 2 / n M Φ(0) ω ⋆ (cid:16) ωω ⋆ (cid:17) ∝ ω , ω ≪ ω ⋆ . (5.25)We see that at low frequencies the dependence of g III ( ω ) (Eq. (5.24)) differs from ω dependence, the DOS in the higher minimum is proportional to ω in accordance withRef. . From Eqs. (5.19) and (5.25) it follows that g III ( ω ) /g II ( ω ) ≈ ω ⋆ /ω ≫ ω ≪ ω ⋆ and equal population of both minima the DOS in the higherminimum is larger than the one in the lower minimum. However, as we will see below,including the thermal population factors for the two minima reverses this.For high frequencies, ω ≫ ω ⋆ , the integral in Eq. (5.24) is ( π/ ω ⋆ /ω ) and g III ( ω ) = n M Φ(0) ω, ω ≫ ω ⋆ (5.26)which coincides with Eq. (5.2).So far we disregarded the thermal population factor. Taking it into account we get atemperature weighted DOS˜ g III ( ω, T ) = 2 n Φ(0) M ω √ A ∞ Z Mω / d | k | P [ f ( | k | )] p M ω + | k |
11 + exp(∆ /T ) (5.27)where ∆ is the energy difference between the minima.In the low-frequency case, ω ≪ ω ⋆ , the integral in Eq.. (5.27) is a constant plus some ω -dependent correction. To estimate the constant, we set ω = 0 in the integral. Thehigher minimum then turns into a bending point and˜ g III ( ω, T ) = 2 n Φ(0) M ω √ A ∞ Z d | k | p | k | P ( f ⋆k )1 + exp(∆ /T ) , (5.28)where ∆ = 3 k / A is the energy distance between positions of the minimum and thebending point in the potential U eff ( x ) for f = f ⋆k [see Eq. (3.6)].The result of the integration in Eq. (5.28) depends on which of the two functions P ( f ⋆k ) or [1 + exp(∆ /T )] − decays faster with | k | . The function P ( f ⋆k ) decays with acharacteristic scale | k | = k f = 3 A / ( δf ) / / / (5.29)(for f ⋆k = δf ). The function [1 + exp(∆ /T )] − decays with a characteristic scale | k | = k T = 2 p AT /
3. Both scales become equal at the temperature T = T ⋆ , where T ⋆ is givenby T ⋆ = 278 · / ( δf ) / A / = 38 · / M A ( ω ⋆ ) = 3128 · / ~ ω ⋆ (cid:18) ~ ω ⋆ W (cid:19) . (5.30)Estimates show that T ⋆ is a rather large. For example for ~ ω ⋆ = 40 K and W = 4 K wehave T ⋆ ≈
740 K.Therefore, the low-temperatures case is more realistic. For T ≪ T ⋆ , one has k T ≪ k f so that P ( f ⋆k ) = P (0) and therefore from Eq. (5.28)˜ g III ( ω, T ) = 2 √ / n Φ(0) P (0) M ω A / T / ∞ Z dy √ y
11 + e y ∝ ω T / . (5.31)2The last integral in this equation is equal 1 . ≈
1. Therefore the equation can berewritten in the form˜ g III ( ω, T ) ≈ √ · / π n Φ(0) M (cid:16) ωω ⋆ (cid:17) W ~ (cid:18) TW (cid:19) / . (5.32)Now let us compare g tot ( ω ) and ˜ g III ( ω, T ) for ω ≪ ω ⋆ . Taking into account Eqs. (5.19)and (5.32) we have g tot ( ω )˜ g III ( ω, T ) ≈ ~ ωW (cid:18) WT (cid:19) / . (5.33)Thus for T ≪ W ( ~ ω/W ) we get g tot ( ω ) ≫ g III ( ω ). In the opposite case the contributionof the higher minimum to the DOS dominates. VI. RESONANT SCATTERING OF PHONONS BY HO’s
Taking Eq. (2.4) for the coupling of the quasilocal oscillators to the phonons we get l − , HO = π Λ M ρv g ( ω ) (6.1)where l res , HO is the mean-free path of phonons due to resonant scattering on quasilocalHO’s with a density of states g ( ω ). For low frequencies, below the Boson peak frequency, ω ≪ ω ⋆ , we have from Eq. (5.20) (see also Ref. ) l − , HO = π Cωv (cid:18) ~ ωW (cid:19) ∝ ω (6.2)where C ≃ − ÷ − is the TLS’s dimensionless parameter given by Eqs. (1.4) and(4.17) (see also the estimate, Eq. (4.20)). Its value is well known from the low temper-ature properties of glasses. For high frequencies, above the Boson peak, in the interval ω ⋆ ≪ ω ≪ ω c , we have from Eq. (5.21), g ( ω ) = 3 n M Φ(0) ω . As a result l − , HO = 32 π Λ n ρv Φ(0) ωv ∝ ω. (6.3)Let us compare the last quantity (proportional to ω ) with the inverse wave length ofthe phonons λ − = ω/ πv . We have the ratio λl res , HO = 3 π Λ n ρv Φ(0) . (6.4)Using the estimatesΛ n /ρv ≈ I ≈ M ω c ω , Φ(0) ≈ F ( k c ) = g ( ω c ) / M ω c , (6.5)and ω ≃ /g ( ω ) we have λl res , HO ≈ π g ( ω c ) g ( ω ) ≈ π (cid:18) ω b ω c (cid:19) , (6.6)3the last estimate follows from Eq. (5.11). Thus this ratio is a constant in the interval ω ∗ < ω < ω c and depends only on the characteristic frequency ω c ∝ I and the behaviorof the initial DOS g ( ω ). From the last equation it follows that ratio λ/l res , HO dependson the cube of the ratio of two important frequencies, the Boson peak frequency ω b ≈ ω ∗ and the characteristic frequency ω c . Both of them can be measured on experiment (seeEq. (4.22) and Ref. ).For the weak interaction I which we consider in the paper ω c ≪ ω and if for ω → g ( ω ) also goes to zero sufficiently rapidly, then g ( ω c ) ≪ g ( ω ) and λ ≪ l res , HO , i.e. resonant phonon scattering is also weak. However, due to big numericalcoefficient in Eq. (6.6) in some realistic cases we can have a strong phonon scattering.For example if the initial DOS g ( ω ) ∝ ω and ω c ≈ ω /
3, then λ/l res , HO ≈ π / ≈ . . (6.7)In this case criterion of Ioffe-Regel for the phonons ( l res , HO < λ ) is approximately satisfiedand we have strong phonon scattering above the Boson peak frequency in the interval ω ∗ ≪ ω ≪ ω c (which is not too big since in this case ω c ≈ ω ∗ ).Another interesting case is a flat initial DOS, g ( ω ) ≈ const, then g ( ω c ) ≃ g ( ω )and as follows from Eq. (5.11) ω ∗ ≃ ω c and interval [ ω ∗ , ω c ] shrinks to one point (Bosonpeak frequency, ω b ≃ ω c ) and at the Boson peak frequency we have λl res , HO = 32 π ≈ . (6.8)In this case the criterion of Ioffe-Regel is again satisfied and we have at the Boson peakfrequency ω ∗ ≃ ω c the regime of very strong resonant scattering of phonons on quasilocalharmonic oscillators, independent of the strength of interaction I (and ω c ). However,it is necessary to stress that in this case the strong scattering takes place only in thevicinity of the Boson peak frequency ω c .At higher frequencies the initial DOS g ( ω ) ≃ const and according to Eq. (6.1)the resonance phonon mean free path is also constant and independent of frequency.However the phonon wave length λ ∝ /ω decreases with frequency. Therefore theregime of the weak phonon scattering will recover again at higher frequencies ω ≫ ω c .Similar behavior was observed in Ref. for resonant scattering of phonons on librational(quasilocal) modes in crystals. We can extend the regime of strong scattering to wellabove the Boson peak frequency (up to Debye frequency ω ) only when the initial DOSis a linear function of the frequency, g ( ω ) ∝ ω and interaction I is not too small, ω c > ω /
15 (see Eq. (6.6)).As we already mentioned, we will have a weak resonant scattering of phonons onquasilocal oscillators only when the interaction I is sufficiently weak and the initialdensity of states g ( ω ) decreases to zero sufficiently fast with ω , so that g ( ω c ) ≪ (2 / π ) g ( ω ). In such a case the mean free path of the phonons l res , HO will be muchlarger than their wave length λ in the whole frequency range. In this case phonons arewell defined quasiparticles everywhere.We give here also the relaxation time τ of a HO with frequency ω due to the interactionwith phonons. From Eq. (2.4) we get1 τ = Λ ρv ω πM v = J ω πM v . (6.9)Estimating Λ ≃ E /a, ρv ≃ E /a , M v ≃ E , ~ v/a ≃ ~ ω , (6.10)4where E ≃
10 eV is of the order of atomic energy, a ≃ ω is of the order of Debye frequency we get1 /ωτ ≃ ω/ πω ≪ . (6.11)Therefore HO’s with ω ≪ ω are well defined objects. VII. DISCUSSION
In our previous papers we proposed a mechanism of the Boson peak formation.The essence of the mechanism can be formulated as follows. A vibrational instabilityof the weakly interacting QLV’s (stabilized by the anharmonicity) is responsible forthe Boson peak in glasses and other disordered systems. The instability occurs belowsome frequency ω c proportional to the strength of the interaction I between low andhigh frequency oscillators. Whereas anharmonicity is essential in creating the atomicstructures supporting the Boson peak, the vibrations forming the peak in the inelasticscattering intensity or the reduced density of states are essentially harmonic.The present paper extends these ideas. We show that such seemingly unrelated phe-nomena in glasses (typical for the glassy state and usually treated by separate unrelatedmodels) as the formation of the two-level systems and the Boson peak in the reduceddensity of low-frequency vibrational states g ( ω ) /ω can be explained by the same phys-ical mechanism, namely the vibrational instability of weakly interacting soft harmonicvibrations. These can be seen as localized vibrations with a bilinear interaction withthe extended modes, the sound waves. The resulting exact harmonic eigenmodes arequasilocalized vibrations that have been observed in numerous computer simulationsand have been discussed extensively — see Ref. and the references therein.The instability, which as in all solids is controlled by the anharmonicity, creates anew stable universal spectrum of harmonic vibrations with the Boson peak feature aswell as double-well potentials with a wide distribution of the barriers heights that isdetermined by the strength of the interaction I between the oscillators. Dependingon the barrier height (and temperature) these will lead to tunneling and relaxationaltransitions. To check for the consistency of our theory we calculated the dimensionlessparameter C = P γ /ρv ≈ − for the two-level systems in glasses which is observed inexperiment . The smallness of this parameter is a longstanding puzzle. In our theoryit follows naturally. The physical reason for small value of the parameter C is that onlya small fraction of all created TLS’s can actually tunnel in realistic timescales.We show that the larger is the interaction I between the original harmonic vibrationsthe smaller is parameter C . It reminds partly the ideas of Ref. though we do not havehere the frustrated strong interactions. We prove that for our simple model weakly in-teracting oscillators C = ( W/ ~ ω c ) ∝ I − . Here W is an important characteristic energyin glasses of the order of a few Kelvin. The value of C is independent of the assumedinitial DOS of HO’s g ( ω ) and in this sense it is universal. Varying the characteristicenergy W and interaction I (i.e. the characteristic energy ~ ω c ) for different glasses C lies in the interval from 10 − to 10 − . However, we want to stress that we are not free inthe choice of these two parameters. The energy W is well known from experiments onspecific heat , thermal conductivity and heat release in glasses. As for the charac-teristic frequency ω c it should be of the order or larger than the Boson peak frequency ω b .In the unified approach developed in the present paper the densities of tunnelingstates and of excess vibrational states at the Boson peak frequency are interrelated.5Since the experimental values of C and W are well known for many glasses we canuse this formula to get the important energy ~ ω c = W C − / giving us the onset of thevibrational instability region. For vitreous silica ~ ω c ≈
60 K falling perfectly into theBoson peak range. The same holds for many other glasses . It indicates that the Bosonpeak is indeed placed inside the vibrational instability range.It is instructive to compare the results of the present paper and of Refs. with ourearlier paper where we also discuss the possible origin of the Boson peak. In thispaper we consider low-energy Raman scattering in glasses. As in the present paperand in Refs. we assume that the scattering and the energy transfer are due to theinteraction of the light with the soft potentials in glasses. The density of states of thequasilocalized HO’s, according to Ref. , is proportional to ω for low frequencies andto ω for high frequencies. This behavior qualitatively resembles the one obtained inRef. . However, the phenomenon of vibrational instability was disregarded in and thediscussion of the Boson peak was necessarily somewhat qualitative. Considering thevibrational instability, puts the theory on a more quantitative level and, for instance,allows the determination of the shape of the Boson peak.Let us now compare the results of our paper with previously published importantclass of models of the Boson peak . Experiment has shown that the Boson peakis formed by largely harmonic vibrations. Therefore, in all these models the authorshave considered an Hamiltonian of the form U tot ( x , x , ...x n ) = 12 X i,j = i k ij ( x i − x j ) (7.1)with randomly distributed quasielastic constants k ij . Since this potential energy is purelyharmonic we call all such models of the Boson peak harmonic random matrix (HRM)models. The main difference between the quoted four HRM models is in the differentdistributions of the quasielastic constants k ij .If all quasielastic constants are positive, k ij >
0, then the corresponding dynami-cal matrix (Hessian) is positive-definite and therefore all the eigenvalues are obviouslypositive as well, ω i > i = 1 , , ...n ) excluding those zeroes which come from the trans-lational and rotational invariances. In such a case the system is mechanically stable .As was shown in the system remains to be stable even when some (rather small)fraction of the quasielastic constants k ij in Eq. (7.1) is negative (and small enough). In-creasing the fraction of negative k ij (or their absolute values) authors have approachedthe mechanical stability threshold (when the first negative ω has appears in the spec-trum).The reduced density of states g ( ω ) /ω for Hamiltonian (7.1) usually has a maximumat some frequency ω max for typical values of k ij >
0. Changing the parameters of thedistribution function P ( k ij ) one can shift this maximum to higher or to low frequen-cies. The last case was the main goal of the papers . The biggest red shift of themaximum has been achieved approaching the mechanical stability threshold. In the firsttwo papers the original maximum was due to Van-Hove singularity of the crystallineDOS. Atoms in these papers were placed on a perfect cubic lattice. As a result theBoson peak has been ascribed to the lowest Van Hove singularity shifted due to disor-der. In another two papers atoms were distributed randomly in 3-d space (so-calledEuclidean Random Matrix models) and quasielastic constants k ij depend exponentiallyon the interatomic distances | r i − r j | . Therefore the distribution function of quasielasticconstants in these two cases P ( k ) ∝ /k and has a singularity for k →
0. Thereby theportion of small k was increased compared to Gaussian and box distributions used inthe papers . Obviously the red shift of the original peak in g ( ω ) /ω was much morepronounced in these latter two cases .6Our approach, which continues from our previous papers differs essentially in twoways. First, we postulate that the excess in vibrational modes originates from quasi-localized vibrations. Their existence has been shown in numerous simulations of differenttypes of materials. Such modes can be described as local modes (cores) which weakly in-teract bilinearly with the extended modes (sound waves) and thus with each other. Theexact harmonic eigenvectors are of course extended as in HRM models. Secondly, we donot invoke special distributions for the elements of the dynamical matrix to avoid un-stable vibrations but, on the contrary, show that the generic instability, when controlledby the anharmonicity which is present in all real systems, automatically gives both theTLS and the Boson peak (with a universal shape) without any further assumptions.The essence of the mechanism can be formulated as follows. The randomly distributedweakly interacting QLV become unstable at low frequencies in harmonic approximation.This is the equivalent to the instability in the general HRM models. Instead of assump-tions on distribution functions of interactions I ij we use the always present anharmonicityas the stabilizing factor. The previous vibrational instability of the weakly interactingQLV’s thus becomes responsible for the Boson peak and TLS’s in glasses and other dis-ordered systems. Whereas anharmonicity is essential in creating the atomic structuressupporting the Boson peak, the vibrations forming the peak in the inelastic scatteringintensity or the reduced density of states are essentially harmonic. Comparing with ran-dom matrix models our Eq. (2.2) without the anharmonicity term would correspond tothe case of the unstable random matrix whereas the result of Eq. (2.16) correspond tothe stabilized case. And, importantly, the anharmonicity strength A (thanks to mirrortransformation) does not enter in the expression for the renormalized frequency. So,the anharmonicity reconstructs the spectrum but the final result is independent of thestrength of anharmonicity. The advantage of our approach is that the stabilization isnot a result of an additional assumption but is a benefit of the vibrational instability +anharmonicity which is haunting the alternative approach.Summarizing briefly we can say that the Boson peak in papers was obtainedby a purely harmonic ansatz inside the mechanically stable region of their Hamiltonians.The position and the form of the peak depend strongly on the distribution functionof quasielastic constants P ( k ij ). In our approach the Boson peak is built inside themechanically unstable region of the harmonic potential parameters. Therefore, the role ofanharmonicity as stabilizing factor is crucial. But the form of the Boson peak appears tobe universal and independent of the initial assumptions about interaction I ij or the initialdistribution function g ( ω ) or of the anharmonicity strengths A i . Relating TLS’s andBoson peak parameters in our theory we are able to show that the instability crossoverfrequency ω c given by Eq. (1.14) lies in the Boson peak region. Due to weakness ofthe interaction I the universal reconstruction of the spectrum in our theory takes placein the low frequency range only, leaving high frequency range nearly unchanged. Thisis different in the HRM models where the whole spectral range is completelyreconstructed in the course of the change of the distribution function P ( k ij ).As mentioned before in our theory the estimate for the dimensionless tunnelingstrength C emerges in a naturally (see Eq. (4.20)). Varying the characteristic energy W and the interaction strength I for different glasses C falls into the interval between 10 − and 10 − . We want to mention that other explanations of the smallness of the constant C have been proposed. Burin and Kagan predict a number of universal propertiesof amorphous solids, including the small values of the tunneling strength C , due to aspecial form of interaction of such defect centers with internal degrees of freedom like theTLS’s. It is important that the interaction between such centers falls off with distance r as 1 /r . In this case effects of correlations between many TLS’s ( dipole gap effects)might be important at sufficiently low temperatures.7In our paper we have neglected all such many particle correlation effects betweentunneling TLS’s which could further reduce (or stabilize) the value of C . One should,however, keep in mind that the 1 /r law of interaction is valid as far as the effectsof retardation play no role. In dielectric glasses these effects are determined by thesound velocity. At sufficiently large distances they could play a role that would result inthe variation of the interaction law. We believe that the role of the retardation effectsdeserves a special investigation.Another important difference between our and Burin and Kagan approaches is thatthe parameter C in our theory is a quantum mechanical quantity, C ∝ ~ (see Eq. (4.24)).It disappears in the classical limit ~ →
0. In other words the smallness of C is directlyrelated to the smallness of the quantum mechanical probability for particles tunnelingthrough high barriers in a glass (see also Eq. (4.33)). Differently, the dipole gap effectis purely classical since it is based on the classical dipole-dipole interaction betweenTLS’s. Therefore, it would be very interesting to elucidate which of two mechanisms(or both) dominates and is responsible for the small value of parameter C in glasses. Inparticular it would be very interesting to calculate the relaxation times for many particlecorrelations effects to build the dipole gap.An analysis of the low temperature properties of glasses along the lines of the softpotential model based on a numeric simulation of a Lennard-Jones glass was presentedby Heuer and Silbey . They numerically searched for the energy minima of the glass andconstructed double-well potentials for close minima. By extrapolation to small valuesthey were able to extract the distribution functions for the soft potential parameters.These potentials correspond to our potentials after inclusion of the interaction betweenthe HO’s. The present theory is in agreement with their simulation results.The theory presented in this paper deals with the effects of soft modes produced bydisorder which can be expected to have a broad frequency distribution. In the literaturethe term Boson peak is rather loosely defined. Often it is used for any low frequencymaximum in the reduced DOS. In particular in plastic crystals, see eg. Ref. , soft HOare present even before disorder. Consequently disorder only broadens their sharp DOS.Depending on the strength of this broadening our theory will apply more or less to suchcases. The same applies to TLS which also can be present before disorder.In summary, we have shown that the same physical mechanism is fundamental forsuch seemingly different phenomena as formation of the two-level systems in glasses andthe Boson peak in the reduced density of low-frequency vibrational states g ( ω ) /ω . Inthis way two of the most fundamental properties of glasses are interconnected. Acknowledgments
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