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Tunnelling states and anharmonic resonant modes in glasses
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Solid State Communications, Vol. 56, No. 10, pp. 889-893, 1985. Printed in Great Britain.
TUNNELLING STATES AND ANHARMONIC RESONANT MODES IN GLASSES U. Buchenau Institut fiir FestkOrperforschung, Kernforschungsanlage Jtilich, Postfach 1913, 5170 Jiilich, West Germany
(Received 6 August 1985 by P.H. Dederichs) The tunnelling states in amorphous solids are explained in terms of coupled rotations of molecular or quasimolecular groups. The elastic stresses frozen in at the glass transition lead to a broad distribution of restoring forces with a finite probability for very small and even negative values. In the latter case, the anharmonic terms give rise to an asymmetric doubleminimum potential as required for the tunnelling. The model predicts strongly anharmonie resonant modes which could explain the low temperature glass anomalies above 1 K. For vitreous silica, the calculated densities of tunnelling states and resonant modes agree reasonably well with measured values. THE LOW-TEMPERATURE PROPERTIES of amorphous solids are dominated by specific low-frequency excitations of the glassy state [1 ]. At the low-frequency end, these excitation have been shown to be two-level states [2]. Though there is a well-established tunnelling model [3, 4] for those two.level states, the nature of the tunnelling motion is still unknown. Even less is known about the additional modes at slightly higher frequencies (> 100 GHz), which give rise to the anomalous TO-term in the specific heat [1] and are possibly responsible for the plateau in the thermal conductivity [5, 6] around 20 K. An appealing explanation for these excitations is the crossover from sound waves at lower frequencies to vibrations of fractals, "fractons", which leads to an enhanced vibrational density-of-states in the crossover region [7]. Since the fractons are localized excitations, this would explain the plateau in the thermal conductivity. It is, however, difficult to see how one can get fractals in a continuous random network. An alternative possibility [1, 8] is to look for a common explanation for both tunnelling and low frequency vibrational modes. From this point of view, which will be adopted in this Letter, the low frequency vibrations are resonant modes with the same eigenvector as the tunnelling modes. The type of motion proposed for both tunnelling and resonant modes is shown schematically in Fig. l(a). It consists essentially of a coupled rotation of two neighbouring molecular or quasimolecular units. The picture of coupled rotations emerges from neutron scattering measurements of the low frequency vibrations in vitreous silica [9]. This kind of mode has a low frequency to start with in many glasses. The restoring force for the motion in Fig. 1 depends on the local molecular arrangement and on the local stress pattern. 889
~(tt) \\z
%/
.,2"
v_.,.
(c)
¢ Fig. 1. (a) Coupled rotation of two neighbouring molecular units in a glass (schematic). (b) Component of the elastic dipole force tensor which couples to the rotational motion. (c) Quadmpolar restoring force of the rotational motion. If the surrounding medium exerts compressive forces on the two outer atoms as shown in Fig. l(b), the restoring force is lowered. For low enough restoring force, the local motion decouples from the rotational motion of the neighbouring molecules and becomes a resonant mode [10]. This is an important point; the localization requires strong local deviations from the
890
TUNNELLING STATES AND ANHARMONIC RESONANT MODES IN GLASSES
average force constants. In principle, the localization is never complete because there is always the interaction with the sound waves even at the lowest frequencies. That is why these modes are called resonant rather than localized. For still stronger compression, the total restoring force against the coupled rotation becomes negative and a local structural instability is created. This marks the transition from the resonant modes to the tunnelling regime. These statements can be put on a more quantitative basis using the concept of the static elastic Greens function [11] much in the same way as it is used to calculate the elastic deformations around defects in crystals. The restoring forces against the coupled rotation in Fig. l(a) are given by the rotational potential
Vol. 56, No. 10
Let us begin with the determination of the destabilizing influence of the double force pair of Fig. l(b). (Naturally, at one half of these atom pairs the forces will not point inwards but outwards; here, however, we are especially interested in the destabilizing case). Assuming incompressible bonds, the distance between the two outer atoms is 2rocos 4. Let ~a denote the rotational angle at which the double force disappears. By the simple calculation indicated above one gets
Va(~) = 2rrC44 r3o cos if(cos ~b -- cos ~a) 2.
(6)
This is a symmetric double minimum potential with minima at -+ ~a, i.e. exactly the kind of potential that we need to explain the occurrence of tunnelling states. It is interesting to note that this highly anharmonic potential is obtained without invoking any anharmonic V(~b) = Vbb + Ira+ Vq, (I) atom-atom potential. where Vbb is the bond bending potential of the two The quadrupolar contribution V~ (Fig. l(c)) can be central bonds and Ira and Vq are the elasticenergies of determined accordingly assuming opposite torques the dipolar compressive force of Fig. l(b) and the acting again at the two outer atoms. The calculation of quadrupolar restoring force of Fig. I(c), respectively. the displacement field and the energy is simple because qJ is the rotational angle as defined in Fig. l(c). The there is only one nonzero element of the elastic quaddisplacement vector s(r) of the elastic deformation rupole tensor, namely PI12. This deformation contains can be calculated from the expression both shear and compression. Therefore both Cn and C44 enter the equations. One can simplify the expressions by assuming Cn = 4 C44. Then for small Si(r ) = -- ~" (~Gii/axk)p.i k le
Vo(~b) = (4zr/7)Ca4rg(~O -- ~q)2,
o a/doxkax3Pjk,,
+ 0/2)
(2)
kl
(i,], k, l = 1. . . . . 3). The elastic Greens function G/,i, the dipolar tensor element P,i~ and the quadrupole tensor element P,ikl are given by the following equations (3, 4 and 5) Gi,i(r) = (1/8nr)(1/C44 + 1/Cn)(8i,i - x i x , i / r 2)
+ (2/C44)(xixjj/r2),
(3)
(Cn, C44 elastic constants of the glass)
Pjk = E x~n)ff n),
where ~0q denotes the angle at which the quadrupolar force disappears. Again as in the case of Ca, ~bq will exhibit a distribution which is determined by the stresses frozen in at the glass temperature Tg. The balance between the destabilizing effect of the elastic dipole and the stabilizing restoring force of the elastic quadrupole turns out to depend on the value of the bond length ro. Assuming a relatively strong dipole
Va(O) = 4 k Tg,
(8)
the optimum bond length for a strong negative force constant at ~ = 0 is given by (4)
rl
rg = 49 k rg/(21rC44).
(f~n) force acting at the atom n with the coordinates
x~n)) and
6k, = Y x n>xP) ">.
(7)
(5)
n
By assuming these equations to hold down to atomic distances, one can calculate the displacements for given atomic forces and consequently the corresponding energies. The energies obtained in this simple way are reasonably accurate [11 ]. It is obvious that this approach is especially suited for the elastically isotropic glasses.
(9)
For vitreous silica with Tg = 1450K and C44 = 31 GPa, r0 = 1.5A, close to the S i - O bond length of 1.6A. However, even in the favourable cases where equation (9) holds, the negative contributions to the force constant are not very strong. The harmonic force constant D2
=
(1/2)~2VlS~k2lqj=o
=
--7kTg+D2bb.
(10)
Thus the bond bending force constant D2bb must be either rather weak or even negative in order to explain the tunnelling states and the resonant modes. As shall be seen, it is in fact negative in vitreous silica.
Vol. 56, No. 10
TUNNELLING STATES AND ANHARMONIC RESONANT MODES IN GLASSES
The rotational potential (equation (1)) is conveniently written in the form V(~0) = D l ~ +D2~b 2 + D 4 ~ / 4 ,
(11)
where DI and D2 exhibit more or less uniform distributions around zero. D4 can be considered to have a fixed value given by 04 = 7zrC44rao/12 + D4bb,
(12)
where again D4bb is the bond bending contribution. The effective momentum of inertia of the motion can be calculated from the displacement fields. Since these decrease rapidly with increasing distance from the bond, one gets a good estimate by taking only the motion of the three central atoms into account. One finds that the two outer atoms move opposite to the central atom with 1/6 of the amplitude of the central atom. Therefore 0elf = (36mc + 2mo)r~/49,
(13)
where m e and mo are the masses of the central and the outer atom respectively. It is convenient to express the force constants in units of D0 with
Do = D~/a(h2/Oefr) 1/a.
(14)
A characteristic frequency of the problem is Vo, where
hvo = D~/a(h2/Oerf) z/a.
(15)
In the case D~ = D2 = 0 the level splitting between the two lowest levels turns out to be 1.8 huo. Since ~o is of the order of 1 THz, the resonant modes at 100GHz must already have a strongly negative D2 in order to achieve a lowering of the level spacing. Thus the transition from resonant modes to tunnelling states must be continuous. For D~ < - 3Do and small Dx one can straightforwardly reformulate the well-known equation [4] for the tunnel splitting A o in terms of D2 and ~'o. One obtains Ao = (2/V/-~)hi:od s/4 exp (-- da/2),
(16)
where d=--D2/Do. The basic assumption of the tunnelling model is a uniform distribution in the exponential of equation (16). The arguments given here do not lead directly to a constant distribution function. However, since D2 results from a sum of positive and negative contributions, we will expect a slow variation in the distribution function of d and consequently of d 3/2 with varying d. In this sense, the present explanation confirms the central assumption of the tunnelling model. It is obvious that the modes considered here will exhibit a strong negative mode Griineisen parameter.
891
This explains the negative thermal expansion observed in several glasses at low temperatures [1, 12]. For vitreous silica with Do = 0.2 eV (see below), equations (6) and (16) yield a mode Griineisen parameter o f - - 18 for the tunnelling states with splittings corresponding to I K, close to the experimental value [12] o f - - 2 0 . Another experimental result which finds a natural explanation is the observation of tunnelling states in crystalline materials doped with impurities having permanent elastic dipole moments [13]. These will introduce the irregular stress patterns which are normally absent in a crystal and which in a glass are frozen in at the glass transition. In order to estimate the densities of tunnelling states and resonant modes, one has to specify the bond bending contributions which vary from glass to glass. In vitreous silica, the bond bending potential seems to have a minimum at ~ = 18 ° ( S i - O - S i angle ~0= 144 °) as evidenced by theoretical calculations [14] and by the angles found in the different crystalline modifications [15] of SiO2. The restoring force to that minimum position can be described by a longitudinal force constant of 80N/m between the two Si-atoms [16]. Since the energy is independent on the sign of ~, this is a double minimum potential with a negative value of D2bb and a positive D4bb. From the values of the force constant and the minimum angle one gets D:bb = -- 1.26 eV and D4bt~ = 6.4 eV. From equation (13), h2/O = 0.125 meV. These values inserted in equations (12), (14) and (16) yield Do = 0.2 eV and a characteristic frequency Vo = 1.2 THz. The distribution of rotational potentials was calculated assuming equal a priori probabilities for all possible elastic dipole and elastic quadrupole configurations and then weighing the different configurations according to their Boltzmann factor at the glass temperature. The Boltzmann factor was determined from equation (1) with equations (6) and (7), taking the energy at its minimum with respect to the bond angle. According to this calculation, 8.3% of the S i - O - S i bonds in vitreous silica have a double minimum potential. Fer a rough estimate, this number may be identified with the number of resonant modes below the characteristic frequency ~o. With two bonds per SiO2-unit, this corresponds to 2% of the total vibrational density-of-states in good agreement with experiment [9] (0.8%). A detailed discussion of the potential distribution functions is beyond the scope of this Letter. A fraction of 10 -4 of the bonds exhibited both an asymmetry below 8 meV (88 K) and a barrier height corresponding to tunnel splittings between 1/aK and 1 K. This yields 2 - 1 0 -6 tunnelling states per SiO2 unit with level splittings below 1 K, about twice the value deduced from the specific heat. Thus the model considered here
892
TUNNELLING STATES AND ANHARMONIC RESONANT MODES IN GLASSES
I
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Fig. 2. Relative position of two neighbouring chains in crystalline polyethylene. The chain axes are perpendicular to the drawing plane. The dashed lines mark schematically the hard core interaction contours. The possible coupled rotation is indicated by arrows. reproduces not only the density of the resonant modes, but also the density of tunnelling states. This is achieved without any adjustable parameters; the parameters inserted were values measured either in vitreous silica or in its crystalline modifications. One might question the bond bending potential parameters D2bb and D4~b which were obtained rather indirectly. However, the reported results are not strongly dependent on the exact values of D2~b and D4bb. Only when D2bb becomes positive the tunnelling and resonant mode densities decrease drastically. This result indicates that an inherent asymmetry of the bond is important in order to create the local structural instabilities. The mechanism considered here will naturally apply equally well to the whole class of silica glasses. There is a second class of amorphous solids where it can be argued to apply, namely the amorphous polymers. In the polymer case, it is easy to convince oneself that a given single chain conformation is frozen at low temperatures and that the strong bond bending force constants within the chain are not easily destabilized. In contrast, the forces between different chains are weak. The packing of neighbouring chains resembles the packing in crystalline polymers. Figure 2 shows the crystalline packing in polyethylene. The difference is that in the crystalline case the chains lie side by side for many monomers, while in the amorphous case the chain folds away after only a few monomers to join new neighbours. Two neighbouring chain segments are kept apart at a more or less constant distance by the hard core repulsion of the protons (or eventually side groups). They can, however, rotate around one of the protons as indicated in Fig. 2. This motion is very similar to the motion of Fig. 1. The distance 2ro in Fig. 1 would correspond to the chain distance in Fig. 2. In the case of amorphous polystyrene with Tg = 400 K
Vol. 56, No. 10
and C44 = 2.5GPa one calculates from equation (9) an optimum bond length ro = 2.5 A which is close to half the interchain distance. Moreover, looking at the hard core interaction contours drawn schematically in Fig. 2, one realizes that the rotational potential from the chain interaction will have low-lying maxima with negative values of the harmonic force constant D2b~. Thus one might postulate that the tunnelling states and the resonant modes in the amorphous polymers will be coupled rotational motions of neighbouring chain segments. This explanation should be tested by measuring the inelastic structure factor of the resonant modes with neutrons, quite in the same way as it has been done for vitreous silica, to see whether it can be really described by a coupled rotational motion of the polymer chains. In conclusion, the tunnelling states and the low frequency vibrational modes in glasses have been explained in terms of coupled rotations, taking into account the elastic distortions around the librating units. For vitreous silica, this explanation gives correct orders of magnitude for the densities of both tunnelling states and resonant modes without any adjustable parameter.
Acknowledgements -
Helpful discussions with H. Grimm, S. Hunklinger, J. J~ickle, H.R. Schober and T. Springer are gratefully acknowledged. REFERENCES
1. Amorphous Solids-Low Temperature Properties, 2. 3. 4. 5. 6. 7. 8. 9.
10.
Topics in Current Physics, Vol. 24 (Edited by W.A. Phillips), Springer, Berlin (1981). S. Hunklinger & W. Arnold, Physical Acoustics, Vol. XII, p. 155 (Edited by W.P. Mason & R.N. Thurston), Academic Press, New York (1976). P.W. Anderson, B.I. Halperin & C.M. Varma, Phil. Mag. 25, 1 (1972). W.A. Phillips, J. Low Temp. Phys. 7,351 (1972). M.P. Zaitlin & A.C. Anderson, Phys. Rev. B12, 4487 (1975). J. Jiickle, The Physics of Non-C'~ystalline Solids, p.568, (Edited by G.H. Frischat), Transtech, Aedermannsdorf (1977). B. Derrida, R. Orbach & Kin-Wah Yu, Phys. Rev. B29, 6645 (1984). V.G. Karpov & D.A. Parshin, Pis'ma Zh. Eksp. Teor. Fiz. 38, 536 (1983); JETP Lett. 38, 647 (1983). U. Buehenau, N. Nticker & A.J. Dianoux, Phys. Rev. Lett. 53, 2316 (1984) and U. Buchenau, M. Prager, N. Nticker & A.J. Dianoux (unpublished). A.A. Maradudin, E.W. Montroll, G.H. Weiss & I.P. Ipatova, Theory of Lattice Dynamics in the
Harmonic Approximation, Solid State Physics: Advances in Research and Applications, Academic, 11.
New York, (1971). G. Leibfried & N. Breuer, Point Defects in Metals /, Springer Tracts in Modern Physics, Vol. 81,
Vol. 56, No. 10
12. 13.
TUNNELLING STATES AND ANHARMONIC RESONANT MODES IN GLASSES
p. 138 if, (Edited by G. H6hler), Springer, Berlin, Heidelberg, New York (1978). G.K. White, S.J. Collocott & J.S. Cook, Phys. Rev. B29, 4778 (1984). D.A. Ackerman, D. Moy, R.C. Potter, A.C. Anderson & W.N. Lawless, Phys. Rev. B23, 3886 (1981).
14. 15. 16.
893
M.D. Newton & G.V. Gibbs, Phys. Chem. Minerals, 6,221 (1980). H.D. Megaw, Crystal Structures: A Working Approach, pp.259-269, Saunders, Philadelphia, (1973). T.H.K. Barron, C.C. Huang & A. Pastemak, J. Phys. C9, 3925 (1976).
Solid State Communications, Vol. 56, No. 10, pp. 889-893, 1985. Printed in Great Britain.
TUNNELLING STATES AND ANHARMONIC RESONANT MODES IN GLASSES U. Buchenau Institut fiir FestkOrperforschung, Kernforschungsanlage Jtilich, Postfach 1913, 5170 Jiilich, West Germany
(Received 6 August 1985 by P.H. Dederichs) The tunnelling states in amorphous solids are explained in terms of coupled rotations of molecular or quasimolecular groups. The elastic stresses frozen in at the glass transition lead to a broad distribution of restoring forces with a finite probability for very small and even negative values. In the latter case, the anharmonic terms give rise to an asymmetric doubleminimum potential as required for the tunnelling. The model predicts strongly anharmonie resonant modes which could explain the low temperature glass anomalies above 1 K. For vitreous silica, the calculated densities of tunnelling states and resonant modes agree reasonably well with measured values. THE LOW-TEMPERATURE PROPERTIES of amorphous solids are dominated by specific low-frequency excitations of the glassy state [1 ]. At the low-frequency end, these excitation have been shown to be two-level states [2]. Though there is a well-established tunnelling model [3, 4] for those two.level states, the nature of the tunnelling motion is still unknown. Even less is known about the additional modes at slightly higher frequencies (> 100 GHz), which give rise to the anomalous TO-term in the specific heat [1] and are possibly responsible for the plateau in the thermal conductivity [5, 6] around 20 K. An appealing explanation for these excitations is the crossover from sound waves at lower frequencies to vibrations of fractals, "fractons", which leads to an enhanced vibrational density-of-states in the crossover region [7]. Since the fractons are localized excitations, this would explain the plateau in the thermal conductivity. It is, however, difficult to see how one can get fractals in a continuous random network. An alternative possibility [1, 8] is to look for a common explanation for both tunnelling and low frequency vibrational modes. From this point of view, which will be adopted in this Letter, the low frequency vibrations are resonant modes with the same eigenvector as the tunnelling modes. The type of motion proposed for both tunnelling and resonant modes is shown schematically in Fig. l(a). It consists essentially of a coupled rotation of two neighbouring molecular or quasimolecular units. The picture of coupled rotations emerges from neutron scattering measurements of the low frequency vibrations in vitreous silica [9]. This kind of mode has a low frequency to start with in many glasses. The restoring force for the motion in Fig. 1 depends on the local molecular arrangement and on the local stress pattern. 889
~(tt) \\z
%/
.,2"
v_.,.
(c)
¢ Fig. 1. (a) Coupled rotation of two neighbouring molecular units in a glass (schematic). (b) Component of the elastic dipole force tensor which couples to the rotational motion. (c) Quadmpolar restoring force of the rotational motion. If the surrounding medium exerts compressive forces on the two outer atoms as shown in Fig. l(b), the restoring force is lowered. For low enough restoring force, the local motion decouples from the rotational motion of the neighbouring molecules and becomes a resonant mode [10]. This is an important point; the localization requires strong local deviations from the
890
TUNNELLING STATES AND ANHARMONIC RESONANT MODES IN GLASSES
average force constants. In principle, the localization is never complete because there is always the interaction with the sound waves even at the lowest frequencies. That is why these modes are called resonant rather than localized. For still stronger compression, the total restoring force against the coupled rotation becomes negative and a local structural instability is created. This marks the transition from the resonant modes to the tunnelling regime. These statements can be put on a more quantitative basis using the concept of the static elastic Greens function [11] much in the same way as it is used to calculate the elastic deformations around defects in crystals. The restoring forces against the coupled rotation in Fig. l(a) are given by the rotational potential
Vol. 56, No. 10
Let us begin with the determination of the destabilizing influence of the double force pair of Fig. l(b). (Naturally, at one half of these atom pairs the forces will not point inwards but outwards; here, however, we are especially interested in the destabilizing case). Assuming incompressible bonds, the distance between the two outer atoms is 2rocos 4. Let ~a denote the rotational angle at which the double force disappears. By the simple calculation indicated above one gets
Va(~) = 2rrC44 r3o cos if(cos ~b -- cos ~a) 2.
(6)
This is a symmetric double minimum potential with minima at -+ ~a, i.e. exactly the kind of potential that we need to explain the occurrence of tunnelling states. It is interesting to note that this highly anharmonic potential is obtained without invoking any anharmonic V(~b) = Vbb + Ira+ Vq, (I) atom-atom potential. where Vbb is the bond bending potential of the two The quadrupolar contribution V~ (Fig. l(c)) can be central bonds and Ira and Vq are the elasticenergies of determined accordingly assuming opposite torques the dipolar compressive force of Fig. l(b) and the acting again at the two outer atoms. The calculation of quadrupolar restoring force of Fig. I(c), respectively. the displacement field and the energy is simple because qJ is the rotational angle as defined in Fig. l(c). The there is only one nonzero element of the elastic quaddisplacement vector s(r) of the elastic deformation rupole tensor, namely PI12. This deformation contains can be calculated from the expression both shear and compression. Therefore both Cn and C44 enter the equations. One can simplify the expressions by assuming Cn = 4 C44. Then for small Si(r ) = -- ~" (~Gii/axk)p.i k le
Vo(~b) = (4zr/7)Ca4rg(~O -- ~q)2,
o a/doxkax3Pjk,,
+ 0/2)
(2)
kl
(i,], k, l = 1. . . . . 3). The elastic Greens function G/,i, the dipolar tensor element P,i~ and the quadrupole tensor element P,ikl are given by the following equations (3, 4 and 5) Gi,i(r) = (1/8nr)(1/C44 + 1/Cn)(8i,i - x i x , i / r 2)
+ (2/C44)(xixjj/r2),
(3)
(Cn, C44 elastic constants of the glass)
Pjk = E x~n)ff n),
where ~0q denotes the angle at which the quadrupolar force disappears. Again as in the case of Ca, ~bq will exhibit a distribution which is determined by the stresses frozen in at the glass temperature Tg. The balance between the destabilizing effect of the elastic dipole and the stabilizing restoring force of the elastic quadrupole turns out to depend on the value of the bond length ro. Assuming a relatively strong dipole
Va(O) = 4 k Tg,
(8)
the optimum bond length for a strong negative force constant at ~ = 0 is given by (4)
rl
rg = 49 k rg/(21rC44).
(f~n) force acting at the atom n with the coordinates
x~n)) and
6k, = Y x n>xP) ">.
(7)
(5)
n
By assuming these equations to hold down to atomic distances, one can calculate the displacements for given atomic forces and consequently the corresponding energies. The energies obtained in this simple way are reasonably accurate [11 ]. It is obvious that this approach is especially suited for the elastically isotropic glasses.
(9)
For vitreous silica with Tg = 1450K and C44 = 31 GPa, r0 = 1.5A, close to the S i - O bond length of 1.6A. However, even in the favourable cases where equation (9) holds, the negative contributions to the force constant are not very strong. The harmonic force constant D2
=
(1/2)~2VlS~k2lqj=o
=
--7kTg+D2bb.
(10)
Thus the bond bending force constant D2bb must be either rather weak or even negative in order to explain the tunnelling states and the resonant modes. As shall be seen, it is in fact negative in vitreous silica.
Vol. 56, No. 10
TUNNELLING STATES AND ANHARMONIC RESONANT MODES IN GLASSES
The rotational potential (equation (1)) is conveniently written in the form V(~0) = D l ~ +D2~b 2 + D 4 ~ / 4 ,
(11)
where DI and D2 exhibit more or less uniform distributions around zero. D4 can be considered to have a fixed value given by 04 = 7zrC44rao/12 + D4bb,
(12)
where again D4bb is the bond bending contribution. The effective momentum of inertia of the motion can be calculated from the displacement fields. Since these decrease rapidly with increasing distance from the bond, one gets a good estimate by taking only the motion of the three central atoms into account. One finds that the two outer atoms move opposite to the central atom with 1/6 of the amplitude of the central atom. Therefore 0elf = (36mc + 2mo)r~/49,
(13)
where m e and mo are the masses of the central and the outer atom respectively. It is convenient to express the force constants in units of D0 with
Do = D~/a(h2/Oefr) 1/a.
(14)
A characteristic frequency of the problem is Vo, where
hvo = D~/a(h2/Oerf) z/a.
(15)
In the case D~ = D2 = 0 the level splitting between the two lowest levels turns out to be 1.8 huo. Since ~o is of the order of 1 THz, the resonant modes at 100GHz must already have a strongly negative D2 in order to achieve a lowering of the level spacing. Thus the transition from resonant modes to tunnelling states must be continuous. For D~ < - 3Do and small Dx one can straightforwardly reformulate the well-known equation [4] for the tunnel splitting A o in terms of D2 and ~'o. One obtains Ao = (2/V/-~)hi:od s/4 exp (-- da/2),
(16)
where d=--D2/Do. The basic assumption of the tunnelling model is a uniform distribution in the exponential of equation (16). The arguments given here do not lead directly to a constant distribution function. However, since D2 results from a sum of positive and negative contributions, we will expect a slow variation in the distribution function of d and consequently of d 3/2 with varying d. In this sense, the present explanation confirms the central assumption of the tunnelling model. It is obvious that the modes considered here will exhibit a strong negative mode Griineisen parameter.
891
This explains the negative thermal expansion observed in several glasses at low temperatures [1, 12]. For vitreous silica with Do = 0.2 eV (see below), equations (6) and (16) yield a mode Griineisen parameter o f - - 18 for the tunnelling states with splittings corresponding to I K, close to the experimental value [12] o f - - 2 0 . Another experimental result which finds a natural explanation is the observation of tunnelling states in crystalline materials doped with impurities having permanent elastic dipole moments [13]. These will introduce the irregular stress patterns which are normally absent in a crystal and which in a glass are frozen in at the glass transition. In order to estimate the densities of tunnelling states and resonant modes, one has to specify the bond bending contributions which vary from glass to glass. In vitreous silica, the bond bending potential seems to have a minimum at ~ = 18 ° ( S i - O - S i angle ~0= 144 °) as evidenced by theoretical calculations [14] and by the angles found in the different crystalline modifications [15] of SiO2. The restoring force to that minimum position can be described by a longitudinal force constant of 80N/m between the two Si-atoms [16]. Since the energy is independent on the sign of ~, this is a double minimum potential with a negative value of D2bb and a positive D4bb. From the values of the force constant and the minimum angle one gets D:bb = -- 1.26 eV and D4bt~ = 6.4 eV. From equation (13), h2/O = 0.125 meV. These values inserted in equations (12), (14) and (16) yield Do = 0.2 eV and a characteristic frequency Vo = 1.2 THz. The distribution of rotational potentials was calculated assuming equal a priori probabilities for all possible elastic dipole and elastic quadrupole configurations and then weighing the different configurations according to their Boltzmann factor at the glass temperature. The Boltzmann factor was determined from equation (1) with equations (6) and (7), taking the energy at its minimum with respect to the bond angle. According to this calculation, 8.3% of the S i - O - S i bonds in vitreous silica have a double minimum potential. Fer a rough estimate, this number may be identified with the number of resonant modes below the characteristic frequency ~o. With two bonds per SiO2-unit, this corresponds to 2% of the total vibrational density-of-states in good agreement with experiment [9] (0.8%). A detailed discussion of the potential distribution functions is beyond the scope of this Letter. A fraction of 10 -4 of the bonds exhibited both an asymmetry below 8 meV (88 K) and a barrier height corresponding to tunnel splittings between 1/aK and 1 K. This yields 2 - 1 0 -6 tunnelling states per SiO2 unit with level splittings below 1 K, about twice the value deduced from the specific heat. Thus the model considered here
892
TUNNELLING STATES AND ANHARMONIC RESONANT MODES IN GLASSES
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(~C
oH
Fig. 2. Relative position of two neighbouring chains in crystalline polyethylene. The chain axes are perpendicular to the drawing plane. The dashed lines mark schematically the hard core interaction contours. The possible coupled rotation is indicated by arrows. reproduces not only the density of the resonant modes, but also the density of tunnelling states. This is achieved without any adjustable parameters; the parameters inserted were values measured either in vitreous silica or in its crystalline modifications. One might question the bond bending potential parameters D2bb and D4~b which were obtained rather indirectly. However, the reported results are not strongly dependent on the exact values of D2~b and D4bb. Only when D2bb becomes positive the tunnelling and resonant mode densities decrease drastically. This result indicates that an inherent asymmetry of the bond is important in order to create the local structural instabilities. The mechanism considered here will naturally apply equally well to the whole class of silica glasses. There is a second class of amorphous solids where it can be argued to apply, namely the amorphous polymers. In the polymer case, it is easy to convince oneself that a given single chain conformation is frozen at low temperatures and that the strong bond bending force constants within the chain are not easily destabilized. In contrast, the forces between different chains are weak. The packing of neighbouring chains resembles the packing in crystalline polymers. Figure 2 shows the crystalline packing in polyethylene. The difference is that in the crystalline case the chains lie side by side for many monomers, while in the amorphous case the chain folds away after only a few monomers to join new neighbours. Two neighbouring chain segments are kept apart at a more or less constant distance by the hard core repulsion of the protons (or eventually side groups). They can, however, rotate around one of the protons as indicated in Fig. 2. This motion is very similar to the motion of Fig. 1. The distance 2ro in Fig. 1 would correspond to the chain distance in Fig. 2. In the case of amorphous polystyrene with Tg = 400 K
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and C44 = 2.5GPa one calculates from equation (9) an optimum bond length ro = 2.5 A which is close to half the interchain distance. Moreover, looking at the hard core interaction contours drawn schematically in Fig. 2, one realizes that the rotational potential from the chain interaction will have low-lying maxima with negative values of the harmonic force constant D2b~. Thus one might postulate that the tunnelling states and the resonant modes in the amorphous polymers will be coupled rotational motions of neighbouring chain segments. This explanation should be tested by measuring the inelastic structure factor of the resonant modes with neutrons, quite in the same way as it has been done for vitreous silica, to see whether it can be really described by a coupled rotational motion of the polymer chains. In conclusion, the tunnelling states and the low frequency vibrational modes in glasses have been explained in terms of coupled rotations, taking into account the elastic distortions around the librating units. For vitreous silica, this explanation gives correct orders of magnitude for the densities of both tunnelling states and resonant modes without any adjustable parameter.
Acknowledgements -
Helpful discussions with H. Grimm, S. Hunklinger, J. J~ickle, H.R. Schober and T. Springer are gratefully acknowledged. REFERENCES
1. Amorphous Solids-Low Temperature Properties, 2. 3. 4. 5. 6. 7. 8. 9.
10.
Topics in Current Physics, Vol. 24 (Edited by W.A. Phillips), Springer, Berlin (1981). S. Hunklinger & W. Arnold, Physical Acoustics, Vol. XII, p. 155 (Edited by W.P. Mason & R.N. Thurston), Academic Press, New York (1976). P.W. Anderson, B.I. Halperin & C.M. Varma, Phil. Mag. 25, 1 (1972). W.A. Phillips, J. Low Temp. Phys. 7,351 (1972). M.P. Zaitlin & A.C. Anderson, Phys. Rev. B12, 4487 (1975). J. Jiickle, The Physics of Non-C'~ystalline Solids, p.568, (Edited by G.H. Frischat), Transtech, Aedermannsdorf (1977). B. Derrida, R. Orbach & Kin-Wah Yu, Phys. Rev. B29, 6645 (1984). V.G. Karpov & D.A. Parshin, Pis'ma Zh. Eksp. Teor. Fiz. 38, 536 (1983); JETP Lett. 38, 647 (1983). U. Buehenau, N. Nticker & A.J. Dianoux, Phys. Rev. Lett. 53, 2316 (1984) and U. Buchenau, M. Prager, N. Nticker & A.J. Dianoux (unpublished). A.A. Maradudin, E.W. Montroll, G.H. Weiss & I.P. Ipatova, Theory of Lattice Dynamics in the
Harmonic Approximation, Solid State Physics: Advances in Research and Applications, Academic, 11.
New York, (1971). G. Leibfried & N. Breuer, Point Defects in Metals /, Springer Tracts in Modern Physics, Vol. 81,
Vol. 56, No. 10
12. 13.
TUNNELLING STATES AND ANHARMONIC RESONANT MODES IN GLASSES
p. 138 if, (Edited by G. H6hler), Springer, Berlin, Heidelberg, New York (1978). G.K. White, S.J. Collocott & J.S. Cook, Phys. Rev. B29, 4778 (1984). D.A. Ackerman, D. Moy, R.C. Potter, A.C. Anderson & W.N. Lawless, Phys. Rev. B23, 3886 (1981).
14. 15. 16.
893
M.D. Newton & G.V. Gibbs, Phys. Chem. Minerals, 6,221 (1980). H.D. Megaw, Crystal Structures: A Working Approach, pp.259-269, Saunders, Philadelphia, (1973). T.H.K. Barron, C.C. Huang & A. Pastemak, J. Phys. C9, 3925 (1976).