1

Breakdown of the Universality of Glasses at Ultralow Temperatures– Interplay of Nuclear Spins and Atomic Tunneling Systems

Catania 1999

2

Catania 1999

3

Catania 1999Catania 1999

4

Comment Paper by Clare and Tony 1988

5

Comment Paper by Clare and Tony 1988

6

… a totally unbelievable degree of chance coincidence

7

Specific Heat of Amorphous and Crystalline SiO2

J.C. Lasjaunias et al., Sol. State Commun. 17, 1045 (1975)

broad distribution oflow-energy excitations

R.C. Zeller, R.O. Pohl, Phys. Rev. B 4, 2029 (1971)

8

Specific Heat of Amorphous and Crystalline SiO2

strong coupling to phonons

R.C. Zeller, R.O. Pohl, Phys. Rev. B 4, 2029 (1971)

excitations are localized

9

Universality of Glasses

R. B. Stephens, Phys. Rev. B 8, 2896 (1973)

Thermal conductivity of glasseswithin one order of magnitude

10

W.A. Phillips, J. Low. Temp. Phys. 7, 351 (1972)P.W. Anderson et al., Philos. Mag. 25, 1 (1972)

Atomic Tunneling Systems in Glasses

energy splitting

distribution function

tunnel splitting

elastic, dielectric und thermalproperities

11

Dresden 2003

12

13

“Tunneling Two-Level Systems” Model of the Low-TemperatureProperties of Glasses: Are “Smoking-Gun” Tests Possible?Anthony J. Leggett*,† and Dervis C. Vural*,‡

†Department of Physics, University of Illinois at Urbana−Champaign, 1110 West Green Street, Urbana, Illinois, United States‡Applied Physics, SEAS, Harvard University, 29 Oxford Street, Cambridge, Massachusetts, United States

ABSTRACT: Following a brief review of the “two-level (tunneling) systems” model of thelow-temperature properties of amorphous solids (“glasses”), we ask whether it is in fact theunique explanation of these properties as is usually assumed, concluding that this is notnecessarily the case. We point out that (a) one specific form of the model is alreadyexperimentally refuted and (b) that a definitive test of the model in its most general form,while not yet carried out, would appear to be now experimentally feasible.

1. INTRODUCTIONStructurally amorphous materials (“glasses”) constitute asubstantial fraction of all terrestrial matter, yet our overallunderstanding of their behavior is not at all comparable to thatwhich we have for crystalline matter. While both the transitionto the glassy state and the thermodynamic and responsebehavior at ambient temperatures have some universal featuresthat have been the subject of a huge amount of literature (see,e.g., refs 1 and 2 respectively), a particularly intriguing problemis posed by the behavior of glasses below about 1 K. Ever sincethe pioneering experiments of Pohl and co-workers in the early1970s,3 it has been recognized that in this regime, theproperties of glasses not only are qualitatively different fromthose of crystalline solids but show a remarkable degree ofuniversality. For example, almost without exception, the specificheat of an arbitrary (insulating) amorphous solid well below 1K is approximately linear in T, the thermal conductivity isapproximately quadratic in T, and the ultrasonic behavior isconsistent with a Q factor that at zero temperature isindependent of frequency and surprisingly large (more onthis below).Very soon after the original experiments,3 a plausible model

to explain them was published independently by Phillips4 andby Anderson, Halperin, and Varma.5 This model, which hasbecome known as the “tunneling two-level system” (TTLS)model, postulates that because of the structurally amorphousnature of the system, there exist some entities (single atoms,groups of atoms, or in some cases even single electrons) thathave available two nearly degenerate configurations and cantunnel between them (a more quantitative description is givenin the next section). With a plausible choice of the distributionof the relevant parameters, the TTLS model can account for thequalitatively universal features noted above; furthermore, byanalogy with other well-known examples of “two-state” systemsin, for example, atomic physics and NMR, it naturally predictsvarious nonlinear phenomena such as acoustic saturation andechoes, all of which have been at least qualitatively verified byexperiment.6 The original model has undergone considerableelaboration over the last 40 years; in particular, an interesting

series of papers7 by Peter Wolynes and his collaborators hasattempted to explain the generic existence of the postulatedtwo-level systems (TLSs) as a natural consequence of processesoccurring in the glass transition. These successes havepersuaded the overwhelming majority of the relevantcommunity that the TTLS model is the unique explanationof the low-temperature properties of glasses.Viewed from the above perspective, the aim of the present

article may seem rather quixotic: to ask whether the TTLSexplanation is indeed as unique as it is usually taken to be andto try to suggest definitive ways of testing the model against aparticular (loosely defined) class of alternative hypotheses.Obviously, to do this, it is necessary to define exactly what wemean by the “TTLS model” (and, in particular, what itexcludes), and this will be done in section 2. In section 3, wesketch some reasons for skepticism about the model andintroduce a (very generic) class of alternative hypotheses, andin section 4, we propose an experiment that we believe shoulddiscriminate unambiguously between this class and the TTLSmodel. Throughout, we confine ourselves to insulating glassesand concentrate on the linear ultrasonic properties, which webelieve are among the most unambiguously predictedconsequences of the model.

2. DEFINITION OF THE TTLS MODEL; SOME SIMPLECONSEQUENCES

There exist in the literature a number of good accounts of theTTLS model and its most important experimental predic-tions.6,8 For present purposes, a sufficient definition runsroughly as follows: For the purpose of calculating the low-energy states of glasses (those relevant to the equilibrium andnear-equilibrium properties well below 1 K), an adequateeffective Hamiltonian is of the form

Special Issue: Peter G. Wolynes Festschrift

Received: March 4, 2013Revised: August 6, 2013Published: August 7, 2013

Article

pubs.acs.org/JPCB

© 2013 American Chemical Society 12966 dx.doi.org/10.1021/jp402222g | J. Phys. Chem. B 2013, 117, 12966−12971

Is there an unambiguous test for TLS in glasses?

The Cindarella Problem of Glasses

Universality of glasses cannot be explained with the Standard TLS Model

(although it is itself no contradiction to it).

At the same time universality is the very reason, why it is so hard to

determine the microscopic nature of the excitations in glasses.

Unexpected Magnetic Field Dependence

S. Ludwig. C. E., S. Hunklinger, P. Strehlow, Phys. Rev. Lett. 88, 75501 (2002)

Dipole Echoes

Duran

BK7

a-BaO-Al2O3-SiO2

15 mK

15 mK

50 mK

BK7 and Duran: M. Wohlfahrt, P. Strehlow, C. E., S. Hunklinger, Europhys. Lett. 56, 690 (2001)

a-BaO- Al2O3- SiO2: R. Haueisen, G. Weiss, PhysicaB 316-317, 555 (2002)

Dielectric Susceptibility

10 mK

16

Nuclear Quadrupole Moment is Important

no effect for a-SiO2because no quadrupole moment

nuclear quadrupole moment sees electric field gradient in the two wells

hyperfine splitting of tunneling levels

multi-level systems

magnetic field causes an additionalZeeman splitting of nuclear levels

A. Würger, A. Fleischmann, C. E., Phys. Rev. Lett. 89, 237601 (2002)

EEQ

17

hydrogen

Glycerol

P. Nagel, A. Fleischmann, S. Hunklinger, C. E., Phys. Rev. Lett. 92, 245511-1 (2004)proof of the quadrupole model

deuterium atom

D C C C D

D D D

OD OD ODH C C C H

H H H

OH OH OH

Proof: Isotope Effect H D

18

Zero Magnetic Field

A. Würger, JLTP 137, 143 (2004)

D.A. Parshin, JLTP 137, 233 (2004)

Partially Deuterated Glycerol

19

A. Bartkowiak, M. Bazrafshan, C. Fischer, A. Fleischmann, C. E., Phys. Rev. Lett. 110 (2013)

W. Schnauss, F. Fujara, H. Sillescu, J. Chem. Phys. 97, 1378 (1992).

Beating in case of d3 disappears faster local enviroment ?

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Microscopic Modeling

Tunneling angle 2Θ = 17� M. Bazrafshan, PhD Thesis 2008

A. Bartkowiak, M. Bazrafshan, C. Fischer, A. Fleischmann, C. E., Phys. Rev. Lett. 110 (2013)

First measurement of microscopic properities of TLS in Glass

21

Dielectric Susceptibility

Can nuclear spins explain the magnetic field dependence of the dielectric susceptibility?

Coupling to Electric and Elastic Fields

resonant processes

relaxational processes

modulation of D

T < 1 K one-phonon relaxation

rot

~a⇥~b ? ~a (1)

AParallelogramm = |~a⇥~b| (2)

ADreieck =1

2|~a⇥~b| (3)

Tmin / ⌫1/3

(4)

Tmin / ⌫1/7

(5)

~c (6)

⌧1 = A

✓E

�0

◆21

E3tanh

✓E

2kBT

◆(7)

g =

p14 h = | ~MA| ' 3,08 A =

1

2

p14 3,08 = 5,7 FE (8)

|~c| =p13 (9)

|~b| =p13 (10)

|~a| =p14 (11)

~a⇥~b ? ~a (12)

~a⇥~b ? ~a (13)

~a⇥~b ? ~a (14)

~c (15)

A =1

2g h (16)

~a⇥~b ? ~a (17)

~c (18)

1

Dielectric Susceptibility in the STM

Real part after integration of tunnling parametern distribution

Different Frequencies

Relaxation Rateone phonon

two phonon

Samples

Name Type of Glass Quadrupole Moment Q

CVAc Polymer glass None

Herasil Pure quartz glass None

BK7 Multi-component glass No large Q

N-KZFS11 Multi-component glass 181Ta, Q =3.3 barn

HY-1 Multi-component glass 165Ho, Q =3.5 barn

Dielectric Susceptibility of Standard Glasses

26

Multi-component glass BK7

A. Luck, A. Fleischmann, A. Reiser, C. E., J. Phys. C 568, 032013 (2014)

Herasil

C.E., C. Bechinger, M. v. Schickfus, Phonons 89,474 (1989)

334 9. Tunneling Systems

.01 0.1 1Temperature T / K

0

2

4

6

8

104δv

/v

14.0 kHz5.03 kHz2.52 kHz1.03 kHz

Vitreoussilica

Temperature T / K

0

2

4

105δε’/ε’

480 Hz2 kHz10 kHz50 kHz

BK7

0.05 0.1 0.2 0.5 1

Fig. 9.42. (a) Relative change in the sound velocity of vitreous silica as a function oftemperature [441]. (b) Relative change in the dielectric constant of the borosilicateglass (BK7) versus temperature [442]

There is an additional observation that we would like to mention withoutgoing into a detailed discussion. Below 100 mK, the real and imaginary partsof the elastic and dielectric susceptibility exhibit pronounced non-linearities.The reason is that, with decreasing temperature, tunneling systems withsmaller and smaller energy splitting become important. For these systems theassumption originally made in Sect. 9.1.2, namely, that changes caused by theexternal fields are small compared to the energy splitting in zero field, is nolonger fulfilled. Fields of moderate strength already cause changes δ¢ & E.We do not discuss this interesting aspect further but refer the reader to theliterature [443, 444].

Non-equilibrium Phenomena – Memory EÆect The temporarily appli-cation of a strong dc electric field across a capacitor with a glassy dielectricresults in a sudden jump of the capacitance (which is proportional to thedielectric constant of the glass in this measurement) and a subsequent slowrelaxation towards its starting value [445, 446, 447]. As an example we showin Fig. 9.43a the time evolution of the capacitance of a capacitor filled witha-SiOx after the sudden application of a large dc electric field at 50 mK. Theslow relaxation observed in this experiment is logarithmic in the time elapsedsince the field was applied.

Another interesting observation can be made when slowly sweeping thedc field after the sample has been kept in a constant field for a long time.In this case the capacitance shows a minimum at the field at which thesample was kept prior the sweep experiment. Fig. 9.42b shows the responseto such a sweep of a capacitor filled with photoresist (an amorphous polymer)which initially had been cooled down without an electric field and then keptat a field of F = 5MV/m for about 2 hours. Clearly, a minimum of thecapacitance was found at zero field, at which the sample was kept for a long

Logarithmic temperature dependence

shift of minimum temperature with frequency

0.01 0.1 1Temperature T [K]

0

2

4

6

8

105

∆ε’/ε’

120Hz600Hz1kHz8kHz16 kHz

Herasil

0.1 1Temperature T [K]

0

5

10

15

104

∆ε’/ε’

10 kHz20 kHz50 kHz100 kHz

N-BK7

Frequency Dependence of Minimum in DK

27

one phonon process:

two phonon process:

102 104 106 108Frequency ν [Hz]

100

1000

TemperatureT m

in[mK]

Herasil

∝ ν1/3N-BK7

PVAc

∝ ν1/3

Dielectric Susceptibility of Glasses With Nuclear Quadrupols

28

A. Luck, A. Fleischmann, A. Reiser, C. Enss, J. Phys. C 568, 032013 (2014)

Multi-component glass containing Tatalum

29

Multi-component glass containing Holmium

HY-1 and N-KZFS11 different from other glasses

no shift of minimum temperature with frequency at low frequencies

A. Luck, A. Fleischmann, A. Reiser, C. Enss, J. Phys. C 568, 032013 (2014)

Multi-component glass containing Tatalum

Dielectric Susceptibility of Glasses With Nuclear Quadrupols

Frequency Dependence of Minimum in DK

30

one phonon process:

two phonon process:

102 104 106 108Frequency ν [Hz]

100

1000

TemperatureT m

in[mK]

Herasil

∝ ν1/3N-BK7

PVAc

∝ ν1/3

Frequency Dependence of Minimum in DK

31

one phonon process:

two phonon process:

102 104 106 108Frequency ν [Hz]

100

1000

TemperatureT m

in[mK]

N-KZFS11

ν1/3∝

Herasil

∝ ν1/3N-BK7

PVAc

∝ ν1/3

Frequency Dependence of Minimum in DK

32

one phonon process:

two phonon process:

102 104 106 108Frequency ν [Hz]

100

1000

TemperatureT m

in[mK]

N-KZFS11

ν1/3∝

HY-1

∝ ν1/3

Herasil

∝ ν1/3N-BK7

PVAc

∝ ν1/3

33

Nuclear Quadrupols

TLS Phonons

Relaxation Into Nuclear Spin Bath

34

Relaxation Into Nuclear Spin Bath

35

Atomic Tunneling Systems in Glasses

o Nuclear spins are important in glasses at low temperatures

o The universality of glasses breaks down at very low temperatures

o This allows for material dependent studies

o TLS model can be underpinned microscopically at least for some materials

o Still no microscopic explanation for the universality in glasses

There are still many fundamental questions regarding glasses

to think about and work on

Summary

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Happy Birthday